Session.name = Polynomial_Interpolation
Loading theory "Adhoc_Overloading" (required by "Monad_Syntax")
loading_theory = Adhoc_Overloading
Loading theory "Infinite_Set" (required by "Polynomial")
loading_theory = Infinite_Set
signature ADHOC_OVERLOADING =
  sig
    val generic_add_overloaded: string -> Context.generic -> Context.generic
    val generic_add_variant:
       string -> term -> Context.generic -> Context.generic
    val generic_remove_overloaded:
       string -> Context.generic -> Context.generic
    val generic_remove_variant:
       string -> term -> Context.generic -> Context.generic
    val is_overloaded: Proof.context -> string -> bool
    val show_variants: bool Config.T
  end
structure Adhoc_Overloading: ADHOC_OVERLOADING
command_timing = :file=~~/src/Tools/Adhoc_Overloading.thyoffset=276name=ML_fileelapsed=0.062
### theory "Adhoc_Overloading"
### 0.064s elapsed time, 0.128s cpu time, 0.000s GC time
Loading theory "Monad_Syntax"
loading_theory = Monad_Syntax
command_timing = :file=~~/src/HOL/Library/Infinite_Set.thyoffset=125name=theoryelapsed=0.093
command_timing = :file=~~/src/HOL/Library/Infinite_Set.thyoffset=7707name=textelapsed=0.011
command_timing = :file=~~/src/HOL/Library/Monad_Syntax.thyoffset=148name=theoryelapsed=0.225
ML_statistics = :now=1489705801.25tasks_ready=53tasks_pending=1tasks_running=2tasks_passive=0tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=0size_allocation=436207616size_allocation_free=435936632size_heap=436207616size_heap_free_last_full_GC=0size_heap_free_last_GC=0threads_in_ML=0threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=0.008time_GC=0user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
### theory "Monad_Syntax"
### 0.252s elapsed time, 0.572s cpu time, 0.000s GC time
Loading theory "More_List" (required by "Polynomial")
loading_theory = More_List
consts
  enumerate :: "'a set \<Rightarrow> nat \<Rightarrow> 'a"
command_timing = :file=~~/src/HOL/Library/Infinite_Set.thyoffset=7837name=primrecelapsed=0.094
command_timing = :file=~~/src/HOL/Library/More_List.thyoffset=131name=theoryelapsed=0.091
### theory "Infinite_Set"
### 0.413s elapsed time, 0.896s cpu time, 0.000s GC time
Loading theory "Sqrt_Babylonian_Auxiliary"
loading_theory = Sqrt_Babylonian_Auxiliary
command_timing = :file=~~/src/HOL/Library/More_List.thyoffset=2105name=definitionelapsed=0.011
command_timing = :file=~~/afp/thys/Sqrt_Babylonian/Sqrt_Babylonian_Auxiliary.thyoffset=980name=theoryelapsed=0.145
### theory "More_List"
### 0.332s elapsed time, 0.668s cpu time, 0.000s GC time
Loading theory "Polynomial"
loading_theory = Polynomial
ML_statistics = :now=1489705801.75tasks_ready=123tasks_pending=1tasks_running=3tasks_passive=0tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=0size_allocation=1064304640size_allocation_free=103245200size_heap=1064304640size_heap_free_last_full_GC=0size_heap_free_last_GC=0threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=3.468time_GC=0user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
class linordered_idom = ring_char_0 + idom_abs_sgn +
  linordered_ring_strict + linordered_semidom +
  linordered_semiring_1_strict + ordered_comm_ring + ordered_ring_abs +
  assumes
    "sgn_if":
      "\<And>x.
          sgn x =
          (if x = (0::'a) then 0::'a
           else if (0::'a) < x then 1::'a else - (1::'a))"
command_timing = :file=~~/afp/thys/Sqrt_Babylonian/Sqrt_Babylonian_Auxiliary.thyoffset=4622name=contextelapsed=0.536
command_timing = :file=~~/afp/thys/Sqrt_Babylonian/Sqrt_Babylonian_Auxiliary.thyoffset=4653name=lemmaelapsed=0.019
command_timing = :file=~~/afp/thys/Sqrt_Babylonian/Sqrt_Babylonian_Auxiliary.thyoffset=4776name=lemmaelapsed=0.019
command_timing = :file=~~/afp/thys/Sqrt_Babylonian/Sqrt_Babylonian_Auxiliary.thyoffset=4922name=lemmaelapsed=0.019
### theory "Sqrt_Babylonian_Auxiliary"
### 0.817s elapsed time, 1.636s cpu time, 0.000s GC time
command_timing = :file=~~/src/HOL/Library/Infinite_Set.thyoffset=776name=byelapsed=0.015
ML_statistics = :now=1489705802.26tasks_ready=109tasks_pending=1tasks_running=2tasks_passive=0tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=0size_allocation=1607467008size_allocation_free=173399512size_heap=1607467008size_heap_free_last_full_GC=0size_heap_free_last_GC=0threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=4.268time_GC=0user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/src/HOL/Library/Infinite_Set.thyoffset=942name=byelapsed=0.015
command_timing = :file=~~/src/HOL/Library/Infinite_Set.thyoffset=8341name=byelapsed=0.014
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=226name=theoryelapsed=0.644
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=430name=definitionelapsed=0.015
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=1439name=typedefelapsed=0.014
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=1439name=typedefelapsed=0.037
command_timing = :file=~~/src/HOL/Library/Infinite_Set.thyoffset=10816name=byelapsed=0.010
command_timing = :file=~~/src/HOL/Library/Infinite_Set.thyoffset=10630name=byelapsed=0.016
command_timing = :file=~~/src/HOL/Library/Infinite_Set.thyoffset=10406name=byelapsed=0.011
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=1567name=setup_liftingelapsed=0.057
command_timing = :file=~~/src/HOL/Library/Infinite_Set.thyoffset=10574name=byelapsed=0.018
command_timing = :file=~~/src/HOL/Library/Infinite_Set.thyoffset=10483name=byelapsed=0.012
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=1912name=definitionelapsed=0.011
instantiation
  poly :: (zero) zero
  zero_poly == zero_class.zero :: 'a poly
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=2679name=lift_definitionelapsed=0.015
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=4699name=lift_definitionelapsed=0.015
command_timing = :file=~~/src/HOL/Library/More_List.thyoffset=3456name=byelapsed=0.140
command_timing = :file=~~/src/HOL/Library/More_List.thyoffset=3605name=byelapsed=0.010
### Legacy feature! Old 'def' command -- use 'define' instead
ML_statistics = :now=1489705802.88tasks_ready=74tasks_pending=1tasks_running=3tasks_passive=1tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=1size_allocation=2097152000size_allocation_free=2097152000size_heap=2141192192size_heap_free_last_full_GC=0size_heap_free_last_GC=2100223968threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=5.404time_GC=0.232user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/src/HOL/Library/More_List.thyoffset=6586name=byelapsed=0.069
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=7942name=quickcheck_generatorelapsed=0.310
command_timing = :file=~~/src/HOL/Library/More_List.thyoffset=6917name=byelapsed=0.012
consts
  Poly :: "'a list \<Rightarrow> 'a poly"
instantiation
  poly :: ({zero,equal}) equal
  equal_poly == equal_class.equal ::
    'a poly \<Rightarrow> 'a poly \<Rightarrow> bool
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=12363name=definitionelapsed=0.012
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=15109name=lift_definitionelapsed=0.012
instantiation
  poly :: (comm_monoid_add) comm_monoid_add
  plus_poly == plus :: 'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=17219name=lift_definitionelapsed=0.016
instantiation
  poly :: (cancel_comm_monoid_add) cancel_comm_monoid_add
  minus_poly == minus :: 'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=9993name=byelapsed=0.041
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=17890name=lift_definitionelapsed=0.016
instantiation
  poly :: (ab_group_add) ab_group_add
  uminus_poly == uminus :: 'a poly \<Rightarrow> 'a poly
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=9915name=byelapsed=0.022
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=18483name=lift_definitionelapsed=0.015
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=11029name=byelapsed=0.040
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=1239name=byelapsed=0.014
ML_statistics = :now=1489705803.38tasks_ready=105tasks_pending=1tasks_running=2tasks_passive=0tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=1size_allocation=2097152000size_allocation_free=1396959360size_heap=2141192192size_heap_free_last_full_GC=0size_heap_free_last_GC=2100223968threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=6.208time_GC=0.232user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
Found termination order: "(\<lambda>p. length (snd p)) <*mlex*> {}"
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=21481name=funelapsed=0.097
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=24428name=lift_definitionelapsed=0.011
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=16584name=byelapsed=0.030
instantiation
  poly :: (comm_semiring_0) comm_semiring_0
  times_poly == times :: 'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=27423name=definitionelapsed=0.013
instantiation
  poly :: (comm_semiring_1) comm_semiring_1
  one_poly == one_class.one :: 'a poly
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=30212name=definitionelapsed=0.010
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=21321name=byelapsed=0.010
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=23967name=byelapsed=0.019
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=28977name=byelapsed=0.025
instantiation
  poly :: (linordered_idom) linordered_idom
  sgn_poly == sgn :: 'a poly \<Rightarrow> 'a poly
  abs_poly == abs :: 'a poly \<Rightarrow> 'a poly
  less_eq_poly == less_eq ::
    'a poly \<Rightarrow> 'a poly \<Rightarrow> bool
  less_poly == less :: 'a poly \<Rightarrow> 'a poly \<Rightarrow> bool
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=45160name=definitionelapsed=0.011
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=45201name=definitionelapsed=0.012
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=45250name=definitionelapsed=0.011
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=45308name=definitionelapsed=0.012
ML_statistics = :now=1489705803.88tasks_ready=117tasks_pending=1tasks_running=4tasks_passive=2tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=1size_allocation=2097152000size_allocation_free=661199792size_heap=2141192192size_heap_free_last_full_GC=0size_heap_free_last_GC=2100223968threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=7.016time_GC=0.232user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=46711name=definitionelapsed=0.010
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=29733name=byelapsed=0.027
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=29949name=applyelapsed=0.022
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=51023name=definitionelapsed=0.011
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=34183name=byelapsed=0.020
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=38015name=byelapsed=0.018
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=59360name=definitionelapsed=0.011
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=66147name=definitionelapsed=0.010
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=42533name=byelapsed=0.018
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=67660name=definitionelapsed=0.010
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=12101name=byelapsed=0.020
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=47474name=byelapsed=0.011
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=47247name=byelapsed=0.015
ML_statistics = :now=1489705804.38tasks_ready=149tasks_pending=1tasks_running=3tasks_passive=1tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=1size_allocation=2690646016size_allocation_free=6379408size_heap=2734686208size_heap_free_last_full_GC=0size_heap_free_last_GC=2100223968threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=8.628time_GC=0.232user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=74451name=functionelapsed=0.072
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=74451name=functionelapsed=0.012
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=49457name=byelapsed=0.025
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=51280name=applyelapsed=0.011
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=31363name=byelapsed=0.015
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=52569name=applyelapsed=0.034
Found termination order: "(\<lambda>p. length (snd p)) <*mlex*> {}"
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=75365name=funelapsed=0.122
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=52681name=applyelapsed=0.037
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=53769name=applyelapsed=0.089
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=84758name=lemmaelapsed=0.013
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=54284name=byelapsed=0.015
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=88288name=definitionelapsed=0.011
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=88981name=lemmaelapsed=0.010
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=89959name=definitionelapsed=0.012
ML_statistics = :now=1489705804.88tasks_ready=204tasks_pending=1tasks_running=4tasks_passive=2tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=1size_allocation=3376414720size_allocation_free=156219904size_heap=3420454912size_heap_free_last_full_GC=0size_heap_free_last_GC=2100223968threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=9.424time_GC=0.232user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=95154name=definitionelapsed=0.010
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=54630name=byelapsed=0.126
instantiation
  poly :: (idom_divide) idom_divide
  divide_poly == divide ::
    'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=54472name=byelapsed=0.017
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=56003name=applyelapsed=0.011
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=59688name=byelapsed=0.013
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=60664name=haveelapsed=0.011
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=61530name=byelapsed=0.011
Found termination order:
  "(\<lambda>p. size (snd (snd (snd (snd (snd p)))))) <*mlex*> {}"
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=97961name=funelapsed=0.291
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=98387name=definitionelapsed=0.023
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=98600name=lemmaelapsed=0.012
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=66348name=byelapsed=0.072
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=67876name=byelapsed=0.020
ML_statistics = :now=1489705805.38tasks_ready=209tasks_pending=1tasks_running=4tasks_passive=1tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=1size_allocation=4020240384size_allocation_free=261568368size_heap=4064280576size_heap_free_last_full_GC=0size_heap_free_last_GC=2100223968threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=10.22time_GC=0.232user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=69534name=byelapsed=0.120
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=70859name=byelapsed=0.183
Found termination order:
  "(\<lambda>p. size (snd (snd (snd (snd (snd p)))))) <*mlex*> {}"
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=106350name=funelapsed=0.440
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=106727name=definitionelapsed=0.012
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=73117name=qedelapsed=0.075
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=106972name=lemmaelapsed=0.011
instantiation
  poly :: ({idom_divide,semidom_divide_unit_factor}) normalization_semidom
  normalize_poly == normalize :: 'a poly \<Rightarrow> 'a poly
  unit_factor_poly == unit_factor :: 'a poly \<Rightarrow> 'a poly
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=114313name=definitionelapsed=0.012
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=114427name=definitionelapsed=0.012
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=73015name=byelapsed=0.207
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=72876name=byelapsed=0.024
ML_statistics = :now=1489705806.03tasks_ready=224tasks_pending=1tasks_running=7tasks_passive=4tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=2size_allocation=4109369344size_allocation_free=3238736264size_heap=4132438016size_heap_free_last_full_GC=0size_heap_free_last_GC=4111999720threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=11.56time_GC=0.56user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=72618name=byelapsed=0.471
ML_statistics = :now=1489705806.53tasks_ready=222tasks_pending=1tasks_running=6tasks_passive=4tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=2size_allocation=4109369344size_allocation_free=2417377520size_heap=4132438016size_heap_free_last_full_GC=0size_heap_free_last_GC=4111999720threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=12.364time_GC=0.56user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=72430name=byelapsed=0.141
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=75258name=byelapsed=0.012
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=78675name=applyelapsed=0.012
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=78826name=applyelapsed=0.012
class field_unit_factor = field + unit_factor +
  assumes "unit_factor_field": "unit_factor = id"
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=116615name=classelapsed=0.861
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=81497name=byelapsed=0.019
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=82438name=byelapsed=0.018
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=116725name=subclasselapsed=0.066
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=86162name=byelapsed=0.038
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=86041name=byelapsed=0.035
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=85824name=applyelapsed=0.014
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=85428name=byelapsed=0.020
ML_statistics = :now=1489705807.03tasks_ready=183tasks_pending=1tasks_running=3tasks_passive=1tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=2size_allocation=4109369344size_allocation_free=1727303256size_heap=4132438016size_heap_free_last_full_GC=0size_heap_free_last_GC=4111999720threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=13.164time_GC=0.56user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=86932name=byelapsed=0.014
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=93785name=byelapsed=0.065
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=35540name=byelapsed=0.011
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=35437name=byelapsed=0.011
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=116725name=subclasselapsed=0.460
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=95096name=byelapsed=0.011
Proofs for inductive predicate(s) "eucl_rel_poly"
  Proving monotonicity ...
  Proving the introduction rules ...
  Proving the elimination rules ...
  Proving the induction rule ...
  Proving the simplification rules ...
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=118214name=inductiveelapsed=0.044
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=118976name=lemmaelapsed=0.012
instantiation
  poly :: (field) semidom_modulo
  modulo_poly == modulo ::
    'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=121792name=definitionelapsed=0.018
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=102833name=byelapsed=0.019
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=124977name=lemmaelapsed=0.012
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=100715name=byelapsed=0.011
ML_statistics = :now=1489705807.53tasks_ready=200tasks_pending=1tasks_running=6tasks_passive=4tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=2size_allocation=4109369344size_allocation_free=1354316760size_heap=4132438016size_heap_free_last_full_GC=0size_heap_free_last_GC=4111999720threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=13.976time_GC=0.56user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=101905name=byelapsed=0.029
Found termination order: "(\<lambda>p. length (snd p)) <*mlex*> {}"
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=129975name=funelapsed=0.117
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=101559name=qedelapsed=0.271
Found termination order:
  "(\<lambda>p. size (snd (snd (snd (snd p))))) <*mlex*> {}"
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=101361name=byelapsed=0.031
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=130210name=funelapsed=0.234
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=100844name=byelapsed=0.017
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=99747name=byelapsed=0.012
ML_statistics = :now=1489705808.03tasks_ready=242tasks_pending=1tasks_running=8tasks_passive=6tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=2size_allocation=4109369344size_allocation_free=26455496size_heap=4132438016size_heap_free_last_full_GC=0size_heap_free_last_GC=4111999720threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=15.58time_GC=0.56user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
Found termination order:
  "(\<lambda>p. size (snd (snd (snd p)))) <*mlex*> {}"
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=130630name=funelapsed=0.313
Found termination order:
  "(\<lambda>p. size (snd (snd (snd p)))) <*mlex*> {}"
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=130969name=funelapsed=0.187
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=100194name=byelapsed=0.519
Found termination order: "(\<lambda>p. size (snd (snd p))) <*mlex*> {}"
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=131336name=funelapsed=0.147
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=99834name=byelapsed=0.092
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=131637name=definitionelapsed=0.016
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=99121name=byelapsed=0.017
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=131915name=definitionelapsed=0.014
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=4631name=byelapsed=0.030
ML_statistics = :now=1489705808.64tasks_ready=269tasks_pending=1tasks_running=2tasks_passive=0tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=3size_allocation=4028628992size_allocation_free=3304580872size_heap=4154458112size_heap_free_last_full_GC=0size_heap_free_last_GC=4031540352threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=16.416time_GC=0.812user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=134898name=lemmaelapsed=0.014
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=140975name=lemmaelapsed=0.014
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=110235name=byelapsed=0.033
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=109994name=byelapsed=0.042
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=107261name=proofelapsed=0.010
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=109349name=qedelapsed=0.010
Found termination order:
  "(\<lambda>p. size (snd (snd (snd (snd p))))) <*mlex*> {}"
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=142884name=funelapsed=0.260
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=143709name=definitionelapsed=0.015
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=144274name=lemmaelapsed=0.011
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=144793name=definitionelapsed=0.019
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=145795name=lemmaelapsed=0.011
ML_statistics = :now=1489705809.14tasks_ready=293tasks_pending=1tasks_running=6tasks_passive=4tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=3size_allocation=4028628992size_allocation_free=2638104760size_heap=4154458112size_heap_free_last_full_GC=0size_heap_free_last_GC=4031540352threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=17.22time_GC=0.812user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=148864name=no_notationelapsed=0.109
### theory "Polynomial"
### 7.446s elapsed time, 14.844s cpu time, 0.812s GC time
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=109033name=qedelapsed=0.265
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=108850name=byelapsed=0.044
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=108762name=byelapsed=0.027
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=111855name=haveelapsed=0.011
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=108214name=byelapsed=0.014
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=112069name=haveelapsed=0.014
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=107710name=byelapsed=0.021
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=112307name=alsoelapsed=0.025
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=112498name=byelapsed=0.011
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=112028name=byelapsed=0.010
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=116315name=byelapsed=0.010
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=106193name=byelapsed=0.011
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=118675name=byelapsed=0.035
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=124947name=byelapsed=0.013
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=126969name=byelapsed=0.039
ML_statistics = :now=1489705809.64tasks_ready=231tasks_pending=1tasks_running=2tasks_passive=0tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=3size_allocation=4028628992size_allocation_free=2116520120size_heap=4154458112size_heap_free_last_full_GC=0size_heap_free_last_GC=4031540352threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=17.94time_GC=0.812user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=128710name=applyelapsed=0.013
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=128882name=applyelapsed=0.012
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=128918name=applyelapsed=0.020
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=129004name=applyelapsed=0.017
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=129055name=applyelapsed=0.012
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=129103name=applyelapsed=0.020
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=139224name=byelapsed=0.014
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=136519name=byelapsed=0.032
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=139824name=byelapsed=0.032
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=135372name=byelapsed=0.010
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=142828name=byelapsed=0.011
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=136317name=byelapsed=0.011
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=136176name=byelapsed=0.023
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=136037name=byelapsed=0.039
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=141941name=byelapsed=0.011
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=135742name=byelapsed=0.039
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=134551name=byelapsed=0.023
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=134308name=byelapsed=0.012
ML_statistics = :now=1489705810.14tasks_ready=141tasks_pending=1tasks_running=8tasks_passive=6tasks_urgent=0workers_total=4workers_active=2workers_waiting=2full_GCs=0partial_GCs=3size_allocation=4028628992size_allocation_free=1735214592size_heap=4154458112size_heap_free_last_full_GC=0size_heap_free_last_GC=4031540352threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=18.496time_GC=0.812user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=134094name=byelapsed=0.012
command_timing = :file=~~/afp/thys/Sqrt_Babylonian/Sqrt_Babylonian_Auxiliary.thyoffset=4427name=byelapsed=0.016
ML_statistics = :now=1489705810.75tasks_ready=74tasks_pending=1tasks_running=4tasks_passive=2tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=4size_allocation=4010803200size_allocation_free=4010803200size_heap=4148166656size_heap_free_last_full_GC=0size_heap_free_last_GC=4013483032threads_in_ML=1threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=19.996time_GC=1.1user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/afp/thys/Sqrt_Babylonian/Sqrt_Babylonian_Auxiliary.thyoffset=4193name=byelapsed=0.627
### Legacy feature! Old 'def' command -- use 'define' instead
### Legacy feature! Old 'def' command -- use 'define' instead
### Legacy feature! Old 'def' command -- use 'define' instead
command_timing = :file=~~/afp/thys/Sqrt_Babylonian/Sqrt_Babylonian_Auxiliary.thyoffset=3762name=byelapsed=0.036
command_timing = :file=~~/afp/thys/Sqrt_Babylonian/Sqrt_Babylonian_Auxiliary.thyoffset=2703name=byelapsed=0.014
command_timing = :file=~~/afp/thys/Sqrt_Babylonian/Sqrt_Babylonian_Auxiliary.thyoffset=3357name=byelapsed=0.013
command_timing = :file=~~/afp/thys/Sqrt_Babylonian/Sqrt_Babylonian_Auxiliary.thyoffset=2603name=byelapsed=0.018
command_timing = :file=~~/src/HOL/Library/Infinite_Set.thyoffset=1477name=applyelapsed=0.045
command_timing = :file=~~/src/HOL/Library/Infinite_Set.thyoffset=2155name=byelapsed=0.088
ML_statistics = :now=1489705811.25tasks_ready=69tasks_pending=1tasks_running=3tasks_passive=1tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=4size_allocation=4010803200size_allocation_free=2485556720size_heap=4148166656size_heap_free_last_full_GC=0size_heap_free_last_GC=4013483032threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=20.8time_GC=1.1user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/src/HOL/Library/Infinite_Set.thyoffset=2404name=byelapsed=0.132
command_timing = :file=~~/src/HOL/Library/Infinite_Set.thyoffset=11756name=byelapsed=1.306
command_timing = :file=~~/src/HOL/Library/Infinite_Set.thyoffset=12211name=applyelapsed=0.200
command_timing = :file=~~/src/HOL/Library/Infinite_Set.thyoffset=12248name=applyelapsed=0.061
command_timing = :file=~~/src/HOL/Library/More_List.thyoffset=7876name=byelapsed=0.099
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=8582name=byelapsed=0.061
ML_statistics = :now=1489705811.76tasks_ready=65tasks_pending=1tasks_running=4tasks_passive=2tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=4size_allocation=4010803200size_allocation_free=1349322288size_heap=4148166656size_heap_free_last_full_GC=0size_heap_free_last_GC=4013483032threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=21.6time_GC=1.1user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=22372name=byelapsed=0.063
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=22020name=byelapsed=0.014
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=44191name=applyelapsed=0.079
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=68849name=byelapsed=0.019
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=68822name=byelapsed=0.018
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=68441name=byelapsed=0.126
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=65305name=byelapsed=0.177
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=77363name=byelapsed=0.012
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=65193name=byelapsed=0.019
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=40992name=byelapsed=0.012
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=77085name=byelapsed=0.023
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=76942name=byelapsed=0.013
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=76060name=byelapsed=0.019
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=64642name=byelapsed=0.044
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=78158name=byelapsed=0.025
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=64326name=byelapsed=0.011
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=77908name=byelapsed=0.027
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=77572name=byelapsed=0.011
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=81066name=byelapsed=0.016
ML_statistics = :now=1489705812.26tasks_ready=57tasks_pending=1tasks_running=13tasks_passive=10tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=4size_allocation=4010803200size_allocation_free=646330808size_heap=4148166656size_heap_free_last_full_GC=0size_heap_free_last_GC=4013483032threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=22.4time_GC=1.1user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=63278name=byelapsed=0.038
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=80929name=byelapsed=0.021
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=80717name=byelapsed=0.022
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=20264name=byelapsed=0.037
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=82986name=applyelapsed=0.099
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=82760name=byelapsed=0.151
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=83512name=byelapsed=0.034
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=83357name=byelapsed=0.029
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=84680name=byelapsed=0.042
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=114202name=byelapsed=0.065
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=84547name=byelapsed=0.020
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=146198name=byelapsed=0.010
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=84451name=byelapsed=0.013
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=84064name=byelapsed=0.013
command_timing = :file=~~/src/HOL/Library/Infinite_Set.thyoffset=2686name=byelapsed=0.019
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=146821name=henceelapsed=0.015
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=146535name=byelapsed=0.049
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=147209name=byelapsed=0.067
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=143611name=byelapsed=0.013
ML_statistics = :now=1489705812.76tasks_ready=44tasks_pending=1tasks_running=3tasks_passive=0tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=5size_allocation=3959422976size_allocation_free=3360876392size_heap=4155506688size_heap_free_last_full_GC=0size_heap_free_last_GC=3961493288threads_in_ML=1threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=23.964time_GC=1.144user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/src/HOL/Library/Infinite_Set.thyoffset=2864name=byelapsed=0.016
command_timing = :file=~~/src/HOL/Library/More_List.thyoffset=1105name=byelapsed=0.033
command_timing = :file=~~/src/HOL/Library/More_List.thyoffset=5441name=byelapsed=0.024
command_timing = :file=~~/afp/thys/Sqrt_Babylonian/Sqrt_Babylonian_Auxiliary.thyoffset=1156name=byelapsed=0.018
command_timing = :file=~~/afp/thys/Sqrt_Babylonian/Sqrt_Babylonian_Auxiliary.thyoffset=1233name=byelapsed=0.011
command_timing = :file=~~/afp/thys/Sqrt_Babylonian/Sqrt_Babylonian_Auxiliary.thyoffset=4562name=byelapsed=0.032
command_timing = :file=~~/afp/thys/Sqrt_Babylonian/Sqrt_Babylonian_Auxiliary.thyoffset=2035name=byelapsed=0.045
command_timing = :file=~~/src/HOL/Library/More_List.thyoffset=8394name=byelapsed=0.017
command_timing = :file=~~/src/HOL/Library/More_List.thyoffset=7222name=byelapsed=0.035
command_timing = :file=~~/src/HOL/Library/More_List.thyoffset=8811name=byelapsed=0.038
command_timing = :file=~~/afp/thys/Sqrt_Babylonian/Sqrt_Babylonian_Auxiliary.thyoffset=4752name=byelapsed=0.014
command_timing = :file=~~/src/HOL/Library/More_List.thyoffset=10401name=byelapsed=0.014
command_timing = :file=~~/afp/thys/Sqrt_Babylonian/Sqrt_Babylonian_Auxiliary.thyoffset=4864name=byelapsed=0.035
command_timing = :file=~~/afp/thys/Sqrt_Babylonian/Sqrt_Babylonian_Auxiliary.thyoffset=5012name=byelapsed=0.011
command_timing = :file=~~/src/HOL/Library/More_List.thyoffset=10334name=byelapsed=0.015
command_timing = :file=~~/afp/thys/Sqrt_Babylonian/Sqrt_Babylonian_Auxiliary.thyoffset=5426name=byelapsed=0.010
command_timing = :file=~~/src/HOL/Library/More_List.thyoffset=10098name=byelapsed=0.014
command_timing = :file=~~/afp/thys/Sqrt_Babylonian/Sqrt_Babylonian_Auxiliary.thyoffset=5582name=byelapsed=0.010
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=23414name=byelapsed=0.010
command_timing = :file=~~/afp/thys/Sqrt_Babylonian/Sqrt_Babylonian_Auxiliary.thyoffset=5846name=byelapsed=0.014
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=32016name=byelapsed=0.011
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=23666name=byelapsed=0.011
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=44562name=byelapsed=0.010
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=45400name=proofelapsed=0.011
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=47974name=byelapsed=0.024
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=48073name=byelapsed=0.034
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=62076name=byelapsed=0.014
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=48439name=applyelapsed=0.030
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=62254name=byelapsed=0.013
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=68124name=byelapsed=0.011
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=67298name=byelapsed=0.019
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=70449name=byelapsed=0.019
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=70105name=byelapsed=0.035
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=88258name=byelapsed=0.010
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=73789name=byelapsed=0.012
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=88845name=applyelapsed=0.023
ML_statistics = :now=1489705813.26tasks_ready=8tasks_pending=1tasks_running=2tasks_passive=0tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=5size_allocation=3959422976size_allocation_free=2725940856size_heap=4155506688size_heap_free_last_full_GC=0size_heap_free_last_GC=3961493288threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=24.76time_GC=1.144user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=88924name=applyelapsed=0.018
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=94336name=byelapsed=0.017
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=110571name=byelapsed=0.024
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=122444name=byelapsed=0.010
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=92347name=byelapsed=0.012
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=125209name=byelapsed=0.042
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=124151name=byelapsed=0.017
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=136438name=byelapsed=0.016
command_timing = :file=~~/src/HOL/Library/Polynomial.thyoffset=148737name=byelapsed=0.030
Loading theory "Cancellation" (required by "Polynomial_Interpolation" via "Newton_Interpolation" via "Ring_Hom_Poly" via "Euclidean_Algorithm" via "Factorial_Ring" via "Multiset")
Loading theory "Divmod_Int"
loading_theory = Cancellation
loading_theory = Divmod_Int
ML_statistics = :now=1489705813.76tasks_ready=6tasks_pending=11tasks_running=2tasks_passive=0tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=5size_allocation=3959422976size_allocation_free=2154863552size_heap=4155506688size_heap_free_last_full_GC=0size_heap_free_last_GC=3961493288threads_in_ML=1threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=25.484time_GC=1.144user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/src/HOL/Library/Cancellation.thyoffset=261name=theoryelapsed=0.084
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Divmod_Int.thyoffset=192name=theoryelapsed=0.109
### theory "Divmod_Int"
### 0.123s elapsed time, 0.284s cpu time, 0.000s GC time
Loading theory "Fraction_Field" (required by "Polynomial_Interpolation" via "Newton_Interpolation" via "Ring_Hom_Poly" via "Missing_Polynomial" via "Polynomial_Factorial" via "Normalized_Fraction")
loading_theory = Fraction_Field
command_timing = :file=~~/src/HOL/Library/Cancellation.thyoffset=301name=named_theoremselapsed=0.109
command_timing = :file=~~/src/HOL/Library/Fraction_Field.thyoffset=248name=theoryelapsed=0.098
command_timing = :file=~~/src/HOL/Library/Cancellation.thyoffset=507name=named_theoremselapsed=0.108
command_timing = :file=~~/src/HOL/Library/Cancellation.thyoffset=718name=named_theoremselapsed=0.107
command_timing = :file=~~/src/HOL/Library/Fraction_Field.thyoffset=390name=contextelapsed=0.193
command_timing = :file=~~/src/HOL/Library/Fraction_Field.thyoffset=410name=definitionelapsed=0.026
### ML warning (line 57 of "~~/src/HOL/Library/Cancellation/cancel.ML"):
### Pattern is not exhaustive.
signature CANCEL = sig val proc: Proof.context -> cterm -> thm option end
functor Cancel_Fun (Data: CANCEL_NUMERALS_DATA): CANCEL
signature CANCEL_DATA =
  sig
    val dest_coeff: term -> int * term
    val dest_sum: term -> term list
    val find_first_coeff: term -> term list -> int * term list
    val mk_coeff: int * term -> term
    val mk_sum: typ -> term list -> term
    val norm_ss1: simpset
    val norm_ss2: simpset
    val norm_tac: Proof.context -> tactic
    val numeral_simp_tac: Proof.context -> tactic
    val prove_conv:
       tactic list -> Proof.context -> thm list -> term * term -> thm option
    val simplify_meta_eq: Proof.context -> thm -> thm
    val trans_tac: Proof.context -> thm option -> tactic
  end
structure Cancel_Data: CANCEL_DATA
command_timing = :file=~~/src/HOL/Library/Cancellation.thyoffset=3528name=ML_fileelapsed=0.026
command_timing = :file=~~/src/HOL/Library/Fraction_Field.thyoffset=1583name=quotient_typeelapsed=0.075
signature CANCEL_SIMPROCS =
  sig
    val diff_cancel: Proof.context -> cterm -> thm option
    val eq_cancel: Proof.context -> cterm -> thm option
    val less_cancel: Proof.context -> cterm -> thm option
    val less_eq_cancel: Proof.context -> cterm -> thm option
  end
structure Cancel_Simprocs: CANCEL_SIMPROCS
command_timing = :file=~~/src/HOL/Library/Cancellation.thyoffset=3566name=ML_fileelapsed=0.054
### theory "Cancellation"
### 0.552s elapsed time, 1.140s cpu time, 0.000s GC time
Loading theory "Multiset" (required by "Polynomial_Interpolation" via "Newton_Interpolation" via "Ring_Hom_Poly" via "Euclidean_Algorithm" via "Factorial_Ring")
loading_theory = Multiset
command_timing = :file=~~/src/HOL/Library/Fraction_Field.thyoffset=1745name=lift_definitionelapsed=0.011
ML_statistics = :now=1489705814.26tasks_ready=28tasks_pending=11tasks_running=2tasks_passive=0tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=5size_allocation=3959422976size_allocation_free=1790976512size_heap=4155506688size_heap_free_last_full_GC=0size_heap_free_last_GC=3961493288threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=26.184time_GC=1.144user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
instantiation
  fract :: (idom) comm_ring_1
  uminus_fract == uminus :: 'a fract \<Rightarrow> 'a fract
  one_fract == one_class.one :: 'a fract
  times_fract == times ::
    'a fract \<Rightarrow> 'a fract \<Rightarrow> 'a fract
  zero_fract == zero_class.zero :: 'a fract
  minus_fract == minus ::
    'a fract \<Rightarrow> 'a fract \<Rightarrow> 'a fract
  plus_fract == plus ::
    'a fract \<Rightarrow> 'a fract \<Rightarrow> 'a fract
command_timing = :file=~~/src/HOL/Library/Fraction_Field.thyoffset=2319name=lift_definitionelapsed=0.017
command_timing = :file=~~/src/HOL/Library/Fraction_Field.thyoffset=2437name=lift_definitionelapsed=0.011
command_timing = :file=~~/src/HOL/Library/Fraction_Field.thyoffset=2553name=lift_definitionelapsed=0.016
command_timing = :file=~~/src/HOL/Library/Fraction_Field.thyoffset=2829name=lift_definitionelapsed=0.013
command_timing = :file=~~/src/HOL/Library/Fraction_Field.thyoffset=3140name=definitionelapsed=0.010
command_timing = :file=~~/src/HOL/Library/Fraction_Field.thyoffset=3333name=lift_definitionelapsed=0.020
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=338name=theoryelapsed=0.118
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=480name=typedefelapsed=0.024
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=655name=setup_liftingelapsed=0.043
instantiation
  multiset :: (type) cancel_comm_monoid_add
  zero_multiset == zero_class.zero :: 'a multiset
  minus_multiset == minus ::
    'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset
  plus_multiset == plus ::
    'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=2038name=lift_definitionelapsed=0.011
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=2193name=lift_definitionelapsed=0.018
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=2331name=lift_definitionelapsed=0.018
command_timing = :file=~~/src/HOL/Library/Fraction_Field.thyoffset=5457name=contextelapsed=0.204
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=2837name=lift_definitionelapsed=0.013
command_timing = :file=~~/src/HOL/Library/Fraction_Field.thyoffset=5477name=subclasselapsed=0.037
ML_statistics = :now=1489705814.76tasks_ready=108tasks_pending=11tasks_running=2tasks_passive=0tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=5size_allocation=3959422976size_allocation_free=499666272size_heap=4155506688size_heap_free_last_full_GC=0size_heap_free_last_GC=3961493288threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=27.78time_GC=1.144user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/src/HOL/Library/Fraction_Field.thyoffset=5477name=subclasselapsed=0.130
instantiation
  fract :: (idom) field
  inverse_fract == inverse :: 'a fract \<Rightarrow> 'a fract
  divide_fract == divide ::
    'a fract \<Rightarrow> 'a fract \<Rightarrow> 'a fract
command_timing = :file=~~/src/HOL/Library/Fraction_Field.thyoffset=5560name=lift_definitionelapsed=0.015
command_timing = :file=~~/src/HOL/Library/Fraction_Field.thyoffset=5797name=definitionelapsed=0.017
instantiation
  fract :: (linordered_idom) linorder
  less_eq_fract == less_eq ::
    'a fract \<Rightarrow> 'a fract \<Rightarrow> bool
  less_fract == less :: 'a fract \<Rightarrow> 'a fract \<Rightarrow> bool
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=14181name=definitionelapsed=0.011
command_timing = :file=~~/src/HOL/Library/Fraction_Field.thyoffset=7946name=lift_definitionelapsed=0.010
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=14309name=definitionelapsed=0.010
command_timing = :file=~~/src/HOL/Library/Fraction_Field.thyoffset=7946name=lift_definitionelapsed=0.019
command_timing = :file=~~/src/HOL/Library/Fraction_Field.thyoffset=8138name=definitionelapsed=0.010
instantiation
  fract :: (linordered_idom) linordered_field
  sgn_fract == sgn :: 'a fract \<Rightarrow> 'a fract
  abs_fract == abs :: 'a fract \<Rightarrow> 'a fract
command_timing = :file=~~/src/HOL/Library/Fraction_Field.thyoffset=10870name=definitionelapsed=0.010
command_timing = :file=~~/src/HOL/Library/Fraction_Field.thyoffset=10946name=definitionelapsed=0.017
instantiation
  fract :: (linordered_idom) distrib_lattice
  inf_fract == inf :: 'a fract \<Rightarrow> 'a fract \<Rightarrow> 'a fract
  sup_fract == sup :: 'a fract \<Rightarrow> 'a fract \<Rightarrow> 'a fract
command_timing = :file=~~/src/HOL/Library/Fraction_Field.thyoffset=12827name=definitionelapsed=0.012
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=14923name=interpretationelapsed=0.073
command_timing = :file=~~/src/HOL/Library/Fraction_Field.thyoffset=12904name=definitionelapsed=0.012
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=15188name=interpretationelapsed=0.019
### theory "Fraction_Field"
### 1.193s elapsed time, 2.376s cpu time, 0.000s GC time
Loading theory "Improved_Code_Equations"
loading_theory = Improved_Code_Equations
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=19849name=definitionelapsed=0.013
### Partially applied constant "Multiset.inf_subset_mset" on left hand side of equation, in theorem:
### semilattice_inf.Inf_fin op \<inter># (set (?x # ?xs)) \<equiv>
### fold op \<inter># ?xs ?x
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=20003name=interpretationelapsed=0.215
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=20352name=definitionelapsed=0.011
### Partially applied constant "Multiset.sup_subset_mset" on left hand side of equation, in theorem:
### semilattice_sup.Sup_fin op \<union># (set (?x # ?xs)) \<equiv>
### fold op \<union># ?xs ?x
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=20515name=interpretationelapsed=0.021
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=20865name=interpretationelapsed=0.016
ML_statistics = :now=1489705815.42tasks_ready=202tasks_pending=11tasks_running=2tasks_passive=0tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=6size_allocation=3991928832size_allocation_free=3991928832size_heap=4140826624size_heap_free_last_full_GC=0size_heap_free_last_GC=3993382760threads_in_ML=1threads_total=7threads_wait_condvar=2threads_wait_IO=0threads_wait_mutex=1threads_wait_signal=1time_CPU=28.548time_GC=1.484user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
Found termination order: "(\<lambda>p. size (fst p)) <*mlex*> {}"
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=27271name=funelapsed=0.081
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Improved_Code_Equations.thyoffset=206name=theoryelapsed=0.602
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Improved_Code_Equations.thyoffset=498name=lemmaelapsed=0.018
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Improved_Code_Equations.thyoffset=1442name=definitionelapsed=0.010
signature MULTISET_SIMPROCS =
  sig
    val subset_cancel_msets: Proof.context -> cterm -> thm option
    val subseteq_cancel_msets: Proof.context -> cterm -> thm option
  end
structure Multiset_Simprocs: MULTISET_SIMPROCS
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=29674name=ML_fileelapsed=0.037
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Improved_Code_Equations.thyoffset=1607name=lemmaelapsed=0.021
### Code generator: dropping subsumed code equation
### divmod_integer ?k ?l \<equiv>
### if ?k = 0 then (0, 0)
### else if ?l = 0 then (0, ?k)
###      else (apsnd \<circ> op * \<circ> sgn) ?l
###            (if sgn ?k = sgn ?l then Code_Numeral.divmod_abs ?k ?l
###             else let (r, s) = Code_Numeral.divmod_abs ?k ?l
###                  in if s = 0 then (- r, 0)
###                     else (- r - 1, \<bar>?l\<bar> - s))
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=30372name=simproc_setupelapsed=0.010
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=30656name=simproc_setupelapsed=0.011
### Code generator: dropping subsumed code equation
### Divides.divmod_nat ?m ?n \<equiv>
### if ?n = 0 \<or> ?m < ?n then (0, ?m)
### else let (q, y) = Divides.divmod_nat (?m - ?n) ?n in (Suc q, y)
### Code generator: dropping subsumed code equation
### Suc ?n choose Suc ?k \<equiv> ?n choose ?k + (?n choose Suc ?k)
### Code generator: dropping subsumed code equation
### 0 choose Suc ?k \<equiv> 0
### Code generator: dropping subsumed code equation
### ?n choose 0 \<equiv> 1
### theory "Improved_Code_Equations"
### 0.697s elapsed time, 1.360s cpu time, 0.340s GC time
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=30964name=simproc_setupelapsed=0.010
Loading theory "Neville_Aitken_Interpolation" (required by "Polynomial_Interpolation")
loading_theory = Neville_Aitken_Interpolation
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=31277name=simproc_setupelapsed=0.011
instantiation
  multiset :: (type) Inf
  Inf_multiset == Inf :: 'a multiset set \<Rightarrow> 'a multiset
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=31662name=lift_definitionelapsed=0.020
instantiation
  multiset :: (type) Sup
  Sup_multiset == Sup :: 'a multiset set \<Rightarrow> 'a multiset
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=32678name=definitionelapsed=0.014
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=35075name=interpretationelapsed=0.018
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=39863name=lift_definitionelapsed=0.014
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Neville_Aitken_Interpolation.thyoffset=354name=theoryelapsed=0.189
ML_statistics = :now=1489705815.92tasks_ready=290tasks_pending=11tasks_running=2tasks_passive=0tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=6size_allocation=3991928832size_allocation_free=3403418192size_heap=4140826624size_heap_free_last_full_GC=0size_heap_free_last_GC=3993382760threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=29.356time_GC=1.484user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
instantiation
  multiset :: (type) size
  size_multiset == size :: 'a multiset \<Rightarrow> nat
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=43400name=definitionelapsed=0.010
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Neville_Aitken_Interpolation.thyoffset=592name=functionelapsed=0.107
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Neville_Aitken_Interpolation.thyoffset=592name=functionelapsed=0.016
locale comp_fun_commute =
  fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" 
  assumes "comp_fun_commute f"
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=54754name=textelapsed=0.013
consts
  mset :: "'a list \<Rightarrow> 'a multiset"
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=55974name=primrecelapsed=0.010
Found termination order: "size_list (\<lambda>p. size (snd p)) <*mlex*> {}"
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Neville_Aitken_Interpolation.thyoffset=3917name=funelapsed=0.217
class linorder = order +
  assumes "linear": "\<And>x y. x \<le> y \<or> y \<le> x"
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=62246name=contextelapsed=0.087
ML_statistics = :now=1489705816.42tasks_ready=431tasks_pending=11tasks_running=2tasks_passive=0tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=6size_allocation=3991928832size_allocation_free=2910349672size_heap=4140826624size_heap_free_last_full_GC=0size_heap_free_last_GC=3993382760threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=30.156time_GC=1.484user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=62270name=definitionelapsed=0.023
locale comm_monoid_mset =
  fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" 
    and z :: "'a" 
  assumes "comm_monoid_mset op \<^bold>* \<^bold>1"
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=66606name=localeelapsed=0.017
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=66833name=definitionelapsed=0.012
Found termination order: "length <*mlex*> {}"
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Neville_Aitken_Interpolation.thyoffset=4442name=funelapsed=0.353
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Neville_Aitken_Interpolation.thyoffset=4712name=lemmaelapsed=0.012
class comm_monoid_add = ab_semigroup_add + monoid_add +
  assumes "add_0": "\<And>a. (0::'a) + a = a"
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=68954name=contextelapsed=0.046
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Neville_Aitken_Interpolation.thyoffset=7238name=definitionelapsed=0.013
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=68985name=sublocaleelapsed=0.017
### theory "Neville_Aitken_Interpolation"
### 1.009s elapsed time, 2.016s cpu time, 0.000s GC time
Loading theory "Missing_Unsorted" (required by "Polynomial_Interpolation" via "Newton_Interpolation" via "Ring_Hom_Poly" via "Missing_Polynomial")
loading_theory = Missing_Unsorted
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=68985name=sublocaleelapsed=0.019
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=69065name=lemmaelapsed=0.043
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=69065name=lemmaelapsed=0.027
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Unsorted.thyoffset=346name=theoryelapsed=0.153
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=71372name=lemmaelapsed=0.053
ML_statistics = :now=1489705816.92tasks_ready=507tasks_pending=11tasks_running=2tasks_passive=0tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=6size_allocation=3991928832size_allocation_free=1756251672size_heap=4140826624size_heap_free_last_full_GC=0size_heap_free_last_GC=3993382760threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=31.76time_GC=1.484user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
class comm_monoid_mult = ab_semigroup_mult + monoid_mult + dvd +
  assumes "mult_1": "\<And>a. (1::'a) * a = a"
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=72541name=contextelapsed=0.055
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=72573name=sublocaleelapsed=0.016
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=72573name=sublocaleelapsed=0.021
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=74004name=lemmaelapsed=0.033
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=74335name=lemmaelapsed=0.034
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=74488name=lemmaelapsed=0.100
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=74584name=lemmaelapsed=0.108
ML_statistics = :now=1489705817.43tasks_ready=523tasks_pending=11tasks_running=2tasks_passive=0tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=6size_allocation=3991928832size_allocation_free=1073498864size_heap=4140826624size_heap_free_last_full_GC=0size_heap_free_last_GC=3993382760threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=32.56time_GC=1.484user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=74847name=lemmaelapsed=0.105
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=75026name=lemmaelapsed=0.111
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=75174name=lemmaelapsed=0.117
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=75341name=lemmaelapsed=0.156
ML_statistics = :now=1489705818.16tasks_ready=530tasks_pending=11tasks_running=2tasks_passive=0tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=7size_allocation=3893362688size_allocation_free=3893362688size_heap=4120903680size_heap_free_last_full_GC=0size_heap_free_last_GC=3894441728threads_in_ML=1threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=33.716time_GC=1.964user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
class ring_div_field = field + modulo +
  assumes "mod": "\<And>x y. x mod y = (if y = (0::'a) then x else (0::'a))"
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Unsorted.thyoffset=8652name=classelapsed=1.310
class linorder = order +
  assumes "linear": "\<And>x y. x \<le> y \<or> y \<le> x"
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=75781name=contextelapsed=0.359
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Unsorted.thyoffset=8759name=subclasselapsed=0.063
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=80724name=definitionelapsed=0.023
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=80899name=lemmaelapsed=0.011
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=81212name=lemmaelapsed=0.018
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=82884name=definitionelapsed=0.011
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=92354name=definitionelapsed=0.012
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=92563name=definitionelapsed=0.016
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=94227name=lemmaelapsed=0.017
instantiation
  multiset :: (preorder) order
  less_eq_multiset == less_eq ::
    'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool
  less_multiset == less ::
    'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=94674name=definitionelapsed=0.013
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=94790name=definitionelapsed=0.011
instantiation
  multiset :: (preorder) ordered_ab_semigroup_add
Proofs for inductive predicate(s) "pw_leq"
  Proving monotonicity ...
  Proving the introduction rules ...
  Proving the elimination rules ...
  Proving the induction rule ...
  Proving the simplification rules ...
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=98043name=inductiveelapsed=0.033
ML_statistics = :now=1489705818.66tasks_ready=581tasks_pending=11tasks_running=2tasks_passive=0tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=7size_allocation=3893362688size_allocation_free=3283482472size_heap=4120903680size_heap_free_last_full_GC=0size_heap_free_last_GC=3894441728threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=1threads_wait_signal=1time_CPU=34.524time_GC=1.964user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=100718name=setupelapsed=0.014
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=105207name=code_datatypeelapsed=0.019
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Unsorted.thyoffset=8759name=subclasselapsed=0.481
### Missing patterns in function definition:
### max_list_non_empty [] = undefined
Found termination order: "length <*mlex*> {}"
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Unsorted.thyoffset=13063name=funelapsed=0.118
Found termination order: "(\<lambda>p. length (fst p)) <*mlex*> {}"
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=107240name=funelapsed=0.186
instantiation
  multiset :: (equal) equal
  equal_multiset == equal_class.equal ::
    'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool
### theory "Missing_Unsorted"
### 2.312s elapsed time, 4.596s cpu time, 0.480s GC time
Loading theory "Is_Rat_To_Rat"
loading_theory = Is_Rat_To_Rat
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=109155name=definitionelapsed=0.012
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=109905name=definitionelapsed=0.064
instantiation
  multiset :: (random) random
  random_multiset == random_class.random ::
    natural
    \<Rightarrow> natural \<times> natural
                  \<Rightarrow> ('a multiset \<times>
                                 (unit \<Rightarrow> term)) \<times>
                                natural \<times> natural
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=110239name=definitionelapsed=0.014
ML_statistics = :now=1489705819.16tasks_ready=673tasks_pending=11tasks_running=2tasks_passive=0tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=7size_allocation=3893362688size_allocation_free=2806238288size_heap=4120903680size_heap_free_last_full_GC=0size_heap_free_last_GC=3894441728threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=35.324time_GC=1.964user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=110357name=no_notationelapsed=0.116
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Is_Rat_To_Rat.thyoffset=498name=theoryelapsed=0.242
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=110392name=no_notationelapsed=0.114
instantiation
  multiset :: (full_exhaustive) full_exhaustive
  full_exhaustive_multiset == full_exhaustive_class.full_exhaustive ::
    ('a multiset \<times> (unit \<Rightarrow> term)
     \<Rightarrow> (bool \<times> term list) option)
    \<Rightarrow> natural \<Rightarrow> (bool \<times> term list) option
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=110495name=definitionelapsed=0.015
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=110806name=definitionelapsed=0.011
Proofs for inductive predicate(s) "pred_mset"
  Proving monotonicity ...
  Proving the introduction rules ...
  Proving the elimination rules ...
  Proving the induction rule ...
  Proving the simplification rules ...
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=114186name=inductiveelapsed=0.030
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=114325name=bnfelapsed=0.055
Proofs for inductive predicate(s) "rel_mset'"
  Proving monotonicity ...
  Proving the introduction rules ...
  Proving the elimination rules ...
  Proving the induction rule ...
  Proving the simplification rules ...
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=116461name=inductiveelapsed=0.032
### theory "Multiset"
### 5.344s elapsed time, 10.624s cpu time, 0.820s GC time
Loading theory "Factorial_Ring" (required by "Polynomial_Interpolation" via "Newton_Interpolation" via "Ring_Hom_Poly" via "Euclidean_Algorithm")
loading_theory = Factorial_Ring
ML_statistics = :now=1489705819.66tasks_ready=744tasks_pending=10tasks_running=2tasks_passive=0tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=7size_allocation=3893362688size_allocation_free=2412879984size_heap=4120903680size_heap_free_last_full_GC=0size_heap_free_last_GC=3894441728threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=36.124time_GC=1.964user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=179name=theoryelapsed=0.386
ML_statistics = :now=1489705820.16tasks_ready=746tasks_pending=10tasks_running=2tasks_passive=0tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=7size_allocation=3893362688size_allocation_free=1153515608size_heap=4120903680size_heap_free_last_full_GC=0size_heap_free_last_GC=3894441728threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=37.728time_GC=1.964user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=320name=contextelapsed=0.145
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=351name=definitionelapsed=0.026
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=966name=definitionelapsed=0.027
class is_rat = field_char_0 +
  fixes is_rat :: "'a \<Rightarrow> bool" 
    and to_rat :: "'a \<Rightarrow> rat" 
  assumes
    "is_rat":
      "\<And>x.
          is_rat x =
          (x \<in> field_char_0.Rats op / (1::'a) op + (0::'a) op -)"
    and
    "to_rat":
      "\<And>x.
          to_rat x =
          (if x \<in> field_char_0.Rats op / (1::'a) op + (0::'a) op -
           then THE y.
                   x = field_char_0.of_rat op / (1::'a) op + (0::'a) op - y
           else 0)"
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Is_Rat_To_Rat.thyoffset=594name=classelapsed=1.171
instantiation
  rat :: is_rat
  is_rat_rat == is_rat :: rat \<Rightarrow> bool
  to_rat_rat == to_rat :: rat \<Rightarrow> rat
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Is_Rat_To_Rat.thyoffset=1006name=definitionelapsed=0.012
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Is_Rat_To_Rat.thyoffset=1048name=definitionelapsed=0.011
instantiation
  real :: is_rat
  is_rat_real == is_rat :: real \<Rightarrow> bool
  to_rat_real == to_rat :: real \<Rightarrow> rat
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Is_Rat_To_Rat.thyoffset=1356name=definitionelapsed=0.014
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Is_Rat_To_Rat.thyoffset=1403name=definitionelapsed=0.015
instantiation
  complex :: is_rat
  is_rat_complex == is_rat :: complex \<Rightarrow> bool
  to_rat_complex == to_rat :: complex \<Rightarrow> rat
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Is_Rat_To_Rat.thyoffset=3296name=definitionelapsed=0.013
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Is_Rat_To_Rat.thyoffset=3368name=definitionelapsed=0.018
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=2363name=contextelapsed=0.211
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Is_Rat_To_Rat.thyoffset=4068name=definitionelapsed=0.012
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Is_Rat_To_Rat.thyoffset=4156name=definitionelapsed=0.011
### theory "Is_Rat_To_Rat"
### 1.572s elapsed time, 3.144s cpu time, 0.000s GC time
Loading theory "Ring_Hom" (required by "Polynomial_Interpolation" via "Newton_Interpolation" via "Ring_Hom_Poly")
loading_theory = Ring_Hom
ML_statistics = :now=1489705820.67tasks_ready=776tasks_pending=10tasks_running=2tasks_passive=0tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=7size_allocation=3893362688size_allocation_free=420413216size_heap=4120903680size_heap_free_last_full_GC=0size_heap_free_last_GC=3894441728threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=38.536time_GC=1.964user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=2945name=lemmaelapsed=0.103
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Ring_Hom.thyoffset=154name=theoryelapsed=0.104
locale semiring_hom =
  fixes hom :: "'a \<Rightarrow> 'b" 
  assumes "semiring_hom hom"
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Ring_Hom.thyoffset=199name=localeelapsed=0.012
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=2945name=lemmaelapsed=0.103
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Ring_Hom.thyoffset=453name=named_theoremselapsed=0.109
locale ring_hom =
  fixes hom :: "'a \<Rightarrow> 'b" 
  assumes "ring_hom hom"
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Ring_Hom.thyoffset=1856name=lemmaelapsed=0.221
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=7307name=lemmaelapsed=0.225
locale inj_semiring_hom =
  fixes hom :: "'a \<Rightarrow> 'b" 
  assumes "inj_semiring_hom hom"
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Ring_Hom.thyoffset=2089name=localeelapsed=0.011
locale inj_ring_hom =
  fixes hom :: "'a \<Rightarrow> 'b" 
  assumes "inj_ring_hom hom"
locale idom_hom =
  fixes hom :: "'a \<Rightarrow> 'b" 
  assumes "idom_hom hom"
locale field_hom =
  fixes hom :: "'a \<Rightarrow> 'b" 
  assumes "field_hom hom"
locale inj_field_hom =
  fixes hom :: "'a \<Rightarrow> 'b" 
  assumes "inj_field_hom hom"
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Ring_Hom.thyoffset=3193name=localeelapsed=0.011
locale inj_field_hom_0 =
  fixes hom :: "'a \<Rightarrow> 'b" 
  assumes "inj_field_hom_0 hom"
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Ring_Hom.thyoffset=3331name=localeelapsed=0.012
### theory "Ring_Hom"
### 0.568s elapsed time, 1.116s cpu time, 0.408s GC time
command_timing = :file=~~/src/HOL/Library/Fraction_Field.thyoffset=2249name=byelapsed=0.024
ML_statistics = :now=1489705821.36tasks_ready=783tasks_pending=10tasks_running=3tasks_passive=1tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=8size_allocation=3801088000size_allocation_free=3801088000size_heap=4133486592size_heap_free_last_full_GC=0size_heap_free_last_GC=3803089280threads_in_ML=1threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=39.18time_GC=2.372user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/src/HOL/Library/Fraction_Field.thyoffset=2811name=byelapsed=0.015
command_timing = :file=~~/src/HOL/Library/Fraction_Field.thyoffset=2676name=byelapsed=0.020
command_timing = :file=~~/src/HOL/Library/Fraction_Field.thyoffset=4854name=byelapsed=0.010
class normalization_semidom = algebraic_semidom +
  semidom_divide_unit_factor +
  fixes normalize :: "'a \<Rightarrow> 'a" 
  assumes
    "unit_factor_mult_normalize": "\<And>a. unit_factor a * normalize a = a"
    and "normalize_0": "normalize (0::'a) = (0::'a)"
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=11343name=contextelapsed=0.207
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=12633name=definitionelapsed=0.026
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=14569name=lemmaelapsed=0.010
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=16910name=lemmaelapsed=0.011
ML_statistics = :now=1489705821.86tasks_ready=738tasks_pending=10tasks_running=4tasks_passive=2tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=8size_allocation=3801088000size_allocation_free=3028863216size_heap=4133486592size_heap_free_last_full_GC=0size_heap_free_last_GC=3803089280threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=39.992time_GC=2.372user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
class semiring_gcd = gcd + normalization_semidom +
  assumes "gcd_dvd1": "\<And>a b. gcd a b dvd a"
    and "gcd_dvd2": "\<And>a b. gcd a b dvd b"
    and
    "gcd_greatest":
      "\<And>c a b.
          \<lbrakk>c dvd a; c dvd b\<rbrakk>
          \<Longrightarrow> c dvd gcd a b"
    and "normalize_gcd": "\<And>a b. normalize (gcd a b) = gcd a b"
    and "lcm_gcd": "\<And>a b. lcm a b = normalize (a * b) div gcd a b"
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=19108name=contextelapsed=0.297
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=11833name=byelapsed=0.412
command_timing = :file=~~/src/HOL/Library/Fraction_Field.thyoffset=5779name=byelapsed=0.017
command_timing = :file=~~/src/HOL/Library/Fraction_Field.thyoffset=5671name=byelapsed=0.022
command_timing = :file=~~/src/HOL/Library/Fraction_Field.thyoffset=3555name=byelapsed=0.020
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=9935name=byelapsed=0.034
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=9670name=byelapsed=0.019
ML_statistics = :now=1489705822.36tasks_ready=729tasks_pending=10tasks_running=2tasks_passive=0tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=8size_allocation=3801088000size_allocation_free=1433088936size_heap=4133486592size_heap_free_last_full_GC=0size_heap_free_last_GC=3803089280threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=41.596time_GC=2.372user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/src/HOL/Library/Fraction_Field.thyoffset=8901name=byelapsed=0.011
command_timing = :file=~~/src/HOL/Library/Fraction_Field.thyoffset=9153name=byelapsed=0.010
command_timing = :file=~~/src/HOL/Library/Fraction_Field.thyoffset=9461name=byelapsed=0.015
command_timing = :file=~~/src/HOL/Library/Fraction_Field.thyoffset=8314name=byelapsed=0.018
command_timing = :file=~~/src/HOL/Library/Fraction_Field.thyoffset=8091name=byelapsed=0.010
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=15015name=byelapsed=0.026
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=18045name=applyelapsed=0.010
ML_statistics = :now=1489705822.86tasks_ready=692tasks_pending=10tasks_running=3tasks_passive=1tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=8size_allocation=3801088000size_allocation_free=750318024size_heap=4133486592size_heap_free_last_full_GC=0size_heap_free_last_GC=3803089280threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=42.4time_GC=2.372user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
class factorial_semiring = normalization_semidom +
  assumes
    "prime_factorization_exists":
      "\<And>x.
          x \<noteq> (0::'a) \<Longrightarrow>
          \<exists>A.
             (\<forall>x. x \<in># A \<longrightarrow> prime_elem x) \<and>
             prod_mset A = normalize x"
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=21094name=classelapsed=0.865
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=18585name=applyelapsed=0.380
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=18671name=applyelapsed=0.045
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=21297name=lemmaelapsed=0.145
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=20731name=byelapsed=0.027
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=20669name=byelapsed=0.084
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=20219name=byelapsed=0.034
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=20157name=byelapsed=0.017
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=21210name=byelapsed=0.069
ML_statistics = :now=1489705823.43tasks_ready=676tasks_pending=10tasks_running=2tasks_passive=0tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=9size_allocation=3752853504size_allocation_free=3752853504size_heap=4156555264size_heap_free_last_full_GC=0size_heap_free_last_GC=3754888080threads_in_ML=1threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=43.292time_GC=2.508user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
class factorial_semiring = normalization_semidom +
  assumes
    "prime_factorization_exists":
      "\<And>x.
          x \<noteq> (0::'a) \<Longrightarrow>
          \<exists>A.
             (\<forall>x. x \<in># A \<longrightarrow> prime_elem x) \<and>
             prod_mset A = normalize x"
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=24250name=contextelapsed=0.306
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=25392name=byelapsed=0.038
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=28394name=definitionelapsed=0.034
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Improved_Code_Equations.thyoffset=3804name=byelapsed=0.010
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Improved_Code_Equations.thyoffset=2133name=byelapsed=0.088
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Improved_Code_Equations.thyoffset=1153name=byelapsed=0.087
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=36671name=lift_definitionelapsed=0.050
ML_statistics = :now=1489705823.93tasks_ready=659tasks_pending=10tasks_running=3tasks_passive=1tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=9size_allocation=3752853504size_allocation_free=2988710280size_heap=4156555264size_heap_free_last_full_GC=0size_heap_free_last_GC=3754888080threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=44.088time_GC=2.508user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=37226name=abbreviationelapsed=0.011
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=34559name=byelapsed=0.013
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=38657name=byelapsed=0.013
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=38481name=byelapsed=0.013
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=44339name=applyelapsed=0.018
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=44126name=byelapsed=0.023
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=46648name=byelapsed=0.015
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=52676name=definitionelapsed=0.028
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=52843name=definitionelapsed=0.028
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=46884name=byelapsed=0.017
ML_statistics = :now=1489705824.43tasks_ready=645tasks_pending=10tasks_running=2tasks_passive=0tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=9size_allocation=3752853504size_allocation_free=1491032040size_heap=4156555264size_heap_free_last_full_GC=0size_heap_free_last_GC=3754888080threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=45.692time_GC=2.508user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=52972name=definitionelapsed=0.030
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=53081name=definitionelapsed=0.032
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Neville_Aitken_Interpolation.thyoffset=3729name=byelapsed=0.010
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Neville_Aitken_Interpolation.thyoffset=3321name=byelapsed=0.064
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Neville_Aitken_Interpolation.thyoffset=3057name=byelapsed=0.028
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=62300name=lemmaselapsed=0.015
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Neville_Aitken_Interpolation.thyoffset=875name=byelapsed=0.054
### Introduced fixed type variable(s): 'd, 'e, 'f in "f__" or "g__"
### Introduced fixed type variable(s): 'h in "A__"
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=41407name=byelapsed=0.015
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=58130name=byelapsed=0.010
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=57726name=byelapsed=0.018
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=57538name=byelapsed=0.011
ML_statistics = :now=1489705824.93tasks_ready=618tasks_pending=10tasks_running=2tasks_passive=0tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=9size_allocation=3752853504size_allocation_free=610231192size_heap=4156555264size_heap_free_last_full_GC=0size_heap_free_last_GC=3754888080threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=46.5time_GC=2.508user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=61071name=byelapsed=0.016
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=60023name=byelapsed=0.010
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=67295name=byelapsed=0.092
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=67752name=byelapsed=0.012
ML_statistics = :now=1489705825.51tasks_ready=570tasks_pending=10tasks_running=7tasks_passive=5tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=10size_allocation=3769630720size_allocation_free=3769630720size_heap=4167041024size_heap_free_last_full_GC=0size_heap_free_last_GC=3771841584threads_in_ML=1threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=47.1time_GC=2.664user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
class factorial_semiring_gcd = factorial_semiring + Gcd +
  assumes "gcd_eq_gcd_factorial": "\<And>a b. gcd a b = gcd_factorial a b"
    and "lcm_eq_lcm_factorial": "\<And>a b. lcm a b = lcm_factorial a b"
    and "Gcd_eq_Gcd_factorial": "\<And>A. Gcd A = Gcd_factorial A"
    and "Lcm_eq_Lcm_factorial": "\<And>A. Lcm A = Lcm_factorial A"
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=62581name=classelapsed=0.891
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=63828name=subclasselapsed=0.044
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Neville_Aitken_Interpolation.thyoffset=2316name=byelapsed=0.345
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Neville_Aitken_Interpolation.thyoffset=2156name=byelapsed=0.015
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Neville_Aitken_Interpolation.thyoffset=1924name=byelapsed=0.072
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Neville_Aitken_Interpolation.thyoffset=6981name=byelapsed=0.010
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Neville_Aitken_Interpolation.thyoffset=6759name=qedelapsed=0.013
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Neville_Aitken_Interpolation.thyoffset=6568name=byelapsed=0.021
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Neville_Aitken_Interpolation.thyoffset=5426name=haveelapsed=0.010
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Neville_Aitken_Interpolation.thyoffset=6206name=qedelapsed=0.012
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=63828name=subclasselapsed=0.287
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Neville_Aitken_Interpolation.thyoffset=5662name=byelapsed=0.044
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Neville_Aitken_Interpolation.thyoffset=5289name=byelapsed=0.020
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=63960name=subclasselapsed=0.047
ML_statistics = :now=1489705826.02tasks_ready=558tasks_pending=10tasks_running=3tasks_passive=0tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=10size_allocation=3769630720size_allocation_free=2393231304size_heap=4167041024size_heap_free_last_full_GC=0size_heap_free_last_GC=3771841584threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=48.704time_GC=2.664user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=70242name=byelapsed=0.029
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=68249name=applyelapsed=0.014
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=63960name=subclasselapsed=0.276
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=64092name=lemmaelapsed=0.014
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=64092name=lemmaelapsed=0.011
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=65074name=lemmaelapsed=0.010
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=75885name=byelapsed=0.011
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=78555name=byelapsed=0.023
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=65854name=byelapsed=0.012
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=76817name=byelapsed=0.015
ML_statistics = :now=1489705826.52tasks_ready=517tasks_pending=10tasks_running=6tasks_passive=4tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=10size_allocation=3769630720size_allocation_free=1738777464size_heap=4167041024size_heap_free_last_full_GC=0size_heap_free_last_GC=3771841584threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=49.508time_GC=2.664user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=76623name=byelapsed=0.095
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=76535name=byelapsed=0.023
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=58839name=byelapsed=0.052
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=80374name=byelapsed=0.040
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=79857name=byelapsed=0.016
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=79511name=byelapsed=0.020
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=80231name=byelapsed=0.037
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=80139name=byelapsed=0.041
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=79238name=byelapsed=0.080
ML_statistics = :now=1489705827.02tasks_ready=513tasks_pending=10tasks_running=4tasks_passive=2tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=10size_allocation=3769630720size_allocation_free=1048682856size_heap=4167041024size_heap_free_last_full_GC=0size_heap_free_last_GC=3771841584threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=50.312time_GC=2.664user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=79153name=byelapsed=0.059
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=84755name=byelapsed=0.015
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=13469name=byelapsed=0.019
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=84237name=byelapsed=0.018
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=96369name=byelapsed=0.049
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=99267name=byelapsed=0.136
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=98784name=byelapsed=0.011
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=98690name=byelapsed=0.010
ML_statistics = :now=1489705827.58tasks_ready=488tasks_pending=10tasks_running=3tasks_passive=1tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=11size_allocation=3737124864size_allocation_free=3737124864size_heap=4178575360size_heap_free_last_full_GC=0size_heap_free_last_GC=3739275048threads_in_ML=1threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=51.52time_GC=2.78user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=97944name=byelapsed=0.036
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=89458name=byelapsed=0.022
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=89357name=byelapsed=0.025
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=89010name=byelapsed=0.098
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=97706name=byelapsed=0.013
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=66996name=classelapsed=1.575
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=67060name=subclasselapsed=0.050
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=117309name=byelapsed=0.049
### Introduced fixed type variable(s): 'd, 'e, 'f in "f__" or "g__"
### Introduced fixed type variable(s): 'd, 'e in "X__" or "f__" or "g__"
### Introduced fixed type variable(s): 'd, 'e in "f__"
### Introduced fixed type variable(s): 'd in "X__"
### Introduced fixed type variable(s): 'd, 'e, 'f in "R__" or "S__"
### Introduced fixed type variable(s): 'd, 'e in "R__"
### Ignoring duplicate rewrite rule:
### mset (map ?f1 ?xs1) \<equiv> image_mset ?f1 (mset ?xs1)
### Ignoring duplicate rewrite rule:
### mset (map ?f1 ?xs1) \<equiv> image_mset ?f1 (mset ?xs1)
### Introduced fixed type variable(s): 'd in "z__"
### Introduced fixed type variable(s): 'd in "P__"
ML_statistics = :now=1489705828.09tasks_ready=428tasks_pending=10tasks_running=6tasks_passive=4tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=11size_allocation=3737124864size_allocation_free=2951082088size_heap=4178575360size_heap_free_last_full_GC=0size_heap_free_last_GC=3739275048threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=52.324time_GC=2.78user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=116406name=byelapsed=0.012
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=115404name=applyelapsed=0.014
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=114054name=byelapsed=0.054
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=113830name=byelapsed=0.029
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=67060name=subclasselapsed=0.306
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=67082name=subclasselapsed=0.054
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=113139name=byelapsed=0.069
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=112252name=byelapsed=0.055
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=112957name=byelapsed=0.041
ML_statistics = :now=1489705828.59tasks_ready=424tasks_pending=10tasks_running=12tasks_passive=10tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=11size_allocation=3737124864size_allocation_free=2487484136size_heap=4178575360size_heap_free_last_full_GC=0size_heap_free_last_GC=3739275048threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=53.132time_GC=2.78user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=67082name=subclasselapsed=0.311
### theory "Factorial_Ring"
### 8.932s elapsed time, 17.828s cpu time, 0.816s GC time
Loading theory "Euclidean_Algorithm" (required by "Polynomial_Interpolation" via "Newton_Interpolation" via "Ring_Hom_Poly")
loading_theory = Euclidean_Algorithm
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=112769name=byelapsed=0.186
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=112043name=byelapsed=0.016
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=111659name=byelapsed=0.017
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=119025name=byelapsed=0.010
command_timing = :file=~~/src/HOL/Number_Theory/Euclidean_Algorithm.thyoffset=170name=theoryelapsed=0.105
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=121644name=byelapsed=0.014
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=120068name=byelapsed=0.028
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=119487name=byelapsed=0.011
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=119372name=byelapsed=0.078
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=119931name=byelapsed=0.015
class euclidean_semiring = semidom_modulo + normalization_semidom +
  fixes euclidean_size :: "'a \<Rightarrow> nat" 
  assumes "size_0": "euclidean_size (0::'a) = 0"
  assumes
    "mod_size_less":
      "\<And>b a.
          b \<noteq> (0::'a) \<Longrightarrow>
          euclidean_size (a mod b) < euclidean_size b"
  assumes
    "size_mult_mono":
      "\<And>b a.
          b \<noteq> (0::'a) \<Longrightarrow>
          euclidean_size a \<le> euclidean_size (a * b)"
command_timing = :file=~~/src/HOL/Number_Theory/Euclidean_Algorithm.thyoffset=350name=contextelapsed=0.278
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=1898name=byelapsed=0.012
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Is_Rat_To_Rat.thyoffset=1100name=byelapsed=0.010
*** Failed to finish proof (line 50 of "~~/afp/thys/Polynomial_Interpolation/Is_Rat_To_Rat.thy"):
*** goal (2 subgoals):
***  1. Complex 0 0 = 0
***  2. \<And>n.
***        of_nat n = Complex (real n) 0 \<Longrightarrow>
***        1 + Complex (real n) 0 = Complex (1 + real n) 0
*** At command "by" (line 50 of "~~/afp/thys/Polynomial_Interpolation/Is_Rat_To_Rat.thy")
*** Failed to finish proof (line 70 of "~~/afp/thys/Polynomial_Interpolation/Is_Rat_To_Rat.thy"):
*** goal (1 subgoal):
***  1. of_int z = - Complex (- real_of_int z) 0 \<Longrightarrow>
***     - Complex (- real_of_int z) 0 = Complex (real_of_int z) 0
*** At command "by" (line 70 of "~~/afp/thys/Polynomial_Interpolation/Is_Rat_To_Rat.thy")
*** Failed to apply initial proof method (line 79 of "~~/afp/thys/Polynomial_Interpolation/Is_Rat_To_Rat.thy"):
*** goal (1 subgoal):
***  1. Complex (real_of_int d / real_of_int n) 0 =
***     Complex (real_of_int d) 0 / Complex (real_of_int n) 0
*** At command "by" (line 79 of "~~/afp/thys/Polynomial_Interpolation/Is_Rat_To_Rat.thy")
ML_statistics = :now=1489705829.09tasks_ready=390tasks_pending=9tasks_running=2tasks_passive=0tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=11size_allocation=3737124864size_allocation_free=1285682496size_heap=4178575360size_heap_free_last_full_GC=0size_heap_free_last_GC=3739275048threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=54.736time_GC=2.78user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
*** Failed to finish proof (line 91 of "~~/afp/thys/Polynomial_Interpolation/Is_Rat_To_Rat.thy"):
*** goal (1 subgoal):
***  1. of_rat q = Complex (real_of_rat q) 0
*** At command "by" (line 91 of "~~/afp/thys/Polynomial_Interpolation/Is_Rat_To_Rat.thy")
command_timing = :file=~~/src/HOL/Number_Theory/Euclidean_Algorithm.thyoffset=409name=functionelapsed=0.254
command_timing = :file=~~/src/HOL/Number_Theory/Euclidean_Algorithm.thyoffset=409name=functionelapsed=0.049
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=5323name=byelapsed=0.020
command_timing = :file=~~/src/HOL/Number_Theory/Euclidean_Algorithm.thyoffset=536name=terminationelapsed=0.042
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Ring_Hom.thyoffset=2791name=byelapsed=0.023
command_timing = :file=~~/src/HOL/Number_Theory/Euclidean_Algorithm.thyoffset=1291name=definitionelapsed=0.031
command_timing = :file=~~/src/HOL/Number_Theory/Euclidean_Algorithm.thyoffset=1385name=definitionelapsed=0.039
command_timing = :file=~~/src/HOL/Number_Theory/Euclidean_Algorithm.thyoffset=1780name=definitionelapsed=0.031
command_timing = :file=~~/src/HOL/Number_Theory/Euclidean_Algorithm.thyoffset=2667name=interpretationelapsed=0.015
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=9510name=byelapsed=0.043
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=9391name=byelapsed=0.016
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=9145name=byelapsed=0.016
command_timing = :file=~~/src/HOL/Number_Theory/Euclidean_Algorithm.thyoffset=2667name=interpretationelapsed=0.038
command_timing = :file=~~/src/HOL/Number_Theory/Euclidean_Algorithm.thyoffset=5967name=interpretationelapsed=0.016
ML_statistics = :now=1489705829.59tasks_ready=373tasks_pending=9tasks_running=5tasks_passive=3tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=11size_allocation=3737124864size_allocation_free=483814888size_heap=4178575360size_heap_free_last_full_GC=0size_heap_free_last_GC=3739275048threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=55.544time_GC=2.78user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/src/HOL/Number_Theory/Euclidean_Algorithm.thyoffset=5967name=interpretationelapsed=0.016
command_timing = :file=~~/src/HOL/Number_Theory/Euclidean_Algorithm.thyoffset=6096name=subclasselapsed=0.046
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=8922name=byelapsed=0.231
command_timing = :file=~~/src/HOL/Number_Theory/Euclidean_Algorithm.thyoffset=6096name=subclasselapsed=0.415
ML_statistics = :now=1489705830.19tasks_ready=379tasks_pending=9tasks_running=8tasks_passive=6tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=12size_allocation=3733979136size_allocation_free=3733979136size_heap=4170186752size_heap_free_last_full_GC=0size_heap_free_last_GC=3735860312threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=56.204time_GC=2.98user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=8442name=byelapsed=0.440
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=1595name=byelapsed=0.013
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=7712name=byelapsed=0.011
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=14195name=byelapsed=0.044
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=10160name=byelapsed=0.013
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=16110name=byelapsed=0.040
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=15965name=byelapsed=0.019
ML_statistics = :now=1489705830.69tasks_ready=350tasks_pending=9tasks_running=6tasks_passive=4tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=12size_allocation=3733979136size_allocation_free=2815370000size_heap=4170186752size_heap_free_last_full_GC=0size_heap_free_last_GC=3735860312threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=57.008time_GC=2.98user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
class euclidean_semiring_gcd = euclidean_semiring + Gcd +
  assumes "gcd_eucl": "euclidean_semiring_class.gcd = gcd"
    and "lcm_eucl": "euclidean_semiring_class.lcm = lcm"
  assumes "Gcd_eucl": "euclidean_semiring_class.Gcd = Gcd"
    and "Lcm_eucl": "euclidean_semiring_class.Lcm = Lcm"
command_timing = :file=~~/src/HOL/Number_Theory/Euclidean_Algorithm.thyoffset=9847name=classelapsed=0.880
command_timing = :file=~~/src/HOL/Number_Theory/Euclidean_Algorithm.thyoffset=10138name=subclasselapsed=0.046
ML_statistics = :now=1489705831.19tasks_ready=352tasks_pending=9tasks_running=6tasks_passive=4tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=12size_allocation=3733979136size_allocation_free=1141128944size_heap=4170186752size_heap_free_last_full_GC=0size_heap_free_last_GC=3735860312threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=58.612time_GC=2.98user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=15901name=byelapsed=0.683
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=15484name=byelapsed=0.011
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=15816name=byelapsed=0.013
command_timing = :file=~~/src/HOL/Number_Theory/Euclidean_Algorithm.thyoffset=10138name=subclasselapsed=0.297
command_timing = :file=~~/src/HOL/Number_Theory/Euclidean_Algorithm.thyoffset=10240name=subclasselapsed=0.049
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=23824name=byelapsed=0.012
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=23738name=byelapsed=0.015
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=23083name=byelapsed=0.013
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=22849name=byelapsed=0.022
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=24806name=byelapsed=0.011
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=25464name=byelapsed=0.015
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=25344name=byelapsed=0.015
command_timing = :file=~~/src/HOL/Number_Theory/Euclidean_Algorithm.thyoffset=10240name=subclasselapsed=0.298
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=18091name=byelapsed=0.024
ML_statistics = :now=1489705831.69tasks_ready=340tasks_pending=9tasks_running=9tasks_passive=6tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=12size_allocation=3733979136size_allocation_free=643984112size_heap=4170186752size_heap_free_last_full_GC=0size_heap_free_last_GC=3735860312threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=59.416time_GC=2.98user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/src/HOL/Number_Theory/Euclidean_Algorithm.thyoffset=10389name=subclasselapsed=0.053
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=17353name=byelapsed=0.026
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=16796name=byelapsed=0.021
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=27038name=byelapsed=0.010
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=25938name=byelapsed=0.018
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=25741name=byelapsed=0.012
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=18763name=byelapsed=0.027
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=65559name=byelapsed=0.023
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=30103name=byelapsed=0.010
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=29824name=byelapsed=0.012
command_timing = :file=~~/src/HOL/Number_Theory/Euclidean_Algorithm.thyoffset=10389name=subclasselapsed=0.309
command_timing = :file=~~/src/HOL/Number_Theory/Euclidean_Algorithm.thyoffset=10933name=lemmaelapsed=0.015
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=29975name=byelapsed=0.012
command_timing = :file=~~/src/HOL/Number_Theory/Euclidean_Algorithm.thyoffset=11085name=lemmaelapsed=0.014
command_timing = :file=~~/src/HOL/Number_Theory/Euclidean_Algorithm.thyoffset=11211name=lemmaelapsed=0.084
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=31881name=byelapsed=0.099
command_timing = :file=~~/src/HOL/Number_Theory/Euclidean_Algorithm.thyoffset=11575name=lemmaelapsed=0.020
command_timing = :file=~~/src/HOL/Number_Theory/Euclidean_Algorithm.thyoffset=11723name=lemmaelapsed=0.018
command_timing = :file=~~/src/HOL/Number_Theory/Euclidean_Algorithm.thyoffset=12210name=lemmaelapsed=0.016
ML_statistics = :now=1489705832.23tasks_ready=323tasks_pending=9tasks_running=4tasks_passive=2tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=13size_allocation=3702521856size_allocation_free=3702521856size_heap=4180672512size_heap_free_last_full_GC=0size_heap_free_last_GC=3704975280threads_in_ML=1threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=60.476time_GC=3.088user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/src/HOL/Number_Theory/Euclidean_Algorithm.thyoffset=12399name=lemmaelapsed=0.015
command_timing = :file=~~/src/HOL/Number_Theory/Euclidean_Algorithm.thyoffset=12739name=lemmaelapsed=0.014
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=31620name=byelapsed=0.107
command_timing = :file=~~/src/HOL/Number_Theory/Euclidean_Algorithm.thyoffset=12898name=lemmaelapsed=0.011
command_timing = :file=~~/src/HOL/Number_Theory/Euclidean_Algorithm.thyoffset=13466name=lemmaelapsed=0.011
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=33528name=qedelapsed=0.012
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=33469name=byelapsed=0.011
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=33205name=byelapsed=0.011
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=35292name=byelapsed=0.011
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=37684name=byelapsed=0.010
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=40977name=byelapsed=0.014
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=40373name=qedelapsed=0.015
ML_statistics = :now=1489705832.73tasks_ready=300tasks_pending=9tasks_running=6tasks_passive=4tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=13size_allocation=3702521856size_allocation_free=2972895184size_heap=4180672512size_heap_free_last_full_GC=0size_heap_free_last_GC=3704975280threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=61.28time_GC=3.088user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=34330name=byelapsed=0.014
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=40590name=byelapsed=0.028
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=45956name=byelapsed=0.011
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=45152name=byelapsed=0.018
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=45442name=byelapsed=0.012
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=46326name=byelapsed=0.010
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=42019name=byelapsed=0.014
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=17779name=byelapsed=0.011
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=51199name=byelapsed=0.010
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=25505name=byelapsed=0.022
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=56031name=qedelapsed=0.019
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=55919name=byelapsed=0.014
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=55671name=byelapsed=0.017
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=57502name=byelapsed=0.010
ML_statistics = :now=1489705833.23tasks_ready=267tasks_pending=9tasks_running=2tasks_passive=0tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=13size_allocation=3702521856size_allocation_free=2274650576size_heap=4180672512size_heap_free_last_full_GC=0size_heap_free_last_GC=3704975280threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=62.084time_GC=3.088user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=26455name=byelapsed=0.047
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=58584name=byelapsed=0.020
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=54903name=byelapsed=0.013
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=60566name=byelapsed=0.095
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=55525name=byelapsed=0.010
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=55334name=byelapsed=0.021
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=56485name=byelapsed=0.030
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=56983name=byelapsed=0.012
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=57147name=qedelapsed=0.015
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=62248name=qedelapsed=0.016
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=62135name=byelapsed=0.013
ML_statistics = :now=1489705833.73tasks_ready=254tasks_pending=9tasks_running=6tasks_passive=5tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=13size_allocation=3702521856size_allocation_free=934450240size_heap=4180672512size_heap_free_last_full_GC=0size_heap_free_last_GC=3704975280threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=63.684time_GC=3.088user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=61998name=byelapsed=0.015
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=61669name=byelapsed=0.016
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=61239name=qedelapsed=0.035
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=60415name=qedelapsed=0.052
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=60382name=byelapsed=0.055
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=59652name=byelapsed=0.012
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=59514name=byelapsed=0.018
command_timing = :file=~~/src/HOL/Number_Theory/Euclidean_Algorithm.thyoffset=15077name=classelapsed=1.741
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=58352name=byelapsed=0.010
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=58414name=byelapsed=0.012
command_timing = :file=~~/src/HOL/Number_Theory/Euclidean_Algorithm.thyoffset=15141name=subclasselapsed=0.052
command_timing = :file=~~/src/HOL/Number_Theory/Euclidean_Algorithm.thyoffset=2490name=byelapsed=0.013
command_timing = :file=~~/src/HOL/Number_Theory/Euclidean_Algorithm.thyoffset=2050name=byelapsed=0.017
ML_statistics = :now=1489705834.23tasks_ready=234tasks_pending=9tasks_running=5tasks_passive=2tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=13size_allocation=3702521856size_allocation_free=307289128size_heap=4180672512size_heap_free_last_full_GC=0size_heap_free_last_GC=3704975280threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=64.484time_GC=3.088user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/src/HOL/Number_Theory/Euclidean_Algorithm.thyoffset=5849name=byelapsed=0.025
command_timing = :file=~~/src/HOL/Number_Theory/Euclidean_Algorithm.thyoffset=5089name=henceelapsed=0.056
command_timing = :file=~~/src/HOL/Number_Theory/Euclidean_Algorithm.thyoffset=15141name=subclasselapsed=0.389
command_timing = :file=~~/src/HOL/Number_Theory/Euclidean_Algorithm.thyoffset=5360name=byelapsed=0.016
command_timing = :file=~~/src/HOL/Number_Theory/Euclidean_Algorithm.thyoffset=4170name=byelapsed=0.025
command_timing = :file=~~/src/HOL/Number_Theory/Euclidean_Algorithm.thyoffset=15168name=subclasselapsed=0.051
command_timing = :file=~~/src/HOL/Number_Theory/Euclidean_Algorithm.thyoffset=3824name=byelapsed=0.022
command_timing = :file=~~/src/HOL/Number_Theory/Euclidean_Algorithm.thyoffset=3114name=byelapsed=0.031
ML_statistics = :now=1489705834.75tasks_ready=234tasks_pending=9tasks_running=3tasks_passive=1tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=14size_allocation=3699376128size_allocation_free=3699376128size_heap=4179623936size_heap_free_last_full_GC=0size_heap_free_last_GC=3701967392threads_in_ML=1threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=65.004time_GC=3.184user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=19495name=byelapsed=0.015
command_timing = :file=~~/src/HOL/Number_Theory/Euclidean_Algorithm.thyoffset=15168name=subclasselapsed=0.322
command_timing = :file=~~/src/HOL/Number_Theory/Euclidean_Algorithm.thyoffset=15189name=subclasselapsed=0.052
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Unsorted.thyoffset=8779name=byelapsed=0.015
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Neville_Aitken_Interpolation.thyoffset=8813name=qedelapsed=0.010
command_timing = :file=~~/src/HOL/Number_Theory/Euclidean_Algorithm.thyoffset=15189name=subclasselapsed=0.325
ML_statistics = :now=1489705835.25tasks_ready=99tasks_pending=9tasks_running=4tasks_passive=2tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=14size_allocation=3699376128size_allocation_free=2575499312size_heap=4179623936size_heap_free_last_full_GC=0size_heap_free_last_GC=3701967392threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=66.608time_GC=3.184user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Neville_Aitken_Interpolation.thyoffset=8628name=byelapsed=0.228
ML_statistics = :now=1489705835.75tasks_ready=110tasks_pending=9tasks_running=3tasks_passive=1tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=14size_allocation=3699376128size_allocation_free=1331787848size_heap=4179623936size_heap_free_last_full_GC=0size_heap_free_last_GC=3701967392threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=67.408time_GC=3.184user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/src/HOL/Number_Theory/Euclidean_Algorithm.thyoffset=15221name=functionelapsed=0.656
command_timing = :file=~~/src/HOL/Number_Theory/Euclidean_Algorithm.thyoffset=15221name=functionelapsed=0.189
command_timing = :file=~~/src/HOL/Number_Theory/Euclidean_Algorithm.thyoffset=15532name=terminationelapsed=0.066
ML_statistics = :now=1489705836.29tasks_ready=112tasks_pending=9tasks_running=3tasks_passive=1tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=15size_allocation=3683647488size_allocation_free=3683647488size_heap=4174381056size_heap_free_last_full_GC=0size_heap_free_last_GC=3686100904threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=68.26time_GC=3.288user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/src/HOL/Number_Theory/Euclidean_Algorithm.thyoffset=15532name=terminationelapsed=0.194
command_timing = :file=~~/src/HOL/Number_Theory/Euclidean_Algorithm.thyoffset=15645name=abbreviationelapsed=0.031
command_timing = :file=~~/src/HOL/Number_Theory/Euclidean_Algorithm.thyoffset=15758name=lemmaelapsed=0.014
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Neville_Aitken_Interpolation.thyoffset=8414name=byelapsed=1.081
command_timing = :file=~~/src/HOL/Number_Theory/Euclidean_Algorithm.thyoffset=15546name=byelapsed=0.010
command_timing = :file=~~/src/HOL/Number_Theory/Euclidean_Algorithm.thyoffset=15758name=lemmaelapsed=0.024
command_timing = :file=~~/src/HOL/Number_Theory/Euclidean_Algorithm.thyoffset=18048name=declareelapsed=0.021
command_timing = :file=~~/src/HOL/Number_Theory/Euclidean_Algorithm.thyoffset=18089name=definitionelapsed=0.077
command_timing = :file=~~/src/HOL/Number_Theory/Euclidean_Algorithm.thyoffset=18209name=lemmaelapsed=0.020
command_timing = :file=~~/src/HOL/Number_Theory/Euclidean_Algorithm.thyoffset=18357name=lemmaelapsed=0.021
command_timing = :file=~~/src/HOL/Number_Theory/Euclidean_Algorithm.thyoffset=18510name=lemmaelapsed=0.020
command_timing = :file=~~/src/HOL/Number_Theory/Euclidean_Algorithm.thyoffset=18733name=lemmaelapsed=0.021
command_timing = :file=~~/src/HOL/Number_Theory/Euclidean_Algorithm.thyoffset=18890name=lemmaelapsed=0.022
command_timing = :file=~~/src/HOL/Number_Theory/Euclidean_Algorithm.thyoffset=19253name=declareelapsed=0.021
### theory "Euclidean_Algorithm"
### 8.034s elapsed time, 16.004s cpu time, 0.508s GC time
Loading theory "Normalized_Fraction" (required by "Polynomial_Interpolation" via "Newton_Interpolation" via "Ring_Hom_Poly" via "Missing_Polynomial" via "Polynomial_Factorial")
loading_theory = Normalized_Fraction
ML_statistics = :now=1489705836.79tasks_ready=108tasks_pending=7tasks_running=3tasks_passive=1tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=15size_allocation=3683647488size_allocation_free=2601939360size_heap=4174381056size_heap_free_last_full_GC=0size_heap_free_last_GC=3686100904threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=69.056time_GC=3.288user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/src/HOL/Library/Normalized_Fraction.thyoffset=86name=theoryelapsed=0.319
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=96016name=byelapsed=0.572
command_timing = :file=~~/src/HOL/Library/Normalized_Fraction.thyoffset=346name=definitionelapsed=0.013
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=95733name=byelapsed=0.011
command_timing = :file=~~/src/HOL/Library/Normalized_Fraction.thyoffset=4446name=lift_definitionelapsed=0.011
command_timing = :file=~~/src/HOL/Library/Normalized_Fraction.thyoffset=6278name=lemmaelapsed=0.015
locale fract_as_normalized_quot
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=95127name=byelapsed=0.100
command_timing = :file=~~/src/HOL/Library/Normalized_Fraction.thyoffset=6808name=setup_liftingelapsed=0.029
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=88180name=byelapsed=0.010
command_timing = :file=~~/src/HOL/Library/Normalized_Fraction.thyoffset=7709name=lemmaelapsed=0.016
command_timing = :file=~~/src/HOL/Library/Normalized_Fraction.thyoffset=8696name=lemmaelapsed=0.010
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=88332name=byelapsed=0.058
ML_statistics = :now=1489705837.29tasks_ready=149tasks_pending=7tasks_running=7tasks_passive=5tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=15size_allocation=3683647488size_allocation_free=1867835968size_heap=4174381056size_heap_free_last_full_GC=0size_heap_free_last_GC=3686100904threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=69.856time_GC=3.288user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
### theory "Normalized_Fraction"
### 0.630s elapsed time, 1.260s cpu time, 0.000s GC time
Loading theory "Field_as_Ring" (required by "Polynomial_Interpolation" via "Newton_Interpolation" via "Ring_Hom_Poly" via "Missing_Polynomial" via "Polynomial_Factorial")
loading_theory = Field_as_Ring
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=87962name=byelapsed=0.056
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=87435name=byelapsed=0.101
command_timing = :file=~~/src/HOL/Number_Theory/Euclidean_Algorithm.thyoffset=15524name=byelapsed=0.024
command_timing = :file=~~/src/HOL/Number_Theory/Euclidean_Algorithm.thyoffset=18031name=byelapsed=0.013
command_timing = :file=~~/src/HOL/Number_Theory/Euclidean_Algorithm.thyoffset=16187name=proofelapsed=0.010
### Ignoring duplicate rewrite rule:
### ?a1 \<noteq> (0::'a) \<Longrightarrow>
### gcd ?a1 (?b1 mod ?a1) \<equiv> gcd ?a1 ?b1
command_timing = :file=~~/src/HOL/Number_Theory/Euclidean_Algorithm.thyoffset=17832name=qedelapsed=0.018
command_timing = :file=~~/src/HOL/Library/Normalized_Fraction.thyoffset=899name=byelapsed=0.017
command_timing = :file=~~/src/HOL/Library/Normalized_Fraction.thyoffset=1624name=byelapsed=0.016
command_timing = :file=~~/src/HOL/Library/Field_as_Ring.thyoffset=80name=theoryelapsed=0.468
command_timing = :file=~~/src/HOL/Library/Normalized_Fraction.thyoffset=2343name=byelapsed=0.026
command_timing = :file=~~/src/HOL/Library/Normalized_Fraction.thyoffset=3680name=byelapsed=0.017
ML_statistics = :now=1489705837.79tasks_ready=128tasks_pending=7tasks_running=3tasks_passive=1tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=15size_allocation=3683647488size_allocation_free=700032376size_heap=4174381056size_heap_free_last_full_GC=0size_heap_free_last_GC=3686100904threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=71.46time_GC=3.288user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/src/HOL/Library/Normalized_Fraction.thyoffset=4003name=byelapsed=0.024
command_timing = :file=~~/src/HOL/Library/Normalized_Fraction.thyoffset=3071name=byelapsed=0.027
command_timing = :file=~~/src/HOL/Library/Normalized_Fraction.thyoffset=5640name=byelapsed=0.015
command_timing = :file=~~/src/HOL/Library/Normalized_Fraction.thyoffset=5022name=byelapsed=0.011
command_timing = :file=~~/src/HOL/Library/Normalized_Fraction.thyoffset=7230name=byelapsed=0.019
class field = division_ring + idom_divide +
  assumes
    "field_inverse":
      "\<And>a.
          a \<noteq> (0::'a) \<Longrightarrow> inverse a * a = (1::'a)"
  assumes "field_divide_inverse": "\<And>a b. a / b = a * inverse b"
  assumes "field_inverse_zero": "inverse (0::'a) = (0::'a)"
command_timing = :file=~~/src/HOL/Library/Field_as_Ring.thyoffset=181name=contextelapsed=0.232
command_timing = :file=~~/src/HOL/Library/Normalized_Fraction.thyoffset=7012name=byelapsed=0.021
command_timing = :file=~~/src/HOL/Library/Normalized_Fraction.thyoffset=5998name=byelapsed=0.011
command_timing = :file=~~/src/HOL/Library/Field_as_Ring.thyoffset=202name=subclasselapsed=0.065
command_timing = :file=~~/src/HOL/Library/Fraction_Field.thyoffset=4411name=byelapsed=0.013
command_timing = :file=~~/src/HOL/Library/Fraction_Field.thyoffset=4002name=byelapsed=0.087
command_timing = :file=~~/src/HOL/Library/Fraction_Field.thyoffset=11149name=byelapsed=0.090
ML_statistics = :now=1489705838.34tasks_ready=111tasks_pending=7tasks_running=5tasks_passive=3tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=16size_allocation=3659530240size_allocation_free=3659530240size_heap=4172283904size_heap_free_last_full_GC=0size_heap_free_last_GC=3662202544threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=72.216time_GC=3.424user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/src/HOL/Library/Field_as_Ring.thyoffset=202name=subclasselapsed=0.287
command_timing = :file=~~/src/HOL/Library/Fraction_Field.thyoffset=12604name=byelapsed=0.021
command_timing = :file=~~/src/HOL/Library/Field_as_Ring.thyoffset=227name=definitionelapsed=0.028
command_timing = :file=~~/src/HOL/Library/Field_as_Ring.thyoffset=331name=definitionelapsed=0.025
command_timing = :file=~~/src/HOL/Library/Fraction_Field.thyoffset=11830name=byelapsed=0.016
command_timing = :file=~~/src/HOL/Library/Fraction_Field.thyoffset=11766name=byelapsed=0.014
command_timing = :file=~~/src/HOL/Library/Field_as_Ring.thyoffset=416name=definitionelapsed=0.029
command_timing = :file=~~/src/HOL/Library/Field_as_Ring.thyoffset=531name=definitionelapsed=0.030
instantiation
  real :: unique_euclidean_ring
  uniqueness_constraint_real == uniqueness_constraint ::
    real \<Rightarrow> real \<Rightarrow> bool
  euclidean_size_real == euclidean_size :: real \<Rightarrow> nat
  modulo_real == modulo :: real \<Rightarrow> real \<Rightarrow> real
  normalize_real == normalize :: real \<Rightarrow> real
  unit_factor_real == unit_factor :: real \<Rightarrow> real
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=22815name=byelapsed=0.049
command_timing = :file=~~/src/HOL/Library/Field_as_Ring.thyoffset=686name=definitionelapsed=0.013
command_timing = :file=~~/src/HOL/Library/Field_as_Ring.thyoffset=754name=definitionelapsed=0.011
command_timing = :file=~~/src/HOL/Library/Field_as_Ring.thyoffset=826name=definitionelapsed=0.013
command_timing = :file=~~/src/HOL/Library/Field_as_Ring.thyoffset=904name=definitionelapsed=0.015
command_timing = :file=~~/src/HOL/Library/Field_as_Ring.thyoffset=982name=definitionelapsed=0.013
instantiation
  real :: euclidean_ring_gcd
  Gcd_real == Gcd :: real set \<Rightarrow> real
  Lcm_real == Lcm :: real set \<Rightarrow> real
  gcd_real == gcd :: real \<Rightarrow> real \<Rightarrow> real
  lcm_real == lcm :: real \<Rightarrow> real \<Rightarrow> real
command_timing = :file=~~/src/HOL/Library/Field_as_Ring.thyoffset=1183name=definitionelapsed=0.013
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=22943name=byelapsed=0.085
command_timing = :file=~~/src/HOL/Library/Field_as_Ring.thyoffset=1272name=definitionelapsed=0.013
command_timing = :file=~~/src/HOL/Library/Field_as_Ring.thyoffset=1361name=definitionelapsed=0.013
command_timing = :file=~~/src/HOL/Library/Field_as_Ring.thyoffset=1446name=definitionelapsed=0.013
instantiation
  rat :: unique_euclidean_ring
  uniqueness_constraint_rat == uniqueness_constraint ::
    rat \<Rightarrow> rat \<Rightarrow> bool
  euclidean_size_rat == euclidean_size :: rat \<Rightarrow> nat
  modulo_rat == modulo :: rat \<Rightarrow> rat \<Rightarrow> rat
  normalize_rat == normalize :: rat \<Rightarrow> rat
  unit_factor_rat == unit_factor :: rat \<Rightarrow> rat
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=25855name=byelapsed=0.051
command_timing = :file=~~/src/HOL/Library/Field_as_Ring.thyoffset=1677name=definitionelapsed=0.013
command_timing = :file=~~/src/HOL/Library/Field_as_Ring.thyoffset=1743name=definitionelapsed=0.012
command_timing = :file=~~/src/HOL/Library/Field_as_Ring.thyoffset=1813name=definitionelapsed=0.013
command_timing = :file=~~/src/HOL/Library/Field_as_Ring.thyoffset=1889name=definitionelapsed=0.015
command_timing = :file=~~/src/HOL/Library/Field_as_Ring.thyoffset=1964name=definitionelapsed=0.014
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=26107name=byelapsed=0.055
instantiation
  rat :: euclidean_ring_gcd
  Gcd_rat == Gcd :: rat set \<Rightarrow> rat
  Lcm_rat == Lcm :: rat set \<Rightarrow> rat
  gcd_rat == gcd :: rat \<Rightarrow> rat \<Rightarrow> rat
  lcm_rat == lcm :: rat \<Rightarrow> rat \<Rightarrow> rat
command_timing = :file=~~/src/HOL/Library/Field_as_Ring.thyoffset=2162name=definitionelapsed=0.014
command_timing = :file=~~/src/HOL/Library/Field_as_Ring.thyoffset=2246name=definitionelapsed=0.015
command_timing = :file=~~/src/HOL/Library/Field_as_Ring.thyoffset=2330name=definitionelapsed=0.014
command_timing = :file=~~/src/HOL/Library/Field_as_Ring.thyoffset=2411name=definitionelapsed=0.014
instantiation
  complex :: unique_euclidean_ring
  uniqueness_constraint_complex == uniqueness_constraint ::
    complex \<Rightarrow> complex \<Rightarrow> bool
  euclidean_size_complex == euclidean_size :: complex \<Rightarrow> nat
  modulo_complex == modulo ::
    complex \<Rightarrow> complex \<Rightarrow> complex
  normalize_complex == normalize :: complex \<Rightarrow> complex
  unit_factor_complex == unit_factor :: complex \<Rightarrow> complex
command_timing = :file=~~/src/HOL/Library/Field_as_Ring.thyoffset=2638name=definitionelapsed=0.015
command_timing = :file=~~/src/HOL/Library/Field_as_Ring.thyoffset=2712name=definitionelapsed=0.013
command_timing = :file=~~/src/HOL/Library/Field_as_Ring.thyoffset=2790name=definitionelapsed=0.015
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=39679name=byelapsed=0.109
command_timing = :file=~~/src/HOL/Library/Field_as_Ring.thyoffset=2874name=definitionelapsed=0.017
command_timing = :file=~~/src/HOL/Library/Field_as_Ring.thyoffset=2961name=definitionelapsed=0.016
instantiation
  complex :: euclidean_ring_gcd
  Gcd_complex == Gcd :: complex set \<Rightarrow> complex
  Lcm_complex == Lcm :: complex set \<Rightarrow> complex
  gcd_complex == gcd :: complex \<Rightarrow> complex \<Rightarrow> complex
  lcm_complex == lcm :: complex \<Rightarrow> complex \<Rightarrow> complex
command_timing = :file=~~/src/HOL/Library/Field_as_Ring.thyoffset=3171name=definitionelapsed=0.015
ML_statistics = :now=1489705838.84tasks_ready=106tasks_pending=7tasks_running=3tasks_passive=1tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=16size_allocation=3659530240size_allocation_free=2941678088size_heap=4172283904size_heap_free_last_full_GC=0size_heap_free_last_GC=3662202544threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=73.02time_GC=3.424user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/src/HOL/Library/Field_as_Ring.thyoffset=3275name=definitionelapsed=0.015
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=53788name=byelapsed=0.059
command_timing = :file=~~/src/HOL/Library/Field_as_Ring.thyoffset=3379name=definitionelapsed=0.015
command_timing = :file=~~/src/HOL/Library/Field_as_Ring.thyoffset=3476name=definitionelapsed=0.016
### theory "Field_as_Ring"
### 1.600s elapsed time, 3.188s cpu time, 0.136s GC time
Loading theory "Polynomial_Factorial" (required by "Polynomial_Interpolation" via "Newton_Interpolation" via "Ring_Hom_Poly" via "Missing_Polynomial")
loading_theory = Polynomial_Factorial
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=55801name=byelapsed=0.163
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=59887name=byelapsed=0.091
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=60731name=byelapsed=0.111
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=60662name=byelapsed=0.011
ML_statistics = :now=1489705839.34tasks_ready=103tasks_pending=6tasks_running=2tasks_passive=0tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=16size_allocation=3659530240size_allocation_free=2407851616size_heap=4172283904size_heap_free_last_full_GC=0size_heap_free_last_GC=3662202544threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=73.82time_GC=3.424user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Unsorted.thyoffset=2430name=byelapsed=0.075
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=81164name=byelapsed=0.113
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=82609name=byelapsed=0.085
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=91498name=byelapsed=0.012
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=91433name=byelapsed=0.091
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=91072name=byelapsed=0.011
ML_statistics = :now=1489705839.85tasks_ready=103tasks_pending=6tasks_running=5tasks_passive=3tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=16size_allocation=3659530240size_allocation_free=1572240184size_heap=4172283904size_heap_free_last_full_GC=0size_heap_free_last_GC=3662202544threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=75.428time_GC=3.424user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=91166name=byelapsed=0.094
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=90879name=byelapsed=0.016
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=90710name=byelapsed=0.027
command_timing = :file=~~/src/HOL/Library/Polynomial_Factorial.thyoffset=212name=theoryelapsed=1.070
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=90514name=byelapsed=0.034
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=90388name=byelapsed=0.021
command_timing = :file=~~/src/HOL/Library/Polynomial_Factorial.thyoffset=8512name=definitionelapsed=0.017
command_timing = :file=~~/src/HOL/Library/Polynomial_Factorial.thyoffset=8756name=definitionelapsed=0.016
command_timing = :file=~~/src/HOL/Library/Normalized_Fraction.thyoffset=9782name=byelapsed=0.021
command_timing = :file=~~/src/HOL/Library/Polynomial_Factorial.thyoffset=3543name=byelapsed=0.021
command_timing = :file=~~/src/HOL/Library/Polynomial_Factorial.thyoffset=5865name=byelapsed=0.012
command_timing = :file=~~/src/HOL/Library/Polynomial_Factorial.thyoffset=5714name=byelapsed=0.013
command_timing = :file=~~/src/HOL/Library/Polynomial_Factorial.thyoffset=20119name=definitionelapsed=0.010
command_timing = :file=~~/src/HOL/Library/Polynomial_Factorial.thyoffset=20252name=definitionelapsed=0.012
command_timing = :file=~~/src/HOL/Library/Polynomial_Factorial.thyoffset=9420name=byelapsed=0.010
command_timing = :file=~~/src/HOL/Library/Polynomial_Factorial.thyoffset=20521name=interpretationelapsed=0.024
command_timing = :file=~~/src/HOL/Library/Polynomial_Factorial.thyoffset=11670name=byelapsed=0.011
ML_statistics = :now=1489705840.35tasks_ready=112tasks_pending=6tasks_running=3tasks_passive=1tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=16size_allocation=3659530240size_allocation_free=1061530416size_heap=4172283904size_heap_free_last_full_GC=0size_heap_free_last_GC=3662202544threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=76.232time_GC=3.424user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/src/HOL/Library/Polynomial_Factorial.thyoffset=11386name=byelapsed=0.015
command_timing = :file=~~/src/HOL/Library/Polynomial_Factorial.thyoffset=10229name=byelapsed=0.029
### Partially applied constant "Groups.times_class.times" on left hand side of equation, in theorem:
### field_poly.Lcm ?A \<equiv>
### if \<exists>l. l \<noteq> 0 \<and> (\<forall>a\<in>?A. dvd.dvd op * a l)
### then Let (SOME l.
###              l \<noteq> 0 \<and>
###              (\<forall>a\<in>?A. dvd.dvd op * a l) \<and>
###              euclidean_size_field_poly l =
###              (LEAST n.
###                  \<exists>l.
###                     l \<noteq> 0 \<and>
###                     (\<forall>a\<in>?A. dvd.dvd op * a l) \<and>
###                     euclidean_size_field_poly l = n))
###       normalize_field_poly
### else 0
### Partially applied constant "Groups.times_class.times" on left hand side of equation, in theorem:
### field_poly.Gcd ?A \<equiv> field_poly.Lcm {d. Ball ?A (dvd.dvd op * d)}
### Partially applied constant "Groups.times_class.times" on left hand side of equation, in theorem:
### field_poly.Gcd (set ?xs) \<equiv> fold field_poly.gcd ?xs 0
### Partially applied constant "Groups.times_class.times" on left hand side of equation, in theorem:
### field_poly.Lcm (set ?xs) \<equiv> fold field_poly.lcm ?xs 1
command_timing = :file=~~/src/HOL/Library/Polynomial_Factorial.thyoffset=20521name=interpretationelapsed=0.219
command_timing = :file=~~/src/HOL/Library/Polynomial_Factorial.thyoffset=14587name=byelapsed=0.015
instantiation
  poly :: (factorial_ring_gcd) factorial_ring_gcd
  Gcd_poly == Gcd :: 'a poly set \<Rightarrow> 'a poly
  Lcm_poly == Lcm :: 'a poly set \<Rightarrow> 'a poly
  gcd_poly == gcd :: 'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly
  lcm_poly == lcm :: 'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly
command_timing = :file=~~/src/HOL/Library/Polynomial_Factorial.thyoffset=34945name=definitionelapsed=0.017
command_timing = :file=~~/src/HOL/Library/Polynomial_Factorial.thyoffset=14452name=byelapsed=0.015
command_timing = :file=~~/src/HOL/Library/Polynomial_Factorial.thyoffset=35046name=definitionelapsed=0.017
command_timing = :file=~~/src/HOL/Library/Polynomial_Factorial.thyoffset=13980name=byelapsed=0.011
command_timing = :file=~~/src/HOL/Library/Polynomial_Factorial.thyoffset=35149name=definitionelapsed=0.017
command_timing = :file=~~/src/HOL/Library/Polynomial_Factorial.thyoffset=35243name=definitionelapsed=0.017
instantiation
  poly :: ({factorial_ring_gcd,field}) unique_euclidean_ring
  uniqueness_constraint_poly == uniqueness_constraint ::
    'a poly \<Rightarrow> 'a poly \<Rightarrow> bool
  euclidean_size_poly == euclidean_size :: 'a poly \<Rightarrow> nat
command_timing = :file=~~/src/HOL/Library/Polynomial_Factorial.thyoffset=35515name=definitionelapsed=0.019
command_timing = :file=~~/src/HOL/Library/Polynomial_Factorial.thyoffset=35636name=definitionelapsed=0.020
command_timing = :file=~~/src/HOL/Library/Polynomial_Factorial.thyoffset=38659name=functionelapsed=0.075
command_timing = :file=~~/src/HOL/Library/Polynomial_Factorial.thyoffset=20915name=proofelapsed=0.012
command_timing = :file=~~/src/HOL/Library/Polynomial_Factorial.thyoffset=38659name=functionelapsed=0.012
ML_statistics = :now=1489705840.85tasks_ready=137tasks_pending=6tasks_running=3tasks_passive=1tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=16size_allocation=3659530240size_allocation_free=440725504size_heap=4172283904size_heap_free_last_full_GC=0size_heap_free_last_GC=3662202544threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=77.032time_GC=3.424user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/src/HOL/Library/Polynomial_Factorial.thyoffset=40036name=definitionelapsed=0.014
command_timing = :file=~~/src/HOL/Library/Polynomial_Factorial.thyoffset=22040name=qedelapsed=0.019
command_timing = :file=~~/src/HOL/Library/Polynomial_Factorial.thyoffset=21927name=byelapsed=0.012
command_timing = :file=~~/src/HOL/Library/Polynomial_Factorial.thyoffset=21693name=byelapsed=0.030
command_timing = :file=~~/src/HOL/Library/Polynomial_Factorial.thyoffset=22880name=haveelapsed=0.127
command_timing = :file=~~/src/HOL/Library/Polynomial_Factorial.thyoffset=26159name=byelapsed=0.011
command_timing = :file=~~/src/HOL/Library/Polynomial_Factorial.thyoffset=26078name=byelapsed=0.012
command_timing = :file=~~/src/HOL/Library/Polynomial_Factorial.thyoffset=24258name=byelapsed=0.015
command_timing = :file=~~/src/HOL/Library/Polynomial_Factorial.thyoffset=25151name=byelapsed=0.031
command_timing = :file=~~/src/HOL/Library/Polynomial_Factorial.thyoffset=19861name=byelapsed=0.017
command_timing = :file=~~/src/HOL/Library/Polynomial_Factorial.thyoffset=7191name=byelapsed=0.010
command_timing = :file=~~/src/HOL/Library/Polynomial_Factorial.thyoffset=7024name=byelapsed=0.010
ML_statistics = :now=1489705841.44tasks_ready=131tasks_pending=6tasks_running=2tasks_passive=0tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=17size_allocation=3635412992size_allocation_free=3099061136size_heap=4155506688size_heap_free_last_full_GC=0size_heap_free_last_GC=3638729104threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=78.492time_GC=3.648user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/src/HOL/Library/Polynomial_Factorial.thyoffset=9229name=byelapsed=0.017
command_timing = :file=~~/src/HOL/Library/Polynomial_Factorial.thyoffset=28582name=byelapsed=0.037
command_timing = :file=~~/src/HOL/Library/Polynomial_Factorial.thyoffset=29783name=byelapsed=0.019
command_timing = :file=~~/src/HOL/Library/Polynomial_Factorial.thyoffset=29726name=byelapsed=0.011
command_timing = :file=~~/src/HOL/Library/Polynomial_Factorial.thyoffset=30235name=byelapsed=0.010
command_timing = :file=~~/src/HOL/Library/Polynomial_Factorial.thyoffset=34684name=byelapsed=0.014
command_timing = :file=~~/src/HOL/Library/Polynomial_Factorial.thyoffset=32978name=byelapsed=0.023
command_timing = :file=~~/src/HOL/Library/Polynomial_Factorial.thyoffset=32915name=byelapsed=0.010
command_timing = :file=~~/src/HOL/Number_Theory/Euclidean_Algorithm.thyoffset=13805name=interpretelapsed=0.014
ML_statistics = :now=1489705841.94tasks_ready=118tasks_pending=6tasks_running=3tasks_passive=1tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=17size_allocation=3635412992size_allocation_free=2610321752size_heap=4155506688size_heap_free_last_full_GC=0size_heap_free_last_GC=3638729104threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=79.296time_GC=3.648user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/src/HOL/Number_Theory/Euclidean_Algorithm.thyoffset=14030name=byelapsed=0.109
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=91844name=byelapsed=0.070
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=93551name=byelapsed=0.060
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=93321name=byelapsed=0.013
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=93019name=byelapsed=0.119
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=93187name=byelapsed=0.023
ML_statistics = :now=1489705842.45tasks_ready=98tasks_pending=6tasks_running=2tasks_passive=0tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=17size_allocation=3635412992size_allocation_free=1988486696size_heap=4155506688size_heap_free_last_full_GC=0size_heap_free_last_GC=3638729104threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=80.096time_GC=3.648user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Unsorted.thyoffset=9083name=byelapsed=0.049
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Unsorted.thyoffset=9214name=byelapsed=0.049
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Unsorted.thyoffset=9346name=byelapsed=0.049
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Unsorted.thyoffset=9478name=byelapsed=0.048
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=106557name=byelapsed=0.069
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Unsorted.thyoffset=14679name=byelapsed=0.017
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Unsorted.thyoffset=14431name=byelapsed=0.016
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Unsorted.thyoffset=14190name=byelapsed=0.016
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=108903name=byelapsed=0.012
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=108776name=byelapsed=0.026
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Is_Rat_To_Rat.thyoffset=4327name=byelapsed=0.075
command_timing = :file=~~/src/HOL/Number_Theory/Euclidean_Algorithm.thyoffset=19386name=interpretelapsed=0.014
ML_statistics = :now=1489705842.95tasks_ready=89tasks_pending=6tasks_running=2tasks_passive=0tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=17size_allocation=3635412992size_allocation_free=1425785480size_heap=4155506688size_heap_free_last_full_GC=0size_heap_free_last_GC=3638729104threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=80.896time_GC=3.648user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/src/HOL/Number_Theory/Euclidean_Algorithm.thyoffset=19619name=byelapsed=0.092
### Rule already declared as introduction (intro)
### (\<And>x. ?f x = ?g x) \<Longrightarrow> ?f = ?g
command_timing = :file=~~/src/HOL/Number_Theory/Euclidean_Algorithm.thyoffset=20097name=byelapsed=0.016
command_timing = :file=~~/src/HOL/Number_Theory/Euclidean_Algorithm.thyoffset=20039name=byelapsed=0.027
command_timing = :file=~~/src/HOL/Number_Theory/Euclidean_Algorithm.thyoffset=20587name=interpretelapsed=0.015
command_timing = :file=~~/src/HOL/Number_Theory/Euclidean_Algorithm.thyoffset=20820name=byelapsed=0.095
### Rule already declared as introduction (intro)
### (\<And>x. ?f x = ?g x) \<Longrightarrow> ?f = ?g
command_timing = :file=~~/src/HOL/Number_Theory/Euclidean_Algorithm.thyoffset=21608name=byelapsed=0.013
command_timing = :file=~~/src/HOL/Number_Theory/Euclidean_Algorithm.thyoffset=21540name=byelapsed=0.011
command_timing = :file=~~/src/HOL/Number_Theory/Euclidean_Algorithm.thyoffset=21341name=byelapsed=0.011
command_timing = :file=~~/src/HOL/Library/Normalized_Fraction.thyoffset=8580name=byelapsed=0.060
command_timing = :file=~~/src/HOL/Library/Normalized_Fraction.thyoffset=8470name=byelapsed=0.044
ML_statistics = :now=1489705843.45tasks_ready=89tasks_pending=6tasks_running=5tasks_passive=2tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=17size_allocation=3635412992size_allocation_free=186263696size_heap=4155506688size_heap_free_last_full_GC=0size_heap_free_last_GC=3638729104threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=82.504time_GC=3.648user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/src/HOL/Library/Normalized_Fraction.thyoffset=8412name=byelapsed=0.044
command_timing = :file=~~/src/HOL/Library/Normalized_Fraction.thyoffset=8328name=byelapsed=0.018
command_timing = :file=~~/src/HOL/Library/Normalized_Fraction.thyoffset=8108name=byelapsed=0.026
find_theorems
  "_ div _ = (0::?'a)"

found 16 theorem(s):
Fields.division_ring_class.division_ring_divide_zero:
  ?a / (0::?'a) = (0::?'a)
Rings.semidom_divide_class.div_0: (0::?'a) div ?a = (0::?'a)
Rings.semidom_divide_class.div_by_0: ?a div (0::?'a) = (0::?'a)
Divides.semiring_div_class.mod_div_trivial: ?a mod ?b div ?b = (0::?'a)
Divides.semiring_div_parity_class.one_div_two_eq_zero:
  (1::?'a) div (2::?'a) = (0::?'a)
Fields.linordered_field_class.one_divide_eq_0_iff:
  ((1::?'a) / ?a = (0::?'a)) = (?a = (0::?'a))
Rings.normalization_semidom_class.inv_unit_factor_eq_0_iff:
  ((1::?'a) div unit_factor ?a = (0::?'a)) = (?a = (0::?'a))
Divides.div_less: ?m < ?n \<Longrightarrow> ?m div ?n = 0
Fields.field_class.divide_eq_0_iff:
  (?a / ?b = (0::?'a)) = (?a = (0::?'a) \<or> ?b = (0::?'a))
Divides.div_eq_0_iff: (?a div ?b = 0) = (?a < ?b \<or> ?b = 0)
Rings.semidom_divide_class.dvd_div_eq_0_iff:
  ?b dvd ?a \<Longrightarrow> (?a div ?b = (0::?'a)) = (?a = (0::?'a))
Rings.algebraic_semidom_class.unit_div_eq_0_iff:
  is_unit ?b \<Longrightarrow> (?a div ?b = (0::?'a)) = (?a = (0::?'a))
Divides.semiring_numeral_div_class.div_less:
  \<lbrakk>(0::?'a) \<le> ?a; ?a < ?b\<rbrakk>
  \<Longrightarrow> ?a div ?b = (0::?'a)
Divides.div_pos_pos_trivial:
  \<lbrakk>0 \<le> ?a; ?a < ?b\<rbrakk> \<Longrightarrow> ?a div ?b = 0
Divides.div_neg_neg_trivial:
  \<lbrakk>?a \<le> 0; ?b < ?a\<rbrakk> \<Longrightarrow> ?a div ?b = 0
Divides.zdiv_eq_0_iff:
  (?i div ?k = 0) =
  (?k = 0 \<or> 0 \<le> ?i \<and> ?i < ?k \<or> ?i \<le> 0 \<and> ?k < ?i)
command_timing = :file=~~/src/HOL/Library/Normalized_Fraction.thyoffset=9876name=find_theoremselapsed=0.239
command_timing = :file=~~/src/HOL/Library/Normalized_Fraction.thyoffset=11590name=byelapsed=0.060
command_timing = :file=~~/src/HOL/Library/Normalized_Fraction.thyoffset=12229name=byelapsed=0.063
command_timing = :file=~~/src/HOL/Library/Normalized_Fraction.thyoffset=13739name=byelapsed=0.043
ML_statistics = :now=1489705844tasks_ready=81tasks_pending=6tasks_running=5tasks_passive=3tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=18size_allocation=3579838464size_allocation_free=3579838464size_heap=4171235328size_heap_free_last_full_GC=0size_heap_free_last_GC=3581941008threads_in_ML=1threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=82.776time_GC=3.764user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/src/HOL/Library/Normalized_Fraction.thyoffset=11076name=byelapsed=0.027
command_timing = :file=~~/src/HOL/Library/Normalized_Fraction.thyoffset=10709name=byelapsed=0.029
command_timing = :file=~~/src/HOL/Library/Normalized_Fraction.thyoffset=10444name=byelapsed=0.021
command_timing = :file=~~/src/HOL/Library/Normalized_Fraction.thyoffset=9376name=byelapsed=0.030
command_timing = :file=~~/src/HOL/Library/Field_as_Ring.thyoffset=1053name=byelapsed=0.063
command_timing = :file=~~/src/HOL/Library/Field_as_Ring.thyoffset=2033name=byelapsed=0.060
command_timing = :file=~~/src/HOL/Library/Field_as_Ring.thyoffset=3038name=byelapsed=0.058
command_timing = :file=~~/src/HOL/Library/Polynomial_Factorial.thyoffset=35768name=byelapsed=0.153
command_timing = :file=~~/src/HOL/Library/Polynomial_Factorial.thyoffset=39950name=byelapsed=0.016
ML_statistics = :now=1489705844.5tasks_ready=76tasks_pending=6tasks_running=5tasks_passive=2tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=18size_allocation=3579838464size_allocation_free=2454088272size_heap=4171235328size_heap_free_last_full_GC=0size_heap_free_last_GC=3581941008threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=84.376time_GC=3.764user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/src/HOL/Library/Polynomial_Factorial.thyoffset=39872name=byelapsed=0.045
command_timing = :file=~~/src/HOL/Library/Polynomial_Factorial.thyoffset=39679name=byelapsed=0.021
command_timing = :file=~~/src/HOL/Library/Polynomial_Factorial.thyoffset=38891name=byelapsed=0.013
command_timing = :file=~~/src/HOL/Library/Fraction_Field.thyoffset=3644name=byelapsed=0.013
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=2479name=byelapsed=0.030
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=10188name=byelapsed=0.013
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=23075name=byelapsed=0.034
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=23744name=byelapsed=0.016
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=23996name=byelapsed=0.016
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=26553name=byelapsed=0.047
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=26669name=byelapsed=0.012
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Improved_Code_Equations.thyoffset=4048name=byelapsed=0.022
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=43139name=byelapsed=0.012
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=46165name=byelapsed=0.030
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=51169name=byelapsed=0.010
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=53144name=byelapsed=0.013
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=53940name=byelapsed=0.016
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=54068name=byelapsed=0.038
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=56238name=byelapsed=0.015
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=58566name=byelapsed=0.017
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=59155name=byelapsed=0.010
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=63474name=byelapsed=0.014
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=63933name=byelapsed=0.016
ML_statistics = :now=1489705845tasks_ready=50tasks_pending=6tasks_running=2tasks_passive=0tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=18size_allocation=3579838464size_allocation_free=1901388288size_heap=4171235328size_heap_free_last_full_GC=0size_heap_free_last_GC=3581941008threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=85.18time_GC=3.764user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=64454name=byelapsed=0.013
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=64859name=byelapsed=0.011
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=66091name=byelapsed=0.019
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=70008name=byelapsed=0.017
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Unsorted.thyoffset=6949name=byelapsed=0.011
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=72183name=byelapsed=0.011
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=72273name=byelapsed=0.011
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Unsorted.thyoffset=8356name=byelapsed=0.010
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=80675name=byelapsed=0.018
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=82246name=byelapsed=0.016
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=82755name=byelapsed=0.028
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=94070name=byelapsed=0.020
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=93926name=byelapsed=0.011
ML_statistics = :now=1489705845.51tasks_ready=36tasks_pending=6tasks_running=3tasks_passive=1tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=18size_allocation=3579838464size_allocation_free=754671632size_heap=4171235328size_heap_free_last_full_GC=0size_heap_free_last_GC=3581941008threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=85.98time_GC=3.764user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=94597name=byelapsed=0.010
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=94501name=byelapsed=0.080
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Unsorted.thyoffset=8939name=byelapsed=0.015
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=106159name=byelapsed=0.017
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Unsorted.thyoffset=12441name=byelapsed=0.010
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Unsorted.thyoffset=13755name=byelapsed=0.021
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Unsorted.thyoffset=15156name=byelapsed=0.028
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Unsorted.thyoffset=15728name=byelapsed=0.051
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Unsorted.thyoffset=15908name=byelapsed=0.017
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=109062name=byelapsed=0.012
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Unsorted.thyoffset=16075name=byelapsed=0.011
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=109377name=byelapsed=0.025
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=118719name=byelapsed=0.010
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=118578name=byelapsed=0.012
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=118446name=byelapsed=0.012
ML_statistics = :now=1489705846.03tasks_ready=24tasks_pending=6tasks_running=2tasks_passive=0tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=19size_allocation=3587178496size_allocation_free=2462080720size_heap=4184866816size_heap_free_last_full_GC=0size_heap_free_last_GC=3588934456threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=87.308time_GC=3.856user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=120863name=byelapsed=0.022
command_timing = :file=~~/src/HOL/Library/Multiset.thyoffset=120726name=byelapsed=0.014
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Is_Rat_To_Rat.thyoffset=3480name=proofelapsed=0.017
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Is_Rat_To_Rat.thyoffset=4664name=byelapsed=0.023
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Is_Rat_To_Rat.thyoffset=4850name=byelapsed=0.011
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Ring_Hom.thyoffset=1106name=byelapsed=0.010
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Ring_Hom.thyoffset=1237name=byelapsed=0.010
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=10901name=byelapsed=0.023
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=20825name=byelapsed=0.012
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=41676name=byelapsed=0.013
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=47174name=byelapsed=0.010
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=50399name=byelapsed=0.011
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=51696name=byelapsed=0.020
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=52184name=byelapsed=0.017
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=52061name=byelapsed=0.017
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=64878name=byelapsed=0.022
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=64591name=byelapsed=0.016
ML_statistics = :now=1489705846.53tasks_ready=8tasks_pending=6tasks_running=2tasks_passive=0tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=19size_allocation=3587178496size_allocation_free=1267645048size_heap=4184866816size_heap_free_last_full_GC=0size_heap_free_last_GC=3588934456threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=88.136time_GC=3.856user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=66291name=byelapsed=0.017
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=66143name=byelapsed=0.015
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=66784name=byelapsed=0.017
command_timing = :file=~~/src/HOL/Number_Theory/Factorial_Ring.thyoffset=66636name=byelapsed=0.015
command_timing = :file=~~/src/HOL/Number_Theory/Euclidean_Algorithm.thyoffset=9716name=byelapsed=0.023
command_timing = :file=~~/src/HOL/Number_Theory/Euclidean_Algorithm.thyoffset=12873name=byelapsed=0.011
command_timing = :file=~~/src/HOL/Library/Normalized_Fraction.thyoffset=3533name=byelapsed=0.050
command_timing = :file=~~/src/HOL/Library/Normalized_Fraction.thyoffset=5463name=byelapsed=0.010
command_timing = :file=~~/src/HOL/Library/Normalized_Fraction.thyoffset=5342name=byelapsed=0.016
command_timing = :file=~~/src/HOL/Library/Normalized_Fraction.thyoffset=9996name=byelapsed=0.014
command_timing = :file=~~/src/HOL/Library/Normalized_Fraction.thyoffset=12712name=byelapsed=0.011
command_timing = :file=~~/src/HOL/Library/Polynomial_Factorial.thyoffset=40414name=byelapsed=0.012
ML_statistics = :now=1489705847.11tasks_ready=0tasks_pending=5tasks_running=1tasks_passive=0tasks_urgent=0workers_total=4workers_active=1workers_waiting=0full_GCs=0partial_GCs=20size_allocation=3586129920size_allocation_free=3586129920size_heap=4169138176size_heap_free_last_full_GC=0size_heap_free_last_GC=3588820360threads_in_ML=1threads_total=7threads_wait_condvar=4threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=89.132time_GC=3.988user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
ML_statistics = :now=1489705847.61tasks_ready=0tasks_pending=5tasks_running=1tasks_passive=0tasks_urgent=0workers_total=4workers_active=1workers_waiting=0full_GCs=0partial_GCs=20size_allocation=3586129920size_allocation_free=2994280040size_heap=4169138176size_heap_free_last_full_GC=0size_heap_free_last_GC=3588820360threads_in_ML=1threads_total=7threads_wait_condvar=4threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=89.536time_GC=3.988user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
ML_statistics = :now=1489705848.11tasks_ready=0tasks_pending=5tasks_running=1tasks_passive=0tasks_urgent=0workers_total=4workers_active=1workers_waiting=0full_GCs=0partial_GCs=20size_allocation=3586129920size_allocation_free=2589949856size_heap=4169138176size_heap_free_last_full_GC=0size_heap_free_last_GC=3588820360threads_in_ML=1threads_total=7threads_wait_condvar=4threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=89.936time_GC=3.988user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
ML_statistics = :now=1489705848.61tasks_ready=0tasks_pending=5tasks_running=1tasks_passive=0tasks_urgent=0workers_total=4workers_active=1workers_waiting=0full_GCs=0partial_GCs=20size_allocation=3586129920size_allocation_free=1913635272size_heap=4169138176size_heap_free_last_full_GC=0size_heap_free_last_GC=3588820360threads_in_ML=1threads_total=7threads_wait_condvar=4threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=90.736time_GC=3.988user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
ML_statistics = :now=1489705849.11tasks_ready=0tasks_pending=5tasks_running=1tasks_passive=0tasks_urgent=0workers_total=4workers_active=1workers_waiting=0full_GCs=0partial_GCs=20size_allocation=3586129920size_allocation_free=1147034768size_heap=4169138176size_heap_free_last_full_GC=0size_heap_free_last_GC=3588820360threads_in_ML=1threads_total=7threads_wait_condvar=4threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=91.14time_GC=3.988user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
ML_statistics = :now=1489705849.61tasks_ready=0tasks_pending=5tasks_running=1tasks_passive=0tasks_urgent=0workers_total=4workers_active=1workers_waiting=0full_GCs=0partial_GCs=20size_allocation=3586129920size_allocation_free=416085856size_heap=4169138176size_heap_free_last_full_GC=0size_heap_free_last_GC=3588820360threads_in_ML=1threads_total=7threads_wait_condvar=4threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=91.54time_GC=3.988user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
ML_statistics = :now=1489705850.19tasks_ready=0tasks_pending=5tasks_running=1tasks_passive=0tasks_urgent=0workers_total=4workers_active=1workers_waiting=0full_GCs=0partial_GCs=21size_allocation=3545235456size_allocation_free=2720264640size_heap=4161798144size_heap_free_last_full_GC=0size_heap_free_last_GC=3546085648threads_in_ML=1threads_total=7threads_wait_condvar=4threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=92.164time_GC=4.196user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
ML_statistics = :now=1489705850.7tasks_ready=0tasks_pending=5tasks_running=1tasks_passive=0tasks_urgent=0workers_total=4workers_active=1workers_waiting=0full_GCs=0partial_GCs=21size_allocation=3545235456size_allocation_free=2009238632size_heap=4161798144size_heap_free_last_full_GC=0size_heap_free_last_GC=3546085648threads_in_ML=1threads_total=7threads_wait_condvar=4threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=92.564time_GC=4.196user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
ML_statistics = :now=1489705851.2tasks_ready=0tasks_pending=5tasks_running=1tasks_passive=0tasks_urgent=0workers_total=4workers_active=1workers_waiting=0full_GCs=0partial_GCs=21size_allocation=3545235456size_allocation_free=1428236128size_heap=4161798144size_heap_free_last_full_GC=0size_heap_free_last_GC=3546085648threads_in_ML=1threads_total=7threads_wait_condvar=4threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=92.968time_GC=4.196user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
ML_statistics = :now=1489705851.7tasks_ready=0tasks_pending=5tasks_running=1tasks_passive=0tasks_urgent=0workers_total=4workers_active=1workers_waiting=0full_GCs=0partial_GCs=21size_allocation=3545235456size_allocation_free=940556800size_heap=4161798144size_heap_free_last_full_GC=0size_heap_free_last_GC=3546085648threads_in_ML=1threads_total=7threads_wait_condvar=4threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=93.368time_GC=4.196user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/src/HOL/Library/Polynomial_Factorial.thyoffset=40851name=textelapsed=11.255
### theory "Polynomial_Factorial"
### 13.199s elapsed time, 21.240s cpu time, 0.772s GC time
Loading theory "Missing_Polynomial" (required by "Polynomial_Interpolation" via "Newton_Interpolation" via "Ring_Hom_Poly")
loading_theory = Missing_Polynomial
ML_statistics = :now=1489705852.2tasks_ready=0tasks_pending=4tasks_running=1tasks_passive=0tasks_urgent=0workers_total=4workers_active=1workers_waiting=0full_GCs=0partial_GCs=21size_allocation=3545235456size_allocation_free=213949528size_heap=4161798144size_heap_free_last_full_GC=0size_heap_free_last_GC=3546085648threads_in_ML=1threads_total=7threads_wait_condvar=4threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=94.168time_GC=4.196user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=310name=theoryelapsed=0.514
ML_statistics = :now=1489705852.7tasks_ready=8tasks_pending=4tasks_running=3tasks_passive=1tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=21size_allocation=3545235456size_allocation_free=134166448size_heap=4161798144size_heap_free_last_full_GC=0size_heap_free_last_GC=3546085648threads_in_ML=1threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=94.572time_GC=4.196user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=596name=byelapsed=0.052
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=1392name=byelapsed=0.044
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=4470name=lemmaelapsed=0.046
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=4871name=lemmaelapsed=0.031
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=4849name=byelapsed=0.013
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=6659name=lemmaelapsed=0.010
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=6898name=lemmaelapsed=0.010
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=7008name=applyelapsed=0.013
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=8898name=lemmaelapsed=0.014
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=7060name=applyelapsed=0.032
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=10392name=henceelapsed=0.013
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=16209name=lemmaelapsed=0.025
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=20545name=definitionelapsed=0.013
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=9915name=byelapsed=0.046
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=9748name=byelapsed=0.014
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=23721name=lemmaelapsed=0.011
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=9331name=byelapsed=0.017
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=27316name=definitionelapsed=0.010
ML_statistics = :now=1489705853.23tasks_ready=51tasks_pending=4tasks_running=2tasks_passive=0tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=22size_allocation=3507486720size_allocation_free=2976272912size_heap=4171235328size_heap_free_last_full_GC=0size_heap_free_last_GC=3509832072threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=95.4time_GC=4.244user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=717name=byelapsed=0.030
Found termination order: "size_list (\<lambda>p. size (fst p)) <*mlex*> {}"
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=31322name=funelapsed=0.120
### theory "Missing_Polynomial"
### 1.327s elapsed time, 1.880s cpu time, 0.048s GC time
Loading theory "Lagrange_Interpolation" (required by "Polynomial_Interpolation")
loading_theory = Lagrange_Interpolation
Loading theory "Ring_Hom_Poly" (required by "Polynomial_Interpolation" via "Newton_Interpolation")
loading_theory = Ring_Hom_Poly
ML_statistics = :now=1489705853.73tasks_ready=62tasks_pending=3tasks_running=2tasks_passive=0tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=22size_allocation=3507486720size_allocation_free=2284244752size_heap=4171235328size_heap_free_last_full_GC=0size_heap_free_last_GC=3509832072threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=96.2time_GC=4.244user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Lagrange_Interpolation.thyoffset=354name=theoryelapsed=0.284
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Lagrange_Interpolation.thyoffset=421name=definitionelapsed=0.015
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Lagrange_Interpolation.thyoffset=635name=definitionelapsed=0.014
### theory "Lagrange_Interpolation"
### 0.349s elapsed time, 0.700s cpu time, 0.000s GC time
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=5799name=byelapsed=0.021
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=5719name=byelapsed=0.015
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=40213name=byelapsed=0.011
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=39943name=byelapsed=0.018
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=11943name=byelapsed=0.011
ML_statistics = :now=1489705854.23tasks_ready=48tasks_pending=3tasks_running=5tasks_passive=3tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=22size_allocation=3507486720size_allocation_free=1932872976size_heap=4171235328size_heap_free_last_full_GC=0size_heap_free_last_GC=3509832072threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=97.004time_GC=4.244user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=11909name=byelapsed=0.156
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Ring_Hom_Poly.thyoffset=154name=theoryelapsed=0.832
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=11809name=qedelapsed=0.017
locale ring_hom_pcompose =
  fixes p :: "'a poly" 
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Ring_Hom_Poly.thyoffset=872name=definitionelapsed=0.010
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=11514name=byelapsed=0.053
locale semiring_hom =
  fixes hom :: "'a \<Rightarrow> 'b" 
  assumes "semiring_hom hom"
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=11443name=byelapsed=0.060
locale map_poly_semiring_hom =
  fixes hom :: "'a \<Rightarrow> 'b" 
  assumes "map_poly_semiring_hom hom"
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Ring_Hom_Poly.thyoffset=7554name=localeelapsed=0.013
locale ring_hom =
  fixes hom :: "'a \<Rightarrow> 'b" 
  assumes "ring_hom hom"
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=11201name=byelapsed=0.059
locale map_poly_ring_hom =
  fixes hom :: "'a \<Rightarrow> 'b" 
  assumes "map_poly_ring_hom hom"
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Ring_Hom_Poly.thyoffset=8148name=localeelapsed=0.015
locale inj_semiring_hom =
  fixes hom :: "'a \<Rightarrow> 'b" 
  assumes "inj_semiring_hom hom"
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=11117name=byelapsed=0.054
locale inj_ring_hom_map_poly =
  fixes hom :: "'a \<Rightarrow> 'b" 
  assumes "inj_ring_hom_map_poly hom"
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Ring_Hom_Poly.thyoffset=10419name=localeelapsed=0.017
locale idom_hom =
  fixes hom :: "'a \<Rightarrow> 'b" 
  assumes "idom_hom hom"
locale inj_field_hom' =
  fixes hom :: "'a \<Rightarrow> 'b" 
  assumes "inj_field_hom' hom"
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Ring_Hom_Poly.thyoffset=10801name=localeelapsed=0.027
locale inj_field_hom =
  fixes hom :: "'a \<Rightarrow> 'b" 
  assumes "inj_field_hom hom"
locale inj_field_hom' =
  fixes hom :: "'a \<Rightarrow> 'b" 
  assumes "inj_field_hom' hom"
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Ring_Hom_Poly.thyoffset=2477name=byelapsed=0.019
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Ring_Hom_Poly.thyoffset=12602name=definitionelapsed=0.011
ML_statistics = :now=1489705854.73tasks_ready=87tasks_pending=3tasks_running=6tasks_passive=4tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=22size_allocation=3507486720size_allocation_free=1438371224size_heap=4171235328size_heap_free_last_full_GC=0size_heap_free_last_GC=3509832072threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=97.804time_GC=4.244user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Ring_Hom_Poly.thyoffset=1815name=byelapsed=0.018
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Ring_Hom_Poly.thyoffset=13068name=interpretationelapsed=0.011
### theory "Ring_Hom_Poly"
### 1.228s elapsed time, 2.460s cpu time, 0.000s GC time
Loading theory "Newton_Interpolation" (required by "Polynomial_Interpolation")
loading_theory = Newton_Interpolation
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Ring_Hom_Poly.thyoffset=5840name=byelapsed=0.010
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Ring_Hom_Poly.thyoffset=8323name=byelapsed=0.014
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Ring_Hom_Poly.thyoffset=9428name=byelapsed=0.021
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Ring_Hom_Poly.thyoffset=9776name=byelapsed=0.030
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Ring_Hom_Poly.thyoffset=11895name=byelapsed=0.016
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Ring_Hom_Poly.thyoffset=11518name=byelapsed=0.010
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=29071name=byelapsed=0.018
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Ring_Hom_Poly.thyoffset=3847name=byelapsed=0.032
ML_statistics = :now=1489705855.23tasks_ready=49tasks_pending=2tasks_running=3tasks_passive=1tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=22size_allocation=3507486720size_allocation_free=225055176size_heap=4171235328size_heap_free_last_full_GC=0size_heap_free_last_GC=3509832072threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=99.412time_GC=4.244user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=16603name=byelapsed=0.494
ML_statistics = :now=1489705855.96tasks_ready=48tasks_pending=2tasks_running=3tasks_passive=1tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=23size_allocation=3524263936size_allocation_free=3524263936size_heap=4140826624size_heap_free_last_full_GC=0size_heap_free_last_GC=3526219304threads_in_ML=1threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=99.8time_GC=4.684user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=16581name=byelapsed=0.375
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=16541name=byelapsed=0.422
ML_statistics = :now=1489705856.46tasks_ready=45tasks_pending=2tasks_running=2tasks_passive=0tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=23size_allocation=3524263936size_allocation_free=2537979616size_heap=4140826624size_heap_free_last_full_GC=0size_heap_free_last_GC=3526219304threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=101.416time_GC=4.684user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=370name=theoryelapsed=1.902
Found termination order: "(\<lambda>p. length (snd p)) <*mlex*> {}"
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=882name=funelapsed=0.175
### Missing patterns in function definition:
### \<And>v va b c. divided_differences_impl (v # va) b c [] = undefined
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=854name=byelapsed=0.496
ML_statistics = :now=1489705856.96tasks_ready=53tasks_pending=2tasks_running=3tasks_passive=1tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=23size_allocation=3524263936size_allocation_free=1851164496size_heap=4140826624size_heap_free_last_full_GC=0size_heap_free_last_GC=3526219304threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=102.216time_GC=4.684user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
Found termination order:
  "(\<lambda>p. length (snd (snd (snd p)))) <*mlex*> {}"
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=5451name=funelapsed=0.177
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=5275name=byelapsed=0.035
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=4373name=byelapsed=0.016
Found termination order: "(\<lambda>p. length (snd p)) <*mlex*> {}"
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=5761name=funelapsed=0.184
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=6113name=definitionelapsed=0.014
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=6242name=definitionelapsed=0.011
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=6371name=definitionelapsed=0.011
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=6407name=definitionelapsed=0.011
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=6442name=definitionelapsed=0.012
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=6635name=functionelapsed=0.107
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=6635name=functionelapsed=0.013
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=6858name=definitionelapsed=0.012
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=6920name=definitionelapsed=0.012
ML_statistics = :now=1489705857.47tasks_ready=92tasks_pending=2tasks_running=9tasks_passive=7tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=23size_allocation=3524263936size_allocation_free=1145894984size_heap=4140826624size_heap_free_last_full_GC=0size_heap_free_last_GC=3526219304threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=103.02time_GC=4.684user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=6962name=functionelapsed=0.103
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=6962name=functionelapsed=0.013
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=3971name=byelapsed=0.426
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=7192name=definitionelapsed=0.012
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=7275name=definitionelapsed=0.016
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=7335name=definitionelapsed=0.012
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=3460name=proofelapsed=0.060
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=9135name=definitionelapsed=0.014
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=10798name=lemmaelapsed=0.012
### Missing patterns in function definition:
### \<And>v va b c. divided_differences_impl_int (v # va) b c [] = undefined
Found termination order:
  "(\<lambda>p. size_list (nat \<circ> abs) (snd (snd (snd p)))) <*mlex*>
   {}"
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=21942name=funelapsed=0.206
ML_statistics = :now=1489705858.08tasks_ready=135tasks_pending=2tasks_running=12tasks_passive=9tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=24size_allocation=3425697792size_allocation_free=3425697792size_heap=4158652416size_heap_free_last_full_GC=0size_heap_free_last_GC=3427394472threads_in_ML=1threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=104.516time_GC=4.9user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
Found termination order:
  "(\<lambda>p. size_list (nat \<circ> abs) (snd p)) <*mlex*> {}"
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=22359name=funelapsed=0.320
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=22767name=definitionelapsed=0.012
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=26977name=lemmaelapsed=0.010
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=28635name=lemmaelapsed=0.010
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=32824name=definitionelapsed=0.015
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=34670name=definitionelapsed=0.011
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=34867name=definitionelapsed=0.013
### theory "Newton_Interpolation"
### 3.623s elapsed time, 7.204s cpu time, 0.656s GC time
Loading theory "Polynomial_Interpolation"
loading_theory = Polynomial_Interpolation
ML_statistics = :now=1489705858.58tasks_ready=160tasks_pending=1tasks_running=12tasks_passive=9tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=24size_allocation=3425697792size_allocation_free=2656462296size_heap=4158652416size_heap_free_last_full_GC=0size_heap_free_last_GC=3427394472threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=105.32time_GC=4.9user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
ML_statistics = :now=1489705859.08tasks_ready=159tasks_pending=1tasks_running=12tasks_passive=9tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=24size_allocation=3425697792size_allocation_free=2154744272size_heap=4158652416size_heap_free_last_full_GC=0size_heap_free_last_GC=3427394472threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=106.12time_GC=4.9user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=3549name=byelapsed=1.558
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=3315name=byelapsed=0.081
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=3218name=byelapsed=0.011
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=4872name=byelapsed=0.011
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=2366name=byelapsed=0.025
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=2210name=byelapsed=0.122
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=8625name=byelapsed=0.013
ML_statistics = :now=1489705859.58tasks_ready=130tasks_pending=1tasks_running=7tasks_passive=5tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=24size_allocation=3425697792size_allocation_free=1705720120size_heap=4158652416size_heap_free_last_full_GC=0size_heap_free_last_GC=3427394472threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=106.924time_GC=4.9user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=7113name=byelapsed=0.033
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=16035name=qedelapsed=0.016
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=16023name=byelapsed=0.026
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Polynomial_Interpolation.thyoffset=609name=theoryelapsed=1.472
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=15902name=byelapsed=0.201
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=15363name=byelapsed=0.020
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=15178name=byelapsed=0.020
ML_statistics = :now=1489705860.08tasks_ready=176tasks_pending=1tasks_running=8tasks_passive=6tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=24size_allocation=3425697792size_allocation_free=772375384size_heap=4158652416size_heap_free_last_full_GC=0size_heap_free_last_GC=3427394472threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=108.528time_GC=4.9user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=14976name=byelapsed=0.015
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=11828name=byelapsed=0.016
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=10702name=byelapsed=0.022
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=11721name=byelapsed=0.090
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=6802name=byelapsed=0.065
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=14827name=byelapsed=0.070
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=13428name=byelapsed=0.010
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=13747name=haveelapsed=0.010
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=14687name=byelapsed=0.012
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=14728name=qedelapsed=0.044
ML_statistics = :now=1489705860.58tasks_ready=226tasks_pending=1tasks_running=9tasks_passive=7tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=24size_allocation=3425697792size_allocation_free=181917144size_heap=4158652416size_heap_free_last_full_GC=0size_heap_free_last_GC=3427394472threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=109.332time_GC=4.9user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=14718name=byelapsed=0.149
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=14423name=byelapsed=0.019
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=12871name=byelapsed=0.098
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=12653name=byelapsed=0.137
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Polynomial_Interpolation.thyoffset=761name=datatypeelapsed=1.115
Found termination order: "{}"
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Polynomial_Interpolation.thyoffset=832name=funelapsed=0.087
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=12350name=byelapsed=0.099
ML_statistics = :now=1489705861.15tasks_ready=249tasks_pending=1tasks_running=10tasks_passive=8tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=25size_allocation=3408920576size_allocation_free=2636533480size_heap=4160749568size_heap_free_last_full_GC=0size_heap_free_last_GC=3411031488threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=110.344time_GC=5.112user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=14184name=byelapsed=0.162
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=14077name=byelapsed=0.011
Found termination order: "{}"
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=13945name=byelapsed=0.021
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Polynomial_Interpolation.thyoffset=1120name=funelapsed=0.207
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=13711name=byelapsed=0.014
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Polynomial_Interpolation.thyoffset=1541name=lemmaelapsed=0.011
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Polynomial_Interpolation.thyoffset=3429name=lemmaelapsed=0.012
### theory "Polynomial_Interpolation"
### 2.973s elapsed time, 5.936s cpu time, 0.212s GC time
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=13563name=byelapsed=0.104
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=13290name=byelapsed=0.036
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=29139name=byelapsed=0.011
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=31379name=qedelapsed=0.013
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=13114name=byelapsed=0.051
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=29734name=byelapsed=0.012
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=11533name=byelapsed=0.034
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=12126name=byelapsed=0.070
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=10318name=byelapsed=0.018
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=10178name=byelapsed=0.021
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=30883name=byelapsed=0.083
ML_statistics = :now=1489705861.65tasks_ready=251tasks_pending=1tasks_running=7tasks_passive=5tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=25size_allocation=3408920576size_allocation_free=2059820896size_heap=4160749568size_heap_free_last_full_GC=0size_heap_free_last_GC=3411031488threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=111.148time_GC=5.112user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
ML_statistics = :now=1489705862.16tasks_ready=251tasks_pending=1tasks_running=7tasks_passive=6tasks_urgent=0workers_total=4workers_active=1workers_waiting=1full_GCs=0partial_GCs=25size_allocation=3408920576size_allocation_free=1401460840size_heap=4160749568size_heap_free_last_full_GC=0size_heap_free_last_GC=3411031488threads_in_ML=1threads_total=7threads_wait_condvar=4threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=111.732time_GC=5.112user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=30604name=byelapsed=0.506
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=29917name=byelapsed=0.016
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=28625name=qedelapsed=0.018
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=21468name=byelapsed=0.013
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=20847name=byelapsed=0.017
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=20581name=byelapsed=0.019
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=20376name=byelapsed=0.014
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=20271name=byelapsed=0.014
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=18739name=haveelapsed=0.010
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=19846name=byelapsed=0.016
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=19584name=byelapsed=0.039
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=19492name=byelapsed=0.056
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=28606name=byelapsed=0.344
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=27981name=byelapsed=0.016
ML_statistics = :now=1489705862.66tasks_ready=250tasks_pending=1tasks_running=23tasks_passive=20tasks_urgent=0workers_total=4workers_active=2workers_waiting=1full_GCs=0partial_GCs=25size_allocation=3408920576size_allocation_free=812327712size_heap=4160749568size_heap_free_last_full_GC=0size_heap_free_last_GC=3411031488threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=112.276time_GC=5.112user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=19220name=byelapsed=0.064
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=18954name=byelapsed=0.010
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=18547name=byelapsed=0.012
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=18089name=byelapsed=0.030
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=23812name=byelapsed=0.040
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=28169name=byelapsed=0.016
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=27882name=byelapsed=0.010
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=17580name=byelapsed=0.108
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=29247name=byelapsed=0.190
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=36076name=byelapsed=0.043
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=30448name=byelapsed=0.015
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=30062name=byelapsed=0.016
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=29994name=byelapsed=0.010
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=35539name=byelapsed=0.055
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Lagrange_Interpolation.thyoffset=3968name=byelapsed=0.010
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=26555name=byelapsed=0.021
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=5003name=byelapsed=0.013
ML_statistics = :now=1489705863.2tasks_ready=146tasks_pending=1tasks_running=12tasks_passive=10tasks_urgent=0workers_total=4workers_active=2workers_waiting=1full_GCs=0partial_GCs=26size_allocation=3377463296size_allocation_free=3377463296size_heap=4173332480size_heap_free_last_full_GC=0size_heap_free_last_GC=3379249880threads_in_ML=1threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=113.248time_GC=5.232user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Lagrange_Interpolation.thyoffset=2123name=byelapsed=0.111
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=15176name=byelapsed=0.071
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=1028name=byelapsed=0.120
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=24689name=byelapsed=0.565
ML_statistics = :now=1489705863.7tasks_ready=46tasks_pending=1tasks_running=10tasks_passive=8tasks_urgent=0workers_total=4workers_active=2workers_waiting=1full_GCs=0partial_GCs=26size_allocation=3377463296size_allocation_free=1908869304size_heap=4173332480size_heap_free_last_full_GC=0size_heap_free_last_GC=3379249880threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=114.86time_GC=5.232user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=24590name=byelapsed=0.017
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=8593name=byelapsed=0.053
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=12966name=byelapsed=0.024
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=12249name=byelapsed=0.129
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=12899name=byelapsed=0.030
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=12828name=byelapsed=0.012
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=17613name=byelapsed=0.085
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=19346name=byelapsed=0.075
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=14832name=byelapsed=0.287
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=24064name=byelapsed=0.010
ML_statistics = :now=1489705864.21tasks_ready=39tasks_pending=1tasks_running=3tasks_passive=1tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=26size_allocation=3377463296size_allocation_free=1299684368size_heap=4173332480size_heap_free_last_full_GC=0size_heap_free_last_GC=3379249880threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=115.664time_GC=5.232user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=30101name=byelapsed=0.067
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=25287name=byelapsed=0.010
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=26406name=byelapsed=0.024
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=26265name=byelapsed=0.020
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=33902name=byelapsed=0.145
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=26184name=byelapsed=0.013
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=25933name=byelapsed=0.016
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=24358name=byelapsed=0.016
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=25634name=byelapsed=0.051
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=25585name=byelapsed=0.025
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=34116name=byelapsed=0.127
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=25390name=byelapsed=0.020
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=22608name=byelapsed=0.019
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=26808name=byelapsed=0.033
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=21874name=byelapsed=0.011
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=34843name=byelapsed=0.057
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=34368name=byelapsed=0.101
ML_statistics = :now=1489705864.71tasks_ready=30tasks_pending=1tasks_running=3tasks_passive=1tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=26size_allocation=3377463296size_allocation_free=664621872size_heap=4173332480size_heap_free_last_full_GC=0size_heap_free_last_GC=3379249880threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=116.464time_GC=5.232user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=36726name=byelapsed=0.118
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Lagrange_Interpolation.thyoffset=1869name=byelapsed=0.032
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Lagrange_Interpolation.thyoffset=1767name=byelapsed=0.023
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Lagrange_Interpolation.thyoffset=1701name=byelapsed=0.061
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Lagrange_Interpolation.thyoffset=1236name=byelapsed=0.037
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Ring_Hom_Poly.thyoffset=712name=byelapsed=0.059
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Ring_Hom_Poly.thyoffset=5060name=byelapsed=0.013
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=34705name=byelapsed=0.480
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Ring_Hom_Poly.thyoffset=4935name=byelapsed=0.058
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Polynomial_Interpolation.thyoffset=5749name=byelapsed=0.022
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Polynomial_Interpolation.thyoffset=2637name=byelapsed=0.104
ML_statistics = :now=1489705865.24tasks_ready=25tasks_pending=1tasks_running=5tasks_passive=2tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=27size_allocation=3370123264size_allocation_free=2374373984size_heap=4182769664size_heap_free_last_full_GC=0size_heap_free_last_GC=3372212088threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=117.844time_GC=5.304user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Polynomial_Interpolation.thyoffset=1848name=byelapsed=0.035
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Polynomial_Interpolation.thyoffset=7422name=byelapsed=0.013
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=37823name=byelapsed=0.011
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Newton_Interpolation.thyoffset=37688name=byelapsed=0.012
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Polynomial_Interpolation.thyoffset=9081name=byelapsed=0.011
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Polynomial_Interpolation.thyoffset=5497name=byelapsed=0.081
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Polynomial_Interpolation.thyoffset=7161name=byelapsed=0.085
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Polynomial_Interpolation.thyoffset=8954name=byelapsed=0.042
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Polynomial_Interpolation.thyoffset=6816name=byelapsed=0.089
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Polynomial_Interpolation.thyoffset=3831name=byelapsed=0.017
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=4176name=byelapsed=0.010
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=5313name=byelapsed=0.010
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=6203name=byelapsed=0.014
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=3071name=byelapsed=0.016
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=14307name=byelapsed=0.018
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=14093name=byelapsed=0.031
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=14668name=byelapsed=0.034
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=23015name=byelapsed=0.010
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=21689name=byelapsed=0.010
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=21598name=byelapsed=0.018
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=23608name=byelapsed=0.010
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=31146name=byelapsed=0.010
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=33129name=byelapsed=0.013
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=33728name=byelapsed=0.010
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=35105name=byelapsed=0.027
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Ring_Hom_Poly.thyoffset=1102name=byelapsed=0.012
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Missing_Polynomial.thyoffset=35575name=byelapsed=0.011
ML_statistics = :now=1489705865.74tasks_ready=1tasks_pending=1tasks_running=5tasks_passive=1tasks_urgent=0workers_total=4workers_active=2workers_waiting=0full_GCs=0partial_GCs=27size_allocation=3370123264size_allocation_free=1782339096size_heap=4182769664size_heap_free_last_full_GC=0size_heap_free_last_GC=3372212088threads_in_ML=2threads_total=7threads_wait_condvar=3threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=118.668time_GC=5.304user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Ring_Hom_Poly.thyoffset=12500name=byelapsed=0.013
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Ring_Hom_Poly.thyoffset=10767name=byelapsed=0.027
command_timing = :file=~~/afp/thys/Polynomial_Interpolation/Polynomial_Interpolation.thyoffset=4746name=byelapsed=0.014
ML_statistics = :now=1489705866.24tasks_ready=0tasks_pending=0tasks_running=1tasks_passive=0tasks_urgent=0workers_total=4workers_active=1workers_waiting=0full_GCs=0partial_GCs=27size_allocation=3370123264size_allocation_free=1119228064size_heap=4182769664size_heap_free_last_full_GC=0size_heap_free_last_GC=3372212088threads_in_ML=1threads_total=7threads_wait_condvar=4threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=119.46time_GC=5.304user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
ML_statistics = :now=1489705866.78tasks_ready=0tasks_pending=0tasks_running=1tasks_passive=0tasks_urgent=0workers_total=4workers_active=1workers_waiting=0full_GCs=0partial_GCs=28size_allocation=3374317568size_allocation_free=3374317568size_heap=4173332480size_heap_free_last_full_GC=0size_heap_free_last_GC=3376655160threads_in_ML=1threads_total=7threads_wait_condvar=4threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=120.076time_GC=5.432user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
ML_statistics = :now=1489705867.28tasks_ready=0tasks_pending=0tasks_running=1tasks_passive=0tasks_urgent=0workers_total=4workers_active=1workers_waiting=0full_GCs=0partial_GCs=28size_allocation=3374317568size_allocation_free=2556599224size_heap=4173332480size_heap_free_last_full_GC=0size_heap_free_last_GC=3376655160threads_in_ML=1threads_total=7threads_wait_condvar=4threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=120.48time_GC=5.432user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
ML_statistics = :now=1489705867.79tasks_ready=0tasks_pending=0tasks_running=1tasks_passive=0tasks_urgent=0workers_total=4workers_active=1workers_waiting=0full_GCs=0partial_GCs=28size_allocation=3374317568size_allocation_free=1734424544size_heap=4173332480size_heap_free_last_full_GC=0size_heap_free_last_GC=3376655160threads_in_ML=1threads_total=7threads_wait_condvar=4threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=120.884time_GC=5.432user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
ML_statistics = :now=1489705868.29tasks_ready=0tasks_pending=0tasks_running=1tasks_passive=0tasks_urgent=0workers_total=4workers_active=1workers_waiting=0full_GCs=0partial_GCs=28size_allocation=3374317568size_allocation_free=82318712size_heap=4173332480size_heap_free_last_full_GC=0size_heap_free_last_GC=3376655160threads_in_ML=1threads_total=7threads_wait_condvar=4threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=121.684time_GC=5.432user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
ML_statistics = :now=1489705868.9tasks_ready=0tasks_pending=0tasks_running=1tasks_passive=0tasks_urgent=0workers_total=4workers_active=1workers_waiting=0full_GCs=0partial_GCs=29size_allocation=3340763136size_allocation_free=2827363936size_heap=4143972352size_heap_free_last_full_GC=0size_heap_free_last_GC=3342713528threads_in_ML=1threads_total=7threads_wait_condvar=4threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=122.16time_GC=5.684user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
ML_statistics = :now=1489705869.4tasks_ready=0tasks_pending=0tasks_running=1tasks_passive=0tasks_urgent=0workers_total=4workers_active=1workers_waiting=0full_GCs=0partial_GCs=29size_allocation=3340763136size_allocation_free=2382532120size_heap=4143972352size_heap_free_last_full_GC=0size_heap_free_last_GC=3342713528threads_in_ML=1threads_total=7threads_wait_condvar=4threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=122.564time_GC=5.684user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
ML_statistics = :now=1489705869.9tasks_ready=0tasks_pending=0tasks_running=1tasks_passive=0tasks_urgent=0workers_total=4workers_active=1workers_waiting=0full_GCs=0partial_GCs=29size_allocation=3340763136size_allocation_free=1983982200size_heap=4143972352size_heap_free_last_full_GC=0size_heap_free_last_GC=3342713528threads_in_ML=1threads_total=7threads_wait_condvar=4threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=122.964time_GC=5.684user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
ML_statistics = :now=1489705870.4tasks_ready=0tasks_pending=0tasks_running=1tasks_passive=0tasks_urgent=0workers_total=4workers_active=1workers_waiting=0full_GCs=0partial_GCs=29size_allocation=3340763136size_allocation_free=1194837720size_heap=4143972352size_heap_free_last_full_GC=0size_heap_free_last_GC=3342713528threads_in_ML=1threads_total=7threads_wait_condvar=4threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=123.364time_GC=5.684user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
ML_statistics = :now=1489705870.91tasks_ready=0tasks_pending=0tasks_running=1tasks_passive=0tasks_urgent=0workers_total=4workers_active=1workers_waiting=0full_GCs=0partial_GCs=30size_allocation=3262119936size_allocation_free=3262119936size_heap=4150263808size_heap_free_last_full_GC=0size_heap_free_last_GC=3264431160threads_in_ML=1threads_total=7threads_wait_condvar=4threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=123.984time_GC=5.744user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
ML_statistics = :now=1489705871.41tasks_ready=0tasks_pending=0tasks_running=1tasks_passive=0tasks_urgent=0workers_total=4workers_active=1workers_waiting=0full_GCs=0partial_GCs=30size_allocation=3262119936size_allocation_free=2449644464size_heap=4150263808size_heap_free_last_full_GC=0size_heap_free_last_GC=3264431160threads_in_ML=1threads_total=7threads_wait_condvar=4threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=124.384time_GC=5.744user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
ML_statistics = :now=1489705871.91tasks_ready=0tasks_pending=0tasks_running=1tasks_passive=0tasks_urgent=0workers_total=4workers_active=1workers_waiting=0full_GCs=0partial_GCs=30size_allocation=3262119936size_allocation_free=1620129792size_heap=4150263808size_heap_free_last_full_GC=0size_heap_free_last_GC=3264431160threads_in_ML=1threads_total=7threads_wait_condvar=4threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=124.788time_GC=5.744user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
ML_statistics = :now=1489705872.41tasks_ready=0tasks_pending=0tasks_running=1tasks_passive=0tasks_urgent=0workers_total=4workers_active=1workers_waiting=0full_GCs=0partial_GCs=30size_allocation=3262119936size_allocation_free=665874992size_heap=4150263808size_heap_free_last_full_GC=0size_heap_free_last_GC=3264431160threads_in_ML=1threads_total=7threads_wait_condvar=4threads_wait_IO=0threads_wait_mutex=0threads_wait_signal=1time_CPU=125.588time_GC=5.744user_counter0=0user_counter1=0user_counter2=0user_counter3=0user_counter4=0user_counter5=0user_counter6=0user_counter7=0
isabelle document -c -o 'pdf' -n 'document' -t '' /media/data/jenkins/workspace/afp-repo-afp/browser_info/AFP/Polynomial_Interpolation/document 2>&1
isabelle document -c -o 'pdf' -n 'outline' -t '/proof,/ML' /media/data/jenkins/workspace/afp-repo-afp/browser_info/AFP/Polynomial_Interpolation/outline 2>&1
This is pdfTeX, Version 3.14159265-2.6-1.40.16 (TeX Live 2015/Debian) (preloaded format=pdflatex)
 restricted \write18 enabled.
entering extended mode
LaTeX2e <2016/02/01>
Babel <3.9q> and hyphenation patterns for 81 language(s) loaded.

(./root.tex (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls
Document Class: article 2014/09/29 v1.4h Standard LaTeX document class
(/usr/share/texlive/texmf-dist/tex/latex/base/size11.clo)) (./isabelle.sty
(./comment.sty Excluding comment 'comment') Including comment 'isadelimtheory'
Including comment 'isatagtheory' Including comment 'isadelimproof'
Including comment 'isatagproof' Including comment 'isadelimML'
Including comment 'isatagML' Including comment 'isadelimvisible'
Including comment 'isatagvisible' Excluding comment 'isadeliminvisible'
Excluding comment 'isataginvisible') (./isabelletags.sty
Including comment 'isadelimproof' Excluding comment 'isatagproof'
Including comment 'isadelimML' Excluding comment 'isatagML') (./isabellesym.sty
) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty
(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty))
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty
For additional information on amsmath, use the `?' option.
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty))
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty)
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty))
(/usr/share/texlive/texmf-dist/tex/latex/tools/xspace.sty) (./pdfsetup.sty
(/usr/share/texlive/texmf-dist/tex/latex/graphics/color.sty
(/usr/share/texlive/texmf-dist/tex/latex/latexconfig/color.cfg)
(/usr/share/texlive/texmf-dist/tex/latex/pdftex-def/pdftex.def
(/usr/share/texlive/texmf-dist/tex/generic/oberdiek/infwarerr.sty)
(/usr/share/texlive/texmf-dist/tex/generic/oberdiek/ltxcmds.sty))))
(/usr/share/texlive/texmf-dist/tex/latex/hyperref/hyperref.sty
(/usr/share/texlive/texmf-dist/tex/generic/oberdiek/hobsub-hyperref.sty
(/usr/share/texlive/texmf-dist/tex/generic/oberdiek/hobsub-generic.sty))
(/usr/share/texlive/texmf-dist/tex/latex/graphics/keyval.sty)
(/usr/share/texlive/texmf-dist/tex/generic/ifxetex/ifxetex.sty)
(/usr/share/texlive/texmf-dist/tex/latex/oberdiek/auxhook.sty)
(/usr/share/texlive/texmf-dist/tex/latex/oberdiek/kvoptions.sty)
(/usr/share/texlive/texmf-dist/tex/latex/hyperref/pd1enc.def)
(/usr/share/texlive/texmf-dist/tex/latex/latexconfig/hyperref.cfg)
(/usr/share/texlive/texmf-dist/tex/latex/url/url.sty))

Package hyperref Message: Driver (autodetected): hpdftex.

(/usr/share/texlive/texmf-dist/tex/latex/hyperref/hpdftex.def
(/usr/share/texlive/texmf-dist/tex/latex/oberdiek/rerunfilecheck.sty))
No file root.aux.
(/usr/share/texlive/texmf-dist/tex/context/base/supp-pdf.mkii
[Loading MPS to PDF converter (version 2006.09.02).]
) (/usr/share/texlive/texmf-dist/tex/latex/hyperref/nameref.sty
(/usr/share/texlive/texmf-dist/tex/generic/oberdiek/gettitlestring.sty))
(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd)
(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd)
(/usr/share/texlive/texmf-dist/tex/latex/base/omscmr.fd)

LaTeX Warning: Citation `interpolation' on page 1 undefined on input line 55.

[1{/var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map}] (./session.tex
(./More_List.tex) (./Infinite_Set.tex) (./Polynomial.tex)
(./Adhoc_Overloading.tex) (./Monad_Syntax.tex) (./Sqrt_Babylonian_Auxiliary.tex
) (./Is_Rat_To_Rat.tex) (./Divmod_Int.tex
Overfull \hbox (12.96191pt too wide) in paragraph at lines 41--43
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 divmod[]int[]code$\OT1/cmr/m/n/10 [
$\OT1/cmr/m/it/10 code$\OT1/cmr/m/n/10 ]$$:$ \OT1/cmr/m/it/10 divmod[]int m n $
\OT1/cmr/m/n/10 =$ \OT1/cmr/m/it/10 map[]prod int[]of[]integer int[]of[]integer

) (./Improved_Code_Equations.tex [2]

Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 33.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 33.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 33.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 33.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 33.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 33.

[3]

Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 128.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 128.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 128.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 128.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 128.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 128.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `math shift' on input line 128.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\mathord' on input line 128.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `math shift' on input line 128.


Overfull \hbox (32.11456pt too wide) in paragraph at lines 141--143
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 divmod[]nat[]code$\OT1/cmr/m/n/10 :
$ \OT1/cmr/m/it/10 Di-vides$\OML/cmm/m/it/10 :$\OT1/cmr/m/it/10 divmod[]nat m n
 $\OT1/cmr/m/n/10 =$ \OT1/cmr/m/it/10 map[]prod nat[]of[]integer nat[]of[]integ
er

Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 221.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 221.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 221.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 221.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 221.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 221.

[4]) (./Cancellation.tex
Overfull \hbox (15.88786pt too wide) in paragraph at lines 30--31
[][]  \OT1/cmr/m/it/10 can-ce-la-tion sim-proc$\OML/cmm/m/it/10 :$ \OT1/cmr/m/i
t/10 Nor-mal-i-sa-tion of []iterate[]add[] back to the nor-male representation[
] 

Overfull \hbox (16.82057pt too wide) in paragraph at lines 81--83
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 iterate[]add[]distrib$\OT1/cmr/m/n/
10 [$\OT1/cmr/m/it/10 simp$\OT1/cmr/m/n/10 ]$$:$ []\OT1/cmr/m/it/10 iterate[]ad
d $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 m$\OT1/cmr/m/n/10 +$\OT1/cmr/m/it/10 n$\O
T1/cmr/m/n/10 )$ \OT1/cmr/m/it/10 a $\OT1/cmr/m/n/10 =$ \OT1/cmr/m/it/10 iterat
e[]add m a $\OT1/cmr/m/n/10 +$ \OT1/cmr/m/it/10 iterate[]add
[5]
Overfull \hbox (85.96669pt too wide) in paragraph at lines 175--176
[][]  \OT1/cmr/m/it/10 j $\OMS/cmsy/m/n/10 ^^T$ $\OT1/cmr/m/n/10 ($\OT1/cmr/m/i
t/10 i$\OT1/cmr/m/n/10 :$$:$\OT1/cmr/m/it/10 nat$\OT1/cmr/m/n/10 )$ $=[]\OMS/cm
sy/m/n/10 )$ $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 iterate[]add i $\OT1/cmr/m/n/1
0 ($\OT1/cmr/m/it/10 u$\OT1/cmr/m/n/10 :$$:$ $ [] $\OT1/cmr/m/it/10 a $\OT1/cmr
/m/n/10 :$$:$ $\OMS/cmsy/m/n/10 f$\OT1/cmr/m/it/10 cancel[]comm[]monoid[]add$\O
ML/cmm/m/it/10 ;$ \OT1/cmr/m/it/10 ordered[]ab[]semigroup[]add[]imp[]le$\OMS/cm
sy/m/n/10 g$$\OT1/cmr/m/n/10 )$

Overfull \hbox (85.96669pt too wide) in paragraph at lines 194--195
[][]  \OT1/cmr/m/it/10 i $\OMS/cmsy/m/n/10 ^^T$ $\OT1/cmr/m/n/10 ($\OT1/cmr/m/i
t/10 j$\OT1/cmr/m/n/10 :$$:$\OT1/cmr/m/it/10 nat$\OT1/cmr/m/n/10 )$ $=[]\OMS/cm
sy/m/n/10 )$ $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 iterate[]add i $\OT1/cmr/m/n/1
0 ($\OT1/cmr/m/it/10 u$\OT1/cmr/m/n/10 :$$:$ $ [] $\OT1/cmr/m/it/10 a $\OT1/cmr
/m/n/10 :$$:$ $\OMS/cmsy/m/n/10 f$\OT1/cmr/m/it/10 cancel[]comm[]monoid[]add$\O
ML/cmm/m/it/10 ;$ \OT1/cmr/m/it/10 ordered[]ab[]semigroup[]add[]imp[]le$\OMS/cm
sy/m/n/10 g$$\OT1/cmr/m/n/10 )$

Overfull \hbox (85.96669pt too wide) in paragraph at lines 213--214
[][]  \OT1/cmr/m/it/10 j $\OMS/cmsy/m/n/10 ^^T$ $\OT1/cmr/m/n/10 ($\OT1/cmr/m/i
t/10 i$\OT1/cmr/m/n/10 :$$:$\OT1/cmr/m/it/10 nat$\OT1/cmr/m/n/10 )$ $=[]\OMS/cm
sy/m/n/10 )$ $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 iterate[]add i $\OT1/cmr/m/n/1
0 ($\OT1/cmr/m/it/10 u$\OT1/cmr/m/n/10 :$$:$ $ [] $\OT1/cmr/m/it/10 a $\OT1/cmr
/m/n/10 :$$:$ $\OMS/cmsy/m/n/10 f$\OT1/cmr/m/it/10 cancel[]comm[]monoid[]add$\O
ML/cmm/m/it/10 ;$ \OT1/cmr/m/it/10 ordered[]ab[]semigroup[]add[]imp[]le$\OMS/cm
sy/m/n/10 g$$\OT1/cmr/m/n/10 )$

Overfull \hbox (85.96669pt too wide) in paragraph at lines 232--233
[][]  \OT1/cmr/m/it/10 i $\OMS/cmsy/m/n/10 ^^T$ $\OT1/cmr/m/n/10 ($\OT1/cmr/m/i
t/10 j$\OT1/cmr/m/n/10 :$$:$\OT1/cmr/m/it/10 nat$\OT1/cmr/m/n/10 )$ $=[]\OMS/cm
sy/m/n/10 )$ $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 iterate[]add i $\OT1/cmr/m/n/1
0 ($\OT1/cmr/m/it/10 u$\OT1/cmr/m/n/10 :$$:$ $ [] $\OT1/cmr/m/it/10 a $\OT1/cmr
/m/n/10 :$$:$ $\OMS/cmsy/m/n/10 f$\OT1/cmr/m/it/10 cancel[]comm[]monoid[]add$\O
ML/cmm/m/it/10 ;$ \OT1/cmr/m/it/10 ordered[]ab[]semigroup[]add[]imp[]le$\OMS/cm
sy/m/n/10 g$$\OT1/cmr/m/n/10 )$
) (./Multiset.tex [6] [7] [8] [9] [10]
Overfull \hbox (13.45732pt too wide) in paragraph at lines 783--785
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 set[]mset[]add[]mset[]insert $\OT1/
cmr/m/n/10 [$\OT1/cmr/m/it/10 simp$\OT1/cmr/m/n/10 ]$$:$ []\OT1/cmr/m/it/10 set
[]mset $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 add[]mset a A$\OT1/cmr/m/n/10 )$ $=$
 \OT1/cmr/m/it/10 in-sert a $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 set[]mset
[11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
Overfull \hbox (20.85715pt too wide) in paragraph at lines 3518--3520
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 replicate[]mset[]Suc $\OT1/cmr/m/n/
10 [$\OT1/cmr/m/it/10 simp$\OT1/cmr/m/n/10 ]$$:$ \OT1/cmr/m/it/10 replicate[]ms
et $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 Suc n$\OT1/cmr/m/n/10 )$ \OT1/cmr/m/it/1
0 x $\OT1/cmr/m/n/10 =$ \OT1/cmr/m/it/10 add[]mset x $\OT1/cmr/m/n/10 ($\OT1/cm
r/m/it/10 replicate[]mset
[22] [23] [24] [25]
Overfull \hbox (7.00906pt too wide) in paragraph at lines 4821--4823
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 filter[]diff[]mset $\OT1/cmr/m/n/10
 [$\OT1/cmr/m/it/10 simp$\OT1/cmr/m/n/10 ]$$:$ \OT1/cmr/m/it/10 filter[]mset P 
$\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 M $\OMS/cmsy/m/n/10 ^^@$ \OT1/cmr/m/it/10 N
$\OT1/cmr/m/n/10 )$ $=$ \OT1/cmr/m/it/10 filter[]mset P M $\OMS/cmsy/m/n/10 ^^@
$ \OT1/cmr/m/it/10 filter[]mset

Overfull \hbox (11.80153pt too wide) in paragraph at lines 4857--4859
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 filter[]sup[]mset$\OT1/cmr/m/n/10 [
$\OT1/cmr/m/it/10 simp$\OT1/cmr/m/n/10 ]$$:$ \OT1/cmr/m/it/10 filter[]mset P $\
OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 A $\OMS/cmsy/m/n/10 [$$\OT1/cmr/m/n/10 #$ \OT
1/cmr/m/it/10 B$\OT1/cmr/m/n/10 )$ $=$ \OT1/cmr/m/it/10 filter[]mset P A $\OMS/
cmsy/m/n/10 [$$\OT1/cmr/m/n/10 #$ \OT1/cmr/m/it/10 filter[]mset
[26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40]
[41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52]
Overfull \hbox (5.43213pt too wide) in paragraph at lines 12056--12057
[][]  \OT1/cmr/m/it/10 full[]exhaustive[]multiset f i $\OT1/cmr/m/n/10 =$ \OT1/
cmr/m/it/10 Quickcheck[]Exhaustive$\OML/cmm/m/it/10 :$\OT1/cmr/m/it/10 full[]ex
haustive $\OT1/cmr/m/n/10 ($$\OML/cmm/m/it/10 ^^U$\OT1/cmr/m/it/10 xs$\OML/cmm/
m/it/10 :$ \OT1/cmr/m/it/10 f $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 msetify
[53] [54]) (./Factorial_Ring.tex [55] [56] [57] [58] [59] [60] [61] [62]
Overfull \hbox (11.16814pt too wide) in paragraph at lines 2290--2290
[]\OT1/cmr/bx/n/12 Factorial semir-ings: al-ge-braic struc-tures with unique pr
ime
[63] [64] [65] [66] [67] [68] [69]
Overfull \hbox (6.21616pt too wide) in paragraph at lines 4656--4657
[][]  \OT1/cmr/bx/n/10 shows   \OT1/cmr/m/it/10 prime[]factorization $\OT1/cmr/
m/n/10 ($\OT1/cmr/m/it/10 a div b$\OT1/cmr/m/n/10 )$ $=$ \OT1/cmr/m/it/10 prime
[]factorization a $\OMS/cmsy/m/n/10 ^^@$ \OT1/cmr/m/it/10 prime[]factorization
[70] [71] [72] [73]
Overfull \hbox (14.46059pt too wide) in paragraph at lines 5797--5798
[][]  \OT1/cmr/bx/n/10 shows   \OT1/cmr/m/it/10 prime[]factorization $\OT1/cmr/
m/n/10 ($\OT1/cmr/m/it/10 gcd a b$\OT1/cmr/m/n/10 )$ $=$ \OT1/cmr/m/it/10 prime
[]factorization a $\OMS/cmsy/m/n/10 \$$\OT1/cmr/m/n/10 #$ \OT1/cmr/m/it/10 prim
e[]factorization

Overfull \hbox (15.99394pt too wide) in paragraph at lines 5817--5818
[][]  \OT1/cmr/bx/n/10 shows   \OT1/cmr/m/it/10 prime[]factorization $\OT1/cmr/
m/n/10 ($\OT1/cmr/m/it/10 lcm a b$\OT1/cmr/m/n/10 )$ $=$ \OT1/cmr/m/it/10 prime
[]factorization a $\OMS/cmsy/m/n/10 [$$\OT1/cmr/m/n/10 #$ \OT1/cmr/m/it/10 prim
e[]factorization
[74]) (./Euclidean_Algorithm.tex
Overfull \hbox (36.91122pt too wide) in paragraph at lines 6--6
[]\OT1/cmr/bx/n/14.4 Abstract eu-clidean al-go-rithm in eu-clidean (semi)rings 

[75] [76] [77]
Overfull \hbox (36.0064pt too wide) in paragraph at lines 1171--1172
[][]  \OT1/cmr/m/it/10 OF-CLASS$\OT1/cmr/m/n/10 ($$ [] $\OT1/cmr/m/it/10 a$\OT1
/cmr/m/n/10 :$$:$$\OMS/cmsy/m/n/10 f$\OT1/cmr/m/it/10 factorial[]semiring[]gcd$
\OML/cmm/m/it/10 ;$ \OT1/cmr/m/it/10 euclidean[]semiring$\OMS/cmsy/m/n/10 g$$\O
ML/cmm/m/it/10 ;$ \OT1/cmr/m/it/10 euclidean[]semiring[]gcd[]class$\OT1/cmr/m/n
/10 )$[] 
[78] [79]) (./Ring_Hom.tex [80] [81]) (./Fraction_Field.tex [82] [83] [84]
[85] [86] [87]) (./Normalized_Fraction.tex [88]
Overfull \hbox (25.4372pt too wide) in paragraph at lines 205--207
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 normalize[]quot[]in[]normalized[]fr
acts $\OT1/cmr/m/n/10 [$\OT1/cmr/m/it/10 simp$\OT1/cmr/m/n/10 ]$$:$ \OT1/cmr/m/
it/10 normalize[]quot x $\OMS/cmsy/m/n/10 2$ \OT1/cmr/m/it/10 normalized[]fract
s[] 

Overfull \hbox (26.27673pt too wide) in paragraph at lines 311--313
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 normalize[]quot[]idem $\OT1/cmr/m/n
/10 [$\OT1/cmr/m/it/10 simp$\OT1/cmr/m/n/10 ]$$:$ \OT1/cmr/m/it/10 normalize[]q
uot $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 normalize[]quot x$\OT1/cmr/m/n/10 )$ $=
$ \OT1/cmr/m/it/10 normalize[]quot
[89]
Overfull \hbox (3.45944pt too wide) in paragraph at lines 510--512
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 quot[]of[]fract[]in[]normalized[]fr
acts $\OT1/cmr/m/n/10 [$\OT1/cmr/m/it/10 simp$\OT1/cmr/m/n/10 ]$$:$ \OT1/cmr/m/
it/10 quot[]of[]fract x $\OMS/cmsy/m/n/10 2$ \OT1/cmr/m/it/10 normalized[]fract
s[] 
[90] [91]) (./Field_as_Ring.tex [92] [93] [94]) (./Polynomial_Factorial.tex
[95] [96]
Overfull \hbox (30.61302pt too wide) in paragraph at lines 511--513
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 normalize[]snd[]quot[]of[]fract$\OT
1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 nor-mal-ize $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/1
0 snd $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 quot[]of[]fract x$\OT1/cmr/m/n/10 )$$
)$ $=$ \OT1/cmr/m/it/10 snd $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 quot[]of[]fract


Overfull \hbox (3.69643pt too wide) in paragraph at lines 539--541
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 fract[]poly[]smult $\OT1/cmr/m/n/10
 [$\OT1/cmr/m/it/10 simp$\OT1/cmr/m/n/10 ]$$:$ \OT1/cmr/m/it/10 fract[]poly $\O
T1/cmr/m/n/10 ($\OT1/cmr/m/it/10 smult c p$\OT1/cmr/m/n/10 )$ $=$ \OT1/cmr/m/it
/10 smult $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 to[]fract c$\OT1/cmr/m/n/10 )$ $(
$\OT1/cmr/m/it/10 fract[]poly

Overfull \hbox (9.21042pt too wide) in paragraph at lines 725--726
[][]  \OT1/cmr/m/it/10 prod[]mset $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 image[]ms
et $\OT1/cmr/m/n/10 ($$\OML/cmm/m/it/10 ^^U$\OT1/cmr/m/it/10 x$\OML/cmm/m/it/10
 :$ \OT1/cmr/m/it/10 fract[]poly $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 f x$\OT1/c
mr/m/n/10 )$$)$ \OT1/cmr/m/it/10 A$\OT1/cmr/m/n/10 )$ $=$ \OT1/cmr/m/it/10 frac
t[]poly $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 prod[]mset $\OT1/cmr/m/n/10 ($\OT1/
cmr/m/it/10 image[]mset
[97] [98]
Overfull \hbox (9.28085pt too wide) in paragraph at lines 1049--1051
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 fract[]content[]fract[]poly $\OT1/c
mr/m/n/10 [$\OT1/cmr/m/it/10 simp$\OT1/cmr/m/n/10 ]$$:$ \OT1/cmr/m/it/10 fract[
]content $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 fract[]poly p$\OT1/cmr/m/n/10 )$ $
=$ \OT1/cmr/m/it/10 to[]fract $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 content
[99] [100] [101] [102] [103]
Overfull \hbox (31.4pt too wide) in paragraph at lines 3027--3029
[][]\OT1/cmr/bx/n/10 lemmas \OT1/cmr/m/it/10 Gcd[]poly[]set[]eq[]fold $\OT1/cmr
/m/n/10 [$\OT1/cmr/m/it/10 code$\OT1/cmr/m/n/10 ]$ $=$ \OT1/cmr/m/it/10 Gcd[]se
t[]eq[]fold $\OT1/cmr/m/n/10 [$\OT1/cmr/bx/n/10 where \OT1/cmr/m/it/10 ?$ [] $a
 $\OT1/cmr/m/n/10 =$ $ [] $\OT1/cmr/m/it/10 a $\OT1/cmr/m/n/10 :$$:$ \OT1/cmr/m
/it/10 factorial[]ring[]gcd

Overfull \hbox (34.6055pt too wide) in paragraph at lines 3029--3032
[][]\OT1/cmr/bx/n/10 lemmas \OT1/cmr/m/it/10 Lcm[]poly[]set[]eq[]fold $\OT1/cmr
/m/n/10 [$\OT1/cmr/m/it/10 code$\OT1/cmr/m/n/10 ]$ $=$ \OT1/cmr/m/it/10 Lcm[]se
t[]eq[]fold $\OT1/cmr/m/n/10 [$\OT1/cmr/bx/n/10 where \OT1/cmr/m/it/10 ?$ [] $a
 $\OT1/cmr/m/n/10 =$ $ [] $\OT1/cmr/m/it/10 a $\OT1/cmr/m/n/10 :$$:$ \OT1/cmr/m
/it/10 factorial[]ring[]gcd
) (./Missing_Unsorted.tex [104] [105] [106] [107]) (./Missing_Polynomial.tex
[108] [109] [110] [111]
Overfull \hbox (26.31035pt too wide) in paragraph at lines 1194--1196
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 degree[]prod[]sum[]lt[]n$\OT1/cmr/m
/n/10 :$ \OT1/cmr/bx/n/10 as-sumes $[]$ \OT1/cmr/m/it/10 i$\OML/cmm/m/it/10 :$ 
\OT1/cmr/m/it/10 i $\OML/cmm/m/it/10 <$ \OT1/cmr/m/it/10 n $\OT1/cmr/m/n/10 =[]
\OMS/cmsy/m/n/10 )$ \OT1/cmr/m/it/10 de-gree $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/1
0 f i $\OT1/cmr/m/n/10 :$$:$ $ [] $\OT1/cmr/m/it/10 a $\OT1/cmr/m/n/10 :$$:$ \O
T1/cmr/m/it/10 comm[]semiring[]1
[112] [113] [114] [115]

Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `math shift' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `math shift' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `math shift' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\mathord' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `math shift' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `math shift' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\mathord' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `math shift' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `math shift' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\mskip' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\mskip' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `math shift' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `math shift' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `math shift' on input line 2890.

[116] [117]
Overfull \hbox (17.23799pt too wide) in paragraph at lines 3444--3445
[][]  \OT1/cmr/m/it/10 nor-mal-ize $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 f $\OT1/
cmr/m/n/10 :$$:$ $ [] $\OT1/cmr/m/it/10 a $\OT1/cmr/m/n/10 :$$:$ $\OMS/cmsy/m/n
/10 f$\OT1/cmr/m/it/10 normalization[]semidom$\OML/cmm/m/it/10 ;$\OT1/cmr/m/it/
10 field$\OMS/cmsy/m/n/10 g$ \OT1/cmr/m/it/10 poly$\OT1/cmr/m/n/10 )$ $=$ \OT1/
cmr/m/it/10 smult $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 inverse $\OT1/cmr/m/n/10 
($\OT1/cmr/m/it/10 unit[]factor
[118]) (./Ring_Hom_Poly.tex [119] [120] [121] [122] [123] [124])
(./Newton_Interpolation.tex
Overfull \hbox (17.22385pt too wide) in paragraph at lines 50--51
[][]  \OT1/cmr/m/it/10 horner[]composition $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 
map hom cs$\OT1/cmr/m/n/10 )$ $($\OT1/cmr/m/it/10 map hom xs$\OT1/cmr/m/n/10 )$
 $=$ \OT1/cmr/m/it/10 map[]poly hom $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 horner[
]composition
[125] [126] [127] [128]
Overfull \hbox (4.60117pt too wide) in paragraph at lines 1779--1780
[][]  $\OT1/cmr/m/n/10 =[]\OMS/cmsy/m/n/10 )$ \OT1/cmr/m/it/10 newton[]coeffici
ents[]main $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 map rat[]of[]int xs$\OT1/cmr/m/n
/10 )$ $($\OT1/cmr/m/it/10 map rat[]of[]int gs$\OT1/cmr/m/n/10 )$ $($\OT1/cmr/m
/it/10 map rat[]of[]int
[129]) (./Lagrange_Interpolation.tex [130]) (./Neville_Aitken_Interpolation.tex
[131]
Overfull \hbox (0.26402pt too wide) in paragraph at lines 318--320
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 length[]neville[]aitken[]merge$\OT1
/cmr/m/n/10 [$\OT1/cmr/m/it/10 termination[]simp$\OT1/cmr/m/n/10 ]$$:$ \OT1/cmr
/m/it/10 length $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 neville[]aitken[]merge
[132]) (./Polynomial_Interpolation.tex [133] [134]
Overfull \hbox (26.8093pt too wide) in paragraph at lines 468--469
[][]  \OT1/cmr/m/it/10 interpolation[]poly[]int[]Some$\OT1/cmr/m/n/10 [$\OT1/cm
r/bx/n/10 where \OT1/cmr/m/it/10 alg $\OT1/cmr/m/n/10 =$ \OT1/cmr/m/it/10 New-t
on$\OML/cmm/m/it/10 ;$ \OT1/cmr/m/it/10 un-folded interpolation[]poly[]int$\OML
/cmm/m/it/10 :$\OT1/cmr/m/it/10 simps$\OT1/cmr/m/n/10 ]$[] 

Overfull \hbox (26.06485pt too wide) in paragraph at lines 472--474
[][]  \OT1/cmr/m/it/10 interpolation[]poly[]int[]None$\OT1/cmr/m/n/10 [$\OT1/cm
r/bx/n/10 where \OT1/cmr/m/it/10 alg $\OT1/cmr/m/n/10 =$ \OT1/cmr/m/it/10 New-t
on$\OML/cmm/m/it/10 ;$ \OT1/cmr/m/it/10 un-folded interpolation[]poly[]int$\OML
/cmm/m/it/10 :$\OT1/cmr/m/it/10 simps$\OT1/cmr/m/n/10 ]$ 
[135]))
No file root.bbl.
[136] (./root.aux)

Package rerunfilecheck Warning: File `root.out' has changed.
(rerunfilecheck)                Rerun to get outlines right
(rerunfilecheck)                or use package `bookmark'.


LaTeX Warning: There were undefined references.

 )
(see the transcript file for additional information)</usr/share/texlive/texmf-d
ist/fonts/type1/public/amsfonts/cm/cmbsy10.pfb></usr/share/texlive/texmf-dist/f
onts/type1/public/amsfonts/cm/cmbx10.pfb></usr/share/texlive/texmf-dist/fonts/t
ype1/public/amsfonts/cm/cmbx12.pfb></usr/share/texlive/texmf-dist/fonts/type1/p
ublic/amsfonts/cm/cmex10.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/
amsfonts/cm/cmmi10.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfon
ts/cm/cmmi12.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/
cmmi7.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi8.p
fb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmr10.pfb></us
r/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmr12.pfb></usr/share
/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmr17.pfb></usr/share/texliv
e/texmf-dist/fonts/type1/public/amsfonts/cm/cmr7.pfb></usr/share/texlive/texmf-
dist/fonts/type1/public/amsfonts/cm/cmr8.pfb></usr/share/texlive/texmf-dist/fon
ts/type1/public/amsfonts/cm/cmr9.pfb></usr/share/texlive/texmf-dist/fonts/type1
/public/amsfonts/cm/cmss10.pfb></usr/share/texlive/texmf-dist/fonts/type1/publi
c/amsfonts/cm/cmsy10.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsf
onts/cm/cmsy5.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm
/cmsy6.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy7.
pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy8.pfb></u
sr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmti10.pfb></usr/sha
re/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmti12.pfb></usr/share/tex
live/texmf-dist/fonts/type1/public/amsfonts/symbols/msam10.pfb>
Output written on root.pdf (136 pages, 489890 bytes).
Transcript written on root.log.
This is BibTeX, Version 0.99d (TeX Live 2015/Debian)
The top-level auxiliary file: root.aux
The style file: abbrv.bst
Database file #1: root.bib
This is pdfTeX, Version 3.14159265-2.6-1.40.16 (TeX Live 2015/Debian) (preloaded format=pdflatex)
 restricted \write18 enabled.
entering extended mode
LaTeX2e <2016/02/01>
Babel <3.9q> and hyphenation patterns for 81 language(s) loaded.

(./root.tex (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls
Document Class: article 2014/09/29 v1.4h Standard LaTeX document class
(/usr/share/texlive/texmf-dist/tex/latex/base/size11.clo)) (./isabelle.sty
(./comment.sty Excluding comment 'comment') Including comment 'isadelimtheory'
Including comment 'isatagtheory' Including comment 'isadelimproof'
Including comment 'isatagproof' Including comment 'isadelimML'
Including comment 'isatagML' Including comment 'isadelimvisible'
Including comment 'isatagvisible' Excluding comment 'isadeliminvisible'
Excluding comment 'isataginvisible') (./isabelletags.sty
Including comment 'isadelimproof' Excluding comment 'isatagproof'
Including comment 'isadelimML' Excluding comment 'isatagML') (./isabellesym.sty
) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty
(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty))
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty
For additional information on amsmath, use the `?' option.
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty))
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty)
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty))
(/usr/share/texlive/texmf-dist/tex/latex/tools/xspace.sty) (./pdfsetup.sty
(/usr/share/texlive/texmf-dist/tex/latex/graphics/color.sty
(/usr/share/texlive/texmf-dist/tex/latex/latexconfig/color.cfg)
(/usr/share/texlive/texmf-dist/tex/latex/pdftex-def/pdftex.def
(/usr/share/texlive/texmf-dist/tex/generic/oberdiek/infwarerr.sty)
(/usr/share/texlive/texmf-dist/tex/generic/oberdiek/ltxcmds.sty))))
(/usr/share/texlive/texmf-dist/tex/latex/hyperref/hyperref.sty
(/usr/share/texlive/texmf-dist/tex/generic/oberdiek/hobsub-hyperref.sty
(/usr/share/texlive/texmf-dist/tex/generic/oberdiek/hobsub-generic.sty))
(/usr/share/texlive/texmf-dist/tex/latex/graphics/keyval.sty)
(/usr/share/texlive/texmf-dist/tex/generic/ifxetex/ifxetex.sty)
(/usr/share/texlive/texmf-dist/tex/latex/oberdiek/auxhook.sty)
(/usr/share/texlive/texmf-dist/tex/latex/oberdiek/kvoptions.sty)
(/usr/share/texlive/texmf-dist/tex/latex/hyperref/pd1enc.def)
(/usr/share/texlive/texmf-dist/tex/latex/latexconfig/hyperref.cfg)
(/usr/share/texlive/texmf-dist/tex/latex/url/url.sty))

Package hyperref Message: Driver (autodetected): hpdftex.

(/usr/share/texlive/texmf-dist/tex/latex/hyperref/hpdftex.def
(/usr/share/texlive/texmf-dist/tex/latex/oberdiek/rerunfilecheck.sty))
(./root.aux) (/usr/share/texlive/texmf-dist/tex/context/base/supp-pdf.mkii
[Loading MPS to PDF converter (version 2006.09.02).]
) (/usr/share/texlive/texmf-dist/tex/latex/hyperref/nameref.sty
(/usr/share/texlive/texmf-dist/tex/generic/oberdiek/gettitlestring.sty))
(./root.out) (./root.out)
(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd)
(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd)
(/usr/share/texlive/texmf-dist/tex/latex/base/omscmr.fd) (./root.toc [1{/var/li
b/texmf/fonts/map/pdftex/updmap/pdftex.map}] [2]) [3]

LaTeX Warning: Citation `interpolation' on page 4 undefined on input line 55.

(./session.tex (./More_List.tex) (./Infinite_Set.tex) (./Polynomial.tex)
(./Adhoc_Overloading.tex) (./Monad_Syntax.tex) (./Sqrt_Babylonian_Auxiliary.tex
) (./Is_Rat_To_Rat.tex) (./Divmod_Int.tex [4]
Overfull \hbox (12.96191pt too wide) in paragraph at lines 41--43
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 divmod[]int[]code$\OT1/cmr/m/n/10 [
$\OT1/cmr/m/it/10 code$\OT1/cmr/m/n/10 ]$$:$ \OT1/cmr/m/it/10 divmod[]int m n $
\OT1/cmr/m/n/10 =$ \OT1/cmr/m/it/10 map[]prod int[]of[]integer int[]of[]integer

) (./Improved_Code_Equations.tex

Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 33.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 33.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 33.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 33.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 33.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 33.

[5]

Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 128.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 128.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 128.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 128.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 128.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 128.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `math shift' on input line 128.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\mathord' on input line 128.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `math shift' on input line 128.


Overfull \hbox (32.11456pt too wide) in paragraph at lines 141--143
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 divmod[]nat[]code$\OT1/cmr/m/n/10 :
$ \OT1/cmr/m/it/10 Di-vides$\OML/cmm/m/it/10 :$\OT1/cmr/m/it/10 divmod[]nat m n
 $\OT1/cmr/m/n/10 =$ \OT1/cmr/m/it/10 map[]prod nat[]of[]integer nat[]of[]integ
er
[6]

Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 221.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 221.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 221.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 221.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 221.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 221.

) (./Cancellation.tex
Overfull \hbox (15.88786pt too wide) in paragraph at lines 30--31
[][]  \OT1/cmr/m/it/10 can-ce-la-tion sim-proc$\OML/cmm/m/it/10 :$ \OT1/cmr/m/i
t/10 Nor-mal-i-sa-tion of []iterate[]add[] back to the nor-male representation[
] 
[7]
Overfull \hbox (16.82057pt too wide) in paragraph at lines 81--83
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 iterate[]add[]distrib$\OT1/cmr/m/n/
10 [$\OT1/cmr/m/it/10 simp$\OT1/cmr/m/n/10 ]$$:$ []\OT1/cmr/m/it/10 iterate[]ad
d $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 m$\OT1/cmr/m/n/10 +$\OT1/cmr/m/it/10 n$\O
T1/cmr/m/n/10 )$ \OT1/cmr/m/it/10 a $\OT1/cmr/m/n/10 =$ \OT1/cmr/m/it/10 iterat
e[]add m a $\OT1/cmr/m/n/10 +$ \OT1/cmr/m/it/10 iterate[]add

Overfull \hbox (85.96669pt too wide) in paragraph at lines 175--176
[][]  \OT1/cmr/m/it/10 j $\OMS/cmsy/m/n/10 ^^T$ $\OT1/cmr/m/n/10 ($\OT1/cmr/m/i
t/10 i$\OT1/cmr/m/n/10 :$$:$\OT1/cmr/m/it/10 nat$\OT1/cmr/m/n/10 )$ $=[]\OMS/cm
sy/m/n/10 )$ $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 iterate[]add i $\OT1/cmr/m/n/1
0 ($\OT1/cmr/m/it/10 u$\OT1/cmr/m/n/10 :$$:$ $ [] $\OT1/cmr/m/it/10 a $\OT1/cmr
/m/n/10 :$$:$ $\OMS/cmsy/m/n/10 f$\OT1/cmr/m/it/10 cancel[]comm[]monoid[]add$\O
ML/cmm/m/it/10 ;$ \OT1/cmr/m/it/10 ordered[]ab[]semigroup[]add[]imp[]le$\OMS/cm
sy/m/n/10 g$$\OT1/cmr/m/n/10 )$

Overfull \hbox (85.96669pt too wide) in paragraph at lines 194--195
[][]  \OT1/cmr/m/it/10 i $\OMS/cmsy/m/n/10 ^^T$ $\OT1/cmr/m/n/10 ($\OT1/cmr/m/i
t/10 j$\OT1/cmr/m/n/10 :$$:$\OT1/cmr/m/it/10 nat$\OT1/cmr/m/n/10 )$ $=[]\OMS/cm
sy/m/n/10 )$ $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 iterate[]add i $\OT1/cmr/m/n/1
0 ($\OT1/cmr/m/it/10 u$\OT1/cmr/m/n/10 :$$:$ $ [] $\OT1/cmr/m/it/10 a $\OT1/cmr
/m/n/10 :$$:$ $\OMS/cmsy/m/n/10 f$\OT1/cmr/m/it/10 cancel[]comm[]monoid[]add$\O
ML/cmm/m/it/10 ;$ \OT1/cmr/m/it/10 ordered[]ab[]semigroup[]add[]imp[]le$\OMS/cm
sy/m/n/10 g$$\OT1/cmr/m/n/10 )$

Overfull \hbox (85.96669pt too wide) in paragraph at lines 213--214
[][]  \OT1/cmr/m/it/10 j $\OMS/cmsy/m/n/10 ^^T$ $\OT1/cmr/m/n/10 ($\OT1/cmr/m/i
t/10 i$\OT1/cmr/m/n/10 :$$:$\OT1/cmr/m/it/10 nat$\OT1/cmr/m/n/10 )$ $=[]\OMS/cm
sy/m/n/10 )$ $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 iterate[]add i $\OT1/cmr/m/n/1
0 ($\OT1/cmr/m/it/10 u$\OT1/cmr/m/n/10 :$$:$ $ [] $\OT1/cmr/m/it/10 a $\OT1/cmr
/m/n/10 :$$:$ $\OMS/cmsy/m/n/10 f$\OT1/cmr/m/it/10 cancel[]comm[]monoid[]add$\O
ML/cmm/m/it/10 ;$ \OT1/cmr/m/it/10 ordered[]ab[]semigroup[]add[]imp[]le$\OMS/cm
sy/m/n/10 g$$\OT1/cmr/m/n/10 )$

Overfull \hbox (85.96669pt too wide) in paragraph at lines 232--233
[][]  \OT1/cmr/m/it/10 i $\OMS/cmsy/m/n/10 ^^T$ $\OT1/cmr/m/n/10 ($\OT1/cmr/m/i
t/10 j$\OT1/cmr/m/n/10 :$$:$\OT1/cmr/m/it/10 nat$\OT1/cmr/m/n/10 )$ $=[]\OMS/cm
sy/m/n/10 )$ $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 iterate[]add i $\OT1/cmr/m/n/1
0 ($\OT1/cmr/m/it/10 u$\OT1/cmr/m/n/10 :$$:$ $ [] $\OT1/cmr/m/it/10 a $\OT1/cmr
/m/n/10 :$$:$ $\OMS/cmsy/m/n/10 f$\OT1/cmr/m/it/10 cancel[]comm[]monoid[]add$\O
ML/cmm/m/it/10 ;$ \OT1/cmr/m/it/10 ordered[]ab[]semigroup[]add[]imp[]le$\OMS/cm
sy/m/n/10 g$$\OT1/cmr/m/n/10 )$
[8]) (./Multiset.tex [9] [10] [11] [12]
Overfull \hbox (13.45732pt too wide) in paragraph at lines 783--785
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 set[]mset[]add[]mset[]insert $\OT1/
cmr/m/n/10 [$\OT1/cmr/m/it/10 simp$\OT1/cmr/m/n/10 ]$$:$ []\OT1/cmr/m/it/10 set
[]mset $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 add[]mset a A$\OT1/cmr/m/n/10 )$ $=$
 \OT1/cmr/m/it/10 in-sert a $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 set[]mset
[13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]
Overfull \hbox (20.85715pt too wide) in paragraph at lines 3518--3520
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 replicate[]mset[]Suc $\OT1/cmr/m/n/
10 [$\OT1/cmr/m/it/10 simp$\OT1/cmr/m/n/10 ]$$:$ \OT1/cmr/m/it/10 replicate[]ms
et $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 Suc n$\OT1/cmr/m/n/10 )$ \OT1/cmr/m/it/1
0 x $\OT1/cmr/m/n/10 =$ \OT1/cmr/m/it/10 add[]mset x $\OT1/cmr/m/n/10 ($\OT1/cm
r/m/it/10 replicate[]mset
[24] [25] [26] [27]
Overfull \hbox (7.00906pt too wide) in paragraph at lines 4821--4823
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 filter[]diff[]mset $\OT1/cmr/m/n/10
 [$\OT1/cmr/m/it/10 simp$\OT1/cmr/m/n/10 ]$$:$ \OT1/cmr/m/it/10 filter[]mset P 
$\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 M $\OMS/cmsy/m/n/10 ^^@$ \OT1/cmr/m/it/10 N
$\OT1/cmr/m/n/10 )$ $=$ \OT1/cmr/m/it/10 filter[]mset P M $\OMS/cmsy/m/n/10 ^^@
$ \OT1/cmr/m/it/10 filter[]mset
[28]
Overfull \hbox (11.80153pt too wide) in paragraph at lines 4857--4859
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 filter[]sup[]mset$\OT1/cmr/m/n/10 [
$\OT1/cmr/m/it/10 simp$\OT1/cmr/m/n/10 ]$$:$ \OT1/cmr/m/it/10 filter[]mset P $\
OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 A $\OMS/cmsy/m/n/10 [$$\OT1/cmr/m/n/10 #$ \OT
1/cmr/m/it/10 B$\OT1/cmr/m/n/10 )$ $=$ \OT1/cmr/m/it/10 filter[]mset P A $\OMS/
cmsy/m/n/10 [$$\OT1/cmr/m/n/10 #$ \OT1/cmr/m/it/10 filter[]mset
[29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43]
[44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54]
Overfull \hbox (5.43213pt too wide) in paragraph at lines 12056--12057
[][]  \OT1/cmr/m/it/10 full[]exhaustive[]multiset f i $\OT1/cmr/m/n/10 =$ \OT1/
cmr/m/it/10 Quickcheck[]Exhaustive$\OML/cmm/m/it/10 :$\OT1/cmr/m/it/10 full[]ex
haustive $\OT1/cmr/m/n/10 ($$\OML/cmm/m/it/10 ^^U$\OT1/cmr/m/it/10 xs$\OML/cmm/
m/it/10 :$ \OT1/cmr/m/it/10 f $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 msetify
[55] [56] [57]) (./Factorial_Ring.tex [58] [59] [60] [61] [62] [63] [64]
[65]
Overfull \hbox (11.16814pt too wide) in paragraph at lines 2290--2290
[]\OT1/cmr/bx/n/12 Factorial semir-ings: al-ge-braic struc-tures with unique pr
ime
[66] [67] [68] [69] [70] [71] [72]
Overfull \hbox (6.21616pt too wide) in paragraph at lines 4656--4657
[][]  \OT1/cmr/bx/n/10 shows   \OT1/cmr/m/it/10 prime[]factorization $\OT1/cmr/
m/n/10 ($\OT1/cmr/m/it/10 a div b$\OT1/cmr/m/n/10 )$ $=$ \OT1/cmr/m/it/10 prime
[]factorization a $\OMS/cmsy/m/n/10 ^^@$ \OT1/cmr/m/it/10 prime[]factorization
[73] [74] [75]
Overfull \hbox (14.46059pt too wide) in paragraph at lines 5797--5798
[][]  \OT1/cmr/bx/n/10 shows   \OT1/cmr/m/it/10 prime[]factorization $\OT1/cmr/
m/n/10 ($\OT1/cmr/m/it/10 gcd a b$\OT1/cmr/m/n/10 )$ $=$ \OT1/cmr/m/it/10 prime
[]factorization a $\OMS/cmsy/m/n/10 \$$\OT1/cmr/m/n/10 #$ \OT1/cmr/m/it/10 prim
e[]factorization

Overfull \hbox (15.99394pt too wide) in paragraph at lines 5817--5818
[][]  \OT1/cmr/bx/n/10 shows   \OT1/cmr/m/it/10 prime[]factorization $\OT1/cmr/
m/n/10 ($\OT1/cmr/m/it/10 lcm a b$\OT1/cmr/m/n/10 )$ $=$ \OT1/cmr/m/it/10 prime
[]factorization a $\OMS/cmsy/m/n/10 [$$\OT1/cmr/m/n/10 #$ \OT1/cmr/m/it/10 prim
e[]factorization
[76] [77]) (./Euclidean_Algorithm.tex
Overfull \hbox (36.91122pt too wide) in paragraph at lines 6--6
[]\OT1/cmr/bx/n/14.4 Abstract eu-clidean al-go-rithm in eu-clidean (semi)rings 

[78] [79] [80]
Overfull \hbox (36.0064pt too wide) in paragraph at lines 1171--1172
[][]  \OT1/cmr/m/it/10 OF-CLASS$\OT1/cmr/m/n/10 ($$ [] $\OT1/cmr/m/it/10 a$\OT1
/cmr/m/n/10 :$$:$$\OMS/cmsy/m/n/10 f$\OT1/cmr/m/it/10 factorial[]semiring[]gcd$
\OML/cmm/m/it/10 ;$ \OT1/cmr/m/it/10 euclidean[]semiring$\OMS/cmsy/m/n/10 g$$\O
ML/cmm/m/it/10 ;$ \OT1/cmr/m/it/10 euclidean[]semiring[]gcd[]class$\OT1/cmr/m/n
/10 )$[] 
[81]) (./Ring_Hom.tex [82] [83] [84]) (./Fraction_Field.tex [85] [86] [87]
[88] [89]) (./Normalized_Fraction.tex [90]
Overfull \hbox (25.4372pt too wide) in paragraph at lines 205--207
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 normalize[]quot[]in[]normalized[]fr
acts $\OT1/cmr/m/n/10 [$\OT1/cmr/m/it/10 simp$\OT1/cmr/m/n/10 ]$$:$ \OT1/cmr/m/
it/10 normalize[]quot x $\OMS/cmsy/m/n/10 2$ \OT1/cmr/m/it/10 normalized[]fract
s[] 
[91]
Overfull \hbox (26.27673pt too wide) in paragraph at lines 311--313
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 normalize[]quot[]idem $\OT1/cmr/m/n
/10 [$\OT1/cmr/m/it/10 simp$\OT1/cmr/m/n/10 ]$$:$ \OT1/cmr/m/it/10 normalize[]q
uot $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 normalize[]quot x$\OT1/cmr/m/n/10 )$ $=
$ \OT1/cmr/m/it/10 normalize[]quot

Overfull \hbox (3.45944pt too wide) in paragraph at lines 510--512
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 quot[]of[]fract[]in[]normalized[]fr
acts $\OT1/cmr/m/n/10 [$\OT1/cmr/m/it/10 simp$\OT1/cmr/m/n/10 ]$$:$ \OT1/cmr/m/
it/10 quot[]of[]fract x $\OMS/cmsy/m/n/10 2$ \OT1/cmr/m/it/10 normalized[]fract
s[] 
[92] [93] [94]) (./Field_as_Ring.tex [95] [96]) (./Polynomial_Factorial.tex
[97] [98]
Overfull \hbox (30.61302pt too wide) in paragraph at lines 511--513
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 normalize[]snd[]quot[]of[]fract$\OT
1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 nor-mal-ize $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/1
0 snd $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 quot[]of[]fract x$\OT1/cmr/m/n/10 )$$
)$ $=$ \OT1/cmr/m/it/10 snd $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 quot[]of[]fract


Overfull \hbox (3.69643pt too wide) in paragraph at lines 539--541
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 fract[]poly[]smult $\OT1/cmr/m/n/10
 [$\OT1/cmr/m/it/10 simp$\OT1/cmr/m/n/10 ]$$:$ \OT1/cmr/m/it/10 fract[]poly $\O
T1/cmr/m/n/10 ($\OT1/cmr/m/it/10 smult c p$\OT1/cmr/m/n/10 )$ $=$ \OT1/cmr/m/it
/10 smult $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 to[]fract c$\OT1/cmr/m/n/10 )$ $(
$\OT1/cmr/m/it/10 fract[]poly
[99]
Overfull \hbox (9.21042pt too wide) in paragraph at lines 725--726
[][]  \OT1/cmr/m/it/10 prod[]mset $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 image[]ms
et $\OT1/cmr/m/n/10 ($$\OML/cmm/m/it/10 ^^U$\OT1/cmr/m/it/10 x$\OML/cmm/m/it/10
 :$ \OT1/cmr/m/it/10 fract[]poly $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 f x$\OT1/c
mr/m/n/10 )$$)$ \OT1/cmr/m/it/10 A$\OT1/cmr/m/n/10 )$ $=$ \OT1/cmr/m/it/10 frac
t[]poly $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 prod[]mset $\OT1/cmr/m/n/10 ($\OT1/
cmr/m/it/10 image[]mset
[100]
Overfull \hbox (9.28085pt too wide) in paragraph at lines 1049--1051
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 fract[]content[]fract[]poly $\OT1/c
mr/m/n/10 [$\OT1/cmr/m/it/10 simp$\OT1/cmr/m/n/10 ]$$:$ \OT1/cmr/m/it/10 fract[
]content $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 fract[]poly p$\OT1/cmr/m/n/10 )$ $
=$ \OT1/cmr/m/it/10 to[]fract $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 content
[101] [102] [103] [104] [105] [106]
Overfull \hbox (31.4pt too wide) in paragraph at lines 3027--3029
[][]\OT1/cmr/bx/n/10 lemmas \OT1/cmr/m/it/10 Gcd[]poly[]set[]eq[]fold $\OT1/cmr
/m/n/10 [$\OT1/cmr/m/it/10 code$\OT1/cmr/m/n/10 ]$ $=$ \OT1/cmr/m/it/10 Gcd[]se
t[]eq[]fold $\OT1/cmr/m/n/10 [$\OT1/cmr/bx/n/10 where \OT1/cmr/m/it/10 ?$ [] $a
 $\OT1/cmr/m/n/10 =$ $ [] $\OT1/cmr/m/it/10 a $\OT1/cmr/m/n/10 :$$:$ \OT1/cmr/m
/it/10 factorial[]ring[]gcd

Overfull \hbox (34.6055pt too wide) in paragraph at lines 3029--3032
[][]\OT1/cmr/bx/n/10 lemmas \OT1/cmr/m/it/10 Lcm[]poly[]set[]eq[]fold $\OT1/cmr
/m/n/10 [$\OT1/cmr/m/it/10 code$\OT1/cmr/m/n/10 ]$ $=$ \OT1/cmr/m/it/10 Lcm[]se
t[]eq[]fold $\OT1/cmr/m/n/10 [$\OT1/cmr/bx/n/10 where \OT1/cmr/m/it/10 ?$ [] $a
 $\OT1/cmr/m/n/10 =$ $ [] $\OT1/cmr/m/it/10 a $\OT1/cmr/m/n/10 :$$:$ \OT1/cmr/m
/it/10 factorial[]ring[]gcd
) (./Missing_Unsorted.tex [107] [108] [109] [110]) (./Missing_Polynomial.tex
[111] [112] [113]
Overfull \hbox (26.31035pt too wide) in paragraph at lines 1194--1196
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 degree[]prod[]sum[]lt[]n$\OT1/cmr/m
/n/10 :$ \OT1/cmr/bx/n/10 as-sumes $[]$ \OT1/cmr/m/it/10 i$\OML/cmm/m/it/10 :$ 
\OT1/cmr/m/it/10 i $\OML/cmm/m/it/10 <$ \OT1/cmr/m/it/10 n $\OT1/cmr/m/n/10 =[]
\OMS/cmsy/m/n/10 )$ \OT1/cmr/m/it/10 de-gree $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/1
0 f i $\OT1/cmr/m/n/10 :$$:$ $ [] $\OT1/cmr/m/it/10 a $\OT1/cmr/m/n/10 :$$:$ \O
T1/cmr/m/it/10 comm[]semiring[]1
[114] [115] [116] [117]

Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `math shift' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `math shift' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `math shift' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\mathord' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `math shift' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `math shift' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\mathord' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `math shift' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `math shift' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\mskip' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\mskip' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `math shift' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `math shift' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `math shift' on input line 2890.

[118] [119]
Overfull \hbox (17.23799pt too wide) in paragraph at lines 3444--3445
[][]  \OT1/cmr/m/it/10 nor-mal-ize $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 f $\OT1/
cmr/m/n/10 :$$:$ $ [] $\OT1/cmr/m/it/10 a $\OT1/cmr/m/n/10 :$$:$ $\OMS/cmsy/m/n
/10 f$\OT1/cmr/m/it/10 normalization[]semidom$\OML/cmm/m/it/10 ;$\OT1/cmr/m/it/
10 field$\OMS/cmsy/m/n/10 g$ \OT1/cmr/m/it/10 poly$\OT1/cmr/m/n/10 )$ $=$ \OT1/
cmr/m/it/10 smult $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 inverse $\OT1/cmr/m/n/10 
($\OT1/cmr/m/it/10 unit[]factor
[120] [121]) (./Ring_Hom_Poly.tex [122] [123] [124] [125] [126])
(./Newton_Interpolation.tex [127]
Overfull \hbox (17.22385pt too wide) in paragraph at lines 50--51
[][]  \OT1/cmr/m/it/10 horner[]composition $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 
map hom cs$\OT1/cmr/m/n/10 )$ $($\OT1/cmr/m/it/10 map hom xs$\OT1/cmr/m/n/10 )$
 $=$ \OT1/cmr/m/it/10 map[]poly hom $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 horner[
]composition
[128] [129] [130]
Overfull \hbox (4.60117pt too wide) in paragraph at lines 1779--1780
[][]  $\OT1/cmr/m/n/10 =[]\OMS/cmsy/m/n/10 )$ \OT1/cmr/m/it/10 newton[]coeffici
ents[]main $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 map rat[]of[]int xs$\OT1/cmr/m/n
/10 )$ $($\OT1/cmr/m/it/10 map rat[]of[]int gs$\OT1/cmr/m/n/10 )$ $($\OT1/cmr/m
/it/10 map rat[]of[]int
[131] [132]) (./Lagrange_Interpolation.tex [133])
(./Neville_Aitken_Interpolation.tex [134]
Overfull \hbox (0.26402pt too wide) in paragraph at lines 318--320
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 length[]neville[]aitken[]merge$\OT1
/cmr/m/n/10 [$\OT1/cmr/m/it/10 termination[]simp$\OT1/cmr/m/n/10 ]$$:$ \OT1/cmr
/m/it/10 length $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 neville[]aitken[]merge
[135]) (./Polynomial_Interpolation.tex [136] [137]
Overfull \hbox (26.8093pt too wide) in paragraph at lines 468--469
[][]  \OT1/cmr/m/it/10 interpolation[]poly[]int[]Some$\OT1/cmr/m/n/10 [$\OT1/cm
r/bx/n/10 where \OT1/cmr/m/it/10 alg $\OT1/cmr/m/n/10 =$ \OT1/cmr/m/it/10 New-t
on$\OML/cmm/m/it/10 ;$ \OT1/cmr/m/it/10 un-folded interpolation[]poly[]int$\OML
/cmm/m/it/10 :$\OT1/cmr/m/it/10 simps$\OT1/cmr/m/n/10 ]$[] 

Overfull \hbox (26.06485pt too wide) in paragraph at lines 472--474
[][]  \OT1/cmr/m/it/10 interpolation[]poly[]int[]None$\OT1/cmr/m/n/10 [$\OT1/cm
r/bx/n/10 where \OT1/cmr/m/it/10 alg $\OT1/cmr/m/n/10 =$ \OT1/cmr/m/it/10 New-t
on$\OML/cmm/m/it/10 ;$ \OT1/cmr/m/it/10 un-folded interpolation[]poly[]int$\OML
/cmm/m/it/10 :$\OT1/cmr/m/it/10 simps$\OT1/cmr/m/n/10 ]$ 
)) (./root.bbl) [138] (./root.aux)

LaTeX Warning: There were undefined references.


LaTeX Warning: Label(s) may have changed. Rerun to get cross-references right.

 )
(see the transcript file for additional information)</usr/share/texlive/texmf-d
ist/fonts/type1/public/amsfonts/cm/cmbsy10.pfb></usr/share/texlive/texmf-dist/f
onts/type1/public/amsfonts/cm/cmbx10.pfb></usr/share/texlive/texmf-dist/fonts/t
ype1/public/amsfonts/cm/cmbx12.pfb></usr/share/texlive/texmf-dist/fonts/type1/p
ublic/amsfonts/cm/cmex10.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/
amsfonts/cm/cmmi10.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfon
ts/cm/cmmi12.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/
cmmi7.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi8.p
fb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmr10.pfb></us
r/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmr12.pfb></usr/share
/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmr17.pfb></usr/share/texliv
e/texmf-dist/fonts/type1/public/amsfonts/cm/cmr7.pfb></usr/share/texlive/texmf-
dist/fonts/type1/public/amsfonts/cm/cmr8.pfb></usr/share/texlive/texmf-dist/fon
ts/type1/public/amsfonts/cm/cmr9.pfb></usr/share/texlive/texmf-dist/fonts/type1
/public/amsfonts/cm/cmss10.pfb></usr/share/texlive/texmf-dist/fonts/type1/publi
c/amsfonts/cm/cmsy10.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsf
onts/cm/cmsy5.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm
/cmsy6.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy7.
pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy8.pfb></u
sr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmti10.pfb></usr/sha
re/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmti12.pfb></usr/share/tex
live/texmf-dist/fonts/type1/public/amsfonts/symbols/msam10.pfb>
Output written on root.pdf (138 pages, 505217 bytes).
Transcript written on root.log.
This is pdfTeX, Version 3.14159265-2.6-1.40.16 (TeX Live 2015/Debian) (preloaded format=pdflatex)
 restricted \write18 enabled.
entering extended mode
LaTeX2e <2016/02/01>
Babel <3.9q> and hyphenation patterns for 81 language(s) loaded.

(./root.tex (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls
Document Class: article 2014/09/29 v1.4h Standard LaTeX document class
(/usr/share/texlive/texmf-dist/tex/latex/base/size11.clo)) (./isabelle.sty
(./comment.sty Excluding comment 'comment') Including comment 'isadelimtheory'
Including comment 'isatagtheory' Including comment 'isadelimproof'
Including comment 'isatagproof' Including comment 'isadelimML'
Including comment 'isatagML' Including comment 'isadelimvisible'
Including comment 'isatagvisible' Excluding comment 'isadeliminvisible'
Excluding comment 'isataginvisible') (./isabelletags.sty
Including comment 'isadelimproof' Excluding comment 'isatagproof'
Including comment 'isadelimML' Excluding comment 'isatagML') (./isabellesym.sty
) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty
(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty))
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty
For additional information on amsmath, use the `?' option.
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty))
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty)
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty))
(/usr/share/texlive/texmf-dist/tex/latex/tools/xspace.sty) (./pdfsetup.sty
(/usr/share/texlive/texmf-dist/tex/latex/graphics/color.sty
(/usr/share/texlive/texmf-dist/tex/latex/latexconfig/color.cfg)
(/usr/share/texlive/texmf-dist/tex/latex/pdftex-def/pdftex.def
(/usr/share/texlive/texmf-dist/tex/generic/oberdiek/infwarerr.sty)
(/usr/share/texlive/texmf-dist/tex/generic/oberdiek/ltxcmds.sty))))
(/usr/share/texlive/texmf-dist/tex/latex/hyperref/hyperref.sty
(/usr/share/texlive/texmf-dist/tex/generic/oberdiek/hobsub-hyperref.sty
(/usr/share/texlive/texmf-dist/tex/generic/oberdiek/hobsub-generic.sty))
(/usr/share/texlive/texmf-dist/tex/latex/graphics/keyval.sty)
(/usr/share/texlive/texmf-dist/tex/generic/ifxetex/ifxetex.sty)
(/usr/share/texlive/texmf-dist/tex/latex/oberdiek/auxhook.sty)
(/usr/share/texlive/texmf-dist/tex/latex/oberdiek/kvoptions.sty)
(/usr/share/texlive/texmf-dist/tex/latex/hyperref/pd1enc.def)
(/usr/share/texlive/texmf-dist/tex/latex/latexconfig/hyperref.cfg)
(/usr/share/texlive/texmf-dist/tex/latex/url/url.sty))

Package hyperref Message: Driver (autodetected): hpdftex.

(/usr/share/texlive/texmf-dist/tex/latex/hyperref/hpdftex.def
(/usr/share/texlive/texmf-dist/tex/latex/oberdiek/rerunfilecheck.sty))
(./root.aux) (/usr/share/texlive/texmf-dist/tex/context/base/supp-pdf.mkii
[Loading MPS to PDF converter (version 2006.09.02).]
) (/usr/share/texlive/texmf-dist/tex/latex/hyperref/nameref.sty
(/usr/share/texlive/texmf-dist/tex/generic/oberdiek/gettitlestring.sty))
(./root.out) (./root.out)
(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd)
(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd)
(/usr/share/texlive/texmf-dist/tex/latex/base/omscmr.fd) (./root.toc [1{/var/li
b/texmf/fonts/map/pdftex/updmap/pdftex.map}] [2]) [3] (./session.tex
(./More_List.tex) (./Infinite_Set.tex) (./Polynomial.tex)
(./Adhoc_Overloading.tex) (./Monad_Syntax.tex) (./Sqrt_Babylonian_Auxiliary.tex
) (./Is_Rat_To_Rat.tex) (./Divmod_Int.tex [4]
Overfull \hbox (12.96191pt too wide) in paragraph at lines 41--43
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 divmod[]int[]code$\OT1/cmr/m/n/10 [
$\OT1/cmr/m/it/10 code$\OT1/cmr/m/n/10 ]$$:$ \OT1/cmr/m/it/10 divmod[]int m n $
\OT1/cmr/m/n/10 =$ \OT1/cmr/m/it/10 map[]prod int[]of[]integer int[]of[]integer

) (./Improved_Code_Equations.tex

Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 33.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 33.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 33.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 33.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 33.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 33.

[5]

Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 128.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 128.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 128.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 128.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 128.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 128.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `math shift' on input line 128.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\mathord' on input line 128.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `math shift' on input line 128.


Overfull \hbox (32.11456pt too wide) in paragraph at lines 141--143
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 divmod[]nat[]code$\OT1/cmr/m/n/10 :
$ \OT1/cmr/m/it/10 Di-vides$\OML/cmm/m/it/10 :$\OT1/cmr/m/it/10 divmod[]nat m n
 $\OT1/cmr/m/n/10 =$ \OT1/cmr/m/it/10 map[]prod nat[]of[]integer nat[]of[]integ
er
[6]

Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 221.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 221.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 221.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 221.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 221.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 221.

) (./Cancellation.tex
Overfull \hbox (15.88786pt too wide) in paragraph at lines 30--31
[][]  \OT1/cmr/m/it/10 can-ce-la-tion sim-proc$\OML/cmm/m/it/10 :$ \OT1/cmr/m/i
t/10 Nor-mal-i-sa-tion of []iterate[]add[] back to the nor-male representation[
] 
[7]
Overfull \hbox (16.82057pt too wide) in paragraph at lines 81--83
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 iterate[]add[]distrib$\OT1/cmr/m/n/
10 [$\OT1/cmr/m/it/10 simp$\OT1/cmr/m/n/10 ]$$:$ []\OT1/cmr/m/it/10 iterate[]ad
d $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 m$\OT1/cmr/m/n/10 +$\OT1/cmr/m/it/10 n$\O
T1/cmr/m/n/10 )$ \OT1/cmr/m/it/10 a $\OT1/cmr/m/n/10 =$ \OT1/cmr/m/it/10 iterat
e[]add m a $\OT1/cmr/m/n/10 +$ \OT1/cmr/m/it/10 iterate[]add

Overfull \hbox (85.96669pt too wide) in paragraph at lines 175--176
[][]  \OT1/cmr/m/it/10 j $\OMS/cmsy/m/n/10 ^^T$ $\OT1/cmr/m/n/10 ($\OT1/cmr/m/i
t/10 i$\OT1/cmr/m/n/10 :$$:$\OT1/cmr/m/it/10 nat$\OT1/cmr/m/n/10 )$ $=[]\OMS/cm
sy/m/n/10 )$ $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 iterate[]add i $\OT1/cmr/m/n/1
0 ($\OT1/cmr/m/it/10 u$\OT1/cmr/m/n/10 :$$:$ $ [] $\OT1/cmr/m/it/10 a $\OT1/cmr
/m/n/10 :$$:$ $\OMS/cmsy/m/n/10 f$\OT1/cmr/m/it/10 cancel[]comm[]monoid[]add$\O
ML/cmm/m/it/10 ;$ \OT1/cmr/m/it/10 ordered[]ab[]semigroup[]add[]imp[]le$\OMS/cm
sy/m/n/10 g$$\OT1/cmr/m/n/10 )$

Overfull \hbox (85.96669pt too wide) in paragraph at lines 194--195
[][]  \OT1/cmr/m/it/10 i $\OMS/cmsy/m/n/10 ^^T$ $\OT1/cmr/m/n/10 ($\OT1/cmr/m/i
t/10 j$\OT1/cmr/m/n/10 :$$:$\OT1/cmr/m/it/10 nat$\OT1/cmr/m/n/10 )$ $=[]\OMS/cm
sy/m/n/10 )$ $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 iterate[]add i $\OT1/cmr/m/n/1
0 ($\OT1/cmr/m/it/10 u$\OT1/cmr/m/n/10 :$$:$ $ [] $\OT1/cmr/m/it/10 a $\OT1/cmr
/m/n/10 :$$:$ $\OMS/cmsy/m/n/10 f$\OT1/cmr/m/it/10 cancel[]comm[]monoid[]add$\O
ML/cmm/m/it/10 ;$ \OT1/cmr/m/it/10 ordered[]ab[]semigroup[]add[]imp[]le$\OMS/cm
sy/m/n/10 g$$\OT1/cmr/m/n/10 )$

Overfull \hbox (85.96669pt too wide) in paragraph at lines 213--214
[][]  \OT1/cmr/m/it/10 j $\OMS/cmsy/m/n/10 ^^T$ $\OT1/cmr/m/n/10 ($\OT1/cmr/m/i
t/10 i$\OT1/cmr/m/n/10 :$$:$\OT1/cmr/m/it/10 nat$\OT1/cmr/m/n/10 )$ $=[]\OMS/cm
sy/m/n/10 )$ $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 iterate[]add i $\OT1/cmr/m/n/1
0 ($\OT1/cmr/m/it/10 u$\OT1/cmr/m/n/10 :$$:$ $ [] $\OT1/cmr/m/it/10 a $\OT1/cmr
/m/n/10 :$$:$ $\OMS/cmsy/m/n/10 f$\OT1/cmr/m/it/10 cancel[]comm[]monoid[]add$\O
ML/cmm/m/it/10 ;$ \OT1/cmr/m/it/10 ordered[]ab[]semigroup[]add[]imp[]le$\OMS/cm
sy/m/n/10 g$$\OT1/cmr/m/n/10 )$

Overfull \hbox (85.96669pt too wide) in paragraph at lines 232--233
[][]  \OT1/cmr/m/it/10 i $\OMS/cmsy/m/n/10 ^^T$ $\OT1/cmr/m/n/10 ($\OT1/cmr/m/i
t/10 j$\OT1/cmr/m/n/10 :$$:$\OT1/cmr/m/it/10 nat$\OT1/cmr/m/n/10 )$ $=[]\OMS/cm
sy/m/n/10 )$ $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 iterate[]add i $\OT1/cmr/m/n/1
0 ($\OT1/cmr/m/it/10 u$\OT1/cmr/m/n/10 :$$:$ $ [] $\OT1/cmr/m/it/10 a $\OT1/cmr
/m/n/10 :$$:$ $\OMS/cmsy/m/n/10 f$\OT1/cmr/m/it/10 cancel[]comm[]monoid[]add$\O
ML/cmm/m/it/10 ;$ \OT1/cmr/m/it/10 ordered[]ab[]semigroup[]add[]imp[]le$\OMS/cm
sy/m/n/10 g$$\OT1/cmr/m/n/10 )$
[8]) (./Multiset.tex [9] [10] [11] [12]
Overfull \hbox (13.45732pt too wide) in paragraph at lines 783--785
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 set[]mset[]add[]mset[]insert $\OT1/
cmr/m/n/10 [$\OT1/cmr/m/it/10 simp$\OT1/cmr/m/n/10 ]$$:$ []\OT1/cmr/m/it/10 set
[]mset $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 add[]mset a A$\OT1/cmr/m/n/10 )$ $=$
 \OT1/cmr/m/it/10 in-sert a $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 set[]mset
[13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]
Overfull \hbox (20.85715pt too wide) in paragraph at lines 3518--3520
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 replicate[]mset[]Suc $\OT1/cmr/m/n/
10 [$\OT1/cmr/m/it/10 simp$\OT1/cmr/m/n/10 ]$$:$ \OT1/cmr/m/it/10 replicate[]ms
et $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 Suc n$\OT1/cmr/m/n/10 )$ \OT1/cmr/m/it/1
0 x $\OT1/cmr/m/n/10 =$ \OT1/cmr/m/it/10 add[]mset x $\OT1/cmr/m/n/10 ($\OT1/cm
r/m/it/10 replicate[]mset
[24] [25] [26] [27]
Overfull \hbox (7.00906pt too wide) in paragraph at lines 4821--4823
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 filter[]diff[]mset $\OT1/cmr/m/n/10
 [$\OT1/cmr/m/it/10 simp$\OT1/cmr/m/n/10 ]$$:$ \OT1/cmr/m/it/10 filter[]mset P 
$\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 M $\OMS/cmsy/m/n/10 ^^@$ \OT1/cmr/m/it/10 N
$\OT1/cmr/m/n/10 )$ $=$ \OT1/cmr/m/it/10 filter[]mset P M $\OMS/cmsy/m/n/10 ^^@
$ \OT1/cmr/m/it/10 filter[]mset
[28]
Overfull \hbox (11.80153pt too wide) in paragraph at lines 4857--4859
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 filter[]sup[]mset$\OT1/cmr/m/n/10 [
$\OT1/cmr/m/it/10 simp$\OT1/cmr/m/n/10 ]$$:$ \OT1/cmr/m/it/10 filter[]mset P $\
OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 A $\OMS/cmsy/m/n/10 [$$\OT1/cmr/m/n/10 #$ \OT
1/cmr/m/it/10 B$\OT1/cmr/m/n/10 )$ $=$ \OT1/cmr/m/it/10 filter[]mset P A $\OMS/
cmsy/m/n/10 [$$\OT1/cmr/m/n/10 #$ \OT1/cmr/m/it/10 filter[]mset
[29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43]
[44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54]
Overfull \hbox (5.43213pt too wide) in paragraph at lines 12056--12057
[][]  \OT1/cmr/m/it/10 full[]exhaustive[]multiset f i $\OT1/cmr/m/n/10 =$ \OT1/
cmr/m/it/10 Quickcheck[]Exhaustive$\OML/cmm/m/it/10 :$\OT1/cmr/m/it/10 full[]ex
haustive $\OT1/cmr/m/n/10 ($$\OML/cmm/m/it/10 ^^U$\OT1/cmr/m/it/10 xs$\OML/cmm/
m/it/10 :$ \OT1/cmr/m/it/10 f $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 msetify
[55] [56] [57]) (./Factorial_Ring.tex [58] [59] [60] [61] [62] [63] [64]
[65]
Overfull \hbox (11.16814pt too wide) in paragraph at lines 2290--2290
[]\OT1/cmr/bx/n/12 Factorial semir-ings: al-ge-braic struc-tures with unique pr
ime
[66] [67] [68] [69] [70] [71] [72]
Overfull \hbox (6.21616pt too wide) in paragraph at lines 4656--4657
[][]  \OT1/cmr/bx/n/10 shows   \OT1/cmr/m/it/10 prime[]factorization $\OT1/cmr/
m/n/10 ($\OT1/cmr/m/it/10 a div b$\OT1/cmr/m/n/10 )$ $=$ \OT1/cmr/m/it/10 prime
[]factorization a $\OMS/cmsy/m/n/10 ^^@$ \OT1/cmr/m/it/10 prime[]factorization
[73] [74] [75]
Overfull \hbox (14.46059pt too wide) in paragraph at lines 5797--5798
[][]  \OT1/cmr/bx/n/10 shows   \OT1/cmr/m/it/10 prime[]factorization $\OT1/cmr/
m/n/10 ($\OT1/cmr/m/it/10 gcd a b$\OT1/cmr/m/n/10 )$ $=$ \OT1/cmr/m/it/10 prime
[]factorization a $\OMS/cmsy/m/n/10 \$$\OT1/cmr/m/n/10 #$ \OT1/cmr/m/it/10 prim
e[]factorization

Overfull \hbox (15.99394pt too wide) in paragraph at lines 5817--5818
[][]  \OT1/cmr/bx/n/10 shows   \OT1/cmr/m/it/10 prime[]factorization $\OT1/cmr/
m/n/10 ($\OT1/cmr/m/it/10 lcm a b$\OT1/cmr/m/n/10 )$ $=$ \OT1/cmr/m/it/10 prime
[]factorization a $\OMS/cmsy/m/n/10 [$$\OT1/cmr/m/n/10 #$ \OT1/cmr/m/it/10 prim
e[]factorization
[76] [77]) (./Euclidean_Algorithm.tex
Overfull \hbox (36.91122pt too wide) in paragraph at lines 6--6
[]\OT1/cmr/bx/n/14.4 Abstract eu-clidean al-go-rithm in eu-clidean (semi)rings 

[78] [79] [80]
Overfull \hbox (36.0064pt too wide) in paragraph at lines 1171--1172
[][]  \OT1/cmr/m/it/10 OF-CLASS$\OT1/cmr/m/n/10 ($$ [] $\OT1/cmr/m/it/10 a$\OT1
/cmr/m/n/10 :$$:$$\OMS/cmsy/m/n/10 f$\OT1/cmr/m/it/10 factorial[]semiring[]gcd$
\OML/cmm/m/it/10 ;$ \OT1/cmr/m/it/10 euclidean[]semiring$\OMS/cmsy/m/n/10 g$$\O
ML/cmm/m/it/10 ;$ \OT1/cmr/m/it/10 euclidean[]semiring[]gcd[]class$\OT1/cmr/m/n
/10 )$[] 
[81]) (./Ring_Hom.tex [82] [83] [84]) (./Fraction_Field.tex [85] [86] [87]
[88] [89]) (./Normalized_Fraction.tex [90]
Overfull \hbox (25.4372pt too wide) in paragraph at lines 205--207
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 normalize[]quot[]in[]normalized[]fr
acts $\OT1/cmr/m/n/10 [$\OT1/cmr/m/it/10 simp$\OT1/cmr/m/n/10 ]$$:$ \OT1/cmr/m/
it/10 normalize[]quot x $\OMS/cmsy/m/n/10 2$ \OT1/cmr/m/it/10 normalized[]fract
s[] 
[91]
Overfull \hbox (26.27673pt too wide) in paragraph at lines 311--313
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 normalize[]quot[]idem $\OT1/cmr/m/n
/10 [$\OT1/cmr/m/it/10 simp$\OT1/cmr/m/n/10 ]$$:$ \OT1/cmr/m/it/10 normalize[]q
uot $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 normalize[]quot x$\OT1/cmr/m/n/10 )$ $=
$ \OT1/cmr/m/it/10 normalize[]quot

Overfull \hbox (3.45944pt too wide) in paragraph at lines 510--512
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 quot[]of[]fract[]in[]normalized[]fr
acts $\OT1/cmr/m/n/10 [$\OT1/cmr/m/it/10 simp$\OT1/cmr/m/n/10 ]$$:$ \OT1/cmr/m/
it/10 quot[]of[]fract x $\OMS/cmsy/m/n/10 2$ \OT1/cmr/m/it/10 normalized[]fract
s[] 
[92] [93] [94]) (./Field_as_Ring.tex [95] [96]) (./Polynomial_Factorial.tex
[97] [98]
Overfull \hbox (30.61302pt too wide) in paragraph at lines 511--513
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 normalize[]snd[]quot[]of[]fract$\OT
1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 nor-mal-ize $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/1
0 snd $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 quot[]of[]fract x$\OT1/cmr/m/n/10 )$$
)$ $=$ \OT1/cmr/m/it/10 snd $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 quot[]of[]fract


Overfull \hbox (3.69643pt too wide) in paragraph at lines 539--541
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 fract[]poly[]smult $\OT1/cmr/m/n/10
 [$\OT1/cmr/m/it/10 simp$\OT1/cmr/m/n/10 ]$$:$ \OT1/cmr/m/it/10 fract[]poly $\O
T1/cmr/m/n/10 ($\OT1/cmr/m/it/10 smult c p$\OT1/cmr/m/n/10 )$ $=$ \OT1/cmr/m/it
/10 smult $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 to[]fract c$\OT1/cmr/m/n/10 )$ $(
$\OT1/cmr/m/it/10 fract[]poly
[99]
Overfull \hbox (9.21042pt too wide) in paragraph at lines 725--726
[][]  \OT1/cmr/m/it/10 prod[]mset $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 image[]ms
et $\OT1/cmr/m/n/10 ($$\OML/cmm/m/it/10 ^^U$\OT1/cmr/m/it/10 x$\OML/cmm/m/it/10
 :$ \OT1/cmr/m/it/10 fract[]poly $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 f x$\OT1/c
mr/m/n/10 )$$)$ \OT1/cmr/m/it/10 A$\OT1/cmr/m/n/10 )$ $=$ \OT1/cmr/m/it/10 frac
t[]poly $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 prod[]mset $\OT1/cmr/m/n/10 ($\OT1/
cmr/m/it/10 image[]mset
[100]
Overfull \hbox (9.28085pt too wide) in paragraph at lines 1049--1051
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 fract[]content[]fract[]poly $\OT1/c
mr/m/n/10 [$\OT1/cmr/m/it/10 simp$\OT1/cmr/m/n/10 ]$$:$ \OT1/cmr/m/it/10 fract[
]content $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 fract[]poly p$\OT1/cmr/m/n/10 )$ $
=$ \OT1/cmr/m/it/10 to[]fract $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 content
[101] [102] [103] [104] [105] [106]
Overfull \hbox (31.4pt too wide) in paragraph at lines 3027--3029
[][]\OT1/cmr/bx/n/10 lemmas \OT1/cmr/m/it/10 Gcd[]poly[]set[]eq[]fold $\OT1/cmr
/m/n/10 [$\OT1/cmr/m/it/10 code$\OT1/cmr/m/n/10 ]$ $=$ \OT1/cmr/m/it/10 Gcd[]se
t[]eq[]fold $\OT1/cmr/m/n/10 [$\OT1/cmr/bx/n/10 where \OT1/cmr/m/it/10 ?$ [] $a
 $\OT1/cmr/m/n/10 =$ $ [] $\OT1/cmr/m/it/10 a $\OT1/cmr/m/n/10 :$$:$ \OT1/cmr/m
/it/10 factorial[]ring[]gcd

Overfull \hbox (34.6055pt too wide) in paragraph at lines 3029--3032
[][]\OT1/cmr/bx/n/10 lemmas \OT1/cmr/m/it/10 Lcm[]poly[]set[]eq[]fold $\OT1/cmr
/m/n/10 [$\OT1/cmr/m/it/10 code$\OT1/cmr/m/n/10 ]$ $=$ \OT1/cmr/m/it/10 Lcm[]se
t[]eq[]fold $\OT1/cmr/m/n/10 [$\OT1/cmr/bx/n/10 where \OT1/cmr/m/it/10 ?$ [] $a
 $\OT1/cmr/m/n/10 =$ $ [] $\OT1/cmr/m/it/10 a $\OT1/cmr/m/n/10 :$$:$ \OT1/cmr/m
/it/10 factorial[]ring[]gcd
) (./Missing_Unsorted.tex [107] [108] [109] [110]) (./Missing_Polynomial.tex
[111] [112] [113]
Overfull \hbox (26.31035pt too wide) in paragraph at lines 1194--1196
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 degree[]prod[]sum[]lt[]n$\OT1/cmr/m
/n/10 :$ \OT1/cmr/bx/n/10 as-sumes $[]$ \OT1/cmr/m/it/10 i$\OML/cmm/m/it/10 :$ 
\OT1/cmr/m/it/10 i $\OML/cmm/m/it/10 <$ \OT1/cmr/m/it/10 n $\OT1/cmr/m/n/10 =[]
\OMS/cmsy/m/n/10 )$ \OT1/cmr/m/it/10 de-gree $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/1
0 f i $\OT1/cmr/m/n/10 :$$:$ $ [] $\OT1/cmr/m/it/10 a $\OT1/cmr/m/n/10 :$$:$ \O
T1/cmr/m/it/10 comm[]semiring[]1
[114] [115] [116] [117]

Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `math shift' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `math shift' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `math shift' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\mathord' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `math shift' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `math shift' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\mathord' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `math shift' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `math shift' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\mskip' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\mskip' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `math shift' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `math shift' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `math shift' on input line 2890.

[118] [119]
Overfull \hbox (17.23799pt too wide) in paragraph at lines 3444--3445
[][]  \OT1/cmr/m/it/10 nor-mal-ize $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 f $\OT1/
cmr/m/n/10 :$$:$ $ [] $\OT1/cmr/m/it/10 a $\OT1/cmr/m/n/10 :$$:$ $\OMS/cmsy/m/n
/10 f$\OT1/cmr/m/it/10 normalization[]semidom$\OML/cmm/m/it/10 ;$\OT1/cmr/m/it/
10 field$\OMS/cmsy/m/n/10 g$ \OT1/cmr/m/it/10 poly$\OT1/cmr/m/n/10 )$ $=$ \OT1/
cmr/m/it/10 smult $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 inverse $\OT1/cmr/m/n/10 
($\OT1/cmr/m/it/10 unit[]factor
[120] [121]) (./Ring_Hom_Poly.tex [122] [123] [124] [125] [126])
(./Newton_Interpolation.tex [127]
Overfull \hbox (17.22385pt too wide) in paragraph at lines 50--51
[][]  \OT1/cmr/m/it/10 horner[]composition $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 
map hom cs$\OT1/cmr/m/n/10 )$ $($\OT1/cmr/m/it/10 map hom xs$\OT1/cmr/m/n/10 )$
 $=$ \OT1/cmr/m/it/10 map[]poly hom $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 horner[
]composition
[128] [129] [130]
Overfull \hbox (4.60117pt too wide) in paragraph at lines 1779--1780
[][]  $\OT1/cmr/m/n/10 =[]\OMS/cmsy/m/n/10 )$ \OT1/cmr/m/it/10 newton[]coeffici
ents[]main $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 map rat[]of[]int xs$\OT1/cmr/m/n
/10 )$ $($\OT1/cmr/m/it/10 map rat[]of[]int gs$\OT1/cmr/m/n/10 )$ $($\OT1/cmr/m
/it/10 map rat[]of[]int
[131] [132]) (./Lagrange_Interpolation.tex [133])
(./Neville_Aitken_Interpolation.tex [134]
Overfull \hbox (0.26402pt too wide) in paragraph at lines 318--320
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 length[]neville[]aitken[]merge$\OT1
/cmr/m/n/10 [$\OT1/cmr/m/it/10 termination[]simp$\OT1/cmr/m/n/10 ]$$:$ \OT1/cmr
/m/it/10 length $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 neville[]aitken[]merge
[135]) (./Polynomial_Interpolation.tex [136] [137]
Overfull \hbox (26.8093pt too wide) in paragraph at lines 468--469
[][]  \OT1/cmr/m/it/10 interpolation[]poly[]int[]Some$\OT1/cmr/m/n/10 [$\OT1/cm
r/bx/n/10 where \OT1/cmr/m/it/10 alg $\OT1/cmr/m/n/10 =$ \OT1/cmr/m/it/10 New-t
on$\OML/cmm/m/it/10 ;$ \OT1/cmr/m/it/10 un-folded interpolation[]poly[]int$\OML
/cmm/m/it/10 :$\OT1/cmr/m/it/10 simps$\OT1/cmr/m/n/10 ]$[] 

Overfull \hbox (26.06485pt too wide) in paragraph at lines 472--474
[][]  \OT1/cmr/m/it/10 interpolation[]poly[]int[]None$\OT1/cmr/m/n/10 [$\OT1/cm
r/bx/n/10 where \OT1/cmr/m/it/10 alg $\OT1/cmr/m/n/10 =$ \OT1/cmr/m/it/10 New-t
on$\OML/cmm/m/it/10 ;$ \OT1/cmr/m/it/10 un-folded interpolation[]poly[]int$\OML
/cmm/m/it/10 :$\OT1/cmr/m/it/10 simps$\OT1/cmr/m/n/10 ]$ 
)) (./root.bbl) [138] (./root.aux) )
(see the transcript file for additional information)</usr/share/texlive/texmf-d
ist/fonts/type1/public/amsfonts/cm/cmbsy10.pfb></usr/share/texlive/texmf-dist/f
onts/type1/public/amsfonts/cm/cmbx10.pfb></usr/share/texlive/texmf-dist/fonts/t
ype1/public/amsfonts/cm/cmbx12.pfb></usr/share/texlive/texmf-dist/fonts/type1/p
ublic/amsfonts/cm/cmex10.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/
amsfonts/cm/cmmi10.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfon
ts/cm/cmmi12.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/
cmmi7.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi8.p
fb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmr10.pfb></us
r/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmr12.pfb></usr/share
/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmr17.pfb></usr/share/texliv
e/texmf-dist/fonts/type1/public/amsfonts/cm/cmr7.pfb></usr/share/texlive/texmf-
dist/fonts/type1/public/amsfonts/cm/cmr8.pfb></usr/share/texlive/texmf-dist/fon
ts/type1/public/amsfonts/cm/cmr9.pfb></usr/share/texlive/texmf-dist/fonts/type1
/public/amsfonts/cm/cmss10.pfb></usr/share/texlive/texmf-dist/fonts/type1/publi
c/amsfonts/cm/cmsy10.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsf
onts/cm/cmsy5.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm
/cmsy6.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy7.
pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy8.pfb></u
sr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmti10.pfb></usr/sha
re/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmti12.pfb></usr/share/tex
live/texmf-dist/fonts/type1/public/amsfonts/symbols/msam10.pfb>
Output written on root.pdf (138 pages, 504970 bytes).
Transcript written on root.log.

This is pdfTeX, Version 3.14159265-2.6-1.40.16 (TeX Live 2015/Debian) (preloaded format=pdflatex)
 restricted \write18 enabled.
entering extended mode
LaTeX2e <2016/02/01>
Babel <3.9q> and hyphenation patterns for 81 language(s) loaded.

(./root.tex (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls
Document Class: article 2014/09/29 v1.4h Standard LaTeX document class
(/usr/share/texlive/texmf-dist/tex/latex/base/size11.clo)) (./isabelle.sty
(./comment.sty Excluding comment 'comment') Including comment 'isadelimtheory'
Including comment 'isatagtheory' Including comment 'isadelimproof'
Including comment 'isatagproof' Including comment 'isadelimML'
Including comment 'isatagML' Including comment 'isadelimvisible'
Including comment 'isatagvisible' Excluding comment 'isadeliminvisible'
Excluding comment 'isataginvisible') (./isabelletags.sty) (./isabellesym.sty)
(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty
(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty))
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty
For additional information on amsmath, use the `?' option.
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty))
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty)
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty))
(/usr/share/texlive/texmf-dist/tex/latex/tools/xspace.sty) (./pdfsetup.sty
(/usr/share/texlive/texmf-dist/tex/latex/graphics/color.sty
(/usr/share/texlive/texmf-dist/tex/latex/latexconfig/color.cfg)
(/usr/share/texlive/texmf-dist/tex/latex/pdftex-def/pdftex.def
(/usr/share/texlive/texmf-dist/tex/generic/oberdiek/infwarerr.sty)
(/usr/share/texlive/texmf-dist/tex/generic/oberdiek/ltxcmds.sty))))
(/usr/share/texlive/texmf-dist/tex/latex/hyperref/hyperref.sty
(/usr/share/texlive/texmf-dist/tex/generic/oberdiek/hobsub-hyperref.sty
(/usr/share/texlive/texmf-dist/tex/generic/oberdiek/hobsub-generic.sty))
(/usr/share/texlive/texmf-dist/tex/latex/graphics/keyval.sty)
(/usr/share/texlive/texmf-dist/tex/generic/ifxetex/ifxetex.sty)
(/usr/share/texlive/texmf-dist/tex/latex/oberdiek/auxhook.sty)
(/usr/share/texlive/texmf-dist/tex/latex/oberdiek/kvoptions.sty)
(/usr/share/texlive/texmf-dist/tex/latex/hyperref/pd1enc.def)
(/usr/share/texlive/texmf-dist/tex/latex/latexconfig/hyperref.cfg)
(/usr/share/texlive/texmf-dist/tex/latex/url/url.sty))

Package hyperref Message: Driver (autodetected): hpdftex.

(/usr/share/texlive/texmf-dist/tex/latex/hyperref/hpdftex.def
(/usr/share/texlive/texmf-dist/tex/latex/oberdiek/rerunfilecheck.sty))
No file root.aux.
(/usr/share/texlive/texmf-dist/tex/context/base/supp-pdf.mkii
[Loading MPS to PDF converter (version 2006.09.02).]
) (/usr/share/texlive/texmf-dist/tex/latex/hyperref/nameref.sty
(/usr/share/texlive/texmf-dist/tex/generic/oberdiek/gettitlestring.sty))
(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd)
(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd)
(/usr/share/texlive/texmf-dist/tex/latex/base/omscmr.fd)

LaTeX Warning: Citation `interpolation' on page 1 undefined on input line 55.

[1{/var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map}] (./session.tex
(./More_List.tex) (./Infinite_Set.tex) (./Polynomial.tex)
(./Adhoc_Overloading.tex) (./Monad_Syntax.tex) (./Sqrt_Babylonian_Auxiliary.tex
) (./Is_Rat_To_Rat.tex) (./Divmod_Int.tex
Overfull \hbox (12.96191pt too wide) in paragraph at lines 41--43
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 divmod[]int[]code$\OT1/cmr/m/n/10 [
$\OT1/cmr/m/it/10 code$\OT1/cmr/m/n/10 ]$$:$ \OT1/cmr/m/it/10 divmod[]int m n $
\OT1/cmr/m/n/10 =$ \OT1/cmr/m/it/10 map[]prod int[]of[]integer int[]of[]integer

[2]) (./Improved_Code_Equations.tex

Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 33.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 33.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 33.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 33.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 33.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 33.

[3]
Overfull \hbox (12.29745pt too wide) in paragraph at lines 108--115
[][]   \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 cases l $\OT1/cm
r/m/n/10 =$ \OT1/cmr/m/it/10 0$\OT1/cmr/m/n/10 ;$ \OT1/cmr/m/it/10 cases l $\OM
L/cmm/m/it/10 <$ \OT1/cmr/m/it/10 0$\OT1/cmr/m/n/10 ;$ \OT1/cmr/m/it/10 cases l
 $\OML/cmm/m/it/10 >$ \OT1/cmr/m/it/10 0$\OT1/cmr/m/n/10 ;$ \OT1/cmr/m/it/10 au
to split$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 prod$\OML/cmm/m/it/10 :$\OT1/cmr/m
/it/10 splits simp$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 divmod[]abs$ [] $[]def

Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 128.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 128.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 128.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 128.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 128.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 128.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `math shift' on input line 128.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\mathord' on input line 128.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `math shift' on input line 128.


Overfull \hbox (32.11456pt too wide) in paragraph at lines 141--143
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 divmod[]nat[]code$\OT1/cmr/m/n/10 :
$ \OT1/cmr/m/it/10 Di-vides$\OML/cmm/m/it/10 :$\OT1/cmr/m/it/10 divmod[]nat m n
 $\OT1/cmr/m/n/10 =$ \OT1/cmr/m/it/10 map[]prod nat[]of[]integer nat[]of[]integ
er
[4]

Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 221.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 221.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 221.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 221.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 221.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 221.

) (./Cancellation.tex
Overfull \hbox (15.88786pt too wide) in paragraph at lines 30--31
[][]  \OT1/cmr/m/it/10 can-ce-la-tion sim-proc$\OML/cmm/m/it/10 :$ \OT1/cmr/m/i
t/10 Nor-mal-i-sa-tion of []iterate[]add[] back to the nor-male representation[
] 
[5]
Overfull \hbox (16.82057pt too wide) in paragraph at lines 81--83
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 iterate[]add[]distrib$\OT1/cmr/m/n/
10 [$\OT1/cmr/m/it/10 simp$\OT1/cmr/m/n/10 ]$$:$ []\OT1/cmr/m/it/10 iterate[]ad
d $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 m$\OT1/cmr/m/n/10 +$\OT1/cmr/m/it/10 n$\O
T1/cmr/m/n/10 )$ \OT1/cmr/m/it/10 a $\OT1/cmr/m/n/10 =$ \OT1/cmr/m/it/10 iterat
e[]add m a $\OT1/cmr/m/n/10 +$ \OT1/cmr/m/it/10 iterate[]add

Overfull \hbox (4.55688pt too wide) in paragraph at lines 138--151
[][]  \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 auto dest$\OT1/cm
r/m/n/10 !$$:$ \OT1/cmr/m/it/10 le[]Suc[]ex add[]right[]imp[]eq simp$\OT1/cmr/m
/n/10 :$ \OT1/cmr/m/it/10 ab[]semigroup[]add[]class$\OML/cmm/m/it/10 :$\OT1/cmr
/m/it/10 add[]ac$\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 1$\OT1/cmr/m/n/10 )$$)$[] 

Overfull \hbox (4.55688pt too wide) in paragraph at lines 157--170
[][]  \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 auto dest$\OT1/cm
r/m/n/10 !$$:$ \OT1/cmr/m/it/10 le[]Suc[]ex add[]right[]imp[]eq simp$\OT1/cmr/m
/n/10 :$ \OT1/cmr/m/it/10 ab[]semigroup[]add[]class$\OML/cmm/m/it/10 :$\OT1/cmr
/m/it/10 add[]ac$\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 1$\OT1/cmr/m/n/10 )$$)$[] 

Overfull \hbox (85.96669pt too wide) in paragraph at lines 175--176
[][]  \OT1/cmr/m/it/10 j $\OMS/cmsy/m/n/10 ^^T$ $\OT1/cmr/m/n/10 ($\OT1/cmr/m/i
t/10 i$\OT1/cmr/m/n/10 :$$:$\OT1/cmr/m/it/10 nat$\OT1/cmr/m/n/10 )$ $=[]\OMS/cm
sy/m/n/10 )$ $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 iterate[]add i $\OT1/cmr/m/n/1
0 ($\OT1/cmr/m/it/10 u$\OT1/cmr/m/n/10 :$$:$ $ [] $\OT1/cmr/m/it/10 a $\OT1/cmr
/m/n/10 :$$:$ $\OMS/cmsy/m/n/10 f$\OT1/cmr/m/it/10 cancel[]comm[]monoid[]add$\O
ML/cmm/m/it/10 ;$ \OT1/cmr/m/it/10 ordered[]ab[]semigroup[]add[]imp[]le$\OMS/cm
sy/m/n/10 g$$\OT1/cmr/m/n/10 )$

Overfull \hbox (4.55688pt too wide) in paragraph at lines 176--189
[][]  \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 auto dest$\OT1/cm
r/m/n/10 !$$:$ \OT1/cmr/m/it/10 le[]Suc[]ex add[]right[]imp[]eq simp$\OT1/cmr/m
/n/10 :$ \OT1/cmr/m/it/10 ab[]semigroup[]add[]class$\OML/cmm/m/it/10 :$\OT1/cmr
/m/it/10 add[]ac$\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 1$\OT1/cmr/m/n/10 )$$)$[] 

Overfull \hbox (85.96669pt too wide) in paragraph at lines 194--195
[][]  \OT1/cmr/m/it/10 i $\OMS/cmsy/m/n/10 ^^T$ $\OT1/cmr/m/n/10 ($\OT1/cmr/m/i
t/10 j$\OT1/cmr/m/n/10 :$$:$\OT1/cmr/m/it/10 nat$\OT1/cmr/m/n/10 )$ $=[]\OMS/cm
sy/m/n/10 )$ $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 iterate[]add i $\OT1/cmr/m/n/1
0 ($\OT1/cmr/m/it/10 u$\OT1/cmr/m/n/10 :$$:$ $ [] $\OT1/cmr/m/it/10 a $\OT1/cmr
/m/n/10 :$$:$ $\OMS/cmsy/m/n/10 f$\OT1/cmr/m/it/10 cancel[]comm[]monoid[]add$\O
ML/cmm/m/it/10 ;$ \OT1/cmr/m/it/10 ordered[]ab[]semigroup[]add[]imp[]le$\OMS/cm
sy/m/n/10 g$$\OT1/cmr/m/n/10 )$

Overfull \hbox (4.55688pt too wide) in paragraph at lines 195--208
[][]  \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 auto dest$\OT1/cm
r/m/n/10 !$$:$ \OT1/cmr/m/it/10 le[]Suc[]ex add[]right[]imp[]eq simp$\OT1/cmr/m
/n/10 :$ \OT1/cmr/m/it/10 ab[]semigroup[]add[]class$\OML/cmm/m/it/10 :$\OT1/cmr
/m/it/10 add[]ac$\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 1$\OT1/cmr/m/n/10 )$$)$[] 

Overfull \hbox (85.96669pt too wide) in paragraph at lines 213--214
[][]  \OT1/cmr/m/it/10 j $\OMS/cmsy/m/n/10 ^^T$ $\OT1/cmr/m/n/10 ($\OT1/cmr/m/i
t/10 i$\OT1/cmr/m/n/10 :$$:$\OT1/cmr/m/it/10 nat$\OT1/cmr/m/n/10 )$ $=[]\OMS/cm
sy/m/n/10 )$ $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 iterate[]add i $\OT1/cmr/m/n/1
0 ($\OT1/cmr/m/it/10 u$\OT1/cmr/m/n/10 :$$:$ $ [] $\OT1/cmr/m/it/10 a $\OT1/cmr
/m/n/10 :$$:$ $\OMS/cmsy/m/n/10 f$\OT1/cmr/m/it/10 cancel[]comm[]monoid[]add$\O
ML/cmm/m/it/10 ;$ \OT1/cmr/m/it/10 ordered[]ab[]semigroup[]add[]imp[]le$\OMS/cm
sy/m/n/10 g$$\OT1/cmr/m/n/10 )$

Overfull \hbox (4.55688pt too wide) in paragraph at lines 214--227
[][]  \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 auto dest$\OT1/cm
r/m/n/10 !$$:$ \OT1/cmr/m/it/10 le[]Suc[]ex add[]right[]imp[]eq simp$\OT1/cmr/m
/n/10 :$ \OT1/cmr/m/it/10 ab[]semigroup[]add[]class$\OML/cmm/m/it/10 :$\OT1/cmr
/m/it/10 add[]ac$\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 1$\OT1/cmr/m/n/10 )$$)$[] 

Overfull \hbox (85.96669pt too wide) in paragraph at lines 232--233
[][]  \OT1/cmr/m/it/10 i $\OMS/cmsy/m/n/10 ^^T$ $\OT1/cmr/m/n/10 ($\OT1/cmr/m/i
t/10 j$\OT1/cmr/m/n/10 :$$:$\OT1/cmr/m/it/10 nat$\OT1/cmr/m/n/10 )$ $=[]\OMS/cm
sy/m/n/10 )$ $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 iterate[]add i $\OT1/cmr/m/n/1
0 ($\OT1/cmr/m/it/10 u$\OT1/cmr/m/n/10 :$$:$ $ [] $\OT1/cmr/m/it/10 a $\OT1/cmr
/m/n/10 :$$:$ $\OMS/cmsy/m/n/10 f$\OT1/cmr/m/it/10 cancel[]comm[]monoid[]add$\O
ML/cmm/m/it/10 ;$ \OT1/cmr/m/it/10 ordered[]ab[]semigroup[]add[]imp[]le$\OMS/cm
sy/m/n/10 g$$\OT1/cmr/m/n/10 )$

Overfull \hbox (4.55688pt too wide) in paragraph at lines 233--246
[][]  \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 auto dest$\OT1/cm
r/m/n/10 !$$:$ \OT1/cmr/m/it/10 le[]Suc[]ex add[]right[]imp[]eq simp$\OT1/cmr/m
/n/10 :$ \OT1/cmr/m/it/10 ab[]semigroup[]add[]class$\OML/cmm/m/it/10 :$\OT1/cmr
/m/it/10 add[]ac$\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 1$\OT1/cmr/m/n/10 )$$)$[] 
[6]
Overfull \hbox (4.55688pt too wide) in paragraph at lines 252--265
[][]  \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 auto dest$\OT1/cm
r/m/n/10 !$$:$ \OT1/cmr/m/it/10 le[]Suc[]ex add[]right[]imp[]eq simp$\OT1/cmr/m
/n/10 :$ \OT1/cmr/m/it/10 ab[]semigroup[]add[]class$\OML/cmm/m/it/10 :$\OT1/cmr
/m/it/10 add[]ac$\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 1$\OT1/cmr/m/n/10 )$$)$[] 

Overfull \hbox (4.55688pt too wide) in paragraph at lines 271--288
[][]  \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 auto dest$\OT1/cm
r/m/n/10 !$$:$ \OT1/cmr/m/it/10 le[]Suc[]ex add[]right[]imp[]eq simp$\OT1/cmr/m
/n/10 :$ \OT1/cmr/m/it/10 ab[]semigroup[]add[]class$\OML/cmm/m/it/10 :$\OT1/cmr
/m/it/10 add[]ac$\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 1$\OT1/cmr/m/n/10 )$$)$ 
) (./Multiset.tex [7] [8] [9] [10] [11]
Overfull \hbox (13.45732pt too wide) in paragraph at lines 783--785
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 set[]mset[]add[]mset[]insert $\OT1/
cmr/m/n/10 [$\OT1/cmr/m/it/10 simp$\OT1/cmr/m/n/10 ]$$:$ []\OT1/cmr/m/it/10 set
[]mset $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 add[]mset a A$\OT1/cmr/m/n/10 )$ $=$
 \OT1/cmr/m/it/10 in-sert a $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 set[]mset
[12] [13] [14] [15]
Overfull \hbox (33.36261pt too wide) in paragraph at lines 1710--1715
[][]    \OT1/cmr/bx/n/10 more-over from []\OT1/cmr/m/it/10 ?lhs[] \OT1/cmr/bx/n
/10 have \OT1/cmr/m/it/10 M $\OT1/cmr/m/n/10 =$ \OT1/cmr/m/it/10 add[]mset b N 
$\OMS/cmsy/m/n/10 ^^@$ $f$$\OT1/cmr/m/n/10 #$\OT1/cmr/m/it/10 a$\OT1/cmr/m/n/10
 #$$\OMS/cmsy/m/n/10 g$ \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10
 rule union[]single[]eq[]diff$\OT1/cmr/m/n/10 )$[] 
[16] [17] [18] [19] [20] [21] [22]
Overfull \hbox (31.97931pt too wide) in paragraph at lines 2987--2995
[][]  \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 metis add[]mset[]
add[]single add[]mset[]diff[]bothsides diff[]subset[]eq[]self multiset[]inter[]
def[] 
[23] [24]
Overfull \hbox (20.85715pt too wide) in paragraph at lines 3518--3520
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 replicate[]mset[]Suc $\OT1/cmr/m/n/
10 [$\OT1/cmr/m/it/10 simp$\OT1/cmr/m/n/10 ]$$:$ \OT1/cmr/m/it/10 replicate[]ms
et $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 Suc n$\OT1/cmr/m/n/10 )$ \OT1/cmr/m/it/1
0 x $\OT1/cmr/m/n/10 =$ \OT1/cmr/m/it/10 add[]mset x $\OT1/cmr/m/n/10 ($\OT1/cm
r/m/it/10 replicate[]mset
[25] [26]
Overfull \hbox (51.96812pt too wide) in paragraph at lines 3798--3805
[][]  \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp add$\OT1/cmr
/m/n/10 :$ \OT1/cmr/m/it/10 iterate[]add[]eq[]add[]iff1 mset[]subseteq[]add[]if
f1$\OT1/cmr/m/n/10 [$\OT1/cmr/m/it/10 unfolded repeat[]mset[]iterate[]add$\OT1/
cmr/m/n/10 ]$$)$[] 

Overfull \hbox (51.96812pt too wide) in paragraph at lines 3819--3826
[][]  \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp add$\OT1/cmr
/m/n/10 :$ \OT1/cmr/m/it/10 iterate[]add[]eq[]add[]iff2 mset[]subseteq[]add[]if
f2$\OT1/cmr/m/n/10 [$\OT1/cmr/m/it/10 unfolded repeat[]mset[]iterate[]add$\OT1/
cmr/m/n/10 ]$$)$[] 
[27]
Overfull \hbox (1.90573pt too wide) in paragraph at lines 3975--3978
[][]  \OT1/cmr/bx/n/10 thus $\OT1/cmr/m/n/10 ($$\OML/cmm/m/it/10 ^^U$\OT1/cmr/m
/it/10 i$\OML/cmm/m/it/10 :$ \OT1/cmr/m/it/10 if A $\OT1/cmr/m/n/10 =$ $\OMS/cm
sy/m/n/10 f$$g$ \OT1/cmr/m/it/10 then 0 else INF f$\OT1/cmr/m/n/10 :$\OT1/cmr/m
/it/10 A$\OML/cmm/m/it/10 :$ \OT1/cmr/m/it/10 f i$\OT1/cmr/m/n/10 )$ $\OMS/cmsy
/m/n/10 2$ \OT1/cmr/m/it/10 mul-ti-set \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\
OT1/cmr/m/it/10 simp add$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 multiset[]def$\OT1
/cmr/m/n/10 )$[] 
[28] [29]
Overfull \hbox (19.41994pt too wide) in paragraph at lines 4233--4238
[][]  \OT1/cmr/bx/n/10 more-over from \OT1/cmr/m/it/10 assms \OT1/cmr/bx/n/10 h
ave \OT1/cmr/m/it/10 fi-nite $[] $ \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/
cmr/m/it/10 rule bdd[]above[]multiset[]imp[]finite[]support$\OT1/cmr/m/n/10 )$[
] 

Overfull \hbox (39.6184pt too wide) in paragraph at lines 4256--4270
[][]  \OT1/cmr/bx/n/10 us-ing \OT1/cmr/m/it/10 assms \OT1/cmr/bx/n/10 by $\OT1/
cmr/m/n/10 ($\OT1/cmr/m/it/10 simp add$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 Sup[
]multiset[]def Abs[]multiset[]inverse Sup[]multiset[]in[]multiset$\OT1/cmr/m/n/
10 )$[] 

Overfull \hbox (55.51436pt too wide) in paragraph at lines 4331--4336
[][]    \OT1/cmr/bx/n/10 also from \OT1/cmr/m/it/10 nonempty \OT1/cmr/bx/n/10 h
ave $[] $ $\OT1/cmr/m/n/10 =$ \OT1/cmr/m/it/10 count $\OT1/cmr/m/n/10 ($\OT1/cm
r/m/it/10 Inf A$\OT1/cmr/m/n/10 )$ \OT1/cmr/m/it/10 x \OT1/cmr/bx/n/10 by $\OT1
/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp add$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 cou
nt[]Inf[]multiset[]nonempty$\OT1/cmr/m/n/10 )$[] 
[30] [31]
Overfull \hbox (5.59465pt too wide) in paragraph at lines 4613--4617
[][]  \OT1/cmr/bx/n/10 from \OT1/cmr/m/it/10 assms \OT1/cmr/bx/n/10 have \OT1/c
mr/m/it/10 set[]mset $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 Sup A$\OT1/cmr/m/n/10 
)$ $=$ $($$[] $\OT1/cmr/m/it/10 X$\OMS/cmsy/m/n/10 2$\OT1/cmr/m/it/10 A$\OML/cm
m/m/it/10 :$ \OT1/cmr/m/it/10 set[]mset X$\OT1/cmr/m/n/10 )$ \OT1/cmr/bx/n/10 b
y $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 rule set[]mset[]Sup$\OT1/cmr/m/n/10 )$[] 

[32]
Overfull \hbox (7.00906pt too wide) in paragraph at lines 4821--4823
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 filter[]diff[]mset $\OT1/cmr/m/n/10
 [$\OT1/cmr/m/it/10 simp$\OT1/cmr/m/n/10 ]$$:$ \OT1/cmr/m/it/10 filter[]mset P 
$\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 M $\OMS/cmsy/m/n/10 ^^@$ \OT1/cmr/m/it/10 N
$\OT1/cmr/m/n/10 )$ $=$ \OT1/cmr/m/it/10 filter[]mset P M $\OMS/cmsy/m/n/10 ^^@
$ \OT1/cmr/m/it/10 filter[]mset
[33]
Overfull \hbox (11.80153pt too wide) in paragraph at lines 4857--4859
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 filter[]sup[]mset$\OT1/cmr/m/n/10 [
$\OT1/cmr/m/it/10 simp$\OT1/cmr/m/n/10 ]$$:$ \OT1/cmr/m/it/10 filter[]mset P $\
OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 A $\OMS/cmsy/m/n/10 [$$\OT1/cmr/m/n/10 #$ \OT
1/cmr/m/it/10 B$\OT1/cmr/m/n/10 )$ $=$ \OT1/cmr/m/it/10 filter[]mset P A $\OMS/
cmsy/m/n/10 [$$\OT1/cmr/m/n/10 #$ \OT1/cmr/m/it/10 filter[]mset
[34] [35]
Overfull \hbox (30.90695pt too wide) in paragraph at lines 5235--5243
[][]\OT1/cmr/bx/n/10 apply $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp add$\OT1/cm
r/m/n/10 :$ \OT1/cmr/m/it/10 size[]multiset[]def sum[]Un[]nat sum$\OML/cmm/m/it
/10 :$\OT1/cmr/m/it/10 distrib sum[]wcount[]Int wcount[]union$\OT1/cmr/m/n/10 )
$[] 

Overfull \hbox (55.98346pt too wide) in paragraph at lines 5260--5274
[][]  \OT1/cmr/bx/n/10 un-fold-ing \OT1/cmr/m/it/10 add[]mset[]add[]single$\OT1
/cmr/m/n/10 [$\OT1/cmr/m/it/10 of [] M$\OT1/cmr/m/n/10 ]$ \OT1/cmr/m/it/10 size
[]multiset[]union \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 auto 
simp$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 size[]multiset[]single$\OT1/cmr/m/n/10
 )$[] 
[36] [37] [38] [39] [40]
Overfull \hbox (44.37408pt too wide) in paragraph at lines 6145--6159
[][]  \OT1/cmr/bx/n/10 un-fold-ing \OT1/cmr/m/it/10 image[]mset[]union add[]mse
t[]add[]single$\OT1/cmr/m/n/10 [$\OT1/cmr/m/it/10 of a M$\OT1/cmr/m/n/10 ]$ \OT
1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp add$\OT1/cmr/m/n/10 :
$ \OT1/cmr/m/it/10 image[]mset[]single$\OT1/cmr/m/n/10 )$[] 
[41]
Overfull \hbox (9.85997pt too wide) in paragraph at lines 6333--6341
[][]  \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 metis $\OT1/cmr/m
/n/10 ($\OT1/cmr/m/it/10 no[]types$\OT1/cmr/m/n/10 )$ \OT1/cmr/m/it/10 Diff[]eq
[]empty[]iff[]mset image[]mset[]Diff image[]mset[]is[]empty[]iff[] 
[42] [43] [44] [45] [46] [47] [48]
Overfull \hbox (4.1574pt too wide) in paragraph at lines 7718--7731
[][]  \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 auto simp add$\OT
1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 mset[]subset[]eqI$\OT1/cmr/m/n/10 )$ $($\OT1/c
mr/m/it/10 metis count[]replicate[]mset subseteq[]mset[]def$\OT1/cmr/m/n/10 )$[
] 
[49] [50] [51] [52]
Overfull \hbox (14.65363pt too wide) in paragraph at lines 8409--8410
[][]      $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp[]all add$\OT1/cmr/m/n/10 :$ 
\OT1/cmr/m/it/10 size[]multiset[]overloaded[]eq not[]in[]iff sum$\OML/cmm/m/it/
10 :$\OT1/cmr/m/it/10 If[]cases Diff[]eq$\OT1/cmr/m/n/10 [$\OT1/cmr/m/it/10 sym
metric$\OT1/cmr/m/n/10 ]$[] 
[53]
Overfull \hbox (20.93939pt too wide) in paragraph at lines 8552--8565
[][]  \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 subst sum[]mset$\
OML/cmm/m/it/10 :$\OT1/cmr/m/it/10 commute$\OT1/cmr/m/n/10 )$ $($\OT1/cmr/m/it/
10 simp add$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 sum[]mset[]distrib[]left sum[]m
set[]distrib[]right$\OT1/cmr/m/n/10 )$[] 
[54] [55] [56] [57] [58]
Overfull \hbox (15.84276pt too wide) in paragraph at lines 9390--9393
[][]    \OT1/cmr/bx/n/10 un-fold-ing \OT1/cmr/m/it/10 filter[]sort \OT1/cmr/bx/
n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 rule properties[]for[]sort[]key$\OT
1/cmr/m/n/10 )$ $($\OT1/cmr/m/it/10 auto in-tro$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/i
t/10 sorted[]map[]same$\OT1/cmr/m/n/10 )$[] 
[59] [60] [61] [62] [63] [64] [65]
Overfull \hbox (23.14763pt too wide) in paragraph at lines 10339--10342
[][]      \OT1/cmr/bx/n/10 us-ing $\OMS/cmsy/m/n/10 ^^C$$\OT1/cmr/m/n/10 ($\OT1
/cmr/m/it/10 1$\OML/cmm/m/it/10 ;$\OT1/cmr/m/it/10 2$\OT1/cmr/m/n/10 )$ \OT1/cm
r/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 metis add[]mset[]remove[]trivi
al[]If insert[]iff set[]mset[]add[]mset[]insert
[66]
Overfull \hbox (18.23695pt too wide) in paragraph at lines 10471--10473
[][]    $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 M $\OMS/cmsy/m/n/10 ^^@$ \OT1/cmr/m
/it/10 M $\OMS/cmsy/m/n/10 \$$\OT1/cmr/m/n/10 #$ \OT1/cmr/m/it/10 N$\OT1/cmr/m/
n/10 )$ $\OMS/cmsy/m/n/10 \$$\OT1/cmr/m/n/10 #$ $($\OT1/cmr/m/it/10 N $\OMS/cms
y/m/n/10 ^^@$ \OT1/cmr/m/it/10 M $\OMS/cmsy/m/n/10 \$$\OT1/cmr/m/n/10 #$ \OT1/c
mr/m/it/10 N$\OT1/cmr/m/n/10 )$ $=$ $\OMS/cmsy/m/n/10 f$$\OT1/cmr/m/n/10 #$$\OM
S/cmsy/m/n/10 g$ \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 auto s
imp$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 count[]inject$\OT1/cmr/m/n/10 [$\OT1/cm
r/m/it/10 symmetric$\OT1/cmr/m/n/10 ]$$)$[] 

Overfull \hbox (24.56795pt too wide) in paragraph at lines 10531--10535
[][]    \OT1/cmr/bx/n/10 then ob-tain \OT1/cmr/m/it/10 y \OT1/cmr/bx/n/10 where
 \OT1/cmr/m/it/10 count N y $\OMS/cmsy/m/n/10 6\OT1/cmr/m/n/10 =$ \OT1/cmr/m/it
/10 count M y \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 auto simp
$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 count[]inject$\OT1/cmr/m/n/10 [$\OT1/cmr/m
/it/10 symmetric$\OT1/cmr/m/n/10 ]$$)$[] 
[67]
Overfull \hbox (0.73877pt too wide) in paragraph at lines 10629--10633
[][]    \OT1/cmr/bx/n/10 then have \OT1/cmr/m/it/10 MM$\OT1/cmr/m/n/10 :$ $($\O
T1/cmr/m/it/10 M$\OML/cmm/m/it/10 ;$ \OT1/cmr/m/it/10 M$\OT1/cmr/m/n/10 )$ $\OM
S/cmsy/m/n/10 2$ \OT1/cmr/m/it/10 mult $\OMS/cmsy/m/n/10 f$$\OT1/cmr/m/n/10 ($\
OT1/cmr/m/it/10 x$\OML/cmm/m/it/10 ;$ \OT1/cmr/m/it/10 y$\OT1/cmr/m/n/10 )$$\OM
L/cmm/m/it/10 :$ \OT1/cmr/m/it/10 x $\OML/cmm/m/it/10 <$ \OT1/cmr/m/it/10 y$\OM
S/cmsy/m/n/10 g$ \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp a
dd$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 less[]multiset[]def$\OT1/cmr/m/n/10 )$[]
 
[68] [69]
Overfull \hbox (55.81471pt too wide) in paragraph at lines 10887--10894
[][]\OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 rule one[]step[]imp
lies[]mult$\OT1/cmr/m/n/10 )$ $($\OT1/cmr/m/it/10 auto simp add$\OT1/cmr/m/n/10
 :$ \OT1/cmr/m/it/10 max[]strict[]def pair[]less[]def elim$\OT1/cmr/m/n/10 !$$:
$\OT1/cmr/m/it/10 max[]ext$\OML/cmm/m/it/10 :$\OT1/cmr/m/it/10 cases$\OT1/cmr/m
/n/10 )$[] 

Overfull \hbox (54.71396pt too wide) in paragraph at lines 10909--10916
[][]\OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 auto simp add$\OT1/
cmr/m/n/10 :$ \OT1/cmr/m/it/10 pair[]less[]def max[]strict[]def elim$\OT1/cmr/m
/n/10 !$$:$\OT1/cmr/m/it/10 max[]ext$\OML/cmm/m/it/10 :$\OT1/cmr/m/it/10 cases 
in-tro$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 one[]step[]implies[]mult$\OT1/cmr/m/
n/10 )$[] 
[70] [71]
Overfull \hbox (13.03835pt too wide) in paragraph at lines 11158--11159
[][]        $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 map $\OT1/cmr/m/n/10 ($\OT1/cmr
/m/it/10 fn t $\OT1/cmr/m/n/10 =$$\OML/cmm/m/it/10 >$ \OT1/cmr/m/it/10 t RS eq[
]reflection$\OT1/cmr/m/n/10 )$ $($$@$$\OMS/cmsy/m/n/10 f$\OT1/cmr/m/it/10 thms 
ac[]simps$\OMS/cmsy/m/n/10 g$ $\OT1/cmr/m/n/10 @$ $@$$\OMS/cmsy/m/n/10 f$\OT1/c
mr/m/it/10 thms empty[]neutral$\OMS/cmsy/m/n/10 g$$\OT1/cmr/m/n/10 )$$)$[] 
[72] [73] [74] [75] [76] [77]
Overfull \hbox (5.43213pt too wide) in paragraph at lines 12056--12057
[][]  \OT1/cmr/m/it/10 full[]exhaustive[]multiset f i $\OT1/cmr/m/n/10 =$ \OT1/
cmr/m/it/10 Quickcheck[]Exhaustive$\OML/cmm/m/it/10 :$\OT1/cmr/m/it/10 full[]ex
haustive $\OT1/cmr/m/n/10 ($$\OML/cmm/m/it/10 ^^U$\OT1/cmr/m/it/10 xs$\OML/cmm/
m/it/10 :$ \OT1/cmr/m/it/10 f $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 msetify
[78]
Overfull \hbox (21.37383pt too wide) in paragraph at lines 12215--12217
[][]    \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 metis append[]e
q[]append[]conv append[]take[]drop[]id diff[]Suc[]Suc Cons[]nth[]drop[]Suc

Overfull \hbox (17.54051pt too wide) in paragraph at lines 12222--12224
[][]    \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 metis add[]Suc[
]right append[]take[]drop[]id length[]Cons length[]append less[]eq[]Suc[]le

Overfull \hbox (26.54095pt too wide) in paragraph at lines 12229--12231
[][]    \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 metis add[]Suc[
]right append[]take[]drop[]id length[]Cons length[]append mset[]eq[]length$\OT1
/cmr/m/n/10 )$[] 
[79]
Overfull \hbox (19.33397pt too wide) in paragraph at lines 12350--12353
[][]    \OT1/cmr/bx/n/10 un-fold-ing \OT1/cmr/m/it/10 comp[]def \OT1/cmr/bx/n/1
0 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 rule ext$\OT1/cmr/m/n/10 )$ $($\OT1/cm
r/m/it/10 simp add$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 comp[]def image[]mset$\O
ML/cmm/m/it/10 :$\OT1/cmr/m/it/10 compositionality$\OT1/cmr/m/n/10 )$[] 
[80] [81] [82]
Overfull \hbox (6.28758pt too wide) in paragraph at lines 12638--12642
[][]    \OT1/cmr/bx/n/10 then ob-tain \OT1/cmr/m/it/10 N1 b \OT1/cmr/bx/n/10 wh
ere \OT1/cmr/m/it/10 N$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 N $\OT1/cmr/m/n/10 =
$ \OT1/cmr/m/it/10 add[]mset b N1 \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/c
mr/m/it/10 metis multi[]nonempty[]split$\OT1/cmr/m/n/10 )$[] 
[83] [84]
Overfull \hbox (23.3852pt too wide) in paragraph at lines 12982--12983
[][]  \OT1/cmr/m/it/10 BNF[]LFP[]Size$\OML/cmm/m/it/10 :$\OT1/cmr/m/it/10 regis
ter[]size[]global $\OT1/cmr/m/n/10 @$$\OMS/cmsy/m/n/10 f$\OT1/cmr/m/it/10 type[
]name mul-ti-set$\OMS/cmsy/m/n/10 g$ $\OT1/cmr/m/n/10 @$$\OMS/cmsy/m/n/10 f$\OT
1/cmr/m/it/10 const[]name size[]multiset$\OMS/cmsy/m/n/10 g$[] 
) (./Factorial_Ring.tex [85] [86] [87]
Overfull \hbox (9.44902pt too wide) in paragraph at lines 435--449
[][]  \OT1/cmr/bx/n/10 us-ing \OT1/cmr/m/it/10 unit[]imp[]no[]irreducible[]divi
sors$\OT1/cmr/m/n/10 [$\OT1/cmr/m/it/10 OF assms$\OT1/cmr/m/n/10 ($\OT1/cmr/m/i
t/10 1$\OT1/cmr/m/n/10 )$ \OT1/cmr/m/it/10 prime[]elem[]imp[]irreducible$\OT1/c
mr/m/n/10 [$\OT1/cmr/m/it/10 OF
[88] [89]
Overfull \hbox (26.81523pt too wide) in paragraph at lines 815--820
[][]  \OT1/cmr/bx/n/10 more-over from \OT1/cmr/m/it/10 assms \OT1/cmr/bx/n/10 h
ave $\OMS/cmsy/m/n/10 :$\OT1/cmr/m/it/10 is[]unit p \OT1/cmr/bx/n/10 by $\OT1/c
mr/m/n/10 ($\OT1/cmr/m/it/10 simp add$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 prime
[]elem[]def is[]unit[]power[]iff$\OT1/cmr/m/n/10 )$[] 

Overfull \hbox (10.79758pt too wide) in paragraph at lines 887--900
[][]  \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 auto simp$\OT1/cm
r/m/n/10 :$ \OT1/cmr/m/it/10 prime[]elem[]def mult$\OML/cmm/m/it/10 :$\OT1/cmr/
m/it/10 commute$\OT1/cmr/m/n/10 [$\OT1/cmr/m/it/10 of a$\OT1/cmr/m/n/10 ]$ \OT1
/cmr/m/it/10 is[]unit[]mult[]iff mult[]unit[]dvd[]iff$\OT1/cmr/m/n/10 )$[] 
[90]
Overfull \hbox (12.0489pt too wide) in paragraph at lines 955--957
[][]      \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 metis dvd[]mu
lt[]right dvd[]times[]left[]cancel[]iff mult$\OML/cmm/m/it/10 :$\OT1/cmr/m/it/1
0 left[]commute mult[]zero[]left$\OT1/cmr/m/n/10 )$[] 
[91]
Overfull \hbox (5.12607pt too wide) in paragraph at lines 1111--1125
[][]  \OT1/cmr/bx/n/10 us-ing \OT1/cmr/m/it/10 assms \OT1/cmr/bx/n/10 by $\OT1/
cmr/m/n/10 ($\OT1/cmr/m/it/10 induct n$\OT1/cmr/m/n/10 )$ $($\OT1/cmr/m/it/10 a
uto dest$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 prime[]elem[]not[]unit prime[]elem
[]dvd[]multD$\OT1/cmr/m/n/10 )$[] 
[92] [93] [94]
Overfull \hbox (11.69632pt too wide) in paragraph at lines 1749--1753
[][]  \OT1/cmr/bx/n/10 from \OT1/cmr/m/it/10 assms \OT1/cmr/bx/n/10 have \OT1/c
mr/m/it/10 ir-re-ducible q \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it
/10 simp add$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 prime[]elem[]imp[]irreducible 
prime[]def$\OT1/cmr/m/n/10 )$[] 
[95]
Overfull \hbox (30.63289pt too wide) in paragraph at lines 1904--1918
[][]  \OT1/cmr/bx/n/10 us-ing \OT1/cmr/m/it/10 assms \OT1/cmr/bx/n/10 by $\OT1/
cmr/m/n/10 ($\OT1/cmr/m/it/10 auto in-tro$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 p
rod[]mset[]subset[]imp[]dvd prod[]mset[]primes[]dvd[]imp[]subset$\OT1/cmr/m/n/1
0 )$[] 

Overfull \hbox (11.4095pt too wide) in paragraph at lines 1978--1980
[][]    \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 auto simp$\OT1/
cmr/m/n/10 :$ \OT1/cmr/m/it/10 A1[]def A2[]def Mul-ti-set$\OML/cmm/m/it/10 :$\O
T1/cmr/m/it/10 subset[]eq[]diff[]conv Mul-ti-set$\OML/cmm/m/it/10 :$\OT1/cmr/m/
it/10 union[]commute$\OT1/cmr/m/n/10 )$[] 
[96]
Overfull \hbox (1.77489pt too wide) in paragraph at lines 1993--1995
[][]    \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 subst $\OT1/cmr
/m/n/10 ($\OT1/cmr/m/it/10 asm$\OT1/cmr/m/n/10 )$ $($\OT1/cmr/m/it/10 1 2$\OT1/
cmr/m/n/10 )$ \OT1/cmr/m/it/10 is[]unit[]prod[]mset[]primes[]iff$\OT1/cmr/m/n/1
0 )$ $($\OT1/cmr/m/it/10 auto dest$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 Mul-ti-s
et$\OML/cmm/m/it/10 :$\OT1/cmr/m/it/10 in[]diffD$\OT1/cmr/m/n/10 )$[] 

Overfull \hbox (1.20242pt too wide) in paragraph at lines 2037--2041
[][]    \OT1/cmr/bx/n/10 from \OT1/cmr/m/it/10 x \OT1/cmr/bx/n/10 have \OT1/cmr
/m/it/10 x ^ n dvd prod[]mset A ^ n \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1
/cmr/m/it/10 intro dvd[]power[]same dvd[]prod[]mset$\OT1/cmr/m/n/10 )$[] 
[97]
Overfull \hbox (11.16814pt too wide) in paragraph at lines 2290--2290
[]\OT1/cmr/bx/n/12 Factorial semir-ings: al-ge-braic struc-tures with unique pr
ime
[98]
Overfull \hbox (6.13147pt too wide) in paragraph at lines 2337--2341
[][]      \OT1/cmr/bx/n/10 with []$\OMS/cmsy/m/n/10 :$\OT1/cmr/m/it/10 is[]unit
 a[] less$\OML/cmm/m/it/10 :$\OT1/cmr/m/it/10 prems \OT1/cmr/bx/n/10 have \OT1/
cmr/m/it/10 ir-re-ducible a \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/i
t/10 auto simp$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 irreducible[]altdef$\OT1/cmr
/m/n/10 )$[] 
[99]
Overfull \hbox (5.4238pt too wide) in paragraph at lines 2451--2455
[][]      \OT1/cmr/bx/n/10 from \OT1/cmr/m/it/10 A B \OT1/cmr/bx/n/10 show \OT1
/cmr/m/it/10 ?the-sis \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 i
ntro exI$\OT1/cmr/m/n/10 [$\OT1/cmr/m/it/10 of [] A $\OT1/cmr/m/n/10 +$ \OT1/cm
r/m/it/10 B$\OT1/cmr/m/n/10 ]$$)$ $($\OT1/cmr/m/it/10 auto simp$\OT1/cmr/m/n/10
 :$ \OT1/cmr/m/it/10 c normalize[]mult$\OT1/cmr/m/n/10 )$[] 
[100]
Overfull \hbox (7.38255pt too wide) in paragraph at lines 2584--2586
[][]    \OT1/cmr/bx/n/10 from \OT1/cmr/m/it/10 dvd prod[]mset[]primes[]irreduci
ble[]imp[]prime$\OT1/cmr/m/n/10 [$\OT1/cmr/m/it/10 of A B C$\OML/cmm/m/it/10 ;$
 \OT1/cmr/m/it/10 OF this ABC$\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 1$\OML/cmm/m/i
t/10 ;$\OT1/cmr/m/it/10 3$\OML/cmm/m/it/10 ;$\OT1/cmr/m/it/10 5$\OT1/cmr/m/n/10
 )$$]$

Overfull \hbox (6.55193pt too wide) in paragraph at lines 2685--2690
[][]    \OT1/cmr/bx/n/10 fi-nally show \OT1/cmr/m/it/10 p $\OMS/cmsy/m/n/10 2$$
\OT1/cmr/m/n/10 #$ \OT1/cmr/m/it/10 A \OT1/cmr/bx/n/10 us-ing \OT1/cmr/m/it/10 
p A \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 subst $\OT1/cmr/m/n
/10 ($\OT1/cmr/m/it/10 asm$\OT1/cmr/m/n/10 )$ \OT1/cmr/m/it/10 prime[]dvd[]prod
[]mset[]primes[]iff$\OT1/cmr/m/n/10 )$[] 
[101] [102] [103]
Overfull \hbox (35.03365pt too wide) in paragraph at lines 3203--3217
[][]  \OT1/cmr/bx/n/10 us-ing \OT1/cmr/m/it/10 assms \OT1/cmr/bx/n/10 by $\OT1/
cmr/m/n/10 ($\OT1/cmr/m/it/10 subst prime[]elem[]multiplicity[]eq[]zero[]iff$\O
T1/cmr/m/n/10 )$ $($\OT1/cmr/m/it/10 auto dest$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it
/10 primes[]dvd[]imp[]eq$\OT1/cmr/m/n/10 )$
[104]
Overfull \hbox (31.33105pt too wide) in paragraph at lines 3329--3334
[][]    \OT1/cmr/bx/n/10 more-over from \OT1/cmr/m/it/10 assms$\OT1/cmr/m/n/10 
($\OT1/cmr/m/it/10 2$\OT1/cmr/m/n/10 )$ \OT1/cmr/m/it/10 xp \OT1/cmr/bx/n/10 ha
ve \OT1/cmr/m/it/10 mul-ti-plic-ity p x $\OML/cmm/m/it/10 <$ \OT1/cmr/m/it/10 S
uc n \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 intro multiplicity
[]lessI$\OT1/cmr/m/n/10 )$[] 
[105] [106]
Overfull \hbox (42.299pt too wide) in paragraph at lines 3591--3593
[][]    \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 subst mult$\OML
/cmm/m/it/10 :$\OT1/cmr/m/it/10 commute$\OT1/cmr/m/n/10 )$ $($\OT1/cmr/m/it/10 
simp add$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 mult[]unit[]dvd[]iff power[]mult[]
distrib is[]unit[]power[]iff$\OT1/cmr/m/n/10 )$[] 
[107]
Overfull \hbox (2.52225pt too wide) in paragraph at lines 3870--3872
[][]    \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 intro multiplic
ity[]geI $\OT1/cmr/m/n/10 )$ $($\OT1/cmr/m/it/10 auto in-tro$\OT1/cmr/m/n/10 :$
 \OT1/cmr/m/it/10 dvd[]trans$\OT1/cmr/m/n/10 [$\OT1/cmr/m/it/10 OF multiplicity
[]dvd$ [] $ assms$\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 1$\OT1/cmr/m/n/10 )$$]$$)$
[] 
[108]
Overfull \hbox (4.71416pt too wide) in paragraph at lines 3931--3936
[][]      \OT1/cmr/bx/n/10 also from \OT1/cmr/m/it/10 x p \OT1/cmr/bx/n/10 have
 $[] $ $\OT1/cmr/m/n/10 =$ \OT1/cmr/m/it/10 0 $\OMS/cmsy/m/n/10  []!$ $:$\OT1/c
mr/m/it/10 p dvd x \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp
 add$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 multiplicity[]eq[]zero[]iff$\OT1/cmr/m
/n/10 )$[] 

Overfull \hbox (0.98625pt too wide) in paragraph at lines 4066--4068
[][]  \OT1/cmr/bx/n/10 thus \OT1/cmr/m/it/10 count $\OT1/cmr/m/n/10 ($\OT1/cmr/
m/it/10 prime[]factorization $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 p $\OMS/cmsy/m
/n/10 ^^C$ \OT1/cmr/m/it/10 x$\OT1/cmr/m/n/10 )$$)$ \OT1/cmr/m/it/10 q $\OT1/cm
r/m/n/10 =$ \OT1/cmr/m/it/10 count $\OT1/cmr/m/n/10 ($$\OMS/cmsy/m/n/10 f$$\OT1
/cmr/m/n/10 #$\OT1/cmr/m/it/10 p$\OT1/cmr/m/n/10 #$$\OMS/cmsy/m/n/10 g$ $\OT1/c
mr/m/n/10 +$ \OT1/cmr/m/it/10 prime[]factorization
[109] [110]
Overfull \hbox (15.81303pt too wide) in paragraph at lines 4304--4306
[][]    \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp[]all add$\
OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 mult[]ac prime[]factorization[]times[]prime 
Mul-ti-set$\OML/cmm/m/it/10 :$\OT1/cmr/m/it/10 union[]commute$\OT1/cmr/m/n/10 )
$[] 

Overfull \hbox (40.95099pt too wide) in paragraph at lines 4355--4359
[][]  \OT1/cmr/bx/n/10 with \OT1/cmr/m/it/10 assms \OT1/cmr/bx/n/10 show \OT1/c
mr/m/it/10 nor-mal-ize x $\OT1/cmr/m/n/10 =$ \OT1/cmr/m/it/10 nor-mal-ize y \OT
1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp add$\OT1/cmr/m/n/10 :
$ \OT1/cmr/m/it/10 prod[]mset[]prime[]factorization$\OT1/cmr/m/n/10 )$[] 
[111]
Overfull \hbox (44.55727pt too wide) in paragraph at lines 4390--4392
[][]    \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 rule prime[]fac
torization[]prod[]mset[]primes$\OT1/cmr/m/n/10 )$ $($\OT1/cmr/m/it/10 simp[]all
 add$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 in[]prime[]factors[]imp[]prime$\OT1/cm
r/m/n/10 )$[] 

Overfull \hbox (20.85785pt too wide) in paragraph at lines 4392--4395
[][]  \OT1/cmr/bx/n/10 also have \OT1/cmr/m/it/10 prime[]factorization $\OT1/cm
r/m/n/10 ($\OT1/cmr/m/it/10 prod[]mset $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 prim
e[]factorization x $\OT1/cmr/m/n/10 +$ \OT1/cmr/m/it/10 prime[]factorization

Overfull \hbox (33.42188pt too wide) in paragraph at lines 4396--4398
[][]    \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 rule prime[]fac
torization[]prod[]mset[]primes$\OT1/cmr/m/n/10 )$ $($\OT1/cmr/m/it/10 auto simp
$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 in[]prime[]factors[]imp[]prime$\OT1/cmr/m/
n/10 )$[] 

Overfull \hbox (25.28094pt too wide) in paragraph at lines 4520--4521
[][]                   $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp[]all add$\OT1/c
mr/m/n/10 :$ \OT1/cmr/m/it/10 prime[]elem[]multiplicity[]prod[]mset[]distrib su
m[]unfold[]sum[]mset
[112]
Overfull \hbox (16.5629pt too wide) in paragraph at lines 4545--4558
[][]  \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 subst prime[]elem
[]multiplicity[]power[]distrib$\OT1/cmr/m/n/10 )$ $($\OT1/cmr/m/it/10 auto simp
$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 prime[]multiplicity[]other$\OT1/cmr/m/n/10
 )$[] 

Overfull \hbox (50.5364pt too wide) in paragraph at lines 4572--4578
[][]     $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp[]all add$\OT1/cmr/m/n/10 :$ \
OT1/cmr/m/it/10 prime[]factorization[]mult prime[]factorization[]prime Mul-ti-s
et$\OML/cmm/m/it/10 :$\OT1/cmr/m/it/10 union[]commute$\OT1/cmr/m/n/10 )$[] 

Overfull \hbox (22.83113pt too wide) in paragraph at lines 4611--4613
[][]  \OT1/cmr/bx/n/10 have \OT1/cmr/m/it/10 x dvd y $\OMS/cmsy/m/n/10  []!$ \O
T1/cmr/m/it/10 prod[]mset $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 prime[]factorizat
ion x$\OT1/cmr/m/n/10 )$ \OT1/cmr/m/it/10 dvd prod[]mset $\OT1/cmr/m/n/10 ($\OT
1/cmr/m/it/10 prime[]factorization

Overfull \hbox (6.21616pt too wide) in paragraph at lines 4656--4657
[][]  \OT1/cmr/bx/n/10 shows   \OT1/cmr/m/it/10 prime[]factorization $\OT1/cmr/
m/n/10 ($\OT1/cmr/m/it/10 a div b$\OT1/cmr/m/n/10 )$ $=$ \OT1/cmr/m/it/10 prime
[]factorization a $\OMS/cmsy/m/n/10 ^^@$ \OT1/cmr/m/it/10 prime[]factorization
[113]
Overfull \hbox (12.16023pt too wide) in paragraph at lines 4752--4753
[][]      $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp[]all add$\OT1/cmr/m/n/10 :$ 
\OT1/cmr/m/it/10 assms count[]prime[]factorization[]prime in[]prime[]factors[]i
mp[]prime$\OT1/cmr/m/n/10 )$[] 

Overfull \hbox (24.46587pt too wide) in paragraph at lines 4794--4797
[][]    \OT1/cmr/bx/n/10 un-fold-ing \OT1/cmr/m/it/10 A[]def \OT1/cmr/bx/n/10 b
y $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 subst mul-ti-set$\OML/cmm/m/it/10 :$\OT1/
cmr/m/it/10 Abs[]multiset[]inverse$\OT1/cmr/m/n/10 )$ $($\OT1/cmr/m/it/10 simp[
]all add$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 multiset[]def$\OT1/cmr/m/n/10 )$[]
 

Overfull \hbox (1.20328pt too wide) in paragraph at lines 4806--4810
[][]  \OT1/cmr/bx/n/10 also have $[] $ $\OT1/cmr/m/n/10 =$ \OT1/cmr/m/it/10 pro
d[]mset A \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp add$\OT1
/cmr/m/n/10 :$ \OT1/cmr/m/it/10 prod[]mset[]multiplicity S[]eq set[]mset[]A

Overfull \hbox (29.18195pt too wide) in paragraph at lines 4827--4830
[][]  \OT1/cmr/bx/n/10 also have \OT1/cmr/m/it/10 prime[]factorization $\OT1/cm
r/m/n/10 ($\OT1/cmr/m/it/10 prod[]mset $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 prim
e[]factorization n$\OT1/cmr/m/n/10 )$$)$ $=$ \OT1/cmr/m/it/10 prime[]factorizat
ion
[114]
Overfull \hbox (56.44945pt too wide) in paragraph at lines 4864--4866
[][]      \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 intro image[]
mset[]cong$\OT1/cmr/m/n/10 )$ $($\OT1/cmr/m/it/10 auto simp$\OT1/cmr/m/n/10 :$ 
\OT1/cmr/m/it/10 set[]mset[]A multiplicity[]self prime[]multiplicity[]other$\OT
1/cmr/m/n/10 )$[] 

Overfull \hbox (5.04556pt too wide) in paragraph at lines 4944--4948
[][]  \OT1/cmr/bx/n/10 with \OT1/cmr/m/it/10 assms False \OT1/cmr/bx/n/10 show 
\OT1/cmr/m/it/10 ?the-sis \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/
10 subst $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 asm$\OT1/cmr/m/n/10 )$ \OT1/cmr/m/
it/10 prime[]factorization[]subset[]iff[]dvd$\OT1/cmr/m/n/10 )$[] 
[115]
Overfull \hbox (1.69742pt too wide) in paragraph at lines 5127--5128
[][]          \OT1/cmr/m/it/10 prime[]factorization $\OT1/cmr/m/n/10 ($\OT1/cmr
/m/it/10 prod[]mset $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 prime[]factorization a 
$\OMS/cmsy/m/n/10 \$$\OT1/cmr/m/n/10 #$ \OT1/cmr/m/it/10 prime[]factorization
[116]
Overfull \hbox (1.69742pt too wide) in paragraph at lines 5163--5164
[][]          \OT1/cmr/m/it/10 prime[]factorization $\OT1/cmr/m/n/10 ($\OT1/cmr
/m/it/10 prod[]mset $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 prime[]factorization a 
$\OMS/cmsy/m/n/10 [$$\OT1/cmr/m/n/10 #$ \OT1/cmr/m/it/10 prime[]factorization
[117]
Overfull \hbox (4.25296pt too wide) in paragraph at lines 5397--5398
[][]           \OT1/cmr/m/it/10 prime[]factorization $\OT1/cmr/m/n/10 ($\OT1/cm
r/m/it/10 prod[]mset $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 prime[]factorization a
 $\OMS/cmsy/m/n/10 \$$\OT1/cmr/m/n/10 #$ \OT1/cmr/m/it/10 prime[]factorization
[118]
Overfull \hbox (3.08865pt too wide) in paragraph at lines 5538--5541
[][]  \OT1/cmr/bx/n/10 with \OT1/cmr/m/it/10 assms \OT1/cmr/bx/n/10 have \OT1/c
mr/m/it/10 prime[]factorization $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 Gcd[]factor
ial A$\OT1/cmr/m/n/10 )$ $=$ \OT1/cmr/m/it/10 Inf $\OT1/cmr/m/n/10 ($\OT1/cmr/m
/it/10 prime[]factorization
[119] [120]
Overfull \hbox (14.46059pt too wide) in paragraph at lines 5797--5798
[][]  \OT1/cmr/bx/n/10 shows   \OT1/cmr/m/it/10 prime[]factorization $\OT1/cmr/
m/n/10 ($\OT1/cmr/m/it/10 gcd a b$\OT1/cmr/m/n/10 )$ $=$ \OT1/cmr/m/it/10 prime
[]factorization a $\OMS/cmsy/m/n/10 \$$\OT1/cmr/m/n/10 #$ \OT1/cmr/m/it/10 prim
e[]factorization

Overfull \hbox (15.99394pt too wide) in paragraph at lines 5817--5818
[][]  \OT1/cmr/bx/n/10 shows   \OT1/cmr/m/it/10 prime[]factorization $\OT1/cmr/
m/n/10 ($\OT1/cmr/m/it/10 lcm a b$\OT1/cmr/m/n/10 )$ $=$ \OT1/cmr/m/it/10 prime
[]factorization a $\OMS/cmsy/m/n/10 [$$\OT1/cmr/m/n/10 #$ \OT1/cmr/m/it/10 prim
e[]factorization

Overfull \hbox (43.35431pt too wide) in paragraph at lines 5846--5853
[][]  \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp add$\OT1/cmr
/m/n/10 :$ \OT1/cmr/m/it/10 prime[]factorization[]Gcd[]factorial Gcd[]eq[]Gcd[]
factorial Gcd[]factorial[]eq[]0[]iff$\OT1/cmr/m/n/10 )$[] 
[121]
Overfull \hbox (49.76532pt too wide) in paragraph at lines 5868--5875
[][]  \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp add$\OT1/cmr
/m/n/10 :$ \OT1/cmr/m/it/10 prime[]factorization[]Lcm[]factorial Lcm[]eq[]Lcm[]
factorial Lcm[]factorial[]eq[]0[]iff$\OT1/cmr/m/n/10 )$[] 

Overfull \hbox (55.76273pt too wide) in paragraph at lines 5989--5991
[][]    \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp add$\OT1/c
mr/m/n/10 :$ \OT1/cmr/m/it/10 count[]prime[]factorization[]prime $\OT1/cmr/m/n/
10 [$\OT1/cmr/m/it/10 symmetric$\OT1/cmr/m/n/10 ]$ \OT1/cmr/m/it/10 prime[]fact
orization[]gcd[]factorial$\OT1/cmr/m/n/10 )$[] 
[122]
Overfull \hbox (57.29608pt too wide) in paragraph at lines 6002--6004
[][]    \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp add$\OT1/c
mr/m/n/10 :$ \OT1/cmr/m/it/10 count[]prime[]factorization[]prime $\OT1/cmr/m/n/
10 [$\OT1/cmr/m/it/10 symmetric$\OT1/cmr/m/n/10 ]$ \OT1/cmr/m/it/10 prime[]fact
orization[]lcm[]factorial$\OT1/cmr/m/n/10 )$[] 
[123]) (./Euclidean_Algorithm.tex
Overfull \hbox (36.91122pt too wide) in paragraph at lines 6--6
[]\OT1/cmr/bx/n/14.4 Abstract eu-clidean al-go-rithm in eu-clidean (semi)rings 

[124] [125] [126] [127] [128] [129] [130] [131]
Overfull \hbox (36.0064pt too wide) in paragraph at lines 1171--1172
[][]  \OT1/cmr/m/it/10 OF-CLASS$\OT1/cmr/m/n/10 ($$ [] $\OT1/cmr/m/it/10 a$\OT1
/cmr/m/n/10 :$$:$$\OMS/cmsy/m/n/10 f$\OT1/cmr/m/it/10 factorial[]semiring[]gcd$
\OML/cmm/m/it/10 ;$ \OT1/cmr/m/it/10 euclidean[]semiring$\OMS/cmsy/m/n/10 g$$\O
ML/cmm/m/it/10 ;$ \OT1/cmr/m/it/10 euclidean[]semiring[]gcd[]class$\OT1/cmr/m/n
/10 )$[] 

Overfull \hbox (2.401pt too wide) in paragraph at lines 1231--1233
[][]    \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp add$\OT1/c
mr/m/n/10 :$ \OT1/cmr/m/it/10 fun[]eq[]iff Euclidean[]Algorithm$\OML/cmm/m/it/1
0 :$\OT1/cmr/m/it/10 lcm[]def semiring[]gcd[]class$\OML/cmm/m/it/10 :$\OT1/cmr/
m/it/10 lcm[]gcd$\OT1/cmr/m/n/10 )$[] 

Overfull \hbox (13.21527pt too wide) in paragraph at lines 1235--1237
[][]    \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp add$\OT1/c
mr/m/n/10 :$ \OT1/cmr/m/it/10 fun[]eq[]iff Euclidean[]Algorithm$\OML/cmm/m/it/1
0 :$\OT1/cmr/m/it/10 Gcd[]def semiring[]Gcd[]class$\OML/cmm/m/it/10 :$\OT1/cmr/
m/it/10 Gcd[]Lcm$\OT1/cmr/m/n/10 )$[] 
[132] [133]
Overfull \hbox (4.18866pt too wide) in paragraph at lines 1454--1456
[][]      \OT1/cmr/bx/n/10 qed $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 auto simp$\O
T1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 gcd[]mod[]right algebra[]simps minus[]mod[]eq
[]div[]mult $\OT1/cmr/m/n/10 [$\OT1/cmr/m/it/10 symmetric$\OT1/cmr/m/n/10 ]$
[134]
Overfull \hbox (2.401pt too wide) in paragraph at lines 1668--1670
[][]    \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp add$\OT1/c
mr/m/n/10 :$ \OT1/cmr/m/it/10 fun[]eq[]iff Euclidean[]Algorithm$\OML/cmm/m/it/1
0 :$\OT1/cmr/m/it/10 lcm[]def semiring[]gcd[]class$\OML/cmm/m/it/10 :$\OT1/cmr/
m/it/10 lcm[]gcd$\OT1/cmr/m/n/10 )$[] 

Overfull \hbox (13.21527pt too wide) in paragraph at lines 1672--1674
[][]    \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp add$\OT1/c
mr/m/n/10 :$ \OT1/cmr/m/it/10 fun[]eq[]iff Euclidean[]Algorithm$\OML/cmm/m/it/1
0 :$\OT1/cmr/m/it/10 Gcd[]def semiring[]Gcd[]class$\OML/cmm/m/it/10 :$\OT1/cmr/
m/it/10 Gcd[]Lcm$\OT1/cmr/m/n/10 )$[] 
[135]
Overfull \hbox (2.401pt too wide) in paragraph at lines 1771--1773
[][]    \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp add$\OT1/c
mr/m/n/10 :$ \OT1/cmr/m/it/10 fun[]eq[]iff Euclidean[]Algorithm$\OML/cmm/m/it/1
0 :$\OT1/cmr/m/it/10 lcm[]def semiring[]gcd[]class$\OML/cmm/m/it/10 :$\OT1/cmr/
m/it/10 lcm[]gcd$\OT1/cmr/m/n/10 )$[] 

Overfull \hbox (13.21527pt too wide) in paragraph at lines 1775--1777
[][]    \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp add$\OT1/c
mr/m/n/10 :$ \OT1/cmr/m/it/10 fun[]eq[]iff Euclidean[]Algorithm$\OML/cmm/m/it/1
0 :$\OT1/cmr/m/it/10 Gcd[]def semiring[]Gcd[]class$\OML/cmm/m/it/10 :$\OT1/cmr/
m/it/10 Gcd[]Lcm$\OT1/cmr/m/n/10 )$[] 
) (./Ring_Hom.tex [136] [137] [138]) (./Fraction_Field.tex [139] [140] [141]
[142] [143] [144] [145] [146] [147] [148]) (./Normalized_Fraction.tex [149]
Overfull \hbox (25.4372pt too wide) in paragraph at lines 205--207
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 normalize[]quot[]in[]normalized[]fr
acts $\OT1/cmr/m/n/10 [$\OT1/cmr/m/it/10 simp$\OT1/cmr/m/n/10 ]$$:$ \OT1/cmr/m/
it/10 normalize[]quot x $\OMS/cmsy/m/n/10 2$ \OT1/cmr/m/it/10 normalized[]fract
s[] 
[150]
Overfull \hbox (11.43802pt too wide) in paragraph at lines 251--253
[][]    \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 intro coprime[]
crossproduct$ [] $$\OT1/cmr/m/n/10 )$ $($\OT1/cmr/m/it/10 simp[]all add$\OT1/cm
r/m/n/10 :$ \OT1/cmr/m/it/10 x[]def y[]def coprime[]normalize[]quot$\OT1/cmr/m/
n/10 )$[] 

Overfull \hbox (1.46599pt too wide) in paragraph at lines 276--290
[][]  \OT1/cmr/bx/n/10 us-ing \OT1/cmr/m/it/10 assms \OT1/cmr/bx/n/10 by $\OT1/
cmr/m/n/10 ($\OT1/cmr/m/it/10 cases x$\OML/cmm/m/it/10 ;$ \OT1/cmr/m/it/10 case
s y$\OML/cmm/m/it/10 ;$ \OT1/cmr/m/it/10 hyp-subst$\OT1/cmr/m/n/10 )$ $($\OT1/c
mr/m/it/10 subst normalize[]quot[]eq[]iff$\OML/cmm/m/it/10 ;$ \OT1/cmr/m/it/10 
simp[]all$\OT1/cmr/m/n/10 )$[] 

Overfull \hbox (26.27673pt too wide) in paragraph at lines 311--313
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 normalize[]quot[]idem $\OT1/cmr/m/n
/10 [$\OT1/cmr/m/it/10 simp$\OT1/cmr/m/n/10 ]$$:$ \OT1/cmr/m/it/10 normalize[]q
uot $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 normalize[]quot x$\OT1/cmr/m/n/10 )$ $=
$ \OT1/cmr/m/it/10 normalize[]quot
[151]
Overfull \hbox (2.21165pt too wide) in paragraph at lines 476--480
[][]  \OT1/cmr/bx/n/10 thus \OT1/cmr/m/it/10 ?the-sis \OT1/cmr/bx/n/10 un-fold-
ing \OT1/cmr/m/it/10 quot[]to[]fract[]def \OT1/cmr/bx/n/10 by \OT1/cmr/m/it/10 
trans-fer $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp add$\OT1/cmr/m/n/10 :$ \OT1/
cmr/m/it/10 case[]prod[]unfold$\OT1/cmr/m/n/10 )$[] 

Overfull \hbox (3.45944pt too wide) in paragraph at lines 510--512
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 quot[]of[]fract[]in[]normalized[]fr
acts $\OT1/cmr/m/n/10 [$\OT1/cmr/m/it/10 simp$\OT1/cmr/m/n/10 ]$$:$ \OT1/cmr/m/
it/10 quot[]of[]fract x $\OMS/cmsy/m/n/10 2$ \OT1/cmr/m/it/10 normalized[]fract
s[] 
[152]
Overfull \hbox (12.708pt too wide) in paragraph at lines 742--746
[][]  \OT1/cmr/bx/n/10 from \OT1/cmr/m/it/10 assms \OT1/cmr/bx/n/10 have \OT1/c
mr/m/it/10 co-prime e f co-prime g h \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT
1/cmr/m/it/10 simp[]all add$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 coprime[]normal
ize[]quot$\OT1/cmr/m/n/10 )$[] 

Overfull \hbox (21.90312pt too wide) in paragraph at lines 749--751
[][]    \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp add$\OT1/c
mr/m/n/10 :$ \OT1/cmr/m/it/10 normalized[]fracts[]def unit[]factor[]mult coprim
e[]mul[]eq coprime[]mul[]eq$ [] $$\OT1/cmr/m/n/10 )$[] 
[153]
Overfull \hbox (36.04117pt too wide) in paragraph at lines 842--855
[][]  \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 auto simp$\OT1/cm
r/m/n/10 :$ \OT1/cmr/m/it/10 normalize[]quot[]def case[]prod[]unfold Let[]def d
iv[]mult[]unit2 dvd[]div[]eq[]0[]iff$\OT1/cmr/m/n/10 )$[] 

Overfull \hbox (15.3068pt too wide) in paragraph at lines 907--910
[][]    \OT1/cmr/bx/n/10 us-ing \OT1/cmr/m/it/10 assms$\OT1/cmr/m/n/10 ($\OT1/c
mr/m/it/10 1$\OML/cmm/m/it/10 ;$\OT1/cmr/m/it/10 2$\OT1/cmr/m/n/10 )$ \OT1/cmr/
m/it/10 d \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 auto simp$\OT
1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 normalized[]fracts[]def gcd[]div[]unit1 gcd[]d
iv[]unit2
[154]
Overfull \hbox (49.1144pt too wide) in paragraph at lines 971--975
[][]  \OT1/cmr/bx/n/10 from \OT1/cmr/m/it/10 assms \OT1/cmr/bx/n/10 have \OT1/c
mr/m/it/10 co-prime x$ [] $ y$ [] $ unit[]factor y$ [] $ $\OT1/cmr/m/n/10 =$ \O
T1/cmr/m/it/10 1 \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp[]
all add$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 coprime[]normalize[]quot$\OT1/cmr/m
/n/10 )$[] 

Overfull \hbox (29.09537pt too wide) in paragraph at lines 980--981
[][]       $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 auto simp$\OT1/cmr/m/n/10 :$ \OT
1/cmr/m/it/10 normalized[]fracts[]def unit[]div[]mult[]swap unit[]div[]commute 
unit[]div[]cancel[] 

Overfull \hbox (49.1144pt too wide) in paragraph at lines 1013--1017
[][]  \OT1/cmr/bx/n/10 from \OT1/cmr/m/it/10 assms \OT1/cmr/bx/n/10 have \OT1/c
mr/m/it/10 co-prime x$ [] $ y$ [] $ unit[]factor y$ [] $ $\OT1/cmr/m/n/10 =$ \O
T1/cmr/m/it/10 1 \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp[]
all add$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 coprime[]normalize[]quot$\OT1/cmr/m
/n/10 )$[] 

Overfull \hbox (29.09537pt too wide) in paragraph at lines 1022--1023
[][]       $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 auto simp$\OT1/cmr/m/n/10 :$ \OT
1/cmr/m/it/10 normalized[]fracts[]def unit[]div[]mult[]swap unit[]div[]commute 
unit[]div[]cancel[] 
[155]) (./Field_as_Ring.tex [156]
Overfull \hbox (21.73355pt too wide) in paragraph at lines 116--132
[][]\OT1/cmr/bx/n/10 instance by \OT1/cmr/m/it/10 stan-dard $\OT1/cmr/m/n/10 ($
\OT1/cmr/m/it/10 simp[]all add$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 gcd[]real[]d
ef lcm[]real[]def Gcd[]real[]def Lcm[]real[]def$\OT1/cmr/m/n/10 )$[] 
[157]
Overfull \hbox (8.4448pt too wide) in paragraph at lines 190--206
[][]\OT1/cmr/bx/n/10 instance by \OT1/cmr/m/it/10 stan-dard $\OT1/cmr/m/n/10 ($
\OT1/cmr/m/it/10 simp[]all add$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 gcd[]rat[]de
f lcm[]rat[]def Gcd[]rat[]def Lcm[]rat[]def$\OT1/cmr/m/n/10 )$[] 

Overfull \hbox (15.7032pt too wide) in paragraph at lines 264--280
[][]\OT1/cmr/bx/n/10 instance by \OT1/cmr/m/it/10 stan-dard $\OT1/cmr/m/n/10 ($
\OT1/cmr/m/it/10 simp[]all add$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 gcd[]complex
[]def lcm[]complex[]def Gcd[]complex[]def
) (./Polynomial_Factorial.tex [158]
Overfull \hbox (2.32649pt too wide) in paragraph at lines 83--87
[][]    \OT1/cmr/bx/n/10 then ob-tain \OT1/cmr/m/it/10 a$ [] $ b$ [] $ \OT1/cmr
/bx/n/10 where \OT1/cmr/m/it/10 ab$ [] $$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 a 
$\OT1/cmr/m/n/10 =$ $[$$:$\OT1/cmr/m/it/10 a$ [] $$\OT1/cmr/m/n/10 :$$]$ \OT1/c
mr/m/it/10 b $\OT1/cmr/m/n/10 =$ $[$$:$\OT1/cmr/m/it/10 b$ [] $$\OT1/cmr/m/n/10
 :$$]$ \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 auto elim$\OT1/c
mr/m/n/10 !$$:$ \OT1/cmr/m/it/10 degree[]eq[]zeroE$\OT1/cmr/m/n/10 )$[] 
[159] [160]
Overfull \hbox (30.61302pt too wide) in paragraph at lines 511--513
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 normalize[]snd[]quot[]of[]fract$\OT
1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 nor-mal-ize $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/1
0 snd $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 quot[]of[]fract x$\OT1/cmr/m/n/10 )$$
)$ $=$ \OT1/cmr/m/it/10 snd $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 quot[]of[]fract


Overfull \hbox (3.69643pt too wide) in paragraph at lines 539--541
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 fract[]poly[]smult $\OT1/cmr/m/n/10
 [$\OT1/cmr/m/it/10 simp$\OT1/cmr/m/n/10 ]$$:$ \OT1/cmr/m/it/10 fract[]poly $\O
T1/cmr/m/n/10 ($\OT1/cmr/m/it/10 smult c p$\OT1/cmr/m/n/10 )$ $=$ \OT1/cmr/m/it
/10 smult $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 to[]fract c$\OT1/cmr/m/n/10 )$ $(
$\OT1/cmr/m/it/10 fract[]poly
[161]
Overfull \hbox (9.21042pt too wide) in paragraph at lines 725--726
[][]  \OT1/cmr/m/it/10 prod[]mset $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 image[]ms
et $\OT1/cmr/m/n/10 ($$\OML/cmm/m/it/10 ^^U$\OT1/cmr/m/it/10 x$\OML/cmm/m/it/10
 :$ \OT1/cmr/m/it/10 fract[]poly $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 f x$\OT1/c
mr/m/n/10 )$$)$ \OT1/cmr/m/it/10 A$\OT1/cmr/m/n/10 )$ $=$ \OT1/cmr/m/it/10 frac
t[]poly $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 prod[]mset $\OT1/cmr/m/n/10 ($\OT1/
cmr/m/it/10 image[]mset
[162]
Overfull \hbox (5.10414pt too wide) in paragraph at lines 850--853
[][]  \OT1/cmr/bx/n/10 hence \OT1/cmr/m/it/10 fract[]poly $\OT1/cmr/m/n/10 ($\O
T1/cmr/m/it/10 smult a q$\OT1/cmr/m/n/10 )$ $=$ \OT1/cmr/m/it/10 fract[]poly $\
OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 smult b p$\OT1/cmr/m/n/10 )$ \OT1/cmr/bx/n/10
 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp del$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/
it/10 fract[]poly[]eq[]iff$\OT1/cmr/m/n/10 )$[] 
[163]
Overfull \hbox (9.28085pt too wide) in paragraph at lines 1049--1051
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 fract[]content[]fract[]poly $\OT1/c
mr/m/n/10 [$\OT1/cmr/m/it/10 simp$\OT1/cmr/m/n/10 ]$$:$ \OT1/cmr/m/it/10 fract[
]content $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 fract[]poly p$\OT1/cmr/m/n/10 )$ $
=$ \OT1/cmr/m/it/10 to[]fract $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 content
[164]
Overfull \hbox (54.7458pt too wide) in paragraph at lines 1113--1115
[][]    \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 rule that$\OT1/
cmr/m/n/10 [$\OT1/cmr/m/it/10 OF content[]times[]primitive[]part[]fract $\OT1/c
mr/m/n/10 [$\OT1/cmr/m/it/10 symmetric$\OT1/cmr/m/n/10 ]$ \OT1/cmr/m/it/10 cont
ent[]primitive[]part[]fract$\OT1/cmr/m/n/10 ]$$)$[] 
[165]
Overfull \hbox (14.12436pt too wide) in paragraph at lines 1301--1305
[][]    \OT1/cmr/bx/n/10 from \OT1/cmr/m/it/10 A \OT1/cmr/bx/n/10 and \OT1/cmr/
m/it/10 this \OT1/cmr/bx/n/10 have $\OT1/cmr/m/n/10 [$$:$\OT1/cmr/m/it/10 c$\OT
1/cmr/m/n/10 :$$]$ \OT1/cmr/m/it/10 dvd $\OT1/cmr/m/n/10 [$$:$\OT1/cmr/m/it/10 
a$\OT1/cmr/m/n/10 :$$]$ $\OMS/cmsy/m/n/10 _$ $\OT1/cmr/m/n/10 [$$:$\OT1/cmr/m/i
t/10 c$\OT1/cmr/m/n/10 :$$]$ \OT1/cmr/m/it/10 dvd $\OT1/cmr/m/n/10 [$$:$\OT1/cm
r/m/it/10 b$\OT1/cmr/m/n/10 :$$]$ \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/c
mr/m/it/10 rule prime[]elem[]dvd[]multD$\OT1/cmr/m/n/10 )$[] 
[166]
Overfull \hbox (17.28046pt too wide) in paragraph at lines 1365--1370
[][]    \OT1/cmr/bx/n/10 more-over from []\OT1/cmr/m/it/10 prime p[] \OT1/cmr/b
x/n/10 have \OT1/cmr/m/it/10 prime[]elem $\OT1/cmr/m/n/10 [$$:$\OT1/cmr/m/it/10
 p$\OT1/cmr/m/n/10 :$$]$ \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/1
0 simp add$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 lift[]prime[]elem[]poly$\OT1/cmr
/m/n/10 )$[] 
[167] [168]
Overfull \hbox (8.21928pt too wide) in paragraph at lines 1744--1746
[][]    \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 elim is[]unit[]
polyE$\OT1/cmr/m/n/10 )$ $($\OT1/cmr/m/it/10 auto simp$\OT1/cmr/m/n/10 :$ \OT1/
cmr/m/it/10 unit[]factor[]field[]poly[]def monom[]0 one[]poly[]def
[169]
Overfull \hbox (10.56847pt too wide) in paragraph at lines 1781--1782
[][]      $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 auto simp add$\OT1/cmr/m/n/10 :$ 
\OT1/cmr/m/it/10 unit[]factor[]field[]poly[]def euclidean[]size[]field[]poly[]d
ef div[]poly[]less$\OT1/cmr/m/n/10 )$[] 

Overfull \hbox (6.67152pt too wide) in paragraph at lines 1842--1844
[][]    \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 intro ext$\OT1/
cmr/m/n/10 )$ $($\OT1/cmr/m/it/10 simp add$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 
comm[]monoid[]mult$\OML/cmm/m/it/10 :$\OT1/cmr/m/it/10 prod[]mset[]def$\OT1/cmr
/m/n/10 [$\OT1/cmr/m/it/10 OF A$\OT1/cmr/m/n/10 ]$ \OT1/cmr/m/it/10 prod[]mset[
]def$\OT1/cmr/m/n/10 )$[] 

Overfull \hbox (0.43723pt too wide) in paragraph at lines 1882--1884
[][]      \OT1/cmr/m/it/10 comm[]semiring[]1$\OML/cmm/m/it/10 :$\OT1/cmr/m/it/1
0 prime[]elem[]def$\OT1/cmr/m/n/10 [$\OT1/cmr/m/it/10 OF A$\OT1/cmr/m/n/10 ]$ \
OT1/cmr/m/it/10 normalization[]semidom$\OML/cmm/m/it/10 :$\OT1/cmr/m/it/10 prim
e[]def$\OT1/cmr/m/n/10 [$\OT1/cmr/m/it/10 OF
[170] [171]
Overfull \hbox (3.74106pt too wide) in paragraph at lines 2045--2048
[][]   \OT1/cmr/bx/n/10 hence \OT1/cmr/m/it/10 q$ [] $ dvd 1 $\OMS/cmsy/m/n/10 
_$ \OT1/cmr/m/it/10 r$ [] $ dvd 1 \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/c
mr/m/it/10 auto dest$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 dvd[]mult[]right simp 
del$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 mult[]pCons[]left$\OT1/cmr/m/n/10 )$[] 


Overfull \hbox (41.16667pt too wide) in paragraph at lines 2143--2147
[][]  \OT1/cmr/bx/n/10 from \OT1/cmr/m/it/10 ir-red \OT1/cmr/bx/n/10 have \OT1/
cmr/m/it/10 prime$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 prime[]elem $\OT1/cmr/m/n
/10 ($\OT1/cmr/m/it/10 fract[]poly p$\OT1/cmr/m/n/10 )$ \OT1/cmr/bx/n/10 by $\O
T1/cmr/m/n/10 ($\OT1/cmr/m/it/10 rule field[]poly[]irreducible[]imp[]prime$\OT1
/cmr/m/n/10 )$[] 
[172] [173] [174]
Overfull \hbox (2.86989pt too wide) in paragraph at lines 2444--2446
[][]    \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp add$\OT1/c
mr/m/n/10 :$ \OT1/cmr/m/it/10 unit[]factor[]field[]poly[]def monom[]0 degree[]m
ap[]poly coeff[]map[]poly$\OT1/cmr/m/n/10 )$[] 

Overfull \hbox (14.1927pt too wide) in paragraph at lines 2465--2467
[][]    \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 subst prod[]mse
t$\OML/cmm/m/it/10 :$\OT1/cmr/m/it/10 distrib$\OT1/cmr/m/n/10 )$ $($\OT1/cmr/m/
it/10 simp[]all add$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 prod[]mset[]fract[]poly
 prod[]mset[]const[]poly

Overfull \hbox (47.79518pt too wide) in paragraph at lines 2535--2538
[][]    \OT1/cmr/bx/n/10 us-ing \OT1/cmr/m/it/10 that \OT1/cmr/bx/n/10 by $\OT1
/cmr/m/n/10 ($\OT1/cmr/m/it/10 auto simp$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 A[
]def prime[]elem[]primitive[]part[]fract prime[]elem[]imp[]irreducible
[175]
Overfull \hbox (14.65166pt too wide) in paragraph at lines 2569--2571
[][]  \OT1/cmr/bx/n/10 have $\OMS/cmsy/m/n/10 9 $\OT1/cmr/m/it/10 A$\OML/cmm/m/
it/10 :$ $\OT1/cmr/m/n/10 ($$\OMS/cmsy/m/n/10 8 $\OT1/cmr/m/it/10 p$\OML/cmm/m/
it/10 :$ \OT1/cmr/m/it/10 p $\OMS/cmsy/m/n/10 2$$\OT1/cmr/m/n/10 #$ \OT1/cmr/m/
it/10 A $[][]\OMS/cmsy/m/n/10 !$ \OT1/cmr/m/it/10 prime[]elem p$\OT1/cmr/m/n/10
 )$ $\OMS/cmsy/m/n/10 ^$ \OT1/cmr/m/it/10 prod[]mset A $\OT1/cmr/m/n/10 =$ \OT1
/cmr/m/it/10 nor-mal-ize $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 primitive[]part

Overfull \hbox (7.14774pt too wide) in paragraph at lines 2587--2589
[][]    \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 auto simp$\OT1/
cmr/m/n/10 :$ \OT1/cmr/m/it/10 B[]def in-tro$\OT1/cmr/m/n/10 !$$:$ \OT1/cmr/m/i
t/10 lift[]prime[]elem[]poly dest$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 in[]prime
[]factors[]imp[]prime$\OT1/cmr/m/n/10 )$[] 

Overfull \hbox (28.37808pt too wide) in paragraph at lines 2646--2662
[][]\OT1/cmr/bx/n/10 instance by \OT1/cmr/m/it/10 stan-dard $\OT1/cmr/m/n/10 ($
\OT1/cmr/m/it/10 simp[]all add$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 gcd[]poly[]d
ef lcm[]poly[]def Gcd[]poly[]def Lcm[]poly[]def$\OT1/cmr/m/n/10 )$[] 
[176]
Overfull \hbox (15.0717pt too wide) in paragraph at lines 2688--2689
[][]   $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 auto simp$\OT1/cmr/m/n/10 :$ \OT1/cm
r/m/it/10 euclidean[]size[]poly[]def Rings$\OML/cmm/m/it/10 :$\OT1/cmr/m/it/10 
div[]mult[]mod[]eq div[]poly[]less degree[]mult[]eq

Overfull \hbox (10.64919pt too wide) in paragraph at lines 2763--2765
[][]  \OT1/cmr/bx/n/10 thus \OT1/cmr/m/it/10 d dvd smult $\OT1/cmr/m/n/10 ($\OT
1/cmr/m/it/10 gcd $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 content p$\OT1/cmr/m/n/10
 )$ $($\OT1/cmr/m/it/10 content q$\OT1/cmr/m/n/10 )$$)$ $($\OT1/cmr/m/it/10 gcd
 $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 primitive[]part p$\OT1/cmr/m/n/10 )$ $($\O
T1/cmr/m/it/10 primitive[]part
[177] [178]
Overfull \hbox (35.85698pt too wide) in paragraph at lines 2991--3004
[][]  \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp add$\OT1/cmr
/m/n/10 :$ \OT1/cmr/m/it/10 gcd[]poly[]code[]def gcd[]poly[]code[]aux[]correct 
gcd[]poly[]decompose $\OT1/cmr/m/n/10 [$\OT1/cmr/m/it/10 symmetric$\OT1/cmr/m/n
/10 ]$$)$[] 

Overfull \hbox (31.4pt too wide) in paragraph at lines 3027--3029
[][]\OT1/cmr/bx/n/10 lemmas \OT1/cmr/m/it/10 Gcd[]poly[]set[]eq[]fold $\OT1/cmr
/m/n/10 [$\OT1/cmr/m/it/10 code$\OT1/cmr/m/n/10 ]$ $=$ \OT1/cmr/m/it/10 Gcd[]se
t[]eq[]fold $\OT1/cmr/m/n/10 [$\OT1/cmr/bx/n/10 where \OT1/cmr/m/it/10 ?$ [] $a
 $\OT1/cmr/m/n/10 =$ $ [] $\OT1/cmr/m/it/10 a $\OT1/cmr/m/n/10 :$$:$ \OT1/cmr/m
/it/10 factorial[]ring[]gcd

Overfull \hbox (34.6055pt too wide) in paragraph at lines 3029--3032
[][]\OT1/cmr/bx/n/10 lemmas \OT1/cmr/m/it/10 Lcm[]poly[]set[]eq[]fold $\OT1/cmr
/m/n/10 [$\OT1/cmr/m/it/10 code$\OT1/cmr/m/n/10 ]$ $=$ \OT1/cmr/m/it/10 Lcm[]se
t[]eq[]fold $\OT1/cmr/m/n/10 [$\OT1/cmr/bx/n/10 where \OT1/cmr/m/it/10 ?$ [] $a
 $\OT1/cmr/m/n/10 =$ $ [] $\OT1/cmr/m/it/10 a $\OT1/cmr/m/n/10 :$$:$ \OT1/cmr/m
/it/10 factorial[]ring[]gcd
) (./Missing_Unsorted.tex [179]
Overfull \hbox (22.26895pt too wide) in paragraph at lines 177--182
[][]  \OT1/cmr/bx/n/10 also have $[] $ $\OML/cmm/m/it/10 <$ \OT1/cmr/m/it/10 bb
b $\OMS/cmsy/m/n/10 ^^C$ \OT1/cmr/m/it/10 of[]nat n \OT1/cmr/bx/n/10 us-ing \OT
1/cmr/m/it/10 lt bbb \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 me
tis mult$\OML/cmm/m/it/10 :$\OT1/cmr/m/it/10 commute pos[]divide[]less[]eq$\OT1
/cmr/m/n/10 )$[] 
[180] [181] [182] [183] [184] [185] [186] [187]) (./Missing_Polynomial.tex
[188]
Overfull \hbox (19.45976pt too wide) in paragraph at lines 104--117
[][]  \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 metis Suc[]1 Suc[
]neq[]Zero degree1[]coeffs degree[]pCons[]eq[]if nat$\OML/cmm/m/it/10 :$\OT1/cm
r/m/it/10 inject pCons[]cases$\OT1/cmr/m/n/10 )$[] 
[189] [190] [191] [192] [193] [194] [195]
Overfull \hbox (26.31035pt too wide) in paragraph at lines 1194--1196
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 degree[]prod[]sum[]lt[]n$\OT1/cmr/m
/n/10 :$ \OT1/cmr/bx/n/10 as-sumes $[]$ \OT1/cmr/m/it/10 i$\OML/cmm/m/it/10 :$ 
\OT1/cmr/m/it/10 i $\OML/cmm/m/it/10 <$ \OT1/cmr/m/it/10 n $\OT1/cmr/m/n/10 =[]
\OMS/cmsy/m/n/10 )$ \OT1/cmr/m/it/10 de-gree $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/1
0 f i $\OT1/cmr/m/n/10 :$$:$ $ [] $\OT1/cmr/m/it/10 a $\OT1/cmr/m/n/10 :$$:$ \O
T1/cmr/m/it/10 comm[]semiring[]1
[196] [197] [198] [199] [200] [201] [202] [203] [204] [205] [206]

Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `math shift' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `math shift' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `math shift' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\mathord' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `math shift' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `math shift' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\mathord' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `math shift' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `math shift' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\mskip' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\mskip' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `math shift' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `math shift' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `math shift' on input line 2890.

[207] [208]
Overfull \hbox (46.84029pt too wide) in paragraph at lines 3269--3271
[][]  \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 metis $\OT1/cmr/m
/n/10 ($\OT1/cmr/m/it/10 no[]types$\OML/cmm/m/it/10 ;$ \OT1/cmr/m/it/10 lift-in
g$\OT1/cmr/m/n/10 )$ \OT1/cmr/m/it/10 coeff[]smult degree[]smult[]eq ir-re-duci
bleD$\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 1$\OT1/cmr/m/n/10 )$ \OT1/cmr/m/it/10 i
rreducible[]dvd[]smult
[209]
Overfull \hbox (17.23799pt too wide) in paragraph at lines 3444--3445
[][]  \OT1/cmr/m/it/10 nor-mal-ize $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 f $\OT1/
cmr/m/n/10 :$$:$ $ [] $\OT1/cmr/m/it/10 a $\OT1/cmr/m/n/10 :$$:$ $\OMS/cmsy/m/n
/10 f$\OT1/cmr/m/it/10 normalization[]semidom$\OML/cmm/m/it/10 ;$\OT1/cmr/m/it/
10 field$\OMS/cmsy/m/n/10 g$ \OT1/cmr/m/it/10 poly$\OT1/cmr/m/n/10 )$ $=$ \OT1/
cmr/m/it/10 smult $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 inverse $\OT1/cmr/m/n/10 
($\OT1/cmr/m/it/10 unit[]factor
[210] [211]
Overfull \hbox (4.89697pt too wide) in paragraph at lines 3798--3802
[][]    \OT1/cmr/bx/n/10 have \OT1/cmr/m/it/10 id$\OT1/cmr/m/n/10 :$ \OT1/cmr/m
/it/10 n $\OMS/cmsy/m/n/10 ^^C$ \OT1/cmr/m/it/10 Suc ii $\OT1/cmr/m/n/10 +$ \OT
1/cmr/m/it/10 j $\OMS/cmsy/m/n/10 ^^@$ \OT1/cmr/m/it/10 n $\OT1/cmr/m/n/10 =$ \
OT1/cmr/m/it/10 n $\OMS/cmsy/m/n/10 ^^C$ \OT1/cmr/m/it/10 ii $\OT1/cmr/m/n/10 +
$ \OT1/cmr/m/it/10 j \OT1/cmr/bx/n/10 us-ing \OT1/cmr/m/it/10 n \OT1/cmr/bx/n/1
0 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp add$\OT1/cmr/m/n/10 :$ \OT1/cmr/m
/it/10 diff[]mult[]distrib2$\OT1/cmr/m/n/10 )$[] 
) (./Ring_Hom_Poly.tex [212]
Overfull \hbox (12.6962pt too wide) in paragraph at lines 73--87
[][]  \OT1/cmr/bx/n/10 us-ing \OT1/cmr/m/it/10 pcompose[]add pcompose[]mult pco
mpose[]1 pcompose[]uminus \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/
10 unfold[]locales$\OML/cmm/m/it/10 ;$
[213] [214] [215] [216] [217] [218] [219] [220]) (./Newton_Interpolation.tex
Overfull \hbox (17.22385pt too wide) in paragraph at lines 50--51
[][]  \OT1/cmr/m/it/10 horner[]composition $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 
map hom cs$\OT1/cmr/m/n/10 )$ $($\OT1/cmr/m/it/10 map hom xs$\OT1/cmr/m/n/10 )$
 $=$ \OT1/cmr/m/it/10 map[]poly hom $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 horner[
]composition
[221] [222] [223] [224] [225] [226] [227] [228] [229] [230] [231]
Overfull \hbox (2.43918pt too wide) in paragraph at lines 1490--1492
[][]    \OT1/cmr/bx/n/10 un-fold-ing \OT1/cmr/m/it/10 newton[]poly[]impl[]def n
ewton[]coefficients rev[]rev[]ident newton[]poly[]def
[232] [233] [234]
Overfull \hbox (4.60117pt too wide) in paragraph at lines 1779--1780
[][]  $\OT1/cmr/m/n/10 =[]\OMS/cmsy/m/n/10 )$ \OT1/cmr/m/it/10 newton[]coeffici
ents[]main $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 map rat[]of[]int xs$\OT1/cmr/m/n
/10 )$ $($\OT1/cmr/m/it/10 map rat[]of[]int gs$\OT1/cmr/m/n/10 )$ $($\OT1/cmr/m
/it/10 map rat[]of[]int
[235] [236] [237] [238]
Overfull \hbox (17.51727pt too wide) in paragraph at lines 2158--2160
[][]    \OT1/cmr/bx/n/10 have \OT1/cmr/m/it/10 id$\OT1/cmr/m/n/10 :$ \OT1/cmr/m
/it/10 horner[]composition $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 rev ?nc$\OT1/cmr
/m/n/10 )$ $($\OT1/cmr/m/it/10 ?mri xs$\OT1/cmr/m/n/10 )$ $=$ \OT1/cmr/m/it/10 
map[]poly ?ri $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 horner[]composition
[239]
Overfull \hbox (6.79475pt too wide) in paragraph at lines 2349--2351
[][]    \OT1/cmr/bx/n/10 un-fold-ing \OT1/cmr/m/it/10 newton[]interpolation[]po
ly[]def  newton[]interpolation[]poly[]int[]def Let[]def
[240]) (./Lagrange_Interpolation.tex [241]
Overfull \hbox (5.18915pt too wide) in paragraph at lines 177--179
[][]    \OT1/cmr/bx/n/10 un-fold-ing \OT1/cmr/m/it/10 lagrange[]basis[]poly[]de
f Let[]def poly[]prod[]list prod[]list[]map[]remove1$\OT1/cmr/m/n/10 [$\OT1/cmr
/m/it/10 OF

Overfull \hbox (10.95041pt too wide) in paragraph at lines 211--214
[][]    \OT1/cmr/bx/n/10 show \OT1/cmr/m/it/10 poly p x $\OT1/cmr/m/n/10 =$ \OT
1/cmr/m/it/10 y \OT1/cmr/bx/n/10 un-fold-ing \OT1/cmr/m/it/10 p lagrange[]inter
polation[]poly[]def Let[]def poly[]sum[]list

Overfull \hbox (33.61983pt too wide) in paragraph at lines 214--216
[][]    \OT1/cmr/bx/n/10 proof $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 subst sum[]l
ist[]map[]remove1$\OT1/cmr/m/n/10 [$\OT1/cmr/m/it/10 OF xy$\OT1/cmr/m/n/10 ]$$\
OML/cmm/m/it/10 ;$ \OT1/cmr/m/it/10 un-fold split poly[]smult lagrange[]basis[]
poly[]1$\OML/cmm/m/it/10 ;$[] 

Overfull \hbox (17.11581pt too wide) in paragraph at lines 219--221
[][]      \OT1/cmr/bx/n/10 as-sume \OT1/cmr/m/it/10 v $\OMS/cmsy/m/n/10 2$ \OT1
/cmr/m/it/10 set $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 map $\OT1/cmr/m/n/10 ($$\O
ML/cmm/m/it/10 ^^U$\OT1/cmr/m/it/10 xa$\OML/cmm/m/it/10 :$ \OT1/cmr/m/it/10 pol
y $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 case xa of $\OT1/cmr/m/n/10 ($\OT1/cmr/m/
it/10 xj$\OML/cmm/m/it/10 ;$ \OT1/cmr/m/it/10 yj$\OT1/cmr/m/n/10 )$ $\OMS/cmsy/
m/n/10 )$ \OT1/cmr/m/it/10 smult yj $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 lagrang
e[]basis[]poly
[242]) (./Neville_Aitken_Interpolation.tex [243] [244]
Overfull \hbox (0.26402pt too wide) in paragraph at lines 318--320
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 length[]neville[]aitken[]merge$\OT1
/cmr/m/n/10 [$\OT1/cmr/m/it/10 termination[]simp$\OT1/cmr/m/n/10 ]$$:$ \OT1/cmr
/m/it/10 length $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 neville[]aitken[]merge
[245]
Overfull \hbox (30.07036pt too wide) in paragraph at lines 441--444
[][]  \OT1/cmr/bx/n/10 show \OT1/cmr/m/it/10 ?case \OT1/cmr/bx/n/10 un-fold-ing
 \OT1/cmr/m/it/10 neville[]aitken[]impl[]main$\OML/cmm/m/it/10 :$\OT1/cmr/m/it/
10 simps 1$\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 1$\OT1/cmr/m/n/10 )$$[$\OT1/cmr/m
/it/10 OF neville[]aitken[]merge$\OT1/cmr/m/n/10 [$\OT1/cmr/m/it/10 OF
[246] [247]) (./Polynomial_Interpolation.tex [248] [249] [250] [251]
Overfull \hbox (26.8093pt too wide) in paragraph at lines 468--469
[][]  \OT1/cmr/m/it/10 interpolation[]poly[]int[]Some$\OT1/cmr/m/n/10 [$\OT1/cm
r/bx/n/10 where \OT1/cmr/m/it/10 alg $\OT1/cmr/m/n/10 =$ \OT1/cmr/m/it/10 New-t
on$\OML/cmm/m/it/10 ;$ \OT1/cmr/m/it/10 un-folded interpolation[]poly[]int$\OML
/cmm/m/it/10 :$\OT1/cmr/m/it/10 simps$\OT1/cmr/m/n/10 ]$[] 

Overfull \hbox (26.06485pt too wide) in paragraph at lines 472--474
[][]  \OT1/cmr/m/it/10 interpolation[]poly[]int[]None$\OT1/cmr/m/n/10 [$\OT1/cm
r/bx/n/10 where \OT1/cmr/m/it/10 alg $\OT1/cmr/m/n/10 =$ \OT1/cmr/m/it/10 New-t
on$\OML/cmm/m/it/10 ;$ \OT1/cmr/m/it/10 un-folded interpolation[]poly[]int$\OML
/cmm/m/it/10 :$\OT1/cmr/m/it/10 simps$\OT1/cmr/m/n/10 ]$ 
[252]))
No file root.bbl.
[253] (./root.aux)

Package rerunfilecheck Warning: File `root.out' has changed.
(rerunfilecheck)                Rerun to get outlines right
(rerunfilecheck)                or use package `bookmark'.


LaTeX Warning: There were undefined references.

 )
(see the transcript file for additional information)</usr/share/texlive/texmf-d
ist/fonts/type1/public/amsfonts/cm/cmbsy10.pfb></usr/share/texlive/texmf-dist/f
onts/type1/public/amsfonts/cm/cmbx10.pfb></usr/share/texlive/texmf-dist/fonts/t
ype1/public/amsfonts/cm/cmbx12.pfb></usr/share/texlive/texmf-dist/fonts/type1/p
ublic/amsfonts/cm/cmex10.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/
amsfonts/cm/cmmi10.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfon
ts/cm/cmmi12.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/
cmmi7.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi8.p
fb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmr10.pfb></us
r/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmr12.pfb></usr/share
/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmr17.pfb></usr/share/texliv
e/texmf-dist/fonts/type1/public/amsfonts/cm/cmr7.pfb></usr/share/texlive/texmf-
dist/fonts/type1/public/amsfonts/cm/cmr8.pfb></usr/share/texlive/texmf-dist/fon
ts/type1/public/amsfonts/cm/cmr9.pfb></usr/share/texlive/texmf-dist/fonts/type1
/public/amsfonts/cm/cmss10.pfb></usr/share/texlive/texmf-dist/fonts/type1/publi
c/amsfonts/cm/cmsy10.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsf
onts/cm/cmsy5.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm
/cmsy6.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy7.
pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy8.pfb></u
sr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmti10.pfb></usr/sha
re/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmti12.pfb></usr/share/tex
live/texmf-dist/fonts/type1/public/amsfonts/symbols/msam10.pfb>
Output written on root.pdf (253 pages, 816114 bytes).
Transcript written on root.log.
This is BibTeX, Version 0.99d (TeX Live 2015/Debian)
The top-level auxiliary file: root.aux
The style file: abbrv.bst
Database file #1: root.bib
This is pdfTeX, Version 3.14159265-2.6-1.40.16 (TeX Live 2015/Debian) (preloaded format=pdflatex)
 restricted \write18 enabled.
entering extended mode
LaTeX2e <2016/02/01>
Babel <3.9q> and hyphenation patterns for 81 language(s) loaded.

(./root.tex (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls
Document Class: article 2014/09/29 v1.4h Standard LaTeX document class
(/usr/share/texlive/texmf-dist/tex/latex/base/size11.clo)) (./isabelle.sty
(./comment.sty Excluding comment 'comment') Including comment 'isadelimtheory'
Including comment 'isatagtheory' Including comment 'isadelimproof'
Including comment 'isatagproof' Including comment 'isadelimML'
Including comment 'isatagML' Including comment 'isadelimvisible'
Including comment 'isatagvisible' Excluding comment 'isadeliminvisible'
Excluding comment 'isataginvisible') (./isabelletags.sty) (./isabellesym.sty)
(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty
(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty))
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty
For additional information on amsmath, use the `?' option.
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty))
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty)
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty))
(/usr/share/texlive/texmf-dist/tex/latex/tools/xspace.sty) (./pdfsetup.sty
(/usr/share/texlive/texmf-dist/tex/latex/graphics/color.sty
(/usr/share/texlive/texmf-dist/tex/latex/latexconfig/color.cfg)
(/usr/share/texlive/texmf-dist/tex/latex/pdftex-def/pdftex.def
(/usr/share/texlive/texmf-dist/tex/generic/oberdiek/infwarerr.sty)
(/usr/share/texlive/texmf-dist/tex/generic/oberdiek/ltxcmds.sty))))
(/usr/share/texlive/texmf-dist/tex/latex/hyperref/hyperref.sty
(/usr/share/texlive/texmf-dist/tex/generic/oberdiek/hobsub-hyperref.sty
(/usr/share/texlive/texmf-dist/tex/generic/oberdiek/hobsub-generic.sty))
(/usr/share/texlive/texmf-dist/tex/latex/graphics/keyval.sty)
(/usr/share/texlive/texmf-dist/tex/generic/ifxetex/ifxetex.sty)
(/usr/share/texlive/texmf-dist/tex/latex/oberdiek/auxhook.sty)
(/usr/share/texlive/texmf-dist/tex/latex/oberdiek/kvoptions.sty)
(/usr/share/texlive/texmf-dist/tex/latex/hyperref/pd1enc.def)
(/usr/share/texlive/texmf-dist/tex/latex/latexconfig/hyperref.cfg)
(/usr/share/texlive/texmf-dist/tex/latex/url/url.sty))

Package hyperref Message: Driver (autodetected): hpdftex.

(/usr/share/texlive/texmf-dist/tex/latex/hyperref/hpdftex.def
(/usr/share/texlive/texmf-dist/tex/latex/oberdiek/rerunfilecheck.sty))
(./root.aux) (/usr/share/texlive/texmf-dist/tex/context/base/supp-pdf.mkii
[Loading MPS to PDF converter (version 2006.09.02).]
) (/usr/share/texlive/texmf-dist/tex/latex/hyperref/nameref.sty
(/usr/share/texlive/texmf-dist/tex/generic/oberdiek/gettitlestring.sty))
(./root.out) (./root.out)
(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd)
(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd)
(/usr/share/texlive/texmf-dist/tex/latex/base/omscmr.fd) (./root.toc [1{/var/li
b/texmf/fonts/map/pdftex/updmap/pdftex.map}] [2]) [3]

LaTeX Warning: Citation `interpolation' on page 4 undefined on input line 55.

(./session.tex (./More_List.tex) (./Infinite_Set.tex) (./Polynomial.tex)
(./Adhoc_Overloading.tex) (./Monad_Syntax.tex) (./Sqrt_Babylonian_Auxiliary.tex
) (./Is_Rat_To_Rat.tex) (./Divmod_Int.tex [4]
Overfull \hbox (12.96191pt too wide) in paragraph at lines 41--43
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 divmod[]int[]code$\OT1/cmr/m/n/10 [
$\OT1/cmr/m/it/10 code$\OT1/cmr/m/n/10 ]$$:$ \OT1/cmr/m/it/10 divmod[]int m n $
\OT1/cmr/m/n/10 =$ \OT1/cmr/m/it/10 map[]prod int[]of[]integer int[]of[]integer

) (./Improved_Code_Equations.tex

Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 33.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 33.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 33.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 33.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 33.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 33.

[5]
Overfull \hbox (12.29745pt too wide) in paragraph at lines 108--115
[][]   \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 cases l $\OT1/cm
r/m/n/10 =$ \OT1/cmr/m/it/10 0$\OT1/cmr/m/n/10 ;$ \OT1/cmr/m/it/10 cases l $\OM
L/cmm/m/it/10 <$ \OT1/cmr/m/it/10 0$\OT1/cmr/m/n/10 ;$ \OT1/cmr/m/it/10 cases l
 $\OML/cmm/m/it/10 >$ \OT1/cmr/m/it/10 0$\OT1/cmr/m/n/10 ;$ \OT1/cmr/m/it/10 au
to split$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 prod$\OML/cmm/m/it/10 :$\OT1/cmr/m
/it/10 splits simp$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 divmod[]abs$ [] $[]def

Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 128.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 128.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 128.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 128.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 128.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 128.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `math shift' on input line 128.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\mathord' on input line 128.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `math shift' on input line 128.

[6]
Overfull \hbox (32.11456pt too wide) in paragraph at lines 141--143
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 divmod[]nat[]code$\OT1/cmr/m/n/10 :
$ \OT1/cmr/m/it/10 Di-vides$\OML/cmm/m/it/10 :$\OT1/cmr/m/it/10 divmod[]nat m n
 $\OT1/cmr/m/n/10 =$ \OT1/cmr/m/it/10 map[]prod nat[]of[]integer nat[]of[]integ
er

Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 221.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 221.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 221.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 221.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 221.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 221.

) (./Cancellation.tex [7]
Overfull \hbox (15.88786pt too wide) in paragraph at lines 30--31
[][]  \OT1/cmr/m/it/10 can-ce-la-tion sim-proc$\OML/cmm/m/it/10 :$ \OT1/cmr/m/i
t/10 Nor-mal-i-sa-tion of []iterate[]add[] back to the nor-male representation[
] 

Overfull \hbox (16.82057pt too wide) in paragraph at lines 81--83
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 iterate[]add[]distrib$\OT1/cmr/m/n/
10 [$\OT1/cmr/m/it/10 simp$\OT1/cmr/m/n/10 ]$$:$ []\OT1/cmr/m/it/10 iterate[]ad
d $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 m$\OT1/cmr/m/n/10 +$\OT1/cmr/m/it/10 n$\O
T1/cmr/m/n/10 )$ \OT1/cmr/m/it/10 a $\OT1/cmr/m/n/10 =$ \OT1/cmr/m/it/10 iterat
e[]add m a $\OT1/cmr/m/n/10 +$ \OT1/cmr/m/it/10 iterate[]add

Overfull \hbox (4.55688pt too wide) in paragraph at lines 138--151
[][]  \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 auto dest$\OT1/cm
r/m/n/10 !$$:$ \OT1/cmr/m/it/10 le[]Suc[]ex add[]right[]imp[]eq simp$\OT1/cmr/m
/n/10 :$ \OT1/cmr/m/it/10 ab[]semigroup[]add[]class$\OML/cmm/m/it/10 :$\OT1/cmr
/m/it/10 add[]ac$\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 1$\OT1/cmr/m/n/10 )$$)$[] 

Overfull \hbox (4.55688pt too wide) in paragraph at lines 157--170
[][]  \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 auto dest$\OT1/cm
r/m/n/10 !$$:$ \OT1/cmr/m/it/10 le[]Suc[]ex add[]right[]imp[]eq simp$\OT1/cmr/m
/n/10 :$ \OT1/cmr/m/it/10 ab[]semigroup[]add[]class$\OML/cmm/m/it/10 :$\OT1/cmr
/m/it/10 add[]ac$\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 1$\OT1/cmr/m/n/10 )$$)$[] 
[8]
Overfull \hbox (85.96669pt too wide) in paragraph at lines 175--176
[][]  \OT1/cmr/m/it/10 j $\OMS/cmsy/m/n/10 ^^T$ $\OT1/cmr/m/n/10 ($\OT1/cmr/m/i
t/10 i$\OT1/cmr/m/n/10 :$$:$\OT1/cmr/m/it/10 nat$\OT1/cmr/m/n/10 )$ $=[]\OMS/cm
sy/m/n/10 )$ $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 iterate[]add i $\OT1/cmr/m/n/1
0 ($\OT1/cmr/m/it/10 u$\OT1/cmr/m/n/10 :$$:$ $ [] $\OT1/cmr/m/it/10 a $\OT1/cmr
/m/n/10 :$$:$ $\OMS/cmsy/m/n/10 f$\OT1/cmr/m/it/10 cancel[]comm[]monoid[]add$\O
ML/cmm/m/it/10 ;$ \OT1/cmr/m/it/10 ordered[]ab[]semigroup[]add[]imp[]le$\OMS/cm
sy/m/n/10 g$$\OT1/cmr/m/n/10 )$

Overfull \hbox (4.55688pt too wide) in paragraph at lines 176--189
[][]  \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 auto dest$\OT1/cm
r/m/n/10 !$$:$ \OT1/cmr/m/it/10 le[]Suc[]ex add[]right[]imp[]eq simp$\OT1/cmr/m
/n/10 :$ \OT1/cmr/m/it/10 ab[]semigroup[]add[]class$\OML/cmm/m/it/10 :$\OT1/cmr
/m/it/10 add[]ac$\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 1$\OT1/cmr/m/n/10 )$$)$[] 

Overfull \hbox (85.96669pt too wide) in paragraph at lines 194--195
[][]  \OT1/cmr/m/it/10 i $\OMS/cmsy/m/n/10 ^^T$ $\OT1/cmr/m/n/10 ($\OT1/cmr/m/i
t/10 j$\OT1/cmr/m/n/10 :$$:$\OT1/cmr/m/it/10 nat$\OT1/cmr/m/n/10 )$ $=[]\OMS/cm
sy/m/n/10 )$ $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 iterate[]add i $\OT1/cmr/m/n/1
0 ($\OT1/cmr/m/it/10 u$\OT1/cmr/m/n/10 :$$:$ $ [] $\OT1/cmr/m/it/10 a $\OT1/cmr
/m/n/10 :$$:$ $\OMS/cmsy/m/n/10 f$\OT1/cmr/m/it/10 cancel[]comm[]monoid[]add$\O
ML/cmm/m/it/10 ;$ \OT1/cmr/m/it/10 ordered[]ab[]semigroup[]add[]imp[]le$\OMS/cm
sy/m/n/10 g$$\OT1/cmr/m/n/10 )$

Overfull \hbox (4.55688pt too wide) in paragraph at lines 195--208
[][]  \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 auto dest$\OT1/cm
r/m/n/10 !$$:$ \OT1/cmr/m/it/10 le[]Suc[]ex add[]right[]imp[]eq simp$\OT1/cmr/m
/n/10 :$ \OT1/cmr/m/it/10 ab[]semigroup[]add[]class$\OML/cmm/m/it/10 :$\OT1/cmr
/m/it/10 add[]ac$\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 1$\OT1/cmr/m/n/10 )$$)$[] 

Overfull \hbox (85.96669pt too wide) in paragraph at lines 213--214
[][]  \OT1/cmr/m/it/10 j $\OMS/cmsy/m/n/10 ^^T$ $\OT1/cmr/m/n/10 ($\OT1/cmr/m/i
t/10 i$\OT1/cmr/m/n/10 :$$:$\OT1/cmr/m/it/10 nat$\OT1/cmr/m/n/10 )$ $=[]\OMS/cm
sy/m/n/10 )$ $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 iterate[]add i $\OT1/cmr/m/n/1
0 ($\OT1/cmr/m/it/10 u$\OT1/cmr/m/n/10 :$$:$ $ [] $\OT1/cmr/m/it/10 a $\OT1/cmr
/m/n/10 :$$:$ $\OMS/cmsy/m/n/10 f$\OT1/cmr/m/it/10 cancel[]comm[]monoid[]add$\O
ML/cmm/m/it/10 ;$ \OT1/cmr/m/it/10 ordered[]ab[]semigroup[]add[]imp[]le$\OMS/cm
sy/m/n/10 g$$\OT1/cmr/m/n/10 )$

Overfull \hbox (4.55688pt too wide) in paragraph at lines 214--227
[][]  \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 auto dest$\OT1/cm
r/m/n/10 !$$:$ \OT1/cmr/m/it/10 le[]Suc[]ex add[]right[]imp[]eq simp$\OT1/cmr/m
/n/10 :$ \OT1/cmr/m/it/10 ab[]semigroup[]add[]class$\OML/cmm/m/it/10 :$\OT1/cmr
/m/it/10 add[]ac$\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 1$\OT1/cmr/m/n/10 )$$)$[] 

Overfull \hbox (85.96669pt too wide) in paragraph at lines 232--233
[][]  \OT1/cmr/m/it/10 i $\OMS/cmsy/m/n/10 ^^T$ $\OT1/cmr/m/n/10 ($\OT1/cmr/m/i
t/10 j$\OT1/cmr/m/n/10 :$$:$\OT1/cmr/m/it/10 nat$\OT1/cmr/m/n/10 )$ $=[]\OMS/cm
sy/m/n/10 )$ $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 iterate[]add i $\OT1/cmr/m/n/1
0 ($\OT1/cmr/m/it/10 u$\OT1/cmr/m/n/10 :$$:$ $ [] $\OT1/cmr/m/it/10 a $\OT1/cmr
/m/n/10 :$$:$ $\OMS/cmsy/m/n/10 f$\OT1/cmr/m/it/10 cancel[]comm[]monoid[]add$\O
ML/cmm/m/it/10 ;$ \OT1/cmr/m/it/10 ordered[]ab[]semigroup[]add[]imp[]le$\OMS/cm
sy/m/n/10 g$$\OT1/cmr/m/n/10 )$

Overfull \hbox (4.55688pt too wide) in paragraph at lines 233--246
[][]  \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 auto dest$\OT1/cm
r/m/n/10 !$$:$ \OT1/cmr/m/it/10 le[]Suc[]ex add[]right[]imp[]eq simp$\OT1/cmr/m
/n/10 :$ \OT1/cmr/m/it/10 ab[]semigroup[]add[]class$\OML/cmm/m/it/10 :$\OT1/cmr
/m/it/10 add[]ac$\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 1$\OT1/cmr/m/n/10 )$$)$[] 

Overfull \hbox (4.55688pt too wide) in paragraph at lines 252--265
[][]  \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 auto dest$\OT1/cm
r/m/n/10 !$$:$ \OT1/cmr/m/it/10 le[]Suc[]ex add[]right[]imp[]eq simp$\OT1/cmr/m
/n/10 :$ \OT1/cmr/m/it/10 ab[]semigroup[]add[]class$\OML/cmm/m/it/10 :$\OT1/cmr
/m/it/10 add[]ac$\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 1$\OT1/cmr/m/n/10 )$$)$[] 

Overfull \hbox (4.55688pt too wide) in paragraph at lines 271--288
[][]  \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 auto dest$\OT1/cm
r/m/n/10 !$$:$ \OT1/cmr/m/it/10 le[]Suc[]ex add[]right[]imp[]eq simp$\OT1/cmr/m
/n/10 :$ \OT1/cmr/m/it/10 ab[]semigroup[]add[]class$\OML/cmm/m/it/10 :$\OT1/cmr
/m/it/10 add[]ac$\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 1$\OT1/cmr/m/n/10 )$$)$ 
) (./Multiset.tex [9] [10] [11] [12] [13]
Overfull \hbox (13.45732pt too wide) in paragraph at lines 783--785
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 set[]mset[]add[]mset[]insert $\OT1/
cmr/m/n/10 [$\OT1/cmr/m/it/10 simp$\OT1/cmr/m/n/10 ]$$:$ []\OT1/cmr/m/it/10 set
[]mset $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 add[]mset a A$\OT1/cmr/m/n/10 )$ $=$
 \OT1/cmr/m/it/10 in-sert a $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 set[]mset
[14] [15] [16] [17]
Overfull \hbox (33.36261pt too wide) in paragraph at lines 1710--1715
[][]    \OT1/cmr/bx/n/10 more-over from []\OT1/cmr/m/it/10 ?lhs[] \OT1/cmr/bx/n
/10 have \OT1/cmr/m/it/10 M $\OT1/cmr/m/n/10 =$ \OT1/cmr/m/it/10 add[]mset b N 
$\OMS/cmsy/m/n/10 ^^@$ $f$$\OT1/cmr/m/n/10 #$\OT1/cmr/m/it/10 a$\OT1/cmr/m/n/10
 #$$\OMS/cmsy/m/n/10 g$ \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10
 rule union[]single[]eq[]diff$\OT1/cmr/m/n/10 )$[] 
[18] [19] [20] [21] [22] [23] [24]
Overfull \hbox (31.97931pt too wide) in paragraph at lines 2987--2995
[][]  \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 metis add[]mset[]
add[]single add[]mset[]diff[]bothsides diff[]subset[]eq[]self multiset[]inter[]
def[] 
[25] [26] [27]
Overfull \hbox (20.85715pt too wide) in paragraph at lines 3518--3520
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 replicate[]mset[]Suc $\OT1/cmr/m/n/
10 [$\OT1/cmr/m/it/10 simp$\OT1/cmr/m/n/10 ]$$:$ \OT1/cmr/m/it/10 replicate[]ms
et $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 Suc n$\OT1/cmr/m/n/10 )$ \OT1/cmr/m/it/1
0 x $\OT1/cmr/m/n/10 =$ \OT1/cmr/m/it/10 add[]mset x $\OT1/cmr/m/n/10 ($\OT1/cm
r/m/it/10 replicate[]mset
[28]
Overfull \hbox (51.96812pt too wide) in paragraph at lines 3798--3805
[][]  \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp add$\OT1/cmr
/m/n/10 :$ \OT1/cmr/m/it/10 iterate[]add[]eq[]add[]iff1 mset[]subseteq[]add[]if
f1$\OT1/cmr/m/n/10 [$\OT1/cmr/m/it/10 unfolded repeat[]mset[]iterate[]add$\OT1/
cmr/m/n/10 ]$$)$[] 

Overfull \hbox (51.96812pt too wide) in paragraph at lines 3819--3826
[][]  \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp add$\OT1/cmr
/m/n/10 :$ \OT1/cmr/m/it/10 iterate[]add[]eq[]add[]iff2 mset[]subseteq[]add[]if
f2$\OT1/cmr/m/n/10 [$\OT1/cmr/m/it/10 unfolded repeat[]mset[]iterate[]add$\OT1/
cmr/m/n/10 ]$$)$[] 
[29] [30]
Overfull \hbox (1.90573pt too wide) in paragraph at lines 3975--3978
[][]  \OT1/cmr/bx/n/10 thus $\OT1/cmr/m/n/10 ($$\OML/cmm/m/it/10 ^^U$\OT1/cmr/m
/it/10 i$\OML/cmm/m/it/10 :$ \OT1/cmr/m/it/10 if A $\OT1/cmr/m/n/10 =$ $\OMS/cm
sy/m/n/10 f$$g$ \OT1/cmr/m/it/10 then 0 else INF f$\OT1/cmr/m/n/10 :$\OT1/cmr/m
/it/10 A$\OML/cmm/m/it/10 :$ \OT1/cmr/m/it/10 f i$\OT1/cmr/m/n/10 )$ $\OMS/cmsy
/m/n/10 2$ \OT1/cmr/m/it/10 mul-ti-set \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\
OT1/cmr/m/it/10 simp add$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 multiset[]def$\OT1
/cmr/m/n/10 )$[] 
[31]
Overfull \hbox (19.41994pt too wide) in paragraph at lines 4233--4238
[][]  \OT1/cmr/bx/n/10 more-over from \OT1/cmr/m/it/10 assms \OT1/cmr/bx/n/10 h
ave \OT1/cmr/m/it/10 fi-nite $[] $ \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/
cmr/m/it/10 rule bdd[]above[]multiset[]imp[]finite[]support$\OT1/cmr/m/n/10 )$[
] 

Overfull \hbox (39.6184pt too wide) in paragraph at lines 4256--4270
[][]  \OT1/cmr/bx/n/10 us-ing \OT1/cmr/m/it/10 assms \OT1/cmr/bx/n/10 by $\OT1/
cmr/m/n/10 ($\OT1/cmr/m/it/10 simp add$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 Sup[
]multiset[]def Abs[]multiset[]inverse Sup[]multiset[]in[]multiset$\OT1/cmr/m/n/
10 )$[] 
[32]
Overfull \hbox (55.51436pt too wide) in paragraph at lines 4331--4336
[][]    \OT1/cmr/bx/n/10 also from \OT1/cmr/m/it/10 nonempty \OT1/cmr/bx/n/10 h
ave $[] $ $\OT1/cmr/m/n/10 =$ \OT1/cmr/m/it/10 count $\OT1/cmr/m/n/10 ($\OT1/cm
r/m/it/10 Inf A$\OT1/cmr/m/n/10 )$ \OT1/cmr/m/it/10 x \OT1/cmr/bx/n/10 by $\OT1
/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp add$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 cou
nt[]Inf[]multiset[]nonempty$\OT1/cmr/m/n/10 )$[] 
[33] [34]
Overfull \hbox (5.59465pt too wide) in paragraph at lines 4613--4617
[][]  \OT1/cmr/bx/n/10 from \OT1/cmr/m/it/10 assms \OT1/cmr/bx/n/10 have \OT1/c
mr/m/it/10 set[]mset $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 Sup A$\OT1/cmr/m/n/10 
)$ $=$ $($$[] $\OT1/cmr/m/it/10 X$\OMS/cmsy/m/n/10 2$\OT1/cmr/m/it/10 A$\OML/cm
m/m/it/10 :$ \OT1/cmr/m/it/10 set[]mset X$\OT1/cmr/m/n/10 )$ \OT1/cmr/bx/n/10 b
y $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 rule set[]mset[]Sup$\OT1/cmr/m/n/10 )$[] 

[35]
Overfull \hbox (7.00906pt too wide) in paragraph at lines 4821--4823
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 filter[]diff[]mset $\OT1/cmr/m/n/10
 [$\OT1/cmr/m/it/10 simp$\OT1/cmr/m/n/10 ]$$:$ \OT1/cmr/m/it/10 filter[]mset P 
$\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 M $\OMS/cmsy/m/n/10 ^^@$ \OT1/cmr/m/it/10 N
$\OT1/cmr/m/n/10 )$ $=$ \OT1/cmr/m/it/10 filter[]mset P M $\OMS/cmsy/m/n/10 ^^@
$ \OT1/cmr/m/it/10 filter[]mset

Overfull \hbox (11.80153pt too wide) in paragraph at lines 4857--4859
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 filter[]sup[]mset$\OT1/cmr/m/n/10 [
$\OT1/cmr/m/it/10 simp$\OT1/cmr/m/n/10 ]$$:$ \OT1/cmr/m/it/10 filter[]mset P $\
OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 A $\OMS/cmsy/m/n/10 [$$\OT1/cmr/m/n/10 #$ \OT
1/cmr/m/it/10 B$\OT1/cmr/m/n/10 )$ $=$ \OT1/cmr/m/it/10 filter[]mset P A $\OMS/
cmsy/m/n/10 [$$\OT1/cmr/m/n/10 #$ \OT1/cmr/m/it/10 filter[]mset
[36] [37]
Overfull \hbox (30.90695pt too wide) in paragraph at lines 5235--5243
[][]\OT1/cmr/bx/n/10 apply $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp add$\OT1/cm
r/m/n/10 :$ \OT1/cmr/m/it/10 size[]multiset[]def sum[]Un[]nat sum$\OML/cmm/m/it
/10 :$\OT1/cmr/m/it/10 distrib sum[]wcount[]Int wcount[]union$\OT1/cmr/m/n/10 )
$[] 

Overfull \hbox (55.98346pt too wide) in paragraph at lines 5260--5274
[][]  \OT1/cmr/bx/n/10 un-fold-ing \OT1/cmr/m/it/10 add[]mset[]add[]single$\OT1
/cmr/m/n/10 [$\OT1/cmr/m/it/10 of [] M$\OT1/cmr/m/n/10 ]$ \OT1/cmr/m/it/10 size
[]multiset[]union \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 auto 
simp$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 size[]multiset[]single$\OT1/cmr/m/n/10
 )$[] 
[38] [39] [40] [41] [42]
Overfull \hbox (44.37408pt too wide) in paragraph at lines 6145--6159
[][]  \OT1/cmr/bx/n/10 un-fold-ing \OT1/cmr/m/it/10 image[]mset[]union add[]mse
t[]add[]single$\OT1/cmr/m/n/10 [$\OT1/cmr/m/it/10 of a M$\OT1/cmr/m/n/10 ]$ \OT
1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp add$\OT1/cmr/m/n/10 :
$ \OT1/cmr/m/it/10 image[]mset[]single$\OT1/cmr/m/n/10 )$[] 
[43]
Overfull \hbox (9.85997pt too wide) in paragraph at lines 6333--6341
[][]  \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 metis $\OT1/cmr/m
/n/10 ($\OT1/cmr/m/it/10 no[]types$\OT1/cmr/m/n/10 )$ \OT1/cmr/m/it/10 Diff[]eq
[]empty[]iff[]mset image[]mset[]Diff image[]mset[]is[]empty[]iff[] 
[44] [45] [46] [47] [48] [49] [50] [51]
Overfull \hbox (4.1574pt too wide) in paragraph at lines 7718--7731
[][]  \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 auto simp add$\OT
1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 mset[]subset[]eqI$\OT1/cmr/m/n/10 )$ $($\OT1/c
mr/m/it/10 metis count[]replicate[]mset subseteq[]mset[]def$\OT1/cmr/m/n/10 )$[
] 
[52] [53] [54]
Overfull \hbox (14.65363pt too wide) in paragraph at lines 8409--8410
[][]      $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp[]all add$\OT1/cmr/m/n/10 :$ 
\OT1/cmr/m/it/10 size[]multiset[]overloaded[]eq not[]in[]iff sum$\OML/cmm/m/it/
10 :$\OT1/cmr/m/it/10 If[]cases Diff[]eq$\OT1/cmr/m/n/10 [$\OT1/cmr/m/it/10 sym
metric$\OT1/cmr/m/n/10 ]$[] 
[55]
Overfull \hbox (20.93939pt too wide) in paragraph at lines 8552--8565
[][]  \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 subst sum[]mset$\
OML/cmm/m/it/10 :$\OT1/cmr/m/it/10 commute$\OT1/cmr/m/n/10 )$ $($\OT1/cmr/m/it/
10 simp add$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 sum[]mset[]distrib[]left sum[]m
set[]distrib[]right$\OT1/cmr/m/n/10 )$[] 
[56] [57] [58] [59] [60]
Overfull \hbox (15.84276pt too wide) in paragraph at lines 9390--9393
[][]    \OT1/cmr/bx/n/10 un-fold-ing \OT1/cmr/m/it/10 filter[]sort \OT1/cmr/bx/
n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 rule properties[]for[]sort[]key$\OT
1/cmr/m/n/10 )$ $($\OT1/cmr/m/it/10 auto in-tro$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/i
t/10 sorted[]map[]same$\OT1/cmr/m/n/10 )$[] 
[61] [62] [63] [64] [65] [66] [67]
Overfull \hbox (23.14763pt too wide) in paragraph at lines 10339--10342
[][]      \OT1/cmr/bx/n/10 us-ing $\OMS/cmsy/m/n/10 ^^C$$\OT1/cmr/m/n/10 ($\OT1
/cmr/m/it/10 1$\OML/cmm/m/it/10 ;$\OT1/cmr/m/it/10 2$\OT1/cmr/m/n/10 )$ \OT1/cm
r/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 metis add[]mset[]remove[]trivi
al[]If insert[]iff set[]mset[]add[]mset[]insert
[68]
Overfull \hbox (18.23695pt too wide) in paragraph at lines 10471--10473
[][]    $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 M $\OMS/cmsy/m/n/10 ^^@$ \OT1/cmr/m
/it/10 M $\OMS/cmsy/m/n/10 \$$\OT1/cmr/m/n/10 #$ \OT1/cmr/m/it/10 N$\OT1/cmr/m/
n/10 )$ $\OMS/cmsy/m/n/10 \$$\OT1/cmr/m/n/10 #$ $($\OT1/cmr/m/it/10 N $\OMS/cms
y/m/n/10 ^^@$ \OT1/cmr/m/it/10 M $\OMS/cmsy/m/n/10 \$$\OT1/cmr/m/n/10 #$ \OT1/c
mr/m/it/10 N$\OT1/cmr/m/n/10 )$ $=$ $\OMS/cmsy/m/n/10 f$$\OT1/cmr/m/n/10 #$$\OM
S/cmsy/m/n/10 g$ \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 auto s
imp$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 count[]inject$\OT1/cmr/m/n/10 [$\OT1/cm
r/m/it/10 symmetric$\OT1/cmr/m/n/10 ]$$)$[] 
[69]
Overfull \hbox (24.56795pt too wide) in paragraph at lines 10531--10535
[][]    \OT1/cmr/bx/n/10 then ob-tain \OT1/cmr/m/it/10 y \OT1/cmr/bx/n/10 where
 \OT1/cmr/m/it/10 count N y $\OMS/cmsy/m/n/10 6\OT1/cmr/m/n/10 =$ \OT1/cmr/m/it
/10 count M y \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 auto simp
$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 count[]inject$\OT1/cmr/m/n/10 [$\OT1/cmr/m
/it/10 symmetric$\OT1/cmr/m/n/10 ]$$)$[] 

Overfull \hbox (0.73877pt too wide) in paragraph at lines 10629--10633
[][]    \OT1/cmr/bx/n/10 then have \OT1/cmr/m/it/10 MM$\OT1/cmr/m/n/10 :$ $($\O
T1/cmr/m/it/10 M$\OML/cmm/m/it/10 ;$ \OT1/cmr/m/it/10 M$\OT1/cmr/m/n/10 )$ $\OM
S/cmsy/m/n/10 2$ \OT1/cmr/m/it/10 mult $\OMS/cmsy/m/n/10 f$$\OT1/cmr/m/n/10 ($\
OT1/cmr/m/it/10 x$\OML/cmm/m/it/10 ;$ \OT1/cmr/m/it/10 y$\OT1/cmr/m/n/10 )$$\OM
L/cmm/m/it/10 :$ \OT1/cmr/m/it/10 x $\OML/cmm/m/it/10 <$ \OT1/cmr/m/it/10 y$\OM
S/cmsy/m/n/10 g$ \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp a
dd$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 less[]multiset[]def$\OT1/cmr/m/n/10 )$[]
 
[70] [71]
Overfull \hbox (55.81471pt too wide) in paragraph at lines 10887--10894
[][]\OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 rule one[]step[]imp
lies[]mult$\OT1/cmr/m/n/10 )$ $($\OT1/cmr/m/it/10 auto simp add$\OT1/cmr/m/n/10
 :$ \OT1/cmr/m/it/10 max[]strict[]def pair[]less[]def elim$\OT1/cmr/m/n/10 !$$:
$\OT1/cmr/m/it/10 max[]ext$\OML/cmm/m/it/10 :$\OT1/cmr/m/it/10 cases$\OT1/cmr/m
/n/10 )$[] 

Overfull \hbox (54.71396pt too wide) in paragraph at lines 10909--10916
[][]\OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 auto simp add$\OT1/
cmr/m/n/10 :$ \OT1/cmr/m/it/10 pair[]less[]def max[]strict[]def elim$\OT1/cmr/m
/n/10 !$$:$\OT1/cmr/m/it/10 max[]ext$\OML/cmm/m/it/10 :$\OT1/cmr/m/it/10 cases 
in-tro$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 one[]step[]implies[]mult$\OT1/cmr/m/
n/10 )$[] 
[72] [73]
Overfull \hbox (13.03835pt too wide) in paragraph at lines 11158--11159
[][]        $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 map $\OT1/cmr/m/n/10 ($\OT1/cmr
/m/it/10 fn t $\OT1/cmr/m/n/10 =$$\OML/cmm/m/it/10 >$ \OT1/cmr/m/it/10 t RS eq[
]reflection$\OT1/cmr/m/n/10 )$ $($$@$$\OMS/cmsy/m/n/10 f$\OT1/cmr/m/it/10 thms 
ac[]simps$\OMS/cmsy/m/n/10 g$ $\OT1/cmr/m/n/10 @$ $@$$\OMS/cmsy/m/n/10 f$\OT1/c
mr/m/it/10 thms empty[]neutral$\OMS/cmsy/m/n/10 g$$\OT1/cmr/m/n/10 )$$)$[] 
[74] [75] [76] [77] [78] [79]
Overfull \hbox (5.43213pt too wide) in paragraph at lines 12056--12057
[][]  \OT1/cmr/m/it/10 full[]exhaustive[]multiset f i $\OT1/cmr/m/n/10 =$ \OT1/
cmr/m/it/10 Quickcheck[]Exhaustive$\OML/cmm/m/it/10 :$\OT1/cmr/m/it/10 full[]ex
haustive $\OT1/cmr/m/n/10 ($$\OML/cmm/m/it/10 ^^U$\OT1/cmr/m/it/10 xs$\OML/cmm/
m/it/10 :$ \OT1/cmr/m/it/10 f $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 msetify
[80] [81]
Overfull \hbox (21.37383pt too wide) in paragraph at lines 12215--12217
[][]    \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 metis append[]e
q[]append[]conv append[]take[]drop[]id diff[]Suc[]Suc Cons[]nth[]drop[]Suc

Overfull \hbox (17.54051pt too wide) in paragraph at lines 12222--12224
[][]    \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 metis add[]Suc[
]right append[]take[]drop[]id length[]Cons length[]append less[]eq[]Suc[]le

Overfull \hbox (26.54095pt too wide) in paragraph at lines 12229--12231
[][]    \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 metis add[]Suc[
]right append[]take[]drop[]id length[]Cons length[]append mset[]eq[]length$\OT1
/cmr/m/n/10 )$[] 
[82]
Overfull \hbox (19.33397pt too wide) in paragraph at lines 12350--12353
[][]    \OT1/cmr/bx/n/10 un-fold-ing \OT1/cmr/m/it/10 comp[]def \OT1/cmr/bx/n/1
0 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 rule ext$\OT1/cmr/m/n/10 )$ $($\OT1/cm
r/m/it/10 simp add$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 comp[]def image[]mset$\O
ML/cmm/m/it/10 :$\OT1/cmr/m/it/10 compositionality$\OT1/cmr/m/n/10 )$[] 
[83] [84]
Overfull \hbox (6.28758pt too wide) in paragraph at lines 12638--12642
[][]    \OT1/cmr/bx/n/10 then ob-tain \OT1/cmr/m/it/10 N1 b \OT1/cmr/bx/n/10 wh
ere \OT1/cmr/m/it/10 N$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 N $\OT1/cmr/m/n/10 =
$ \OT1/cmr/m/it/10 add[]mset b N1 \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/c
mr/m/it/10 metis multi[]nonempty[]split$\OT1/cmr/m/n/10 )$[] 
[85] [86]
Overfull \hbox (23.3852pt too wide) in paragraph at lines 12982--12983
[][]  \OT1/cmr/m/it/10 BNF[]LFP[]Size$\OML/cmm/m/it/10 :$\OT1/cmr/m/it/10 regis
ter[]size[]global $\OT1/cmr/m/n/10 @$$\OMS/cmsy/m/n/10 f$\OT1/cmr/m/it/10 type[
]name mul-ti-set$\OMS/cmsy/m/n/10 g$ $\OT1/cmr/m/n/10 @$$\OMS/cmsy/m/n/10 f$\OT
1/cmr/m/it/10 const[]name size[]multiset$\OMS/cmsy/m/n/10 g$[] 
[87]) (./Factorial_Ring.tex [88] [89]
Overfull \hbox (9.44902pt too wide) in paragraph at lines 435--449
[][]  \OT1/cmr/bx/n/10 us-ing \OT1/cmr/m/it/10 unit[]imp[]no[]irreducible[]divi
sors$\OT1/cmr/m/n/10 [$\OT1/cmr/m/it/10 OF assms$\OT1/cmr/m/n/10 ($\OT1/cmr/m/i
t/10 1$\OT1/cmr/m/n/10 )$ \OT1/cmr/m/it/10 prime[]elem[]imp[]irreducible$\OT1/c
mr/m/n/10 [$\OT1/cmr/m/it/10 OF
[90] [91]
Overfull \hbox (26.81523pt too wide) in paragraph at lines 815--820
[][]  \OT1/cmr/bx/n/10 more-over from \OT1/cmr/m/it/10 assms \OT1/cmr/bx/n/10 h
ave $\OMS/cmsy/m/n/10 :$\OT1/cmr/m/it/10 is[]unit p \OT1/cmr/bx/n/10 by $\OT1/c
mr/m/n/10 ($\OT1/cmr/m/it/10 simp add$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 prime
[]elem[]def is[]unit[]power[]iff$\OT1/cmr/m/n/10 )$[] 
[92]
Overfull \hbox (10.79758pt too wide) in paragraph at lines 887--900
[][]  \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 auto simp$\OT1/cm
r/m/n/10 :$ \OT1/cmr/m/it/10 prime[]elem[]def mult$\OML/cmm/m/it/10 :$\OT1/cmr/
m/it/10 commute$\OT1/cmr/m/n/10 [$\OT1/cmr/m/it/10 of a$\OT1/cmr/m/n/10 ]$ \OT1
/cmr/m/it/10 is[]unit[]mult[]iff mult[]unit[]dvd[]iff$\OT1/cmr/m/n/10 )$[] 

Overfull \hbox (12.0489pt too wide) in paragraph at lines 955--957
[][]      \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 metis dvd[]mu
lt[]right dvd[]times[]left[]cancel[]iff mult$\OML/cmm/m/it/10 :$\OT1/cmr/m/it/1
0 left[]commute mult[]zero[]left$\OT1/cmr/m/n/10 )$[] 
[93]
Overfull \hbox (5.12607pt too wide) in paragraph at lines 1111--1125
[][]  \OT1/cmr/bx/n/10 us-ing \OT1/cmr/m/it/10 assms \OT1/cmr/bx/n/10 by $\OT1/
cmr/m/n/10 ($\OT1/cmr/m/it/10 induct n$\OT1/cmr/m/n/10 )$ $($\OT1/cmr/m/it/10 a
uto dest$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 prime[]elem[]not[]unit prime[]elem
[]dvd[]multD$\OT1/cmr/m/n/10 )$[] 
[94] [95] [96]
Overfull \hbox (11.69632pt too wide) in paragraph at lines 1749--1753
[][]  \OT1/cmr/bx/n/10 from \OT1/cmr/m/it/10 assms \OT1/cmr/bx/n/10 have \OT1/c
mr/m/it/10 ir-re-ducible q \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it
/10 simp add$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 prime[]elem[]imp[]irreducible 
prime[]def$\OT1/cmr/m/n/10 )$[] 
[97]
Overfull \hbox (30.63289pt too wide) in paragraph at lines 1904--1918
[][]  \OT1/cmr/bx/n/10 us-ing \OT1/cmr/m/it/10 assms \OT1/cmr/bx/n/10 by $\OT1/
cmr/m/n/10 ($\OT1/cmr/m/it/10 auto in-tro$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 p
rod[]mset[]subset[]imp[]dvd prod[]mset[]primes[]dvd[]imp[]subset$\OT1/cmr/m/n/1
0 )$[] 
[98]
Overfull \hbox (11.4095pt too wide) in paragraph at lines 1978--1980
[][]    \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 auto simp$\OT1/
cmr/m/n/10 :$ \OT1/cmr/m/it/10 A1[]def A2[]def Mul-ti-set$\OML/cmm/m/it/10 :$\O
T1/cmr/m/it/10 subset[]eq[]diff[]conv Mul-ti-set$\OML/cmm/m/it/10 :$\OT1/cmr/m/
it/10 union[]commute$\OT1/cmr/m/n/10 )$[] 

Overfull \hbox (1.77489pt too wide) in paragraph at lines 1993--1995
[][]    \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 subst $\OT1/cmr
/m/n/10 ($\OT1/cmr/m/it/10 asm$\OT1/cmr/m/n/10 )$ $($\OT1/cmr/m/it/10 1 2$\OT1/
cmr/m/n/10 )$ \OT1/cmr/m/it/10 is[]unit[]prod[]mset[]primes[]iff$\OT1/cmr/m/n/1
0 )$ $($\OT1/cmr/m/it/10 auto dest$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 Mul-ti-s
et$\OML/cmm/m/it/10 :$\OT1/cmr/m/it/10 in[]diffD$\OT1/cmr/m/n/10 )$[] 

Overfull \hbox (1.20242pt too wide) in paragraph at lines 2037--2041
[][]    \OT1/cmr/bx/n/10 from \OT1/cmr/m/it/10 x \OT1/cmr/bx/n/10 have \OT1/cmr
/m/it/10 x ^ n dvd prod[]mset A ^ n \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1
/cmr/m/it/10 intro dvd[]power[]same dvd[]prod[]mset$\OT1/cmr/m/n/10 )$[] 
[99] [100]
Overfull \hbox (11.16814pt too wide) in paragraph at lines 2290--2290
[]\OT1/cmr/bx/n/12 Factorial semir-ings: al-ge-braic struc-tures with unique pr
ime

Overfull \hbox (6.13147pt too wide) in paragraph at lines 2337--2341
[][]      \OT1/cmr/bx/n/10 with []$\OMS/cmsy/m/n/10 :$\OT1/cmr/m/it/10 is[]unit
 a[] less$\OML/cmm/m/it/10 :$\OT1/cmr/m/it/10 prems \OT1/cmr/bx/n/10 have \OT1/
cmr/m/it/10 ir-re-ducible a \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/i
t/10 auto simp$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 irreducible[]altdef$\OT1/cmr
/m/n/10 )$[] 
[101]
Overfull \hbox (5.4238pt too wide) in paragraph at lines 2451--2455
[][]      \OT1/cmr/bx/n/10 from \OT1/cmr/m/it/10 A B \OT1/cmr/bx/n/10 show \OT1
/cmr/m/it/10 ?the-sis \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 i
ntro exI$\OT1/cmr/m/n/10 [$\OT1/cmr/m/it/10 of [] A $\OT1/cmr/m/n/10 +$ \OT1/cm
r/m/it/10 B$\OT1/cmr/m/n/10 ]$$)$ $($\OT1/cmr/m/it/10 auto simp$\OT1/cmr/m/n/10
 :$ \OT1/cmr/m/it/10 c normalize[]mult$\OT1/cmr/m/n/10 )$[] 
[102]
Overfull \hbox (7.38255pt too wide) in paragraph at lines 2584--2586
[][]    \OT1/cmr/bx/n/10 from \OT1/cmr/m/it/10 dvd prod[]mset[]primes[]irreduci
ble[]imp[]prime$\OT1/cmr/m/n/10 [$\OT1/cmr/m/it/10 of A B C$\OML/cmm/m/it/10 ;$
 \OT1/cmr/m/it/10 OF this ABC$\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 1$\OML/cmm/m/i
t/10 ;$\OT1/cmr/m/it/10 3$\OML/cmm/m/it/10 ;$\OT1/cmr/m/it/10 5$\OT1/cmr/m/n/10
 )$$]$
[103]
Overfull \hbox (6.55193pt too wide) in paragraph at lines 2685--2690
[][]    \OT1/cmr/bx/n/10 fi-nally show \OT1/cmr/m/it/10 p $\OMS/cmsy/m/n/10 2$$
\OT1/cmr/m/n/10 #$ \OT1/cmr/m/it/10 A \OT1/cmr/bx/n/10 us-ing \OT1/cmr/m/it/10 
p A \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 subst $\OT1/cmr/m/n
/10 ($\OT1/cmr/m/it/10 asm$\OT1/cmr/m/n/10 )$ \OT1/cmr/m/it/10 prime[]dvd[]prod
[]mset[]primes[]iff$\OT1/cmr/m/n/10 )$[] 
[104] [105] [106]
Overfull \hbox (35.03365pt too wide) in paragraph at lines 3203--3217
[][]  \OT1/cmr/bx/n/10 us-ing \OT1/cmr/m/it/10 assms \OT1/cmr/bx/n/10 by $\OT1/
cmr/m/n/10 ($\OT1/cmr/m/it/10 subst prime[]elem[]multiplicity[]eq[]zero[]iff$\O
T1/cmr/m/n/10 )$ $($\OT1/cmr/m/it/10 auto dest$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it
/10 primes[]dvd[]imp[]eq$\OT1/cmr/m/n/10 )$

Overfull \hbox (31.33105pt too wide) in paragraph at lines 3329--3334
[][]    \OT1/cmr/bx/n/10 more-over from \OT1/cmr/m/it/10 assms$\OT1/cmr/m/n/10 
($\OT1/cmr/m/it/10 2$\OT1/cmr/m/n/10 )$ \OT1/cmr/m/it/10 xp \OT1/cmr/bx/n/10 ha
ve \OT1/cmr/m/it/10 mul-ti-plic-ity p x $\OML/cmm/m/it/10 <$ \OT1/cmr/m/it/10 S
uc n \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 intro multiplicity
[]lessI$\OT1/cmr/m/n/10 )$[] 
[107] [108]
Overfull \hbox (42.299pt too wide) in paragraph at lines 3591--3593
[][]    \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 subst mult$\OML
/cmm/m/it/10 :$\OT1/cmr/m/it/10 commute$\OT1/cmr/m/n/10 )$ $($\OT1/cmr/m/it/10 
simp add$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 mult[]unit[]dvd[]iff power[]mult[]
distrib is[]unit[]power[]iff$\OT1/cmr/m/n/10 )$[] 
[109] [110]
Overfull \hbox (2.52225pt too wide) in paragraph at lines 3870--3872
[][]    \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 intro multiplic
ity[]geI $\OT1/cmr/m/n/10 )$ $($\OT1/cmr/m/it/10 auto in-tro$\OT1/cmr/m/n/10 :$
 \OT1/cmr/m/it/10 dvd[]trans$\OT1/cmr/m/n/10 [$\OT1/cmr/m/it/10 OF multiplicity
[]dvd$ [] $ assms$\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 1$\OT1/cmr/m/n/10 )$$]$$)$
[] 

Overfull \hbox (4.71416pt too wide) in paragraph at lines 3931--3936
[][]      \OT1/cmr/bx/n/10 also from \OT1/cmr/m/it/10 x p \OT1/cmr/bx/n/10 have
 $[] $ $\OT1/cmr/m/n/10 =$ \OT1/cmr/m/it/10 0 $\OMS/cmsy/m/n/10  []!$ $:$\OT1/c
mr/m/it/10 p dvd x \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp
 add$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 multiplicity[]eq[]zero[]iff$\OT1/cmr/m
/n/10 )$[] 
[111]
Overfull \hbox (0.98625pt too wide) in paragraph at lines 4066--4068
[][]  \OT1/cmr/bx/n/10 thus \OT1/cmr/m/it/10 count $\OT1/cmr/m/n/10 ($\OT1/cmr/
m/it/10 prime[]factorization $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 p $\OMS/cmsy/m
/n/10 ^^C$ \OT1/cmr/m/it/10 x$\OT1/cmr/m/n/10 )$$)$ \OT1/cmr/m/it/10 q $\OT1/cm
r/m/n/10 =$ \OT1/cmr/m/it/10 count $\OT1/cmr/m/n/10 ($$\OMS/cmsy/m/n/10 f$$\OT1
/cmr/m/n/10 #$\OT1/cmr/m/it/10 p$\OT1/cmr/m/n/10 #$$\OMS/cmsy/m/n/10 g$ $\OT1/c
mr/m/n/10 +$ \OT1/cmr/m/it/10 prime[]factorization
[112]
Overfull \hbox (15.81303pt too wide) in paragraph at lines 4304--4306
[][]    \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp[]all add$\
OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 mult[]ac prime[]factorization[]times[]prime 
Mul-ti-set$\OML/cmm/m/it/10 :$\OT1/cmr/m/it/10 union[]commute$\OT1/cmr/m/n/10 )
$[] 
[113]
Overfull \hbox (40.95099pt too wide) in paragraph at lines 4355--4359
[][]  \OT1/cmr/bx/n/10 with \OT1/cmr/m/it/10 assms \OT1/cmr/bx/n/10 show \OT1/c
mr/m/it/10 nor-mal-ize x $\OT1/cmr/m/n/10 =$ \OT1/cmr/m/it/10 nor-mal-ize y \OT
1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp add$\OT1/cmr/m/n/10 :
$ \OT1/cmr/m/it/10 prod[]mset[]prime[]factorization$\OT1/cmr/m/n/10 )$[] 

Overfull \hbox (44.55727pt too wide) in paragraph at lines 4390--4392
[][]    \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 rule prime[]fac
torization[]prod[]mset[]primes$\OT1/cmr/m/n/10 )$ $($\OT1/cmr/m/it/10 simp[]all
 add$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 in[]prime[]factors[]imp[]prime$\OT1/cm
r/m/n/10 )$[] 

Overfull \hbox (20.85785pt too wide) in paragraph at lines 4392--4395
[][]  \OT1/cmr/bx/n/10 also have \OT1/cmr/m/it/10 prime[]factorization $\OT1/cm
r/m/n/10 ($\OT1/cmr/m/it/10 prod[]mset $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 prim
e[]factorization x $\OT1/cmr/m/n/10 +$ \OT1/cmr/m/it/10 prime[]factorization

Overfull \hbox (33.42188pt too wide) in paragraph at lines 4396--4398
[][]    \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 rule prime[]fac
torization[]prod[]mset[]primes$\OT1/cmr/m/n/10 )$ $($\OT1/cmr/m/it/10 auto simp
$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 in[]prime[]factors[]imp[]prime$\OT1/cmr/m/
n/10 )$[] 
[114]
Overfull \hbox (25.28094pt too wide) in paragraph at lines 4520--4521
[][]                   $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp[]all add$\OT1/c
mr/m/n/10 :$ \OT1/cmr/m/it/10 prime[]elem[]multiplicity[]prod[]mset[]distrib su
m[]unfold[]sum[]mset

Overfull \hbox (16.5629pt too wide) in paragraph at lines 4545--4558
[][]  \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 subst prime[]elem
[]multiplicity[]power[]distrib$\OT1/cmr/m/n/10 )$ $($\OT1/cmr/m/it/10 auto simp
$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 prime[]multiplicity[]other$\OT1/cmr/m/n/10
 )$[] 

Overfull \hbox (50.5364pt too wide) in paragraph at lines 4572--4578
[][]     $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp[]all add$\OT1/cmr/m/n/10 :$ \
OT1/cmr/m/it/10 prime[]factorization[]mult prime[]factorization[]prime Mul-ti-s
et$\OML/cmm/m/it/10 :$\OT1/cmr/m/it/10 union[]commute$\OT1/cmr/m/n/10 )$[] 

Overfull \hbox (22.83113pt too wide) in paragraph at lines 4611--4613
[][]  \OT1/cmr/bx/n/10 have \OT1/cmr/m/it/10 x dvd y $\OMS/cmsy/m/n/10  []!$ \O
T1/cmr/m/it/10 prod[]mset $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 prime[]factorizat
ion x$\OT1/cmr/m/n/10 )$ \OT1/cmr/m/it/10 dvd prod[]mset $\OT1/cmr/m/n/10 ($\OT
1/cmr/m/it/10 prime[]factorization
[115]
Overfull \hbox (6.21616pt too wide) in paragraph at lines 4656--4657
[][]  \OT1/cmr/bx/n/10 shows   \OT1/cmr/m/it/10 prime[]factorization $\OT1/cmr/
m/n/10 ($\OT1/cmr/m/it/10 a div b$\OT1/cmr/m/n/10 )$ $=$ \OT1/cmr/m/it/10 prime
[]factorization a $\OMS/cmsy/m/n/10 ^^@$ \OT1/cmr/m/it/10 prime[]factorization

Overfull \hbox (12.16023pt too wide) in paragraph at lines 4752--4753
[][]      $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp[]all add$\OT1/cmr/m/n/10 :$ 
\OT1/cmr/m/it/10 assms count[]prime[]factorization[]prime in[]prime[]factors[]i
mp[]prime$\OT1/cmr/m/n/10 )$[] 

Overfull \hbox (24.46587pt too wide) in paragraph at lines 4794--4797
[][]    \OT1/cmr/bx/n/10 un-fold-ing \OT1/cmr/m/it/10 A[]def \OT1/cmr/bx/n/10 b
y $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 subst mul-ti-set$\OML/cmm/m/it/10 :$\OT1/
cmr/m/it/10 Abs[]multiset[]inverse$\OT1/cmr/m/n/10 )$ $($\OT1/cmr/m/it/10 simp[
]all add$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 multiset[]def$\OT1/cmr/m/n/10 )$[]
 
[116]
Overfull \hbox (1.20328pt too wide) in paragraph at lines 4806--4810
[][]  \OT1/cmr/bx/n/10 also have $[] $ $\OT1/cmr/m/n/10 =$ \OT1/cmr/m/it/10 pro
d[]mset A \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp add$\OT1
/cmr/m/n/10 :$ \OT1/cmr/m/it/10 prod[]mset[]multiplicity S[]eq set[]mset[]A

Overfull \hbox (29.18195pt too wide) in paragraph at lines 4827--4830
[][]  \OT1/cmr/bx/n/10 also have \OT1/cmr/m/it/10 prime[]factorization $\OT1/cm
r/m/n/10 ($\OT1/cmr/m/it/10 prod[]mset $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 prim
e[]factorization n$\OT1/cmr/m/n/10 )$$)$ $=$ \OT1/cmr/m/it/10 prime[]factorizat
ion

Overfull \hbox (56.44945pt too wide) in paragraph at lines 4864--4866
[][]      \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 intro image[]
mset[]cong$\OT1/cmr/m/n/10 )$ $($\OT1/cmr/m/it/10 auto simp$\OT1/cmr/m/n/10 :$ 
\OT1/cmr/m/it/10 set[]mset[]A multiplicity[]self prime[]multiplicity[]other$\OT
1/cmr/m/n/10 )$[] 
[117]
Overfull \hbox (5.04556pt too wide) in paragraph at lines 4944--4948
[][]  \OT1/cmr/bx/n/10 with \OT1/cmr/m/it/10 assms False \OT1/cmr/bx/n/10 show 
\OT1/cmr/m/it/10 ?the-sis \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/
10 subst $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 asm$\OT1/cmr/m/n/10 )$ \OT1/cmr/m/
it/10 prime[]factorization[]subset[]iff[]dvd$\OT1/cmr/m/n/10 )$[] 
[118]
Overfull \hbox (1.69742pt too wide) in paragraph at lines 5127--5128
[][]          \OT1/cmr/m/it/10 prime[]factorization $\OT1/cmr/m/n/10 ($\OT1/cmr
/m/it/10 prod[]mset $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 prime[]factorization a 
$\OMS/cmsy/m/n/10 \$$\OT1/cmr/m/n/10 #$ \OT1/cmr/m/it/10 prime[]factorization

Overfull \hbox (1.69742pt too wide) in paragraph at lines 5163--5164
[][]          \OT1/cmr/m/it/10 prime[]factorization $\OT1/cmr/m/n/10 ($\OT1/cmr
/m/it/10 prod[]mset $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 prime[]factorization a 
$\OMS/cmsy/m/n/10 [$$\OT1/cmr/m/n/10 #$ \OT1/cmr/m/it/10 prime[]factorization
[119] [120]
Overfull \hbox (4.25296pt too wide) in paragraph at lines 5397--5398
[][]           \OT1/cmr/m/it/10 prime[]factorization $\OT1/cmr/m/n/10 ($\OT1/cm
r/m/it/10 prod[]mset $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 prime[]factorization a
 $\OMS/cmsy/m/n/10 \$$\OT1/cmr/m/n/10 #$ \OT1/cmr/m/it/10 prime[]factorization
[121]
Overfull \hbox (3.08865pt too wide) in paragraph at lines 5538--5541
[][]  \OT1/cmr/bx/n/10 with \OT1/cmr/m/it/10 assms \OT1/cmr/bx/n/10 have \OT1/c
mr/m/it/10 prime[]factorization $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 Gcd[]factor
ial A$\OT1/cmr/m/n/10 )$ $=$ \OT1/cmr/m/it/10 Inf $\OT1/cmr/m/n/10 ($\OT1/cmr/m
/it/10 prime[]factorization
[122] [123]
Overfull \hbox (14.46059pt too wide) in paragraph at lines 5797--5798
[][]  \OT1/cmr/bx/n/10 shows   \OT1/cmr/m/it/10 prime[]factorization $\OT1/cmr/
m/n/10 ($\OT1/cmr/m/it/10 gcd a b$\OT1/cmr/m/n/10 )$ $=$ \OT1/cmr/m/it/10 prime
[]factorization a $\OMS/cmsy/m/n/10 \$$\OT1/cmr/m/n/10 #$ \OT1/cmr/m/it/10 prim
e[]factorization

Overfull \hbox (15.99394pt too wide) in paragraph at lines 5817--5818
[][]  \OT1/cmr/bx/n/10 shows   \OT1/cmr/m/it/10 prime[]factorization $\OT1/cmr/
m/n/10 ($\OT1/cmr/m/it/10 lcm a b$\OT1/cmr/m/n/10 )$ $=$ \OT1/cmr/m/it/10 prime
[]factorization a $\OMS/cmsy/m/n/10 [$$\OT1/cmr/m/n/10 #$ \OT1/cmr/m/it/10 prim
e[]factorization

Overfull \hbox (43.35431pt too wide) in paragraph at lines 5846--5853
[][]  \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp add$\OT1/cmr
/m/n/10 :$ \OT1/cmr/m/it/10 prime[]factorization[]Gcd[]factorial Gcd[]eq[]Gcd[]
factorial Gcd[]factorial[]eq[]0[]iff$\OT1/cmr/m/n/10 )$[] 

Overfull \hbox (49.76532pt too wide) in paragraph at lines 5868--5875
[][]  \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp add$\OT1/cmr
/m/n/10 :$ \OT1/cmr/m/it/10 prime[]factorization[]Lcm[]factorial Lcm[]eq[]Lcm[]
factorial Lcm[]factorial[]eq[]0[]iff$\OT1/cmr/m/n/10 )$[] 
[124]
Overfull \hbox (55.76273pt too wide) in paragraph at lines 5989--5991
[][]    \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp add$\OT1/c
mr/m/n/10 :$ \OT1/cmr/m/it/10 count[]prime[]factorization[]prime $\OT1/cmr/m/n/
10 [$\OT1/cmr/m/it/10 symmetric$\OT1/cmr/m/n/10 ]$ \OT1/cmr/m/it/10 prime[]fact
orization[]gcd[]factorial$\OT1/cmr/m/n/10 )$[] 

Overfull \hbox (57.29608pt too wide) in paragraph at lines 6002--6004
[][]    \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp add$\OT1/c
mr/m/n/10 :$ \OT1/cmr/m/it/10 count[]prime[]factorization[]prime $\OT1/cmr/m/n/
10 [$\OT1/cmr/m/it/10 symmetric$\OT1/cmr/m/n/10 ]$ \OT1/cmr/m/it/10 prime[]fact
orization[]lcm[]factorial$\OT1/cmr/m/n/10 )$[] 
[125]) (./Euclidean_Algorithm.tex
Overfull \hbox (36.91122pt too wide) in paragraph at lines 6--6
[]\OT1/cmr/bx/n/14.4 Abstract eu-clidean al-go-rithm in eu-clidean (semi)rings 

[126] [127] [128] [129] [130] [131] [132] [133]
Overfull \hbox (36.0064pt too wide) in paragraph at lines 1171--1172
[][]  \OT1/cmr/m/it/10 OF-CLASS$\OT1/cmr/m/n/10 ($$ [] $\OT1/cmr/m/it/10 a$\OT1
/cmr/m/n/10 :$$:$$\OMS/cmsy/m/n/10 f$\OT1/cmr/m/it/10 factorial[]semiring[]gcd$
\OML/cmm/m/it/10 ;$ \OT1/cmr/m/it/10 euclidean[]semiring$\OMS/cmsy/m/n/10 g$$\O
ML/cmm/m/it/10 ;$ \OT1/cmr/m/it/10 euclidean[]semiring[]gcd[]class$\OT1/cmr/m/n
/10 )$[] 
[134]
Overfull \hbox (2.401pt too wide) in paragraph at lines 1231--1233
[][]    \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp add$\OT1/c
mr/m/n/10 :$ \OT1/cmr/m/it/10 fun[]eq[]iff Euclidean[]Algorithm$\OML/cmm/m/it/1
0 :$\OT1/cmr/m/it/10 lcm[]def semiring[]gcd[]class$\OML/cmm/m/it/10 :$\OT1/cmr/
m/it/10 lcm[]gcd$\OT1/cmr/m/n/10 )$[] 

Overfull \hbox (13.21527pt too wide) in paragraph at lines 1235--1237
[][]    \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp add$\OT1/c
mr/m/n/10 :$ \OT1/cmr/m/it/10 fun[]eq[]iff Euclidean[]Algorithm$\OML/cmm/m/it/1
0 :$\OT1/cmr/m/it/10 Gcd[]def semiring[]Gcd[]class$\OML/cmm/m/it/10 :$\OT1/cmr/
m/it/10 Gcd[]Lcm$\OT1/cmr/m/n/10 )$[] 
[135]
Overfull \hbox (4.18866pt too wide) in paragraph at lines 1454--1456
[][]      \OT1/cmr/bx/n/10 qed $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 auto simp$\O
T1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 gcd[]mod[]right algebra[]simps minus[]mod[]eq
[]div[]mult $\OT1/cmr/m/n/10 [$\OT1/cmr/m/it/10 symmetric$\OT1/cmr/m/n/10 ]$
[136] [137]
Overfull \hbox (2.401pt too wide) in paragraph at lines 1668--1670
[][]    \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp add$\OT1/c
mr/m/n/10 :$ \OT1/cmr/m/it/10 fun[]eq[]iff Euclidean[]Algorithm$\OML/cmm/m/it/1
0 :$\OT1/cmr/m/it/10 lcm[]def semiring[]gcd[]class$\OML/cmm/m/it/10 :$\OT1/cmr/
m/it/10 lcm[]gcd$\OT1/cmr/m/n/10 )$[] 

Overfull \hbox (13.21527pt too wide) in paragraph at lines 1672--1674
[][]    \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp add$\OT1/c
mr/m/n/10 :$ \OT1/cmr/m/it/10 fun[]eq[]iff Euclidean[]Algorithm$\OML/cmm/m/it/1
0 :$\OT1/cmr/m/it/10 Gcd[]def semiring[]Gcd[]class$\OML/cmm/m/it/10 :$\OT1/cmr/
m/it/10 Gcd[]Lcm$\OT1/cmr/m/n/10 )$[] 
[138]
Overfull \hbox (2.401pt too wide) in paragraph at lines 1771--1773
[][]    \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp add$\OT1/c
mr/m/n/10 :$ \OT1/cmr/m/it/10 fun[]eq[]iff Euclidean[]Algorithm$\OML/cmm/m/it/1
0 :$\OT1/cmr/m/it/10 lcm[]def semiring[]gcd[]class$\OML/cmm/m/it/10 :$\OT1/cmr/
m/it/10 lcm[]gcd$\OT1/cmr/m/n/10 )$[] 

Overfull \hbox (13.21527pt too wide) in paragraph at lines 1775--1777
[][]    \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp add$\OT1/c
mr/m/n/10 :$ \OT1/cmr/m/it/10 fun[]eq[]iff Euclidean[]Algorithm$\OML/cmm/m/it/1
0 :$\OT1/cmr/m/it/10 Gcd[]def semiring[]Gcd[]class$\OML/cmm/m/it/10 :$\OT1/cmr/
m/it/10 Gcd[]Lcm$\OT1/cmr/m/n/10 )$[] 
) (./Ring_Hom.tex [139] [140]) (./Fraction_Field.tex [141] [142] [143] [144]
[145] [146] [147] [148] [149] [150]) (./Normalized_Fraction.tex [151] [152]
Overfull \hbox (25.4372pt too wide) in paragraph at lines 205--207
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 normalize[]quot[]in[]normalized[]fr
acts $\OT1/cmr/m/n/10 [$\OT1/cmr/m/it/10 simp$\OT1/cmr/m/n/10 ]$$:$ \OT1/cmr/m/
it/10 normalize[]quot x $\OMS/cmsy/m/n/10 2$ \OT1/cmr/m/it/10 normalized[]fract
s[] 

Overfull \hbox (11.43802pt too wide) in paragraph at lines 251--253
[][]    \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 intro coprime[]
crossproduct$ [] $$\OT1/cmr/m/n/10 )$ $($\OT1/cmr/m/it/10 simp[]all add$\OT1/cm
r/m/n/10 :$ \OT1/cmr/m/it/10 x[]def y[]def coprime[]normalize[]quot$\OT1/cmr/m/
n/10 )$[] 

Overfull \hbox (1.46599pt too wide) in paragraph at lines 276--290
[][]  \OT1/cmr/bx/n/10 us-ing \OT1/cmr/m/it/10 assms \OT1/cmr/bx/n/10 by $\OT1/
cmr/m/n/10 ($\OT1/cmr/m/it/10 cases x$\OML/cmm/m/it/10 ;$ \OT1/cmr/m/it/10 case
s y$\OML/cmm/m/it/10 ;$ \OT1/cmr/m/it/10 hyp-subst$\OT1/cmr/m/n/10 )$ $($\OT1/c
mr/m/it/10 subst normalize[]quot[]eq[]iff$\OML/cmm/m/it/10 ;$ \OT1/cmr/m/it/10 
simp[]all$\OT1/cmr/m/n/10 )$[] 

Overfull \hbox (26.27673pt too wide) in paragraph at lines 311--313
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 normalize[]quot[]idem $\OT1/cmr/m/n
/10 [$\OT1/cmr/m/it/10 simp$\OT1/cmr/m/n/10 ]$$:$ \OT1/cmr/m/it/10 normalize[]q
uot $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 normalize[]quot x$\OT1/cmr/m/n/10 )$ $=
$ \OT1/cmr/m/it/10 normalize[]quot
[153]
Overfull \hbox (2.21165pt too wide) in paragraph at lines 476--480
[][]  \OT1/cmr/bx/n/10 thus \OT1/cmr/m/it/10 ?the-sis \OT1/cmr/bx/n/10 un-fold-
ing \OT1/cmr/m/it/10 quot[]to[]fract[]def \OT1/cmr/bx/n/10 by \OT1/cmr/m/it/10 
trans-fer $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp add$\OT1/cmr/m/n/10 :$ \OT1/
cmr/m/it/10 case[]prod[]unfold$\OT1/cmr/m/n/10 )$[] 

Overfull \hbox (3.45944pt too wide) in paragraph at lines 510--512
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 quot[]of[]fract[]in[]normalized[]fr
acts $\OT1/cmr/m/n/10 [$\OT1/cmr/m/it/10 simp$\OT1/cmr/m/n/10 ]$$:$ \OT1/cmr/m/
it/10 quot[]of[]fract x $\OMS/cmsy/m/n/10 2$ \OT1/cmr/m/it/10 normalized[]fract
s[] 
[154] [155]
Overfull \hbox (12.708pt too wide) in paragraph at lines 742--746
[][]  \OT1/cmr/bx/n/10 from \OT1/cmr/m/it/10 assms \OT1/cmr/bx/n/10 have \OT1/c
mr/m/it/10 co-prime e f co-prime g h \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT
1/cmr/m/it/10 simp[]all add$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 coprime[]normal
ize[]quot$\OT1/cmr/m/n/10 )$[] 

Overfull \hbox (21.90312pt too wide) in paragraph at lines 749--751
[][]    \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp add$\OT1/c
mr/m/n/10 :$ \OT1/cmr/m/it/10 normalized[]fracts[]def unit[]factor[]mult coprim
e[]mul[]eq coprime[]mul[]eq$ [] $$\OT1/cmr/m/n/10 )$[] 

Overfull \hbox (36.04117pt too wide) in paragraph at lines 842--855
[][]  \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 auto simp$\OT1/cm
r/m/n/10 :$ \OT1/cmr/m/it/10 normalize[]quot[]def case[]prod[]unfold Let[]def d
iv[]mult[]unit2 dvd[]div[]eq[]0[]iff$\OT1/cmr/m/n/10 )$[] 
[156]
Overfull \hbox (15.3068pt too wide) in paragraph at lines 907--910
[][]    \OT1/cmr/bx/n/10 us-ing \OT1/cmr/m/it/10 assms$\OT1/cmr/m/n/10 ($\OT1/c
mr/m/it/10 1$\OML/cmm/m/it/10 ;$\OT1/cmr/m/it/10 2$\OT1/cmr/m/n/10 )$ \OT1/cmr/
m/it/10 d \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 auto simp$\OT
1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 normalized[]fracts[]def gcd[]div[]unit1 gcd[]d
iv[]unit2

Overfull \hbox (49.1144pt too wide) in paragraph at lines 971--975
[][]  \OT1/cmr/bx/n/10 from \OT1/cmr/m/it/10 assms \OT1/cmr/bx/n/10 have \OT1/c
mr/m/it/10 co-prime x$ [] $ y$ [] $ unit[]factor y$ [] $ $\OT1/cmr/m/n/10 =$ \O
T1/cmr/m/it/10 1 \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp[]
all add$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 coprime[]normalize[]quot$\OT1/cmr/m
/n/10 )$[] 

Overfull \hbox (29.09537pt too wide) in paragraph at lines 980--981
[][]       $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 auto simp$\OT1/cmr/m/n/10 :$ \OT
1/cmr/m/it/10 normalized[]fracts[]def unit[]div[]mult[]swap unit[]div[]commute 
unit[]div[]cancel[] 

Overfull \hbox (49.1144pt too wide) in paragraph at lines 1013--1017
[][]  \OT1/cmr/bx/n/10 from \OT1/cmr/m/it/10 assms \OT1/cmr/bx/n/10 have \OT1/c
mr/m/it/10 co-prime x$ [] $ y$ [] $ unit[]factor y$ [] $ $\OT1/cmr/m/n/10 =$ \O
T1/cmr/m/it/10 1 \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp[]
all add$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 coprime[]normalize[]quot$\OT1/cmr/m
/n/10 )$[] 

Overfull \hbox (29.09537pt too wide) in paragraph at lines 1022--1023
[][]       $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 auto simp$\OT1/cmr/m/n/10 :$ \OT
1/cmr/m/it/10 normalized[]fracts[]def unit[]div[]mult[]swap unit[]div[]commute 
unit[]div[]cancel[] 
[157]) (./Field_as_Ring.tex [158]
Overfull \hbox (21.73355pt too wide) in paragraph at lines 116--132
[][]\OT1/cmr/bx/n/10 instance by \OT1/cmr/m/it/10 stan-dard $\OT1/cmr/m/n/10 ($
\OT1/cmr/m/it/10 simp[]all add$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 gcd[]real[]d
ef lcm[]real[]def Gcd[]real[]def Lcm[]real[]def$\OT1/cmr/m/n/10 )$[] 
[159]
Overfull \hbox (8.4448pt too wide) in paragraph at lines 190--206
[][]\OT1/cmr/bx/n/10 instance by \OT1/cmr/m/it/10 stan-dard $\OT1/cmr/m/n/10 ($
\OT1/cmr/m/it/10 simp[]all add$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 gcd[]rat[]de
f lcm[]rat[]def Gcd[]rat[]def Lcm[]rat[]def$\OT1/cmr/m/n/10 )$[] 
[160]
Overfull \hbox (15.7032pt too wide) in paragraph at lines 264--280
[][]\OT1/cmr/bx/n/10 instance by \OT1/cmr/m/it/10 stan-dard $\OT1/cmr/m/n/10 ($
\OT1/cmr/m/it/10 simp[]all add$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 gcd[]complex
[]def lcm[]complex[]def Gcd[]complex[]def
) (./Polynomial_Factorial.tex
Overfull \hbox (2.32649pt too wide) in paragraph at lines 83--87
[][]    \OT1/cmr/bx/n/10 then ob-tain \OT1/cmr/m/it/10 a$ [] $ b$ [] $ \OT1/cmr
/bx/n/10 where \OT1/cmr/m/it/10 ab$ [] $$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 a 
$\OT1/cmr/m/n/10 =$ $[$$:$\OT1/cmr/m/it/10 a$ [] $$\OT1/cmr/m/n/10 :$$]$ \OT1/c
mr/m/it/10 b $\OT1/cmr/m/n/10 =$ $[$$:$\OT1/cmr/m/it/10 b$ [] $$\OT1/cmr/m/n/10
 :$$]$ \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 auto elim$\OT1/c
mr/m/n/10 !$$:$ \OT1/cmr/m/it/10 degree[]eq[]zeroE$\OT1/cmr/m/n/10 )$[] 
[161] [162]
Overfull \hbox (30.61302pt too wide) in paragraph at lines 511--513
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 normalize[]snd[]quot[]of[]fract$\OT
1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 nor-mal-ize $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/1
0 snd $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 quot[]of[]fract x$\OT1/cmr/m/n/10 )$$
)$ $=$ \OT1/cmr/m/it/10 snd $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 quot[]of[]fract


Overfull \hbox (3.69643pt too wide) in paragraph at lines 539--541
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 fract[]poly[]smult $\OT1/cmr/m/n/10
 [$\OT1/cmr/m/it/10 simp$\OT1/cmr/m/n/10 ]$$:$ \OT1/cmr/m/it/10 fract[]poly $\O
T1/cmr/m/n/10 ($\OT1/cmr/m/it/10 smult c p$\OT1/cmr/m/n/10 )$ $=$ \OT1/cmr/m/it
/10 smult $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 to[]fract c$\OT1/cmr/m/n/10 )$ $(
$\OT1/cmr/m/it/10 fract[]poly
[163]
Overfull \hbox (9.21042pt too wide) in paragraph at lines 725--726
[][]  \OT1/cmr/m/it/10 prod[]mset $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 image[]ms
et $\OT1/cmr/m/n/10 ($$\OML/cmm/m/it/10 ^^U$\OT1/cmr/m/it/10 x$\OML/cmm/m/it/10
 :$ \OT1/cmr/m/it/10 fract[]poly $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 f x$\OT1/c
mr/m/n/10 )$$)$ \OT1/cmr/m/it/10 A$\OT1/cmr/m/n/10 )$ $=$ \OT1/cmr/m/it/10 frac
t[]poly $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 prod[]mset $\OT1/cmr/m/n/10 ($\OT1/
cmr/m/it/10 image[]mset
[164]
Overfull \hbox (5.10414pt too wide) in paragraph at lines 850--853
[][]  \OT1/cmr/bx/n/10 hence \OT1/cmr/m/it/10 fract[]poly $\OT1/cmr/m/n/10 ($\O
T1/cmr/m/it/10 smult a q$\OT1/cmr/m/n/10 )$ $=$ \OT1/cmr/m/it/10 fract[]poly $\
OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 smult b p$\OT1/cmr/m/n/10 )$ \OT1/cmr/bx/n/10
 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp del$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/
it/10 fract[]poly[]eq[]iff$\OT1/cmr/m/n/10 )$[] 
[165] [166]
Overfull \hbox (9.28085pt too wide) in paragraph at lines 1049--1051
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 fract[]content[]fract[]poly $\OT1/c
mr/m/n/10 [$\OT1/cmr/m/it/10 simp$\OT1/cmr/m/n/10 ]$$:$ \OT1/cmr/m/it/10 fract[
]content $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 fract[]poly p$\OT1/cmr/m/n/10 )$ $
=$ \OT1/cmr/m/it/10 to[]fract $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 content

Overfull \hbox (54.7458pt too wide) in paragraph at lines 1113--1115
[][]    \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 rule that$\OT1/
cmr/m/n/10 [$\OT1/cmr/m/it/10 OF content[]times[]primitive[]part[]fract $\OT1/c
mr/m/n/10 [$\OT1/cmr/m/it/10 symmetric$\OT1/cmr/m/n/10 ]$ \OT1/cmr/m/it/10 cont
ent[]primitive[]part[]fract$\OT1/cmr/m/n/10 ]$$)$[] 
[167] [168]
Overfull \hbox (14.12436pt too wide) in paragraph at lines 1301--1305
[][]    \OT1/cmr/bx/n/10 from \OT1/cmr/m/it/10 A \OT1/cmr/bx/n/10 and \OT1/cmr/
m/it/10 this \OT1/cmr/bx/n/10 have $\OT1/cmr/m/n/10 [$$:$\OT1/cmr/m/it/10 c$\OT
1/cmr/m/n/10 :$$]$ \OT1/cmr/m/it/10 dvd $\OT1/cmr/m/n/10 [$$:$\OT1/cmr/m/it/10 
a$\OT1/cmr/m/n/10 :$$]$ $\OMS/cmsy/m/n/10 _$ $\OT1/cmr/m/n/10 [$$:$\OT1/cmr/m/i
t/10 c$\OT1/cmr/m/n/10 :$$]$ \OT1/cmr/m/it/10 dvd $\OT1/cmr/m/n/10 [$$:$\OT1/cm
r/m/it/10 b$\OT1/cmr/m/n/10 :$$]$ \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/c
mr/m/it/10 rule prime[]elem[]dvd[]multD$\OT1/cmr/m/n/10 )$[] 

Overfull \hbox (17.28046pt too wide) in paragraph at lines 1365--1370
[][]    \OT1/cmr/bx/n/10 more-over from []\OT1/cmr/m/it/10 prime p[] \OT1/cmr/b
x/n/10 have \OT1/cmr/m/it/10 prime[]elem $\OT1/cmr/m/n/10 [$$:$\OT1/cmr/m/it/10
 p$\OT1/cmr/m/n/10 :$$]$ \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/1
0 simp add$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 lift[]prime[]elem[]poly$\OT1/cmr
/m/n/10 )$[] 
[169] [170] [171]
Overfull \hbox (8.21928pt too wide) in paragraph at lines 1744--1746
[][]    \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 elim is[]unit[]
polyE$\OT1/cmr/m/n/10 )$ $($\OT1/cmr/m/it/10 auto simp$\OT1/cmr/m/n/10 :$ \OT1/
cmr/m/it/10 unit[]factor[]field[]poly[]def monom[]0 one[]poly[]def

Overfull \hbox (10.56847pt too wide) in paragraph at lines 1781--1782
[][]      $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 auto simp add$\OT1/cmr/m/n/10 :$ 
\OT1/cmr/m/it/10 unit[]factor[]field[]poly[]def euclidean[]size[]field[]poly[]d
ef div[]poly[]less$\OT1/cmr/m/n/10 )$[] 
[172]
Overfull \hbox (6.67152pt too wide) in paragraph at lines 1842--1844
[][]    \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 intro ext$\OT1/
cmr/m/n/10 )$ $($\OT1/cmr/m/it/10 simp add$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 
comm[]monoid[]mult$\OML/cmm/m/it/10 :$\OT1/cmr/m/it/10 prod[]mset[]def$\OT1/cmr
/m/n/10 [$\OT1/cmr/m/it/10 OF A$\OT1/cmr/m/n/10 ]$ \OT1/cmr/m/it/10 prod[]mset[
]def$\OT1/cmr/m/n/10 )$[] 

Overfull \hbox (0.43723pt too wide) in paragraph at lines 1882--1884
[][]      \OT1/cmr/m/it/10 comm[]semiring[]1$\OML/cmm/m/it/10 :$\OT1/cmr/m/it/1
0 prime[]elem[]def$\OT1/cmr/m/n/10 [$\OT1/cmr/m/it/10 OF A$\OT1/cmr/m/n/10 ]$ \
OT1/cmr/m/it/10 normalization[]semidom$\OML/cmm/m/it/10 :$\OT1/cmr/m/it/10 prim
e[]def$\OT1/cmr/m/n/10 [$\OT1/cmr/m/it/10 OF
[173]
Overfull \hbox (3.74106pt too wide) in paragraph at lines 2045--2048
[][]   \OT1/cmr/bx/n/10 hence \OT1/cmr/m/it/10 q$ [] $ dvd 1 $\OMS/cmsy/m/n/10 
_$ \OT1/cmr/m/it/10 r$ [] $ dvd 1 \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/c
mr/m/it/10 auto dest$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 dvd[]mult[]right simp 
del$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 mult[]pCons[]left$\OT1/cmr/m/n/10 )$[] 

[174]
Overfull \hbox (41.16667pt too wide) in paragraph at lines 2143--2147
[][]  \OT1/cmr/bx/n/10 from \OT1/cmr/m/it/10 ir-red \OT1/cmr/bx/n/10 have \OT1/
cmr/m/it/10 prime$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 prime[]elem $\OT1/cmr/m/n
/10 ($\OT1/cmr/m/it/10 fract[]poly p$\OT1/cmr/m/n/10 )$ \OT1/cmr/bx/n/10 by $\O
T1/cmr/m/n/10 ($\OT1/cmr/m/it/10 rule field[]poly[]irreducible[]imp[]prime$\OT1
/cmr/m/n/10 )$[] 
[175] [176]
Overfull \hbox (2.86989pt too wide) in paragraph at lines 2444--2446
[][]    \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp add$\OT1/c
mr/m/n/10 :$ \OT1/cmr/m/it/10 unit[]factor[]field[]poly[]def monom[]0 degree[]m
ap[]poly coeff[]map[]poly$\OT1/cmr/m/n/10 )$[] 

Overfull \hbox (14.1927pt too wide) in paragraph at lines 2465--2467
[][]    \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 subst prod[]mse
t$\OML/cmm/m/it/10 :$\OT1/cmr/m/it/10 distrib$\OT1/cmr/m/n/10 )$ $($\OT1/cmr/m/
it/10 simp[]all add$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 prod[]mset[]fract[]poly
 prod[]mset[]const[]poly
[177]
Overfull \hbox (47.79518pt too wide) in paragraph at lines 2535--2538
[][]    \OT1/cmr/bx/n/10 us-ing \OT1/cmr/m/it/10 that \OT1/cmr/bx/n/10 by $\OT1
/cmr/m/n/10 ($\OT1/cmr/m/it/10 auto simp$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 A[
]def prime[]elem[]primitive[]part[]fract prime[]elem[]imp[]irreducible

Overfull \hbox (14.65166pt too wide) in paragraph at lines 2569--2571
[][]  \OT1/cmr/bx/n/10 have $\OMS/cmsy/m/n/10 9 $\OT1/cmr/m/it/10 A$\OML/cmm/m/
it/10 :$ $\OT1/cmr/m/n/10 ($$\OMS/cmsy/m/n/10 8 $\OT1/cmr/m/it/10 p$\OML/cmm/m/
it/10 :$ \OT1/cmr/m/it/10 p $\OMS/cmsy/m/n/10 2$$\OT1/cmr/m/n/10 #$ \OT1/cmr/m/
it/10 A $[][]\OMS/cmsy/m/n/10 !$ \OT1/cmr/m/it/10 prime[]elem p$\OT1/cmr/m/n/10
 )$ $\OMS/cmsy/m/n/10 ^$ \OT1/cmr/m/it/10 prod[]mset A $\OT1/cmr/m/n/10 =$ \OT1
/cmr/m/it/10 nor-mal-ize $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 primitive[]part

Overfull \hbox (7.14774pt too wide) in paragraph at lines 2587--2589
[][]    \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 auto simp$\OT1/
cmr/m/n/10 :$ \OT1/cmr/m/it/10 B[]def in-tro$\OT1/cmr/m/n/10 !$$:$ \OT1/cmr/m/i
t/10 lift[]prime[]elem[]poly dest$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 in[]prime
[]factors[]imp[]prime$\OT1/cmr/m/n/10 )$[] 
[178]
Overfull \hbox (28.37808pt too wide) in paragraph at lines 2646--2662
[][]\OT1/cmr/bx/n/10 instance by \OT1/cmr/m/it/10 stan-dard $\OT1/cmr/m/n/10 ($
\OT1/cmr/m/it/10 simp[]all add$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 gcd[]poly[]d
ef lcm[]poly[]def Gcd[]poly[]def Lcm[]poly[]def$\OT1/cmr/m/n/10 )$[] 

Overfull \hbox (15.0717pt too wide) in paragraph at lines 2688--2689
[][]   $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 auto simp$\OT1/cmr/m/n/10 :$ \OT1/cm
r/m/it/10 euclidean[]size[]poly[]def Rings$\OML/cmm/m/it/10 :$\OT1/cmr/m/it/10 
div[]mult[]mod[]eq div[]poly[]less degree[]mult[]eq
[179]
Overfull \hbox (10.64919pt too wide) in paragraph at lines 2763--2765
[][]  \OT1/cmr/bx/n/10 thus \OT1/cmr/m/it/10 d dvd smult $\OT1/cmr/m/n/10 ($\OT
1/cmr/m/it/10 gcd $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 content p$\OT1/cmr/m/n/10
 )$ $($\OT1/cmr/m/it/10 content q$\OT1/cmr/m/n/10 )$$)$ $($\OT1/cmr/m/it/10 gcd
 $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 primitive[]part p$\OT1/cmr/m/n/10 )$ $($\O
T1/cmr/m/it/10 primitive[]part
[180]
Overfull \hbox (35.85698pt too wide) in paragraph at lines 2991--3004
[][]  \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp add$\OT1/cmr
/m/n/10 :$ \OT1/cmr/m/it/10 gcd[]poly[]code[]def gcd[]poly[]code[]aux[]correct 
gcd[]poly[]decompose $\OT1/cmr/m/n/10 [$\OT1/cmr/m/it/10 symmetric$\OT1/cmr/m/n
/10 ]$$)$[] 
[181]
Overfull \hbox (31.4pt too wide) in paragraph at lines 3027--3029
[][]\OT1/cmr/bx/n/10 lemmas \OT1/cmr/m/it/10 Gcd[]poly[]set[]eq[]fold $\OT1/cmr
/m/n/10 [$\OT1/cmr/m/it/10 code$\OT1/cmr/m/n/10 ]$ $=$ \OT1/cmr/m/it/10 Gcd[]se
t[]eq[]fold $\OT1/cmr/m/n/10 [$\OT1/cmr/bx/n/10 where \OT1/cmr/m/it/10 ?$ [] $a
 $\OT1/cmr/m/n/10 =$ $ [] $\OT1/cmr/m/it/10 a $\OT1/cmr/m/n/10 :$$:$ \OT1/cmr/m
/it/10 factorial[]ring[]gcd

Overfull \hbox (34.6055pt too wide) in paragraph at lines 3029--3032
[][]\OT1/cmr/bx/n/10 lemmas \OT1/cmr/m/it/10 Lcm[]poly[]set[]eq[]fold $\OT1/cmr
/m/n/10 [$\OT1/cmr/m/it/10 code$\OT1/cmr/m/n/10 ]$ $=$ \OT1/cmr/m/it/10 Lcm[]se
t[]eq[]fold $\OT1/cmr/m/n/10 [$\OT1/cmr/bx/n/10 where \OT1/cmr/m/it/10 ?$ [] $a
 $\OT1/cmr/m/n/10 =$ $ [] $\OT1/cmr/m/it/10 a $\OT1/cmr/m/n/10 :$$:$ \OT1/cmr/m
/it/10 factorial[]ring[]gcd
) (./Missing_Unsorted.tex [182]
Overfull \hbox (22.26895pt too wide) in paragraph at lines 177--182
[][]  \OT1/cmr/bx/n/10 also have $[] $ $\OML/cmm/m/it/10 <$ \OT1/cmr/m/it/10 bb
b $\OMS/cmsy/m/n/10 ^^C$ \OT1/cmr/m/it/10 of[]nat n \OT1/cmr/bx/n/10 us-ing \OT
1/cmr/m/it/10 lt bbb \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 me
tis mult$\OML/cmm/m/it/10 :$\OT1/cmr/m/it/10 commute pos[]divide[]less[]eq$\OT1
/cmr/m/n/10 )$[] 
[183] [184] [185] [186] [187] [188] [189] [190]) (./Missing_Polynomial.tex
[191]
Overfull \hbox (19.45976pt too wide) in paragraph at lines 104--117
[][]  \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 metis Suc[]1 Suc[
]neq[]Zero degree1[]coeffs degree[]pCons[]eq[]if nat$\OML/cmm/m/it/10 :$\OT1/cm
r/m/it/10 inject pCons[]cases$\OT1/cmr/m/n/10 )$[] 
[192] [193] [194] [195] [196] [197]
Overfull \hbox (26.31035pt too wide) in paragraph at lines 1194--1196
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 degree[]prod[]sum[]lt[]n$\OT1/cmr/m
/n/10 :$ \OT1/cmr/bx/n/10 as-sumes $[]$ \OT1/cmr/m/it/10 i$\OML/cmm/m/it/10 :$ 
\OT1/cmr/m/it/10 i $\OML/cmm/m/it/10 <$ \OT1/cmr/m/it/10 n $\OT1/cmr/m/n/10 =[]
\OMS/cmsy/m/n/10 )$ \OT1/cmr/m/it/10 de-gree $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/1
0 f i $\OT1/cmr/m/n/10 :$$:$ $ [] $\OT1/cmr/m/it/10 a $\OT1/cmr/m/n/10 :$$:$ \O
T1/cmr/m/it/10 comm[]semiring[]1
[198] [199] [200] [201] [202] [203] [204] [205] [206] [207] [208]

Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `math shift' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `math shift' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `math shift' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\mathord' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `math shift' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `math shift' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\mathord' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `math shift' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `math shift' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\mskip' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\mskip' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `math shift' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `math shift' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `math shift' on input line 2890.

[209] [210]
Overfull \hbox (46.84029pt too wide) in paragraph at lines 3269--3271
[][]  \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 metis $\OT1/cmr/m
/n/10 ($\OT1/cmr/m/it/10 no[]types$\OML/cmm/m/it/10 ;$ \OT1/cmr/m/it/10 lift-in
g$\OT1/cmr/m/n/10 )$ \OT1/cmr/m/it/10 coeff[]smult degree[]smult[]eq ir-re-duci
bleD$\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 1$\OT1/cmr/m/n/10 )$ \OT1/cmr/m/it/10 i
rreducible[]dvd[]smult
[211]
Overfull \hbox (17.23799pt too wide) in paragraph at lines 3444--3445
[][]  \OT1/cmr/m/it/10 nor-mal-ize $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 f $\OT1/
cmr/m/n/10 :$$:$ $ [] $\OT1/cmr/m/it/10 a $\OT1/cmr/m/n/10 :$$:$ $\OMS/cmsy/m/n
/10 f$\OT1/cmr/m/it/10 normalization[]semidom$\OML/cmm/m/it/10 ;$\OT1/cmr/m/it/
10 field$\OMS/cmsy/m/n/10 g$ \OT1/cmr/m/it/10 poly$\OT1/cmr/m/n/10 )$ $=$ \OT1/
cmr/m/it/10 smult $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 inverse $\OT1/cmr/m/n/10 
($\OT1/cmr/m/it/10 unit[]factor
[212] [213]
Overfull \hbox (4.89697pt too wide) in paragraph at lines 3798--3802
[][]    \OT1/cmr/bx/n/10 have \OT1/cmr/m/it/10 id$\OT1/cmr/m/n/10 :$ \OT1/cmr/m
/it/10 n $\OMS/cmsy/m/n/10 ^^C$ \OT1/cmr/m/it/10 Suc ii $\OT1/cmr/m/n/10 +$ \OT
1/cmr/m/it/10 j $\OMS/cmsy/m/n/10 ^^@$ \OT1/cmr/m/it/10 n $\OT1/cmr/m/n/10 =$ \
OT1/cmr/m/it/10 n $\OMS/cmsy/m/n/10 ^^C$ \OT1/cmr/m/it/10 ii $\OT1/cmr/m/n/10 +
$ \OT1/cmr/m/it/10 j \OT1/cmr/bx/n/10 us-ing \OT1/cmr/m/it/10 n \OT1/cmr/bx/n/1
0 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp add$\OT1/cmr/m/n/10 :$ \OT1/cmr/m
/it/10 diff[]mult[]distrib2$\OT1/cmr/m/n/10 )$[] 
[214]) (./Ring_Hom_Poly.tex
Overfull \hbox (12.6962pt too wide) in paragraph at lines 73--87
[][]  \OT1/cmr/bx/n/10 us-ing \OT1/cmr/m/it/10 pcompose[]add pcompose[]mult pco
mpose[]1 pcompose[]uminus \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/
10 unfold[]locales$\OML/cmm/m/it/10 ;$
[215] [216] [217] [218] [219] [220] [221] [222]) (./Newton_Interpolation.tex
[223]
Overfull \hbox (17.22385pt too wide) in paragraph at lines 50--51
[][]  \OT1/cmr/m/it/10 horner[]composition $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 
map hom cs$\OT1/cmr/m/n/10 )$ $($\OT1/cmr/m/it/10 map hom xs$\OT1/cmr/m/n/10 )$
 $=$ \OT1/cmr/m/it/10 map[]poly hom $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 horner[
]composition
[224] [225] [226] [227] [228] [229] [230] [231] [232] [233] [234]
Overfull \hbox (2.43918pt too wide) in paragraph at lines 1490--1492
[][]    \OT1/cmr/bx/n/10 un-fold-ing \OT1/cmr/m/it/10 newton[]poly[]impl[]def n
ewton[]coefficients rev[]rev[]ident newton[]poly[]def
[235] [236] [237]
Overfull \hbox (4.60117pt too wide) in paragraph at lines 1779--1780
[][]  $\OT1/cmr/m/n/10 =[]\OMS/cmsy/m/n/10 )$ \OT1/cmr/m/it/10 newton[]coeffici
ents[]main $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 map rat[]of[]int xs$\OT1/cmr/m/n
/10 )$ $($\OT1/cmr/m/it/10 map rat[]of[]int gs$\OT1/cmr/m/n/10 )$ $($\OT1/cmr/m
/it/10 map rat[]of[]int
[238] [239] [240]
Overfull \hbox (17.51727pt too wide) in paragraph at lines 2158--2160
[][]    \OT1/cmr/bx/n/10 have \OT1/cmr/m/it/10 id$\OT1/cmr/m/n/10 :$ \OT1/cmr/m
/it/10 horner[]composition $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 rev ?nc$\OT1/cmr
/m/n/10 )$ $($\OT1/cmr/m/it/10 ?mri xs$\OT1/cmr/m/n/10 )$ $=$ \OT1/cmr/m/it/10 
map[]poly ?ri $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 horner[]composition
[241] [242]
Overfull \hbox (6.79475pt too wide) in paragraph at lines 2349--2351
[][]    \OT1/cmr/bx/n/10 un-fold-ing \OT1/cmr/m/it/10 newton[]interpolation[]po
ly[]def  newton[]interpolation[]poly[]int[]def Let[]def
) (./Lagrange_Interpolation.tex [243]
Overfull \hbox (5.18915pt too wide) in paragraph at lines 177--179
[][]    \OT1/cmr/bx/n/10 un-fold-ing \OT1/cmr/m/it/10 lagrange[]basis[]poly[]de
f Let[]def poly[]prod[]list prod[]list[]map[]remove1$\OT1/cmr/m/n/10 [$\OT1/cmr
/m/it/10 OF
[244]
Overfull \hbox (10.95041pt too wide) in paragraph at lines 211--214
[][]    \OT1/cmr/bx/n/10 show \OT1/cmr/m/it/10 poly p x $\OT1/cmr/m/n/10 =$ \OT
1/cmr/m/it/10 y \OT1/cmr/bx/n/10 un-fold-ing \OT1/cmr/m/it/10 p lagrange[]inter
polation[]poly[]def Let[]def poly[]sum[]list

Overfull \hbox (33.61983pt too wide) in paragraph at lines 214--216
[][]    \OT1/cmr/bx/n/10 proof $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 subst sum[]l
ist[]map[]remove1$\OT1/cmr/m/n/10 [$\OT1/cmr/m/it/10 OF xy$\OT1/cmr/m/n/10 ]$$\
OML/cmm/m/it/10 ;$ \OT1/cmr/m/it/10 un-fold split poly[]smult lagrange[]basis[]
poly[]1$\OML/cmm/m/it/10 ;$[] 

Overfull \hbox (17.11581pt too wide) in paragraph at lines 219--221
[][]      \OT1/cmr/bx/n/10 as-sume \OT1/cmr/m/it/10 v $\OMS/cmsy/m/n/10 2$ \OT1
/cmr/m/it/10 set $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 map $\OT1/cmr/m/n/10 ($$\O
ML/cmm/m/it/10 ^^U$\OT1/cmr/m/it/10 xa$\OML/cmm/m/it/10 :$ \OT1/cmr/m/it/10 pol
y $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 case xa of $\OT1/cmr/m/n/10 ($\OT1/cmr/m/
it/10 xj$\OML/cmm/m/it/10 ;$ \OT1/cmr/m/it/10 yj$\OT1/cmr/m/n/10 )$ $\OMS/cmsy/
m/n/10 )$ \OT1/cmr/m/it/10 smult yj $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 lagrang
e[]basis[]poly
) (./Neville_Aitken_Interpolation.tex [245] [246] [247]
Overfull \hbox (0.26402pt too wide) in paragraph at lines 318--320
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 length[]neville[]aitken[]merge$\OT1
/cmr/m/n/10 [$\OT1/cmr/m/it/10 termination[]simp$\OT1/cmr/m/n/10 ]$$:$ \OT1/cmr
/m/it/10 length $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 neville[]aitken[]merge
[248]
Overfull \hbox (30.07036pt too wide) in paragraph at lines 441--444
[][]  \OT1/cmr/bx/n/10 show \OT1/cmr/m/it/10 ?case \OT1/cmr/bx/n/10 un-fold-ing
 \OT1/cmr/m/it/10 neville[]aitken[]impl[]main$\OML/cmm/m/it/10 :$\OT1/cmr/m/it/
10 simps 1$\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 1$\OT1/cmr/m/n/10 )$$[$\OT1/cmr/m
/it/10 OF neville[]aitken[]merge$\OT1/cmr/m/n/10 [$\OT1/cmr/m/it/10 OF
[249]) (./Polynomial_Interpolation.tex [250] [251] [252] [253]
Overfull \hbox (26.8093pt too wide) in paragraph at lines 468--469
[][]  \OT1/cmr/m/it/10 interpolation[]poly[]int[]Some$\OT1/cmr/m/n/10 [$\OT1/cm
r/bx/n/10 where \OT1/cmr/m/it/10 alg $\OT1/cmr/m/n/10 =$ \OT1/cmr/m/it/10 New-t
on$\OML/cmm/m/it/10 ;$ \OT1/cmr/m/it/10 un-folded interpolation[]poly[]int$\OML
/cmm/m/it/10 :$\OT1/cmr/m/it/10 simps$\OT1/cmr/m/n/10 ]$[] 

Overfull \hbox (26.06485pt too wide) in paragraph at lines 472--474
[][]  \OT1/cmr/m/it/10 interpolation[]poly[]int[]None$\OT1/cmr/m/n/10 [$\OT1/cm
r/bx/n/10 where \OT1/cmr/m/it/10 alg $\OT1/cmr/m/n/10 =$ \OT1/cmr/m/it/10 New-t
on$\OML/cmm/m/it/10 ;$ \OT1/cmr/m/it/10 un-folded interpolation[]poly[]int$\OML
/cmm/m/it/10 :$\OT1/cmr/m/it/10 simps$\OT1/cmr/m/n/10 ]$ 
[254])) (./root.bbl) [255] (./root.aux)

LaTeX Warning: There were undefined references.


LaTeX Warning: Label(s) may have changed. Rerun to get cross-references right.

 )
(see the transcript file for additional information)</usr/share/texlive/texmf-d
ist/fonts/type1/public/amsfonts/cm/cmbsy10.pfb></usr/share/texlive/texmf-dist/f
onts/type1/public/amsfonts/cm/cmbx10.pfb></usr/share/texlive/texmf-dist/fonts/t
ype1/public/amsfonts/cm/cmbx12.pfb></usr/share/texlive/texmf-dist/fonts/type1/p
ublic/amsfonts/cm/cmex10.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/
amsfonts/cm/cmmi10.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfon
ts/cm/cmmi12.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/
cmmi7.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi8.p
fb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmr10.pfb></us
r/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmr12.pfb></usr/share
/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmr17.pfb></usr/share/texliv
e/texmf-dist/fonts/type1/public/amsfonts/cm/cmr7.pfb></usr/share/texlive/texmf-
dist/fonts/type1/public/amsfonts/cm/cmr8.pfb></usr/share/texlive/texmf-dist/fon
ts/type1/public/amsfonts/cm/cmr9.pfb></usr/share/texlive/texmf-dist/fonts/type1
/public/amsfonts/cm/cmss10.pfb></usr/share/texlive/texmf-dist/fonts/type1/publi
c/amsfonts/cm/cmsy10.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsf
onts/cm/cmsy5.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm
/cmsy6.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy7.
pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy8.pfb></u
sr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmti10.pfb></usr/sha
re/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmti12.pfb></usr/share/tex
live/texmf-dist/fonts/type1/public/amsfonts/symbols/msam10.pfb>
Output written on root.pdf (255 pages, 831947 bytes).
Transcript written on root.log.
This is pdfTeX, Version 3.14159265-2.6-1.40.16 (TeX Live 2015/Debian) (preloaded format=pdflatex)
 restricted \write18 enabled.
entering extended mode
LaTeX2e <2016/02/01>
Babel <3.9q> and hyphenation patterns for 81 language(s) loaded.

(./root.tex (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls
Document Class: article 2014/09/29 v1.4h Standard LaTeX document class
(/usr/share/texlive/texmf-dist/tex/latex/base/size11.clo)) (./isabelle.sty
(./comment.sty Excluding comment 'comment') Including comment 'isadelimtheory'
Including comment 'isatagtheory' Including comment 'isadelimproof'
Including comment 'isatagproof' Including comment 'isadelimML'
Including comment 'isatagML' Including comment 'isadelimvisible'
Including comment 'isatagvisible' Excluding comment 'isadeliminvisible'
Excluding comment 'isataginvisible') (./isabelletags.sty) (./isabellesym.sty)
(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty
(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty))
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty
For additional information on amsmath, use the `?' option.
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty))
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty)
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty))
(/usr/share/texlive/texmf-dist/tex/latex/tools/xspace.sty) (./pdfsetup.sty
(/usr/share/texlive/texmf-dist/tex/latex/graphics/color.sty
(/usr/share/texlive/texmf-dist/tex/latex/latexconfig/color.cfg)
(/usr/share/texlive/texmf-dist/tex/latex/pdftex-def/pdftex.def
(/usr/share/texlive/texmf-dist/tex/generic/oberdiek/infwarerr.sty)
(/usr/share/texlive/texmf-dist/tex/generic/oberdiek/ltxcmds.sty))))
(/usr/share/texlive/texmf-dist/tex/latex/hyperref/hyperref.sty
(/usr/share/texlive/texmf-dist/tex/generic/oberdiek/hobsub-hyperref.sty
(/usr/share/texlive/texmf-dist/tex/generic/oberdiek/hobsub-generic.sty))
(/usr/share/texlive/texmf-dist/tex/latex/graphics/keyval.sty)
(/usr/share/texlive/texmf-dist/tex/generic/ifxetex/ifxetex.sty)
(/usr/share/texlive/texmf-dist/tex/latex/oberdiek/auxhook.sty)
(/usr/share/texlive/texmf-dist/tex/latex/oberdiek/kvoptions.sty)
(/usr/share/texlive/texmf-dist/tex/latex/hyperref/pd1enc.def)
(/usr/share/texlive/texmf-dist/tex/latex/latexconfig/hyperref.cfg)
(/usr/share/texlive/texmf-dist/tex/latex/url/url.sty))

Package hyperref Message: Driver (autodetected): hpdftex.

(/usr/share/texlive/texmf-dist/tex/latex/hyperref/hpdftex.def
(/usr/share/texlive/texmf-dist/tex/latex/oberdiek/rerunfilecheck.sty))
(./root.aux) (/usr/share/texlive/texmf-dist/tex/context/base/supp-pdf.mkii
[Loading MPS to PDF converter (version 2006.09.02).]
) (/usr/share/texlive/texmf-dist/tex/latex/hyperref/nameref.sty
(/usr/share/texlive/texmf-dist/tex/generic/oberdiek/gettitlestring.sty))
(./root.out) (./root.out)
(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd)
(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd)
(/usr/share/texlive/texmf-dist/tex/latex/base/omscmr.fd) (./root.toc [1{/var/li
b/texmf/fonts/map/pdftex/updmap/pdftex.map}] [2]) [3] (./session.tex
(./More_List.tex) (./Infinite_Set.tex) (./Polynomial.tex)
(./Adhoc_Overloading.tex) (./Monad_Syntax.tex) (./Sqrt_Babylonian_Auxiliary.tex
) (./Is_Rat_To_Rat.tex) (./Divmod_Int.tex [4]
Overfull \hbox (12.96191pt too wide) in paragraph at lines 41--43
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 divmod[]int[]code$\OT1/cmr/m/n/10 [
$\OT1/cmr/m/it/10 code$\OT1/cmr/m/n/10 ]$$:$ \OT1/cmr/m/it/10 divmod[]int m n $
\OT1/cmr/m/n/10 =$ \OT1/cmr/m/it/10 map[]prod int[]of[]integer int[]of[]integer

) (./Improved_Code_Equations.tex

Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 33.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 33.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 33.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 33.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 33.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 33.

[5]
Overfull \hbox (12.29745pt too wide) in paragraph at lines 108--115
[][]   \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 cases l $\OT1/cm
r/m/n/10 =$ \OT1/cmr/m/it/10 0$\OT1/cmr/m/n/10 ;$ \OT1/cmr/m/it/10 cases l $\OM
L/cmm/m/it/10 <$ \OT1/cmr/m/it/10 0$\OT1/cmr/m/n/10 ;$ \OT1/cmr/m/it/10 cases l
 $\OML/cmm/m/it/10 >$ \OT1/cmr/m/it/10 0$\OT1/cmr/m/n/10 ;$ \OT1/cmr/m/it/10 au
to split$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 prod$\OML/cmm/m/it/10 :$\OT1/cmr/m
/it/10 splits simp$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 divmod[]abs$ [] $[]def

Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 128.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 128.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 128.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 128.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 128.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 128.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `math shift' on input line 128.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\mathord' on input line 128.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `math shift' on input line 128.

[6]
Overfull \hbox (32.11456pt too wide) in paragraph at lines 141--143
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 divmod[]nat[]code$\OT1/cmr/m/n/10 :
$ \OT1/cmr/m/it/10 Di-vides$\OML/cmm/m/it/10 :$\OT1/cmr/m/it/10 divmod[]nat m n
 $\OT1/cmr/m/n/10 =$ \OT1/cmr/m/it/10 map[]prod nat[]of[]integer nat[]of[]integ
er

Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 221.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 221.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 221.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 221.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 221.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 221.

) (./Cancellation.tex [7]
Overfull \hbox (15.88786pt too wide) in paragraph at lines 30--31
[][]  \OT1/cmr/m/it/10 can-ce-la-tion sim-proc$\OML/cmm/m/it/10 :$ \OT1/cmr/m/i
t/10 Nor-mal-i-sa-tion of []iterate[]add[] back to the nor-male representation[
] 

Overfull \hbox (16.82057pt too wide) in paragraph at lines 81--83
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 iterate[]add[]distrib$\OT1/cmr/m/n/
10 [$\OT1/cmr/m/it/10 simp$\OT1/cmr/m/n/10 ]$$:$ []\OT1/cmr/m/it/10 iterate[]ad
d $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 m$\OT1/cmr/m/n/10 +$\OT1/cmr/m/it/10 n$\O
T1/cmr/m/n/10 )$ \OT1/cmr/m/it/10 a $\OT1/cmr/m/n/10 =$ \OT1/cmr/m/it/10 iterat
e[]add m a $\OT1/cmr/m/n/10 +$ \OT1/cmr/m/it/10 iterate[]add

Overfull \hbox (4.55688pt too wide) in paragraph at lines 138--151
[][]  \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 auto dest$\OT1/cm
r/m/n/10 !$$:$ \OT1/cmr/m/it/10 le[]Suc[]ex add[]right[]imp[]eq simp$\OT1/cmr/m
/n/10 :$ \OT1/cmr/m/it/10 ab[]semigroup[]add[]class$\OML/cmm/m/it/10 :$\OT1/cmr
/m/it/10 add[]ac$\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 1$\OT1/cmr/m/n/10 )$$)$[] 

Overfull \hbox (4.55688pt too wide) in paragraph at lines 157--170
[][]  \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 auto dest$\OT1/cm
r/m/n/10 !$$:$ \OT1/cmr/m/it/10 le[]Suc[]ex add[]right[]imp[]eq simp$\OT1/cmr/m
/n/10 :$ \OT1/cmr/m/it/10 ab[]semigroup[]add[]class$\OML/cmm/m/it/10 :$\OT1/cmr
/m/it/10 add[]ac$\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 1$\OT1/cmr/m/n/10 )$$)$[] 
[8]
Overfull \hbox (85.96669pt too wide) in paragraph at lines 175--176
[][]  \OT1/cmr/m/it/10 j $\OMS/cmsy/m/n/10 ^^T$ $\OT1/cmr/m/n/10 ($\OT1/cmr/m/i
t/10 i$\OT1/cmr/m/n/10 :$$:$\OT1/cmr/m/it/10 nat$\OT1/cmr/m/n/10 )$ $=[]\OMS/cm
sy/m/n/10 )$ $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 iterate[]add i $\OT1/cmr/m/n/1
0 ($\OT1/cmr/m/it/10 u$\OT1/cmr/m/n/10 :$$:$ $ [] $\OT1/cmr/m/it/10 a $\OT1/cmr
/m/n/10 :$$:$ $\OMS/cmsy/m/n/10 f$\OT1/cmr/m/it/10 cancel[]comm[]monoid[]add$\O
ML/cmm/m/it/10 ;$ \OT1/cmr/m/it/10 ordered[]ab[]semigroup[]add[]imp[]le$\OMS/cm
sy/m/n/10 g$$\OT1/cmr/m/n/10 )$

Overfull \hbox (4.55688pt too wide) in paragraph at lines 176--189
[][]  \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 auto dest$\OT1/cm
r/m/n/10 !$$:$ \OT1/cmr/m/it/10 le[]Suc[]ex add[]right[]imp[]eq simp$\OT1/cmr/m
/n/10 :$ \OT1/cmr/m/it/10 ab[]semigroup[]add[]class$\OML/cmm/m/it/10 :$\OT1/cmr
/m/it/10 add[]ac$\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 1$\OT1/cmr/m/n/10 )$$)$[] 

Overfull \hbox (85.96669pt too wide) in paragraph at lines 194--195
[][]  \OT1/cmr/m/it/10 i $\OMS/cmsy/m/n/10 ^^T$ $\OT1/cmr/m/n/10 ($\OT1/cmr/m/i
t/10 j$\OT1/cmr/m/n/10 :$$:$\OT1/cmr/m/it/10 nat$\OT1/cmr/m/n/10 )$ $=[]\OMS/cm
sy/m/n/10 )$ $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 iterate[]add i $\OT1/cmr/m/n/1
0 ($\OT1/cmr/m/it/10 u$\OT1/cmr/m/n/10 :$$:$ $ [] $\OT1/cmr/m/it/10 a $\OT1/cmr
/m/n/10 :$$:$ $\OMS/cmsy/m/n/10 f$\OT1/cmr/m/it/10 cancel[]comm[]monoid[]add$\O
ML/cmm/m/it/10 ;$ \OT1/cmr/m/it/10 ordered[]ab[]semigroup[]add[]imp[]le$\OMS/cm
sy/m/n/10 g$$\OT1/cmr/m/n/10 )$

Overfull \hbox (4.55688pt too wide) in paragraph at lines 195--208
[][]  \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 auto dest$\OT1/cm
r/m/n/10 !$$:$ \OT1/cmr/m/it/10 le[]Suc[]ex add[]right[]imp[]eq simp$\OT1/cmr/m
/n/10 :$ \OT1/cmr/m/it/10 ab[]semigroup[]add[]class$\OML/cmm/m/it/10 :$\OT1/cmr
/m/it/10 add[]ac$\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 1$\OT1/cmr/m/n/10 )$$)$[] 

Overfull \hbox (85.96669pt too wide) in paragraph at lines 213--214
[][]  \OT1/cmr/m/it/10 j $\OMS/cmsy/m/n/10 ^^T$ $\OT1/cmr/m/n/10 ($\OT1/cmr/m/i
t/10 i$\OT1/cmr/m/n/10 :$$:$\OT1/cmr/m/it/10 nat$\OT1/cmr/m/n/10 )$ $=[]\OMS/cm
sy/m/n/10 )$ $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 iterate[]add i $\OT1/cmr/m/n/1
0 ($\OT1/cmr/m/it/10 u$\OT1/cmr/m/n/10 :$$:$ $ [] $\OT1/cmr/m/it/10 a $\OT1/cmr
/m/n/10 :$$:$ $\OMS/cmsy/m/n/10 f$\OT1/cmr/m/it/10 cancel[]comm[]monoid[]add$\O
ML/cmm/m/it/10 ;$ \OT1/cmr/m/it/10 ordered[]ab[]semigroup[]add[]imp[]le$\OMS/cm
sy/m/n/10 g$$\OT1/cmr/m/n/10 )$

Overfull \hbox (4.55688pt too wide) in paragraph at lines 214--227
[][]  \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 auto dest$\OT1/cm
r/m/n/10 !$$:$ \OT1/cmr/m/it/10 le[]Suc[]ex add[]right[]imp[]eq simp$\OT1/cmr/m
/n/10 :$ \OT1/cmr/m/it/10 ab[]semigroup[]add[]class$\OML/cmm/m/it/10 :$\OT1/cmr
/m/it/10 add[]ac$\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 1$\OT1/cmr/m/n/10 )$$)$[] 

Overfull \hbox (85.96669pt too wide) in paragraph at lines 232--233
[][]  \OT1/cmr/m/it/10 i $\OMS/cmsy/m/n/10 ^^T$ $\OT1/cmr/m/n/10 ($\OT1/cmr/m/i
t/10 j$\OT1/cmr/m/n/10 :$$:$\OT1/cmr/m/it/10 nat$\OT1/cmr/m/n/10 )$ $=[]\OMS/cm
sy/m/n/10 )$ $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 iterate[]add i $\OT1/cmr/m/n/1
0 ($\OT1/cmr/m/it/10 u$\OT1/cmr/m/n/10 :$$:$ $ [] $\OT1/cmr/m/it/10 a $\OT1/cmr
/m/n/10 :$$:$ $\OMS/cmsy/m/n/10 f$\OT1/cmr/m/it/10 cancel[]comm[]monoid[]add$\O
ML/cmm/m/it/10 ;$ \OT1/cmr/m/it/10 ordered[]ab[]semigroup[]add[]imp[]le$\OMS/cm
sy/m/n/10 g$$\OT1/cmr/m/n/10 )$

Overfull \hbox (4.55688pt too wide) in paragraph at lines 233--246
[][]  \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 auto dest$\OT1/cm
r/m/n/10 !$$:$ \OT1/cmr/m/it/10 le[]Suc[]ex add[]right[]imp[]eq simp$\OT1/cmr/m
/n/10 :$ \OT1/cmr/m/it/10 ab[]semigroup[]add[]class$\OML/cmm/m/it/10 :$\OT1/cmr
/m/it/10 add[]ac$\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 1$\OT1/cmr/m/n/10 )$$)$[] 

Overfull \hbox (4.55688pt too wide) in paragraph at lines 252--265
[][]  \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 auto dest$\OT1/cm
r/m/n/10 !$$:$ \OT1/cmr/m/it/10 le[]Suc[]ex add[]right[]imp[]eq simp$\OT1/cmr/m
/n/10 :$ \OT1/cmr/m/it/10 ab[]semigroup[]add[]class$\OML/cmm/m/it/10 :$\OT1/cmr
/m/it/10 add[]ac$\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 1$\OT1/cmr/m/n/10 )$$)$[] 

Overfull \hbox (4.55688pt too wide) in paragraph at lines 271--288
[][]  \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 auto dest$\OT1/cm
r/m/n/10 !$$:$ \OT1/cmr/m/it/10 le[]Suc[]ex add[]right[]imp[]eq simp$\OT1/cmr/m
/n/10 :$ \OT1/cmr/m/it/10 ab[]semigroup[]add[]class$\OML/cmm/m/it/10 :$\OT1/cmr
/m/it/10 add[]ac$\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 1$\OT1/cmr/m/n/10 )$$)$ 
) (./Multiset.tex [9] [10] [11] [12] [13]
Overfull \hbox (13.45732pt too wide) in paragraph at lines 783--785
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 set[]mset[]add[]mset[]insert $\OT1/
cmr/m/n/10 [$\OT1/cmr/m/it/10 simp$\OT1/cmr/m/n/10 ]$$:$ []\OT1/cmr/m/it/10 set
[]mset $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 add[]mset a A$\OT1/cmr/m/n/10 )$ $=$
 \OT1/cmr/m/it/10 in-sert a $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 set[]mset
[14] [15] [16] [17]
Overfull \hbox (33.36261pt too wide) in paragraph at lines 1710--1715
[][]    \OT1/cmr/bx/n/10 more-over from []\OT1/cmr/m/it/10 ?lhs[] \OT1/cmr/bx/n
/10 have \OT1/cmr/m/it/10 M $\OT1/cmr/m/n/10 =$ \OT1/cmr/m/it/10 add[]mset b N 
$\OMS/cmsy/m/n/10 ^^@$ $f$$\OT1/cmr/m/n/10 #$\OT1/cmr/m/it/10 a$\OT1/cmr/m/n/10
 #$$\OMS/cmsy/m/n/10 g$ \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10
 rule union[]single[]eq[]diff$\OT1/cmr/m/n/10 )$[] 
[18] [19] [20] [21] [22] [23] [24]
Overfull \hbox (31.97931pt too wide) in paragraph at lines 2987--2995
[][]  \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 metis add[]mset[]
add[]single add[]mset[]diff[]bothsides diff[]subset[]eq[]self multiset[]inter[]
def[] 
[25] [26] [27]
Overfull \hbox (20.85715pt too wide) in paragraph at lines 3518--3520
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 replicate[]mset[]Suc $\OT1/cmr/m/n/
10 [$\OT1/cmr/m/it/10 simp$\OT1/cmr/m/n/10 ]$$:$ \OT1/cmr/m/it/10 replicate[]ms
et $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 Suc n$\OT1/cmr/m/n/10 )$ \OT1/cmr/m/it/1
0 x $\OT1/cmr/m/n/10 =$ \OT1/cmr/m/it/10 add[]mset x $\OT1/cmr/m/n/10 ($\OT1/cm
r/m/it/10 replicate[]mset
[28]
Overfull \hbox (51.96812pt too wide) in paragraph at lines 3798--3805
[][]  \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp add$\OT1/cmr
/m/n/10 :$ \OT1/cmr/m/it/10 iterate[]add[]eq[]add[]iff1 mset[]subseteq[]add[]if
f1$\OT1/cmr/m/n/10 [$\OT1/cmr/m/it/10 unfolded repeat[]mset[]iterate[]add$\OT1/
cmr/m/n/10 ]$$)$[] 

Overfull \hbox (51.96812pt too wide) in paragraph at lines 3819--3826
[][]  \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp add$\OT1/cmr
/m/n/10 :$ \OT1/cmr/m/it/10 iterate[]add[]eq[]add[]iff2 mset[]subseteq[]add[]if
f2$\OT1/cmr/m/n/10 [$\OT1/cmr/m/it/10 unfolded repeat[]mset[]iterate[]add$\OT1/
cmr/m/n/10 ]$$)$[] 
[29] [30]
Overfull \hbox (1.90573pt too wide) in paragraph at lines 3975--3978
[][]  \OT1/cmr/bx/n/10 thus $\OT1/cmr/m/n/10 ($$\OML/cmm/m/it/10 ^^U$\OT1/cmr/m
/it/10 i$\OML/cmm/m/it/10 :$ \OT1/cmr/m/it/10 if A $\OT1/cmr/m/n/10 =$ $\OMS/cm
sy/m/n/10 f$$g$ \OT1/cmr/m/it/10 then 0 else INF f$\OT1/cmr/m/n/10 :$\OT1/cmr/m
/it/10 A$\OML/cmm/m/it/10 :$ \OT1/cmr/m/it/10 f i$\OT1/cmr/m/n/10 )$ $\OMS/cmsy
/m/n/10 2$ \OT1/cmr/m/it/10 mul-ti-set \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\
OT1/cmr/m/it/10 simp add$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 multiset[]def$\OT1
/cmr/m/n/10 )$[] 
[31]
Overfull \hbox (19.41994pt too wide) in paragraph at lines 4233--4238
[][]  \OT1/cmr/bx/n/10 more-over from \OT1/cmr/m/it/10 assms \OT1/cmr/bx/n/10 h
ave \OT1/cmr/m/it/10 fi-nite $[] $ \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/
cmr/m/it/10 rule bdd[]above[]multiset[]imp[]finite[]support$\OT1/cmr/m/n/10 )$[
] 

Overfull \hbox (39.6184pt too wide) in paragraph at lines 4256--4270
[][]  \OT1/cmr/bx/n/10 us-ing \OT1/cmr/m/it/10 assms \OT1/cmr/bx/n/10 by $\OT1/
cmr/m/n/10 ($\OT1/cmr/m/it/10 simp add$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 Sup[
]multiset[]def Abs[]multiset[]inverse Sup[]multiset[]in[]multiset$\OT1/cmr/m/n/
10 )$[] 
[32]
Overfull \hbox (55.51436pt too wide) in paragraph at lines 4331--4336
[][]    \OT1/cmr/bx/n/10 also from \OT1/cmr/m/it/10 nonempty \OT1/cmr/bx/n/10 h
ave $[] $ $\OT1/cmr/m/n/10 =$ \OT1/cmr/m/it/10 count $\OT1/cmr/m/n/10 ($\OT1/cm
r/m/it/10 Inf A$\OT1/cmr/m/n/10 )$ \OT1/cmr/m/it/10 x \OT1/cmr/bx/n/10 by $\OT1
/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp add$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 cou
nt[]Inf[]multiset[]nonempty$\OT1/cmr/m/n/10 )$[] 
[33] [34]
Overfull \hbox (5.59465pt too wide) in paragraph at lines 4613--4617
[][]  \OT1/cmr/bx/n/10 from \OT1/cmr/m/it/10 assms \OT1/cmr/bx/n/10 have \OT1/c
mr/m/it/10 set[]mset $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 Sup A$\OT1/cmr/m/n/10 
)$ $=$ $($$[] $\OT1/cmr/m/it/10 X$\OMS/cmsy/m/n/10 2$\OT1/cmr/m/it/10 A$\OML/cm
m/m/it/10 :$ \OT1/cmr/m/it/10 set[]mset X$\OT1/cmr/m/n/10 )$ \OT1/cmr/bx/n/10 b
y $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 rule set[]mset[]Sup$\OT1/cmr/m/n/10 )$[] 

[35]
Overfull \hbox (7.00906pt too wide) in paragraph at lines 4821--4823
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 filter[]diff[]mset $\OT1/cmr/m/n/10
 [$\OT1/cmr/m/it/10 simp$\OT1/cmr/m/n/10 ]$$:$ \OT1/cmr/m/it/10 filter[]mset P 
$\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 M $\OMS/cmsy/m/n/10 ^^@$ \OT1/cmr/m/it/10 N
$\OT1/cmr/m/n/10 )$ $=$ \OT1/cmr/m/it/10 filter[]mset P M $\OMS/cmsy/m/n/10 ^^@
$ \OT1/cmr/m/it/10 filter[]mset

Overfull \hbox (11.80153pt too wide) in paragraph at lines 4857--4859
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 filter[]sup[]mset$\OT1/cmr/m/n/10 [
$\OT1/cmr/m/it/10 simp$\OT1/cmr/m/n/10 ]$$:$ \OT1/cmr/m/it/10 filter[]mset P $\
OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 A $\OMS/cmsy/m/n/10 [$$\OT1/cmr/m/n/10 #$ \OT
1/cmr/m/it/10 B$\OT1/cmr/m/n/10 )$ $=$ \OT1/cmr/m/it/10 filter[]mset P A $\OMS/
cmsy/m/n/10 [$$\OT1/cmr/m/n/10 #$ \OT1/cmr/m/it/10 filter[]mset
[36] [37]
Overfull \hbox (30.90695pt too wide) in paragraph at lines 5235--5243
[][]\OT1/cmr/bx/n/10 apply $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp add$\OT1/cm
r/m/n/10 :$ \OT1/cmr/m/it/10 size[]multiset[]def sum[]Un[]nat sum$\OML/cmm/m/it
/10 :$\OT1/cmr/m/it/10 distrib sum[]wcount[]Int wcount[]union$\OT1/cmr/m/n/10 )
$[] 

Overfull \hbox (55.98346pt too wide) in paragraph at lines 5260--5274
[][]  \OT1/cmr/bx/n/10 un-fold-ing \OT1/cmr/m/it/10 add[]mset[]add[]single$\OT1
/cmr/m/n/10 [$\OT1/cmr/m/it/10 of [] M$\OT1/cmr/m/n/10 ]$ \OT1/cmr/m/it/10 size
[]multiset[]union \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 auto 
simp$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 size[]multiset[]single$\OT1/cmr/m/n/10
 )$[] 
[38] [39] [40] [41] [42]
Overfull \hbox (44.37408pt too wide) in paragraph at lines 6145--6159
[][]  \OT1/cmr/bx/n/10 un-fold-ing \OT1/cmr/m/it/10 image[]mset[]union add[]mse
t[]add[]single$\OT1/cmr/m/n/10 [$\OT1/cmr/m/it/10 of a M$\OT1/cmr/m/n/10 ]$ \OT
1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp add$\OT1/cmr/m/n/10 :
$ \OT1/cmr/m/it/10 image[]mset[]single$\OT1/cmr/m/n/10 )$[] 
[43]
Overfull \hbox (9.85997pt too wide) in paragraph at lines 6333--6341
[][]  \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 metis $\OT1/cmr/m
/n/10 ($\OT1/cmr/m/it/10 no[]types$\OT1/cmr/m/n/10 )$ \OT1/cmr/m/it/10 Diff[]eq
[]empty[]iff[]mset image[]mset[]Diff image[]mset[]is[]empty[]iff[] 
[44] [45] [46] [47] [48] [49] [50] [51]
Overfull \hbox (4.1574pt too wide) in paragraph at lines 7718--7731
[][]  \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 auto simp add$\OT
1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 mset[]subset[]eqI$\OT1/cmr/m/n/10 )$ $($\OT1/c
mr/m/it/10 metis count[]replicate[]mset subseteq[]mset[]def$\OT1/cmr/m/n/10 )$[
] 
[52] [53] [54]
Overfull \hbox (14.65363pt too wide) in paragraph at lines 8409--8410
[][]      $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp[]all add$\OT1/cmr/m/n/10 :$ 
\OT1/cmr/m/it/10 size[]multiset[]overloaded[]eq not[]in[]iff sum$\OML/cmm/m/it/
10 :$\OT1/cmr/m/it/10 If[]cases Diff[]eq$\OT1/cmr/m/n/10 [$\OT1/cmr/m/it/10 sym
metric$\OT1/cmr/m/n/10 ]$[] 
[55]
Overfull \hbox (20.93939pt too wide) in paragraph at lines 8552--8565
[][]  \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 subst sum[]mset$\
OML/cmm/m/it/10 :$\OT1/cmr/m/it/10 commute$\OT1/cmr/m/n/10 )$ $($\OT1/cmr/m/it/
10 simp add$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 sum[]mset[]distrib[]left sum[]m
set[]distrib[]right$\OT1/cmr/m/n/10 )$[] 
[56] [57] [58] [59] [60]
Overfull \hbox (15.84276pt too wide) in paragraph at lines 9390--9393
[][]    \OT1/cmr/bx/n/10 un-fold-ing \OT1/cmr/m/it/10 filter[]sort \OT1/cmr/bx/
n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 rule properties[]for[]sort[]key$\OT
1/cmr/m/n/10 )$ $($\OT1/cmr/m/it/10 auto in-tro$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/i
t/10 sorted[]map[]same$\OT1/cmr/m/n/10 )$[] 
[61] [62] [63] [64] [65] [66] [67]
Overfull \hbox (23.14763pt too wide) in paragraph at lines 10339--10342
[][]      \OT1/cmr/bx/n/10 us-ing $\OMS/cmsy/m/n/10 ^^C$$\OT1/cmr/m/n/10 ($\OT1
/cmr/m/it/10 1$\OML/cmm/m/it/10 ;$\OT1/cmr/m/it/10 2$\OT1/cmr/m/n/10 )$ \OT1/cm
r/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 metis add[]mset[]remove[]trivi
al[]If insert[]iff set[]mset[]add[]mset[]insert
[68]
Overfull \hbox (18.23695pt too wide) in paragraph at lines 10471--10473
[][]    $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 M $\OMS/cmsy/m/n/10 ^^@$ \OT1/cmr/m
/it/10 M $\OMS/cmsy/m/n/10 \$$\OT1/cmr/m/n/10 #$ \OT1/cmr/m/it/10 N$\OT1/cmr/m/
n/10 )$ $\OMS/cmsy/m/n/10 \$$\OT1/cmr/m/n/10 #$ $($\OT1/cmr/m/it/10 N $\OMS/cms
y/m/n/10 ^^@$ \OT1/cmr/m/it/10 M $\OMS/cmsy/m/n/10 \$$\OT1/cmr/m/n/10 #$ \OT1/c
mr/m/it/10 N$\OT1/cmr/m/n/10 )$ $=$ $\OMS/cmsy/m/n/10 f$$\OT1/cmr/m/n/10 #$$\OM
S/cmsy/m/n/10 g$ \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 auto s
imp$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 count[]inject$\OT1/cmr/m/n/10 [$\OT1/cm
r/m/it/10 symmetric$\OT1/cmr/m/n/10 ]$$)$[] 
[69]
Overfull \hbox (24.56795pt too wide) in paragraph at lines 10531--10535
[][]    \OT1/cmr/bx/n/10 then ob-tain \OT1/cmr/m/it/10 y \OT1/cmr/bx/n/10 where
 \OT1/cmr/m/it/10 count N y $\OMS/cmsy/m/n/10 6\OT1/cmr/m/n/10 =$ \OT1/cmr/m/it
/10 count M y \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 auto simp
$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 count[]inject$\OT1/cmr/m/n/10 [$\OT1/cmr/m
/it/10 symmetric$\OT1/cmr/m/n/10 ]$$)$[] 

Overfull \hbox (0.73877pt too wide) in paragraph at lines 10629--10633
[][]    \OT1/cmr/bx/n/10 then have \OT1/cmr/m/it/10 MM$\OT1/cmr/m/n/10 :$ $($\O
T1/cmr/m/it/10 M$\OML/cmm/m/it/10 ;$ \OT1/cmr/m/it/10 M$\OT1/cmr/m/n/10 )$ $\OM
S/cmsy/m/n/10 2$ \OT1/cmr/m/it/10 mult $\OMS/cmsy/m/n/10 f$$\OT1/cmr/m/n/10 ($\
OT1/cmr/m/it/10 x$\OML/cmm/m/it/10 ;$ \OT1/cmr/m/it/10 y$\OT1/cmr/m/n/10 )$$\OM
L/cmm/m/it/10 :$ \OT1/cmr/m/it/10 x $\OML/cmm/m/it/10 <$ \OT1/cmr/m/it/10 y$\OM
S/cmsy/m/n/10 g$ \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp a
dd$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 less[]multiset[]def$\OT1/cmr/m/n/10 )$[]
 
[70] [71]
Overfull \hbox (55.81471pt too wide) in paragraph at lines 10887--10894
[][]\OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 rule one[]step[]imp
lies[]mult$\OT1/cmr/m/n/10 )$ $($\OT1/cmr/m/it/10 auto simp add$\OT1/cmr/m/n/10
 :$ \OT1/cmr/m/it/10 max[]strict[]def pair[]less[]def elim$\OT1/cmr/m/n/10 !$$:
$\OT1/cmr/m/it/10 max[]ext$\OML/cmm/m/it/10 :$\OT1/cmr/m/it/10 cases$\OT1/cmr/m
/n/10 )$[] 

Overfull \hbox (54.71396pt too wide) in paragraph at lines 10909--10916
[][]\OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 auto simp add$\OT1/
cmr/m/n/10 :$ \OT1/cmr/m/it/10 pair[]less[]def max[]strict[]def elim$\OT1/cmr/m
/n/10 !$$:$\OT1/cmr/m/it/10 max[]ext$\OML/cmm/m/it/10 :$\OT1/cmr/m/it/10 cases 
in-tro$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 one[]step[]implies[]mult$\OT1/cmr/m/
n/10 )$[] 
[72] [73]
Overfull \hbox (13.03835pt too wide) in paragraph at lines 11158--11159
[][]        $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 map $\OT1/cmr/m/n/10 ($\OT1/cmr
/m/it/10 fn t $\OT1/cmr/m/n/10 =$$\OML/cmm/m/it/10 >$ \OT1/cmr/m/it/10 t RS eq[
]reflection$\OT1/cmr/m/n/10 )$ $($$@$$\OMS/cmsy/m/n/10 f$\OT1/cmr/m/it/10 thms 
ac[]simps$\OMS/cmsy/m/n/10 g$ $\OT1/cmr/m/n/10 @$ $@$$\OMS/cmsy/m/n/10 f$\OT1/c
mr/m/it/10 thms empty[]neutral$\OMS/cmsy/m/n/10 g$$\OT1/cmr/m/n/10 )$$)$[] 
[74] [75] [76] [77] [78] [79]
Overfull \hbox (5.43213pt too wide) in paragraph at lines 12056--12057
[][]  \OT1/cmr/m/it/10 full[]exhaustive[]multiset f i $\OT1/cmr/m/n/10 =$ \OT1/
cmr/m/it/10 Quickcheck[]Exhaustive$\OML/cmm/m/it/10 :$\OT1/cmr/m/it/10 full[]ex
haustive $\OT1/cmr/m/n/10 ($$\OML/cmm/m/it/10 ^^U$\OT1/cmr/m/it/10 xs$\OML/cmm/
m/it/10 :$ \OT1/cmr/m/it/10 f $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 msetify
[80] [81]
Overfull \hbox (21.37383pt too wide) in paragraph at lines 12215--12217
[][]    \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 metis append[]e
q[]append[]conv append[]take[]drop[]id diff[]Suc[]Suc Cons[]nth[]drop[]Suc

Overfull \hbox (17.54051pt too wide) in paragraph at lines 12222--12224
[][]    \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 metis add[]Suc[
]right append[]take[]drop[]id length[]Cons length[]append less[]eq[]Suc[]le

Overfull \hbox (26.54095pt too wide) in paragraph at lines 12229--12231
[][]    \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 metis add[]Suc[
]right append[]take[]drop[]id length[]Cons length[]append mset[]eq[]length$\OT1
/cmr/m/n/10 )$[] 
[82]
Overfull \hbox (19.33397pt too wide) in paragraph at lines 12350--12353
[][]    \OT1/cmr/bx/n/10 un-fold-ing \OT1/cmr/m/it/10 comp[]def \OT1/cmr/bx/n/1
0 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 rule ext$\OT1/cmr/m/n/10 )$ $($\OT1/cm
r/m/it/10 simp add$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 comp[]def image[]mset$\O
ML/cmm/m/it/10 :$\OT1/cmr/m/it/10 compositionality$\OT1/cmr/m/n/10 )$[] 
[83] [84]
Overfull \hbox (6.28758pt too wide) in paragraph at lines 12638--12642
[][]    \OT1/cmr/bx/n/10 then ob-tain \OT1/cmr/m/it/10 N1 b \OT1/cmr/bx/n/10 wh
ere \OT1/cmr/m/it/10 N$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 N $\OT1/cmr/m/n/10 =
$ \OT1/cmr/m/it/10 add[]mset b N1 \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/c
mr/m/it/10 metis multi[]nonempty[]split$\OT1/cmr/m/n/10 )$[] 
[85] [86]
Overfull \hbox (23.3852pt too wide) in paragraph at lines 12982--12983
[][]  \OT1/cmr/m/it/10 BNF[]LFP[]Size$\OML/cmm/m/it/10 :$\OT1/cmr/m/it/10 regis
ter[]size[]global $\OT1/cmr/m/n/10 @$$\OMS/cmsy/m/n/10 f$\OT1/cmr/m/it/10 type[
]name mul-ti-set$\OMS/cmsy/m/n/10 g$ $\OT1/cmr/m/n/10 @$$\OMS/cmsy/m/n/10 f$\OT
1/cmr/m/it/10 const[]name size[]multiset$\OMS/cmsy/m/n/10 g$[] 
[87]) (./Factorial_Ring.tex [88] [89]
Overfull \hbox (9.44902pt too wide) in paragraph at lines 435--449
[][]  \OT1/cmr/bx/n/10 us-ing \OT1/cmr/m/it/10 unit[]imp[]no[]irreducible[]divi
sors$\OT1/cmr/m/n/10 [$\OT1/cmr/m/it/10 OF assms$\OT1/cmr/m/n/10 ($\OT1/cmr/m/i
t/10 1$\OT1/cmr/m/n/10 )$ \OT1/cmr/m/it/10 prime[]elem[]imp[]irreducible$\OT1/c
mr/m/n/10 [$\OT1/cmr/m/it/10 OF
[90] [91]
Overfull \hbox (26.81523pt too wide) in paragraph at lines 815--820
[][]  \OT1/cmr/bx/n/10 more-over from \OT1/cmr/m/it/10 assms \OT1/cmr/bx/n/10 h
ave $\OMS/cmsy/m/n/10 :$\OT1/cmr/m/it/10 is[]unit p \OT1/cmr/bx/n/10 by $\OT1/c
mr/m/n/10 ($\OT1/cmr/m/it/10 simp add$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 prime
[]elem[]def is[]unit[]power[]iff$\OT1/cmr/m/n/10 )$[] 
[92]
Overfull \hbox (10.79758pt too wide) in paragraph at lines 887--900
[][]  \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 auto simp$\OT1/cm
r/m/n/10 :$ \OT1/cmr/m/it/10 prime[]elem[]def mult$\OML/cmm/m/it/10 :$\OT1/cmr/
m/it/10 commute$\OT1/cmr/m/n/10 [$\OT1/cmr/m/it/10 of a$\OT1/cmr/m/n/10 ]$ \OT1
/cmr/m/it/10 is[]unit[]mult[]iff mult[]unit[]dvd[]iff$\OT1/cmr/m/n/10 )$[] 

Overfull \hbox (12.0489pt too wide) in paragraph at lines 955--957
[][]      \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 metis dvd[]mu
lt[]right dvd[]times[]left[]cancel[]iff mult$\OML/cmm/m/it/10 :$\OT1/cmr/m/it/1
0 left[]commute mult[]zero[]left$\OT1/cmr/m/n/10 )$[] 
[93]
Overfull \hbox (5.12607pt too wide) in paragraph at lines 1111--1125
[][]  \OT1/cmr/bx/n/10 us-ing \OT1/cmr/m/it/10 assms \OT1/cmr/bx/n/10 by $\OT1/
cmr/m/n/10 ($\OT1/cmr/m/it/10 induct n$\OT1/cmr/m/n/10 )$ $($\OT1/cmr/m/it/10 a
uto dest$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 prime[]elem[]not[]unit prime[]elem
[]dvd[]multD$\OT1/cmr/m/n/10 )$[] 
[94] [95] [96]
Overfull \hbox (11.69632pt too wide) in paragraph at lines 1749--1753
[][]  \OT1/cmr/bx/n/10 from \OT1/cmr/m/it/10 assms \OT1/cmr/bx/n/10 have \OT1/c
mr/m/it/10 ir-re-ducible q \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it
/10 simp add$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 prime[]elem[]imp[]irreducible 
prime[]def$\OT1/cmr/m/n/10 )$[] 
[97]
Overfull \hbox (30.63289pt too wide) in paragraph at lines 1904--1918
[][]  \OT1/cmr/bx/n/10 us-ing \OT1/cmr/m/it/10 assms \OT1/cmr/bx/n/10 by $\OT1/
cmr/m/n/10 ($\OT1/cmr/m/it/10 auto in-tro$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 p
rod[]mset[]subset[]imp[]dvd prod[]mset[]primes[]dvd[]imp[]subset$\OT1/cmr/m/n/1
0 )$[] 
[98]
Overfull \hbox (11.4095pt too wide) in paragraph at lines 1978--1980
[][]    \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 auto simp$\OT1/
cmr/m/n/10 :$ \OT1/cmr/m/it/10 A1[]def A2[]def Mul-ti-set$\OML/cmm/m/it/10 :$\O
T1/cmr/m/it/10 subset[]eq[]diff[]conv Mul-ti-set$\OML/cmm/m/it/10 :$\OT1/cmr/m/
it/10 union[]commute$\OT1/cmr/m/n/10 )$[] 

Overfull \hbox (1.77489pt too wide) in paragraph at lines 1993--1995
[][]    \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 subst $\OT1/cmr
/m/n/10 ($\OT1/cmr/m/it/10 asm$\OT1/cmr/m/n/10 )$ $($\OT1/cmr/m/it/10 1 2$\OT1/
cmr/m/n/10 )$ \OT1/cmr/m/it/10 is[]unit[]prod[]mset[]primes[]iff$\OT1/cmr/m/n/1
0 )$ $($\OT1/cmr/m/it/10 auto dest$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 Mul-ti-s
et$\OML/cmm/m/it/10 :$\OT1/cmr/m/it/10 in[]diffD$\OT1/cmr/m/n/10 )$[] 

Overfull \hbox (1.20242pt too wide) in paragraph at lines 2037--2041
[][]    \OT1/cmr/bx/n/10 from \OT1/cmr/m/it/10 x \OT1/cmr/bx/n/10 have \OT1/cmr
/m/it/10 x ^ n dvd prod[]mset A ^ n \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1
/cmr/m/it/10 intro dvd[]power[]same dvd[]prod[]mset$\OT1/cmr/m/n/10 )$[] 
[99] [100]
Overfull \hbox (11.16814pt too wide) in paragraph at lines 2290--2290
[]\OT1/cmr/bx/n/12 Factorial semir-ings: al-ge-braic struc-tures with unique pr
ime

Overfull \hbox (6.13147pt too wide) in paragraph at lines 2337--2341
[][]      \OT1/cmr/bx/n/10 with []$\OMS/cmsy/m/n/10 :$\OT1/cmr/m/it/10 is[]unit
 a[] less$\OML/cmm/m/it/10 :$\OT1/cmr/m/it/10 prems \OT1/cmr/bx/n/10 have \OT1/
cmr/m/it/10 ir-re-ducible a \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/i
t/10 auto simp$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 irreducible[]altdef$\OT1/cmr
/m/n/10 )$[] 
[101]
Overfull \hbox (5.4238pt too wide) in paragraph at lines 2451--2455
[][]      \OT1/cmr/bx/n/10 from \OT1/cmr/m/it/10 A B \OT1/cmr/bx/n/10 show \OT1
/cmr/m/it/10 ?the-sis \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 i
ntro exI$\OT1/cmr/m/n/10 [$\OT1/cmr/m/it/10 of [] A $\OT1/cmr/m/n/10 +$ \OT1/cm
r/m/it/10 B$\OT1/cmr/m/n/10 ]$$)$ $($\OT1/cmr/m/it/10 auto simp$\OT1/cmr/m/n/10
 :$ \OT1/cmr/m/it/10 c normalize[]mult$\OT1/cmr/m/n/10 )$[] 
[102]
Overfull \hbox (7.38255pt too wide) in paragraph at lines 2584--2586
[][]    \OT1/cmr/bx/n/10 from \OT1/cmr/m/it/10 dvd prod[]mset[]primes[]irreduci
ble[]imp[]prime$\OT1/cmr/m/n/10 [$\OT1/cmr/m/it/10 of A B C$\OML/cmm/m/it/10 ;$
 \OT1/cmr/m/it/10 OF this ABC$\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 1$\OML/cmm/m/i
t/10 ;$\OT1/cmr/m/it/10 3$\OML/cmm/m/it/10 ;$\OT1/cmr/m/it/10 5$\OT1/cmr/m/n/10
 )$$]$
[103]
Overfull \hbox (6.55193pt too wide) in paragraph at lines 2685--2690
[][]    \OT1/cmr/bx/n/10 fi-nally show \OT1/cmr/m/it/10 p $\OMS/cmsy/m/n/10 2$$
\OT1/cmr/m/n/10 #$ \OT1/cmr/m/it/10 A \OT1/cmr/bx/n/10 us-ing \OT1/cmr/m/it/10 
p A \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 subst $\OT1/cmr/m/n
/10 ($\OT1/cmr/m/it/10 asm$\OT1/cmr/m/n/10 )$ \OT1/cmr/m/it/10 prime[]dvd[]prod
[]mset[]primes[]iff$\OT1/cmr/m/n/10 )$[] 
[104] [105] [106]
Overfull \hbox (35.03365pt too wide) in paragraph at lines 3203--3217
[][]  \OT1/cmr/bx/n/10 us-ing \OT1/cmr/m/it/10 assms \OT1/cmr/bx/n/10 by $\OT1/
cmr/m/n/10 ($\OT1/cmr/m/it/10 subst prime[]elem[]multiplicity[]eq[]zero[]iff$\O
T1/cmr/m/n/10 )$ $($\OT1/cmr/m/it/10 auto dest$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it
/10 primes[]dvd[]imp[]eq$\OT1/cmr/m/n/10 )$

Overfull \hbox (31.33105pt too wide) in paragraph at lines 3329--3334
[][]    \OT1/cmr/bx/n/10 more-over from \OT1/cmr/m/it/10 assms$\OT1/cmr/m/n/10 
($\OT1/cmr/m/it/10 2$\OT1/cmr/m/n/10 )$ \OT1/cmr/m/it/10 xp \OT1/cmr/bx/n/10 ha
ve \OT1/cmr/m/it/10 mul-ti-plic-ity p x $\OML/cmm/m/it/10 <$ \OT1/cmr/m/it/10 S
uc n \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 intro multiplicity
[]lessI$\OT1/cmr/m/n/10 )$[] 
[107] [108]
Overfull \hbox (42.299pt too wide) in paragraph at lines 3591--3593
[][]    \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 subst mult$\OML
/cmm/m/it/10 :$\OT1/cmr/m/it/10 commute$\OT1/cmr/m/n/10 )$ $($\OT1/cmr/m/it/10 
simp add$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 mult[]unit[]dvd[]iff power[]mult[]
distrib is[]unit[]power[]iff$\OT1/cmr/m/n/10 )$[] 
[109] [110]
Overfull \hbox (2.52225pt too wide) in paragraph at lines 3870--3872
[][]    \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 intro multiplic
ity[]geI $\OT1/cmr/m/n/10 )$ $($\OT1/cmr/m/it/10 auto in-tro$\OT1/cmr/m/n/10 :$
 \OT1/cmr/m/it/10 dvd[]trans$\OT1/cmr/m/n/10 [$\OT1/cmr/m/it/10 OF multiplicity
[]dvd$ [] $ assms$\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 1$\OT1/cmr/m/n/10 )$$]$$)$
[] 

Overfull \hbox (4.71416pt too wide) in paragraph at lines 3931--3936
[][]      \OT1/cmr/bx/n/10 also from \OT1/cmr/m/it/10 x p \OT1/cmr/bx/n/10 have
 $[] $ $\OT1/cmr/m/n/10 =$ \OT1/cmr/m/it/10 0 $\OMS/cmsy/m/n/10  []!$ $:$\OT1/c
mr/m/it/10 p dvd x \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp
 add$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 multiplicity[]eq[]zero[]iff$\OT1/cmr/m
/n/10 )$[] 
[111]
Overfull \hbox (0.98625pt too wide) in paragraph at lines 4066--4068
[][]  \OT1/cmr/bx/n/10 thus \OT1/cmr/m/it/10 count $\OT1/cmr/m/n/10 ($\OT1/cmr/
m/it/10 prime[]factorization $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 p $\OMS/cmsy/m
/n/10 ^^C$ \OT1/cmr/m/it/10 x$\OT1/cmr/m/n/10 )$$)$ \OT1/cmr/m/it/10 q $\OT1/cm
r/m/n/10 =$ \OT1/cmr/m/it/10 count $\OT1/cmr/m/n/10 ($$\OMS/cmsy/m/n/10 f$$\OT1
/cmr/m/n/10 #$\OT1/cmr/m/it/10 p$\OT1/cmr/m/n/10 #$$\OMS/cmsy/m/n/10 g$ $\OT1/c
mr/m/n/10 +$ \OT1/cmr/m/it/10 prime[]factorization
[112]
Overfull \hbox (15.81303pt too wide) in paragraph at lines 4304--4306
[][]    \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp[]all add$\
OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 mult[]ac prime[]factorization[]times[]prime 
Mul-ti-set$\OML/cmm/m/it/10 :$\OT1/cmr/m/it/10 union[]commute$\OT1/cmr/m/n/10 )
$[] 
[113]
Overfull \hbox (40.95099pt too wide) in paragraph at lines 4355--4359
[][]  \OT1/cmr/bx/n/10 with \OT1/cmr/m/it/10 assms \OT1/cmr/bx/n/10 show \OT1/c
mr/m/it/10 nor-mal-ize x $\OT1/cmr/m/n/10 =$ \OT1/cmr/m/it/10 nor-mal-ize y \OT
1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp add$\OT1/cmr/m/n/10 :
$ \OT1/cmr/m/it/10 prod[]mset[]prime[]factorization$\OT1/cmr/m/n/10 )$[] 

Overfull \hbox (44.55727pt too wide) in paragraph at lines 4390--4392
[][]    \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 rule prime[]fac
torization[]prod[]mset[]primes$\OT1/cmr/m/n/10 )$ $($\OT1/cmr/m/it/10 simp[]all
 add$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 in[]prime[]factors[]imp[]prime$\OT1/cm
r/m/n/10 )$[] 

Overfull \hbox (20.85785pt too wide) in paragraph at lines 4392--4395
[][]  \OT1/cmr/bx/n/10 also have \OT1/cmr/m/it/10 prime[]factorization $\OT1/cm
r/m/n/10 ($\OT1/cmr/m/it/10 prod[]mset $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 prim
e[]factorization x $\OT1/cmr/m/n/10 +$ \OT1/cmr/m/it/10 prime[]factorization

Overfull \hbox (33.42188pt too wide) in paragraph at lines 4396--4398
[][]    \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 rule prime[]fac
torization[]prod[]mset[]primes$\OT1/cmr/m/n/10 )$ $($\OT1/cmr/m/it/10 auto simp
$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 in[]prime[]factors[]imp[]prime$\OT1/cmr/m/
n/10 )$[] 
[114]
Overfull \hbox (25.28094pt too wide) in paragraph at lines 4520--4521
[][]                   $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp[]all add$\OT1/c
mr/m/n/10 :$ \OT1/cmr/m/it/10 prime[]elem[]multiplicity[]prod[]mset[]distrib su
m[]unfold[]sum[]mset

Overfull \hbox (16.5629pt too wide) in paragraph at lines 4545--4558
[][]  \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 subst prime[]elem
[]multiplicity[]power[]distrib$\OT1/cmr/m/n/10 )$ $($\OT1/cmr/m/it/10 auto simp
$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 prime[]multiplicity[]other$\OT1/cmr/m/n/10
 )$[] 

Overfull \hbox (50.5364pt too wide) in paragraph at lines 4572--4578
[][]     $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp[]all add$\OT1/cmr/m/n/10 :$ \
OT1/cmr/m/it/10 prime[]factorization[]mult prime[]factorization[]prime Mul-ti-s
et$\OML/cmm/m/it/10 :$\OT1/cmr/m/it/10 union[]commute$\OT1/cmr/m/n/10 )$[] 

Overfull \hbox (22.83113pt too wide) in paragraph at lines 4611--4613
[][]  \OT1/cmr/bx/n/10 have \OT1/cmr/m/it/10 x dvd y $\OMS/cmsy/m/n/10  []!$ \O
T1/cmr/m/it/10 prod[]mset $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 prime[]factorizat
ion x$\OT1/cmr/m/n/10 )$ \OT1/cmr/m/it/10 dvd prod[]mset $\OT1/cmr/m/n/10 ($\OT
1/cmr/m/it/10 prime[]factorization
[115]
Overfull \hbox (6.21616pt too wide) in paragraph at lines 4656--4657
[][]  \OT1/cmr/bx/n/10 shows   \OT1/cmr/m/it/10 prime[]factorization $\OT1/cmr/
m/n/10 ($\OT1/cmr/m/it/10 a div b$\OT1/cmr/m/n/10 )$ $=$ \OT1/cmr/m/it/10 prime
[]factorization a $\OMS/cmsy/m/n/10 ^^@$ \OT1/cmr/m/it/10 prime[]factorization

Overfull \hbox (12.16023pt too wide) in paragraph at lines 4752--4753
[][]      $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp[]all add$\OT1/cmr/m/n/10 :$ 
\OT1/cmr/m/it/10 assms count[]prime[]factorization[]prime in[]prime[]factors[]i
mp[]prime$\OT1/cmr/m/n/10 )$[] 

Overfull \hbox (24.46587pt too wide) in paragraph at lines 4794--4797
[][]    \OT1/cmr/bx/n/10 un-fold-ing \OT1/cmr/m/it/10 A[]def \OT1/cmr/bx/n/10 b
y $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 subst mul-ti-set$\OML/cmm/m/it/10 :$\OT1/
cmr/m/it/10 Abs[]multiset[]inverse$\OT1/cmr/m/n/10 )$ $($\OT1/cmr/m/it/10 simp[
]all add$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 multiset[]def$\OT1/cmr/m/n/10 )$[]
 
[116]
Overfull \hbox (1.20328pt too wide) in paragraph at lines 4806--4810
[][]  \OT1/cmr/bx/n/10 also have $[] $ $\OT1/cmr/m/n/10 =$ \OT1/cmr/m/it/10 pro
d[]mset A \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp add$\OT1
/cmr/m/n/10 :$ \OT1/cmr/m/it/10 prod[]mset[]multiplicity S[]eq set[]mset[]A

Overfull \hbox (29.18195pt too wide) in paragraph at lines 4827--4830
[][]  \OT1/cmr/bx/n/10 also have \OT1/cmr/m/it/10 prime[]factorization $\OT1/cm
r/m/n/10 ($\OT1/cmr/m/it/10 prod[]mset $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 prim
e[]factorization n$\OT1/cmr/m/n/10 )$$)$ $=$ \OT1/cmr/m/it/10 prime[]factorizat
ion

Overfull \hbox (56.44945pt too wide) in paragraph at lines 4864--4866
[][]      \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 intro image[]
mset[]cong$\OT1/cmr/m/n/10 )$ $($\OT1/cmr/m/it/10 auto simp$\OT1/cmr/m/n/10 :$ 
\OT1/cmr/m/it/10 set[]mset[]A multiplicity[]self prime[]multiplicity[]other$\OT
1/cmr/m/n/10 )$[] 
[117]
Overfull \hbox (5.04556pt too wide) in paragraph at lines 4944--4948
[][]  \OT1/cmr/bx/n/10 with \OT1/cmr/m/it/10 assms False \OT1/cmr/bx/n/10 show 
\OT1/cmr/m/it/10 ?the-sis \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/
10 subst $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 asm$\OT1/cmr/m/n/10 )$ \OT1/cmr/m/
it/10 prime[]factorization[]subset[]iff[]dvd$\OT1/cmr/m/n/10 )$[] 
[118]
Overfull \hbox (1.69742pt too wide) in paragraph at lines 5127--5128
[][]          \OT1/cmr/m/it/10 prime[]factorization $\OT1/cmr/m/n/10 ($\OT1/cmr
/m/it/10 prod[]mset $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 prime[]factorization a 
$\OMS/cmsy/m/n/10 \$$\OT1/cmr/m/n/10 #$ \OT1/cmr/m/it/10 prime[]factorization

Overfull \hbox (1.69742pt too wide) in paragraph at lines 5163--5164
[][]          \OT1/cmr/m/it/10 prime[]factorization $\OT1/cmr/m/n/10 ($\OT1/cmr
/m/it/10 prod[]mset $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 prime[]factorization a 
$\OMS/cmsy/m/n/10 [$$\OT1/cmr/m/n/10 #$ \OT1/cmr/m/it/10 prime[]factorization
[119] [120]
Overfull \hbox (4.25296pt too wide) in paragraph at lines 5397--5398
[][]           \OT1/cmr/m/it/10 prime[]factorization $\OT1/cmr/m/n/10 ($\OT1/cm
r/m/it/10 prod[]mset $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 prime[]factorization a
 $\OMS/cmsy/m/n/10 \$$\OT1/cmr/m/n/10 #$ \OT1/cmr/m/it/10 prime[]factorization
[121]
Overfull \hbox (3.08865pt too wide) in paragraph at lines 5538--5541
[][]  \OT1/cmr/bx/n/10 with \OT1/cmr/m/it/10 assms \OT1/cmr/bx/n/10 have \OT1/c
mr/m/it/10 prime[]factorization $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 Gcd[]factor
ial A$\OT1/cmr/m/n/10 )$ $=$ \OT1/cmr/m/it/10 Inf $\OT1/cmr/m/n/10 ($\OT1/cmr/m
/it/10 prime[]factorization
[122] [123]
Overfull \hbox (14.46059pt too wide) in paragraph at lines 5797--5798
[][]  \OT1/cmr/bx/n/10 shows   \OT1/cmr/m/it/10 prime[]factorization $\OT1/cmr/
m/n/10 ($\OT1/cmr/m/it/10 gcd a b$\OT1/cmr/m/n/10 )$ $=$ \OT1/cmr/m/it/10 prime
[]factorization a $\OMS/cmsy/m/n/10 \$$\OT1/cmr/m/n/10 #$ \OT1/cmr/m/it/10 prim
e[]factorization

Overfull \hbox (15.99394pt too wide) in paragraph at lines 5817--5818
[][]  \OT1/cmr/bx/n/10 shows   \OT1/cmr/m/it/10 prime[]factorization $\OT1/cmr/
m/n/10 ($\OT1/cmr/m/it/10 lcm a b$\OT1/cmr/m/n/10 )$ $=$ \OT1/cmr/m/it/10 prime
[]factorization a $\OMS/cmsy/m/n/10 [$$\OT1/cmr/m/n/10 #$ \OT1/cmr/m/it/10 prim
e[]factorization

Overfull \hbox (43.35431pt too wide) in paragraph at lines 5846--5853
[][]  \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp add$\OT1/cmr
/m/n/10 :$ \OT1/cmr/m/it/10 prime[]factorization[]Gcd[]factorial Gcd[]eq[]Gcd[]
factorial Gcd[]factorial[]eq[]0[]iff$\OT1/cmr/m/n/10 )$[] 

Overfull \hbox (49.76532pt too wide) in paragraph at lines 5868--5875
[][]  \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp add$\OT1/cmr
/m/n/10 :$ \OT1/cmr/m/it/10 prime[]factorization[]Lcm[]factorial Lcm[]eq[]Lcm[]
factorial Lcm[]factorial[]eq[]0[]iff$\OT1/cmr/m/n/10 )$[] 
[124]
Overfull \hbox (55.76273pt too wide) in paragraph at lines 5989--5991
[][]    \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp add$\OT1/c
mr/m/n/10 :$ \OT1/cmr/m/it/10 count[]prime[]factorization[]prime $\OT1/cmr/m/n/
10 [$\OT1/cmr/m/it/10 symmetric$\OT1/cmr/m/n/10 ]$ \OT1/cmr/m/it/10 prime[]fact
orization[]gcd[]factorial$\OT1/cmr/m/n/10 )$[] 

Overfull \hbox (57.29608pt too wide) in paragraph at lines 6002--6004
[][]    \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp add$\OT1/c
mr/m/n/10 :$ \OT1/cmr/m/it/10 count[]prime[]factorization[]prime $\OT1/cmr/m/n/
10 [$\OT1/cmr/m/it/10 symmetric$\OT1/cmr/m/n/10 ]$ \OT1/cmr/m/it/10 prime[]fact
orization[]lcm[]factorial$\OT1/cmr/m/n/10 )$[] 
[125]) (./Euclidean_Algorithm.tex
Overfull \hbox (36.91122pt too wide) in paragraph at lines 6--6
[]\OT1/cmr/bx/n/14.4 Abstract eu-clidean al-go-rithm in eu-clidean (semi)rings 

[126] [127] [128] [129] [130] [131] [132] [133]
Overfull \hbox (36.0064pt too wide) in paragraph at lines 1171--1172
[][]  \OT1/cmr/m/it/10 OF-CLASS$\OT1/cmr/m/n/10 ($$ [] $\OT1/cmr/m/it/10 a$\OT1
/cmr/m/n/10 :$$:$$\OMS/cmsy/m/n/10 f$\OT1/cmr/m/it/10 factorial[]semiring[]gcd$
\OML/cmm/m/it/10 ;$ \OT1/cmr/m/it/10 euclidean[]semiring$\OMS/cmsy/m/n/10 g$$\O
ML/cmm/m/it/10 ;$ \OT1/cmr/m/it/10 euclidean[]semiring[]gcd[]class$\OT1/cmr/m/n
/10 )$[] 
[134]
Overfull \hbox (2.401pt too wide) in paragraph at lines 1231--1233
[][]    \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp add$\OT1/c
mr/m/n/10 :$ \OT1/cmr/m/it/10 fun[]eq[]iff Euclidean[]Algorithm$\OML/cmm/m/it/1
0 :$\OT1/cmr/m/it/10 lcm[]def semiring[]gcd[]class$\OML/cmm/m/it/10 :$\OT1/cmr/
m/it/10 lcm[]gcd$\OT1/cmr/m/n/10 )$[] 

Overfull \hbox (13.21527pt too wide) in paragraph at lines 1235--1237
[][]    \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp add$\OT1/c
mr/m/n/10 :$ \OT1/cmr/m/it/10 fun[]eq[]iff Euclidean[]Algorithm$\OML/cmm/m/it/1
0 :$\OT1/cmr/m/it/10 Gcd[]def semiring[]Gcd[]class$\OML/cmm/m/it/10 :$\OT1/cmr/
m/it/10 Gcd[]Lcm$\OT1/cmr/m/n/10 )$[] 
[135]
Overfull \hbox (4.18866pt too wide) in paragraph at lines 1454--1456
[][]      \OT1/cmr/bx/n/10 qed $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 auto simp$\O
T1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 gcd[]mod[]right algebra[]simps minus[]mod[]eq
[]div[]mult $\OT1/cmr/m/n/10 [$\OT1/cmr/m/it/10 symmetric$\OT1/cmr/m/n/10 ]$
[136] [137]
Overfull \hbox (2.401pt too wide) in paragraph at lines 1668--1670
[][]    \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp add$\OT1/c
mr/m/n/10 :$ \OT1/cmr/m/it/10 fun[]eq[]iff Euclidean[]Algorithm$\OML/cmm/m/it/1
0 :$\OT1/cmr/m/it/10 lcm[]def semiring[]gcd[]class$\OML/cmm/m/it/10 :$\OT1/cmr/
m/it/10 lcm[]gcd$\OT1/cmr/m/n/10 )$[] 

Overfull \hbox (13.21527pt too wide) in paragraph at lines 1672--1674
[][]    \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp add$\OT1/c
mr/m/n/10 :$ \OT1/cmr/m/it/10 fun[]eq[]iff Euclidean[]Algorithm$\OML/cmm/m/it/1
0 :$\OT1/cmr/m/it/10 Gcd[]def semiring[]Gcd[]class$\OML/cmm/m/it/10 :$\OT1/cmr/
m/it/10 Gcd[]Lcm$\OT1/cmr/m/n/10 )$[] 
[138]
Overfull \hbox (2.401pt too wide) in paragraph at lines 1771--1773
[][]    \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp add$\OT1/c
mr/m/n/10 :$ \OT1/cmr/m/it/10 fun[]eq[]iff Euclidean[]Algorithm$\OML/cmm/m/it/1
0 :$\OT1/cmr/m/it/10 lcm[]def semiring[]gcd[]class$\OML/cmm/m/it/10 :$\OT1/cmr/
m/it/10 lcm[]gcd$\OT1/cmr/m/n/10 )$[] 

Overfull \hbox (13.21527pt too wide) in paragraph at lines 1775--1777
[][]    \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp add$\OT1/c
mr/m/n/10 :$ \OT1/cmr/m/it/10 fun[]eq[]iff Euclidean[]Algorithm$\OML/cmm/m/it/1
0 :$\OT1/cmr/m/it/10 Gcd[]def semiring[]Gcd[]class$\OML/cmm/m/it/10 :$\OT1/cmr/
m/it/10 Gcd[]Lcm$\OT1/cmr/m/n/10 )$[] 
) (./Ring_Hom.tex [139] [140]) (./Fraction_Field.tex [141] [142] [143] [144]
[145] [146] [147] [148] [149] [150]) (./Normalized_Fraction.tex [151] [152]
Overfull \hbox (25.4372pt too wide) in paragraph at lines 205--207
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 normalize[]quot[]in[]normalized[]fr
acts $\OT1/cmr/m/n/10 [$\OT1/cmr/m/it/10 simp$\OT1/cmr/m/n/10 ]$$:$ \OT1/cmr/m/
it/10 normalize[]quot x $\OMS/cmsy/m/n/10 2$ \OT1/cmr/m/it/10 normalized[]fract
s[] 

Overfull \hbox (11.43802pt too wide) in paragraph at lines 251--253
[][]    \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 intro coprime[]
crossproduct$ [] $$\OT1/cmr/m/n/10 )$ $($\OT1/cmr/m/it/10 simp[]all add$\OT1/cm
r/m/n/10 :$ \OT1/cmr/m/it/10 x[]def y[]def coprime[]normalize[]quot$\OT1/cmr/m/
n/10 )$[] 

Overfull \hbox (1.46599pt too wide) in paragraph at lines 276--290
[][]  \OT1/cmr/bx/n/10 us-ing \OT1/cmr/m/it/10 assms \OT1/cmr/bx/n/10 by $\OT1/
cmr/m/n/10 ($\OT1/cmr/m/it/10 cases x$\OML/cmm/m/it/10 ;$ \OT1/cmr/m/it/10 case
s y$\OML/cmm/m/it/10 ;$ \OT1/cmr/m/it/10 hyp-subst$\OT1/cmr/m/n/10 )$ $($\OT1/c
mr/m/it/10 subst normalize[]quot[]eq[]iff$\OML/cmm/m/it/10 ;$ \OT1/cmr/m/it/10 
simp[]all$\OT1/cmr/m/n/10 )$[] 

Overfull \hbox (26.27673pt too wide) in paragraph at lines 311--313
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 normalize[]quot[]idem $\OT1/cmr/m/n
/10 [$\OT1/cmr/m/it/10 simp$\OT1/cmr/m/n/10 ]$$:$ \OT1/cmr/m/it/10 normalize[]q
uot $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 normalize[]quot x$\OT1/cmr/m/n/10 )$ $=
$ \OT1/cmr/m/it/10 normalize[]quot
[153]
Overfull \hbox (2.21165pt too wide) in paragraph at lines 476--480
[][]  \OT1/cmr/bx/n/10 thus \OT1/cmr/m/it/10 ?the-sis \OT1/cmr/bx/n/10 un-fold-
ing \OT1/cmr/m/it/10 quot[]to[]fract[]def \OT1/cmr/bx/n/10 by \OT1/cmr/m/it/10 
trans-fer $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp add$\OT1/cmr/m/n/10 :$ \OT1/
cmr/m/it/10 case[]prod[]unfold$\OT1/cmr/m/n/10 )$[] 

Overfull \hbox (3.45944pt too wide) in paragraph at lines 510--512
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 quot[]of[]fract[]in[]normalized[]fr
acts $\OT1/cmr/m/n/10 [$\OT1/cmr/m/it/10 simp$\OT1/cmr/m/n/10 ]$$:$ \OT1/cmr/m/
it/10 quot[]of[]fract x $\OMS/cmsy/m/n/10 2$ \OT1/cmr/m/it/10 normalized[]fract
s[] 
[154] [155]
Overfull \hbox (12.708pt too wide) in paragraph at lines 742--746
[][]  \OT1/cmr/bx/n/10 from \OT1/cmr/m/it/10 assms \OT1/cmr/bx/n/10 have \OT1/c
mr/m/it/10 co-prime e f co-prime g h \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT
1/cmr/m/it/10 simp[]all add$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 coprime[]normal
ize[]quot$\OT1/cmr/m/n/10 )$[] 

Overfull \hbox (21.90312pt too wide) in paragraph at lines 749--751
[][]    \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp add$\OT1/c
mr/m/n/10 :$ \OT1/cmr/m/it/10 normalized[]fracts[]def unit[]factor[]mult coprim
e[]mul[]eq coprime[]mul[]eq$ [] $$\OT1/cmr/m/n/10 )$[] 

Overfull \hbox (36.04117pt too wide) in paragraph at lines 842--855
[][]  \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 auto simp$\OT1/cm
r/m/n/10 :$ \OT1/cmr/m/it/10 normalize[]quot[]def case[]prod[]unfold Let[]def d
iv[]mult[]unit2 dvd[]div[]eq[]0[]iff$\OT1/cmr/m/n/10 )$[] 
[156]
Overfull \hbox (15.3068pt too wide) in paragraph at lines 907--910
[][]    \OT1/cmr/bx/n/10 us-ing \OT1/cmr/m/it/10 assms$\OT1/cmr/m/n/10 ($\OT1/c
mr/m/it/10 1$\OML/cmm/m/it/10 ;$\OT1/cmr/m/it/10 2$\OT1/cmr/m/n/10 )$ \OT1/cmr/
m/it/10 d \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 auto simp$\OT
1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 normalized[]fracts[]def gcd[]div[]unit1 gcd[]d
iv[]unit2

Overfull \hbox (49.1144pt too wide) in paragraph at lines 971--975
[][]  \OT1/cmr/bx/n/10 from \OT1/cmr/m/it/10 assms \OT1/cmr/bx/n/10 have \OT1/c
mr/m/it/10 co-prime x$ [] $ y$ [] $ unit[]factor y$ [] $ $\OT1/cmr/m/n/10 =$ \O
T1/cmr/m/it/10 1 \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp[]
all add$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 coprime[]normalize[]quot$\OT1/cmr/m
/n/10 )$[] 

Overfull \hbox (29.09537pt too wide) in paragraph at lines 980--981
[][]       $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 auto simp$\OT1/cmr/m/n/10 :$ \OT
1/cmr/m/it/10 normalized[]fracts[]def unit[]div[]mult[]swap unit[]div[]commute 
unit[]div[]cancel[] 

Overfull \hbox (49.1144pt too wide) in paragraph at lines 1013--1017
[][]  \OT1/cmr/bx/n/10 from \OT1/cmr/m/it/10 assms \OT1/cmr/bx/n/10 have \OT1/c
mr/m/it/10 co-prime x$ [] $ y$ [] $ unit[]factor y$ [] $ $\OT1/cmr/m/n/10 =$ \O
T1/cmr/m/it/10 1 \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp[]
all add$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 coprime[]normalize[]quot$\OT1/cmr/m
/n/10 )$[] 

Overfull \hbox (29.09537pt too wide) in paragraph at lines 1022--1023
[][]       $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 auto simp$\OT1/cmr/m/n/10 :$ \OT
1/cmr/m/it/10 normalized[]fracts[]def unit[]div[]mult[]swap unit[]div[]commute 
unit[]div[]cancel[] 
[157]) (./Field_as_Ring.tex [158]
Overfull \hbox (21.73355pt too wide) in paragraph at lines 116--132
[][]\OT1/cmr/bx/n/10 instance by \OT1/cmr/m/it/10 stan-dard $\OT1/cmr/m/n/10 ($
\OT1/cmr/m/it/10 simp[]all add$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 gcd[]real[]d
ef lcm[]real[]def Gcd[]real[]def Lcm[]real[]def$\OT1/cmr/m/n/10 )$[] 
[159]
Overfull \hbox (8.4448pt too wide) in paragraph at lines 190--206
[][]\OT1/cmr/bx/n/10 instance by \OT1/cmr/m/it/10 stan-dard $\OT1/cmr/m/n/10 ($
\OT1/cmr/m/it/10 simp[]all add$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 gcd[]rat[]de
f lcm[]rat[]def Gcd[]rat[]def Lcm[]rat[]def$\OT1/cmr/m/n/10 )$[] 
[160]
Overfull \hbox (15.7032pt too wide) in paragraph at lines 264--280
[][]\OT1/cmr/bx/n/10 instance by \OT1/cmr/m/it/10 stan-dard $\OT1/cmr/m/n/10 ($
\OT1/cmr/m/it/10 simp[]all add$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 gcd[]complex
[]def lcm[]complex[]def Gcd[]complex[]def
) (./Polynomial_Factorial.tex
Overfull \hbox (2.32649pt too wide) in paragraph at lines 83--87
[][]    \OT1/cmr/bx/n/10 then ob-tain \OT1/cmr/m/it/10 a$ [] $ b$ [] $ \OT1/cmr
/bx/n/10 where \OT1/cmr/m/it/10 ab$ [] $$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 a 
$\OT1/cmr/m/n/10 =$ $[$$:$\OT1/cmr/m/it/10 a$ [] $$\OT1/cmr/m/n/10 :$$]$ \OT1/c
mr/m/it/10 b $\OT1/cmr/m/n/10 =$ $[$$:$\OT1/cmr/m/it/10 b$ [] $$\OT1/cmr/m/n/10
 :$$]$ \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 auto elim$\OT1/c
mr/m/n/10 !$$:$ \OT1/cmr/m/it/10 degree[]eq[]zeroE$\OT1/cmr/m/n/10 )$[] 
[161] [162]
Overfull \hbox (30.61302pt too wide) in paragraph at lines 511--513
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 normalize[]snd[]quot[]of[]fract$\OT
1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 nor-mal-ize $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/1
0 snd $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 quot[]of[]fract x$\OT1/cmr/m/n/10 )$$
)$ $=$ \OT1/cmr/m/it/10 snd $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 quot[]of[]fract


Overfull \hbox (3.69643pt too wide) in paragraph at lines 539--541
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 fract[]poly[]smult $\OT1/cmr/m/n/10
 [$\OT1/cmr/m/it/10 simp$\OT1/cmr/m/n/10 ]$$:$ \OT1/cmr/m/it/10 fract[]poly $\O
T1/cmr/m/n/10 ($\OT1/cmr/m/it/10 smult c p$\OT1/cmr/m/n/10 )$ $=$ \OT1/cmr/m/it
/10 smult $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 to[]fract c$\OT1/cmr/m/n/10 )$ $(
$\OT1/cmr/m/it/10 fract[]poly
[163]
Overfull \hbox (9.21042pt too wide) in paragraph at lines 725--726
[][]  \OT1/cmr/m/it/10 prod[]mset $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 image[]ms
et $\OT1/cmr/m/n/10 ($$\OML/cmm/m/it/10 ^^U$\OT1/cmr/m/it/10 x$\OML/cmm/m/it/10
 :$ \OT1/cmr/m/it/10 fract[]poly $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 f x$\OT1/c
mr/m/n/10 )$$)$ \OT1/cmr/m/it/10 A$\OT1/cmr/m/n/10 )$ $=$ \OT1/cmr/m/it/10 frac
t[]poly $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 prod[]mset $\OT1/cmr/m/n/10 ($\OT1/
cmr/m/it/10 image[]mset
[164]
Overfull \hbox (5.10414pt too wide) in paragraph at lines 850--853
[][]  \OT1/cmr/bx/n/10 hence \OT1/cmr/m/it/10 fract[]poly $\OT1/cmr/m/n/10 ($\O
T1/cmr/m/it/10 smult a q$\OT1/cmr/m/n/10 )$ $=$ \OT1/cmr/m/it/10 fract[]poly $\
OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 smult b p$\OT1/cmr/m/n/10 )$ \OT1/cmr/bx/n/10
 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp del$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/
it/10 fract[]poly[]eq[]iff$\OT1/cmr/m/n/10 )$[] 
[165] [166]
Overfull \hbox (9.28085pt too wide) in paragraph at lines 1049--1051
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 fract[]content[]fract[]poly $\OT1/c
mr/m/n/10 [$\OT1/cmr/m/it/10 simp$\OT1/cmr/m/n/10 ]$$:$ \OT1/cmr/m/it/10 fract[
]content $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 fract[]poly p$\OT1/cmr/m/n/10 )$ $
=$ \OT1/cmr/m/it/10 to[]fract $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 content

Overfull \hbox (54.7458pt too wide) in paragraph at lines 1113--1115
[][]    \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 rule that$\OT1/
cmr/m/n/10 [$\OT1/cmr/m/it/10 OF content[]times[]primitive[]part[]fract $\OT1/c
mr/m/n/10 [$\OT1/cmr/m/it/10 symmetric$\OT1/cmr/m/n/10 ]$ \OT1/cmr/m/it/10 cont
ent[]primitive[]part[]fract$\OT1/cmr/m/n/10 ]$$)$[] 
[167] [168]
Overfull \hbox (14.12436pt too wide) in paragraph at lines 1301--1305
[][]    \OT1/cmr/bx/n/10 from \OT1/cmr/m/it/10 A \OT1/cmr/bx/n/10 and \OT1/cmr/
m/it/10 this \OT1/cmr/bx/n/10 have $\OT1/cmr/m/n/10 [$$:$\OT1/cmr/m/it/10 c$\OT
1/cmr/m/n/10 :$$]$ \OT1/cmr/m/it/10 dvd $\OT1/cmr/m/n/10 [$$:$\OT1/cmr/m/it/10 
a$\OT1/cmr/m/n/10 :$$]$ $\OMS/cmsy/m/n/10 _$ $\OT1/cmr/m/n/10 [$$:$\OT1/cmr/m/i
t/10 c$\OT1/cmr/m/n/10 :$$]$ \OT1/cmr/m/it/10 dvd $\OT1/cmr/m/n/10 [$$:$\OT1/cm
r/m/it/10 b$\OT1/cmr/m/n/10 :$$]$ \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/c
mr/m/it/10 rule prime[]elem[]dvd[]multD$\OT1/cmr/m/n/10 )$[] 

Overfull \hbox (17.28046pt too wide) in paragraph at lines 1365--1370
[][]    \OT1/cmr/bx/n/10 more-over from []\OT1/cmr/m/it/10 prime p[] \OT1/cmr/b
x/n/10 have \OT1/cmr/m/it/10 prime[]elem $\OT1/cmr/m/n/10 [$$:$\OT1/cmr/m/it/10
 p$\OT1/cmr/m/n/10 :$$]$ \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/1
0 simp add$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 lift[]prime[]elem[]poly$\OT1/cmr
/m/n/10 )$[] 
[169] [170] [171]
Overfull \hbox (8.21928pt too wide) in paragraph at lines 1744--1746
[][]    \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 elim is[]unit[]
polyE$\OT1/cmr/m/n/10 )$ $($\OT1/cmr/m/it/10 auto simp$\OT1/cmr/m/n/10 :$ \OT1/
cmr/m/it/10 unit[]factor[]field[]poly[]def monom[]0 one[]poly[]def

Overfull \hbox (10.56847pt too wide) in paragraph at lines 1781--1782
[][]      $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 auto simp add$\OT1/cmr/m/n/10 :$ 
\OT1/cmr/m/it/10 unit[]factor[]field[]poly[]def euclidean[]size[]field[]poly[]d
ef div[]poly[]less$\OT1/cmr/m/n/10 )$[] 
[172]
Overfull \hbox (6.67152pt too wide) in paragraph at lines 1842--1844
[][]    \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 intro ext$\OT1/
cmr/m/n/10 )$ $($\OT1/cmr/m/it/10 simp add$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 
comm[]monoid[]mult$\OML/cmm/m/it/10 :$\OT1/cmr/m/it/10 prod[]mset[]def$\OT1/cmr
/m/n/10 [$\OT1/cmr/m/it/10 OF A$\OT1/cmr/m/n/10 ]$ \OT1/cmr/m/it/10 prod[]mset[
]def$\OT1/cmr/m/n/10 )$[] 

Overfull \hbox (0.43723pt too wide) in paragraph at lines 1882--1884
[][]      \OT1/cmr/m/it/10 comm[]semiring[]1$\OML/cmm/m/it/10 :$\OT1/cmr/m/it/1
0 prime[]elem[]def$\OT1/cmr/m/n/10 [$\OT1/cmr/m/it/10 OF A$\OT1/cmr/m/n/10 ]$ \
OT1/cmr/m/it/10 normalization[]semidom$\OML/cmm/m/it/10 :$\OT1/cmr/m/it/10 prim
e[]def$\OT1/cmr/m/n/10 [$\OT1/cmr/m/it/10 OF
[173]
Overfull \hbox (3.74106pt too wide) in paragraph at lines 2045--2048
[][]   \OT1/cmr/bx/n/10 hence \OT1/cmr/m/it/10 q$ [] $ dvd 1 $\OMS/cmsy/m/n/10 
_$ \OT1/cmr/m/it/10 r$ [] $ dvd 1 \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/c
mr/m/it/10 auto dest$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 dvd[]mult[]right simp 
del$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 mult[]pCons[]left$\OT1/cmr/m/n/10 )$[] 

[174]
Overfull \hbox (41.16667pt too wide) in paragraph at lines 2143--2147
[][]  \OT1/cmr/bx/n/10 from \OT1/cmr/m/it/10 ir-red \OT1/cmr/bx/n/10 have \OT1/
cmr/m/it/10 prime$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 prime[]elem $\OT1/cmr/m/n
/10 ($\OT1/cmr/m/it/10 fract[]poly p$\OT1/cmr/m/n/10 )$ \OT1/cmr/bx/n/10 by $\O
T1/cmr/m/n/10 ($\OT1/cmr/m/it/10 rule field[]poly[]irreducible[]imp[]prime$\OT1
/cmr/m/n/10 )$[] 
[175] [176]
Overfull \hbox (2.86989pt too wide) in paragraph at lines 2444--2446
[][]    \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp add$\OT1/c
mr/m/n/10 :$ \OT1/cmr/m/it/10 unit[]factor[]field[]poly[]def monom[]0 degree[]m
ap[]poly coeff[]map[]poly$\OT1/cmr/m/n/10 )$[] 

Overfull \hbox (14.1927pt too wide) in paragraph at lines 2465--2467
[][]    \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 subst prod[]mse
t$\OML/cmm/m/it/10 :$\OT1/cmr/m/it/10 distrib$\OT1/cmr/m/n/10 )$ $($\OT1/cmr/m/
it/10 simp[]all add$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 prod[]mset[]fract[]poly
 prod[]mset[]const[]poly
[177]
Overfull \hbox (47.79518pt too wide) in paragraph at lines 2535--2538
[][]    \OT1/cmr/bx/n/10 us-ing \OT1/cmr/m/it/10 that \OT1/cmr/bx/n/10 by $\OT1
/cmr/m/n/10 ($\OT1/cmr/m/it/10 auto simp$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 A[
]def prime[]elem[]primitive[]part[]fract prime[]elem[]imp[]irreducible

Overfull \hbox (14.65166pt too wide) in paragraph at lines 2569--2571
[][]  \OT1/cmr/bx/n/10 have $\OMS/cmsy/m/n/10 9 $\OT1/cmr/m/it/10 A$\OML/cmm/m/
it/10 :$ $\OT1/cmr/m/n/10 ($$\OMS/cmsy/m/n/10 8 $\OT1/cmr/m/it/10 p$\OML/cmm/m/
it/10 :$ \OT1/cmr/m/it/10 p $\OMS/cmsy/m/n/10 2$$\OT1/cmr/m/n/10 #$ \OT1/cmr/m/
it/10 A $[][]\OMS/cmsy/m/n/10 !$ \OT1/cmr/m/it/10 prime[]elem p$\OT1/cmr/m/n/10
 )$ $\OMS/cmsy/m/n/10 ^$ \OT1/cmr/m/it/10 prod[]mset A $\OT1/cmr/m/n/10 =$ \OT1
/cmr/m/it/10 nor-mal-ize $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 primitive[]part

Overfull \hbox (7.14774pt too wide) in paragraph at lines 2587--2589
[][]    \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 auto simp$\OT1/
cmr/m/n/10 :$ \OT1/cmr/m/it/10 B[]def in-tro$\OT1/cmr/m/n/10 !$$:$ \OT1/cmr/m/i
t/10 lift[]prime[]elem[]poly dest$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 in[]prime
[]factors[]imp[]prime$\OT1/cmr/m/n/10 )$[] 
[178]
Overfull \hbox (28.37808pt too wide) in paragraph at lines 2646--2662
[][]\OT1/cmr/bx/n/10 instance by \OT1/cmr/m/it/10 stan-dard $\OT1/cmr/m/n/10 ($
\OT1/cmr/m/it/10 simp[]all add$\OT1/cmr/m/n/10 :$ \OT1/cmr/m/it/10 gcd[]poly[]d
ef lcm[]poly[]def Gcd[]poly[]def Lcm[]poly[]def$\OT1/cmr/m/n/10 )$[] 

Overfull \hbox (15.0717pt too wide) in paragraph at lines 2688--2689
[][]   $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 auto simp$\OT1/cmr/m/n/10 :$ \OT1/cm
r/m/it/10 euclidean[]size[]poly[]def Rings$\OML/cmm/m/it/10 :$\OT1/cmr/m/it/10 
div[]mult[]mod[]eq div[]poly[]less degree[]mult[]eq
[179]
Overfull \hbox (10.64919pt too wide) in paragraph at lines 2763--2765
[][]  \OT1/cmr/bx/n/10 thus \OT1/cmr/m/it/10 d dvd smult $\OT1/cmr/m/n/10 ($\OT
1/cmr/m/it/10 gcd $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 content p$\OT1/cmr/m/n/10
 )$ $($\OT1/cmr/m/it/10 content q$\OT1/cmr/m/n/10 )$$)$ $($\OT1/cmr/m/it/10 gcd
 $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 primitive[]part p$\OT1/cmr/m/n/10 )$ $($\O
T1/cmr/m/it/10 primitive[]part
[180]
Overfull \hbox (35.85698pt too wide) in paragraph at lines 2991--3004
[][]  \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp add$\OT1/cmr
/m/n/10 :$ \OT1/cmr/m/it/10 gcd[]poly[]code[]def gcd[]poly[]code[]aux[]correct 
gcd[]poly[]decompose $\OT1/cmr/m/n/10 [$\OT1/cmr/m/it/10 symmetric$\OT1/cmr/m/n
/10 ]$$)$[] 
[181]
Overfull \hbox (31.4pt too wide) in paragraph at lines 3027--3029
[][]\OT1/cmr/bx/n/10 lemmas \OT1/cmr/m/it/10 Gcd[]poly[]set[]eq[]fold $\OT1/cmr
/m/n/10 [$\OT1/cmr/m/it/10 code$\OT1/cmr/m/n/10 ]$ $=$ \OT1/cmr/m/it/10 Gcd[]se
t[]eq[]fold $\OT1/cmr/m/n/10 [$\OT1/cmr/bx/n/10 where \OT1/cmr/m/it/10 ?$ [] $a
 $\OT1/cmr/m/n/10 =$ $ [] $\OT1/cmr/m/it/10 a $\OT1/cmr/m/n/10 :$$:$ \OT1/cmr/m
/it/10 factorial[]ring[]gcd

Overfull \hbox (34.6055pt too wide) in paragraph at lines 3029--3032
[][]\OT1/cmr/bx/n/10 lemmas \OT1/cmr/m/it/10 Lcm[]poly[]set[]eq[]fold $\OT1/cmr
/m/n/10 [$\OT1/cmr/m/it/10 code$\OT1/cmr/m/n/10 ]$ $=$ \OT1/cmr/m/it/10 Lcm[]se
t[]eq[]fold $\OT1/cmr/m/n/10 [$\OT1/cmr/bx/n/10 where \OT1/cmr/m/it/10 ?$ [] $a
 $\OT1/cmr/m/n/10 =$ $ [] $\OT1/cmr/m/it/10 a $\OT1/cmr/m/n/10 :$$:$ \OT1/cmr/m
/it/10 factorial[]ring[]gcd
) (./Missing_Unsorted.tex [182]
Overfull \hbox (22.26895pt too wide) in paragraph at lines 177--182
[][]  \OT1/cmr/bx/n/10 also have $[] $ $\OML/cmm/m/it/10 <$ \OT1/cmr/m/it/10 bb
b $\OMS/cmsy/m/n/10 ^^C$ \OT1/cmr/m/it/10 of[]nat n \OT1/cmr/bx/n/10 us-ing \OT
1/cmr/m/it/10 lt bbb \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 me
tis mult$\OML/cmm/m/it/10 :$\OT1/cmr/m/it/10 commute pos[]divide[]less[]eq$\OT1
/cmr/m/n/10 )$[] 
[183] [184] [185] [186] [187] [188] [189] [190]) (./Missing_Polynomial.tex
[191]
Overfull \hbox (19.45976pt too wide) in paragraph at lines 104--117
[][]  \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 metis Suc[]1 Suc[
]neq[]Zero degree1[]coeffs degree[]pCons[]eq[]if nat$\OML/cmm/m/it/10 :$\OT1/cm
r/m/it/10 inject pCons[]cases$\OT1/cmr/m/n/10 )$[] 
[192] [193] [194] [195] [196] [197]
Overfull \hbox (26.31035pt too wide) in paragraph at lines 1194--1196
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 degree[]prod[]sum[]lt[]n$\OT1/cmr/m
/n/10 :$ \OT1/cmr/bx/n/10 as-sumes $[]$ \OT1/cmr/m/it/10 i$\OML/cmm/m/it/10 :$ 
\OT1/cmr/m/it/10 i $\OML/cmm/m/it/10 <$ \OT1/cmr/m/it/10 n $\OT1/cmr/m/n/10 =[]
\OMS/cmsy/m/n/10 )$ \OT1/cmr/m/it/10 de-gree $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/1
0 f i $\OT1/cmr/m/n/10 :$$:$ $ [] $\OT1/cmr/m/it/10 a $\OT1/cmr/m/n/10 :$$:$ \O
T1/cmr/m/it/10 comm[]semiring[]1
[198] [199] [200] [201] [202] [203] [204] [205] [206] [207] [208]

Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\sfcode' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `math shift' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `math shift' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `math shift' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\mathord' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `math shift' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `math shift' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\mathord' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `math shift' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `math shift' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\mskip' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `\mskip' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `math shift' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `math shift' on input line 2890.


Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding):
(hyperref)                removing `math shift' on input line 2890.

[209] [210]
Overfull \hbox (46.84029pt too wide) in paragraph at lines 3269--3271
[][]  \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 metis $\OT1/cmr/m
/n/10 ($\OT1/cmr/m/it/10 no[]types$\OML/cmm/m/it/10 ;$ \OT1/cmr/m/it/10 lift-in
g$\OT1/cmr/m/n/10 )$ \OT1/cmr/m/it/10 coeff[]smult degree[]smult[]eq ir-re-duci
bleD$\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 1$\OT1/cmr/m/n/10 )$ \OT1/cmr/m/it/10 i
rreducible[]dvd[]smult
[211]
Overfull \hbox (17.23799pt too wide) in paragraph at lines 3444--3445
[][]  \OT1/cmr/m/it/10 nor-mal-ize $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 f $\OT1/
cmr/m/n/10 :$$:$ $ [] $\OT1/cmr/m/it/10 a $\OT1/cmr/m/n/10 :$$:$ $\OMS/cmsy/m/n
/10 f$\OT1/cmr/m/it/10 normalization[]semidom$\OML/cmm/m/it/10 ;$\OT1/cmr/m/it/
10 field$\OMS/cmsy/m/n/10 g$ \OT1/cmr/m/it/10 poly$\OT1/cmr/m/n/10 )$ $=$ \OT1/
cmr/m/it/10 smult $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 inverse $\OT1/cmr/m/n/10 
($\OT1/cmr/m/it/10 unit[]factor
[212] [213]
Overfull \hbox (4.89697pt too wide) in paragraph at lines 3798--3802
[][]    \OT1/cmr/bx/n/10 have \OT1/cmr/m/it/10 id$\OT1/cmr/m/n/10 :$ \OT1/cmr/m
/it/10 n $\OMS/cmsy/m/n/10 ^^C$ \OT1/cmr/m/it/10 Suc ii $\OT1/cmr/m/n/10 +$ \OT
1/cmr/m/it/10 j $\OMS/cmsy/m/n/10 ^^@$ \OT1/cmr/m/it/10 n $\OT1/cmr/m/n/10 =$ \
OT1/cmr/m/it/10 n $\OMS/cmsy/m/n/10 ^^C$ \OT1/cmr/m/it/10 ii $\OT1/cmr/m/n/10 +
$ \OT1/cmr/m/it/10 j \OT1/cmr/bx/n/10 us-ing \OT1/cmr/m/it/10 n \OT1/cmr/bx/n/1
0 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 simp add$\OT1/cmr/m/n/10 :$ \OT1/cmr/m
/it/10 diff[]mult[]distrib2$\OT1/cmr/m/n/10 )$[] 
[214]) (./Ring_Hom_Poly.tex
Overfull \hbox (12.6962pt too wide) in paragraph at lines 73--87
[][]  \OT1/cmr/bx/n/10 us-ing \OT1/cmr/m/it/10 pcompose[]add pcompose[]mult pco
mpose[]1 pcompose[]uminus \OT1/cmr/bx/n/10 by $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/
10 unfold[]locales$\OML/cmm/m/it/10 ;$
[215] [216] [217] [218] [219] [220] [221] [222]) (./Newton_Interpolation.tex
[223]
Overfull \hbox (17.22385pt too wide) in paragraph at lines 50--51
[][]  \OT1/cmr/m/it/10 horner[]composition $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 
map hom cs$\OT1/cmr/m/n/10 )$ $($\OT1/cmr/m/it/10 map hom xs$\OT1/cmr/m/n/10 )$
 $=$ \OT1/cmr/m/it/10 map[]poly hom $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 horner[
]composition
[224] [225] [226] [227] [228] [229] [230] [231] [232] [233] [234]
Overfull \hbox (2.43918pt too wide) in paragraph at lines 1490--1492
[][]    \OT1/cmr/bx/n/10 un-fold-ing \OT1/cmr/m/it/10 newton[]poly[]impl[]def n
ewton[]coefficients rev[]rev[]ident newton[]poly[]def
[235] [236] [237]
Overfull \hbox (4.60117pt too wide) in paragraph at lines 1779--1780
[][]  $\OT1/cmr/m/n/10 =[]\OMS/cmsy/m/n/10 )$ \OT1/cmr/m/it/10 newton[]coeffici
ents[]main $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 map rat[]of[]int xs$\OT1/cmr/m/n
/10 )$ $($\OT1/cmr/m/it/10 map rat[]of[]int gs$\OT1/cmr/m/n/10 )$ $($\OT1/cmr/m
/it/10 map rat[]of[]int
[238] [239] [240]
Overfull \hbox (17.51727pt too wide) in paragraph at lines 2158--2160
[][]    \OT1/cmr/bx/n/10 have \OT1/cmr/m/it/10 id$\OT1/cmr/m/n/10 :$ \OT1/cmr/m
/it/10 horner[]composition $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 rev ?nc$\OT1/cmr
/m/n/10 )$ $($\OT1/cmr/m/it/10 ?mri xs$\OT1/cmr/m/n/10 )$ $=$ \OT1/cmr/m/it/10 
map[]poly ?ri $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 horner[]composition
[241] [242]
Overfull \hbox (6.79475pt too wide) in paragraph at lines 2349--2351
[][]    \OT1/cmr/bx/n/10 un-fold-ing \OT1/cmr/m/it/10 newton[]interpolation[]po
ly[]def  newton[]interpolation[]poly[]int[]def Let[]def
) (./Lagrange_Interpolation.tex [243]
Overfull \hbox (5.18915pt too wide) in paragraph at lines 177--179
[][]    \OT1/cmr/bx/n/10 un-fold-ing \OT1/cmr/m/it/10 lagrange[]basis[]poly[]de
f Let[]def poly[]prod[]list prod[]list[]map[]remove1$\OT1/cmr/m/n/10 [$\OT1/cmr
/m/it/10 OF
[244]
Overfull \hbox (10.95041pt too wide) in paragraph at lines 211--214
[][]    \OT1/cmr/bx/n/10 show \OT1/cmr/m/it/10 poly p x $\OT1/cmr/m/n/10 =$ \OT
1/cmr/m/it/10 y \OT1/cmr/bx/n/10 un-fold-ing \OT1/cmr/m/it/10 p lagrange[]inter
polation[]poly[]def Let[]def poly[]sum[]list

Overfull \hbox (33.61983pt too wide) in paragraph at lines 214--216
[][]    \OT1/cmr/bx/n/10 proof $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 subst sum[]l
ist[]map[]remove1$\OT1/cmr/m/n/10 [$\OT1/cmr/m/it/10 OF xy$\OT1/cmr/m/n/10 ]$$\
OML/cmm/m/it/10 ;$ \OT1/cmr/m/it/10 un-fold split poly[]smult lagrange[]basis[]
poly[]1$\OML/cmm/m/it/10 ;$[] 

Overfull \hbox (17.11581pt too wide) in paragraph at lines 219--221
[][]      \OT1/cmr/bx/n/10 as-sume \OT1/cmr/m/it/10 v $\OMS/cmsy/m/n/10 2$ \OT1
/cmr/m/it/10 set $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 map $\OT1/cmr/m/n/10 ($$\O
ML/cmm/m/it/10 ^^U$\OT1/cmr/m/it/10 xa$\OML/cmm/m/it/10 :$ \OT1/cmr/m/it/10 pol
y $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 case xa of $\OT1/cmr/m/n/10 ($\OT1/cmr/m/
it/10 xj$\OML/cmm/m/it/10 ;$ \OT1/cmr/m/it/10 yj$\OT1/cmr/m/n/10 )$ $\OMS/cmsy/
m/n/10 )$ \OT1/cmr/m/it/10 smult yj $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 lagrang
e[]basis[]poly
) (./Neville_Aitken_Interpolation.tex [245] [246] [247]
Overfull \hbox (0.26402pt too wide) in paragraph at lines 318--320
[][]\OT1/cmr/bx/n/10 lemma \OT1/cmr/m/it/10 length[]neville[]aitken[]merge$\OT1
/cmr/m/n/10 [$\OT1/cmr/m/it/10 termination[]simp$\OT1/cmr/m/n/10 ]$$:$ \OT1/cmr
/m/it/10 length $\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 neville[]aitken[]merge
[248]
Overfull \hbox (30.07036pt too wide) in paragraph at lines 441--444
[][]  \OT1/cmr/bx/n/10 show \OT1/cmr/m/it/10 ?case \OT1/cmr/bx/n/10 un-fold-ing
 \OT1/cmr/m/it/10 neville[]aitken[]impl[]main$\OML/cmm/m/it/10 :$\OT1/cmr/m/it/
10 simps 1$\OT1/cmr/m/n/10 ($\OT1/cmr/m/it/10 1$\OT1/cmr/m/n/10 )$$[$\OT1/cmr/m
/it/10 OF neville[]aitken[]merge$\OT1/cmr/m/n/10 [$\OT1/cmr/m/it/10 OF
[249]) (./Polynomial_Interpolation.tex [250] [251] [252] [253]
Overfull \hbox (26.8093pt too wide) in paragraph at lines 468--469
[][]  \OT1/cmr/m/it/10 interpolation[]poly[]int[]Some$\OT1/cmr/m/n/10 [$\OT1/cm
r/bx/n/10 where \OT1/cmr/m/it/10 alg $\OT1/cmr/m/n/10 =$ \OT1/cmr/m/it/10 New-t
on$\OML/cmm/m/it/10 ;$ \OT1/cmr/m/it/10 un-folded interpolation[]poly[]int$\OML
/cmm/m/it/10 :$\OT1/cmr/m/it/10 simps$\OT1/cmr/m/n/10 ]$[] 

Overfull \hbox (26.06485pt too wide) in paragraph at lines 472--474
[][]  \OT1/cmr/m/it/10 interpolation[]poly[]int[]None$\OT1/cmr/m/n/10 [$\OT1/cm
r/bx/n/10 where \OT1/cmr/m/it/10 alg $\OT1/cmr/m/n/10 =$ \OT1/cmr/m/it/10 New-t
on$\OML/cmm/m/it/10 ;$ \OT1/cmr/m/it/10 un-folded interpolation[]poly[]int$\OML
/cmm/m/it/10 :$\OT1/cmr/m/it/10 simps$\OT1/cmr/m/n/10 ]$ 
[254])) (./root.bbl) [255] (./root.aux) )
(see the transcript file for additional information)</usr/share/texlive/texmf-d
ist/fonts/type1/public/amsfonts/cm/cmbsy10.pfb></usr/share/texlive/texmf-dist/f
onts/type1/public/amsfonts/cm/cmbx10.pfb></usr/share/texlive/texmf-dist/fonts/t
ype1/public/amsfonts/cm/cmbx12.pfb></usr/share/texlive/texmf-dist/fonts/type1/p
ublic/amsfonts/cm/cmex10.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/
amsfonts/cm/cmmi10.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfon
ts/cm/cmmi12.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/
cmmi7.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi8.p
fb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmr10.pfb></us
r/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmr12.pfb></usr/share
/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmr17.pfb></usr/share/texliv
e/texmf-dist/fonts/type1/public/amsfonts/cm/cmr7.pfb></usr/share/texlive/texmf-
dist/fonts/type1/public/amsfonts/cm/cmr8.pfb></usr/share/texlive/texmf-dist/fon
ts/type1/public/amsfonts/cm/cmr9.pfb></usr/share/texlive/texmf-dist/fonts/type1
/public/amsfonts/cm/cmss10.pfb></usr/share/texlive/texmf-dist/fonts/type1/publi
c/amsfonts/cm/cmsy10.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsf
onts/cm/cmsy5.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm
/cmsy6.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy7.
pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy8.pfb></u
sr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmti10.pfb></usr/sha
re/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmti12.pfb></usr/share/tex
live/texmf-dist/fonts/type1/public/amsfonts/symbols/msam10.pfb>
Output written on root.pdf (255 pages, 831753 bytes).
Transcript written on root.log.

*** Failed to finish proof (line 91 of "~~/afp/thys/Polynomial_Interpolation/Is_Rat_To_Rat.thy"):
*** goal (1 subgoal):
***  1. of_rat q = Complex (real_of_rat q) 0
*** At command "by" (line 91 of "~~/afp/thys/Polynomial_Interpolation/Is_Rat_To_Rat.thy")
*** Failed to apply initial proof method (line 79 of "~~/afp/thys/Polynomial_Interpolation/Is_Rat_To_Rat.thy"):
*** goal (1 subgoal):
***  1. Complex (real_of_int d / real_of_int n) 0 =
***     Complex (real_of_int d) 0 / Complex (real_of_int n) 0
*** At command "by" (line 79 of "~~/afp/thys/Polynomial_Interpolation/Is_Rat_To_Rat.thy")
*** Failed to finish proof (line 70 of "~~/afp/thys/Polynomial_Interpolation/Is_Rat_To_Rat.thy"):
*** goal (1 subgoal):
***  1. of_int z = - Complex (- real_of_int z) 0 \<Longrightarrow>
***     - Complex (- real_of_int z) 0 = Complex (real_of_int z) 0
*** At command "by" (line 70 of "~~/afp/thys/Polynomial_Interpolation/Is_Rat_To_Rat.thy")
*** Failed to finish proof (line 50 of "~~/afp/thys/Polynomial_Interpolation/Is_Rat_To_Rat.thy"):
*** goal (2 subgoals):
***  1. Complex 0 0 = 0
***  2. \<And>n.
***        of_nat n = Complex (real n) 0 \<Longrightarrow>
***        1 + Complex (real n) 0 = Complex (1 + real n) 0
*** At command "by" (line 50 of "~~/afp/thys/Polynomial_Interpolation/Is_Rat_To_Rat.thy")