TPTP Problem File: SLH0795^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : IMP_Compiler_Reuse/0006_Compiler2/prob_00284_015078__6058608_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1457 ( 734 unt; 189 typ; 0 def)
% Number of atoms : 3166 (2019 equ; 0 cnn)
% Maximal formula atoms : 14 ( 2 avg)
% Number of connectives : 10612 ( 337 ~; 41 |; 331 &;8903 @)
% ( 0 <=>;1000 =>; 0 <=; 0 <~>)
% Maximal formula depth : 30 ( 6 avg)
% Number of types : 37 ( 36 usr)
% Number of type conns : 754 ( 754 >; 0 *; 0 +; 0 <<)
% Number of symbols : 156 ( 153 usr; 20 con; 0-3 aty)
% Number of variables : 3675 ( 143 ^;3324 !; 208 ?;3675 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 16:49:08.270
%------------------------------------------------------------------------------
% Could-be-implicit typings (36)
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map_Pr7594757675026762433st_int: ( produc369741286924889929st_int > list_int ) > list_P8654944379044127311st_int > list_list_int ).
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map_Pr6494458893431244577nt_int: ( product_prod_int_int > int ) > list_P5707943133018811711nt_int > list_int ).
thf(sy_c_List_Olist_Omap_001t__Product____Type__Oprod_It__Int__Oint_Mt__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J_J_001t__Compiler__Oinstr,type,
map_Pr7514435936640407687_instr: ( produc6425607678544837394st_int > instr ) > list_P419547167332244514st_int > list_instr ).
thf(sy_c_List_Olist_Omap_001t__Product____Type__Oprod_It__Int__Oint_Mt__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J_J_001t__Int__Oint,type,
map_Pr3224476576807795204nt_int: ( produc6425607678544837394st_int > int ) > list_P419547167332244514st_int > list_int ).
thf(sy_c_List_Olist_Omap_001t__Product____Type__Oprod_It__Int__Oint_Mt__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J_J_001t__List__Olist_It__String__Ochar_J,type,
map_Pr5338895242362685753t_char: ( produc6425607678544837394st_int > list_char ) > list_P419547167332244514st_int > list_list_char ).
thf(sy_c_List_Olist_Omap_001t__Product____Type__Oprod_It__Int__Oint_Mt__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J_J_001t__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J,type,
map_Pr4101632105909876081st_int: ( produc6425607678544837394st_int > produc369741286924889929st_int ) > list_P419547167332244514st_int > list_P8654944379044127311st_int ).
thf(sy_c_List_Olist_Omap_001t__Product____Type__Oprod_It__Int__Oint_Mt__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J_J,type,
map_Pr8625093460839654122st_int: ( produc6425607678544837394st_int > produc6425607678544837394st_int ) > list_P419547167332244514st_int > list_P419547167332244514st_int ).
thf(sy_c_List_Olist_Omap_001t__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__String__Ochar_J_J_Mt__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__Int__Oint_J_J_001t__List__Olist_It__List__Olist_It__String__Ochar_J_J,type,
map_Pr7817743546820071988t_char: ( produc4189061985984568957nt_int > list_list_char ) > list_P7813408393781968525nt_int > list_list_list_char ).
thf(sy_c_List_Olist_Omap_001t__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__String__Ochar_J_J_Mt__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__Int__Oint_J_J_001t__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__Int__Oint_J,type,
map_Pr254812380637628598nt_int: ( produc4189061985984568957nt_int > produc4435102495419491129nt_int ) > list_P7813408393781968525nt_int > list_P98921565119350719nt_int ).
thf(sy_c_List_Onth_001t__Compiler__Oinstr,type,
nth_instr: list_instr > nat > instr ).
thf(sy_c_List_Onth_001t__Int__Oint,type,
nth_int: list_int > nat > int ).
thf(sy_c_List_Onth_001t__List__Olist_It__String__Ochar_J,type,
nth_list_char: list_list_char > nat > list_char ).
thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Int__Oint_Mt__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J_J,type,
nth_Pr8327173284404383691st_int: list_P419547167332244514st_int > nat > produc6425607678544837394st_int ).
thf(sy_c_List_Otake_001t__Compiler__Oinstr,type,
take_instr: nat > list_instr > list_instr ).
thf(sy_c_List_Otake_001t__Int__Oint,type,
take_int: nat > list_int > list_int ).
thf(sy_c_List_Otake_001t__List__Olist_It__String__Ochar_J,type,
take_list_char: nat > list_list_char > list_list_char ).
thf(sy_c_List_Otake_001t__Product____Type__Oprod_It__Int__Oint_Mt__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J_J,type,
take_P1004856038796978596st_int: nat > list_P419547167332244514st_int > list_P419547167332244514st_int ).
thf(sy_c_List_Oupto,type,
upto: int > int > list_int ).
thf(sy_c_List_Oupto__aux,type,
upto_aux: int > int > list_int > list_int ).
thf(sy_c_List_Oupto__rel,type,
upto_rel: product_prod_int_int > product_prod_int_int > $o ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Onat_Ocase__nat_001_Eo,type,
case_nat_o: $o > ( nat > $o ) > nat > $o ).
thf(sy_c_Nat_Onat_Ocase__nat_001t__List__Olist_It__Compiler__Oinstr_J,type,
case_nat_list_instr: list_instr > ( nat > list_instr ) > nat > list_instr ).
thf(sy_c_Nat_Onat_Ocase__nat_001t__List__Olist_It__Int__Oint_J,type,
case_nat_list_int: list_int > ( nat > list_int ) > nat > list_int ).
thf(sy_c_Nat_Onat_Ocase__nat_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J_J_J,type,
case_n3094275698756463160st_int: list_P419547167332244514st_int > ( nat > list_P419547167332244514st_int ) > nat > list_P419547167332244514st_int ).
thf(sy_c_Nat_Onat_Ocase__nat_001t__Nat__Onat,type,
case_nat_nat: nat > ( nat > nat ) > nat > nat ).
thf(sy_c_Nat_Onat_Opred,type,
pred: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
semiri1316708129612266289at_nat: nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Com__Ocom,type,
size_size_com: com > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Compiler__Oinstr,type,
size_size_instr: instr > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Compiler__Oinstr_J,type,
size_size_list_instr: list_instr > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Int__Oint_J,type,
size_size_list_int: list_int > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__String__Ochar_J_J,type,
size_s356637359517785349t_char: list_list_char > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J_J_J,type,
size_s229347708945616566st_int: list_P419547167332244514st_int > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__String__Ochar,type,
size_size_char: char > nat ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Int__Oint,type,
numeral_numeral_int: num > int ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_eq_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Product__Type_OPair_001_062_It__Compiler__Oinstr_M_062_It__Compiler__Oinstr_M_Eo_J_J_001t__List__Olist_It__Compiler__Oinstr_J,type,
produc7818391707689388878_instr: ( instr > instr > $o ) > list_instr > produc7133927873007960030_instr ).
thf(sy_c_Product__Type_OPair_001_062_It__Int__Oint_M_062_It__Int__Oint_M_Eo_J_J_001t__List__Olist_It__Int__Oint_J,type,
produc8618682346314911123st_int: ( int > int > $o ) > list_int > produc5834231552977413017st_int ).
thf(sy_c_Product__Type_OPair_001_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_001t__Int__Oint,type,
produc5790713362662368625nt_int: ( list_char > int ) > int > produc4435102495419491129nt_int ).
thf(sy_c_Product__Type_OPair_001_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_001t__List__Olist_It__Int__Oint_J,type,
produc8650753666468850689st_int: ( list_char > int ) > list_int > produc369741286924889929st_int ).
thf(sy_c_Product__Type_OPair_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J_J_M_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J_J_M_Eo_J_J_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J_J_J,type,
produc554038597315947693st_int: ( produc6425607678544837394st_int > produc6425607678544837394st_int > $o ) > list_P419547167332244514st_int > produc2468426419292222131st_int ).
thf(sy_c_Product__Type_OPair_001t__Compiler__Oinstr_001t__Product____Type__Oprod_It__Int__Oint_Mt__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J_J,type,
produc1484136438817787418st_int: instr > produc6425607678544837394st_int > produc5995290525303592096st_int ).
thf(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Int__Oint,type,
product_Pair_int_int: int > int > product_prod_int_int ).
thf(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J,type,
produc5086643055186798020st_int: int > produc369741286924889929st_int > produc6425607678544837394st_int ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Compiler__Oinstr_J_001t__List__Olist_It__Compiler__Oinstr_J,type,
produc7750586281064920791_instr: list_instr > list_instr > produc8891168062546637159_instr ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Compiler__Oinstr_J_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J_J_J,type,
produc7764413618310988720st_int: list_instr > list_P419547167332244514st_int > produc4400946599758736694st_int ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Int__Oint_J_001t__List__Olist_It__Int__Oint_J,type,
produc364263696895485585st_int: list_int > list_int > produc1186641810826059865st_int ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__List__Olist_It__String__Ochar_J_J_001t__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__Int__Oint_J,type,
produc3965054194175396271nt_int: list_list_char > produc4435102495419491129nt_int > produc4189061985984568957nt_int ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J_J_J_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J_J_J,type,
produc2639222612625641053st_int: list_P419547167332244514st_int > list_P419547167332244514st_int > produc5848496107505245989st_int ).
thf(sy_c_Product__Type_Oprod_Ofst_001_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_001t__Int__Oint,type,
produc9025023058967219549nt_int: produc4435102495419491129nt_int > list_char > int ).
thf(sy_c_Product__Type_Oprod_Ofst_001_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_001t__List__Olist_It__Int__Oint_J,type,
produc2155683374303800813st_int: produc369741286924889929st_int > list_char > int ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Int__Oint_001t__Int__Oint,type,
product_fst_int_int: product_prod_int_int > int ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Int__Oint_001t__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J,type,
produc8941309049508465880st_int: produc6425607678544837394st_int > int ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__List__Olist_It__List__Olist_It__String__Ochar_J_J_001t__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__Int__Oint_J,type,
produc811350159336358595nt_int: produc4189061985984568957nt_int > list_list_char ).
thf(sy_c_Product__Type_Oprod_Osnd_001_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_001t__Int__Oint,type,
produc179445755590504351nt_int: produc4435102495419491129nt_int > int ).
thf(sy_c_Product__Type_Oprod_Osnd_001_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_001t__List__Olist_It__Int__Oint_J,type,
produc6244060863719409455st_int: produc369741286924889929st_int > list_int ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Int__Oint_001t__Int__Oint,type,
product_snd_int_int: product_prod_int_int > int ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Int__Oint_001t__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J,type,
produc1296517004818028822st_int: produc6425607678544837394st_int > produc369741286924889929st_int ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__List__Olist_It__List__Olist_It__String__Ochar_J_J_001t__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__Int__Oint_J,type,
produc30867750519413249nt_int: produc4189061985984568957nt_int > produc4435102495419491129nt_int ).
thf(sy_c_Set_OCollect_001t__Int__Oint,type,
collect_int: ( int > $o ) > set_int ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Int__Oint,type,
set_or4662586982721622107an_int: int > int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
set_or4665077453230672383an_nat: nat > nat > set_nat ).
thf(sy_c_Star_Ostar_001t__Product____Type__Oprod_It__Int__Oint_Mt__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J_J,type,
star_P707599355569300323st_int: ( produc6425607678544837394st_int > produc6425607678544837394st_int > $o ) > produc6425607678544837394st_int > produc6425607678544837394st_int > $o ).
thf(sy_c_String_Ochar_Osize__char,type,
size_char: char > nat ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
accp_P1096762738010456898nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > product_prod_int_int > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__List__Olist_It__Compiler__Oinstr_J_Mt__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J_J_J_J,type,
accp_P1482788143903795053st_int: ( produc4400946599758736694st_int > produc4400946599758736694st_int > $o ) > produc4400946599758736694st_int > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__String__Ochar_J_J_Mt__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__Int__Oint_J_J,type,
accp_P4562858270263085236nt_int: ( produc4189061985984568957nt_int > produc4189061985984568957nt_int > $o ) > produc4189061985984568957nt_int > $o ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_v_P,type,
p: list_instr ).
thf(sy_v_P_H,type,
p2: a > list_instr ).
thf(sy_v_P_H_H,type,
p3: list_instr ).
thf(sy_v_Q,type,
q: a > produc6425607678544837394st_int > produc6425607678544837394st_int > $o ).
thf(sy_v_cfs,type,
cfs: list_P419547167332244514st_int ).
thf(sy_v_k,type,
k: nat ).
thf(sy_v_k_H____,type,
k2: nat ).
thf(sy_v_x,type,
x: a ).
% Relevant facts (1260)
thf(fact_0__092_060open_062k_H_A_092_060in_062_A_123k_O_O_060length_Acfs_125_092_060close_062,axiom,
member_nat @ k2 @ ( set_or4665077453230672383an_nat @ k @ ( size_s229347708945616566st_int @ cfs ) ) ).
% \<open>k' \<in> {k..<length cfs}\<close>
thf(fact_1_B,axiom,
ord_less_nat @ k @ ( size_s229347708945616566st_int @ cfs ) ).
% B
thf(fact_2_C,axiom,
( ( produc8941309049508465880st_int @ ( nth_Pr8327173284404383691st_int @ cfs @ k ) )
= ( semiri1314217659103216013at_int @ ( size_size_list_instr @ p ) ) ) ).
% C
thf(fact_3_prod_Ocollapse,axiom,
! [Prod: produc4189061985984568957nt_int] :
( ( produc3965054194175396271nt_int @ ( produc811350159336358595nt_int @ Prod ) @ ( produc30867750519413249nt_int @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_4_prod_Ocollapse,axiom,
! [Prod: product_prod_int_int] :
( ( product_Pair_int_int @ ( product_fst_int_int @ Prod ) @ ( product_snd_int_int @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_5_prod_Ocollapse,axiom,
! [Prod: produc369741286924889929st_int] :
( ( produc8650753666468850689st_int @ ( produc2155683374303800813st_int @ Prod ) @ ( produc6244060863719409455st_int @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_6_prod_Ocollapse,axiom,
! [Prod: produc4435102495419491129nt_int] :
( ( produc5790713362662368625nt_int @ ( produc9025023058967219549nt_int @ Prod ) @ ( produc179445755590504351nt_int @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_7_prod_Ocollapse,axiom,
! [Prod: produc6425607678544837394st_int] :
( ( produc5086643055186798020st_int @ ( produc8941309049508465880st_int @ Prod ) @ ( produc1296517004818028822st_int @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_8_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_9_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_10_D,axiom,
! [Cfs: list_P419547167332244514st_int] :
( ( execl_all @ ( p2 @ x ) @ Cfs )
=> ( q @ x @ ( nth_Pr8327173284404383691st_int @ Cfs @ zero_zero_nat ) @ ( nth_Pr8327173284404383691st_int @ Cfs @ ( minus_minus_nat @ ( size_s229347708945616566st_int @ Cfs ) @ one_one_nat ) ) ) ) ).
% D
thf(fact_11_prod_Oexhaust__sel,axiom,
! [Prod: produc4189061985984568957nt_int] :
( Prod
= ( produc3965054194175396271nt_int @ ( produc811350159336358595nt_int @ Prod ) @ ( produc30867750519413249nt_int @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_12_prod_Oexhaust__sel,axiom,
! [Prod: product_prod_int_int] :
( Prod
= ( product_Pair_int_int @ ( product_fst_int_int @ Prod ) @ ( product_snd_int_int @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_13_prod_Oexhaust__sel,axiom,
! [Prod: produc369741286924889929st_int] :
( Prod
= ( produc8650753666468850689st_int @ ( produc2155683374303800813st_int @ Prod ) @ ( produc6244060863719409455st_int @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_14_prod_Oexhaust__sel,axiom,
! [Prod: produc4435102495419491129nt_int] :
( Prod
= ( produc5790713362662368625nt_int @ ( produc9025023058967219549nt_int @ Prod ) @ ( produc179445755590504351nt_int @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_15_prod_Oexhaust__sel,axiom,
! [Prod: produc6425607678544837394st_int] :
( Prod
= ( produc5086643055186798020st_int @ ( produc8941309049508465880st_int @ Prod ) @ ( produc1296517004818028822st_int @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_16_exI__realizer,axiom,
! [P: produc4435102495419491129nt_int > list_list_char > $o,Y: produc4435102495419491129nt_int,X: list_list_char] :
( ( P @ Y @ X )
=> ( P @ ( produc30867750519413249nt_int @ ( produc3965054194175396271nt_int @ X @ Y ) ) @ ( produc811350159336358595nt_int @ ( produc3965054194175396271nt_int @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_17_exI__realizer,axiom,
! [P: int > int > $o,Y: int,X: int] :
( ( P @ Y @ X )
=> ( P @ ( product_snd_int_int @ ( product_Pair_int_int @ X @ Y ) ) @ ( product_fst_int_int @ ( product_Pair_int_int @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_18_exI__realizer,axiom,
! [P: list_int > ( list_char > int ) > $o,Y: list_int,X: list_char > int] :
( ( P @ Y @ X )
=> ( P @ ( produc6244060863719409455st_int @ ( produc8650753666468850689st_int @ X @ Y ) ) @ ( produc2155683374303800813st_int @ ( produc8650753666468850689st_int @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_19_exI__realizer,axiom,
! [P: int > ( list_char > int ) > $o,Y: int,X: list_char > int] :
( ( P @ Y @ X )
=> ( P @ ( produc179445755590504351nt_int @ ( produc5790713362662368625nt_int @ X @ Y ) ) @ ( produc9025023058967219549nt_int @ ( produc5790713362662368625nt_int @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_20_exI__realizer,axiom,
! [P: produc369741286924889929st_int > int > $o,Y: produc369741286924889929st_int,X: int] :
( ( P @ Y @ X )
=> ( P @ ( produc1296517004818028822st_int @ ( produc5086643055186798020st_int @ X @ Y ) ) @ ( produc8941309049508465880st_int @ ( produc5086643055186798020st_int @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_21_conjI__realizer,axiom,
! [P: list_list_char > $o,P2: list_list_char,Q: produc4435102495419491129nt_int > $o,Q2: produc4435102495419491129nt_int] :
( ( P @ P2 )
=> ( ( Q @ Q2 )
=> ( ( P @ ( produc811350159336358595nt_int @ ( produc3965054194175396271nt_int @ P2 @ Q2 ) ) )
& ( Q @ ( produc30867750519413249nt_int @ ( produc3965054194175396271nt_int @ P2 @ Q2 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_22_conjI__realizer,axiom,
! [P: int > $o,P2: int,Q: int > $o,Q2: int] :
( ( P @ P2 )
=> ( ( Q @ Q2 )
=> ( ( P @ ( product_fst_int_int @ ( product_Pair_int_int @ P2 @ Q2 ) ) )
& ( Q @ ( product_snd_int_int @ ( product_Pair_int_int @ P2 @ Q2 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_23_conjI__realizer,axiom,
! [P: ( list_char > int ) > $o,P2: list_char > int,Q: list_int > $o,Q2: list_int] :
( ( P @ P2 )
=> ( ( Q @ Q2 )
=> ( ( P @ ( produc2155683374303800813st_int @ ( produc8650753666468850689st_int @ P2 @ Q2 ) ) )
& ( Q @ ( produc6244060863719409455st_int @ ( produc8650753666468850689st_int @ P2 @ Q2 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_24_conjI__realizer,axiom,
! [P: ( list_char > int ) > $o,P2: list_char > int,Q: int > $o,Q2: int] :
( ( P @ P2 )
=> ( ( Q @ Q2 )
=> ( ( P @ ( produc9025023058967219549nt_int @ ( produc5790713362662368625nt_int @ P2 @ Q2 ) ) )
& ( Q @ ( produc179445755590504351nt_int @ ( produc5790713362662368625nt_int @ P2 @ Q2 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_25_conjI__realizer,axiom,
! [P: int > $o,P2: int,Q: produc369741286924889929st_int > $o,Q2: produc369741286924889929st_int] :
( ( P @ P2 )
=> ( ( Q @ Q2 )
=> ( ( P @ ( produc8941309049508465880st_int @ ( produc5086643055186798020st_int @ P2 @ Q2 ) ) )
& ( Q @ ( produc1296517004818028822st_int @ ( produc5086643055186798020st_int @ P2 @ Q2 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_26_surjective__pairing,axiom,
! [T: produc4189061985984568957nt_int] :
( T
= ( produc3965054194175396271nt_int @ ( produc811350159336358595nt_int @ T ) @ ( produc30867750519413249nt_int @ T ) ) ) ).
% surjective_pairing
thf(fact_27_surjective__pairing,axiom,
! [T: product_prod_int_int] :
( T
= ( product_Pair_int_int @ ( product_fst_int_int @ T ) @ ( product_snd_int_int @ T ) ) ) ).
% surjective_pairing
thf(fact_28_surjective__pairing,axiom,
! [T: produc369741286924889929st_int] :
( T
= ( produc8650753666468850689st_int @ ( produc2155683374303800813st_int @ T ) @ ( produc6244060863719409455st_int @ T ) ) ) ).
% surjective_pairing
thf(fact_29_surjective__pairing,axiom,
! [T: produc4435102495419491129nt_int] :
( T
= ( produc5790713362662368625nt_int @ ( produc9025023058967219549nt_int @ T ) @ ( produc179445755590504351nt_int @ T ) ) ) ).
% surjective_pairing
thf(fact_30_surjective__pairing,axiom,
! [T: produc6425607678544837394st_int] :
( T
= ( produc5086643055186798020st_int @ ( produc8941309049508465880st_int @ T ) @ ( produc1296517004818028822st_int @ T ) ) ) ).
% surjective_pairing
thf(fact_31_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: int > produc369741286924889929st_int > $o,X: int,Y: produc369741286924889929st_int,A: produc6425607678544837394st_int] :
( ( P @ X @ Y )
=> ( ( A
= ( produc5086643055186798020st_int @ X @ Y ) )
=> ( P @ ( produc8941309049508465880st_int @ A ) @ ( produc1296517004818028822st_int @ A ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_32_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: ( list_char > int ) > int > $o,X: list_char > int,Y: int,A: produc4435102495419491129nt_int] :
( ( P @ X @ Y )
=> ( ( A
= ( produc5790713362662368625nt_int @ X @ Y ) )
=> ( P @ ( produc9025023058967219549nt_int @ A ) @ ( produc179445755590504351nt_int @ A ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_33_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: ( list_char > int ) > list_int > $o,X: list_char > int,Y: list_int,A: produc369741286924889929st_int] :
( ( P @ X @ Y )
=> ( ( A
= ( produc8650753666468850689st_int @ X @ Y ) )
=> ( P @ ( produc2155683374303800813st_int @ A ) @ ( produc6244060863719409455st_int @ A ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_34_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: int > int > $o,X: int,Y: int,A: product_prod_int_int] :
( ( P @ X @ Y )
=> ( ( A
= ( product_Pair_int_int @ X @ Y ) )
=> ( P @ ( product_fst_int_int @ A ) @ ( product_snd_int_int @ A ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_35_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: list_list_char > produc4435102495419491129nt_int > $o,X: list_list_char,Y: produc4435102495419491129nt_int,A: produc4189061985984568957nt_int] :
( ( P @ X @ Y )
=> ( ( A
= ( produc3965054194175396271nt_int @ X @ Y ) )
=> ( P @ ( produc811350159336358595nt_int @ A ) @ ( produc30867750519413249nt_int @ A ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_36_nth__equalityI,axiom,
! [Xs: list_int,Ys: list_int] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
=> ( ( nth_int @ Xs @ I )
= ( nth_int @ Ys @ I ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_37_nth__equalityI,axiom,
! [Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int] :
( ( ( size_s229347708945616566st_int @ Xs )
= ( size_s229347708945616566st_int @ Ys ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s229347708945616566st_int @ Xs ) )
=> ( ( nth_Pr8327173284404383691st_int @ Xs @ I )
= ( nth_Pr8327173284404383691st_int @ Ys @ I ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_38_nth__equalityI,axiom,
! [Xs: list_instr,Ys: list_instr] :
( ( ( size_size_list_instr @ Xs )
= ( size_size_list_instr @ Ys ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_size_list_instr @ Xs ) )
=> ( ( nth_instr @ Xs @ I )
= ( nth_instr @ Ys @ I ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_39_nth__equalityI,axiom,
! [Xs: list_list_char,Ys: list_list_char] :
( ( ( size_s356637359517785349t_char @ Xs )
= ( size_s356637359517785349t_char @ Ys ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ ( size_s356637359517785349t_char @ Xs ) )
=> ( ( nth_list_char @ Xs @ I )
= ( nth_list_char @ Ys @ I ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_40_Skolem__list__nth,axiom,
! [K: nat,P: nat > int > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ? [X2: int] : ( P @ I2 @ X2 ) ) )
= ( ? [Xs2: list_int] :
( ( ( size_size_list_int @ Xs2 )
= K )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( P @ I2 @ ( nth_int @ Xs2 @ I2 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_41_Skolem__list__nth,axiom,
! [K: nat,P: nat > produc6425607678544837394st_int > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ? [X2: produc6425607678544837394st_int] : ( P @ I2 @ X2 ) ) )
= ( ? [Xs2: list_P419547167332244514st_int] :
( ( ( size_s229347708945616566st_int @ Xs2 )
= K )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( P @ I2 @ ( nth_Pr8327173284404383691st_int @ Xs2 @ I2 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_42_Skolem__list__nth,axiom,
! [K: nat,P: nat > instr > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ? [X2: instr] : ( P @ I2 @ X2 ) ) )
= ( ? [Xs2: list_instr] :
( ( ( size_size_list_instr @ Xs2 )
= K )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( P @ I2 @ ( nth_instr @ Xs2 @ I2 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_43_Skolem__list__nth,axiom,
! [K: nat,P: nat > list_char > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ? [X2: list_char] : ( P @ I2 @ X2 ) ) )
= ( ? [Xs2: list_list_char] :
( ( ( size_s356637359517785349t_char @ Xs2 )
= K )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( P @ I2 @ ( nth_list_char @ Xs2 @ I2 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_44_list__eq__iff__nth__eq,axiom,
( ( ^ [Y2: list_int,Z: list_int] : ( Y2 = Z ) )
= ( ^ [Xs2: list_int,Ys2: list_int] :
( ( ( size_size_list_int @ Xs2 )
= ( size_size_list_int @ Ys2 ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs2 ) )
=> ( ( nth_int @ Xs2 @ I2 )
= ( nth_int @ Ys2 @ I2 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_45_list__eq__iff__nth__eq,axiom,
( ( ^ [Y2: list_P419547167332244514st_int,Z: list_P419547167332244514st_int] : ( Y2 = Z ) )
= ( ^ [Xs2: list_P419547167332244514st_int,Ys2: list_P419547167332244514st_int] :
( ( ( size_s229347708945616566st_int @ Xs2 )
= ( size_s229347708945616566st_int @ Ys2 ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s229347708945616566st_int @ Xs2 ) )
=> ( ( nth_Pr8327173284404383691st_int @ Xs2 @ I2 )
= ( nth_Pr8327173284404383691st_int @ Ys2 @ I2 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_46_list__eq__iff__nth__eq,axiom,
( ( ^ [Y2: list_instr,Z: list_instr] : ( Y2 = Z ) )
= ( ^ [Xs2: list_instr,Ys2: list_instr] :
( ( ( size_size_list_instr @ Xs2 )
= ( size_size_list_instr @ Ys2 ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_instr @ Xs2 ) )
=> ( ( nth_instr @ Xs2 @ I2 )
= ( nth_instr @ Ys2 @ I2 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_47_list__eq__iff__nth__eq,axiom,
( ( ^ [Y2: list_list_char,Z: list_list_char] : ( Y2 = Z ) )
= ( ^ [Xs2: list_list_char,Ys2: list_list_char] :
( ( ( size_s356637359517785349t_char @ Xs2 )
= ( size_s356637359517785349t_char @ Ys2 ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s356637359517785349t_char @ Xs2 ) )
=> ( ( nth_list_char @ Xs2 @ I2 )
= ( nth_list_char @ Ys2 @ I2 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_48_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_49_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_50_old_Oprod_Oinject,axiom,
! [A: list_list_char,B: produc4435102495419491129nt_int,A2: list_list_char,B2: produc4435102495419491129nt_int] :
( ( ( produc3965054194175396271nt_int @ A @ B )
= ( produc3965054194175396271nt_int @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_51_old_Oprod_Oinject,axiom,
! [A: int,B: produc369741286924889929st_int,A2: int,B2: produc369741286924889929st_int] :
( ( ( produc5086643055186798020st_int @ A @ B )
= ( produc5086643055186798020st_int @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_52_old_Oprod_Oinject,axiom,
! [A: int,B: int,A2: int,B2: int] :
( ( ( product_Pair_int_int @ A @ B )
= ( product_Pair_int_int @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_53_old_Oprod_Oinject,axiom,
! [A: list_char > int,B: list_int,A2: list_char > int,B2: list_int] :
( ( ( produc8650753666468850689st_int @ A @ B )
= ( produc8650753666468850689st_int @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_54_old_Oprod_Oinject,axiom,
! [A: list_char > int,B: int,A2: list_char > int,B2: int] :
( ( ( produc5790713362662368625nt_int @ A @ B )
= ( produc5790713362662368625nt_int @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_55_prod_Oinject,axiom,
! [X1: list_list_char,X22: produc4435102495419491129nt_int,Y1: list_list_char,Y22: produc4435102495419491129nt_int] :
( ( ( produc3965054194175396271nt_int @ X1 @ X22 )
= ( produc3965054194175396271nt_int @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_56_prod_Oinject,axiom,
! [X1: int,X22: produc369741286924889929st_int,Y1: int,Y22: produc369741286924889929st_int] :
( ( ( produc5086643055186798020st_int @ X1 @ X22 )
= ( produc5086643055186798020st_int @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_57_prod_Oinject,axiom,
! [X1: int,X22: int,Y1: int,Y22: int] :
( ( ( product_Pair_int_int @ X1 @ X22 )
= ( product_Pair_int_int @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_58_prod_Oinject,axiom,
! [X1: list_char > int,X22: list_int,Y1: list_char > int,Y22: list_int] :
( ( ( produc8650753666468850689st_int @ X1 @ X22 )
= ( produc8650753666468850689st_int @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_59_prod_Oinject,axiom,
! [X1: list_char > int,X22: int,Y1: list_char > int,Y22: int] :
( ( ( produc5790713362662368625nt_int @ X1 @ X22 )
= ( produc5790713362662368625nt_int @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_60_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_61_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1316708129612266289at_nat @ M )
= ( semiri1316708129612266289at_nat @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_62_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_63_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_64_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_65_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_66_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_67_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_68_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1316708129612266289at_nat @ M )
= zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_69_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_int
= ( semiri1314217659103216013at_int @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_70_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_71_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_72_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_73_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_74_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1314217659103216013at_int @ N )
= one_one_int )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_75_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1316708129612266289at_nat @ N )
= one_one_nat )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_76_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_int
= ( semiri1314217659103216013at_int @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_77_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_78_of__nat__1,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% of_nat_1
thf(fact_79_of__nat__1,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% of_nat_1
thf(fact_80_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_81_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_82_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_83_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_84_diff__commute,axiom,
! [I3: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I3 @ K ) @ J ) ) ).
% diff_commute
thf(fact_85_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_86_all__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ! [M2: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( P @ M2 ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
=> ( P @ X3 ) ) ) ) ).
% all_nat_less_eq
thf(fact_87_ex__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ? [M2: nat] :
( ( ord_less_nat @ M2 @ N )
& ( P @ M2 ) ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
& ( P @ X3 ) ) ) ) ).
% ex_nat_less_eq
thf(fact_88_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_89_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_90_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_91_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N2 )
& ~ ( P @ M3 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_92_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_93_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_94_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_95_mem__Collect__eq,axiom,
! [A: int,P: int > $o] :
( ( member_int @ A @ ( collect_int @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_96_Collect__mem__eq,axiom,
! [A3: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_97_Collect__mem__eq,axiom,
! [A3: set_int] :
( ( collect_int
@ ^ [X3: int] : ( member_int @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_98_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_99_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_100_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_101_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_102_Pair__inject,axiom,
! [A: list_list_char,B: produc4435102495419491129nt_int,A2: list_list_char,B2: produc4435102495419491129nt_int] :
( ( ( produc3965054194175396271nt_int @ A @ B )
= ( produc3965054194175396271nt_int @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_103_Pair__inject,axiom,
! [A: int,B: produc369741286924889929st_int,A2: int,B2: produc369741286924889929st_int] :
( ( ( produc5086643055186798020st_int @ A @ B )
= ( produc5086643055186798020st_int @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_104_Pair__inject,axiom,
! [A: int,B: int,A2: int,B2: int] :
( ( ( product_Pair_int_int @ A @ B )
= ( product_Pair_int_int @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_105_Pair__inject,axiom,
! [A: list_char > int,B: list_int,A2: list_char > int,B2: list_int] :
( ( ( produc8650753666468850689st_int @ A @ B )
= ( produc8650753666468850689st_int @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_106_Pair__inject,axiom,
! [A: list_char > int,B: int,A2: list_char > int,B2: int] :
( ( ( produc5790713362662368625nt_int @ A @ B )
= ( produc5790713362662368625nt_int @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_107_prod__cases,axiom,
! [P: produc4189061985984568957nt_int > $o,P2: produc4189061985984568957nt_int] :
( ! [A4: list_list_char,B3: produc4435102495419491129nt_int] : ( P @ ( produc3965054194175396271nt_int @ A4 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_108_prod__cases,axiom,
! [P: produc6425607678544837394st_int > $o,P2: produc6425607678544837394st_int] :
( ! [A4: int,B3: produc369741286924889929st_int] : ( P @ ( produc5086643055186798020st_int @ A4 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_109_prod__cases,axiom,
! [P: product_prod_int_int > $o,P2: product_prod_int_int] :
( ! [A4: int,B3: int] : ( P @ ( product_Pair_int_int @ A4 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_110_prod__cases,axiom,
! [P: produc369741286924889929st_int > $o,P2: produc369741286924889929st_int] :
( ! [A4: list_char > int,B3: list_int] : ( P @ ( produc8650753666468850689st_int @ A4 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_111_prod__cases,axiom,
! [P: produc4435102495419491129nt_int > $o,P2: produc4435102495419491129nt_int] :
( ! [A4: list_char > int,B3: int] : ( P @ ( produc5790713362662368625nt_int @ A4 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_112_surj__pair,axiom,
! [P2: produc4189061985984568957nt_int] :
? [X4: list_list_char,Y3: produc4435102495419491129nt_int] :
( P2
= ( produc3965054194175396271nt_int @ X4 @ Y3 ) ) ).
% surj_pair
thf(fact_113_surj__pair,axiom,
! [P2: produc6425607678544837394st_int] :
? [X4: int,Y3: produc369741286924889929st_int] :
( P2
= ( produc5086643055186798020st_int @ X4 @ Y3 ) ) ).
% surj_pair
thf(fact_114_surj__pair,axiom,
! [P2: product_prod_int_int] :
? [X4: int,Y3: int] :
( P2
= ( product_Pair_int_int @ X4 @ Y3 ) ) ).
% surj_pair
thf(fact_115_surj__pair,axiom,
! [P2: produc369741286924889929st_int] :
? [X4: list_char > int,Y3: list_int] :
( P2
= ( produc8650753666468850689st_int @ X4 @ Y3 ) ) ).
% surj_pair
thf(fact_116_surj__pair,axiom,
! [P2: produc4435102495419491129nt_int] :
? [X4: list_char > int,Y3: int] :
( P2
= ( produc5790713362662368625nt_int @ X4 @ Y3 ) ) ).
% surj_pair
thf(fact_117_old_Oprod_Oexhaust,axiom,
! [Y: produc4189061985984568957nt_int] :
~ ! [A4: list_list_char,B3: produc4435102495419491129nt_int] :
( Y
!= ( produc3965054194175396271nt_int @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_118_old_Oprod_Oexhaust,axiom,
! [Y: produc6425607678544837394st_int] :
~ ! [A4: int,B3: produc369741286924889929st_int] :
( Y
!= ( produc5086643055186798020st_int @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_119_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_int_int] :
~ ! [A4: int,B3: int] :
( Y
!= ( product_Pair_int_int @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_120_old_Oprod_Oexhaust,axiom,
! [Y: produc369741286924889929st_int] :
~ ! [A4: list_char > int,B3: list_int] :
( Y
!= ( produc8650753666468850689st_int @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_121_old_Oprod_Oexhaust,axiom,
! [Y: produc4435102495419491129nt_int] :
~ ! [A4: list_char > int,B3: int] :
( Y
!= ( produc5790713362662368625nt_int @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_122_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_123_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N2 )
& ~ ( P @ M3 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_124_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ( P @ M3 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_125_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_126_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_127_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_128_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_129_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_130_size__neq__size__imp__neq,axiom,
! [X: char,Y: char] :
( ( ( size_size_char @ X )
!= ( size_size_char @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_131_size__neq__size__imp__neq,axiom,
! [X: list_P419547167332244514st_int,Y: list_P419547167332244514st_int] :
( ( ( size_s229347708945616566st_int @ X )
!= ( size_s229347708945616566st_int @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_132_size__neq__size__imp__neq,axiom,
! [X: list_instr,Y: list_instr] :
( ( ( size_size_list_instr @ X )
!= ( size_size_list_instr @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_133_size__neq__size__imp__neq,axiom,
! [X: list_list_char,Y: list_list_char] :
( ( ( size_s356637359517785349t_char @ X )
!= ( size_s356637359517785349t_char @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_134_size__neq__size__imp__neq,axiom,
! [X: com,Y: com] :
( ( ( size_size_com @ X )
!= ( size_size_com @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_135_neq__if__length__neq,axiom,
! [Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int] :
( ( ( size_s229347708945616566st_int @ Xs )
!= ( size_s229347708945616566st_int @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_136_neq__if__length__neq,axiom,
! [Xs: list_instr,Ys: list_instr] :
( ( ( size_size_list_instr @ Xs )
!= ( size_size_list_instr @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_137_neq__if__length__neq,axiom,
! [Xs: list_list_char,Ys: list_list_char] :
( ( ( size_s356637359517785349t_char @ Xs )
!= ( size_s356637359517785349t_char @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_138_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_P419547167332244514st_int] :
( ( size_s229347708945616566st_int @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_139_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_instr] :
( ( size_size_list_instr @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_140_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_list_char] :
( ( size_s356637359517785349t_char @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_141_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_142_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_143_prod__induct3,axiom,
! [P: produc4189061985984568957nt_int > $o,X: produc4189061985984568957nt_int] :
( ! [A4: list_list_char,B3: list_char > int,C: int] : ( P @ ( produc3965054194175396271nt_int @ A4 @ ( produc5790713362662368625nt_int @ B3 @ C ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_144_prod__induct3,axiom,
! [P: produc6425607678544837394st_int > $o,X: produc6425607678544837394st_int] :
( ! [A4: int,B3: list_char > int,C: list_int] : ( P @ ( produc5086643055186798020st_int @ A4 @ ( produc8650753666468850689st_int @ B3 @ C ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_145_prod__cases3,axiom,
! [Y: produc4189061985984568957nt_int] :
~ ! [A4: list_list_char,B3: list_char > int,C: int] :
( Y
!= ( produc3965054194175396271nt_int @ A4 @ ( produc5790713362662368625nt_int @ B3 @ C ) ) ) ).
% prod_cases3
thf(fact_146_prod__cases3,axiom,
! [Y: produc6425607678544837394st_int] :
~ ! [A4: int,B3: list_char > int,C: list_int] :
( Y
!= ( produc5086643055186798020st_int @ A4 @ ( produc8650753666468850689st_int @ B3 @ C ) ) ) ).
% prod_cases3
thf(fact_147_length__induct,axiom,
! [P: list_P419547167332244514st_int > $o,Xs: list_P419547167332244514st_int] :
( ! [Xs3: list_P419547167332244514st_int] :
( ! [Ys3: list_P419547167332244514st_int] :
( ( ord_less_nat @ ( size_s229347708945616566st_int @ Ys3 ) @ ( size_s229347708945616566st_int @ Xs3 ) )
=> ( P @ Ys3 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_148_length__induct,axiom,
! [P: list_instr > $o,Xs: list_instr] :
( ! [Xs3: list_instr] :
( ! [Ys3: list_instr] :
( ( ord_less_nat @ ( size_size_list_instr @ Ys3 ) @ ( size_size_list_instr @ Xs3 ) )
=> ( P @ Ys3 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_149_length__induct,axiom,
! [P: list_list_char > $o,Xs: list_list_char] :
( ! [Xs3: list_list_char] :
( ! [Ys3: list_list_char] :
( ( ord_less_nat @ ( size_s356637359517785349t_char @ Ys3 ) @ ( size_s356637359517785349t_char @ Xs3 ) )
=> ( P @ Ys3 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_150_fst__conv,axiom,
! [X1: int,X22: produc369741286924889929st_int] :
( ( produc8941309049508465880st_int @ ( produc5086643055186798020st_int @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_151_fst__conv,axiom,
! [X1: list_char > int,X22: int] :
( ( produc9025023058967219549nt_int @ ( produc5790713362662368625nt_int @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_152_fst__conv,axiom,
! [X1: list_char > int,X22: list_int] :
( ( produc2155683374303800813st_int @ ( produc8650753666468850689st_int @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_153_fst__conv,axiom,
! [X1: int,X22: int] :
( ( product_fst_int_int @ ( product_Pair_int_int @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_154_fst__conv,axiom,
! [X1: list_list_char,X22: produc4435102495419491129nt_int] :
( ( produc811350159336358595nt_int @ ( produc3965054194175396271nt_int @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_155_fst__eqD,axiom,
! [X: int,Y: produc369741286924889929st_int,A: int] :
( ( ( produc8941309049508465880st_int @ ( produc5086643055186798020st_int @ X @ Y ) )
= A )
=> ( X = A ) ) ).
% fst_eqD
thf(fact_156_fst__eqD,axiom,
! [X: list_char > int,Y: int,A: list_char > int] :
( ( ( produc9025023058967219549nt_int @ ( produc5790713362662368625nt_int @ X @ Y ) )
= A )
=> ( X = A ) ) ).
% fst_eqD
thf(fact_157_fst__eqD,axiom,
! [X: list_char > int,Y: list_int,A: list_char > int] :
( ( ( produc2155683374303800813st_int @ ( produc8650753666468850689st_int @ X @ Y ) )
= A )
=> ( X = A ) ) ).
% fst_eqD
thf(fact_158_fst__eqD,axiom,
! [X: int,Y: int,A: int] :
( ( ( product_fst_int_int @ ( product_Pair_int_int @ X @ Y ) )
= A )
=> ( X = A ) ) ).
% fst_eqD
thf(fact_159_fst__eqD,axiom,
! [X: list_list_char,Y: produc4435102495419491129nt_int,A: list_list_char] :
( ( ( produc811350159336358595nt_int @ ( produc3965054194175396271nt_int @ X @ Y ) )
= A )
=> ( X = A ) ) ).
% fst_eqD
thf(fact_160_snd__conv,axiom,
! [X1: int,X22: produc369741286924889929st_int] :
( ( produc1296517004818028822st_int @ ( produc5086643055186798020st_int @ X1 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_161_snd__conv,axiom,
! [X1: list_char > int,X22: int] :
( ( produc179445755590504351nt_int @ ( produc5790713362662368625nt_int @ X1 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_162_snd__conv,axiom,
! [X1: list_char > int,X22: list_int] :
( ( produc6244060863719409455st_int @ ( produc8650753666468850689st_int @ X1 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_163_snd__conv,axiom,
! [X1: int,X22: int] :
( ( product_snd_int_int @ ( product_Pair_int_int @ X1 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_164_snd__conv,axiom,
! [X1: list_list_char,X22: produc4435102495419491129nt_int] :
( ( produc30867750519413249nt_int @ ( produc3965054194175396271nt_int @ X1 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_165_snd__eqD,axiom,
! [X: int,Y: produc369741286924889929st_int,A: produc369741286924889929st_int] :
( ( ( produc1296517004818028822st_int @ ( produc5086643055186798020st_int @ X @ Y ) )
= A )
=> ( Y = A ) ) ).
% snd_eqD
thf(fact_166_snd__eqD,axiom,
! [X: list_char > int,Y: int,A: int] :
( ( ( produc179445755590504351nt_int @ ( produc5790713362662368625nt_int @ X @ Y ) )
= A )
=> ( Y = A ) ) ).
% snd_eqD
thf(fact_167_snd__eqD,axiom,
! [X: list_char > int,Y: list_int,A: list_int] :
( ( ( produc6244060863719409455st_int @ ( produc8650753666468850689st_int @ X @ Y ) )
= A )
=> ( Y = A ) ) ).
% snd_eqD
thf(fact_168_snd__eqD,axiom,
! [X: int,Y: int,A: int] :
( ( ( product_snd_int_int @ ( product_Pair_int_int @ X @ Y ) )
= A )
=> ( Y = A ) ) ).
% snd_eqD
thf(fact_169_snd__eqD,axiom,
! [X: list_list_char,Y: produc4435102495419491129nt_int,A: produc4435102495419491129nt_int] :
( ( ( produc30867750519413249nt_int @ ( produc3965054194175396271nt_int @ X @ Y ) )
= A )
=> ( Y = A ) ) ).
% snd_eqD
thf(fact_170_prod__eq__iff,axiom,
( ( ^ [Y2: produc6425607678544837394st_int,Z: produc6425607678544837394st_int] : ( Y2 = Z ) )
= ( ^ [S2: produc6425607678544837394st_int,T2: produc6425607678544837394st_int] :
( ( ( produc8941309049508465880st_int @ S2 )
= ( produc8941309049508465880st_int @ T2 ) )
& ( ( produc1296517004818028822st_int @ S2 )
= ( produc1296517004818028822st_int @ T2 ) ) ) ) ) ).
% prod_eq_iff
thf(fact_171_prod__eq__iff,axiom,
( ( ^ [Y2: produc4435102495419491129nt_int,Z: produc4435102495419491129nt_int] : ( Y2 = Z ) )
= ( ^ [S2: produc4435102495419491129nt_int,T2: produc4435102495419491129nt_int] :
( ( ( produc9025023058967219549nt_int @ S2 )
= ( produc9025023058967219549nt_int @ T2 ) )
& ( ( produc179445755590504351nt_int @ S2 )
= ( produc179445755590504351nt_int @ T2 ) ) ) ) ) ).
% prod_eq_iff
thf(fact_172_prod__eq__iff,axiom,
( ( ^ [Y2: produc369741286924889929st_int,Z: produc369741286924889929st_int] : ( Y2 = Z ) )
= ( ^ [S2: produc369741286924889929st_int,T2: produc369741286924889929st_int] :
( ( ( produc2155683374303800813st_int @ S2 )
= ( produc2155683374303800813st_int @ T2 ) )
& ( ( produc6244060863719409455st_int @ S2 )
= ( produc6244060863719409455st_int @ T2 ) ) ) ) ) ).
% prod_eq_iff
thf(fact_173_prod__eq__iff,axiom,
( ( ^ [Y2: product_prod_int_int,Z: product_prod_int_int] : ( Y2 = Z ) )
= ( ^ [S2: product_prod_int_int,T2: product_prod_int_int] :
( ( ( product_fst_int_int @ S2 )
= ( product_fst_int_int @ T2 ) )
& ( ( product_snd_int_int @ S2 )
= ( product_snd_int_int @ T2 ) ) ) ) ) ).
% prod_eq_iff
thf(fact_174_prod__eq__iff,axiom,
( ( ^ [Y2: produc4189061985984568957nt_int,Z: produc4189061985984568957nt_int] : ( Y2 = Z ) )
= ( ^ [S2: produc4189061985984568957nt_int,T2: produc4189061985984568957nt_int] :
( ( ( produc811350159336358595nt_int @ S2 )
= ( produc811350159336358595nt_int @ T2 ) )
& ( ( produc30867750519413249nt_int @ S2 )
= ( produc30867750519413249nt_int @ T2 ) ) ) ) ) ).
% prod_eq_iff
thf(fact_175_exE__realizer_H,axiom,
! [P: produc369741286924889929st_int > int > $o,P2: produc6425607678544837394st_int] :
( ( P @ ( produc1296517004818028822st_int @ P2 ) @ ( produc8941309049508465880st_int @ P2 ) )
=> ~ ! [X4: int,Y3: produc369741286924889929st_int] :
~ ( P @ Y3 @ X4 ) ) ).
% exE_realizer'
thf(fact_176_exE__realizer_H,axiom,
! [P: int > ( list_char > int ) > $o,P2: produc4435102495419491129nt_int] :
( ( P @ ( produc179445755590504351nt_int @ P2 ) @ ( produc9025023058967219549nt_int @ P2 ) )
=> ~ ! [X4: list_char > int,Y3: int] :
~ ( P @ Y3 @ X4 ) ) ).
% exE_realizer'
thf(fact_177_exE__realizer_H,axiom,
! [P: list_int > ( list_char > int ) > $o,P2: produc369741286924889929st_int] :
( ( P @ ( produc6244060863719409455st_int @ P2 ) @ ( produc2155683374303800813st_int @ P2 ) )
=> ~ ! [X4: list_char > int,Y3: list_int] :
~ ( P @ Y3 @ X4 ) ) ).
% exE_realizer'
thf(fact_178_exE__realizer_H,axiom,
! [P: int > int > $o,P2: product_prod_int_int] :
( ( P @ ( product_snd_int_int @ P2 ) @ ( product_fst_int_int @ P2 ) )
=> ~ ! [X4: int,Y3: int] :
~ ( P @ Y3 @ X4 ) ) ).
% exE_realizer'
thf(fact_179_exE__realizer_H,axiom,
! [P: produc4435102495419491129nt_int > list_list_char > $o,P2: produc4189061985984568957nt_int] :
( ( P @ ( produc30867750519413249nt_int @ P2 ) @ ( produc811350159336358595nt_int @ P2 ) )
=> ~ ! [X4: list_list_char,Y3: produc4435102495419491129nt_int] :
~ ( P @ Y3 @ X4 ) ) ).
% exE_realizer'
thf(fact_180_prod__eqI,axiom,
! [P2: produc6425607678544837394st_int,Q2: produc6425607678544837394st_int] :
( ( ( produc8941309049508465880st_int @ P2 )
= ( produc8941309049508465880st_int @ Q2 ) )
=> ( ( ( produc1296517004818028822st_int @ P2 )
= ( produc1296517004818028822st_int @ Q2 ) )
=> ( P2 = Q2 ) ) ) ).
% prod_eqI
thf(fact_181_prod__eqI,axiom,
! [P2: produc4435102495419491129nt_int,Q2: produc4435102495419491129nt_int] :
( ( ( produc9025023058967219549nt_int @ P2 )
= ( produc9025023058967219549nt_int @ Q2 ) )
=> ( ( ( produc179445755590504351nt_int @ P2 )
= ( produc179445755590504351nt_int @ Q2 ) )
=> ( P2 = Q2 ) ) ) ).
% prod_eqI
thf(fact_182_prod__eqI,axiom,
! [P2: produc369741286924889929st_int,Q2: produc369741286924889929st_int] :
( ( ( produc2155683374303800813st_int @ P2 )
= ( produc2155683374303800813st_int @ Q2 ) )
=> ( ( ( produc6244060863719409455st_int @ P2 )
= ( produc6244060863719409455st_int @ Q2 ) )
=> ( P2 = Q2 ) ) ) ).
% prod_eqI
thf(fact_183_prod__eqI,axiom,
! [P2: product_prod_int_int,Q2: product_prod_int_int] :
( ( ( product_fst_int_int @ P2 )
= ( product_fst_int_int @ Q2 ) )
=> ( ( ( product_snd_int_int @ P2 )
= ( product_snd_int_int @ Q2 ) )
=> ( P2 = Q2 ) ) ) ).
% prod_eqI
thf(fact_184_prod__eqI,axiom,
! [P2: produc4189061985984568957nt_int,Q2: produc4189061985984568957nt_int] :
( ( ( produc811350159336358595nt_int @ P2 )
= ( produc811350159336358595nt_int @ Q2 ) )
=> ( ( ( produc30867750519413249nt_int @ P2 )
= ( produc30867750519413249nt_int @ Q2 ) )
=> ( P2 = Q2 ) ) ) ).
% prod_eqI
thf(fact_185_prod_Oexpand,axiom,
! [Prod: produc6425607678544837394st_int,Prod2: produc6425607678544837394st_int] :
( ( ( ( produc8941309049508465880st_int @ Prod )
= ( produc8941309049508465880st_int @ Prod2 ) )
& ( ( produc1296517004818028822st_int @ Prod )
= ( produc1296517004818028822st_int @ Prod2 ) ) )
=> ( Prod = Prod2 ) ) ).
% prod.expand
thf(fact_186_prod_Oexpand,axiom,
! [Prod: produc4435102495419491129nt_int,Prod2: produc4435102495419491129nt_int] :
( ( ( ( produc9025023058967219549nt_int @ Prod )
= ( produc9025023058967219549nt_int @ Prod2 ) )
& ( ( produc179445755590504351nt_int @ Prod )
= ( produc179445755590504351nt_int @ Prod2 ) ) )
=> ( Prod = Prod2 ) ) ).
% prod.expand
thf(fact_187_prod_Oexpand,axiom,
! [Prod: produc369741286924889929st_int,Prod2: produc369741286924889929st_int] :
( ( ( ( produc2155683374303800813st_int @ Prod )
= ( produc2155683374303800813st_int @ Prod2 ) )
& ( ( produc6244060863719409455st_int @ Prod )
= ( produc6244060863719409455st_int @ Prod2 ) ) )
=> ( Prod = Prod2 ) ) ).
% prod.expand
thf(fact_188_prod_Oexpand,axiom,
! [Prod: product_prod_int_int,Prod2: product_prod_int_int] :
( ( ( ( product_fst_int_int @ Prod )
= ( product_fst_int_int @ Prod2 ) )
& ( ( product_snd_int_int @ Prod )
= ( product_snd_int_int @ Prod2 ) ) )
=> ( Prod = Prod2 ) ) ).
% prod.expand
thf(fact_189_prod_Oexpand,axiom,
! [Prod: produc4189061985984568957nt_int,Prod2: produc4189061985984568957nt_int] :
( ( ( ( produc811350159336358595nt_int @ Prod )
= ( produc811350159336358595nt_int @ Prod2 ) )
& ( ( produc30867750519413249nt_int @ Prod )
= ( produc30867750519413249nt_int @ Prod2 ) ) )
=> ( Prod = Prod2 ) ) ).
% prod.expand
thf(fact_190_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_191_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_192_A,axiom,
execl_all @ ( append_instr @ p @ ( append_instr @ ( p2 @ x ) @ p3 ) ) @ cfs ).
% A
thf(fact_193_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
= N ) ) ).
% Suc_diff_1
thf(fact_194_Suc__pred,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
% Suc_pred
thf(fact_195_nth__map,axiom,
! [N: nat,Xs: list_int,F: int > produc6425607678544837394st_int] :
( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
=> ( ( nth_Pr8327173284404383691st_int @ ( map_in8512336359594627332st_int @ F @ Xs ) @ N )
= ( F @ ( nth_int @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_196_nth__map,axiom,
! [N: nat,Xs: list_int,F: int > int] :
( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
=> ( ( nth_int @ ( map_int_int @ F @ Xs ) @ N )
= ( F @ ( nth_int @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_197_nth__map,axiom,
! [N: nat,Xs: list_P419547167332244514st_int,F: produc6425607678544837394st_int > int] :
( ( ord_less_nat @ N @ ( size_s229347708945616566st_int @ Xs ) )
=> ( ( nth_int @ ( map_Pr3224476576807795204nt_int @ F @ Xs ) @ N )
= ( F @ ( nth_Pr8327173284404383691st_int @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_198_nth__map,axiom,
! [N: nat,Xs: list_P419547167332244514st_int,F: produc6425607678544837394st_int > produc6425607678544837394st_int] :
( ( ord_less_nat @ N @ ( size_s229347708945616566st_int @ Xs ) )
=> ( ( nth_Pr8327173284404383691st_int @ ( map_Pr8625093460839654122st_int @ F @ Xs ) @ N )
= ( F @ ( nth_Pr8327173284404383691st_int @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_199_nth__map,axiom,
! [N: nat,Xs: list_instr,F: instr > produc6425607678544837394st_int] :
( ( ord_less_nat @ N @ ( size_size_list_instr @ Xs ) )
=> ( ( nth_Pr8327173284404383691st_int @ ( map_in7259965606695656109st_int @ F @ Xs ) @ N )
= ( F @ ( nth_instr @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_200_nth__map,axiom,
! [N: nat,Xs: list_instr,F: instr > int] :
( ( ord_less_nat @ N @ ( size_size_list_instr @ Xs ) )
=> ( ( nth_int @ ( map_instr_int @ F @ Xs ) @ N )
= ( F @ ( nth_instr @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_201_nth__map,axiom,
! [N: nat,Xs: list_list_char,F: list_char > produc6425607678544837394st_int] :
( ( ord_less_nat @ N @ ( size_s356637359517785349t_char @ Xs ) )
=> ( ( nth_Pr8327173284404383691st_int @ ( map_li6440205260635360185st_int @ F @ Xs ) @ N )
= ( F @ ( nth_list_char @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_202_nth__map,axiom,
! [N: nat,Xs: list_list_char,F: list_char > int] :
( ( ord_less_nat @ N @ ( size_s356637359517785349t_char @ Xs ) )
=> ( ( nth_int @ ( map_list_char_int @ F @ Xs ) @ N )
= ( F @ ( nth_list_char @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_203_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_204_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_205_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_206_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_207_diff__gt__0__iff__gt,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
= ( ord_less_int @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_208_length__drop,axiom,
! [N: nat,Xs: list_P419547167332244514st_int] :
( ( size_s229347708945616566st_int @ ( drop_P7838799151754935884st_int @ N @ Xs ) )
= ( minus_minus_nat @ ( size_s229347708945616566st_int @ Xs ) @ N ) ) ).
% length_drop
thf(fact_209_length__drop,axiom,
! [N: nat,Xs: list_instr] :
( ( size_size_list_instr @ ( drop_instr @ N @ Xs ) )
= ( minus_minus_nat @ ( size_size_list_instr @ Xs ) @ N ) ) ).
% length_drop
thf(fact_210_length__drop,axiom,
! [N: nat,Xs: list_list_char] :
( ( size_s356637359517785349t_char @ ( drop_list_char @ N @ Xs ) )
= ( minus_minus_nat @ ( size_s356637359517785349t_char @ Xs ) @ N ) ) ).
% length_drop
thf(fact_211_nth__take,axiom,
! [I3: nat,N: nat,Xs: list_int] :
( ( ord_less_nat @ I3 @ N )
=> ( ( nth_int @ ( take_int @ N @ Xs ) @ I3 )
= ( nth_int @ Xs @ I3 ) ) ) ).
% nth_take
thf(fact_212_nth__take,axiom,
! [I3: nat,N: nat,Xs: list_P419547167332244514st_int] :
( ( ord_less_nat @ I3 @ N )
=> ( ( nth_Pr8327173284404383691st_int @ ( take_P1004856038796978596st_int @ N @ Xs ) @ I3 )
= ( nth_Pr8327173284404383691st_int @ Xs @ I3 ) ) ) ).
% nth_take
thf(fact_213_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_214_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_215_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_216_append_Oassoc,axiom,
! [A: list_instr,B: list_instr,C2: list_instr] :
( ( append_instr @ ( append_instr @ A @ B ) @ C2 )
= ( append_instr @ A @ ( append_instr @ B @ C2 ) ) ) ).
% append.assoc
thf(fact_217_append_Oassoc,axiom,
! [A: list_P419547167332244514st_int,B: list_P419547167332244514st_int,C2: list_P419547167332244514st_int] :
( ( append5028292981072010935st_int @ ( append5028292981072010935st_int @ A @ B ) @ C2 )
= ( append5028292981072010935st_int @ A @ ( append5028292981072010935st_int @ B @ C2 ) ) ) ).
% append.assoc
thf(fact_218_append_Oassoc,axiom,
! [A: list_int,B: list_int,C2: list_int] :
( ( append_int @ ( append_int @ A @ B ) @ C2 )
= ( append_int @ A @ ( append_int @ B @ C2 ) ) ) ).
% append.assoc
thf(fact_219_append__assoc,axiom,
! [Xs: list_instr,Ys: list_instr,Zs: list_instr] :
( ( append_instr @ ( append_instr @ Xs @ Ys ) @ Zs )
= ( append_instr @ Xs @ ( append_instr @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_220_append__assoc,axiom,
! [Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int,Zs: list_P419547167332244514st_int] :
( ( append5028292981072010935st_int @ ( append5028292981072010935st_int @ Xs @ Ys ) @ Zs )
= ( append5028292981072010935st_int @ Xs @ ( append5028292981072010935st_int @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_221_append__assoc,axiom,
! [Xs: list_int,Ys: list_int,Zs: list_int] :
( ( append_int @ ( append_int @ Xs @ Ys ) @ Zs )
= ( append_int @ Xs @ ( append_int @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_222_append__same__eq,axiom,
! [Ys: list_instr,Xs: list_instr,Zs: list_instr] :
( ( ( append_instr @ Ys @ Xs )
= ( append_instr @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_223_append__same__eq,axiom,
! [Ys: list_P419547167332244514st_int,Xs: list_P419547167332244514st_int,Zs: list_P419547167332244514st_int] :
( ( ( append5028292981072010935st_int @ Ys @ Xs )
= ( append5028292981072010935st_int @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_224_append__same__eq,axiom,
! [Ys: list_int,Xs: list_int,Zs: list_int] :
( ( ( append_int @ Ys @ Xs )
= ( append_int @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_225_same__append__eq,axiom,
! [Xs: list_instr,Ys: list_instr,Zs: list_instr] :
( ( ( append_instr @ Xs @ Ys )
= ( append_instr @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_226_same__append__eq,axiom,
! [Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int,Zs: list_P419547167332244514st_int] :
( ( ( append5028292981072010935st_int @ Xs @ Ys )
= ( append5028292981072010935st_int @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_227_same__append__eq,axiom,
! [Xs: list_int,Ys: list_int,Zs: list_int] :
( ( ( append_int @ Xs @ Ys )
= ( append_int @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_228_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_229_diff__self,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% diff_self
thf(fact_230_diff__0__right,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_0_right
thf(fact_231_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_232_diff__zero,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_zero
thf(fact_233_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_234_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_235_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_236_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_237_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_238_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_less_eq
thf(fact_239_Suc__diff__diff,axiom,
! [M: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_240_diff__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_Suc_Suc
thf(fact_241_length__map,axiom,
! [F: produc6425607678544837394st_int > produc6425607678544837394st_int,Xs: list_P419547167332244514st_int] :
( ( size_s229347708945616566st_int @ ( map_Pr8625093460839654122st_int @ F @ Xs ) )
= ( size_s229347708945616566st_int @ Xs ) ) ).
% length_map
thf(fact_242_length__map,axiom,
! [F: instr > produc6425607678544837394st_int,Xs: list_instr] :
( ( size_s229347708945616566st_int @ ( map_in7259965606695656109st_int @ F @ Xs ) )
= ( size_size_list_instr @ Xs ) ) ).
% length_map
thf(fact_243_length__map,axiom,
! [F: list_char > produc6425607678544837394st_int,Xs: list_list_char] :
( ( size_s229347708945616566st_int @ ( map_li6440205260635360185st_int @ F @ Xs ) )
= ( size_s356637359517785349t_char @ Xs ) ) ).
% length_map
thf(fact_244_length__map,axiom,
! [F: produc6425607678544837394st_int > instr,Xs: list_P419547167332244514st_int] :
( ( size_size_list_instr @ ( map_Pr7514435936640407687_instr @ F @ Xs ) )
= ( size_s229347708945616566st_int @ Xs ) ) ).
% length_map
thf(fact_245_length__map,axiom,
! [F: instr > instr,Xs: list_instr] :
( ( size_size_list_instr @ ( map_instr_instr @ F @ Xs ) )
= ( size_size_list_instr @ Xs ) ) ).
% length_map
thf(fact_246_length__map,axiom,
! [F: list_char > instr,Xs: list_list_char] :
( ( size_size_list_instr @ ( map_list_char_instr @ F @ Xs ) )
= ( size_s356637359517785349t_char @ Xs ) ) ).
% length_map
thf(fact_247_length__map,axiom,
! [F: produc6425607678544837394st_int > list_char,Xs: list_P419547167332244514st_int] :
( ( size_s356637359517785349t_char @ ( map_Pr5338895242362685753t_char @ F @ Xs ) )
= ( size_s229347708945616566st_int @ Xs ) ) ).
% length_map
thf(fact_248_length__map,axiom,
! [F: instr > list_char,Xs: list_instr] :
( ( size_s356637359517785349t_char @ ( map_instr_list_char @ F @ Xs ) )
= ( size_size_list_instr @ Xs ) ) ).
% length_map
thf(fact_249_length__map,axiom,
! [F: list_char > list_char,Xs: list_list_char] :
( ( size_s356637359517785349t_char @ ( map_li116305933131242120t_char @ F @ Xs ) )
= ( size_s356637359517785349t_char @ Xs ) ) ).
% length_map
thf(fact_250_append__eq__append__conv,axiom,
! [Xs: list_int,Ys: list_int,Us: list_int,Vs: list_int] :
( ( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys ) )
| ( ( size_size_list_int @ Us )
= ( size_size_list_int @ Vs ) ) )
=> ( ( ( append_int @ Xs @ Us )
= ( append_int @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_251_append__eq__append__conv,axiom,
! [Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int,Us: list_P419547167332244514st_int,Vs: list_P419547167332244514st_int] :
( ( ( ( size_s229347708945616566st_int @ Xs )
= ( size_s229347708945616566st_int @ Ys ) )
| ( ( size_s229347708945616566st_int @ Us )
= ( size_s229347708945616566st_int @ Vs ) ) )
=> ( ( ( append5028292981072010935st_int @ Xs @ Us )
= ( append5028292981072010935st_int @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_252_append__eq__append__conv,axiom,
! [Xs: list_instr,Ys: list_instr,Us: list_instr,Vs: list_instr] :
( ( ( ( size_size_list_instr @ Xs )
= ( size_size_list_instr @ Ys ) )
| ( ( size_size_list_instr @ Us )
= ( size_size_list_instr @ Vs ) ) )
=> ( ( ( append_instr @ Xs @ Us )
= ( append_instr @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_253_append__eq__append__conv,axiom,
! [Xs: list_list_char,Ys: list_list_char,Us: list_list_char,Vs: list_list_char] :
( ( ( ( size_s356637359517785349t_char @ Xs )
= ( size_s356637359517785349t_char @ Ys ) )
| ( ( size_s356637359517785349t_char @ Us )
= ( size_s356637359517785349t_char @ Vs ) ) )
=> ( ( ( append_list_char @ Xs @ Us )
= ( append_list_char @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_254_map__append,axiom,
! [F: instr > instr,Xs: list_instr,Ys: list_instr] :
( ( map_instr_instr @ F @ ( append_instr @ Xs @ Ys ) )
= ( append_instr @ ( map_instr_instr @ F @ Xs ) @ ( map_instr_instr @ F @ Ys ) ) ) ).
% map_append
thf(fact_255_map__append,axiom,
! [F: instr > produc6425607678544837394st_int,Xs: list_instr,Ys: list_instr] :
( ( map_in7259965606695656109st_int @ F @ ( append_instr @ Xs @ Ys ) )
= ( append5028292981072010935st_int @ ( map_in7259965606695656109st_int @ F @ Xs ) @ ( map_in7259965606695656109st_int @ F @ Ys ) ) ) ).
% map_append
thf(fact_256_map__append,axiom,
! [F: instr > int,Xs: list_instr,Ys: list_instr] :
( ( map_instr_int @ F @ ( append_instr @ Xs @ Ys ) )
= ( append_int @ ( map_instr_int @ F @ Xs ) @ ( map_instr_int @ F @ Ys ) ) ) ).
% map_append
thf(fact_257_map__append,axiom,
! [F: produc6425607678544837394st_int > instr,Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int] :
( ( map_Pr7514435936640407687_instr @ F @ ( append5028292981072010935st_int @ Xs @ Ys ) )
= ( append_instr @ ( map_Pr7514435936640407687_instr @ F @ Xs ) @ ( map_Pr7514435936640407687_instr @ F @ Ys ) ) ) ).
% map_append
thf(fact_258_map__append,axiom,
! [F: produc6425607678544837394st_int > int,Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int] :
( ( map_Pr3224476576807795204nt_int @ F @ ( append5028292981072010935st_int @ Xs @ Ys ) )
= ( append_int @ ( map_Pr3224476576807795204nt_int @ F @ Xs ) @ ( map_Pr3224476576807795204nt_int @ F @ Ys ) ) ) ).
% map_append
thf(fact_259_map__append,axiom,
! [F: int > instr,Xs: list_int,Ys: list_int] :
( ( map_int_instr @ F @ ( append_int @ Xs @ Ys ) )
= ( append_instr @ ( map_int_instr @ F @ Xs ) @ ( map_int_instr @ F @ Ys ) ) ) ).
% map_append
thf(fact_260_map__append,axiom,
! [F: int > produc6425607678544837394st_int,Xs: list_int,Ys: list_int] :
( ( map_in8512336359594627332st_int @ F @ ( append_int @ Xs @ Ys ) )
= ( append5028292981072010935st_int @ ( map_in8512336359594627332st_int @ F @ Xs ) @ ( map_in8512336359594627332st_int @ F @ Ys ) ) ) ).
% map_append
thf(fact_261_map__append,axiom,
! [F: int > int,Xs: list_int,Ys: list_int] :
( ( map_int_int @ F @ ( append_int @ Xs @ Ys ) )
= ( append_int @ ( map_int_int @ F @ Xs ) @ ( map_int_int @ F @ Ys ) ) ) ).
% map_append
thf(fact_262_map__append,axiom,
! [F: produc6425607678544837394st_int > produc6425607678544837394st_int,Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int] :
( ( map_Pr8625093460839654122st_int @ F @ ( append5028292981072010935st_int @ Xs @ Ys ) )
= ( append5028292981072010935st_int @ ( map_Pr8625093460839654122st_int @ F @ Xs ) @ ( map_Pr8625093460839654122st_int @ F @ Ys ) ) ) ).
% map_append
thf(fact_263_drop0,axiom,
( ( drop_P7838799151754935884st_int @ zero_zero_nat )
= ( ^ [X3: list_P419547167332244514st_int] : X3 ) ) ).
% drop0
thf(fact_264_append__take__drop__id,axiom,
! [N: nat,Xs: list_instr] :
( ( append_instr @ ( take_instr @ N @ Xs ) @ ( drop_instr @ N @ Xs ) )
= Xs ) ).
% append_take_drop_id
thf(fact_265_append__take__drop__id,axiom,
! [N: nat,Xs: list_P419547167332244514st_int] :
( ( append5028292981072010935st_int @ ( take_P1004856038796978596st_int @ N @ Xs ) @ ( drop_P7838799151754935884st_int @ N @ Xs ) )
= Xs ) ).
% append_take_drop_id
thf(fact_266_append__take__drop__id,axiom,
! [N: nat,Xs: list_int] :
( ( append_int @ ( take_int @ N @ Xs ) @ ( drop_int @ N @ Xs ) )
= Xs ) ).
% append_take_drop_id
thf(fact_267_take__append,axiom,
! [N: nat,Xs: list_int,Ys: list_int] :
( ( take_int @ N @ ( append_int @ Xs @ Ys ) )
= ( append_int @ ( take_int @ N @ Xs ) @ ( take_int @ ( minus_minus_nat @ N @ ( size_size_list_int @ Xs ) ) @ Ys ) ) ) ).
% take_append
thf(fact_268_take__append,axiom,
! [N: nat,Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int] :
( ( take_P1004856038796978596st_int @ N @ ( append5028292981072010935st_int @ Xs @ Ys ) )
= ( append5028292981072010935st_int @ ( take_P1004856038796978596st_int @ N @ Xs ) @ ( take_P1004856038796978596st_int @ ( minus_minus_nat @ N @ ( size_s229347708945616566st_int @ Xs ) ) @ Ys ) ) ) ).
% take_append
thf(fact_269_take__append,axiom,
! [N: nat,Xs: list_instr,Ys: list_instr] :
( ( take_instr @ N @ ( append_instr @ Xs @ Ys ) )
= ( append_instr @ ( take_instr @ N @ Xs ) @ ( take_instr @ ( minus_minus_nat @ N @ ( size_size_list_instr @ Xs ) ) @ Ys ) ) ) ).
% take_append
thf(fact_270_take__append,axiom,
! [N: nat,Xs: list_list_char,Ys: list_list_char] :
( ( take_list_char @ N @ ( append_list_char @ Xs @ Ys ) )
= ( append_list_char @ ( take_list_char @ N @ Xs ) @ ( take_list_char @ ( minus_minus_nat @ N @ ( size_s356637359517785349t_char @ Xs ) ) @ Ys ) ) ) ).
% take_append
thf(fact_271_drop__append,axiom,
! [N: nat,Xs: list_int,Ys: list_int] :
( ( drop_int @ N @ ( append_int @ Xs @ Ys ) )
= ( append_int @ ( drop_int @ N @ Xs ) @ ( drop_int @ ( minus_minus_nat @ N @ ( size_size_list_int @ Xs ) ) @ Ys ) ) ) ).
% drop_append
thf(fact_272_drop__append,axiom,
! [N: nat,Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int] :
( ( drop_P7838799151754935884st_int @ N @ ( append5028292981072010935st_int @ Xs @ Ys ) )
= ( append5028292981072010935st_int @ ( drop_P7838799151754935884st_int @ N @ Xs ) @ ( drop_P7838799151754935884st_int @ ( minus_minus_nat @ N @ ( size_s229347708945616566st_int @ Xs ) ) @ Ys ) ) ) ).
% drop_append
thf(fact_273_drop__append,axiom,
! [N: nat,Xs: list_instr,Ys: list_instr] :
( ( drop_instr @ N @ ( append_instr @ Xs @ Ys ) )
= ( append_instr @ ( drop_instr @ N @ Xs ) @ ( drop_instr @ ( minus_minus_nat @ N @ ( size_size_list_instr @ Xs ) ) @ Ys ) ) ) ).
% drop_append
thf(fact_274_drop__append,axiom,
! [N: nat,Xs: list_list_char,Ys: list_list_char] :
( ( drop_list_char @ N @ ( append_list_char @ Xs @ Ys ) )
= ( append_list_char @ ( drop_list_char @ N @ Xs ) @ ( drop_list_char @ ( minus_minus_nat @ N @ ( size_s356637359517785349t_char @ Xs ) ) @ Ys ) ) ) ).
% drop_append
thf(fact_275_append__eq__appendI,axiom,
! [Xs: list_instr,Xs1: list_instr,Zs: list_instr,Ys: list_instr,Us: list_instr] :
( ( ( append_instr @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_instr @ Xs1 @ Us ) )
=> ( ( append_instr @ Xs @ Ys )
= ( append_instr @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_276_append__eq__appendI,axiom,
! [Xs: list_P419547167332244514st_int,Xs1: list_P419547167332244514st_int,Zs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int,Us: list_P419547167332244514st_int] :
( ( ( append5028292981072010935st_int @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append5028292981072010935st_int @ Xs1 @ Us ) )
=> ( ( append5028292981072010935st_int @ Xs @ Ys )
= ( append5028292981072010935st_int @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_277_append__eq__appendI,axiom,
! [Xs: list_int,Xs1: list_int,Zs: list_int,Ys: list_int,Us: list_int] :
( ( ( append_int @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_int @ Xs1 @ Us ) )
=> ( ( append_int @ Xs @ Ys )
= ( append_int @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_278_append__eq__map__conv,axiom,
! [Ys: list_instr,Zs: list_instr,F: instr > instr,Xs: list_instr] :
( ( ( append_instr @ Ys @ Zs )
= ( map_instr_instr @ F @ Xs ) )
= ( ? [Us2: list_instr,Vs2: list_instr] :
( ( Xs
= ( append_instr @ Us2 @ Vs2 ) )
& ( Ys
= ( map_instr_instr @ F @ Us2 ) )
& ( Zs
= ( map_instr_instr @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_279_append__eq__map__conv,axiom,
! [Ys: list_instr,Zs: list_instr,F: produc6425607678544837394st_int > instr,Xs: list_P419547167332244514st_int] :
( ( ( append_instr @ Ys @ Zs )
= ( map_Pr7514435936640407687_instr @ F @ Xs ) )
= ( ? [Us2: list_P419547167332244514st_int,Vs2: list_P419547167332244514st_int] :
( ( Xs
= ( append5028292981072010935st_int @ Us2 @ Vs2 ) )
& ( Ys
= ( map_Pr7514435936640407687_instr @ F @ Us2 ) )
& ( Zs
= ( map_Pr7514435936640407687_instr @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_280_append__eq__map__conv,axiom,
! [Ys: list_instr,Zs: list_instr,F: int > instr,Xs: list_int] :
( ( ( append_instr @ Ys @ Zs )
= ( map_int_instr @ F @ Xs ) )
= ( ? [Us2: list_int,Vs2: list_int] :
( ( Xs
= ( append_int @ Us2 @ Vs2 ) )
& ( Ys
= ( map_int_instr @ F @ Us2 ) )
& ( Zs
= ( map_int_instr @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_281_append__eq__map__conv,axiom,
! [Ys: list_P419547167332244514st_int,Zs: list_P419547167332244514st_int,F: instr > produc6425607678544837394st_int,Xs: list_instr] :
( ( ( append5028292981072010935st_int @ Ys @ Zs )
= ( map_in7259965606695656109st_int @ F @ Xs ) )
= ( ? [Us2: list_instr,Vs2: list_instr] :
( ( Xs
= ( append_instr @ Us2 @ Vs2 ) )
& ( Ys
= ( map_in7259965606695656109st_int @ F @ Us2 ) )
& ( Zs
= ( map_in7259965606695656109st_int @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_282_append__eq__map__conv,axiom,
! [Ys: list_P419547167332244514st_int,Zs: list_P419547167332244514st_int,F: int > produc6425607678544837394st_int,Xs: list_int] :
( ( ( append5028292981072010935st_int @ Ys @ Zs )
= ( map_in8512336359594627332st_int @ F @ Xs ) )
= ( ? [Us2: list_int,Vs2: list_int] :
( ( Xs
= ( append_int @ Us2 @ Vs2 ) )
& ( Ys
= ( map_in8512336359594627332st_int @ F @ Us2 ) )
& ( Zs
= ( map_in8512336359594627332st_int @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_283_append__eq__map__conv,axiom,
! [Ys: list_int,Zs: list_int,F: instr > int,Xs: list_instr] :
( ( ( append_int @ Ys @ Zs )
= ( map_instr_int @ F @ Xs ) )
= ( ? [Us2: list_instr,Vs2: list_instr] :
( ( Xs
= ( append_instr @ Us2 @ Vs2 ) )
& ( Ys
= ( map_instr_int @ F @ Us2 ) )
& ( Zs
= ( map_instr_int @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_284_append__eq__map__conv,axiom,
! [Ys: list_int,Zs: list_int,F: produc6425607678544837394st_int > int,Xs: list_P419547167332244514st_int] :
( ( ( append_int @ Ys @ Zs )
= ( map_Pr3224476576807795204nt_int @ F @ Xs ) )
= ( ? [Us2: list_P419547167332244514st_int,Vs2: list_P419547167332244514st_int] :
( ( Xs
= ( append5028292981072010935st_int @ Us2 @ Vs2 ) )
& ( Ys
= ( map_Pr3224476576807795204nt_int @ F @ Us2 ) )
& ( Zs
= ( map_Pr3224476576807795204nt_int @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_285_append__eq__map__conv,axiom,
! [Ys: list_int,Zs: list_int,F: int > int,Xs: list_int] :
( ( ( append_int @ Ys @ Zs )
= ( map_int_int @ F @ Xs ) )
= ( ? [Us2: list_int,Vs2: list_int] :
( ( Xs
= ( append_int @ Us2 @ Vs2 ) )
& ( Ys
= ( map_int_int @ F @ Us2 ) )
& ( Zs
= ( map_int_int @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_286_append__eq__map__conv,axiom,
! [Ys: list_P419547167332244514st_int,Zs: list_P419547167332244514st_int,F: produc6425607678544837394st_int > produc6425607678544837394st_int,Xs: list_P419547167332244514st_int] :
( ( ( append5028292981072010935st_int @ Ys @ Zs )
= ( map_Pr8625093460839654122st_int @ F @ Xs ) )
= ( ? [Us2: list_P419547167332244514st_int,Vs2: list_P419547167332244514st_int] :
( ( Xs
= ( append5028292981072010935st_int @ Us2 @ Vs2 ) )
& ( Ys
= ( map_Pr8625093460839654122st_int @ F @ Us2 ) )
& ( Zs
= ( map_Pr8625093460839654122st_int @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_287_map__eq__append__conv,axiom,
! [F: instr > instr,Xs: list_instr,Ys: list_instr,Zs: list_instr] :
( ( ( map_instr_instr @ F @ Xs )
= ( append_instr @ Ys @ Zs ) )
= ( ? [Us2: list_instr,Vs2: list_instr] :
( ( Xs
= ( append_instr @ Us2 @ Vs2 ) )
& ( Ys
= ( map_instr_instr @ F @ Us2 ) )
& ( Zs
= ( map_instr_instr @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_288_map__eq__append__conv,axiom,
! [F: produc6425607678544837394st_int > instr,Xs: list_P419547167332244514st_int,Ys: list_instr,Zs: list_instr] :
( ( ( map_Pr7514435936640407687_instr @ F @ Xs )
= ( append_instr @ Ys @ Zs ) )
= ( ? [Us2: list_P419547167332244514st_int,Vs2: list_P419547167332244514st_int] :
( ( Xs
= ( append5028292981072010935st_int @ Us2 @ Vs2 ) )
& ( Ys
= ( map_Pr7514435936640407687_instr @ F @ Us2 ) )
& ( Zs
= ( map_Pr7514435936640407687_instr @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_289_map__eq__append__conv,axiom,
! [F: int > instr,Xs: list_int,Ys: list_instr,Zs: list_instr] :
( ( ( map_int_instr @ F @ Xs )
= ( append_instr @ Ys @ Zs ) )
= ( ? [Us2: list_int,Vs2: list_int] :
( ( Xs
= ( append_int @ Us2 @ Vs2 ) )
& ( Ys
= ( map_int_instr @ F @ Us2 ) )
& ( Zs
= ( map_int_instr @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_290_map__eq__append__conv,axiom,
! [F: instr > produc6425607678544837394st_int,Xs: list_instr,Ys: list_P419547167332244514st_int,Zs: list_P419547167332244514st_int] :
( ( ( map_in7259965606695656109st_int @ F @ Xs )
= ( append5028292981072010935st_int @ Ys @ Zs ) )
= ( ? [Us2: list_instr,Vs2: list_instr] :
( ( Xs
= ( append_instr @ Us2 @ Vs2 ) )
& ( Ys
= ( map_in7259965606695656109st_int @ F @ Us2 ) )
& ( Zs
= ( map_in7259965606695656109st_int @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_291_map__eq__append__conv,axiom,
! [F: int > produc6425607678544837394st_int,Xs: list_int,Ys: list_P419547167332244514st_int,Zs: list_P419547167332244514st_int] :
( ( ( map_in8512336359594627332st_int @ F @ Xs )
= ( append5028292981072010935st_int @ Ys @ Zs ) )
= ( ? [Us2: list_int,Vs2: list_int] :
( ( Xs
= ( append_int @ Us2 @ Vs2 ) )
& ( Ys
= ( map_in8512336359594627332st_int @ F @ Us2 ) )
& ( Zs
= ( map_in8512336359594627332st_int @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_292_map__eq__append__conv,axiom,
! [F: instr > int,Xs: list_instr,Ys: list_int,Zs: list_int] :
( ( ( map_instr_int @ F @ Xs )
= ( append_int @ Ys @ Zs ) )
= ( ? [Us2: list_instr,Vs2: list_instr] :
( ( Xs
= ( append_instr @ Us2 @ Vs2 ) )
& ( Ys
= ( map_instr_int @ F @ Us2 ) )
& ( Zs
= ( map_instr_int @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_293_map__eq__append__conv,axiom,
! [F: produc6425607678544837394st_int > int,Xs: list_P419547167332244514st_int,Ys: list_int,Zs: list_int] :
( ( ( map_Pr3224476576807795204nt_int @ F @ Xs )
= ( append_int @ Ys @ Zs ) )
= ( ? [Us2: list_P419547167332244514st_int,Vs2: list_P419547167332244514st_int] :
( ( Xs
= ( append5028292981072010935st_int @ Us2 @ Vs2 ) )
& ( Ys
= ( map_Pr3224476576807795204nt_int @ F @ Us2 ) )
& ( Zs
= ( map_Pr3224476576807795204nt_int @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_294_map__eq__append__conv,axiom,
! [F: int > int,Xs: list_int,Ys: list_int,Zs: list_int] :
( ( ( map_int_int @ F @ Xs )
= ( append_int @ Ys @ Zs ) )
= ( ? [Us2: list_int,Vs2: list_int] :
( ( Xs
= ( append_int @ Us2 @ Vs2 ) )
& ( Ys
= ( map_int_int @ F @ Us2 ) )
& ( Zs
= ( map_int_int @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_295_map__eq__append__conv,axiom,
! [F: produc6425607678544837394st_int > produc6425607678544837394st_int,Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int,Zs: list_P419547167332244514st_int] :
( ( ( map_Pr8625093460839654122st_int @ F @ Xs )
= ( append5028292981072010935st_int @ Ys @ Zs ) )
= ( ? [Us2: list_P419547167332244514st_int,Vs2: list_P419547167332244514st_int] :
( ( Xs
= ( append5028292981072010935st_int @ Us2 @ Vs2 ) )
& ( Ys
= ( map_Pr8625093460839654122st_int @ F @ Us2 ) )
& ( Zs
= ( map_Pr8625093460839654122st_int @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_296_append__eq__append__conv2,axiom,
! [Xs: list_instr,Ys: list_instr,Zs: list_instr,Ts: list_instr] :
( ( ( append_instr @ Xs @ Ys )
= ( append_instr @ Zs @ Ts ) )
= ( ? [Us2: list_instr] :
( ( ( Xs
= ( append_instr @ Zs @ Us2 ) )
& ( ( append_instr @ Us2 @ Ys )
= Ts ) )
| ( ( ( append_instr @ Xs @ Us2 )
= Zs )
& ( Ys
= ( append_instr @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_297_append__eq__append__conv2,axiom,
! [Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int,Zs: list_P419547167332244514st_int,Ts: list_P419547167332244514st_int] :
( ( ( append5028292981072010935st_int @ Xs @ Ys )
= ( append5028292981072010935st_int @ Zs @ Ts ) )
= ( ? [Us2: list_P419547167332244514st_int] :
( ( ( Xs
= ( append5028292981072010935st_int @ Zs @ Us2 ) )
& ( ( append5028292981072010935st_int @ Us2 @ Ys )
= Ts ) )
| ( ( ( append5028292981072010935st_int @ Xs @ Us2 )
= Zs )
& ( Ys
= ( append5028292981072010935st_int @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_298_append__eq__append__conv2,axiom,
! [Xs: list_int,Ys: list_int,Zs: list_int,Ts: list_int] :
( ( ( append_int @ Xs @ Ys )
= ( append_int @ Zs @ Ts ) )
= ( ? [Us2: list_int] :
( ( ( Xs
= ( append_int @ Zs @ Us2 ) )
& ( ( append_int @ Us2 @ Ys )
= Ts ) )
| ( ( ( append_int @ Xs @ Us2 )
= Zs )
& ( Ys
= ( append_int @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_299_take__map,axiom,
! [N: nat,F: produc6425607678544837394st_int > produc6425607678544837394st_int,Xs: list_P419547167332244514st_int] :
( ( take_P1004856038796978596st_int @ N @ ( map_Pr8625093460839654122st_int @ F @ Xs ) )
= ( map_Pr8625093460839654122st_int @ F @ ( take_P1004856038796978596st_int @ N @ Xs ) ) ) ).
% take_map
thf(fact_300_drop__map,axiom,
! [N: nat,F: produc6425607678544837394st_int > produc6425607678544837394st_int,Xs: list_P419547167332244514st_int] :
( ( drop_P7838799151754935884st_int @ N @ ( map_Pr8625093460839654122st_int @ F @ Xs ) )
= ( map_Pr8625093460839654122st_int @ F @ ( drop_P7838799151754935884st_int @ N @ Xs ) ) ) ).
% drop_map
thf(fact_301_append__eq__conv__conj,axiom,
! [Xs: list_int,Ys: list_int,Zs: list_int] :
( ( ( append_int @ Xs @ Ys )
= Zs )
= ( ( Xs
= ( take_int @ ( size_size_list_int @ Xs ) @ Zs ) )
& ( Ys
= ( drop_int @ ( size_size_list_int @ Xs ) @ Zs ) ) ) ) ).
% append_eq_conv_conj
thf(fact_302_append__eq__conv__conj,axiom,
! [Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int,Zs: list_P419547167332244514st_int] :
( ( ( append5028292981072010935st_int @ Xs @ Ys )
= Zs )
= ( ( Xs
= ( take_P1004856038796978596st_int @ ( size_s229347708945616566st_int @ Xs ) @ Zs ) )
& ( Ys
= ( drop_P7838799151754935884st_int @ ( size_s229347708945616566st_int @ Xs ) @ Zs ) ) ) ) ).
% append_eq_conv_conj
thf(fact_303_append__eq__conv__conj,axiom,
! [Xs: list_instr,Ys: list_instr,Zs: list_instr] :
( ( ( append_instr @ Xs @ Ys )
= Zs )
= ( ( Xs
= ( take_instr @ ( size_size_list_instr @ Xs ) @ Zs ) )
& ( Ys
= ( drop_instr @ ( size_size_list_instr @ Xs ) @ Zs ) ) ) ) ).
% append_eq_conv_conj
thf(fact_304_append__eq__conv__conj,axiom,
! [Xs: list_list_char,Ys: list_list_char,Zs: list_list_char] :
( ( ( append_list_char @ Xs @ Ys )
= Zs )
= ( ( Xs
= ( take_list_char @ ( size_s356637359517785349t_char @ Xs ) @ Zs ) )
& ( Ys
= ( drop_list_char @ ( size_s356637359517785349t_char @ Xs ) @ Zs ) ) ) ) ).
% append_eq_conv_conj
thf(fact_305_take__equalityI,axiom,
! [Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int] :
( ! [I: nat] :
( ( take_P1004856038796978596st_int @ I @ Xs )
= ( take_P1004856038796978596st_int @ I @ Ys ) )
=> ( Xs = Ys ) ) ).
% take_equalityI
thf(fact_306_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_307_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_308_drop__take,axiom,
! [N: nat,M: nat,Xs: list_P419547167332244514st_int] :
( ( drop_P7838799151754935884st_int @ N @ ( take_P1004856038796978596st_int @ M @ Xs ) )
= ( take_P1004856038796978596st_int @ ( minus_minus_nat @ M @ N ) @ ( drop_P7838799151754935884st_int @ N @ Xs ) ) ) ).
% drop_take
thf(fact_309_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_310_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_311_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_312_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
= ( X = one_one_int ) ) ).
% one_reorient
thf(fact_313_diff__eq__diff__eq,axiom,
! [A: int,B: int,C2: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C2 @ D ) )
=> ( ( A = B )
= ( C2 = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_314_diff__right__commute,axiom,
! [A: int,C2: int,B: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ C2 ) @ B )
= ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C2 ) ) ).
% diff_right_commute
thf(fact_315_diff__right__commute,axiom,
! [A: nat,C2: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C2 ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C2 ) ) ).
% diff_right_commute
thf(fact_316_nth__append,axiom,
! [N: nat,Xs: list_int,Ys: list_int] :
( ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
=> ( ( nth_int @ ( append_int @ Xs @ Ys ) @ N )
= ( nth_int @ Xs @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
=> ( ( nth_int @ ( append_int @ Xs @ Ys ) @ N )
= ( nth_int @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_int @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_317_nth__append,axiom,
! [N: nat,Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int] :
( ( ( ord_less_nat @ N @ ( size_s229347708945616566st_int @ Xs ) )
=> ( ( nth_Pr8327173284404383691st_int @ ( append5028292981072010935st_int @ Xs @ Ys ) @ N )
= ( nth_Pr8327173284404383691st_int @ Xs @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_s229347708945616566st_int @ Xs ) )
=> ( ( nth_Pr8327173284404383691st_int @ ( append5028292981072010935st_int @ Xs @ Ys ) @ N )
= ( nth_Pr8327173284404383691st_int @ Ys @ ( minus_minus_nat @ N @ ( size_s229347708945616566st_int @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_318_nth__append,axiom,
! [N: nat,Xs: list_instr,Ys: list_instr] :
( ( ( ord_less_nat @ N @ ( size_size_list_instr @ Xs ) )
=> ( ( nth_instr @ ( append_instr @ Xs @ Ys ) @ N )
= ( nth_instr @ Xs @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_size_list_instr @ Xs ) )
=> ( ( nth_instr @ ( append_instr @ Xs @ Ys ) @ N )
= ( nth_instr @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_instr @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_319_nth__append,axiom,
! [N: nat,Xs: list_list_char,Ys: list_list_char] :
( ( ( ord_less_nat @ N @ ( size_s356637359517785349t_char @ Xs ) )
=> ( ( nth_list_char @ ( append_list_char @ Xs @ Ys ) @ N )
= ( nth_list_char @ Xs @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_s356637359517785349t_char @ Xs ) )
=> ( ( nth_list_char @ ( append_list_char @ Xs @ Ys ) @ N )
= ( nth_list_char @ Ys @ ( minus_minus_nat @ N @ ( size_s356637359517785349t_char @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_320_map__eq__imp__length__eq,axiom,
! [F: produc6425607678544837394st_int > produc6425607678544837394st_int,Xs: list_P419547167332244514st_int,G: produc6425607678544837394st_int > produc6425607678544837394st_int,Ys: list_P419547167332244514st_int] :
( ( ( map_Pr8625093460839654122st_int @ F @ Xs )
= ( map_Pr8625093460839654122st_int @ G @ Ys ) )
=> ( ( size_s229347708945616566st_int @ Xs )
= ( size_s229347708945616566st_int @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_321_map__eq__imp__length__eq,axiom,
! [F: produc6425607678544837394st_int > produc6425607678544837394st_int,Xs: list_P419547167332244514st_int,G: instr > produc6425607678544837394st_int,Ys: list_instr] :
( ( ( map_Pr8625093460839654122st_int @ F @ Xs )
= ( map_in7259965606695656109st_int @ G @ Ys ) )
=> ( ( size_s229347708945616566st_int @ Xs )
= ( size_size_list_instr @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_322_map__eq__imp__length__eq,axiom,
! [F: produc6425607678544837394st_int > produc6425607678544837394st_int,Xs: list_P419547167332244514st_int,G: list_char > produc6425607678544837394st_int,Ys: list_list_char] :
( ( ( map_Pr8625093460839654122st_int @ F @ Xs )
= ( map_li6440205260635360185st_int @ G @ Ys ) )
=> ( ( size_s229347708945616566st_int @ Xs )
= ( size_s356637359517785349t_char @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_323_map__eq__imp__length__eq,axiom,
! [F: instr > produc6425607678544837394st_int,Xs: list_instr,G: produc6425607678544837394st_int > produc6425607678544837394st_int,Ys: list_P419547167332244514st_int] :
( ( ( map_in7259965606695656109st_int @ F @ Xs )
= ( map_Pr8625093460839654122st_int @ G @ Ys ) )
=> ( ( size_size_list_instr @ Xs )
= ( size_s229347708945616566st_int @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_324_map__eq__imp__length__eq,axiom,
! [F: list_char > produc6425607678544837394st_int,Xs: list_list_char,G: produc6425607678544837394st_int > produc6425607678544837394st_int,Ys: list_P419547167332244514st_int] :
( ( ( map_li6440205260635360185st_int @ F @ Xs )
= ( map_Pr8625093460839654122st_int @ G @ Ys ) )
=> ( ( size_s356637359517785349t_char @ Xs )
= ( size_s229347708945616566st_int @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_325_drop__0,axiom,
! [Xs: list_P419547167332244514st_int] :
( ( drop_P7838799151754935884st_int @ zero_zero_nat @ Xs )
= Xs ) ).
% drop_0
thf(fact_326_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% not0_implies_Suc
thf(fact_327_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_328_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_329_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_330_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_331_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X4: nat] : ( P @ X4 @ zero_zero_nat )
=> ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
=> ( ! [X4: nat,Y3: nat] :
( ( P @ X4 @ Y3 )
=> ( P @ ( suc @ X4 ) @ ( suc @ Y3 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_332_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_333_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_334_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_335_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_336_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_337_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_338_Nat_OlessE,axiom,
! [I3: nat,K: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( ( K
!= ( suc @ I3 ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_339_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_340_Suc__lessE,axiom,
! [I3: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I3 ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_341_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% Suc_lessI
thf(fact_342_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_343_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_344_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
& ( P @ I2 ) ) )
= ( ( P @ N )
| ? [I2: nat] :
( ( ord_less_nat @ I2 @ N )
& ( P @ I2 ) ) ) ) ).
% Ex_less_Suc
thf(fact_345_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ).
% less_Suc_eq
thf(fact_346_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_347_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
=> ( P @ I2 ) ) )
= ( ( P @ N )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( P @ I2 ) ) ) ) ).
% All_less_Suc
thf(fact_348_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M5: nat] :
( ( M
= ( suc @ M5 ) )
& ( ord_less_nat @ N @ M5 ) ) ) ) ).
% Suc_less_eq2
thf(fact_349_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_350_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_less_SucD
thf(fact_351_less__trans__Suc,axiom,
! [I3: nat,J: nat,K: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I3 ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_352_less__Suc__induct,axiom,
! [I3: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I3 @ J )
=> ( ! [I: nat] : ( P @ I @ ( suc @ I ) )
=> ( ! [I: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ( ord_less_nat @ J2 @ K2 )
=> ( ( P @ I @ J2 )
=> ( ( P @ J2 @ K2 )
=> ( P @ I @ K2 ) ) ) ) )
=> ( P @ I3 @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_353_strict__inc__induct,axiom,
! [I3: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I3 @ J )
=> ( ! [I: nat] :
( ( J
= ( suc @ I ) )
=> ( P @ I ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( P @ ( suc @ I ) )
=> ( P @ I ) ) )
=> ( P @ I3 ) ) ) ) ).
% strict_inc_induct
thf(fact_354_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_355_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I3: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ ( minus_minus_nat @ K @ I3 ) ) ) ) ).
% zero_induct_lemma
thf(fact_356_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_357_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_358_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_359_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_360_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_361_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_362_eq__iff__diff__eq__0,axiom,
( ( ^ [Y2: int,Z: int] : ( Y2 = Z ) )
= ( ^ [A5: int,B4: int] :
( ( minus_minus_int @ A5 @ B4 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_363_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_364_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_365_diff__strict__mono,axiom,
! [A: int,B: int,D: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ D @ C2 )
=> ( ord_less_int @ ( minus_minus_int @ A @ C2 ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_366_diff__eq__diff__less,axiom,
! [A: int,B: int,C2: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C2 @ D ) )
=> ( ( ord_less_int @ A @ B )
= ( ord_less_int @ C2 @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_367_diff__strict__left__mono,axiom,
! [B: int,A: int,C2: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_int @ ( minus_minus_int @ C2 @ A ) @ ( minus_minus_int @ C2 @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_368_diff__strict__right__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( minus_minus_int @ A @ C2 ) @ ( minus_minus_int @ B @ C2 ) ) ) ).
% diff_strict_right_mono
thf(fact_369_pair__list__eqI,axiom,
! [Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int] :
( ( ( map_Pr3224476576807795204nt_int @ produc8941309049508465880st_int @ Xs )
= ( map_Pr3224476576807795204nt_int @ produc8941309049508465880st_int @ Ys ) )
=> ( ( ( map_Pr4101632105909876081st_int @ produc1296517004818028822st_int @ Xs )
= ( map_Pr4101632105909876081st_int @ produc1296517004818028822st_int @ Ys ) )
=> ( Xs = Ys ) ) ) ).
% pair_list_eqI
thf(fact_370_pair__list__eqI,axiom,
! [Xs: list_P98921565119350719nt_int,Ys: list_P98921565119350719nt_int] :
( ( ( map_Pr6267657629804605729ar_int @ produc9025023058967219549nt_int @ Xs )
= ( map_Pr6267657629804605729ar_int @ produc9025023058967219549nt_int @ Ys ) )
=> ( ( ( map_Pr7025006424454111649nt_int @ produc179445755590504351nt_int @ Xs )
= ( map_Pr7025006424454111649nt_int @ produc179445755590504351nt_int @ Ys ) )
=> ( Xs = Ys ) ) ) ).
% pair_list_eqI
thf(fact_371_pair__list__eqI,axiom,
! [Xs: list_P8654944379044127311st_int,Ys: list_P8654944379044127311st_int] :
( ( ( map_Pr4838045029083743153ar_int @ produc2155683374303800813st_int @ Xs )
= ( map_Pr4838045029083743153ar_int @ produc2155683374303800813st_int @ Ys ) )
=> ( ( ( map_Pr7594757675026762433st_int @ produc6244060863719409455st_int @ Xs )
= ( map_Pr7594757675026762433st_int @ produc6244060863719409455st_int @ Ys ) )
=> ( Xs = Ys ) ) ) ).
% pair_list_eqI
thf(fact_372_pair__list__eqI,axiom,
! [Xs: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int] :
( ( ( map_Pr6494458893431244577nt_int @ product_fst_int_int @ Xs )
= ( map_Pr6494458893431244577nt_int @ product_fst_int_int @ Ys ) )
=> ( ( ( map_Pr6494458893431244577nt_int @ product_snd_int_int @ Xs )
= ( map_Pr6494458893431244577nt_int @ product_snd_int_int @ Ys ) )
=> ( Xs = Ys ) ) ) ).
% pair_list_eqI
thf(fact_373_pair__list__eqI,axiom,
! [Xs: list_P7813408393781968525nt_int,Ys: list_P7813408393781968525nt_int] :
( ( ( map_Pr7817743546820071988t_char @ produc811350159336358595nt_int @ Xs )
= ( map_Pr7817743546820071988t_char @ produc811350159336358595nt_int @ Ys ) )
=> ( ( ( map_Pr254812380637628598nt_int @ produc30867750519413249nt_int @ Xs )
= ( map_Pr254812380637628598nt_int @ produc30867750519413249nt_int @ Ys ) )
=> ( Xs = Ys ) ) ) ).
% pair_list_eqI
thf(fact_374_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N3 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_375_lift__Suc__mono__less,axiom,
! [F: nat > int,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N3 )
=> ( ord_less_int @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_376_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_377_lift__Suc__mono__less__iff,axiom,
! [F: nat > int,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_int @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_378_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
!= zero_zero_int ) ).
% of_nat_neq_0
thf(fact_379_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
!= zero_zero_nat ) ).
% of_nat_neq_0
thf(fact_380_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
& ( P @ I2 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I2: nat] :
( ( ord_less_nat @ I2 @ N )
& ( P @ ( suc @ I2 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_381_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M2: nat] :
( N
= ( suc @ M2 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_382_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
=> ( P @ I2 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( P @ ( suc @ I2 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_383_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% gr0_implies_Suc
thf(fact_384_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( M = zero_zero_nat )
| ? [J3: nat] :
( ( M
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_385_Suc__diff__Suc,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
= ( minus_minus_nat @ M @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_386_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_387_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_388_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ M @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_389_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_390_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_391_less__iff__diff__less__0,axiom,
( ord_less_int
= ( ^ [A5: int,B4: int] : ( ord_less_int @ ( minus_minus_int @ A5 @ B4 ) @ zero_zero_int ) ) ) ).
% less_iff_diff_less_0
thf(fact_392_map__equality__iff,axiom,
! [F: int > produc6425607678544837394st_int,Xs: list_int,G: produc6425607678544837394st_int > produc6425607678544837394st_int,Ys: list_P419547167332244514st_int] :
( ( ( map_in8512336359594627332st_int @ F @ Xs )
= ( map_Pr8625093460839654122st_int @ G @ Ys ) )
= ( ( ( size_size_list_int @ Xs )
= ( size_s229347708945616566st_int @ Ys ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s229347708945616566st_int @ Ys ) )
=> ( ( F @ ( nth_int @ Xs @ I2 ) )
= ( G @ ( nth_Pr8327173284404383691st_int @ Ys @ I2 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_393_map__equality__iff,axiom,
! [F: produc6425607678544837394st_int > produc6425607678544837394st_int,Xs: list_P419547167332244514st_int,G: int > produc6425607678544837394st_int,Ys: list_int] :
( ( ( map_Pr8625093460839654122st_int @ F @ Xs )
= ( map_in8512336359594627332st_int @ G @ Ys ) )
= ( ( ( size_s229347708945616566st_int @ Xs )
= ( size_size_list_int @ Ys ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_int @ Ys ) )
=> ( ( F @ ( nth_Pr8327173284404383691st_int @ Xs @ I2 ) )
= ( G @ ( nth_int @ Ys @ I2 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_394_map__equality__iff,axiom,
! [F: produc6425607678544837394st_int > produc6425607678544837394st_int,Xs: list_P419547167332244514st_int,G: produc6425607678544837394st_int > produc6425607678544837394st_int,Ys: list_P419547167332244514st_int] :
( ( ( map_Pr8625093460839654122st_int @ F @ Xs )
= ( map_Pr8625093460839654122st_int @ G @ Ys ) )
= ( ( ( size_s229347708945616566st_int @ Xs )
= ( size_s229347708945616566st_int @ Ys ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s229347708945616566st_int @ Ys ) )
=> ( ( F @ ( nth_Pr8327173284404383691st_int @ Xs @ I2 ) )
= ( G @ ( nth_Pr8327173284404383691st_int @ Ys @ I2 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_395_map__equality__iff,axiom,
! [F: produc6425607678544837394st_int > produc6425607678544837394st_int,Xs: list_P419547167332244514st_int,G: instr > produc6425607678544837394st_int,Ys: list_instr] :
( ( ( map_Pr8625093460839654122st_int @ F @ Xs )
= ( map_in7259965606695656109st_int @ G @ Ys ) )
= ( ( ( size_s229347708945616566st_int @ Xs )
= ( size_size_list_instr @ Ys ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_instr @ Ys ) )
=> ( ( F @ ( nth_Pr8327173284404383691st_int @ Xs @ I2 ) )
= ( G @ ( nth_instr @ Ys @ I2 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_396_map__equality__iff,axiom,
! [F: produc6425607678544837394st_int > produc6425607678544837394st_int,Xs: list_P419547167332244514st_int,G: list_char > produc6425607678544837394st_int,Ys: list_list_char] :
( ( ( map_Pr8625093460839654122st_int @ F @ Xs )
= ( map_li6440205260635360185st_int @ G @ Ys ) )
= ( ( ( size_s229347708945616566st_int @ Xs )
= ( size_s356637359517785349t_char @ Ys ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s356637359517785349t_char @ Ys ) )
=> ( ( F @ ( nth_Pr8327173284404383691st_int @ Xs @ I2 ) )
= ( G @ ( nth_list_char @ Ys @ I2 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_397_map__equality__iff,axiom,
! [F: instr > produc6425607678544837394st_int,Xs: list_instr,G: produc6425607678544837394st_int > produc6425607678544837394st_int,Ys: list_P419547167332244514st_int] :
( ( ( map_in7259965606695656109st_int @ F @ Xs )
= ( map_Pr8625093460839654122st_int @ G @ Ys ) )
= ( ( ( size_size_list_instr @ Xs )
= ( size_s229347708945616566st_int @ Ys ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s229347708945616566st_int @ Ys ) )
=> ( ( F @ ( nth_instr @ Xs @ I2 ) )
= ( G @ ( nth_Pr8327173284404383691st_int @ Ys @ I2 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_398_map__equality__iff,axiom,
! [F: list_char > produc6425607678544837394st_int,Xs: list_list_char,G: produc6425607678544837394st_int > produc6425607678544837394st_int,Ys: list_P419547167332244514st_int] :
( ( ( map_li6440205260635360185st_int @ F @ Xs )
= ( map_Pr8625093460839654122st_int @ G @ Ys ) )
= ( ( ( size_s356637359517785349t_char @ Xs )
= ( size_s229347708945616566st_int @ Ys ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s229347708945616566st_int @ Ys ) )
=> ( ( F @ ( nth_list_char @ Xs @ I2 ) )
= ( G @ ( nth_Pr8327173284404383691st_int @ Ys @ I2 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_399_diff__Suc__less,axiom,
! [N: nat,I3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I3 ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_400_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_401_Suc__pred_H,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( N
= ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_402__092_060open_062Q_Ax_A_Imap_A_Ioff_AP_J_A_Idrop_Ak_A_Itake_A_ISuc_Ak_H_J_Acfs_J_J_A_B_A0_J_A_Imap_A_Ioff_AP_J_A_Idrop_Ak_A_Itake_A_ISuc_Ak_H_J_Acfs_J_J_A_B_A_Ilength_A_Imap_A_Ioff_AP_J_A_Idrop_Ak_A_Itake_A_ISuc_Ak_H_J_Acfs_J_J_J_A_N_A1_J_J_092_060close_062,axiom,
( q @ x
@ ( nth_Pr8327173284404383691st_int
@ ( map_Pr8625093460839654122st_int
@ ^ [Cf: produc6425607678544837394st_int] : ( produc5086643055186798020st_int @ ( minus_minus_int @ ( produc8941309049508465880st_int @ Cf ) @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ p ) ) ) @ ( produc1296517004818028822st_int @ Cf ) )
@ ( drop_P7838799151754935884st_int @ k @ ( take_P1004856038796978596st_int @ ( suc @ k2 ) @ cfs ) ) )
@ zero_zero_nat )
@ ( nth_Pr8327173284404383691st_int
@ ( map_Pr8625093460839654122st_int
@ ^ [Cf: produc6425607678544837394st_int] : ( produc5086643055186798020st_int @ ( minus_minus_int @ ( produc8941309049508465880st_int @ Cf ) @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ p ) ) ) @ ( produc1296517004818028822st_int @ Cf ) )
@ ( drop_P7838799151754935884st_int @ k @ ( take_P1004856038796978596st_int @ ( suc @ k2 ) @ cfs ) ) )
@ ( minus_minus_nat
@ ( size_s229347708945616566st_int
@ ( map_Pr8625093460839654122st_int
@ ^ [Cf: produc6425607678544837394st_int] : ( produc5086643055186798020st_int @ ( minus_minus_int @ ( produc8941309049508465880st_int @ Cf ) @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ p ) ) ) @ ( produc1296517004818028822st_int @ Cf ) )
@ ( drop_P7838799151754935884st_int @ k @ ( take_P1004856038796978596st_int @ ( suc @ k2 ) @ cfs ) ) ) )
@ one_one_nat ) ) ) ).
% \<open>Q x (map (off P) (drop k (take (Suc k') cfs)) ! 0) (map (off P) (drop k (take (Suc k') cfs)) ! (length (map (off P) (drop k (take (Suc k') cfs))) - 1))\<close>
thf(fact_403__092_060open_062map_A_Ioff_AP_J_A_Idrop_Ak_A_Itake_A_ISuc_Ak_H_J_Acfs_J_J_A_B_A_Ilength_A_Imap_A_Ioff_AP_J_A_Idrop_Ak_A_Itake_A_ISuc_Ak_H_J_Acfs_J_J_J_A_N_A1_J_A_061_Aoff_AP_A_Icfs_A_B_Ak_H_J_092_060close_062,axiom,
( ( nth_Pr8327173284404383691st_int
@ ( map_Pr8625093460839654122st_int
@ ^ [Cf: produc6425607678544837394st_int] : ( produc5086643055186798020st_int @ ( minus_minus_int @ ( produc8941309049508465880st_int @ Cf ) @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ p ) ) ) @ ( produc1296517004818028822st_int @ Cf ) )
@ ( drop_P7838799151754935884st_int @ k @ ( take_P1004856038796978596st_int @ ( suc @ k2 ) @ cfs ) ) )
@ ( minus_minus_nat
@ ( size_s229347708945616566st_int
@ ( map_Pr8625093460839654122st_int
@ ^ [Cf: produc6425607678544837394st_int] : ( produc5086643055186798020st_int @ ( minus_minus_int @ ( produc8941309049508465880st_int @ Cf ) @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ p ) ) ) @ ( produc1296517004818028822st_int @ Cf ) )
@ ( drop_P7838799151754935884st_int @ k @ ( take_P1004856038796978596st_int @ ( suc @ k2 ) @ cfs ) ) ) )
@ one_one_nat ) )
= ( produc5086643055186798020st_int @ ( minus_minus_int @ ( produc8941309049508465880st_int @ ( nth_Pr8327173284404383691st_int @ cfs @ k2 ) ) @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ p ) ) ) @ ( produc1296517004818028822st_int @ ( nth_Pr8327173284404383691st_int @ cfs @ k2 ) ) ) ) ).
% \<open>map (off P) (drop k (take (Suc k') cfs)) ! (length (map (off P) (drop k (take (Suc k') cfs))) - 1) = off P (cfs ! k')\<close>
thf(fact_404_Suc__lessI2,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( M
!= ( minus_minus_nat @ N @ one_one_nat ) )
=> ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% Suc_lessI2
thf(fact_405__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062k_H_O_A_092_060lbrakk_062k_H_A_092_060in_062_A_123k_O_O_060length_Acfs_125_059_AQ_Ax_A_Imap_A_Ioff_AP_J_A_Idrop_Ak_A_Itake_A_ISuc_Ak_H_J_Acfs_J_J_A_B_A0_J_A_Imap_A_Ioff_AP_J_A_Idrop_Ak_A_Itake_A_ISuc_Ak_H_J_Acfs_J_J_A_B_A_Ilength_A_Imap_A_Ioff_AP_J_A_Idrop_Ak_A_Itake_A_ISuc_Ak_H_J_Acfs_J_J_J_A_N_A1_J_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [K3: nat] :
( ( member_nat @ K3 @ ( set_or4665077453230672383an_nat @ k @ ( size_s229347708945616566st_int @ cfs ) ) )
=> ~ ( q @ x
@ ( nth_Pr8327173284404383691st_int
@ ( map_Pr8625093460839654122st_int
@ ^ [Cf: produc6425607678544837394st_int] : ( produc5086643055186798020st_int @ ( minus_minus_int @ ( produc8941309049508465880st_int @ Cf ) @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ p ) ) ) @ ( produc1296517004818028822st_int @ Cf ) )
@ ( drop_P7838799151754935884st_int @ k @ ( take_P1004856038796978596st_int @ ( suc @ K3 ) @ cfs ) ) )
@ zero_zero_nat )
@ ( nth_Pr8327173284404383691st_int
@ ( map_Pr8625093460839654122st_int
@ ^ [Cf: produc6425607678544837394st_int] : ( produc5086643055186798020st_int @ ( minus_minus_int @ ( produc8941309049508465880st_int @ Cf ) @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ p ) ) ) @ ( produc1296517004818028822st_int @ Cf ) )
@ ( drop_P7838799151754935884st_int @ k @ ( take_P1004856038796978596st_int @ ( suc @ K3 ) @ cfs ) ) )
@ ( minus_minus_nat
@ ( size_s229347708945616566st_int
@ ( map_Pr8625093460839654122st_int
@ ^ [Cf: produc6425607678544837394st_int] : ( produc5086643055186798020st_int @ ( minus_minus_int @ ( produc8941309049508465880st_int @ Cf ) @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ p ) ) ) @ ( produc1296517004818028822st_int @ Cf ) )
@ ( drop_P7838799151754935884st_int @ k @ ( take_P1004856038796978596st_int @ ( suc @ K3 ) @ cfs ) ) ) )
@ one_one_nat ) ) ) ) ).
% \<open>\<And>thesis. (\<And>k'. \<lbrakk>k' \<in> {k..<length cfs}; Q x (map (off P) (drop k (take (Suc k') cfs)) ! 0) (map (off P) (drop k (take (Suc k') cfs)) ! (length (map (off P) (drop k (take (Suc k') cfs))) - 1))\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_406__092_060open_062_092_060exists_062k_H_092_060in_062_123k_O_O_060length_Acfs_125_O_AP_H_Ax_A_092_060Turnstile_062_Amap_A_Ioff_AP_J_A_Idrop_Ak_A_Itake_A_ISuc_Ak_H_J_Acfs_J_J_092_060box_062_092_060close_062,axiom,
? [X4: nat] :
( ( member_nat @ X4 @ ( set_or4665077453230672383an_nat @ k @ ( size_s229347708945616566st_int @ cfs ) ) )
& ( execl_all @ ( p2 @ x )
@ ( map_Pr8625093460839654122st_int
@ ^ [Cf: produc6425607678544837394st_int] : ( produc5086643055186798020st_int @ ( minus_minus_int @ ( produc8941309049508465880st_int @ Cf ) @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ p ) ) ) @ ( produc1296517004818028822st_int @ Cf ) )
@ ( drop_P7838799151754935884st_int @ k @ ( take_P1004856038796978596st_int @ ( suc @ X4 ) @ cfs ) ) ) ) ) ).
% \<open>\<exists>k'\<in>{k..<length cfs}. P' x \<Turnstile> map (off P) (drop k (take (Suc k') cfs))\<box>\<close>
thf(fact_407_int__ops_I6_J,axiom,
! [A: nat,B: nat] :
( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
= zero_zero_int ) )
& ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ) ).
% int_ops(6)
thf(fact_408_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N2: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% pos_int_cases
thf(fact_409_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
& ( K
= ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_410_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_411_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_412_map__ident,axiom,
( ( map_Pr8625093460839654122st_int
@ ^ [X3: produc6425607678544837394st_int] : X3 )
= ( ^ [Xs2: list_P419547167332244514st_int] : Xs2 ) ) ).
% map_ident
thf(fact_413_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_414_list_Omap__ident,axiom,
! [T: list_P419547167332244514st_int] :
( ( map_Pr8625093460839654122st_int
@ ^ [X3: produc6425607678544837394st_int] : X3
@ T )
= T ) ).
% list.map_ident
thf(fact_415_nat__less__as__int,axiom,
( ord_less_nat
= ( ^ [A5: nat,B4: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_less_as_int
thf(fact_416_linorder__neqE__linordered__idom,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_417_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_418_verit__comp__simplify1_I1_J,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_419_int__int__eq,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% int_int_eq
thf(fact_420_int__if,axiom,
! [P: $o,A: nat,B: nat] :
( ( P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
= ( semiri1314217659103216013at_int @ A ) ) )
& ( ~ P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
= ( semiri1314217659103216013at_int @ B ) ) ) ) ).
% int_if
thf(fact_421_nat__int__comparison_I1_J,axiom,
( ( ^ [Y2: nat,Z: nat] : ( Y2 = Z ) )
= ( ^ [A5: nat,B4: nat] :
( ( semiri1314217659103216013at_int @ A5 )
= ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_422_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus_int @ K @ zero_zero_int )
= K ) ).
% minus_int_code(1)
thf(fact_423_int__less__induct,axiom,
! [I3: int,K: int,P: int > $o] :
( ( ord_less_int @ I3 @ K )
=> ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
=> ( ! [I: int] :
( ( ord_less_int @ I @ K )
=> ( ( P @ I )
=> ( P @ ( minus_minus_int @ I @ one_one_int ) ) ) )
=> ( P @ I3 ) ) ) ) ).
% int_less_induct
thf(fact_424_iexec_Ocases,axiom,
! [X: produc5995290525303592096st_int] :
~ ! [Ins: instr,I: int,S3: list_char > int,Stk: list_int] :
( X
!= ( produc1484136438817787418st_int @ Ins @ ( produc5086643055186798020st_int @ I @ ( produc8650753666468850689st_int @ S3 @ Stk ) ) ) ) ).
% iexec.cases
thf(fact_425_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_426_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_427_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_428_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A5: nat,B4: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_429_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_430_int__diff__cases,axiom,
! [Z2: int] :
~ ! [M4: nat,N2: nat] :
( Z2
!= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M4 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% int_diff_cases
thf(fact_431_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_432_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_433_execl__sub,axiom,
! [P: list_instr,P3: list_instr,P4: list_instr,Cfs2: list_P419547167332244514st_int,M: nat,N: nat] :
( ( execl @ ( append_instr @ P @ ( append_instr @ P3 @ P4 ) ) @ Cfs2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_or4665077453230672383an_nat @ M @ N ) )
=> ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P ) ) @ ( produc8941309049508465880st_int @ ( nth_Pr8327173284404383691st_int @ Cfs2 @ X4 ) ) )
& ( ord_less_int @ ( minus_minus_int @ ( produc8941309049508465880st_int @ ( nth_Pr8327173284404383691st_int @ Cfs2 @ X4 ) ) @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P ) ) ) @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P3 ) ) ) ) )
=> ( execl @ P3
@ ( map_Pr8625093460839654122st_int
@ ^ [Cf: produc6425607678544837394st_int] : ( produc5086643055186798020st_int @ ( minus_minus_int @ ( produc8941309049508465880st_int @ Cf ) @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P ) ) ) @ ( produc1296517004818028822st_int @ Cf ) )
@ ( drop_P7838799151754935884st_int @ M @ ( take_P1004856038796978596st_int @ ( suc @ N ) @ Cfs2 ) ) ) ) ) ) ).
% execl_sub
thf(fact_434_size__char__eq__0,axiom,
( size_size_char
= ( ^ [C3: char] : zero_zero_nat ) ) ).
% size_char_eq_0
thf(fact_435_id__take__nth__drop,axiom,
! [I3: nat,Xs: list_int] :
( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs ) )
=> ( Xs
= ( append_int @ ( take_int @ I3 @ Xs ) @ ( cons_int @ ( nth_int @ Xs @ I3 ) @ ( drop_int @ ( suc @ I3 ) @ Xs ) ) ) ) ) ).
% id_take_nth_drop
thf(fact_436_id__take__nth__drop,axiom,
! [I3: nat,Xs: list_P419547167332244514st_int] :
( ( ord_less_nat @ I3 @ ( size_s229347708945616566st_int @ Xs ) )
=> ( Xs
= ( append5028292981072010935st_int @ ( take_P1004856038796978596st_int @ I3 @ Xs ) @ ( cons_P4151473878978182108st_int @ ( nth_Pr8327173284404383691st_int @ Xs @ I3 ) @ ( drop_P7838799151754935884st_int @ ( suc @ I3 ) @ Xs ) ) ) ) ) ).
% id_take_nth_drop
thf(fact_437_id__take__nth__drop,axiom,
! [I3: nat,Xs: list_instr] :
( ( ord_less_nat @ I3 @ ( size_size_list_instr @ Xs ) )
=> ( Xs
= ( append_instr @ ( take_instr @ I3 @ Xs ) @ ( cons_instr @ ( nth_instr @ Xs @ I3 ) @ ( drop_instr @ ( suc @ I3 ) @ Xs ) ) ) ) ) ).
% id_take_nth_drop
thf(fact_438_id__take__nth__drop,axiom,
! [I3: nat,Xs: list_list_char] :
( ( ord_less_nat @ I3 @ ( size_s356637359517785349t_char @ Xs ) )
=> ( Xs
= ( append_list_char @ ( take_list_char @ I3 @ Xs ) @ ( cons_list_char @ ( nth_list_char @ Xs @ I3 ) @ ( drop_list_char @ ( suc @ I3 ) @ Xs ) ) ) ) ) ).
% id_take_nth_drop
thf(fact_439_eq__snd__iff,axiom,
! [B: produc369741286924889929st_int,P2: produc6425607678544837394st_int] :
( ( B
= ( produc1296517004818028822st_int @ P2 ) )
= ( ? [A5: int] :
( P2
= ( produc5086643055186798020st_int @ A5 @ B ) ) ) ) ).
% eq_snd_iff
thf(fact_440_eq__snd__iff,axiom,
! [B: int,P2: produc4435102495419491129nt_int] :
( ( B
= ( produc179445755590504351nt_int @ P2 ) )
= ( ? [A5: list_char > int] :
( P2
= ( produc5790713362662368625nt_int @ A5 @ B ) ) ) ) ).
% eq_snd_iff
thf(fact_441_eq__snd__iff,axiom,
! [B: list_int,P2: produc369741286924889929st_int] :
( ( B
= ( produc6244060863719409455st_int @ P2 ) )
= ( ? [A5: list_char > int] :
( P2
= ( produc8650753666468850689st_int @ A5 @ B ) ) ) ) ).
% eq_snd_iff
thf(fact_442_eq__snd__iff,axiom,
! [B: int,P2: product_prod_int_int] :
( ( B
= ( product_snd_int_int @ P2 ) )
= ( ? [A5: int] :
( P2
= ( product_Pair_int_int @ A5 @ B ) ) ) ) ).
% eq_snd_iff
thf(fact_443_eq__snd__iff,axiom,
! [B: produc4435102495419491129nt_int,P2: produc4189061985984568957nt_int] :
( ( B
= ( produc30867750519413249nt_int @ P2 ) )
= ( ? [A5: list_list_char] :
( P2
= ( produc3965054194175396271nt_int @ A5 @ B ) ) ) ) ).
% eq_snd_iff
thf(fact_444_sndI,axiom,
! [X: produc6425607678544837394st_int,Y: int,Z2: produc369741286924889929st_int] :
( ( X
= ( produc5086643055186798020st_int @ Y @ Z2 ) )
=> ( ( produc1296517004818028822st_int @ X )
= Z2 ) ) ).
% sndI
thf(fact_445_sndI,axiom,
! [X: produc4435102495419491129nt_int,Y: list_char > int,Z2: int] :
( ( X
= ( produc5790713362662368625nt_int @ Y @ Z2 ) )
=> ( ( produc179445755590504351nt_int @ X )
= Z2 ) ) ).
% sndI
thf(fact_446_sndI,axiom,
! [X: produc369741286924889929st_int,Y: list_char > int,Z2: list_int] :
( ( X
= ( produc8650753666468850689st_int @ Y @ Z2 ) )
=> ( ( produc6244060863719409455st_int @ X )
= Z2 ) ) ).
% sndI
thf(fact_447_sndI,axiom,
! [X: product_prod_int_int,Y: int,Z2: int] :
( ( X
= ( product_Pair_int_int @ Y @ Z2 ) )
=> ( ( product_snd_int_int @ X )
= Z2 ) ) ).
% sndI
thf(fact_448_sndI,axiom,
! [X: produc4189061985984568957nt_int,Y: list_list_char,Z2: produc4435102495419491129nt_int] :
( ( X
= ( produc3965054194175396271nt_int @ Y @ Z2 ) )
=> ( ( produc30867750519413249nt_int @ X )
= Z2 ) ) ).
% sndI
thf(fact_449_list_Oinject,axiom,
! [X21: produc6425607678544837394st_int,X222: list_P419547167332244514st_int,Y21: produc6425607678544837394st_int,Y222: list_P419547167332244514st_int] :
( ( ( cons_P4151473878978182108st_int @ X21 @ X222 )
= ( cons_P4151473878978182108st_int @ Y21 @ Y222 ) )
= ( ( X21 = Y21 )
& ( X222 = Y222 ) ) ) ).
% list.inject
thf(fact_450_list_Oinject,axiom,
! [X21: instr,X222: list_instr,Y21: instr,Y222: list_instr] :
( ( ( cons_instr @ X21 @ X222 )
= ( cons_instr @ Y21 @ Y222 ) )
= ( ( X21 = Y21 )
& ( X222 = Y222 ) ) ) ).
% list.inject
thf(fact_451_list_Oinject,axiom,
! [X21: int,X222: list_int,Y21: int,Y222: list_int] :
( ( ( cons_int @ X21 @ X222 )
= ( cons_int @ Y21 @ Y222 ) )
= ( ( X21 = Y21 )
& ( X222 = Y222 ) ) ) ).
% list.inject
thf(fact_452_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_453_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_454_of__nat__le__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% of_nat_le_iff
thf(fact_455_diff__ge__0__iff__ge,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
= ( ord_less_eq_int @ B @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_456_nth__Cons__0,axiom,
! [X: produc6425607678544837394st_int,Xs: list_P419547167332244514st_int] :
( ( nth_Pr8327173284404383691st_int @ ( cons_P4151473878978182108st_int @ X @ Xs ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_457_nth__Cons__0,axiom,
! [X: instr,Xs: list_instr] :
( ( nth_instr @ ( cons_instr @ X @ Xs ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_458_nth__Cons__0,axiom,
! [X: int,Xs: list_int] :
( ( nth_int @ ( cons_int @ X @ Xs ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_459_nth__Cons__Suc,axiom,
! [X: produc6425607678544837394st_int,Xs: list_P419547167332244514st_int,N: nat] :
( ( nth_Pr8327173284404383691st_int @ ( cons_P4151473878978182108st_int @ X @ Xs ) @ ( suc @ N ) )
= ( nth_Pr8327173284404383691st_int @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_460_nth__Cons__Suc,axiom,
! [X: instr,Xs: list_instr,N: nat] :
( ( nth_instr @ ( cons_instr @ X @ Xs ) @ ( suc @ N ) )
= ( nth_instr @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_461_nth__Cons__Suc,axiom,
! [X: int,Xs: list_int,N: nat] :
( ( nth_int @ ( cons_int @ X @ Xs ) @ ( suc @ N ) )
= ( nth_int @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_462_take__Suc__Cons,axiom,
! [N: nat,X: produc6425607678544837394st_int,Xs: list_P419547167332244514st_int] :
( ( take_P1004856038796978596st_int @ ( suc @ N ) @ ( cons_P4151473878978182108st_int @ X @ Xs ) )
= ( cons_P4151473878978182108st_int @ X @ ( take_P1004856038796978596st_int @ N @ Xs ) ) ) ).
% take_Suc_Cons
thf(fact_463_take__Suc__Cons,axiom,
! [N: nat,X: instr,Xs: list_instr] :
( ( take_instr @ ( suc @ N ) @ ( cons_instr @ X @ Xs ) )
= ( cons_instr @ X @ ( take_instr @ N @ Xs ) ) ) ).
% take_Suc_Cons
thf(fact_464_take__Suc__Cons,axiom,
! [N: nat,X: int,Xs: list_int] :
( ( take_int @ ( suc @ N ) @ ( cons_int @ X @ Xs ) )
= ( cons_int @ X @ ( take_int @ N @ Xs ) ) ) ).
% take_Suc_Cons
thf(fact_465_drop__Suc__Cons,axiom,
! [N: nat,X: produc6425607678544837394st_int,Xs: list_P419547167332244514st_int] :
( ( drop_P7838799151754935884st_int @ ( suc @ N ) @ ( cons_P4151473878978182108st_int @ X @ Xs ) )
= ( drop_P7838799151754935884st_int @ N @ Xs ) ) ).
% drop_Suc_Cons
thf(fact_466_drop__Suc__Cons,axiom,
! [N: nat,X: instr,Xs: list_instr] :
( ( drop_instr @ ( suc @ N ) @ ( cons_instr @ X @ Xs ) )
= ( drop_instr @ N @ Xs ) ) ).
% drop_Suc_Cons
thf(fact_467_drop__Suc__Cons,axiom,
! [N: nat,X: int,Xs: list_int] :
( ( drop_int @ ( suc @ N ) @ ( cons_int @ X @ Xs ) )
= ( drop_int @ N @ Xs ) ) ).
% drop_Suc_Cons
thf(fact_468_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_469_of__nat__le__0__iff,axiom,
! [M: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_470_nth__append__length,axiom,
! [Xs: list_int,X: int,Ys: list_int] :
( ( nth_int @ ( append_int @ Xs @ ( cons_int @ X @ Ys ) ) @ ( size_size_list_int @ Xs ) )
= X ) ).
% nth_append_length
thf(fact_471_nth__append__length,axiom,
! [Xs: list_P419547167332244514st_int,X: produc6425607678544837394st_int,Ys: list_P419547167332244514st_int] :
( ( nth_Pr8327173284404383691st_int @ ( append5028292981072010935st_int @ Xs @ ( cons_P4151473878978182108st_int @ X @ Ys ) ) @ ( size_s229347708945616566st_int @ Xs ) )
= X ) ).
% nth_append_length
thf(fact_472_nth__append__length,axiom,
! [Xs: list_instr,X: instr,Ys: list_instr] :
( ( nth_instr @ ( append_instr @ Xs @ ( cons_instr @ X @ Ys ) ) @ ( size_size_list_instr @ Xs ) )
= X ) ).
% nth_append_length
thf(fact_473_nth__append__length,axiom,
! [Xs: list_list_char,X: list_char,Ys: list_list_char] :
( ( nth_list_char @ ( append_list_char @ Xs @ ( cons_list_char @ X @ Ys ) ) @ ( size_s356637359517785349t_char @ Xs ) )
= X ) ).
% nth_append_length
thf(fact_474_zle__diff1__eq,axiom,
! [W: int,Z2: int] :
( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z2 @ one_one_int ) )
= ( ord_less_int @ W @ Z2 ) ) ).
% zle_diff1_eq
thf(fact_475_nth__Cons__pos,axiom,
! [N: nat,X: produc6425607678544837394st_int,Xs: list_P419547167332244514st_int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_Pr8327173284404383691st_int @ ( cons_P4151473878978182108st_int @ X @ Xs ) @ N )
= ( nth_Pr8327173284404383691st_int @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_476_nth__Cons__pos,axiom,
! [N: nat,X: instr,Xs: list_instr] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_instr @ ( cons_instr @ X @ Xs ) @ N )
= ( nth_instr @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_477_nth__Cons__pos,axiom,
! [N: nat,X: int,Xs: list_int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_int @ ( cons_int @ X @ Xs ) @ N )
= ( nth_int @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_478_verit__comp__simplify1_I2_J,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_479_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_480_verit__la__generic,axiom,
! [A: int,X: int] :
( ( ord_less_eq_int @ A @ X )
| ( A = X )
| ( ord_less_eq_int @ X @ A ) ) ).
% verit_la_generic
thf(fact_481_verit__la__disequality,axiom,
! [A: int,B: int] :
( ( A = B )
| ~ ( ord_less_eq_int @ A @ B )
| ~ ( ord_less_eq_int @ B @ A ) ) ).
% verit_la_disequality
thf(fact_482_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_483_not__Cons__self2,axiom,
! [X: produc6425607678544837394st_int,Xs: list_P419547167332244514st_int] :
( ( cons_P4151473878978182108st_int @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_484_not__Cons__self2,axiom,
! [X: instr,Xs: list_instr] :
( ( cons_instr @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_485_not__Cons__self2,axiom,
! [X: int,Xs: list_int] :
( ( cons_int @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_486_of__nat__mono,axiom,
! [I3: nat,J: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I3 ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).
% of_nat_mono
thf(fact_487_of__nat__mono,axiom,
! [I3: nat,J: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I3 ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).
% of_nat_mono
thf(fact_488_lift__Suc__antimono__le,axiom,
! [F: nat > int,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_int @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_489_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_490_lift__Suc__mono__le,axiom,
! [F: nat > int,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_int @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_491_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_492_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_493_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_494_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_495_verit__comp__simplify1_I3_J,axiom,
! [B2: int,A2: int] :
( ( ~ ( ord_less_eq_int @ B2 @ A2 ) )
= ( ord_less_int @ A2 @ B2 ) ) ).
% verit_comp_simplify1(3)
thf(fact_496_verit__comp__simplify1_I3_J,axiom,
! [B2: nat,A2: nat] :
( ( ~ ( ord_less_eq_nat @ B2 @ A2 ) )
= ( ord_less_nat @ A2 @ B2 ) ) ).
% verit_comp_simplify1(3)
thf(fact_497_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_498_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_499_diff__mono,axiom,
! [A: int,B: int,D: int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ D @ C2 )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C2 ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% diff_mono
thf(fact_500_diff__left__mono,axiom,
! [B: int,A: int,C2: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ ( minus_minus_int @ C2 @ A ) @ ( minus_minus_int @ C2 @ B ) ) ) ).
% diff_left_mono
thf(fact_501_diff__right__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C2 ) @ ( minus_minus_int @ B @ C2 ) ) ) ).
% diff_right_mono
thf(fact_502_diff__eq__diff__less__eq,axiom,
! [A: int,B: int,C2: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C2 @ D ) )
=> ( ( ord_less_eq_int @ A @ B )
= ( ord_less_eq_int @ C2 @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_503_list_Osimps_I9_J,axiom,
! [F: produc6425607678544837394st_int > produc6425607678544837394st_int,X21: produc6425607678544837394st_int,X222: list_P419547167332244514st_int] :
( ( map_Pr8625093460839654122st_int @ F @ ( cons_P4151473878978182108st_int @ X21 @ X222 ) )
= ( cons_P4151473878978182108st_int @ ( F @ X21 ) @ ( map_Pr8625093460839654122st_int @ F @ X222 ) ) ) ).
% list.simps(9)
thf(fact_504_list_Osimps_I9_J,axiom,
! [F: produc6425607678544837394st_int > instr,X21: produc6425607678544837394st_int,X222: list_P419547167332244514st_int] :
( ( map_Pr7514435936640407687_instr @ F @ ( cons_P4151473878978182108st_int @ X21 @ X222 ) )
= ( cons_instr @ ( F @ X21 ) @ ( map_Pr7514435936640407687_instr @ F @ X222 ) ) ) ).
% list.simps(9)
thf(fact_505_list_Osimps_I9_J,axiom,
! [F: produc6425607678544837394st_int > int,X21: produc6425607678544837394st_int,X222: list_P419547167332244514st_int] :
( ( map_Pr3224476576807795204nt_int @ F @ ( cons_P4151473878978182108st_int @ X21 @ X222 ) )
= ( cons_int @ ( F @ X21 ) @ ( map_Pr3224476576807795204nt_int @ F @ X222 ) ) ) ).
% list.simps(9)
thf(fact_506_list_Osimps_I9_J,axiom,
! [F: instr > produc6425607678544837394st_int,X21: instr,X222: list_instr] :
( ( map_in7259965606695656109st_int @ F @ ( cons_instr @ X21 @ X222 ) )
= ( cons_P4151473878978182108st_int @ ( F @ X21 ) @ ( map_in7259965606695656109st_int @ F @ X222 ) ) ) ).
% list.simps(9)
thf(fact_507_list_Osimps_I9_J,axiom,
! [F: instr > instr,X21: instr,X222: list_instr] :
( ( map_instr_instr @ F @ ( cons_instr @ X21 @ X222 ) )
= ( cons_instr @ ( F @ X21 ) @ ( map_instr_instr @ F @ X222 ) ) ) ).
% list.simps(9)
thf(fact_508_list_Osimps_I9_J,axiom,
! [F: instr > int,X21: instr,X222: list_instr] :
( ( map_instr_int @ F @ ( cons_instr @ X21 @ X222 ) )
= ( cons_int @ ( F @ X21 ) @ ( map_instr_int @ F @ X222 ) ) ) ).
% list.simps(9)
thf(fact_509_list_Osimps_I9_J,axiom,
! [F: int > produc6425607678544837394st_int,X21: int,X222: list_int] :
( ( map_in8512336359594627332st_int @ F @ ( cons_int @ X21 @ X222 ) )
= ( cons_P4151473878978182108st_int @ ( F @ X21 ) @ ( map_in8512336359594627332st_int @ F @ X222 ) ) ) ).
% list.simps(9)
thf(fact_510_list_Osimps_I9_J,axiom,
! [F: int > instr,X21: int,X222: list_int] :
( ( map_int_instr @ F @ ( cons_int @ X21 @ X222 ) )
= ( cons_instr @ ( F @ X21 ) @ ( map_int_instr @ F @ X222 ) ) ) ).
% list.simps(9)
thf(fact_511_list_Osimps_I9_J,axiom,
! [F: int > int,X21: int,X222: list_int] :
( ( map_int_int @ F @ ( cons_int @ X21 @ X222 ) )
= ( cons_int @ ( F @ X21 ) @ ( map_int_int @ F @ X222 ) ) ) ).
% list.simps(9)
thf(fact_512_Cons__eq__map__D,axiom,
! [X: produc6425607678544837394st_int,Xs: list_P419547167332244514st_int,F: produc6425607678544837394st_int > produc6425607678544837394st_int,Ys: list_P419547167332244514st_int] :
( ( ( cons_P4151473878978182108st_int @ X @ Xs )
= ( map_Pr8625093460839654122st_int @ F @ Ys ) )
=> ? [Z3: produc6425607678544837394st_int,Zs2: list_P419547167332244514st_int] :
( ( Ys
= ( cons_P4151473878978182108st_int @ Z3 @ Zs2 ) )
& ( X
= ( F @ Z3 ) )
& ( Xs
= ( map_Pr8625093460839654122st_int @ F @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_513_Cons__eq__map__D,axiom,
! [X: produc6425607678544837394st_int,Xs: list_P419547167332244514st_int,F: instr > produc6425607678544837394st_int,Ys: list_instr] :
( ( ( cons_P4151473878978182108st_int @ X @ Xs )
= ( map_in7259965606695656109st_int @ F @ Ys ) )
=> ? [Z3: instr,Zs2: list_instr] :
( ( Ys
= ( cons_instr @ Z3 @ Zs2 ) )
& ( X
= ( F @ Z3 ) )
& ( Xs
= ( map_in7259965606695656109st_int @ F @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_514_Cons__eq__map__D,axiom,
! [X: produc6425607678544837394st_int,Xs: list_P419547167332244514st_int,F: int > produc6425607678544837394st_int,Ys: list_int] :
( ( ( cons_P4151473878978182108st_int @ X @ Xs )
= ( map_in8512336359594627332st_int @ F @ Ys ) )
=> ? [Z3: int,Zs2: list_int] :
( ( Ys
= ( cons_int @ Z3 @ Zs2 ) )
& ( X
= ( F @ Z3 ) )
& ( Xs
= ( map_in8512336359594627332st_int @ F @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_515_Cons__eq__map__D,axiom,
! [X: instr,Xs: list_instr,F: produc6425607678544837394st_int > instr,Ys: list_P419547167332244514st_int] :
( ( ( cons_instr @ X @ Xs )
= ( map_Pr7514435936640407687_instr @ F @ Ys ) )
=> ? [Z3: produc6425607678544837394st_int,Zs2: list_P419547167332244514st_int] :
( ( Ys
= ( cons_P4151473878978182108st_int @ Z3 @ Zs2 ) )
& ( X
= ( F @ Z3 ) )
& ( Xs
= ( map_Pr7514435936640407687_instr @ F @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_516_Cons__eq__map__D,axiom,
! [X: instr,Xs: list_instr,F: instr > instr,Ys: list_instr] :
( ( ( cons_instr @ X @ Xs )
= ( map_instr_instr @ F @ Ys ) )
=> ? [Z3: instr,Zs2: list_instr] :
( ( Ys
= ( cons_instr @ Z3 @ Zs2 ) )
& ( X
= ( F @ Z3 ) )
& ( Xs
= ( map_instr_instr @ F @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_517_Cons__eq__map__D,axiom,
! [X: instr,Xs: list_instr,F: int > instr,Ys: list_int] :
( ( ( cons_instr @ X @ Xs )
= ( map_int_instr @ F @ Ys ) )
=> ? [Z3: int,Zs2: list_int] :
( ( Ys
= ( cons_int @ Z3 @ Zs2 ) )
& ( X
= ( F @ Z3 ) )
& ( Xs
= ( map_int_instr @ F @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_518_Cons__eq__map__D,axiom,
! [X: int,Xs: list_int,F: produc6425607678544837394st_int > int,Ys: list_P419547167332244514st_int] :
( ( ( cons_int @ X @ Xs )
= ( map_Pr3224476576807795204nt_int @ F @ Ys ) )
=> ? [Z3: produc6425607678544837394st_int,Zs2: list_P419547167332244514st_int] :
( ( Ys
= ( cons_P4151473878978182108st_int @ Z3 @ Zs2 ) )
& ( X
= ( F @ Z3 ) )
& ( Xs
= ( map_Pr3224476576807795204nt_int @ F @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_519_Cons__eq__map__D,axiom,
! [X: int,Xs: list_int,F: instr > int,Ys: list_instr] :
( ( ( cons_int @ X @ Xs )
= ( map_instr_int @ F @ Ys ) )
=> ? [Z3: instr,Zs2: list_instr] :
( ( Ys
= ( cons_instr @ Z3 @ Zs2 ) )
& ( X
= ( F @ Z3 ) )
& ( Xs
= ( map_instr_int @ F @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_520_Cons__eq__map__D,axiom,
! [X: int,Xs: list_int,F: int > int,Ys: list_int] :
( ( ( cons_int @ X @ Xs )
= ( map_int_int @ F @ Ys ) )
=> ? [Z3: int,Zs2: list_int] :
( ( Ys
= ( cons_int @ Z3 @ Zs2 ) )
& ( X
= ( F @ Z3 ) )
& ( Xs
= ( map_int_int @ F @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_521_map__eq__Cons__D,axiom,
! [F: produc6425607678544837394st_int > produc6425607678544837394st_int,Xs: list_P419547167332244514st_int,Y: produc6425607678544837394st_int,Ys: list_P419547167332244514st_int] :
( ( ( map_Pr8625093460839654122st_int @ F @ Xs )
= ( cons_P4151473878978182108st_int @ Y @ Ys ) )
=> ? [Z3: produc6425607678544837394st_int,Zs2: list_P419547167332244514st_int] :
( ( Xs
= ( cons_P4151473878978182108st_int @ Z3 @ Zs2 ) )
& ( ( F @ Z3 )
= Y )
& ( ( map_Pr8625093460839654122st_int @ F @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_522_map__eq__Cons__D,axiom,
! [F: instr > produc6425607678544837394st_int,Xs: list_instr,Y: produc6425607678544837394st_int,Ys: list_P419547167332244514st_int] :
( ( ( map_in7259965606695656109st_int @ F @ Xs )
= ( cons_P4151473878978182108st_int @ Y @ Ys ) )
=> ? [Z3: instr,Zs2: list_instr] :
( ( Xs
= ( cons_instr @ Z3 @ Zs2 ) )
& ( ( F @ Z3 )
= Y )
& ( ( map_in7259965606695656109st_int @ F @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_523_map__eq__Cons__D,axiom,
! [F: int > produc6425607678544837394st_int,Xs: list_int,Y: produc6425607678544837394st_int,Ys: list_P419547167332244514st_int] :
( ( ( map_in8512336359594627332st_int @ F @ Xs )
= ( cons_P4151473878978182108st_int @ Y @ Ys ) )
=> ? [Z3: int,Zs2: list_int] :
( ( Xs
= ( cons_int @ Z3 @ Zs2 ) )
& ( ( F @ Z3 )
= Y )
& ( ( map_in8512336359594627332st_int @ F @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_524_map__eq__Cons__D,axiom,
! [F: produc6425607678544837394st_int > instr,Xs: list_P419547167332244514st_int,Y: instr,Ys: list_instr] :
( ( ( map_Pr7514435936640407687_instr @ F @ Xs )
= ( cons_instr @ Y @ Ys ) )
=> ? [Z3: produc6425607678544837394st_int,Zs2: list_P419547167332244514st_int] :
( ( Xs
= ( cons_P4151473878978182108st_int @ Z3 @ Zs2 ) )
& ( ( F @ Z3 )
= Y )
& ( ( map_Pr7514435936640407687_instr @ F @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_525_map__eq__Cons__D,axiom,
! [F: instr > instr,Xs: list_instr,Y: instr,Ys: list_instr] :
( ( ( map_instr_instr @ F @ Xs )
= ( cons_instr @ Y @ Ys ) )
=> ? [Z3: instr,Zs2: list_instr] :
( ( Xs
= ( cons_instr @ Z3 @ Zs2 ) )
& ( ( F @ Z3 )
= Y )
& ( ( map_instr_instr @ F @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_526_map__eq__Cons__D,axiom,
! [F: int > instr,Xs: list_int,Y: instr,Ys: list_instr] :
( ( ( map_int_instr @ F @ Xs )
= ( cons_instr @ Y @ Ys ) )
=> ? [Z3: int,Zs2: list_int] :
( ( Xs
= ( cons_int @ Z3 @ Zs2 ) )
& ( ( F @ Z3 )
= Y )
& ( ( map_int_instr @ F @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_527_map__eq__Cons__D,axiom,
! [F: produc6425607678544837394st_int > int,Xs: list_P419547167332244514st_int,Y: int,Ys: list_int] :
( ( ( map_Pr3224476576807795204nt_int @ F @ Xs )
= ( cons_int @ Y @ Ys ) )
=> ? [Z3: produc6425607678544837394st_int,Zs2: list_P419547167332244514st_int] :
( ( Xs
= ( cons_P4151473878978182108st_int @ Z3 @ Zs2 ) )
& ( ( F @ Z3 )
= Y )
& ( ( map_Pr3224476576807795204nt_int @ F @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_528_map__eq__Cons__D,axiom,
! [F: instr > int,Xs: list_instr,Y: int,Ys: list_int] :
( ( ( map_instr_int @ F @ Xs )
= ( cons_int @ Y @ Ys ) )
=> ? [Z3: instr,Zs2: list_instr] :
( ( Xs
= ( cons_instr @ Z3 @ Zs2 ) )
& ( ( F @ Z3 )
= Y )
& ( ( map_instr_int @ F @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_529_map__eq__Cons__D,axiom,
! [F: int > int,Xs: list_int,Y: int,Ys: list_int] :
( ( ( map_int_int @ F @ Xs )
= ( cons_int @ Y @ Ys ) )
=> ? [Z3: int,Zs2: list_int] :
( ( Xs
= ( cons_int @ Z3 @ Zs2 ) )
& ( ( F @ Z3 )
= Y )
& ( ( map_int_int @ F @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_530_Cons__eq__map__conv,axiom,
! [X: produc6425607678544837394st_int,Xs: list_P419547167332244514st_int,F: produc6425607678544837394st_int > produc6425607678544837394st_int,Ys: list_P419547167332244514st_int] :
( ( ( cons_P4151473878978182108st_int @ X @ Xs )
= ( map_Pr8625093460839654122st_int @ F @ Ys ) )
= ( ? [Z4: produc6425607678544837394st_int,Zs3: list_P419547167332244514st_int] :
( ( Ys
= ( cons_P4151473878978182108st_int @ Z4 @ Zs3 ) )
& ( X
= ( F @ Z4 ) )
& ( Xs
= ( map_Pr8625093460839654122st_int @ F @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_531_Cons__eq__map__conv,axiom,
! [X: produc6425607678544837394st_int,Xs: list_P419547167332244514st_int,F: instr > produc6425607678544837394st_int,Ys: list_instr] :
( ( ( cons_P4151473878978182108st_int @ X @ Xs )
= ( map_in7259965606695656109st_int @ F @ Ys ) )
= ( ? [Z4: instr,Zs3: list_instr] :
( ( Ys
= ( cons_instr @ Z4 @ Zs3 ) )
& ( X
= ( F @ Z4 ) )
& ( Xs
= ( map_in7259965606695656109st_int @ F @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_532_Cons__eq__map__conv,axiom,
! [X: produc6425607678544837394st_int,Xs: list_P419547167332244514st_int,F: int > produc6425607678544837394st_int,Ys: list_int] :
( ( ( cons_P4151473878978182108st_int @ X @ Xs )
= ( map_in8512336359594627332st_int @ F @ Ys ) )
= ( ? [Z4: int,Zs3: list_int] :
( ( Ys
= ( cons_int @ Z4 @ Zs3 ) )
& ( X
= ( F @ Z4 ) )
& ( Xs
= ( map_in8512336359594627332st_int @ F @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_533_Cons__eq__map__conv,axiom,
! [X: instr,Xs: list_instr,F: produc6425607678544837394st_int > instr,Ys: list_P419547167332244514st_int] :
( ( ( cons_instr @ X @ Xs )
= ( map_Pr7514435936640407687_instr @ F @ Ys ) )
= ( ? [Z4: produc6425607678544837394st_int,Zs3: list_P419547167332244514st_int] :
( ( Ys
= ( cons_P4151473878978182108st_int @ Z4 @ Zs3 ) )
& ( X
= ( F @ Z4 ) )
& ( Xs
= ( map_Pr7514435936640407687_instr @ F @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_534_Cons__eq__map__conv,axiom,
! [X: instr,Xs: list_instr,F: instr > instr,Ys: list_instr] :
( ( ( cons_instr @ X @ Xs )
= ( map_instr_instr @ F @ Ys ) )
= ( ? [Z4: instr,Zs3: list_instr] :
( ( Ys
= ( cons_instr @ Z4 @ Zs3 ) )
& ( X
= ( F @ Z4 ) )
& ( Xs
= ( map_instr_instr @ F @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_535_Cons__eq__map__conv,axiom,
! [X: instr,Xs: list_instr,F: int > instr,Ys: list_int] :
( ( ( cons_instr @ X @ Xs )
= ( map_int_instr @ F @ Ys ) )
= ( ? [Z4: int,Zs3: list_int] :
( ( Ys
= ( cons_int @ Z4 @ Zs3 ) )
& ( X
= ( F @ Z4 ) )
& ( Xs
= ( map_int_instr @ F @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_536_Cons__eq__map__conv,axiom,
! [X: int,Xs: list_int,F: produc6425607678544837394st_int > int,Ys: list_P419547167332244514st_int] :
( ( ( cons_int @ X @ Xs )
= ( map_Pr3224476576807795204nt_int @ F @ Ys ) )
= ( ? [Z4: produc6425607678544837394st_int,Zs3: list_P419547167332244514st_int] :
( ( Ys
= ( cons_P4151473878978182108st_int @ Z4 @ Zs3 ) )
& ( X
= ( F @ Z4 ) )
& ( Xs
= ( map_Pr3224476576807795204nt_int @ F @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_537_Cons__eq__map__conv,axiom,
! [X: int,Xs: list_int,F: instr > int,Ys: list_instr] :
( ( ( cons_int @ X @ Xs )
= ( map_instr_int @ F @ Ys ) )
= ( ? [Z4: instr,Zs3: list_instr] :
( ( Ys
= ( cons_instr @ Z4 @ Zs3 ) )
& ( X
= ( F @ Z4 ) )
& ( Xs
= ( map_instr_int @ F @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_538_Cons__eq__map__conv,axiom,
! [X: int,Xs: list_int,F: int > int,Ys: list_int] :
( ( ( cons_int @ X @ Xs )
= ( map_int_int @ F @ Ys ) )
= ( ? [Z4: int,Zs3: list_int] :
( ( Ys
= ( cons_int @ Z4 @ Zs3 ) )
& ( X
= ( F @ Z4 ) )
& ( Xs
= ( map_int_int @ F @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_539_map__eq__Cons__conv,axiom,
! [F: produc6425607678544837394st_int > produc6425607678544837394st_int,Xs: list_P419547167332244514st_int,Y: produc6425607678544837394st_int,Ys: list_P419547167332244514st_int] :
( ( ( map_Pr8625093460839654122st_int @ F @ Xs )
= ( cons_P4151473878978182108st_int @ Y @ Ys ) )
= ( ? [Z4: produc6425607678544837394st_int,Zs3: list_P419547167332244514st_int] :
( ( Xs
= ( cons_P4151473878978182108st_int @ Z4 @ Zs3 ) )
& ( ( F @ Z4 )
= Y )
& ( ( map_Pr8625093460839654122st_int @ F @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_540_map__eq__Cons__conv,axiom,
! [F: instr > produc6425607678544837394st_int,Xs: list_instr,Y: produc6425607678544837394st_int,Ys: list_P419547167332244514st_int] :
( ( ( map_in7259965606695656109st_int @ F @ Xs )
= ( cons_P4151473878978182108st_int @ Y @ Ys ) )
= ( ? [Z4: instr,Zs3: list_instr] :
( ( Xs
= ( cons_instr @ Z4 @ Zs3 ) )
& ( ( F @ Z4 )
= Y )
& ( ( map_in7259965606695656109st_int @ F @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_541_map__eq__Cons__conv,axiom,
! [F: int > produc6425607678544837394st_int,Xs: list_int,Y: produc6425607678544837394st_int,Ys: list_P419547167332244514st_int] :
( ( ( map_in8512336359594627332st_int @ F @ Xs )
= ( cons_P4151473878978182108st_int @ Y @ Ys ) )
= ( ? [Z4: int,Zs3: list_int] :
( ( Xs
= ( cons_int @ Z4 @ Zs3 ) )
& ( ( F @ Z4 )
= Y )
& ( ( map_in8512336359594627332st_int @ F @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_542_map__eq__Cons__conv,axiom,
! [F: produc6425607678544837394st_int > instr,Xs: list_P419547167332244514st_int,Y: instr,Ys: list_instr] :
( ( ( map_Pr7514435936640407687_instr @ F @ Xs )
= ( cons_instr @ Y @ Ys ) )
= ( ? [Z4: produc6425607678544837394st_int,Zs3: list_P419547167332244514st_int] :
( ( Xs
= ( cons_P4151473878978182108st_int @ Z4 @ Zs3 ) )
& ( ( F @ Z4 )
= Y )
& ( ( map_Pr7514435936640407687_instr @ F @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_543_map__eq__Cons__conv,axiom,
! [F: instr > instr,Xs: list_instr,Y: instr,Ys: list_instr] :
( ( ( map_instr_instr @ F @ Xs )
= ( cons_instr @ Y @ Ys ) )
= ( ? [Z4: instr,Zs3: list_instr] :
( ( Xs
= ( cons_instr @ Z4 @ Zs3 ) )
& ( ( F @ Z4 )
= Y )
& ( ( map_instr_instr @ F @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_544_map__eq__Cons__conv,axiom,
! [F: int > instr,Xs: list_int,Y: instr,Ys: list_instr] :
( ( ( map_int_instr @ F @ Xs )
= ( cons_instr @ Y @ Ys ) )
= ( ? [Z4: int,Zs3: list_int] :
( ( Xs
= ( cons_int @ Z4 @ Zs3 ) )
& ( ( F @ Z4 )
= Y )
& ( ( map_int_instr @ F @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_545_map__eq__Cons__conv,axiom,
! [F: produc6425607678544837394st_int > int,Xs: list_P419547167332244514st_int,Y: int,Ys: list_int] :
( ( ( map_Pr3224476576807795204nt_int @ F @ Xs )
= ( cons_int @ Y @ Ys ) )
= ( ? [Z4: produc6425607678544837394st_int,Zs3: list_P419547167332244514st_int] :
( ( Xs
= ( cons_P4151473878978182108st_int @ Z4 @ Zs3 ) )
& ( ( F @ Z4 )
= Y )
& ( ( map_Pr3224476576807795204nt_int @ F @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_546_map__eq__Cons__conv,axiom,
! [F: instr > int,Xs: list_instr,Y: int,Ys: list_int] :
( ( ( map_instr_int @ F @ Xs )
= ( cons_int @ Y @ Ys ) )
= ( ? [Z4: instr,Zs3: list_instr] :
( ( Xs
= ( cons_instr @ Z4 @ Zs3 ) )
& ( ( F @ Z4 )
= Y )
& ( ( map_instr_int @ F @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_547_map__eq__Cons__conv,axiom,
! [F: int > int,Xs: list_int,Y: int,Ys: list_int] :
( ( ( map_int_int @ F @ Xs )
= ( cons_int @ Y @ Ys ) )
= ( ? [Z4: int,Zs3: list_int] :
( ( Xs
= ( cons_int @ Z4 @ Zs3 ) )
& ( ( F @ Z4 )
= Y )
& ( ( map_int_int @ F @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_548_append__Cons,axiom,
! [X: produc6425607678544837394st_int,Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int] :
( ( append5028292981072010935st_int @ ( cons_P4151473878978182108st_int @ X @ Xs ) @ Ys )
= ( cons_P4151473878978182108st_int @ X @ ( append5028292981072010935st_int @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_549_append__Cons,axiom,
! [X: instr,Xs: list_instr,Ys: list_instr] :
( ( append_instr @ ( cons_instr @ X @ Xs ) @ Ys )
= ( cons_instr @ X @ ( append_instr @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_550_append__Cons,axiom,
! [X: int,Xs: list_int,Ys: list_int] :
( ( append_int @ ( cons_int @ X @ Xs ) @ Ys )
= ( cons_int @ X @ ( append_int @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_551_Cons__eq__appendI,axiom,
! [X: produc6425607678544837394st_int,Xs1: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int,Xs: list_P419547167332244514st_int,Zs: list_P419547167332244514st_int] :
( ( ( cons_P4151473878978182108st_int @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append5028292981072010935st_int @ Xs1 @ Zs ) )
=> ( ( cons_P4151473878978182108st_int @ X @ Xs )
= ( append5028292981072010935st_int @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_552_Cons__eq__appendI,axiom,
! [X: instr,Xs1: list_instr,Ys: list_instr,Xs: list_instr,Zs: list_instr] :
( ( ( cons_instr @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_instr @ Xs1 @ Zs ) )
=> ( ( cons_instr @ X @ Xs )
= ( append_instr @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_553_Cons__eq__appendI,axiom,
! [X: int,Xs1: list_int,Ys: list_int,Xs: list_int,Zs: list_int] :
( ( ( cons_int @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_int @ Xs1 @ Zs ) )
=> ( ( cons_int @ X @ Xs )
= ( append_int @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_554_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_555_int__le__induct,axiom,
! [I3: int,K: int,P: int > $o] :
( ( ord_less_eq_int @ I3 @ K )
=> ( ( P @ K )
=> ( ! [I: int] :
( ( ord_less_eq_int @ I @ K )
=> ( ( P @ I )
=> ( P @ ( minus_minus_int @ I @ one_one_int ) ) ) )
=> ( P @ I3 ) ) ) ) ).
% int_le_induct
thf(fact_556_execl__take,axiom,
! [P: list_instr,Cfs2: list_P419547167332244514st_int,N: nat] :
( ( execl @ P @ Cfs2 )
=> ( execl @ P @ ( take_P1004856038796978596st_int @ N @ Cfs2 ) ) ) ).
% execl_take
thf(fact_557_execl__drop,axiom,
! [P: list_instr,Cfs2: list_P419547167332244514st_int,N: nat] :
( ( execl @ P @ Cfs2 )
=> ( execl @ P @ ( drop_P7838799151754935884st_int @ N @ Cfs2 ) ) ) ).
% execl_drop
thf(fact_558_Suc__length__conv,axiom,
! [N: nat,Xs: list_int] :
( ( ( suc @ N )
= ( size_size_list_int @ Xs ) )
= ( ? [Y4: int,Ys2: list_int] :
( ( Xs
= ( cons_int @ Y4 @ Ys2 ) )
& ( ( size_size_list_int @ Ys2 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_559_Suc__length__conv,axiom,
! [N: nat,Xs: list_P419547167332244514st_int] :
( ( ( suc @ N )
= ( size_s229347708945616566st_int @ Xs ) )
= ( ? [Y4: produc6425607678544837394st_int,Ys2: list_P419547167332244514st_int] :
( ( Xs
= ( cons_P4151473878978182108st_int @ Y4 @ Ys2 ) )
& ( ( size_s229347708945616566st_int @ Ys2 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_560_Suc__length__conv,axiom,
! [N: nat,Xs: list_instr] :
( ( ( suc @ N )
= ( size_size_list_instr @ Xs ) )
= ( ? [Y4: instr,Ys2: list_instr] :
( ( Xs
= ( cons_instr @ Y4 @ Ys2 ) )
& ( ( size_size_list_instr @ Ys2 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_561_Suc__length__conv,axiom,
! [N: nat,Xs: list_list_char] :
( ( ( suc @ N )
= ( size_s356637359517785349t_char @ Xs ) )
= ( ? [Y4: list_char,Ys2: list_list_char] :
( ( Xs
= ( cons_list_char @ Y4 @ Ys2 ) )
& ( ( size_s356637359517785349t_char @ Ys2 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_562_length__Suc__conv,axiom,
! [Xs: list_int,N: nat] :
( ( ( size_size_list_int @ Xs )
= ( suc @ N ) )
= ( ? [Y4: int,Ys2: list_int] :
( ( Xs
= ( cons_int @ Y4 @ Ys2 ) )
& ( ( size_size_list_int @ Ys2 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_563_length__Suc__conv,axiom,
! [Xs: list_P419547167332244514st_int,N: nat] :
( ( ( size_s229347708945616566st_int @ Xs )
= ( suc @ N ) )
= ( ? [Y4: produc6425607678544837394st_int,Ys2: list_P419547167332244514st_int] :
( ( Xs
= ( cons_P4151473878978182108st_int @ Y4 @ Ys2 ) )
& ( ( size_s229347708945616566st_int @ Ys2 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_564_length__Suc__conv,axiom,
! [Xs: list_instr,N: nat] :
( ( ( size_size_list_instr @ Xs )
= ( suc @ N ) )
= ( ? [Y4: instr,Ys2: list_instr] :
( ( Xs
= ( cons_instr @ Y4 @ Ys2 ) )
& ( ( size_size_list_instr @ Ys2 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_565_length__Suc__conv,axiom,
! [Xs: list_list_char,N: nat] :
( ( ( size_s356637359517785349t_char @ Xs )
= ( suc @ N ) )
= ( ? [Y4: list_char,Ys2: list_list_char] :
( ( Xs
= ( cons_list_char @ Y4 @ Ys2 ) )
& ( ( size_s356637359517785349t_char @ Ys2 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_566_length__Cons,axiom,
! [X: int,Xs: list_int] :
( ( size_size_list_int @ ( cons_int @ X @ Xs ) )
= ( suc @ ( size_size_list_int @ Xs ) ) ) ).
% length_Cons
thf(fact_567_length__Cons,axiom,
! [X: produc6425607678544837394st_int,Xs: list_P419547167332244514st_int] :
( ( size_s229347708945616566st_int @ ( cons_P4151473878978182108st_int @ X @ Xs ) )
= ( suc @ ( size_s229347708945616566st_int @ Xs ) ) ) ).
% length_Cons
thf(fact_568_length__Cons,axiom,
! [X: instr,Xs: list_instr] :
( ( size_size_list_instr @ ( cons_instr @ X @ Xs ) )
= ( suc @ ( size_size_list_instr @ Xs ) ) ) ).
% length_Cons
thf(fact_569_length__Cons,axiom,
! [X: list_char,Xs: list_list_char] :
( ( size_s356637359517785349t_char @ ( cons_list_char @ X @ Xs ) )
= ( suc @ ( size_s356637359517785349t_char @ Xs ) ) ) ).
% length_Cons
thf(fact_570_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_571_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_572_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_573_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_574_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% zero_less_one_class.zero_le_one
thf(fact_575_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_576_le__iff__diff__le__0,axiom,
( ord_less_eq_int
= ( ^ [A5: int,B4: int] : ( ord_less_eq_int @ ( minus_minus_int @ A5 @ B4 ) @ zero_zero_int ) ) ) ).
% le_iff_diff_le_0
thf(fact_577_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) ) ).
% of_nat_0_le_iff
thf(fact_578_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) ) ).
% of_nat_0_le_iff
thf(fact_579_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ~ ! [N2: nat] :
( K
!= ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% nonneg_int_cases
thf(fact_580_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ? [N2: nat] :
( K
= ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_581_int__one__le__iff__zero__less,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ one_one_int @ Z2 )
= ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% int_one_le_iff_zero_less
thf(fact_582_nth__via__drop,axiom,
! [N: nat,Xs: list_P419547167332244514st_int,Y: produc6425607678544837394st_int,Ys: list_P419547167332244514st_int] :
( ( ( drop_P7838799151754935884st_int @ N @ Xs )
= ( cons_P4151473878978182108st_int @ Y @ Ys ) )
=> ( ( nth_Pr8327173284404383691st_int @ Xs @ N )
= Y ) ) ).
% nth_via_drop
thf(fact_583_nth__via__drop,axiom,
! [N: nat,Xs: list_instr,Y: instr,Ys: list_instr] :
( ( ( drop_instr @ N @ Xs )
= ( cons_instr @ Y @ Ys ) )
=> ( ( nth_instr @ Xs @ N )
= Y ) ) ).
% nth_via_drop
thf(fact_584_nth__via__drop,axiom,
! [N: nat,Xs: list_int,Y: int,Ys: list_int] :
( ( ( drop_int @ N @ Xs )
= ( cons_int @ Y @ Ys ) )
=> ( ( nth_int @ Xs @ N )
= Y ) ) ).
% nth_via_drop
thf(fact_585_nth__Cons_H,axiom,
! [N: nat,X: produc6425607678544837394st_int,Xs: list_P419547167332244514st_int] :
( ( ( N = zero_zero_nat )
=> ( ( nth_Pr8327173284404383691st_int @ ( cons_P4151473878978182108st_int @ X @ Xs ) @ N )
= X ) )
& ( ( N != zero_zero_nat )
=> ( ( nth_Pr8327173284404383691st_int @ ( cons_P4151473878978182108st_int @ X @ Xs ) @ N )
= ( nth_Pr8327173284404383691st_int @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_586_nth__Cons_H,axiom,
! [N: nat,X: instr,Xs: list_instr] :
( ( ( N = zero_zero_nat )
=> ( ( nth_instr @ ( cons_instr @ X @ Xs ) @ N )
= X ) )
& ( ( N != zero_zero_nat )
=> ( ( nth_instr @ ( cons_instr @ X @ Xs ) @ N )
= ( nth_instr @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_587_nth__Cons_H,axiom,
! [N: nat,X: int,Xs: list_int] :
( ( ( N = zero_zero_nat )
=> ( ( nth_int @ ( cons_int @ X @ Xs ) @ N )
= X ) )
& ( ( N != zero_zero_nat )
=> ( ( nth_int @ ( cons_int @ X @ Xs ) @ N )
= ( nth_int @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_588_drop__Cons_H,axiom,
! [N: nat,X: produc6425607678544837394st_int,Xs: list_P419547167332244514st_int] :
( ( ( N = zero_zero_nat )
=> ( ( drop_P7838799151754935884st_int @ N @ ( cons_P4151473878978182108st_int @ X @ Xs ) )
= ( cons_P4151473878978182108st_int @ X @ Xs ) ) )
& ( ( N != zero_zero_nat )
=> ( ( drop_P7838799151754935884st_int @ N @ ( cons_P4151473878978182108st_int @ X @ Xs ) )
= ( drop_P7838799151754935884st_int @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs ) ) ) ) ).
% drop_Cons'
thf(fact_589_drop__Cons_H,axiom,
! [N: nat,X: instr,Xs: list_instr] :
( ( ( N = zero_zero_nat )
=> ( ( drop_instr @ N @ ( cons_instr @ X @ Xs ) )
= ( cons_instr @ X @ Xs ) ) )
& ( ( N != zero_zero_nat )
=> ( ( drop_instr @ N @ ( cons_instr @ X @ Xs ) )
= ( drop_instr @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs ) ) ) ) ).
% drop_Cons'
thf(fact_590_drop__Cons_H,axiom,
! [N: nat,X: int,Xs: list_int] :
( ( ( N = zero_zero_nat )
=> ( ( drop_int @ N @ ( cons_int @ X @ Xs ) )
= ( cons_int @ X @ Xs ) ) )
& ( ( N != zero_zero_nat )
=> ( ( drop_int @ N @ ( cons_int @ X @ Xs ) )
= ( drop_int @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs ) ) ) ) ).
% drop_Cons'
thf(fact_591_nth__non__equal__first__eq,axiom,
! [X: produc6425607678544837394st_int,Y: produc6425607678544837394st_int,Xs: list_P419547167332244514st_int,N: nat] :
( ( X != Y )
=> ( ( ( nth_Pr8327173284404383691st_int @ ( cons_P4151473878978182108st_int @ X @ Xs ) @ N )
= Y )
= ( ( ( nth_Pr8327173284404383691st_int @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
= Y )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_592_nth__non__equal__first__eq,axiom,
! [X: instr,Y: instr,Xs: list_instr,N: nat] :
( ( X != Y )
=> ( ( ( nth_instr @ ( cons_instr @ X @ Xs ) @ N )
= Y )
= ( ( ( nth_instr @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
= Y )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_593_nth__non__equal__first__eq,axiom,
! [X: int,Y: int,Xs: list_int,N: nat] :
( ( X != Y )
=> ( ( ( nth_int @ ( cons_int @ X @ Xs ) @ N )
= Y )
= ( ( ( nth_int @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
= Y )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_594_Cons__nth__drop__Suc,axiom,
! [I3: nat,Xs: list_int] :
( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs ) )
=> ( ( cons_int @ ( nth_int @ Xs @ I3 ) @ ( drop_int @ ( suc @ I3 ) @ Xs ) )
= ( drop_int @ I3 @ Xs ) ) ) ).
% Cons_nth_drop_Suc
thf(fact_595_Cons__nth__drop__Suc,axiom,
! [I3: nat,Xs: list_P419547167332244514st_int] :
( ( ord_less_nat @ I3 @ ( size_s229347708945616566st_int @ Xs ) )
=> ( ( cons_P4151473878978182108st_int @ ( nth_Pr8327173284404383691st_int @ Xs @ I3 ) @ ( drop_P7838799151754935884st_int @ ( suc @ I3 ) @ Xs ) )
= ( drop_P7838799151754935884st_int @ I3 @ Xs ) ) ) ).
% Cons_nth_drop_Suc
thf(fact_596_Cons__nth__drop__Suc,axiom,
! [I3: nat,Xs: list_instr] :
( ( ord_less_nat @ I3 @ ( size_size_list_instr @ Xs ) )
=> ( ( cons_instr @ ( nth_instr @ Xs @ I3 ) @ ( drop_instr @ ( suc @ I3 ) @ Xs ) )
= ( drop_instr @ I3 @ Xs ) ) ) ).
% Cons_nth_drop_Suc
thf(fact_597_Cons__nth__drop__Suc,axiom,
! [I3: nat,Xs: list_list_char] :
( ( ord_less_nat @ I3 @ ( size_s356637359517785349t_char @ Xs ) )
=> ( ( cons_list_char @ ( nth_list_char @ Xs @ I3 ) @ ( drop_list_char @ ( suc @ I3 ) @ Xs ) )
= ( drop_list_char @ I3 @ Xs ) ) ) ).
% Cons_nth_drop_Suc
thf(fact_598_fstI,axiom,
! [X: produc6425607678544837394st_int,Y: int,Z2: produc369741286924889929st_int] :
( ( X
= ( produc5086643055186798020st_int @ Y @ Z2 ) )
=> ( ( produc8941309049508465880st_int @ X )
= Y ) ) ).
% fstI
thf(fact_599_fstI,axiom,
! [X: produc4435102495419491129nt_int,Y: list_char > int,Z2: int] :
( ( X
= ( produc5790713362662368625nt_int @ Y @ Z2 ) )
=> ( ( produc9025023058967219549nt_int @ X )
= Y ) ) ).
% fstI
thf(fact_600_fstI,axiom,
! [X: produc369741286924889929st_int,Y: list_char > int,Z2: list_int] :
( ( X
= ( produc8650753666468850689st_int @ Y @ Z2 ) )
=> ( ( produc2155683374303800813st_int @ X )
= Y ) ) ).
% fstI
thf(fact_601_fstI,axiom,
! [X: product_prod_int_int,Y: int,Z2: int] :
( ( X
= ( product_Pair_int_int @ Y @ Z2 ) )
=> ( ( product_fst_int_int @ X )
= Y ) ) ).
% fstI
thf(fact_602_fstI,axiom,
! [X: produc4189061985984568957nt_int,Y: list_list_char,Z2: produc4435102495419491129nt_int] :
( ( X
= ( produc3965054194175396271nt_int @ Y @ Z2 ) )
=> ( ( produc811350159336358595nt_int @ X )
= Y ) ) ).
% fstI
thf(fact_603_eq__fst__iff,axiom,
! [A: int,P2: produc6425607678544837394st_int] :
( ( A
= ( produc8941309049508465880st_int @ P2 ) )
= ( ? [B4: produc369741286924889929st_int] :
( P2
= ( produc5086643055186798020st_int @ A @ B4 ) ) ) ) ).
% eq_fst_iff
thf(fact_604_eq__fst__iff,axiom,
! [A: list_char > int,P2: produc4435102495419491129nt_int] :
( ( A
= ( produc9025023058967219549nt_int @ P2 ) )
= ( ? [B4: int] :
( P2
= ( produc5790713362662368625nt_int @ A @ B4 ) ) ) ) ).
% eq_fst_iff
thf(fact_605_eq__fst__iff,axiom,
! [A: list_char > int,P2: produc369741286924889929st_int] :
( ( A
= ( produc2155683374303800813st_int @ P2 ) )
= ( ? [B4: list_int] :
( P2
= ( produc8650753666468850689st_int @ A @ B4 ) ) ) ) ).
% eq_fst_iff
thf(fact_606_eq__fst__iff,axiom,
! [A: int,P2: product_prod_int_int] :
( ( A
= ( product_fst_int_int @ P2 ) )
= ( ? [B4: int] :
( P2
= ( product_Pair_int_int @ A @ B4 ) ) ) ) ).
% eq_fst_iff
thf(fact_607_eq__fst__iff,axiom,
! [A: list_list_char,P2: produc4189061985984568957nt_int] :
( ( A
= ( produc811350159336358595nt_int @ P2 ) )
= ( ? [B4: produc4435102495419491129nt_int] :
( P2
= ( produc3965054194175396271nt_int @ A @ B4 ) ) ) ) ).
% eq_fst_iff
thf(fact_608_atLeastLessThan__iff,axiom,
! [I3: nat,L: nat,U: nat] :
( ( member_nat @ I3 @ ( set_or4665077453230672383an_nat @ L @ U ) )
= ( ( ord_less_eq_nat @ L @ I3 )
& ( ord_less_nat @ I3 @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_609_atLeastLessThan__iff,axiom,
! [I3: int,L: int,U: int] :
( ( member_int @ I3 @ ( set_or4662586982721622107an_int @ L @ U ) )
= ( ( ord_less_eq_int @ L @ I3 )
& ( ord_less_int @ I3 @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_610_inth__append,axiom,
! [I3: int,Xs: list_int,Ys: list_int] :
( ( ord_less_eq_int @ zero_zero_int @ I3 )
=> ( ( ( ord_less_int @ I3 @ ( semiri1314217659103216013at_int @ ( size_size_list_int @ Xs ) ) )
=> ( ( inth_int @ ( append_int @ Xs @ Ys ) @ I3 )
= ( inth_int @ Xs @ I3 ) ) )
& ( ~ ( ord_less_int @ I3 @ ( semiri1314217659103216013at_int @ ( size_size_list_int @ Xs ) ) )
=> ( ( inth_int @ ( append_int @ Xs @ Ys ) @ I3 )
= ( inth_int @ Ys @ ( minus_minus_int @ I3 @ ( semiri1314217659103216013at_int @ ( size_size_list_int @ Xs ) ) ) ) ) ) ) ) ).
% inth_append
thf(fact_611_inth__append,axiom,
! [I3: int,Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int] :
( ( ord_less_eq_int @ zero_zero_int @ I3 )
=> ( ( ( ord_less_int @ I3 @ ( semiri1314217659103216013at_int @ ( size_s229347708945616566st_int @ Xs ) ) )
=> ( ( inth_P8039235741264645271st_int @ ( append5028292981072010935st_int @ Xs @ Ys ) @ I3 )
= ( inth_P8039235741264645271st_int @ Xs @ I3 ) ) )
& ( ~ ( ord_less_int @ I3 @ ( semiri1314217659103216013at_int @ ( size_s229347708945616566st_int @ Xs ) ) )
=> ( ( inth_P8039235741264645271st_int @ ( append5028292981072010935st_int @ Xs @ Ys ) @ I3 )
= ( inth_P8039235741264645271st_int @ Ys @ ( minus_minus_int @ I3 @ ( semiri1314217659103216013at_int @ ( size_s229347708945616566st_int @ Xs ) ) ) ) ) ) ) ) ).
% inth_append
thf(fact_612_inth__append,axiom,
! [I3: int,Xs: list_instr,Ys: list_instr] :
( ( ord_less_eq_int @ zero_zero_int @ I3 )
=> ( ( ( ord_less_int @ I3 @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ Xs ) ) )
=> ( ( inth_instr @ ( append_instr @ Xs @ Ys ) @ I3 )
= ( inth_instr @ Xs @ I3 ) ) )
& ( ~ ( ord_less_int @ I3 @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ Xs ) ) )
=> ( ( inth_instr @ ( append_instr @ Xs @ Ys ) @ I3 )
= ( inth_instr @ Ys @ ( minus_minus_int @ I3 @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ Xs ) ) ) ) ) ) ) ) ).
% inth_append
thf(fact_613_inth__append,axiom,
! [I3: int,Xs: list_list_char,Ys: list_list_char] :
( ( ord_less_eq_int @ zero_zero_int @ I3 )
=> ( ( ( ord_less_int @ I3 @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ Xs ) ) )
=> ( ( inth_list_char @ ( append_list_char @ Xs @ Ys ) @ I3 )
= ( inth_list_char @ Xs @ I3 ) ) )
& ( ~ ( ord_less_int @ I3 @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ Xs ) ) )
=> ( ( inth_list_char @ ( append_list_char @ Xs @ Ys ) @ I3 )
= ( inth_list_char @ Ys @ ( minus_minus_int @ I3 @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ Xs ) ) ) ) ) ) ) ) ).
% inth_append
thf(fact_614_inth__map,axiom,
! [I3: int,Xs: list_P419547167332244514st_int,F: produc6425607678544837394st_int > instr] :
( ( ord_less_eq_int @ zero_zero_int @ I3 )
=> ( ( ord_less_int @ I3 @ ( semiri1314217659103216013at_int @ ( size_s229347708945616566st_int @ Xs ) ) )
=> ( ( inth_instr @ ( map_Pr7514435936640407687_instr @ F @ Xs ) @ I3 )
= ( F @ ( inth_P8039235741264645271st_int @ Xs @ I3 ) ) ) ) ) ).
% inth_map
thf(fact_615_inth__map,axiom,
! [I3: int,Xs: list_P419547167332244514st_int,F: produc6425607678544837394st_int > list_char] :
( ( ord_less_eq_int @ zero_zero_int @ I3 )
=> ( ( ord_less_int @ I3 @ ( semiri1314217659103216013at_int @ ( size_s229347708945616566st_int @ Xs ) ) )
=> ( ( inth_list_char @ ( map_Pr5338895242362685753t_char @ F @ Xs ) @ I3 )
= ( F @ ( inth_P8039235741264645271st_int @ Xs @ I3 ) ) ) ) ) ).
% inth_map
thf(fact_616_inth__map,axiom,
! [I3: int,Xs: list_P419547167332244514st_int,F: produc6425607678544837394st_int > produc6425607678544837394st_int] :
( ( ord_less_eq_int @ zero_zero_int @ I3 )
=> ( ( ord_less_int @ I3 @ ( semiri1314217659103216013at_int @ ( size_s229347708945616566st_int @ Xs ) ) )
=> ( ( inth_P8039235741264645271st_int @ ( map_Pr8625093460839654122st_int @ F @ Xs ) @ I3 )
= ( F @ ( inth_P8039235741264645271st_int @ Xs @ I3 ) ) ) ) ) ).
% inth_map
thf(fact_617_inth__map,axiom,
! [I3: int,Xs: list_instr,F: instr > instr] :
( ( ord_less_eq_int @ zero_zero_int @ I3 )
=> ( ( ord_less_int @ I3 @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ Xs ) ) )
=> ( ( inth_instr @ ( map_instr_instr @ F @ Xs ) @ I3 )
= ( F @ ( inth_instr @ Xs @ I3 ) ) ) ) ) ).
% inth_map
thf(fact_618_inth__map,axiom,
! [I3: int,Xs: list_instr,F: instr > list_char] :
( ( ord_less_eq_int @ zero_zero_int @ I3 )
=> ( ( ord_less_int @ I3 @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ Xs ) ) )
=> ( ( inth_list_char @ ( map_instr_list_char @ F @ Xs ) @ I3 )
= ( F @ ( inth_instr @ Xs @ I3 ) ) ) ) ) ).
% inth_map
thf(fact_619_inth__map,axiom,
! [I3: int,Xs: list_list_char,F: list_char > instr] :
( ( ord_less_eq_int @ zero_zero_int @ I3 )
=> ( ( ord_less_int @ I3 @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ Xs ) ) )
=> ( ( inth_instr @ ( map_list_char_instr @ F @ Xs ) @ I3 )
= ( F @ ( inth_list_char @ Xs @ I3 ) ) ) ) ) ).
% inth_map
thf(fact_620_inth__map,axiom,
! [I3: int,Xs: list_list_char,F: list_char > list_char] :
( ( ord_less_eq_int @ zero_zero_int @ I3 )
=> ( ( ord_less_int @ I3 @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ Xs ) ) )
=> ( ( inth_list_char @ ( map_li116305933131242120t_char @ F @ Xs ) @ I3 )
= ( F @ ( inth_list_char @ Xs @ I3 ) ) ) ) ) ).
% inth_map
thf(fact_621_ivl__diff,axiom,
! [I3: nat,N: nat,M: nat] :
( ( ord_less_eq_nat @ I3 @ N )
=> ( ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ I3 @ M ) @ ( set_or4665077453230672383an_nat @ I3 @ N ) )
= ( set_or4665077453230672383an_nat @ N @ M ) ) ) ).
% ivl_diff
thf(fact_622_ivl__diff,axiom,
! [I3: int,N: int,M: int] :
( ( ord_less_eq_int @ I3 @ N )
=> ( ( minus_minus_set_int @ ( set_or4662586982721622107an_int @ I3 @ M ) @ ( set_or4662586982721622107an_int @ I3 @ N ) )
= ( set_or4662586982721622107an_int @ N @ M ) ) ) ).
% ivl_diff
thf(fact_623_ivl__subset,axiom,
! [I3: nat,J: nat,M: nat,N: nat] :
( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ I3 @ J ) @ ( set_or4665077453230672383an_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ J @ I3 )
| ( ( ord_less_eq_nat @ M @ I3 )
& ( ord_less_eq_nat @ J @ N ) ) ) ) ).
% ivl_subset
thf(fact_624_ivl__subset,axiom,
! [I3: int,J: int,M: int,N: int] :
( ( ord_less_eq_set_int @ ( set_or4662586982721622107an_int @ I3 @ J ) @ ( set_or4662586982721622107an_int @ M @ N ) )
= ( ( ord_less_eq_int @ J @ I3 )
| ( ( ord_less_eq_int @ M @ I3 )
& ( ord_less_eq_int @ J @ N ) ) ) ) ).
% ivl_subset
thf(fact_625_size_H__char__eq__0,axiom,
( size_char
= ( ^ [C3: char] : zero_zero_nat ) ) ).
% size'_char_eq_0
thf(fact_626_execl__next,axiom,
! [P: list_instr,Cfs2: list_P419547167332244514st_int,K: nat] :
( ( execl @ P @ Cfs2 )
=> ( ( ord_less_nat @ K @ ( size_s229347708945616566st_int @ Cfs2 ) )
=> ( ( K
!= ( minus_minus_nat @ ( size_s229347708945616566st_int @ Cfs2 ) @ one_one_nat ) )
=> ( ( iexec_pred @ ( produc1484136438817787418st_int @ ( inth_instr @ P @ ( produc8941309049508465880st_int @ ( nth_Pr8327173284404383691st_int @ Cfs2 @ K ) ) ) @ ( nth_Pr8327173284404383691st_int @ Cfs2 @ K ) ) @ ( nth_Pr8327173284404383691st_int @ Cfs2 @ ( suc @ K ) ) )
& ( ord_less_eq_int @ zero_zero_int @ ( produc8941309049508465880st_int @ ( nth_Pr8327173284404383691st_int @ Cfs2 @ K ) ) )
& ( ord_less_int @ ( produc8941309049508465880st_int @ ( nth_Pr8327173284404383691st_int @ Cfs2 @ K ) ) @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P ) ) ) ) ) ) ) ).
% execl_next
thf(fact_627_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_628_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_629_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_630_diff__diff__cancel,axiom,
! [I3: nat,N: nat] :
( ( ord_less_eq_nat @ I3 @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I3 ) )
= I3 ) ) ).
% diff_diff_cancel
thf(fact_631_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_632_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_633_take__all__iff,axiom,
! [N: nat,Xs: list_P419547167332244514st_int] :
( ( ( take_P1004856038796978596st_int @ N @ Xs )
= Xs )
= ( ord_less_eq_nat @ ( size_s229347708945616566st_int @ Xs ) @ N ) ) ).
% take_all_iff
thf(fact_634_take__all__iff,axiom,
! [N: nat,Xs: list_instr] :
( ( ( take_instr @ N @ Xs )
= Xs )
= ( ord_less_eq_nat @ ( size_size_list_instr @ Xs ) @ N ) ) ).
% take_all_iff
thf(fact_635_take__all__iff,axiom,
! [N: nat,Xs: list_list_char] :
( ( ( take_list_char @ N @ Xs )
= Xs )
= ( ord_less_eq_nat @ ( size_s356637359517785349t_char @ Xs ) @ N ) ) ).
% take_all_iff
thf(fact_636_take__all,axiom,
! [Xs: list_P419547167332244514st_int,N: nat] :
( ( ord_less_eq_nat @ ( size_s229347708945616566st_int @ Xs ) @ N )
=> ( ( take_P1004856038796978596st_int @ N @ Xs )
= Xs ) ) ).
% take_all
thf(fact_637_take__all,axiom,
! [Xs: list_instr,N: nat] :
( ( ord_less_eq_nat @ ( size_size_list_instr @ Xs ) @ N )
=> ( ( take_instr @ N @ Xs )
= Xs ) ) ).
% take_all
thf(fact_638_take__all,axiom,
! [Xs: list_list_char,N: nat] :
( ( ord_less_eq_nat @ ( size_s356637359517785349t_char @ Xs ) @ N )
=> ( ( take_list_char @ N @ Xs )
= Xs ) ) ).
% take_all
thf(fact_639_iexec__offset,axiom,
! [Ins2: instr,Pc: int,S: list_char > int,Stk2: list_int,Pc2: int,S4: list_char > int,Stk3: list_int,I3: int] :
( ( iexec_pred @ ( produc1484136438817787418st_int @ Ins2 @ ( produc5086643055186798020st_int @ Pc @ ( produc8650753666468850689st_int @ S @ Stk2 ) ) ) @ ( produc5086643055186798020st_int @ Pc2 @ ( produc8650753666468850689st_int @ S4 @ Stk3 ) ) )
=> ( iexec_pred @ ( produc1484136438817787418st_int @ Ins2 @ ( produc5086643055186798020st_int @ ( minus_minus_int @ Pc @ I3 ) @ ( produc8650753666468850689st_int @ S @ Stk2 ) ) ) @ ( produc5086643055186798020st_int @ ( minus_minus_int @ Pc2 @ I3 ) @ ( produc8650753666468850689st_int @ S4 @ Stk3 ) ) ) ) ).
% iexec_offset
thf(fact_640_subset__Collect__iff,axiom,
! [B5: set_nat,A3: set_nat,P: nat > $o] :
( ( ord_less_eq_set_nat @ B5 @ A3 )
=> ( ( ord_less_eq_set_nat @ B5
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A3 )
& ( P @ X3 ) ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ B5 )
=> ( P @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_641_subset__Collect__iff,axiom,
! [B5: set_int,A3: set_int,P: int > $o] :
( ( ord_less_eq_set_int @ B5 @ A3 )
=> ( ( ord_less_eq_set_int @ B5
@ ( collect_int
@ ^ [X3: int] :
( ( member_int @ X3 @ A3 )
& ( P @ X3 ) ) ) )
= ( ! [X3: int] :
( ( member_int @ X3 @ B5 )
=> ( P @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_642_subset__CollectI,axiom,
! [B5: set_nat,A3: set_nat,Q: nat > $o,P: nat > $o] :
( ( ord_less_eq_set_nat @ B5 @ A3 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ B5 )
=> ( ( Q @ X4 )
=> ( P @ X4 ) ) )
=> ( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ B5 )
& ( Q @ X3 ) ) )
@ ( collect_nat
@ ^ [X3: nat] :
( ( member_nat @ X3 @ A3 )
& ( P @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_643_subset__CollectI,axiom,
! [B5: set_int,A3: set_int,Q: int > $o,P: int > $o] :
( ( ord_less_eq_set_int @ B5 @ A3 )
=> ( ! [X4: int] :
( ( member_int @ X4 @ B5 )
=> ( ( Q @ X4 )
=> ( P @ X4 ) ) )
=> ( ord_less_eq_set_int
@ ( collect_int
@ ^ [X3: int] :
( ( member_int @ X3 @ B5 )
& ( Q @ X3 ) ) )
@ ( collect_int
@ ^ [X3: int] :
( ( member_int @ X3 @ A3 )
& ( P @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_644_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_645_le__trans,axiom,
! [I3: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I3 @ K ) ) ) ).
% le_trans
thf(fact_646_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_647_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_648_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_649_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X4: nat] :
( ( P @ X4 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X4 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_650_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M6: nat] :
( ( P @ X )
=> ( ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M6 ) )
=> ~ ! [M4: nat] :
( ( P @ M4 )
=> ~ ! [X5: nat] :
( ( P @ X5 )
=> ( ord_less_eq_nat @ X5 @ M4 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_651_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_652_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_653_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_654_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_655_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X4: nat] : ( R @ X4 @ X4 )
=> ( ! [X4: nat,Y3: nat,Z3: nat] :
( ( R @ X4 @ Y3 )
=> ( ( R @ Y3 @ Z3 )
=> ( R @ X4 @ Z3 ) ) )
=> ( ! [N2: nat] : ( R @ N2 @ ( suc @ N2 ) )
=> ( R @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_656_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_657_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M3: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N2 )
=> ( P @ M3 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_658_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_659_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_660_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_661_Suc__le__D,axiom,
! [N: nat,M7: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M7 )
=> ? [M4: nat] :
( M7
= ( suc @ M4 ) ) ) ).
% Suc_le_D
thf(fact_662_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_663_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_664_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_665_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I3: nat,J: nat] :
( ! [I: nat,J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ord_less_nat @ ( F @ I ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I3 @ J )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_666_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_667_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_668_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N4: nat] :
( ( ord_less_nat @ M2 @ N4 )
| ( M2 = N4 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_669_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_670_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M2: nat,N4: nat] :
( ( ord_less_eq_nat @ M2 @ N4 )
& ( M2 != N4 ) ) ) ) ).
% nat_less_le
thf(fact_671_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_672_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_673_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_674_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_675_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_676_le__diff__iff_H,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C2 )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C2 @ A ) @ ( minus_minus_nat @ C2 @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_677_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_678_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_679_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_680_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N4: nat] : ( ord_less_eq_nat @ ( suc @ N4 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_681_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% less_Suc_eq_le
thf(fact_682_le__less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% le_less_Suc_eq
thf(fact_683_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_le_lessD
thf(fact_684_inc__induct,axiom,
! [I3: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ( P @ J )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I3 @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) ) ) )
=> ( P @ I3 ) ) ) ) ).
% inc_induct
thf(fact_685_dec__induct,axiom,
! [I3: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ( P @ I3 )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I3 @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_686_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_le_eq
thf(fact_687_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_688_impossible__Cons,axiom,
! [Xs: list_int,Ys: list_int,X: int] :
( ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ ( size_size_list_int @ Ys ) )
=> ( Xs
!= ( cons_int @ X @ Ys ) ) ) ).
% impossible_Cons
thf(fact_689_impossible__Cons,axiom,
! [Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int,X: produc6425607678544837394st_int] :
( ( ord_less_eq_nat @ ( size_s229347708945616566st_int @ Xs ) @ ( size_s229347708945616566st_int @ Ys ) )
=> ( Xs
!= ( cons_P4151473878978182108st_int @ X @ Ys ) ) ) ).
% impossible_Cons
thf(fact_690_impossible__Cons,axiom,
! [Xs: list_instr,Ys: list_instr,X: instr] :
( ( ord_less_eq_nat @ ( size_size_list_instr @ Xs ) @ ( size_size_list_instr @ Ys ) )
=> ( Xs
!= ( cons_instr @ X @ Ys ) ) ) ).
% impossible_Cons
thf(fact_691_impossible__Cons,axiom,
! [Xs: list_list_char,Ys: list_list_char,X: list_char] :
( ( ord_less_eq_nat @ ( size_s356637359517785349t_char @ Xs ) @ ( size_s356637359517785349t_char @ Ys ) )
=> ( Xs
!= ( cons_list_char @ X @ Ys ) ) ) ).
% impossible_Cons
thf(fact_692_Suc__diff__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_693_diff__less__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C2 ) @ ( minus_minus_nat @ B @ C2 ) ) ) ) ).
% diff_less_mono
thf(fact_694_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_695_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B4: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_696_zle__int,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% zle_int
thf(fact_697_nat__leq__as__int,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B4: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_leq_as_int
thf(fact_698_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N )
& ! [I4: nat] :
( ( ord_less_eq_nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_699_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_int] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_int @ Xs ) )
= ( ? [X3: int,Ys2: list_int] :
( ( Xs
= ( cons_int @ X3 @ Ys2 ) )
& ( ord_less_eq_nat @ N @ ( size_size_list_int @ Ys2 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_700_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_P419547167332244514st_int] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_s229347708945616566st_int @ Xs ) )
= ( ? [X3: produc6425607678544837394st_int,Ys2: list_P419547167332244514st_int] :
( ( Xs
= ( cons_P4151473878978182108st_int @ X3 @ Ys2 ) )
& ( ord_less_eq_nat @ N @ ( size_s229347708945616566st_int @ Ys2 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_701_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_instr] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_instr @ Xs ) )
= ( ? [X3: instr,Ys2: list_instr] :
( ( Xs
= ( cons_instr @ X3 @ Ys2 ) )
& ( ord_less_eq_nat @ N @ ( size_size_list_instr @ Ys2 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_702_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_list_char] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_s356637359517785349t_char @ Xs ) )
= ( ? [X3: list_char,Ys2: list_list_char] :
( ( Xs
= ( cons_list_char @ X3 @ Ys2 ) )
& ( ord_less_eq_nat @ N @ ( size_s356637359517785349t_char @ Ys2 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_703_of__nat__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% of_nat_diff
thf(fact_704_of__nat__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).
% of_nat_diff
thf(fact_705_nth__take__lemma,axiom,
! [K: nat,Xs: list_int,Ys: list_int] :
( ( ord_less_eq_nat @ K @ ( size_size_list_int @ Xs ) )
=> ( ( ord_less_eq_nat @ K @ ( size_size_list_int @ Ys ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( nth_int @ Xs @ I )
= ( nth_int @ Ys @ I ) ) )
=> ( ( take_int @ K @ Xs )
= ( take_int @ K @ Ys ) ) ) ) ) ).
% nth_take_lemma
thf(fact_706_nth__take__lemma,axiom,
! [K: nat,Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int] :
( ( ord_less_eq_nat @ K @ ( size_s229347708945616566st_int @ Xs ) )
=> ( ( ord_less_eq_nat @ K @ ( size_s229347708945616566st_int @ Ys ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( nth_Pr8327173284404383691st_int @ Xs @ I )
= ( nth_Pr8327173284404383691st_int @ Ys @ I ) ) )
=> ( ( take_P1004856038796978596st_int @ K @ Xs )
= ( take_P1004856038796978596st_int @ K @ Ys ) ) ) ) ) ).
% nth_take_lemma
thf(fact_707_nth__take__lemma,axiom,
! [K: nat,Xs: list_instr,Ys: list_instr] :
( ( ord_less_eq_nat @ K @ ( size_size_list_instr @ Xs ) )
=> ( ( ord_less_eq_nat @ K @ ( size_size_list_instr @ Ys ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( nth_instr @ Xs @ I )
= ( nth_instr @ Ys @ I ) ) )
=> ( ( take_instr @ K @ Xs )
= ( take_instr @ K @ Ys ) ) ) ) ) ).
% nth_take_lemma
thf(fact_708_nth__take__lemma,axiom,
! [K: nat,Xs: list_list_char,Ys: list_list_char] :
( ( ord_less_eq_nat @ K @ ( size_s356637359517785349t_char @ Xs ) )
=> ( ( ord_less_eq_nat @ K @ ( size_s356637359517785349t_char @ Ys ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( nth_list_char @ Xs @ I )
= ( nth_list_char @ Ys @ I ) ) )
=> ( ( take_list_char @ K @ Xs )
= ( take_list_char @ K @ Ys ) ) ) ) ) ).
% nth_take_lemma
thf(fact_709_append__eq__append__conv__if,axiom,
! [Xs_1: list_int,Xs_2: list_int,Ys_1: list_int,Ys_2: list_int] :
( ( ( append_int @ Xs_1 @ Xs_2 )
= ( append_int @ Ys_1 @ Ys_2 ) )
= ( ( ( ord_less_eq_nat @ ( size_size_list_int @ Xs_1 ) @ ( size_size_list_int @ Ys_1 ) )
=> ( ( Xs_1
= ( take_int @ ( size_size_list_int @ Xs_1 ) @ Ys_1 ) )
& ( Xs_2
= ( append_int @ ( drop_int @ ( size_size_list_int @ Xs_1 ) @ Ys_1 ) @ Ys_2 ) ) ) )
& ( ~ ( ord_less_eq_nat @ ( size_size_list_int @ Xs_1 ) @ ( size_size_list_int @ Ys_1 ) )
=> ( ( ( take_int @ ( size_size_list_int @ Ys_1 ) @ Xs_1 )
= Ys_1 )
& ( ( append_int @ ( drop_int @ ( size_size_list_int @ Ys_1 ) @ Xs_1 ) @ Xs_2 )
= Ys_2 ) ) ) ) ) ).
% append_eq_append_conv_if
thf(fact_710_append__eq__append__conv__if,axiom,
! [Xs_1: list_P419547167332244514st_int,Xs_2: list_P419547167332244514st_int,Ys_1: list_P419547167332244514st_int,Ys_2: list_P419547167332244514st_int] :
( ( ( append5028292981072010935st_int @ Xs_1 @ Xs_2 )
= ( append5028292981072010935st_int @ Ys_1 @ Ys_2 ) )
= ( ( ( ord_less_eq_nat @ ( size_s229347708945616566st_int @ Xs_1 ) @ ( size_s229347708945616566st_int @ Ys_1 ) )
=> ( ( Xs_1
= ( take_P1004856038796978596st_int @ ( size_s229347708945616566st_int @ Xs_1 ) @ Ys_1 ) )
& ( Xs_2
= ( append5028292981072010935st_int @ ( drop_P7838799151754935884st_int @ ( size_s229347708945616566st_int @ Xs_1 ) @ Ys_1 ) @ Ys_2 ) ) ) )
& ( ~ ( ord_less_eq_nat @ ( size_s229347708945616566st_int @ Xs_1 ) @ ( size_s229347708945616566st_int @ Ys_1 ) )
=> ( ( ( take_P1004856038796978596st_int @ ( size_s229347708945616566st_int @ Ys_1 ) @ Xs_1 )
= Ys_1 )
& ( ( append5028292981072010935st_int @ ( drop_P7838799151754935884st_int @ ( size_s229347708945616566st_int @ Ys_1 ) @ Xs_1 ) @ Xs_2 )
= Ys_2 ) ) ) ) ) ).
% append_eq_append_conv_if
thf(fact_711_append__eq__append__conv__if,axiom,
! [Xs_1: list_instr,Xs_2: list_instr,Ys_1: list_instr,Ys_2: list_instr] :
( ( ( append_instr @ Xs_1 @ Xs_2 )
= ( append_instr @ Ys_1 @ Ys_2 ) )
= ( ( ( ord_less_eq_nat @ ( size_size_list_instr @ Xs_1 ) @ ( size_size_list_instr @ Ys_1 ) )
=> ( ( Xs_1
= ( take_instr @ ( size_size_list_instr @ Xs_1 ) @ Ys_1 ) )
& ( Xs_2
= ( append_instr @ ( drop_instr @ ( size_size_list_instr @ Xs_1 ) @ Ys_1 ) @ Ys_2 ) ) ) )
& ( ~ ( ord_less_eq_nat @ ( size_size_list_instr @ Xs_1 ) @ ( size_size_list_instr @ Ys_1 ) )
=> ( ( ( take_instr @ ( size_size_list_instr @ Ys_1 ) @ Xs_1 )
= Ys_1 )
& ( ( append_instr @ ( drop_instr @ ( size_size_list_instr @ Ys_1 ) @ Xs_1 ) @ Xs_2 )
= Ys_2 ) ) ) ) ) ).
% append_eq_append_conv_if
thf(fact_712_append__eq__append__conv__if,axiom,
! [Xs_1: list_list_char,Xs_2: list_list_char,Ys_1: list_list_char,Ys_2: list_list_char] :
( ( ( append_list_char @ Xs_1 @ Xs_2 )
= ( append_list_char @ Ys_1 @ Ys_2 ) )
= ( ( ( ord_less_eq_nat @ ( size_s356637359517785349t_char @ Xs_1 ) @ ( size_s356637359517785349t_char @ Ys_1 ) )
=> ( ( Xs_1
= ( take_list_char @ ( size_s356637359517785349t_char @ Xs_1 ) @ Ys_1 ) )
& ( Xs_2
= ( append_list_char @ ( drop_list_char @ ( size_s356637359517785349t_char @ Xs_1 ) @ Ys_1 ) @ Ys_2 ) ) ) )
& ( ~ ( ord_less_eq_nat @ ( size_s356637359517785349t_char @ Xs_1 ) @ ( size_s356637359517785349t_char @ Ys_1 ) )
=> ( ( ( take_list_char @ ( size_s356637359517785349t_char @ Ys_1 ) @ Xs_1 )
= Ys_1 )
& ( ( append_list_char @ ( drop_list_char @ ( size_s356637359517785349t_char @ Ys_1 ) @ Xs_1 ) @ Xs_2 )
= Ys_2 ) ) ) ) ) ).
% append_eq_append_conv_if
thf(fact_713_atLeastLessThan__subset__iff,axiom,
! [A: nat,B: nat,C2: nat,D: nat] :
( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ A @ B ) @ ( set_or4665077453230672383an_nat @ C2 @ D ) )
=> ( ( ord_less_eq_nat @ B @ A )
| ( ( ord_less_eq_nat @ C2 @ A )
& ( ord_less_eq_nat @ B @ D ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_714_atLeastLessThan__subset__iff,axiom,
! [A: int,B: int,C2: int,D: int] :
( ( ord_less_eq_set_int @ ( set_or4662586982721622107an_int @ A @ B ) @ ( set_or4662586982721622107an_int @ C2 @ D ) )
=> ( ( ord_less_eq_int @ B @ A )
| ( ( ord_less_eq_int @ C2 @ A )
& ( ord_less_eq_int @ B @ D ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_715_atLeastLessThan__eq__iff,axiom,
! [A: nat,B: nat,C2: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C2 @ D )
=> ( ( ( set_or4665077453230672383an_nat @ A @ B )
= ( set_or4665077453230672383an_nat @ C2 @ D ) )
= ( ( A = C2 )
& ( B = D ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_716_atLeastLessThan__eq__iff,axiom,
! [A: int,B: int,C2: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C2 @ D )
=> ( ( ( set_or4662586982721622107an_int @ A @ B )
= ( set_or4662586982721622107an_int @ C2 @ D ) )
= ( ( A = C2 )
& ( B = D ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_717_Ico__eq__Ico,axiom,
! [L: nat,H: nat,L2: nat,H2: nat] :
( ( ( set_or4665077453230672383an_nat @ L @ H )
= ( set_or4665077453230672383an_nat @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_nat @ L @ H )
& ~ ( ord_less_nat @ L2 @ H2 ) ) ) ) ).
% Ico_eq_Ico
thf(fact_718_Ico__eq__Ico,axiom,
! [L: int,H: int,L2: int,H2: int] :
( ( ( set_or4662586982721622107an_int @ L @ H )
= ( set_or4662586982721622107an_int @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_int @ L @ H )
& ~ ( ord_less_int @ L2 @ H2 ) ) ) ) ).
% Ico_eq_Ico
thf(fact_719_atLeastLessThan__inj_I1_J,axiom,
! [A: nat,B: nat,C2: nat,D: nat] :
( ( ( set_or4665077453230672383an_nat @ A @ B )
= ( set_or4665077453230672383an_nat @ C2 @ D ) )
=> ( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C2 @ D )
=> ( A = C2 ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_720_atLeastLessThan__inj_I1_J,axiom,
! [A: int,B: int,C2: int,D: int] :
( ( ( set_or4662586982721622107an_int @ A @ B )
= ( set_or4662586982721622107an_int @ C2 @ D ) )
=> ( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C2 @ D )
=> ( A = C2 ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_721_atLeastLessThan__inj_I2_J,axiom,
! [A: nat,B: nat,C2: nat,D: nat] :
( ( ( set_or4665077453230672383an_nat @ A @ B )
= ( set_or4665077453230672383an_nat @ C2 @ D ) )
=> ( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C2 @ D )
=> ( B = D ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_722_atLeastLessThan__inj_I2_J,axiom,
! [A: int,B: int,C2: int,D: int] :
( ( ( set_or4662586982721622107an_int @ A @ B )
= ( set_or4662586982721622107an_int @ C2 @ D ) )
=> ( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C2 @ D )
=> ( B = D ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_723_inth_Osimps,axiom,
! [I3: int,X: list_char,Xs: list_list_char] :
( ( ( I3 = zero_zero_int )
=> ( ( inth_list_char @ ( cons_list_char @ X @ Xs ) @ I3 )
= X ) )
& ( ( I3 != zero_zero_int )
=> ( ( inth_list_char @ ( cons_list_char @ X @ Xs ) @ I3 )
= ( inth_list_char @ Xs @ ( minus_minus_int @ I3 @ one_one_int ) ) ) ) ) ).
% inth.simps
thf(fact_724_inth_Osimps,axiom,
! [I3: int,X: produc6425607678544837394st_int,Xs: list_P419547167332244514st_int] :
( ( ( I3 = zero_zero_int )
=> ( ( inth_P8039235741264645271st_int @ ( cons_P4151473878978182108st_int @ X @ Xs ) @ I3 )
= X ) )
& ( ( I3 != zero_zero_int )
=> ( ( inth_P8039235741264645271st_int @ ( cons_P4151473878978182108st_int @ X @ Xs ) @ I3 )
= ( inth_P8039235741264645271st_int @ Xs @ ( minus_minus_int @ I3 @ one_one_int ) ) ) ) ) ).
% inth.simps
thf(fact_725_inth_Osimps,axiom,
! [I3: int,X: instr,Xs: list_instr] :
( ( ( I3 = zero_zero_int )
=> ( ( inth_instr @ ( cons_instr @ X @ Xs ) @ I3 )
= X ) )
& ( ( I3 != zero_zero_int )
=> ( ( inth_instr @ ( cons_instr @ X @ Xs ) @ I3 )
= ( inth_instr @ Xs @ ( minus_minus_int @ I3 @ one_one_int ) ) ) ) ) ).
% inth.simps
thf(fact_726_inth_Osimps,axiom,
! [I3: int,X: int,Xs: list_int] :
( ( ( I3 = zero_zero_int )
=> ( ( inth_int @ ( cons_int @ X @ Xs ) @ I3 )
= X ) )
& ( ( I3 != zero_zero_int )
=> ( ( inth_int @ ( cons_int @ X @ Xs ) @ I3 )
= ( inth_int @ Xs @ ( minus_minus_int @ I3 @ one_one_int ) ) ) ) ) ).
% inth.simps
thf(fact_727_execl__last,axiom,
! [P: list_instr,Cfs2: list_P419547167332244514st_int,K: nat] :
( ( execl @ P @ Cfs2 )
=> ( ( ord_less_nat @ K @ ( size_s229347708945616566st_int @ Cfs2 ) )
=> ( ~ ( member_int @ ( produc8941309049508465880st_int @ ( nth_Pr8327173284404383691st_int @ Cfs2 @ K ) ) @ ( set_or4662586982721622107an_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P ) ) ) )
=> ( ( minus_minus_nat @ ( size_s229347708945616566st_int @ Cfs2 ) @ one_one_nat )
= K ) ) ) ) ).
% execl_last
thf(fact_728_zdiff__int__split,axiom,
! [P: int > $o,X: nat,Y: nat] :
( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X @ Y ) ) )
= ( ( ( ord_less_eq_nat @ Y @ X )
=> ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y ) ) ) )
& ( ( ord_less_nat @ X @ Y )
=> ( P @ zero_zero_int ) ) ) ) ).
% zdiff_int_split
thf(fact_729_execl__all__def,axiom,
( execl_all
= ( ^ [P5: list_instr,Cfs3: list_P419547167332244514st_int] :
( ( execl @ P5 @ Cfs3 )
& ( Cfs3 != nil_Pr749310521696113548st_int )
& ( ( produc8941309049508465880st_int @ ( nth_Pr8327173284404383691st_int @ Cfs3 @ zero_zero_nat ) )
= zero_zero_int )
& ~ ( member_int @ ( produc8941309049508465880st_int @ ( nth_Pr8327173284404383691st_int @ Cfs3 @ ( minus_minus_nat @ ( size_s229347708945616566st_int @ Cfs3 ) @ one_one_nat ) ) ) @ ( set_or4662586982721622107an_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P5 ) ) ) ) ) ) ) ).
% execl_all_def
thf(fact_730_exec1__def,axiom,
( exec1
= ( ^ [P5: list_instr,Cf: produc6425607678544837394st_int,Cf2: produc6425607678544837394st_int] :
( ( iexec_pred @ ( produc1484136438817787418st_int @ ( inth_instr @ P5 @ ( produc8941309049508465880st_int @ Cf ) ) @ Cf ) @ Cf2 )
& ( ord_less_eq_int @ zero_zero_int @ ( produc8941309049508465880st_int @ Cf ) )
& ( ord_less_int @ ( produc8941309049508465880st_int @ Cf ) @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P5 ) ) ) ) ) ) ).
% exec1_def
thf(fact_731_to__m__state_Oelims,axiom,
! [X: list_list_char,Xa: list_char > int,Xb: int,Y: int] :
( ( ( to_m_state @ X @ Xa @ Xb )
= Y )
=> ( ( ( ( ord_less_int @ zero_zero_int @ Xb )
& ( ord_less_eq_int @ Xb @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ X ) ) ) )
=> ( Y
= ( Xa @ ( inth_list_char @ X @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ X ) ) @ Xb ) ) ) ) )
& ( ~ ( ( ord_less_int @ zero_zero_int @ Xb )
& ( ord_less_eq_int @ Xb @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ X ) ) ) )
=> ( Y = zero_zero_int ) ) ) ) ).
% to_m_state.elims
thf(fact_732_to__m__state_Osimps,axiom,
( to_m_state
= ( ^ [Xs2: list_list_char,S2: list_char > int,A5: int] :
( if_int
@ ( ( ord_less_int @ zero_zero_int @ A5 )
& ( ord_less_eq_int @ A5 @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ Xs2 ) ) ) )
@ ( S2 @ ( inth_list_char @ Xs2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ Xs2 ) ) @ A5 ) ) )
@ zero_zero_int ) ) ) ).
% to_m_state.simps
thf(fact_733_execl__sub__aux,axiom,
! [Q: list_instr > produc6425607678544837394st_int > $o,P3: list_instr,Pc: int,S: list_char > int,Stk2: list_int,Cfs2: list_P419547167332244514st_int,P: list_instr,F2: list_P419547167332244514st_int > nat > nat > list_P419547167332244514st_int,M: nat,N: nat,Cfs4: list_P419547167332244514st_int] :
( ! [M4: nat,N2: nat] :
( ! [X5: nat] :
( ( member_nat @ X5 @ ( set_or4665077453230672383an_nat @ M4 @ N2 ) )
=> ( Q @ P3 @ ( nth_Pr8327173284404383691st_int @ ( cons_P4151473878978182108st_int @ ( produc5086643055186798020st_int @ Pc @ ( produc8650753666468850689st_int @ S @ Stk2 ) ) @ Cfs2 ) @ X5 ) ) )
=> ( execl @ P3
@ ( map_Pr8625093460839654122st_int
@ ^ [Cf: produc6425607678544837394st_int] : ( produc5086643055186798020st_int @ ( minus_minus_int @ ( produc8941309049508465880st_int @ Cf ) @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P ) ) ) @ ( produc1296517004818028822st_int @ Cf ) )
@ ( case_n3094275698756463160st_int @ ( cons_P4151473878978182108st_int @ ( produc5086643055186798020st_int @ Pc @ ( produc8650753666468850689st_int @ S @ Stk2 ) ) @ ( take_P1004856038796978596st_int @ N2 @ Cfs2 ) )
@ ^ [O: nat] : ( F2 @ Cfs2 @ O @ N2 )
@ M4 ) ) ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_or4665077453230672383an_nat @ M @ ( plus_plus_nat @ ( plus_plus_nat @ N @ M ) @ ( size_s229347708945616566st_int @ Cfs4 ) ) ) )
=> ( Q @ P3 @ ( nth_Pr8327173284404383691st_int @ ( append5028292981072010935st_int @ Cfs4 @ ( cons_P4151473878978182108st_int @ ( produc5086643055186798020st_int @ Pc @ ( produc8650753666468850689st_int @ S @ Stk2 ) ) @ Cfs2 ) ) @ ( minus_minus_nat @ X4 @ M ) ) ) )
=> ( execl @ P3
@ ( cons_P4151473878978182108st_int @ ( produc5086643055186798020st_int @ ( minus_minus_int @ Pc @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P ) ) ) @ ( produc8650753666468850689st_int @ S @ Stk2 ) )
@ ( map_Pr8625093460839654122st_int
@ ^ [Cf: produc6425607678544837394st_int] : ( produc5086643055186798020st_int @ ( minus_minus_int @ ( produc8941309049508465880st_int @ Cf ) @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P ) ) ) @ ( produc1296517004818028822st_int @ Cf ) )
@ ( take_P1004856038796978596st_int @ N @ Cfs2 ) ) ) ) ) ) ).
% execl_sub_aux
thf(fact_734_add__left__cancel,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C2 ) )
= ( B = C2 ) ) ).
% add_left_cancel
thf(fact_735_add__left__cancel,axiom,
! [A: int,B: int,C2: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C2 ) )
= ( B = C2 ) ) ).
% add_left_cancel
thf(fact_736_add__right__cancel,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C2 @ A ) )
= ( B = C2 ) ) ).
% add_right_cancel
thf(fact_737_add__right__cancel,axiom,
! [B: int,A: int,C2: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C2 @ A ) )
= ( B = C2 ) ) ).
% add_right_cancel
thf(fact_738_add__le__cancel__left,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_739_add__le__cancel__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_740_add__le__cancel__right,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_741_add__le__cancel__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_742_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_743_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_744_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y ) )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_745_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_746_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_747_add__cancel__right__right,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ A @ B ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_right
thf(fact_748_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_749_add__cancel__right__left,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ B @ A ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_left
thf(fact_750_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_751_add__cancel__left__right,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_right
thf(fact_752_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_753_add__cancel__left__left,axiom,
! [B: int,A: int] :
( ( ( plus_plus_int @ B @ A )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_left
thf(fact_754_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_755_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_756_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_757_double__eq__0__iff,axiom,
! [A: int] :
( ( ( plus_plus_int @ A @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% double_eq_0_iff
thf(fact_758_add__less__cancel__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_759_add__less__cancel__left,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_760_add__less__cancel__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_761_add__less__cancel__right,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_762_add__diff__cancel,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_763_diff__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_764_add__diff__cancel__left,axiom,
! [C2: int,A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_765_add__diff__cancel__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_766_add__diff__cancel__left_H,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_767_add__diff__cancel__left_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_768_add__diff__cancel__right,axiom,
! [A: int,C2: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_769_add__diff__cancel__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_770_add__diff__cancel__right_H,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_771_add__diff__cancel__right_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_772_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_773_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_774_add__Suc__right,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ M @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc_right
thf(fact_775_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_776_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_777_list_Omap__disc__iff,axiom,
! [F: instr > produc6425607678544837394st_int,A: list_instr] :
( ( ( map_in7259965606695656109st_int @ F @ A )
= nil_Pr749310521696113548st_int )
= ( A = nil_instr ) ) ).
% list.map_disc_iff
thf(fact_778_list_Omap__disc__iff,axiom,
! [F: int > produc6425607678544837394st_int,A: list_int] :
( ( ( map_in8512336359594627332st_int @ F @ A )
= nil_Pr749310521696113548st_int )
= ( A = nil_int ) ) ).
% list.map_disc_iff
thf(fact_779_list_Omap__disc__iff,axiom,
! [F: produc6425607678544837394st_int > instr,A: list_P419547167332244514st_int] :
( ( ( map_Pr7514435936640407687_instr @ F @ A )
= nil_instr )
= ( A = nil_Pr749310521696113548st_int ) ) ).
% list.map_disc_iff
thf(fact_780_list_Omap__disc__iff,axiom,
! [F: instr > instr,A: list_instr] :
( ( ( map_instr_instr @ F @ A )
= nil_instr )
= ( A = nil_instr ) ) ).
% list.map_disc_iff
thf(fact_781_list_Omap__disc__iff,axiom,
! [F: int > instr,A: list_int] :
( ( ( map_int_instr @ F @ A )
= nil_instr )
= ( A = nil_int ) ) ).
% list.map_disc_iff
thf(fact_782_list_Omap__disc__iff,axiom,
! [F: produc6425607678544837394st_int > int,A: list_P419547167332244514st_int] :
( ( ( map_Pr3224476576807795204nt_int @ F @ A )
= nil_int )
= ( A = nil_Pr749310521696113548st_int ) ) ).
% list.map_disc_iff
thf(fact_783_list_Omap__disc__iff,axiom,
! [F: instr > int,A: list_instr] :
( ( ( map_instr_int @ F @ A )
= nil_int )
= ( A = nil_instr ) ) ).
% list.map_disc_iff
thf(fact_784_list_Omap__disc__iff,axiom,
! [F: int > int,A: list_int] :
( ( ( map_int_int @ F @ A )
= nil_int )
= ( A = nil_int ) ) ).
% list.map_disc_iff
thf(fact_785_list_Omap__disc__iff,axiom,
! [F: produc6425607678544837394st_int > produc6425607678544837394st_int,A: list_P419547167332244514st_int] :
( ( ( map_Pr8625093460839654122st_int @ F @ A )
= nil_Pr749310521696113548st_int )
= ( A = nil_Pr749310521696113548st_int ) ) ).
% list.map_disc_iff
thf(fact_786_Nil__is__map__conv,axiom,
! [F: instr > produc6425607678544837394st_int,Xs: list_instr] :
( ( nil_Pr749310521696113548st_int
= ( map_in7259965606695656109st_int @ F @ Xs ) )
= ( Xs = nil_instr ) ) ).
% Nil_is_map_conv
thf(fact_787_Nil__is__map__conv,axiom,
! [F: int > produc6425607678544837394st_int,Xs: list_int] :
( ( nil_Pr749310521696113548st_int
= ( map_in8512336359594627332st_int @ F @ Xs ) )
= ( Xs = nil_int ) ) ).
% Nil_is_map_conv
thf(fact_788_Nil__is__map__conv,axiom,
! [F: produc6425607678544837394st_int > instr,Xs: list_P419547167332244514st_int] :
( ( nil_instr
= ( map_Pr7514435936640407687_instr @ F @ Xs ) )
= ( Xs = nil_Pr749310521696113548st_int ) ) ).
% Nil_is_map_conv
thf(fact_789_Nil__is__map__conv,axiom,
! [F: instr > instr,Xs: list_instr] :
( ( nil_instr
= ( map_instr_instr @ F @ Xs ) )
= ( Xs = nil_instr ) ) ).
% Nil_is_map_conv
thf(fact_790_Nil__is__map__conv,axiom,
! [F: int > instr,Xs: list_int] :
( ( nil_instr
= ( map_int_instr @ F @ Xs ) )
= ( Xs = nil_int ) ) ).
% Nil_is_map_conv
thf(fact_791_Nil__is__map__conv,axiom,
! [F: produc6425607678544837394st_int > int,Xs: list_P419547167332244514st_int] :
( ( nil_int
= ( map_Pr3224476576807795204nt_int @ F @ Xs ) )
= ( Xs = nil_Pr749310521696113548st_int ) ) ).
% Nil_is_map_conv
thf(fact_792_Nil__is__map__conv,axiom,
! [F: instr > int,Xs: list_instr] :
( ( nil_int
= ( map_instr_int @ F @ Xs ) )
= ( Xs = nil_instr ) ) ).
% Nil_is_map_conv
thf(fact_793_Nil__is__map__conv,axiom,
! [F: int > int,Xs: list_int] :
( ( nil_int
= ( map_int_int @ F @ Xs ) )
= ( Xs = nil_int ) ) ).
% Nil_is_map_conv
thf(fact_794_Nil__is__map__conv,axiom,
! [F: produc6425607678544837394st_int > produc6425607678544837394st_int,Xs: list_P419547167332244514st_int] :
( ( nil_Pr749310521696113548st_int
= ( map_Pr8625093460839654122st_int @ F @ Xs ) )
= ( Xs = nil_Pr749310521696113548st_int ) ) ).
% Nil_is_map_conv
thf(fact_795_map__is__Nil__conv,axiom,
! [F: instr > produc6425607678544837394st_int,Xs: list_instr] :
( ( ( map_in7259965606695656109st_int @ F @ Xs )
= nil_Pr749310521696113548st_int )
= ( Xs = nil_instr ) ) ).
% map_is_Nil_conv
thf(fact_796_map__is__Nil__conv,axiom,
! [F: int > produc6425607678544837394st_int,Xs: list_int] :
( ( ( map_in8512336359594627332st_int @ F @ Xs )
= nil_Pr749310521696113548st_int )
= ( Xs = nil_int ) ) ).
% map_is_Nil_conv
thf(fact_797_map__is__Nil__conv,axiom,
! [F: produc6425607678544837394st_int > instr,Xs: list_P419547167332244514st_int] :
( ( ( map_Pr7514435936640407687_instr @ F @ Xs )
= nil_instr )
= ( Xs = nil_Pr749310521696113548st_int ) ) ).
% map_is_Nil_conv
thf(fact_798_map__is__Nil__conv,axiom,
! [F: instr > instr,Xs: list_instr] :
( ( ( map_instr_instr @ F @ Xs )
= nil_instr )
= ( Xs = nil_instr ) ) ).
% map_is_Nil_conv
thf(fact_799_map__is__Nil__conv,axiom,
! [F: int > instr,Xs: list_int] :
( ( ( map_int_instr @ F @ Xs )
= nil_instr )
= ( Xs = nil_int ) ) ).
% map_is_Nil_conv
thf(fact_800_map__is__Nil__conv,axiom,
! [F: produc6425607678544837394st_int > int,Xs: list_P419547167332244514st_int] :
( ( ( map_Pr3224476576807795204nt_int @ F @ Xs )
= nil_int )
= ( Xs = nil_Pr749310521696113548st_int ) ) ).
% map_is_Nil_conv
thf(fact_801_map__is__Nil__conv,axiom,
! [F: instr > int,Xs: list_instr] :
( ( ( map_instr_int @ F @ Xs )
= nil_int )
= ( Xs = nil_instr ) ) ).
% map_is_Nil_conv
thf(fact_802_map__is__Nil__conv,axiom,
! [F: int > int,Xs: list_int] :
( ( ( map_int_int @ F @ Xs )
= nil_int )
= ( Xs = nil_int ) ) ).
% map_is_Nil_conv
thf(fact_803_map__is__Nil__conv,axiom,
! [F: produc6425607678544837394st_int > produc6425607678544837394st_int,Xs: list_P419547167332244514st_int] :
( ( ( map_Pr8625093460839654122st_int @ F @ Xs )
= nil_Pr749310521696113548st_int )
= ( Xs = nil_Pr749310521696113548st_int ) ) ).
% map_is_Nil_conv
thf(fact_804_append_Oright__neutral,axiom,
! [A: list_P419547167332244514st_int] :
( ( append5028292981072010935st_int @ A @ nil_Pr749310521696113548st_int )
= A ) ).
% append.right_neutral
thf(fact_805_append_Oright__neutral,axiom,
! [A: list_instr] :
( ( append_instr @ A @ nil_instr )
= A ) ).
% append.right_neutral
thf(fact_806_append_Oright__neutral,axiom,
! [A: list_int] :
( ( append_int @ A @ nil_int )
= A ) ).
% append.right_neutral
thf(fact_807_append__Nil2,axiom,
! [Xs: list_P419547167332244514st_int] :
( ( append5028292981072010935st_int @ Xs @ nil_Pr749310521696113548st_int )
= Xs ) ).
% append_Nil2
thf(fact_808_append__Nil2,axiom,
! [Xs: list_instr] :
( ( append_instr @ Xs @ nil_instr )
= Xs ) ).
% append_Nil2
thf(fact_809_append__Nil2,axiom,
! [Xs: list_int] :
( ( append_int @ Xs @ nil_int )
= Xs ) ).
% append_Nil2
thf(fact_810_append__self__conv,axiom,
! [Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int] :
( ( ( append5028292981072010935st_int @ Xs @ Ys )
= Xs )
= ( Ys = nil_Pr749310521696113548st_int ) ) ).
% append_self_conv
thf(fact_811_append__self__conv,axiom,
! [Xs: list_instr,Ys: list_instr] :
( ( ( append_instr @ Xs @ Ys )
= Xs )
= ( Ys = nil_instr ) ) ).
% append_self_conv
thf(fact_812_append__self__conv,axiom,
! [Xs: list_int,Ys: list_int] :
( ( ( append_int @ Xs @ Ys )
= Xs )
= ( Ys = nil_int ) ) ).
% append_self_conv
thf(fact_813_self__append__conv,axiom,
! [Y: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int] :
( ( Y
= ( append5028292981072010935st_int @ Y @ Ys ) )
= ( Ys = nil_Pr749310521696113548st_int ) ) ).
% self_append_conv
thf(fact_814_self__append__conv,axiom,
! [Y: list_instr,Ys: list_instr] :
( ( Y
= ( append_instr @ Y @ Ys ) )
= ( Ys = nil_instr ) ) ).
% self_append_conv
thf(fact_815_self__append__conv,axiom,
! [Y: list_int,Ys: list_int] :
( ( Y
= ( append_int @ Y @ Ys ) )
= ( Ys = nil_int ) ) ).
% self_append_conv
thf(fact_816_append__self__conv2,axiom,
! [Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int] :
( ( ( append5028292981072010935st_int @ Xs @ Ys )
= Ys )
= ( Xs = nil_Pr749310521696113548st_int ) ) ).
% append_self_conv2
thf(fact_817_append__self__conv2,axiom,
! [Xs: list_instr,Ys: list_instr] :
( ( ( append_instr @ Xs @ Ys )
= Ys )
= ( Xs = nil_instr ) ) ).
% append_self_conv2
thf(fact_818_append__self__conv2,axiom,
! [Xs: list_int,Ys: list_int] :
( ( ( append_int @ Xs @ Ys )
= Ys )
= ( Xs = nil_int ) ) ).
% append_self_conv2
thf(fact_819_self__append__conv2,axiom,
! [Y: list_P419547167332244514st_int,Xs: list_P419547167332244514st_int] :
( ( Y
= ( append5028292981072010935st_int @ Xs @ Y ) )
= ( Xs = nil_Pr749310521696113548st_int ) ) ).
% self_append_conv2
thf(fact_820_self__append__conv2,axiom,
! [Y: list_instr,Xs: list_instr] :
( ( Y
= ( append_instr @ Xs @ Y ) )
= ( Xs = nil_instr ) ) ).
% self_append_conv2
thf(fact_821_self__append__conv2,axiom,
! [Y: list_int,Xs: list_int] :
( ( Y
= ( append_int @ Xs @ Y ) )
= ( Xs = nil_int ) ) ).
% self_append_conv2
thf(fact_822_Nil__is__append__conv,axiom,
! [Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int] :
( ( nil_Pr749310521696113548st_int
= ( append5028292981072010935st_int @ Xs @ Ys ) )
= ( ( Xs = nil_Pr749310521696113548st_int )
& ( Ys = nil_Pr749310521696113548st_int ) ) ) ).
% Nil_is_append_conv
thf(fact_823_Nil__is__append__conv,axiom,
! [Xs: list_instr,Ys: list_instr] :
( ( nil_instr
= ( append_instr @ Xs @ Ys ) )
= ( ( Xs = nil_instr )
& ( Ys = nil_instr ) ) ) ).
% Nil_is_append_conv
thf(fact_824_Nil__is__append__conv,axiom,
! [Xs: list_int,Ys: list_int] :
( ( nil_int
= ( append_int @ Xs @ Ys ) )
= ( ( Xs = nil_int )
& ( Ys = nil_int ) ) ) ).
% Nil_is_append_conv
thf(fact_825_append__is__Nil__conv,axiom,
! [Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int] :
( ( ( append5028292981072010935st_int @ Xs @ Ys )
= nil_Pr749310521696113548st_int )
= ( ( Xs = nil_Pr749310521696113548st_int )
& ( Ys = nil_Pr749310521696113548st_int ) ) ) ).
% append_is_Nil_conv
thf(fact_826_append__is__Nil__conv,axiom,
! [Xs: list_instr,Ys: list_instr] :
( ( ( append_instr @ Xs @ Ys )
= nil_instr )
= ( ( Xs = nil_instr )
& ( Ys = nil_instr ) ) ) ).
% append_is_Nil_conv
thf(fact_827_append__is__Nil__conv,axiom,
! [Xs: list_int,Ys: list_int] :
( ( ( append_int @ Xs @ Ys )
= nil_int )
= ( ( Xs = nil_int )
& ( Ys = nil_int ) ) ) ).
% append_is_Nil_conv
thf(fact_828_diff__diff__left,axiom,
! [I3: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J ) @ K )
= ( minus_minus_nat @ I3 @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_829_drop__drop,axiom,
! [N: nat,M: nat,Xs: list_P419547167332244514st_int] :
( ( drop_P7838799151754935884st_int @ N @ ( drop_P7838799151754935884st_int @ M @ Xs ) )
= ( drop_P7838799151754935884st_int @ ( plus_plus_nat @ N @ M ) @ Xs ) ) ).
% drop_drop
thf(fact_830_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_831_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_832_le__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel2
thf(fact_833_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_834_le__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel1
thf(fact_835_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_836_add__le__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel2
thf(fact_837_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_838_add__le__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel1
thf(fact_839_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_840_add__less__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_841_add__less__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel1
thf(fact_842_add__less__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_843_add__less__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel2
thf(fact_844_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_845_less__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel1
thf(fact_846_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_847_less__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel2
thf(fact_848_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_849_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_850_le__add__diff__inverse,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_851_le__add__diff__inverse,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_852_le__add__diff__inverse2,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_853_le__add__diff__inverse2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_854_diff__add__zero,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_855_of__nat__add,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_add
thf(fact_856_of__nat__add,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_add
thf(fact_857_length__0__conv,axiom,
! [Xs: list_int] :
( ( ( size_size_list_int @ Xs )
= zero_zero_nat )
= ( Xs = nil_int ) ) ).
% length_0_conv
thf(fact_858_length__0__conv,axiom,
! [Xs: list_P419547167332244514st_int] :
( ( ( size_s229347708945616566st_int @ Xs )
= zero_zero_nat )
= ( Xs = nil_Pr749310521696113548st_int ) ) ).
% length_0_conv
thf(fact_859_length__0__conv,axiom,
! [Xs: list_instr] :
( ( ( size_size_list_instr @ Xs )
= zero_zero_nat )
= ( Xs = nil_instr ) ) ).
% length_0_conv
thf(fact_860_length__0__conv,axiom,
! [Xs: list_list_char] :
( ( ( size_s356637359517785349t_char @ Xs )
= zero_zero_nat )
= ( Xs = nil_list_char ) ) ).
% length_0_conv
thf(fact_861_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_862_append1__eq__conv,axiom,
! [Xs: list_P419547167332244514st_int,X: produc6425607678544837394st_int,Ys: list_P419547167332244514st_int,Y: produc6425607678544837394st_int] :
( ( ( append5028292981072010935st_int @ Xs @ ( cons_P4151473878978182108st_int @ X @ nil_Pr749310521696113548st_int ) )
= ( append5028292981072010935st_int @ Ys @ ( cons_P4151473878978182108st_int @ Y @ nil_Pr749310521696113548st_int ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_863_append1__eq__conv,axiom,
! [Xs: list_instr,X: instr,Ys: list_instr,Y: instr] :
( ( ( append_instr @ Xs @ ( cons_instr @ X @ nil_instr ) )
= ( append_instr @ Ys @ ( cons_instr @ Y @ nil_instr ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_864_append1__eq__conv,axiom,
! [Xs: list_int,X: int,Ys: list_int,Y: int] :
( ( ( append_int @ Xs @ ( cons_int @ X @ nil_int ) )
= ( append_int @ Ys @ ( cons_int @ Y @ nil_int ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_865_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I3 @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I3 @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_866_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I3 )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I3 ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_867_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I3 @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I3 @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_868_length__append,axiom,
! [Xs: list_int,Ys: list_int] :
( ( size_size_list_int @ ( append_int @ Xs @ Ys ) )
= ( plus_plus_nat @ ( size_size_list_int @ Xs ) @ ( size_size_list_int @ Ys ) ) ) ).
% length_append
thf(fact_869_length__append,axiom,
! [Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int] :
( ( size_s229347708945616566st_int @ ( append5028292981072010935st_int @ Xs @ Ys ) )
= ( plus_plus_nat @ ( size_s229347708945616566st_int @ Xs ) @ ( size_s229347708945616566st_int @ Ys ) ) ) ).
% length_append
thf(fact_870_length__append,axiom,
! [Xs: list_instr,Ys: list_instr] :
( ( size_size_list_instr @ ( append_instr @ Xs @ Ys ) )
= ( plus_plus_nat @ ( size_size_list_instr @ Xs ) @ ( size_size_list_instr @ Ys ) ) ) ).
% length_append
thf(fact_871_length__append,axiom,
! [Xs: list_list_char,Ys: list_list_char] :
( ( size_s356637359517785349t_char @ ( append_list_char @ Xs @ Ys ) )
= ( plus_plus_nat @ ( size_s356637359517785349t_char @ Xs ) @ ( size_s356637359517785349t_char @ Ys ) ) ) ).
% length_append
thf(fact_872_take__eq__Nil2,axiom,
! [N: nat,Xs: list_P419547167332244514st_int] :
( ( nil_Pr749310521696113548st_int
= ( take_P1004856038796978596st_int @ N @ Xs ) )
= ( ( N = zero_zero_nat )
| ( Xs = nil_Pr749310521696113548st_int ) ) ) ).
% take_eq_Nil2
thf(fact_873_take__eq__Nil2,axiom,
! [N: nat,Xs: list_instr] :
( ( nil_instr
= ( take_instr @ N @ Xs ) )
= ( ( N = zero_zero_nat )
| ( Xs = nil_instr ) ) ) ).
% take_eq_Nil2
thf(fact_874_take__eq__Nil2,axiom,
! [N: nat,Xs: list_int] :
( ( nil_int
= ( take_int @ N @ Xs ) )
= ( ( N = zero_zero_nat )
| ( Xs = nil_int ) ) ) ).
% take_eq_Nil2
thf(fact_875_take__eq__Nil,axiom,
! [N: nat,Xs: list_P419547167332244514st_int] :
( ( ( take_P1004856038796978596st_int @ N @ Xs )
= nil_Pr749310521696113548st_int )
= ( ( N = zero_zero_nat )
| ( Xs = nil_Pr749310521696113548st_int ) ) ) ).
% take_eq_Nil
thf(fact_876_take__eq__Nil,axiom,
! [N: nat,Xs: list_instr] :
( ( ( take_instr @ N @ Xs )
= nil_instr )
= ( ( N = zero_zero_nat )
| ( Xs = nil_instr ) ) ) ).
% take_eq_Nil
thf(fact_877_take__eq__Nil,axiom,
! [N: nat,Xs: list_int] :
( ( ( take_int @ N @ Xs )
= nil_int )
= ( ( N = zero_zero_nat )
| ( Xs = nil_int ) ) ) ).
% take_eq_Nil
thf(fact_878_take0,axiom,
( ( take_P1004856038796978596st_int @ zero_zero_nat )
= ( ^ [Xs2: list_P419547167332244514st_int] : nil_Pr749310521696113548st_int ) ) ).
% take0
thf(fact_879_take0,axiom,
( ( take_instr @ zero_zero_nat )
= ( ^ [Xs2: list_instr] : nil_instr ) ) ).
% take0
thf(fact_880_take0,axiom,
( ( take_int @ zero_zero_nat )
= ( ^ [Xs2: list_int] : nil_int ) ) ).
% take0
thf(fact_881_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ M ) )
= ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% of_nat_Suc
thf(fact_882_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ M ) )
= ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M ) ) ) ).
% of_nat_Suc
thf(fact_883_length__greater__0__conv,axiom,
! [Xs: list_int] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_int @ Xs ) )
= ( Xs != nil_int ) ) ).
% length_greater_0_conv
thf(fact_884_length__greater__0__conv,axiom,
! [Xs: list_P419547167332244514st_int] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s229347708945616566st_int @ Xs ) )
= ( Xs != nil_Pr749310521696113548st_int ) ) ).
% length_greater_0_conv
thf(fact_885_length__greater__0__conv,axiom,
! [Xs: list_instr] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_instr @ Xs ) )
= ( Xs != nil_instr ) ) ).
% length_greater_0_conv
thf(fact_886_length__greater__0__conv,axiom,
! [Xs: list_list_char] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s356637359517785349t_char @ Xs ) )
= ( Xs != nil_list_char ) ) ).
% length_greater_0_conv
thf(fact_887_diff__Suc__diff__eq2,axiom,
! [K: nat,J: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I3 )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I3 ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_888_diff__Suc__diff__eq1,axiom,
! [K: nat,J: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I3 @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I3 @ K ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_889_nth__append__length__plus,axiom,
! [Xs: list_int,Ys: list_int,N: nat] :
( ( nth_int @ ( append_int @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_size_list_int @ Xs ) @ N ) )
= ( nth_int @ Ys @ N ) ) ).
% nth_append_length_plus
thf(fact_890_nth__append__length__plus,axiom,
! [Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int,N: nat] :
( ( nth_Pr8327173284404383691st_int @ ( append5028292981072010935st_int @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_s229347708945616566st_int @ Xs ) @ N ) )
= ( nth_Pr8327173284404383691st_int @ Ys @ N ) ) ).
% nth_append_length_plus
thf(fact_891_nth__append__length__plus,axiom,
! [Xs: list_instr,Ys: list_instr,N: nat] :
( ( nth_instr @ ( append_instr @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_size_list_instr @ Xs ) @ N ) )
= ( nth_instr @ Ys @ N ) ) ).
% nth_append_length_plus
thf(fact_892_nth__append__length__plus,axiom,
! [Xs: list_list_char,Ys: list_list_char,N: nat] :
( ( nth_list_char @ ( append_list_char @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_s356637359517785349t_char @ Xs ) @ N ) )
= ( nth_list_char @ Ys @ N ) ) ).
% nth_append_length_plus
thf(fact_893_drop__eq__Nil2,axiom,
! [N: nat,Xs: list_int] :
( ( nil_int
= ( drop_int @ N @ Xs ) )
= ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ N ) ) ).
% drop_eq_Nil2
thf(fact_894_drop__eq__Nil2,axiom,
! [N: nat,Xs: list_P419547167332244514st_int] :
( ( nil_Pr749310521696113548st_int
= ( drop_P7838799151754935884st_int @ N @ Xs ) )
= ( ord_less_eq_nat @ ( size_s229347708945616566st_int @ Xs ) @ N ) ) ).
% drop_eq_Nil2
thf(fact_895_drop__eq__Nil2,axiom,
! [N: nat,Xs: list_instr] :
( ( nil_instr
= ( drop_instr @ N @ Xs ) )
= ( ord_less_eq_nat @ ( size_size_list_instr @ Xs ) @ N ) ) ).
% drop_eq_Nil2
thf(fact_896_drop__eq__Nil2,axiom,
! [N: nat,Xs: list_list_char] :
( ( nil_list_char
= ( drop_list_char @ N @ Xs ) )
= ( ord_less_eq_nat @ ( size_s356637359517785349t_char @ Xs ) @ N ) ) ).
% drop_eq_Nil2
thf(fact_897_drop__eq__Nil,axiom,
! [N: nat,Xs: list_int] :
( ( ( drop_int @ N @ Xs )
= nil_int )
= ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ N ) ) ).
% drop_eq_Nil
thf(fact_898_drop__eq__Nil,axiom,
! [N: nat,Xs: list_P419547167332244514st_int] :
( ( ( drop_P7838799151754935884st_int @ N @ Xs )
= nil_Pr749310521696113548st_int )
= ( ord_less_eq_nat @ ( size_s229347708945616566st_int @ Xs ) @ N ) ) ).
% drop_eq_Nil
thf(fact_899_drop__eq__Nil,axiom,
! [N: nat,Xs: list_instr] :
( ( ( drop_instr @ N @ Xs )
= nil_instr )
= ( ord_less_eq_nat @ ( size_size_list_instr @ Xs ) @ N ) ) ).
% drop_eq_Nil
thf(fact_900_drop__eq__Nil,axiom,
! [N: nat,Xs: list_list_char] :
( ( ( drop_list_char @ N @ Xs )
= nil_list_char )
= ( ord_less_eq_nat @ ( size_s356637359517785349t_char @ Xs ) @ N ) ) ).
% drop_eq_Nil
thf(fact_901_drop__all,axiom,
! [Xs: list_int,N: nat] :
( ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ N )
=> ( ( drop_int @ N @ Xs )
= nil_int ) ) ).
% drop_all
thf(fact_902_drop__all,axiom,
! [Xs: list_P419547167332244514st_int,N: nat] :
( ( ord_less_eq_nat @ ( size_s229347708945616566st_int @ Xs ) @ N )
=> ( ( drop_P7838799151754935884st_int @ N @ Xs )
= nil_Pr749310521696113548st_int ) ) ).
% drop_all
thf(fact_903_drop__all,axiom,
! [Xs: list_instr,N: nat] :
( ( ord_less_eq_nat @ ( size_size_list_instr @ Xs ) @ N )
=> ( ( drop_instr @ N @ Xs )
= nil_instr ) ) ).
% drop_all
thf(fact_904_drop__all,axiom,
! [Xs: list_list_char,N: nat] :
( ( ord_less_eq_nat @ ( size_s356637359517785349t_char @ Xs ) @ N )
=> ( ( drop_list_char @ N @ Xs )
= nil_list_char ) ) ).
% drop_all
thf(fact_905_nth__drop,axiom,
! [N: nat,Xs: list_int,I3: nat] :
( ( ord_less_eq_nat @ N @ ( size_size_list_int @ Xs ) )
=> ( ( nth_int @ ( drop_int @ N @ Xs ) @ I3 )
= ( nth_int @ Xs @ ( plus_plus_nat @ N @ I3 ) ) ) ) ).
% nth_drop
thf(fact_906_nth__drop,axiom,
! [N: nat,Xs: list_P419547167332244514st_int,I3: nat] :
( ( ord_less_eq_nat @ N @ ( size_s229347708945616566st_int @ Xs ) )
=> ( ( nth_Pr8327173284404383691st_int @ ( drop_P7838799151754935884st_int @ N @ Xs ) @ I3 )
= ( nth_Pr8327173284404383691st_int @ Xs @ ( plus_plus_nat @ N @ I3 ) ) ) ) ).
% nth_drop
thf(fact_907_nth__drop,axiom,
! [N: nat,Xs: list_instr,I3: nat] :
( ( ord_less_eq_nat @ N @ ( size_size_list_instr @ Xs ) )
=> ( ( nth_instr @ ( drop_instr @ N @ Xs ) @ I3 )
= ( nth_instr @ Xs @ ( plus_plus_nat @ N @ I3 ) ) ) ) ).
% nth_drop
thf(fact_908_nth__drop,axiom,
! [N: nat,Xs: list_list_char,I3: nat] :
( ( ord_less_eq_nat @ N @ ( size_s356637359517785349t_char @ Xs ) )
=> ( ( nth_list_char @ ( drop_list_char @ N @ Xs ) @ I3 )
= ( nth_list_char @ Xs @ ( plus_plus_nat @ N @ I3 ) ) ) ) ).
% nth_drop
thf(fact_909_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C2: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_910_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: int,B: int,C2: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C2 )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_911_is__num__normalize_I1_J,axiom,
! [A: int,B: int,C2: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C2 )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C2 ) ) ) ).
% is_num_normalize(1)
thf(fact_912_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I3: nat,J: nat,K: nat,L: nat] :
( ( ( I3 = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I3 @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_913_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I3: int,J: int,K: int,L: int] :
( ( ( I3 = J )
& ( K = L ) )
=> ( ( plus_plus_int @ I3 @ K )
= ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_914_group__cancel_Oadd1,axiom,
! [A3: nat,K: nat,A: nat,B: nat] :
( ( A3
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A3 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_915_group__cancel_Oadd1,axiom,
! [A3: int,K: int,A: int,B: int] :
( ( A3
= ( plus_plus_int @ K @ A ) )
=> ( ( plus_plus_int @ A3 @ B )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_916_group__cancel_Oadd2,axiom,
! [B5: nat,K: nat,B: nat,A: nat] :
( ( B5
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B5 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_917_group__cancel_Oadd2,axiom,
! [B5: int,K: int,B: int,A: int] :
( ( B5
= ( plus_plus_int @ K @ B ) )
=> ( ( plus_plus_int @ A @ B5 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_918_add_Oassoc,axiom,
! [A: nat,B: nat,C2: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add.assoc
thf(fact_919_add_Oassoc,axiom,
! [A: int,B: int,C2: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C2 )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C2 ) ) ) ).
% add.assoc
thf(fact_920_add_Oleft__cancel,axiom,
! [A: int,B: int,C2: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C2 ) )
= ( B = C2 ) ) ).
% add.left_cancel
thf(fact_921_add_Oright__cancel,axiom,
! [B: int,A: int,C2: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C2 @ A ) )
= ( B = C2 ) ) ).
% add.right_cancel
thf(fact_922_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A5: nat,B4: nat] : ( plus_plus_nat @ B4 @ A5 ) ) ) ).
% add.commute
thf(fact_923_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A5: int,B4: int] : ( plus_plus_int @ B4 @ A5 ) ) ) ).
% add.commute
thf(fact_924_add_Oleft__commute,axiom,
! [B: nat,A: nat,C2: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C2 ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add.left_commute
thf(fact_925_add_Oleft__commute,axiom,
! [B: int,A: int,C2: int] :
( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C2 ) )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C2 ) ) ) ).
% add.left_commute
thf(fact_926_add__left__imp__eq,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C2 ) )
=> ( B = C2 ) ) ).
% add_left_imp_eq
thf(fact_927_add__left__imp__eq,axiom,
! [A: int,B: int,C2: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C2 ) )
=> ( B = C2 ) ) ).
% add_left_imp_eq
thf(fact_928_add__right__imp__eq,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C2 @ A ) )
=> ( B = C2 ) ) ).
% add_right_imp_eq
thf(fact_929_add__right__imp__eq,axiom,
! [B: int,A: int,C2: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C2 @ A ) )
=> ( B = C2 ) ) ).
% add_right_imp_eq
thf(fact_930_execl_Ocases,axiom,
! [X: produc4400946599758736694st_int] :
( ! [P6: list_instr,Cf3: produc6425607678544837394st_int,Cf4: produc6425607678544837394st_int,Cfs5: list_P419547167332244514st_int] :
( X
!= ( produc7764413618310988720st_int @ P6 @ ( cons_P4151473878978182108st_int @ Cf3 @ ( cons_P4151473878978182108st_int @ Cf4 @ Cfs5 ) ) ) )
=> ( ! [P6: list_instr] :
( X
!= ( produc7764413618310988720st_int @ P6 @ nil_Pr749310521696113548st_int ) )
=> ~ ! [P6: list_instr,V: produc6425607678544837394st_int] :
( X
!= ( produc7764413618310988720st_int @ P6 @ ( cons_P4151473878978182108st_int @ V @ nil_Pr749310521696113548st_int ) ) ) ) ) ).
% execl.cases
thf(fact_931_execl_Oelims_I2_J,axiom,
! [X: list_instr,Xa: list_P419547167332244514st_int] :
( ( execl @ X @ Xa )
=> ( ! [Cf3: produc6425607678544837394st_int,Cf4: produc6425607678544837394st_int,Cfs5: list_P419547167332244514st_int] :
( ( Xa
= ( cons_P4151473878978182108st_int @ Cf3 @ ( cons_P4151473878978182108st_int @ Cf4 @ Cfs5 ) ) )
=> ~ ( ( exec1 @ X @ Cf3 @ Cf4 )
& ( execl @ X @ ( cons_P4151473878978182108st_int @ Cf4 @ Cfs5 ) ) ) )
=> ( ( Xa != nil_Pr749310521696113548st_int )
=> ~ ! [V: produc6425607678544837394st_int] :
( Xa
!= ( cons_P4151473878978182108st_int @ V @ nil_Pr749310521696113548st_int ) ) ) ) ) ).
% execl.elims(2)
thf(fact_932_execl_Oelims_I1_J,axiom,
! [X: list_instr,Xa: list_P419547167332244514st_int,Y: $o] :
( ( ( execl @ X @ Xa )
= Y )
=> ( ! [Cf3: produc6425607678544837394st_int,Cf4: produc6425607678544837394st_int,Cfs5: list_P419547167332244514st_int] :
( ( Xa
= ( cons_P4151473878978182108st_int @ Cf3 @ ( cons_P4151473878978182108st_int @ Cf4 @ Cfs5 ) ) )
=> ( Y
= ( ~ ( ( exec1 @ X @ Cf3 @ Cf4 )
& ( execl @ X @ ( cons_P4151473878978182108st_int @ Cf4 @ Cfs5 ) ) ) ) ) )
=> ( ( ( Xa = nil_Pr749310521696113548st_int )
=> ~ Y )
=> ~ ( ? [V: produc6425607678544837394st_int] :
( Xa
= ( cons_P4151473878978182108st_int @ V @ nil_Pr749310521696113548st_int ) )
=> ~ Y ) ) ) ) ).
% execl.elims(1)
thf(fact_933_transpose_Ocases,axiom,
! [X: list_l6422821129637969586st_int] :
( ( X != nil_li4597314861645524892st_int )
=> ( ! [Xss: list_l6422821129637969586st_int] :
( X
!= ( cons_l8648290059330449388st_int @ nil_Pr749310521696113548st_int @ Xss ) )
=> ~ ! [X4: produc6425607678544837394st_int,Xs3: list_P419547167332244514st_int,Xss: list_l6422821129637969586st_int] :
( X
!= ( cons_l8648290059330449388st_int @ ( cons_P4151473878978182108st_int @ X4 @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_934_transpose_Ocases,axiom,
! [X: list_list_instr] :
( ( X != nil_list_instr )
=> ( ! [Xss: list_list_instr] :
( X
!= ( cons_list_instr @ nil_instr @ Xss ) )
=> ~ ! [X4: instr,Xs3: list_instr,Xss: list_list_instr] :
( X
!= ( cons_list_instr @ ( cons_instr @ X4 @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_935_transpose_Ocases,axiom,
! [X: list_list_int] :
( ( X != nil_list_int )
=> ( ! [Xss: list_list_int] :
( X
!= ( cons_list_int @ nil_int @ Xss ) )
=> ~ ! [X4: int,Xs3: list_int,Xss: list_list_int] :
( X
!= ( cons_list_int @ ( cons_int @ X4 @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_936_nat_Ocase__distrib,axiom,
! [H: list_P419547167332244514st_int > list_P419547167332244514st_int,F1: list_P419547167332244514st_int,F22: nat > list_P419547167332244514st_int,Nat: nat] :
( ( H @ ( case_n3094275698756463160st_int @ F1 @ F22 @ Nat ) )
= ( case_n3094275698756463160st_int @ ( H @ F1 )
@ ^ [X3: nat] : ( H @ ( F22 @ X3 ) )
@ Nat ) ) ).
% nat.case_distrib
thf(fact_937_nat_Ocase__distrib,axiom,
! [H: list_P419547167332244514st_int > $o,F1: list_P419547167332244514st_int,F22: nat > list_P419547167332244514st_int,Nat: nat] :
( ( H @ ( case_n3094275698756463160st_int @ F1 @ F22 @ Nat ) )
= ( case_nat_o @ ( H @ F1 )
@ ^ [X3: nat] : ( H @ ( F22 @ X3 ) )
@ Nat ) ) ).
% nat.case_distrib
thf(fact_938_nat_Ocase__distrib,axiom,
! [H: list_P419547167332244514st_int > nat,F1: list_P419547167332244514st_int,F22: nat > list_P419547167332244514st_int,Nat: nat] :
( ( H @ ( case_n3094275698756463160st_int @ F1 @ F22 @ Nat ) )
= ( case_nat_nat @ ( H @ F1 )
@ ^ [X3: nat] : ( H @ ( F22 @ X3 ) )
@ Nat ) ) ).
% nat.case_distrib
thf(fact_939_nat_Ocase__distrib,axiom,
! [H: $o > list_P419547167332244514st_int,F1: $o,F22: nat > $o,Nat: nat] :
( ( H @ ( case_nat_o @ F1 @ F22 @ Nat ) )
= ( case_n3094275698756463160st_int @ ( H @ F1 )
@ ^ [X3: nat] : ( H @ ( F22 @ X3 ) )
@ Nat ) ) ).
% nat.case_distrib
thf(fact_940_nat_Ocase__distrib,axiom,
! [H: $o > $o,F1: $o,F22: nat > $o,Nat: nat] :
( ( H @ ( case_nat_o @ F1 @ F22 @ Nat ) )
= ( case_nat_o @ ( H @ F1 )
@ ^ [X3: nat] : ( H @ ( F22 @ X3 ) )
@ Nat ) ) ).
% nat.case_distrib
thf(fact_941_nat_Ocase__distrib,axiom,
! [H: $o > nat,F1: $o,F22: nat > $o,Nat: nat] :
( ( H @ ( case_nat_o @ F1 @ F22 @ Nat ) )
= ( case_nat_nat @ ( H @ F1 )
@ ^ [X3: nat] : ( H @ ( F22 @ X3 ) )
@ Nat ) ) ).
% nat.case_distrib
thf(fact_942_nat_Ocase__distrib,axiom,
! [H: nat > list_P419547167332244514st_int,F1: nat,F22: nat > nat,Nat: nat] :
( ( H @ ( case_nat_nat @ F1 @ F22 @ Nat ) )
= ( case_n3094275698756463160st_int @ ( H @ F1 )
@ ^ [X3: nat] : ( H @ ( F22 @ X3 ) )
@ Nat ) ) ).
% nat.case_distrib
thf(fact_943_nat_Ocase__distrib,axiom,
! [H: nat > $o,F1: nat,F22: nat > nat,Nat: nat] :
( ( H @ ( case_nat_nat @ F1 @ F22 @ Nat ) )
= ( case_nat_o @ ( H @ F1 )
@ ^ [X3: nat] : ( H @ ( F22 @ X3 ) )
@ Nat ) ) ).
% nat.case_distrib
thf(fact_944_nat_Ocase__distrib,axiom,
! [H: nat > nat,F1: nat,F22: nat > nat,Nat: nat] :
( ( H @ ( case_nat_nat @ F1 @ F22 @ Nat ) )
= ( case_nat_nat @ ( H @ F1 )
@ ^ [X3: nat] : ( H @ ( F22 @ X3 ) )
@ Nat ) ) ).
% nat.case_distrib
thf(fact_945_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I3: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I3 @ J )
& ( K = L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I3 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_946_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I3: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I3 @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_947_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I3: int,J: int,K: int,L: int] :
( ( ( I3 = J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I3 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_948_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I3: nat,J: nat,K: nat,L: nat] :
( ( ( I3 = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_949_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I3: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I3 @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I3 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_950_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I3: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I3 @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_951_add__mono,axiom,
! [A: int,B: int,C2: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C2 @ D )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_mono
thf(fact_952_add__mono,axiom,
! [A: nat,B: nat,C2: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_953_add__left__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) ) ) ).
% add_left_mono
thf(fact_954_add__left__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) ) ) ).
% add_left_mono
thf(fact_955_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C: nat] :
( B
!= ( plus_plus_nat @ A @ C ) ) ) ).
% less_eqE
thf(fact_956_add__right__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) ) ) ).
% add_right_mono
thf(fact_957_add__right__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add_right_mono
thf(fact_958_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B4: nat] :
? [C3: nat] :
( B4
= ( plus_plus_nat @ A5 @ C3 ) ) ) ) ).
% le_iff_add
thf(fact_959_add__le__imp__le__left,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_960_add__le__imp__le__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_961_add__le__imp__le__right,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_962_add__le__imp__le__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_963_add_Ogroup__left__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add.group_left_neutral
thf(fact_964_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_965_add_Ocomm__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.comm_neutral
thf(fact_966_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_967_comm__monoid__add__class_Oadd__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_968_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_969_verit__sum__simplify,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% verit_sum_simplify
thf(fact_970_add__mono__thms__linordered__field_I5_J,axiom,
! [I3: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I3 @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_971_add__mono__thms__linordered__field_I5_J,axiom,
! [I3: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I3 @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I3 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_972_add__mono__thms__linordered__field_I2_J,axiom,
! [I3: nat,J: nat,K: nat,L: nat] :
( ( ( I3 = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_973_add__mono__thms__linordered__field_I2_J,axiom,
! [I3: int,J: int,K: int,L: int] :
( ( ( I3 = J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I3 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_974_add__mono__thms__linordered__field_I1_J,axiom,
! [I3: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I3 @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_975_add__mono__thms__linordered__field_I1_J,axiom,
! [I3: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I3 @ J )
& ( K = L ) )
=> ( ord_less_int @ ( plus_plus_int @ I3 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_976_add__strict__mono,axiom,
! [A: nat,B: nat,C2: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C2 @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_977_add__strict__mono,axiom,
! [A: int,B: int,C2: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C2 @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_978_add__strict__left__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) ) ) ).
% add_strict_left_mono
thf(fact_979_add__strict__left__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) ) ) ).
% add_strict_left_mono
thf(fact_980_add__strict__right__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add_strict_right_mono
thf(fact_981_add__strict__right__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) ) ) ).
% add_strict_right_mono
thf(fact_982_add__less__imp__less__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_983_add__less__imp__less__left,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_984_add__less__imp__less__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_985_add__less__imp__less__right,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_986_group__cancel_Osub1,axiom,
! [A3: int,K: int,A: int,B: int] :
( ( A3
= ( plus_plus_int @ K @ A ) )
=> ( ( minus_minus_int @ A3 @ B )
= ( plus_plus_int @ K @ ( minus_minus_int @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_987_diff__eq__eq,axiom,
! [A: int,B: int,C2: int] :
( ( ( minus_minus_int @ A @ B )
= C2 )
= ( A
= ( plus_plus_int @ C2 @ B ) ) ) ).
% diff_eq_eq
thf(fact_988_eq__diff__eq,axiom,
! [A: int,C2: int,B: int] :
( ( A
= ( minus_minus_int @ C2 @ B ) )
= ( ( plus_plus_int @ A @ B )
= C2 ) ) ).
% eq_diff_eq
thf(fact_989_add__diff__eq,axiom,
! [A: int,B: int,C2: int] :
( ( plus_plus_int @ A @ ( minus_minus_int @ B @ C2 ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ C2 ) ) ).
% add_diff_eq
thf(fact_990_diff__diff__eq2,axiom,
! [A: int,B: int,C2: int] :
( ( minus_minus_int @ A @ ( minus_minus_int @ B @ C2 ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ C2 ) @ B ) ) ).
% diff_diff_eq2
thf(fact_991_diff__add__eq,axiom,
! [A: int,B: int,C2: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ C2 )
= ( minus_minus_int @ ( plus_plus_int @ A @ C2 ) @ B ) ) ).
% diff_add_eq
thf(fact_992_diff__add__eq__diff__diff__swap,axiom,
! [A: int,B: int,C2: int] :
( ( minus_minus_int @ A @ ( plus_plus_int @ B @ C2 ) )
= ( minus_minus_int @ ( minus_minus_int @ A @ C2 ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_993_add__implies__diff,axiom,
! [C2: int,B: int,A: int] :
( ( ( plus_plus_int @ C2 @ B )
= A )
=> ( C2
= ( minus_minus_int @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_994_add__implies__diff,axiom,
! [C2: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ C2 @ B )
= A )
=> ( C2
= ( minus_minus_nat @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_995_diff__diff__eq,axiom,
! [A: int,B: int,C2: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C2 )
= ( minus_minus_int @ A @ ( plus_plus_int @ B @ C2 ) ) ) ).
% diff_diff_eq
thf(fact_996_diff__diff__eq,axiom,
! [A: nat,B: nat,C2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C2 )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% diff_diff_eq
thf(fact_997_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_998_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_999_nat__arith_Osuc1,axiom,
! [A3: nat,K: nat,A: nat] :
( ( A3
= ( plus_plus_nat @ K @ A ) )
=> ( ( suc @ A3 )
= ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_1000_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_1001_add__Suc__shift,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_1002_add__lessD1,axiom,
! [I3: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I3 @ J ) @ K )
=> ( ord_less_nat @ I3 @ K ) ) ).
% add_lessD1
thf(fact_1003_add__less__mono,axiom,
! [I3: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_1004_not__add__less1,axiom,
! [I3: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I3 @ J ) @ I3 ) ).
% not_add_less1
thf(fact_1005_not__add__less2,axiom,
! [J: nat,I3: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I3 ) @ I3 ) ).
% not_add_less2
thf(fact_1006_add__less__mono1,axiom,
! [I3: nat,J: nat,K: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_1007_trans__less__add1,axiom,
! [I3: nat,J: nat,M: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ord_less_nat @ I3 @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_1008_trans__less__add2,axiom,
! [I3: nat,J: nat,M: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ord_less_nat @ I3 @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_1009_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_1010_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N4: nat] :
? [K4: nat] :
( N4
= ( plus_plus_nat @ M2 @ K4 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1011_trans__le__add2,axiom,
! [I3: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ord_less_eq_nat @ I3 @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_1012_trans__le__add1,axiom,
! [I3: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ord_less_eq_nat @ I3 @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_1013_add__le__mono1,axiom,
! [I3: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_1014_add__le__mono,axiom,
! [I3: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_1015_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N2: nat] :
( L
= ( plus_plus_nat @ K @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_1016_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_1017_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_1018_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_1019_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_1020_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_1021_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_1022_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_1023_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_1024_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_1025_list_Odistinct_I1_J,axiom,
! [X21: produc6425607678544837394st_int,X222: list_P419547167332244514st_int] :
( nil_Pr749310521696113548st_int
!= ( cons_P4151473878978182108st_int @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_1026_list_Odistinct_I1_J,axiom,
! [X21: instr,X222: list_instr] :
( nil_instr
!= ( cons_instr @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_1027_list_Odistinct_I1_J,axiom,
! [X21: int,X222: list_int] :
( nil_int
!= ( cons_int @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_1028_list_OdiscI,axiom,
! [List: list_P419547167332244514st_int,X21: produc6425607678544837394st_int,X222: list_P419547167332244514st_int] :
( ( List
= ( cons_P4151473878978182108st_int @ X21 @ X222 ) )
=> ( List != nil_Pr749310521696113548st_int ) ) ).
% list.discI
thf(fact_1029_list_OdiscI,axiom,
! [List: list_instr,X21: instr,X222: list_instr] :
( ( List
= ( cons_instr @ X21 @ X222 ) )
=> ( List != nil_instr ) ) ).
% list.discI
thf(fact_1030_list_OdiscI,axiom,
! [List: list_int,X21: int,X222: list_int] :
( ( List
= ( cons_int @ X21 @ X222 ) )
=> ( List != nil_int ) ) ).
% list.discI
thf(fact_1031_list_Oexhaust,axiom,
! [Y: list_P419547167332244514st_int] :
( ( Y != nil_Pr749310521696113548st_int )
=> ~ ! [X212: produc6425607678544837394st_int,X223: list_P419547167332244514st_int] :
( Y
!= ( cons_P4151473878978182108st_int @ X212 @ X223 ) ) ) ).
% list.exhaust
thf(fact_1032_list_Oexhaust,axiom,
! [Y: list_instr] :
( ( Y != nil_instr )
=> ~ ! [X212: instr,X223: list_instr] :
( Y
!= ( cons_instr @ X212 @ X223 ) ) ) ).
% list.exhaust
thf(fact_1033_list_Oexhaust,axiom,
! [Y: list_int] :
( ( Y != nil_int )
=> ~ ! [X212: int,X223: list_int] :
( Y
!= ( cons_int @ X212 @ X223 ) ) ) ).
% list.exhaust
thf(fact_1034_min__list_Ocases,axiom,
! [X: list_int] :
( ! [X4: int,Xs3: list_int] :
( X
!= ( cons_int @ X4 @ Xs3 ) )
=> ( X = nil_int ) ) ).
% min_list.cases
thf(fact_1035_remdups__adj_Ocases,axiom,
! [X: list_P419547167332244514st_int] :
( ( X != nil_Pr749310521696113548st_int )
=> ( ! [X4: produc6425607678544837394st_int] :
( X
!= ( cons_P4151473878978182108st_int @ X4 @ nil_Pr749310521696113548st_int ) )
=> ~ ! [X4: produc6425607678544837394st_int,Y3: produc6425607678544837394st_int,Xs3: list_P419547167332244514st_int] :
( X
!= ( cons_P4151473878978182108st_int @ X4 @ ( cons_P4151473878978182108st_int @ Y3 @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_1036_remdups__adj_Ocases,axiom,
! [X: list_instr] :
( ( X != nil_instr )
=> ( ! [X4: instr] :
( X
!= ( cons_instr @ X4 @ nil_instr ) )
=> ~ ! [X4: instr,Y3: instr,Xs3: list_instr] :
( X
!= ( cons_instr @ X4 @ ( cons_instr @ Y3 @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_1037_remdups__adj_Ocases,axiom,
! [X: list_int] :
( ( X != nil_int )
=> ( ! [X4: int] :
( X
!= ( cons_int @ X4 @ nil_int ) )
=> ~ ! [X4: int,Y3: int,Xs3: list_int] :
( X
!= ( cons_int @ X4 @ ( cons_int @ Y3 @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_1038_neq__Nil__conv,axiom,
! [Xs: list_P419547167332244514st_int] :
( ( Xs != nil_Pr749310521696113548st_int )
= ( ? [Y4: produc6425607678544837394st_int,Ys2: list_P419547167332244514st_int] :
( Xs
= ( cons_P4151473878978182108st_int @ Y4 @ Ys2 ) ) ) ) ).
% neq_Nil_conv
thf(fact_1039_neq__Nil__conv,axiom,
! [Xs: list_instr] :
( ( Xs != nil_instr )
= ( ? [Y4: instr,Ys2: list_instr] :
( Xs
= ( cons_instr @ Y4 @ Ys2 ) ) ) ) ).
% neq_Nil_conv
thf(fact_1040_neq__Nil__conv,axiom,
! [Xs: list_int] :
( ( Xs != nil_int )
= ( ? [Y4: int,Ys2: list_int] :
( Xs
= ( cons_int @ Y4 @ Ys2 ) ) ) ) ).
% neq_Nil_conv
thf(fact_1041_list__induct2_H,axiom,
! [P: list_P419547167332244514st_int > list_P419547167332244514st_int > $o,Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int] :
( ( P @ nil_Pr749310521696113548st_int @ nil_Pr749310521696113548st_int )
=> ( ! [X4: produc6425607678544837394st_int,Xs3: list_P419547167332244514st_int] : ( P @ ( cons_P4151473878978182108st_int @ X4 @ Xs3 ) @ nil_Pr749310521696113548st_int )
=> ( ! [Y3: produc6425607678544837394st_int,Ys4: list_P419547167332244514st_int] : ( P @ nil_Pr749310521696113548st_int @ ( cons_P4151473878978182108st_int @ Y3 @ Ys4 ) )
=> ( ! [X4: produc6425607678544837394st_int,Xs3: list_P419547167332244514st_int,Y3: produc6425607678544837394st_int,Ys4: list_P419547167332244514st_int] :
( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons_P4151473878978182108st_int @ X4 @ Xs3 ) @ ( cons_P4151473878978182108st_int @ Y3 @ Ys4 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_1042_list__induct2_H,axiom,
! [P: list_P419547167332244514st_int > list_instr > $o,Xs: list_P419547167332244514st_int,Ys: list_instr] :
( ( P @ nil_Pr749310521696113548st_int @ nil_instr )
=> ( ! [X4: produc6425607678544837394st_int,Xs3: list_P419547167332244514st_int] : ( P @ ( cons_P4151473878978182108st_int @ X4 @ Xs3 ) @ nil_instr )
=> ( ! [Y3: instr,Ys4: list_instr] : ( P @ nil_Pr749310521696113548st_int @ ( cons_instr @ Y3 @ Ys4 ) )
=> ( ! [X4: produc6425607678544837394st_int,Xs3: list_P419547167332244514st_int,Y3: instr,Ys4: list_instr] :
( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons_P4151473878978182108st_int @ X4 @ Xs3 ) @ ( cons_instr @ Y3 @ Ys4 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_1043_list__induct2_H,axiom,
! [P: list_P419547167332244514st_int > list_int > $o,Xs: list_P419547167332244514st_int,Ys: list_int] :
( ( P @ nil_Pr749310521696113548st_int @ nil_int )
=> ( ! [X4: produc6425607678544837394st_int,Xs3: list_P419547167332244514st_int] : ( P @ ( cons_P4151473878978182108st_int @ X4 @ Xs3 ) @ nil_int )
=> ( ! [Y3: int,Ys4: list_int] : ( P @ nil_Pr749310521696113548st_int @ ( cons_int @ Y3 @ Ys4 ) )
=> ( ! [X4: produc6425607678544837394st_int,Xs3: list_P419547167332244514st_int,Y3: int,Ys4: list_int] :
( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons_P4151473878978182108st_int @ X4 @ Xs3 ) @ ( cons_int @ Y3 @ Ys4 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_1044_list__induct2_H,axiom,
! [P: list_instr > list_P419547167332244514st_int > $o,Xs: list_instr,Ys: list_P419547167332244514st_int] :
( ( P @ nil_instr @ nil_Pr749310521696113548st_int )
=> ( ! [X4: instr,Xs3: list_instr] : ( P @ ( cons_instr @ X4 @ Xs3 ) @ nil_Pr749310521696113548st_int )
=> ( ! [Y3: produc6425607678544837394st_int,Ys4: list_P419547167332244514st_int] : ( P @ nil_instr @ ( cons_P4151473878978182108st_int @ Y3 @ Ys4 ) )
=> ( ! [X4: instr,Xs3: list_instr,Y3: produc6425607678544837394st_int,Ys4: list_P419547167332244514st_int] :
( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons_instr @ X4 @ Xs3 ) @ ( cons_P4151473878978182108st_int @ Y3 @ Ys4 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_1045_list__induct2_H,axiom,
! [P: list_instr > list_instr > $o,Xs: list_instr,Ys: list_instr] :
( ( P @ nil_instr @ nil_instr )
=> ( ! [X4: instr,Xs3: list_instr] : ( P @ ( cons_instr @ X4 @ Xs3 ) @ nil_instr )
=> ( ! [Y3: instr,Ys4: list_instr] : ( P @ nil_instr @ ( cons_instr @ Y3 @ Ys4 ) )
=> ( ! [X4: instr,Xs3: list_instr,Y3: instr,Ys4: list_instr] :
( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons_instr @ X4 @ Xs3 ) @ ( cons_instr @ Y3 @ Ys4 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_1046_list__induct2_H,axiom,
! [P: list_instr > list_int > $o,Xs: list_instr,Ys: list_int] :
( ( P @ nil_instr @ nil_int )
=> ( ! [X4: instr,Xs3: list_instr] : ( P @ ( cons_instr @ X4 @ Xs3 ) @ nil_int )
=> ( ! [Y3: int,Ys4: list_int] : ( P @ nil_instr @ ( cons_int @ Y3 @ Ys4 ) )
=> ( ! [X4: instr,Xs3: list_instr,Y3: int,Ys4: list_int] :
( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons_instr @ X4 @ Xs3 ) @ ( cons_int @ Y3 @ Ys4 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_1047_list__induct2_H,axiom,
! [P: list_int > list_P419547167332244514st_int > $o,Xs: list_int,Ys: list_P419547167332244514st_int] :
( ( P @ nil_int @ nil_Pr749310521696113548st_int )
=> ( ! [X4: int,Xs3: list_int] : ( P @ ( cons_int @ X4 @ Xs3 ) @ nil_Pr749310521696113548st_int )
=> ( ! [Y3: produc6425607678544837394st_int,Ys4: list_P419547167332244514st_int] : ( P @ nil_int @ ( cons_P4151473878978182108st_int @ Y3 @ Ys4 ) )
=> ( ! [X4: int,Xs3: list_int,Y3: produc6425607678544837394st_int,Ys4: list_P419547167332244514st_int] :
( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons_int @ X4 @ Xs3 ) @ ( cons_P4151473878978182108st_int @ Y3 @ Ys4 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_1048_list__induct2_H,axiom,
! [P: list_int > list_instr > $o,Xs: list_int,Ys: list_instr] :
( ( P @ nil_int @ nil_instr )
=> ( ! [X4: int,Xs3: list_int] : ( P @ ( cons_int @ X4 @ Xs3 ) @ nil_instr )
=> ( ! [Y3: instr,Ys4: list_instr] : ( P @ nil_int @ ( cons_instr @ Y3 @ Ys4 ) )
=> ( ! [X4: int,Xs3: list_int,Y3: instr,Ys4: list_instr] :
( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons_int @ X4 @ Xs3 ) @ ( cons_instr @ Y3 @ Ys4 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_1049_list__induct2_H,axiom,
! [P: list_int > list_int > $o,Xs: list_int,Ys: list_int] :
( ( P @ nil_int @ nil_int )
=> ( ! [X4: int,Xs3: list_int] : ( P @ ( cons_int @ X4 @ Xs3 ) @ nil_int )
=> ( ! [Y3: int,Ys4: list_int] : ( P @ nil_int @ ( cons_int @ Y3 @ Ys4 ) )
=> ( ! [X4: int,Xs3: list_int,Y3: int,Ys4: list_int] :
( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons_int @ X4 @ Xs3 ) @ ( cons_int @ Y3 @ Ys4 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_1050_list__nonempty__induct,axiom,
! [Xs: list_P419547167332244514st_int,P: list_P419547167332244514st_int > $o] :
( ( Xs != nil_Pr749310521696113548st_int )
=> ( ! [X4: produc6425607678544837394st_int] : ( P @ ( cons_P4151473878978182108st_int @ X4 @ nil_Pr749310521696113548st_int ) )
=> ( ! [X4: produc6425607678544837394st_int,Xs3: list_P419547167332244514st_int] :
( ( Xs3 != nil_Pr749310521696113548st_int )
=> ( ( P @ Xs3 )
=> ( P @ ( cons_P4151473878978182108st_int @ X4 @ Xs3 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_1051_list__nonempty__induct,axiom,
! [Xs: list_instr,P: list_instr > $o] :
( ( Xs != nil_instr )
=> ( ! [X4: instr] : ( P @ ( cons_instr @ X4 @ nil_instr ) )
=> ( ! [X4: instr,Xs3: list_instr] :
( ( Xs3 != nil_instr )
=> ( ( P @ Xs3 )
=> ( P @ ( cons_instr @ X4 @ Xs3 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_1052_list__nonempty__induct,axiom,
! [Xs: list_int,P: list_int > $o] :
( ( Xs != nil_int )
=> ( ! [X4: int] : ( P @ ( cons_int @ X4 @ nil_int ) )
=> ( ! [X4: int,Xs3: list_int] :
( ( Xs3 != nil_int )
=> ( ( P @ Xs3 )
=> ( P @ ( cons_int @ X4 @ Xs3 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_1053_successively_Ocases,axiom,
! [X: produc2468426419292222131st_int] :
( ! [P6: produc6425607678544837394st_int > produc6425607678544837394st_int > $o] :
( X
!= ( produc554038597315947693st_int @ P6 @ nil_Pr749310521696113548st_int ) )
=> ( ! [P6: produc6425607678544837394st_int > produc6425607678544837394st_int > $o,X4: produc6425607678544837394st_int] :
( X
!= ( produc554038597315947693st_int @ P6 @ ( cons_P4151473878978182108st_int @ X4 @ nil_Pr749310521696113548st_int ) ) )
=> ~ ! [P6: produc6425607678544837394st_int > produc6425607678544837394st_int > $o,X4: produc6425607678544837394st_int,Y3: produc6425607678544837394st_int,Xs3: list_P419547167332244514st_int] :
( X
!= ( produc554038597315947693st_int @ P6 @ ( cons_P4151473878978182108st_int @ X4 @ ( cons_P4151473878978182108st_int @ Y3 @ Xs3 ) ) ) ) ) ) ).
% successively.cases
thf(fact_1054_successively_Ocases,axiom,
! [X: produc7133927873007960030_instr] :
( ! [P6: instr > instr > $o] :
( X
!= ( produc7818391707689388878_instr @ P6 @ nil_instr ) )
=> ( ! [P6: instr > instr > $o,X4: instr] :
( X
!= ( produc7818391707689388878_instr @ P6 @ ( cons_instr @ X4 @ nil_instr ) ) )
=> ~ ! [P6: instr > instr > $o,X4: instr,Y3: instr,Xs3: list_instr] :
( X
!= ( produc7818391707689388878_instr @ P6 @ ( cons_instr @ X4 @ ( cons_instr @ Y3 @ Xs3 ) ) ) ) ) ) ).
% successively.cases
thf(fact_1055_successively_Ocases,axiom,
! [X: produc5834231552977413017st_int] :
( ! [P6: int > int > $o] :
( X
!= ( produc8618682346314911123st_int @ P6 @ nil_int ) )
=> ( ! [P6: int > int > $o,X4: int] :
( X
!= ( produc8618682346314911123st_int @ P6 @ ( cons_int @ X4 @ nil_int ) ) )
=> ~ ! [P6: int > int > $o,X4: int,Y3: int,Xs3: list_int] :
( X
!= ( produc8618682346314911123st_int @ P6 @ ( cons_int @ X4 @ ( cons_int @ Y3 @ Xs3 ) ) ) ) ) ) ).
% successively.cases
thf(fact_1056_sorted__wrt_Ocases,axiom,
! [X: produc2468426419292222131st_int] :
( ! [P6: produc6425607678544837394st_int > produc6425607678544837394st_int > $o] :
( X
!= ( produc554038597315947693st_int @ P6 @ nil_Pr749310521696113548st_int ) )
=> ~ ! [P6: produc6425607678544837394st_int > produc6425607678544837394st_int > $o,X4: produc6425607678544837394st_int,Ys4: list_P419547167332244514st_int] :
( X
!= ( produc554038597315947693st_int @ P6 @ ( cons_P4151473878978182108st_int @ X4 @ Ys4 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_1057_sorted__wrt_Ocases,axiom,
! [X: produc7133927873007960030_instr] :
( ! [P6: instr > instr > $o] :
( X
!= ( produc7818391707689388878_instr @ P6 @ nil_instr ) )
=> ~ ! [P6: instr > instr > $o,X4: instr,Ys4: list_instr] :
( X
!= ( produc7818391707689388878_instr @ P6 @ ( cons_instr @ X4 @ Ys4 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_1058_sorted__wrt_Ocases,axiom,
! [X: produc5834231552977413017st_int] :
( ! [P6: int > int > $o] :
( X
!= ( produc8618682346314911123st_int @ P6 @ nil_int ) )
=> ~ ! [P6: int > int > $o,X4: int,Ys4: list_int] :
( X
!= ( produc8618682346314911123st_int @ P6 @ ( cons_int @ X4 @ Ys4 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_1059_shuffles_Ocases,axiom,
! [X: produc5848496107505245989st_int] :
( ! [Ys4: list_P419547167332244514st_int] :
( X
!= ( produc2639222612625641053st_int @ nil_Pr749310521696113548st_int @ Ys4 ) )
=> ( ! [Xs3: list_P419547167332244514st_int] :
( X
!= ( produc2639222612625641053st_int @ Xs3 @ nil_Pr749310521696113548st_int ) )
=> ~ ! [X4: produc6425607678544837394st_int,Xs3: list_P419547167332244514st_int,Y3: produc6425607678544837394st_int,Ys4: list_P419547167332244514st_int] :
( X
!= ( produc2639222612625641053st_int @ ( cons_P4151473878978182108st_int @ X4 @ Xs3 ) @ ( cons_P4151473878978182108st_int @ Y3 @ Ys4 ) ) ) ) ) ).
% shuffles.cases
thf(fact_1060_shuffles_Ocases,axiom,
! [X: produc8891168062546637159_instr] :
( ! [Ys4: list_instr] :
( X
!= ( produc7750586281064920791_instr @ nil_instr @ Ys4 ) )
=> ( ! [Xs3: list_instr] :
( X
!= ( produc7750586281064920791_instr @ Xs3 @ nil_instr ) )
=> ~ ! [X4: instr,Xs3: list_instr,Y3: instr,Ys4: list_instr] :
( X
!= ( produc7750586281064920791_instr @ ( cons_instr @ X4 @ Xs3 ) @ ( cons_instr @ Y3 @ Ys4 ) ) ) ) ) ).
% shuffles.cases
thf(fact_1061_shuffles_Ocases,axiom,
! [X: produc1186641810826059865st_int] :
( ! [Ys4: list_int] :
( X
!= ( produc364263696895485585st_int @ nil_int @ Ys4 ) )
=> ( ! [Xs3: list_int] :
( X
!= ( produc364263696895485585st_int @ Xs3 @ nil_int ) )
=> ~ ! [X4: int,Xs3: list_int,Y3: int,Ys4: list_int] :
( X
!= ( produc364263696895485585st_int @ ( cons_int @ X4 @ Xs3 ) @ ( cons_int @ Y3 @ Ys4 ) ) ) ) ) ).
% shuffles.cases
thf(fact_1062_splice_Ocases,axiom,
! [X: produc5848496107505245989st_int] :
( ! [Ys4: list_P419547167332244514st_int] :
( X
!= ( produc2639222612625641053st_int @ nil_Pr749310521696113548st_int @ Ys4 ) )
=> ~ ! [X4: produc6425607678544837394st_int,Xs3: list_P419547167332244514st_int,Ys4: list_P419547167332244514st_int] :
( X
!= ( produc2639222612625641053st_int @ ( cons_P4151473878978182108st_int @ X4 @ Xs3 ) @ Ys4 ) ) ) ).
% splice.cases
thf(fact_1063_splice_Ocases,axiom,
! [X: produc8891168062546637159_instr] :
( ! [Ys4: list_instr] :
( X
!= ( produc7750586281064920791_instr @ nil_instr @ Ys4 ) )
=> ~ ! [X4: instr,Xs3: list_instr,Ys4: list_instr] :
( X
!= ( produc7750586281064920791_instr @ ( cons_instr @ X4 @ Xs3 ) @ Ys4 ) ) ) ).
% splice.cases
thf(fact_1064_splice_Ocases,axiom,
! [X: produc1186641810826059865st_int] :
( ! [Ys4: list_int] :
( X
!= ( produc364263696895485585st_int @ nil_int @ Ys4 ) )
=> ~ ! [X4: int,Xs3: list_int,Ys4: list_int] :
( X
!= ( produc364263696895485585st_int @ ( cons_int @ X4 @ Xs3 ) @ Ys4 ) ) ) ).
% splice.cases
thf(fact_1065_list_Osimps_I8_J,axiom,
! [F: produc6425607678544837394st_int > instr] :
( ( map_Pr7514435936640407687_instr @ F @ nil_Pr749310521696113548st_int )
= nil_instr ) ).
% list.simps(8)
thf(fact_1066_list_Osimps_I8_J,axiom,
! [F: produc6425607678544837394st_int > int] :
( ( map_Pr3224476576807795204nt_int @ F @ nil_Pr749310521696113548st_int )
= nil_int ) ).
% list.simps(8)
thf(fact_1067_list_Osimps_I8_J,axiom,
! [F: instr > produc6425607678544837394st_int] :
( ( map_in7259965606695656109st_int @ F @ nil_instr )
= nil_Pr749310521696113548st_int ) ).
% list.simps(8)
thf(fact_1068_list_Osimps_I8_J,axiom,
! [F: instr > instr] :
( ( map_instr_instr @ F @ nil_instr )
= nil_instr ) ).
% list.simps(8)
thf(fact_1069_list_Osimps_I8_J,axiom,
! [F: instr > int] :
( ( map_instr_int @ F @ nil_instr )
= nil_int ) ).
% list.simps(8)
thf(fact_1070_list_Osimps_I8_J,axiom,
! [F: int > produc6425607678544837394st_int] :
( ( map_in8512336359594627332st_int @ F @ nil_int )
= nil_Pr749310521696113548st_int ) ).
% list.simps(8)
thf(fact_1071_list_Osimps_I8_J,axiom,
! [F: int > instr] :
( ( map_int_instr @ F @ nil_int )
= nil_instr ) ).
% list.simps(8)
thf(fact_1072_list_Osimps_I8_J,axiom,
! [F: int > int] :
( ( map_int_int @ F @ nil_int )
= nil_int ) ).
% list.simps(8)
thf(fact_1073_list_Osimps_I8_J,axiom,
! [F: produc6425607678544837394st_int > produc6425607678544837394st_int] :
( ( map_Pr8625093460839654122st_int @ F @ nil_Pr749310521696113548st_int )
= nil_Pr749310521696113548st_int ) ).
% list.simps(8)
thf(fact_1074_append__Nil,axiom,
! [Ys: list_P419547167332244514st_int] :
( ( append5028292981072010935st_int @ nil_Pr749310521696113548st_int @ Ys )
= Ys ) ).
% append_Nil
thf(fact_1075_append__Nil,axiom,
! [Ys: list_instr] :
( ( append_instr @ nil_instr @ Ys )
= Ys ) ).
% append_Nil
thf(fact_1076_append__Nil,axiom,
! [Ys: list_int] :
( ( append_int @ nil_int @ Ys )
= Ys ) ).
% append_Nil
thf(fact_1077_append_Oleft__neutral,axiom,
! [A: list_P419547167332244514st_int] :
( ( append5028292981072010935st_int @ nil_Pr749310521696113548st_int @ A )
= A ) ).
% append.left_neutral
thf(fact_1078_append_Oleft__neutral,axiom,
! [A: list_instr] :
( ( append_instr @ nil_instr @ A )
= A ) ).
% append.left_neutral
thf(fact_1079_append_Oleft__neutral,axiom,
! [A: list_int] :
( ( append_int @ nil_int @ A )
= A ) ).
% append.left_neutral
thf(fact_1080_eq__Nil__appendI,axiom,
! [Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int] :
( ( Xs = Ys )
=> ( Xs
= ( append5028292981072010935st_int @ nil_Pr749310521696113548st_int @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_1081_eq__Nil__appendI,axiom,
! [Xs: list_instr,Ys: list_instr] :
( ( Xs = Ys )
=> ( Xs
= ( append_instr @ nil_instr @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_1082_eq__Nil__appendI,axiom,
! [Xs: list_int,Ys: list_int] :
( ( Xs = Ys )
=> ( Xs
= ( append_int @ nil_int @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_1083_take__Nil,axiom,
! [N: nat] :
( ( take_P1004856038796978596st_int @ N @ nil_Pr749310521696113548st_int )
= nil_Pr749310521696113548st_int ) ).
% take_Nil
thf(fact_1084_take__Nil,axiom,
! [N: nat] :
( ( take_instr @ N @ nil_instr )
= nil_instr ) ).
% take_Nil
thf(fact_1085_take__Nil,axiom,
! [N: nat] :
( ( take_int @ N @ nil_int )
= nil_int ) ).
% take_Nil
thf(fact_1086_drop__Nil,axiom,
! [N: nat] :
( ( drop_P7838799151754935884st_int @ N @ nil_Pr749310521696113548st_int )
= nil_Pr749310521696113548st_int ) ).
% drop_Nil
thf(fact_1087_drop__Nil,axiom,
! [N: nat] :
( ( drop_instr @ N @ nil_instr )
= nil_instr ) ).
% drop_Nil
thf(fact_1088_drop__Nil,axiom,
! [N: nat] :
( ( drop_int @ N @ nil_int )
= nil_int ) ).
% drop_Nil
thf(fact_1089_execl_Osimps_I2_J,axiom,
! [P: list_instr] : ( execl @ P @ nil_Pr749310521696113548st_int ) ).
% execl.simps(2)
thf(fact_1090_execl_Osimps_I3_J,axiom,
! [P: list_instr,V2: produc6425607678544837394st_int] : ( execl @ P @ ( cons_P4151473878978182108st_int @ V2 @ nil_Pr749310521696113548st_int ) ) ).
% execl.simps(3)
thf(fact_1091_take__Cons,axiom,
! [N: nat,X: instr,Xs: list_instr] :
( ( take_instr @ N @ ( cons_instr @ X @ Xs ) )
= ( case_nat_list_instr @ nil_instr
@ ^ [M2: nat] : ( cons_instr @ X @ ( take_instr @ M2 @ Xs ) )
@ N ) ) ).
% take_Cons
thf(fact_1092_take__Cons,axiom,
! [N: nat,X: int,Xs: list_int] :
( ( take_int @ N @ ( cons_int @ X @ Xs ) )
= ( case_nat_list_int @ nil_int
@ ^ [M2: nat] : ( cons_int @ X @ ( take_int @ M2 @ Xs ) )
@ N ) ) ).
% take_Cons
thf(fact_1093_take__Cons,axiom,
! [N: nat,X: produc6425607678544837394st_int,Xs: list_P419547167332244514st_int] :
( ( take_P1004856038796978596st_int @ N @ ( cons_P4151473878978182108st_int @ X @ Xs ) )
= ( case_n3094275698756463160st_int @ nil_Pr749310521696113548st_int
@ ^ [M2: nat] : ( cons_P4151473878978182108st_int @ X @ ( take_P1004856038796978596st_int @ M2 @ Xs ) )
@ N ) ) ).
% take_Cons
thf(fact_1094_execl_Osimps_I1_J,axiom,
! [P: list_instr,Cf5: produc6425607678544837394st_int,Cf6: produc6425607678544837394st_int,Cfs2: list_P419547167332244514st_int] :
( ( execl @ P @ ( cons_P4151473878978182108st_int @ Cf5 @ ( cons_P4151473878978182108st_int @ Cf6 @ Cfs2 ) ) )
= ( ( exec1 @ P @ Cf5 @ Cf6 )
& ( execl @ P @ ( cons_P4151473878978182108st_int @ Cf6 @ Cfs2 ) ) ) ) ).
% execl.simps(1)
thf(fact_1095_execl_Oelims_I3_J,axiom,
! [X: list_instr,Xa: list_P419547167332244514st_int] :
( ~ ( execl @ X @ Xa )
=> ~ ! [Cf3: produc6425607678544837394st_int,Cf4: produc6425607678544837394st_int,Cfs5: list_P419547167332244514st_int] :
( ( Xa
= ( cons_P4151473878978182108st_int @ Cf3 @ ( cons_P4151473878978182108st_int @ Cf4 @ Cfs5 ) ) )
=> ( ( exec1 @ X @ Cf3 @ Cf4 )
& ( execl @ X @ ( cons_P4151473878978182108st_int @ Cf4 @ Cfs5 ) ) ) ) ) ).
% execl.elims(3)
thf(fact_1096_exec1__appendR,axiom,
! [P: list_instr,C2: produc6425607678544837394st_int,C4: produc6425607678544837394st_int,P3: list_instr] :
( ( exec1 @ P @ C2 @ C4 )
=> ( exec1 @ ( append_instr @ P @ P3 ) @ C2 @ C4 ) ) ).
% exec1_appendR
thf(fact_1097_old_Onat_Osimps_I4_J,axiom,
! [F1: list_P419547167332244514st_int,F22: nat > list_P419547167332244514st_int] :
( ( case_n3094275698756463160st_int @ F1 @ F22 @ zero_zero_nat )
= F1 ) ).
% old.nat.simps(4)
thf(fact_1098_old_Onat_Osimps_I4_J,axiom,
! [F1: $o,F22: nat > $o] :
( ( case_nat_o @ F1 @ F22 @ zero_zero_nat )
= F1 ) ).
% old.nat.simps(4)
thf(fact_1099_old_Onat_Osimps_I4_J,axiom,
! [F1: nat,F22: nat > nat] :
( ( case_nat_nat @ F1 @ F22 @ zero_zero_nat )
= F1 ) ).
% old.nat.simps(4)
thf(fact_1100_old_Onat_Osimps_I5_J,axiom,
! [F1: list_P419547167332244514st_int,F22: nat > list_P419547167332244514st_int,X22: nat] :
( ( case_n3094275698756463160st_int @ F1 @ F22 @ ( suc @ X22 ) )
= ( F22 @ X22 ) ) ).
% old.nat.simps(5)
thf(fact_1101_old_Onat_Osimps_I5_J,axiom,
! [F1: $o,F22: nat > $o,X22: nat] :
( ( case_nat_o @ F1 @ F22 @ ( suc @ X22 ) )
= ( F22 @ X22 ) ) ).
% old.nat.simps(5)
thf(fact_1102_old_Onat_Osimps_I5_J,axiom,
! [F1: nat,F22: nat > nat,X22: nat] :
( ( case_nat_nat @ F1 @ F22 @ ( suc @ X22 ) )
= ( F22 @ X22 ) ) ).
% old.nat.simps(5)
thf(fact_1103_add__decreasing,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ C2 @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ B ) ) ) ).
% add_decreasing
thf(fact_1104_add__decreasing,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C2 @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ B ) ) ) ).
% add_decreasing
thf(fact_1105_add__increasing,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C2 ) ) ) ) ).
% add_increasing
thf(fact_1106_add__increasing,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).
% add_increasing
thf(fact_1107_add__decreasing2,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_eq_int @ C2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ B ) ) ) ).
% add_decreasing2
thf(fact_1108_add__decreasing2,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ B ) ) ) ).
% add_decreasing2
thf(fact_1109_add__increasing2,axiom,
! [C2: int,B: int,A: int] :
( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C2 ) ) ) ) ).
% add_increasing2
thf(fact_1110_add__increasing2,axiom,
! [C2: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).
% add_increasing2
thf(fact_1111_add__nonneg__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_1112_add__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_1113_add__nonpos__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_nonpos
thf(fact_1114_add__nonpos__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_1115_add__nonneg__eq__0__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ( plus_plus_int @ X @ Y )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_1116_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_1117_add__nonpos__eq__0__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ zero_zero_int )
=> ( ( ord_less_eq_int @ Y @ zero_zero_int )
=> ( ( ( plus_plus_int @ X @ Y )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_1118_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_1119_add__mono__thms__linordered__field_I4_J,axiom,
! [I3: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I3 @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I3 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_1120_add__mono__thms__linordered__field_I4_J,axiom,
! [I3: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I3 @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_1121_add__mono__thms__linordered__field_I3_J,axiom,
! [I3: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I3 @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I3 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_1122_add__mono__thms__linordered__field_I3_J,axiom,
! [I3: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I3 @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_1123_add__le__less__mono,axiom,
! [A: int,B: int,C2: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ C2 @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_1124_add__le__less__mono,axiom,
! [A: nat,B: nat,C2: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C2 @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_1125_add__less__le__mono,axiom,
! [A: int,B: int,C2: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ C2 @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_1126_add__less__le__mono,axiom,
! [A: nat,B: nat,C2: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_1127_add__neg__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_1128_add__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_neg_neg
thf(fact_1129_add__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_1130_add__pos__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_1131_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C: nat] :
( ( B
= ( plus_plus_nat @ A @ C ) )
=> ( C = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_1132_pos__add__strict,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).
% pos_add_strict
thf(fact_1133_pos__add__strict,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ C2 )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C2 ) ) ) ) ).
% pos_add_strict
thf(fact_1134_add__less__zeroD,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ ( plus_plus_int @ X @ Y ) @ zero_zero_int )
=> ( ( ord_less_int @ X @ zero_zero_int )
| ( ord_less_int @ Y @ zero_zero_int ) ) ) ).
% add_less_zeroD
thf(fact_1135_add__le__imp__le__diff,axiom,
! [I3: int,K: int,N: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I3 @ K ) @ N )
=> ( ord_less_eq_int @ I3 @ ( minus_minus_int @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_1136_add__le__imp__le__diff,axiom,
! [I3: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ N )
=> ( ord_less_eq_nat @ I3 @ ( minus_minus_nat @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_1137_add__le__add__imp__diff__le,axiom,
! [I3: int,K: int,N: int,J: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I3 @ K ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ I3 @ K ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
=> ( ord_less_eq_int @ ( minus_minus_int @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_1138_add__le__add__imp__diff__le,axiom,
! [I3: nat,K: nat,N: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_1139_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ( ( minus_minus_nat @ B @ A )
= C2 )
= ( B
= ( plus_plus_nat @ C2 @ A ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_1140_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_1141_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ C2 @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C2 @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_1142_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C2 ) @ A )
= ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_1143_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C2 )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C2 ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_1144_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C2 @ B ) @ A )
= ( plus_plus_nat @ C2 @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_1145_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ C2 @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C2 @ B ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_1146_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ ( minus_minus_nat @ B @ A ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_1147_le__add__diff,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ C2 @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C2 ) @ A ) ) ) ).
% le_add_diff
thf(fact_1148_diff__add,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
= B ) ) ).
% diff_add
thf(fact_1149_le__diff__eq,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_eq_int @ A @ ( minus_minus_int @ C2 @ B ) )
= ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ C2 ) ) ).
% le_diff_eq
thf(fact_1150_diff__le__eq,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ ( minus_minus_int @ A @ B ) @ C2 )
= ( ord_less_eq_int @ A @ ( plus_plus_int @ C2 @ B ) ) ) ).
% diff_le_eq
thf(fact_1151_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_1152_less__add__one,axiom,
! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).
% less_add_one
thf(fact_1153_one__is__add,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M @ N ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_1154_add__is__1,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_1155_less__imp__add__positive,axiom,
! [I3: nat,J: nat] :
( ( ord_less_nat @ I3 @ J )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I3 @ K2 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1156_less__natE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ! [Q3: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M @ Q3 ) ) ) ) ).
% less_natE
thf(fact_1157_less__add__Suc1,axiom,
! [I3: nat,M: nat] : ( ord_less_nat @ I3 @ ( suc @ ( plus_plus_nat @ I3 @ M ) ) ) ).
% less_add_Suc1
thf(fact_1158_less__add__Suc2,axiom,
! [I3: nat,M: nat] : ( ord_less_nat @ I3 @ ( suc @ ( plus_plus_nat @ M @ I3 ) ) ) ).
% less_add_Suc2
thf(fact_1159_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M2: nat,N4: nat] :
? [K4: nat] :
( N4
= ( suc @ ( plus_plus_nat @ M2 @ K4 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1160_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ? [K2: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_1161_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M4: nat,N2: nat] :
( ( ord_less_nat @ M4 @ N2 )
=> ( ord_less_nat @ ( F @ M4 ) @ ( F @ N2 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1162_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1163_less__diff__conv,axiom,
! [I3: nat,J: nat,K: nat] :
( ( ord_less_nat @ I3 @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_1164_add__diff__inverse__nat,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_1165_Nat_Ole__imp__diff__is__add,axiom,
! [I3: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I3 @ J )
=> ( ( ( minus_minus_nat @ J @ I3 )
= K )
= ( J
= ( plus_plus_nat @ K @ I3 ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1166_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I3 ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I3 ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1167_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I3 @ J ) @ K )
= ( plus_plus_nat @ I3 @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1168_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I3 @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1169_le__diff__conv,axiom,
! [J: nat,K: nat,I3: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I3 )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I3 @ K ) ) ) ).
% le_diff_conv
thf(fact_1170_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_1171_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1172_Suc__eq__plus1,axiom,
( suc
= ( ^ [N4: nat] : ( plus_plus_nat @ N4 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_1173_nat__diff__split,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P @ zero_zero_nat ) )
& ! [D2: nat] :
( ( A
= ( plus_plus_nat @ B @ D2 ) )
=> ( P @ D2 ) ) ) ) ).
% nat_diff_split
thf(fact_1174_nat__diff__split__asm,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P @ zero_zero_nat ) )
| ? [D2: nat] :
( ( A
= ( plus_plus_nat @ B @ D2 ) )
& ~ ( P @ D2 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1175_less__diff__conv2,axiom,
! [K: nat,J: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I3 )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I3 @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_1176_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M2: nat,N4: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ N4 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N4 ) ) ) ) ) ).
% add_eq_if
thf(fact_1177_imp__le__cong,axiom,
! [X: int,X6: int,P: $o,P3: $o] :
( ( X = X6 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> ( P = P3 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> P )
= ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> P3 ) ) ) ) ).
% imp_le_cong
thf(fact_1178_conj__le__cong,axiom,
! [X: int,X6: int,P: $o,P3: $o] :
( ( X = X6 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> ( P = P3 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
& P )
= ( ( ord_less_eq_int @ zero_zero_int @ X6 )
& P3 ) ) ) ) ).
% conj_le_cong
thf(fact_1179_exec1I,axiom,
! [C4: produc6425607678544837394st_int,P: list_instr,I3: int,S: list_char > int,Stk2: list_int] :
( ( C4
= ( iexec @ ( inth_instr @ P @ I3 ) @ ( produc5086643055186798020st_int @ I3 @ ( produc8650753666468850689st_int @ S @ Stk2 ) ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ I3 )
=> ( ( ord_less_int @ I3 @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P ) ) )
=> ( exec1 @ P @ ( produc5086643055186798020st_int @ I3 @ ( produc8650753666468850689st_int @ S @ Stk2 ) ) @ C4 ) ) ) ) ).
% exec1I
thf(fact_1180_zle__add1__eq__le,axiom,
! [W: int,Z2: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z2 @ one_one_int ) )
= ( ord_less_eq_int @ W @ Z2 ) ) ).
% zle_add1_eq_le
thf(fact_1181_iexec__shift,axiom,
! [N: int,I5: int,S4: list_char > int,Stk3: list_int,Ins2: instr,I3: int,S: list_char > int,Stk2: list_int] :
( ( ( produc5086643055186798020st_int @ ( plus_plus_int @ N @ I5 ) @ ( produc8650753666468850689st_int @ S4 @ Stk3 ) )
= ( iexec @ Ins2 @ ( produc5086643055186798020st_int @ ( plus_plus_int @ N @ I3 ) @ ( produc8650753666468850689st_int @ S @ Stk2 ) ) ) )
= ( ( produc5086643055186798020st_int @ I5 @ ( produc8650753666468850689st_int @ S4 @ Stk3 ) )
= ( iexec @ Ins2 @ ( produc5086643055186798020st_int @ I3 @ ( produc8650753666468850689st_int @ S @ Stk2 ) ) ) ) ) ).
% iexec_shift
thf(fact_1182_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_1183_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus_int @ K @ zero_zero_int )
= K ) ).
% plus_int_code(1)
thf(fact_1184_nat_Odisc__eq__case_I2_J,axiom,
! [Nat: nat] :
( ( Nat != zero_zero_nat )
= ( case_nat_o @ $false
@ ^ [Uu: nat] : $true
@ Nat ) ) ).
% nat.disc_eq_case(2)
thf(fact_1185_nat_Odisc__eq__case_I1_J,axiom,
! [Nat: nat] :
( ( Nat = zero_zero_nat )
= ( case_nat_o @ $true
@ ^ [Uu: nat] : $false
@ Nat ) ) ).
% nat.disc_eq_case(1)
thf(fact_1186_odd__nonzero,axiom,
! [Z2: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_1187_int__ops_I5_J,axiom,
! [A: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A @ B ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(5)
thf(fact_1188_int__plus,axiom,
! [N: nat,M: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N @ M ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% int_plus
thf(fact_1189_zadd__int__left,axiom,
! [M: nat,N: nat,Z2: int] :
( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z2 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) ) @ Z2 ) ) ).
% zadd_int_left
thf(fact_1190_int__ge__induct,axiom,
! [K: int,I3: int,P: int > $o] :
( ( ord_less_eq_int @ K @ I3 )
=> ( ( P @ K )
=> ( ! [I: int] :
( ( ord_less_eq_int @ K @ I )
=> ( ( P @ I )
=> ( P @ ( plus_plus_int @ I @ one_one_int ) ) ) )
=> ( P @ I3 ) ) ) ) ).
% int_ge_induct
thf(fact_1191_zle__iff__zadd,axiom,
( ord_less_eq_int
= ( ^ [W2: int,Z4: int] :
? [N4: nat] :
( Z4
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ N4 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_1192_int__gr__induct,axiom,
! [K: int,I3: int,P: int > $o] :
( ( ord_less_int @ K @ I3 )
=> ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
=> ( ! [I: int] :
( ( ord_less_int @ K @ I )
=> ( ( P @ I )
=> ( P @ ( plus_plus_int @ I @ one_one_int ) ) ) )
=> ( P @ I3 ) ) ) ) ).
% int_gr_induct
thf(fact_1193_zless__add1__eq,axiom,
! [W: int,Z2: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z2 @ one_one_int ) )
= ( ( ord_less_int @ W @ Z2 )
| ( W = Z2 ) ) ) ).
% zless_add1_eq
thf(fact_1194_int__Suc,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ).
% int_Suc
thf(fact_1195_int__ops_I4_J,axiom,
! [A: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ A ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ one_one_int ) ) ).
% int_ops(4)
thf(fact_1196_zless__iff__Suc__zadd,axiom,
( ord_less_int
= ( ^ [W2: int,Z4: int] :
? [N4: nat] :
( Z4
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ ( suc @ N4 ) ) ) ) ) ) ).
% zless_iff_Suc_zadd
thf(fact_1197_odd__less__0__iff,axiom,
! [Z2: int] :
( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 ) @ zero_zero_int )
= ( ord_less_int @ Z2 @ zero_zero_int ) ) ).
% odd_less_0_iff
thf(fact_1198_zless__imp__add1__zle,axiom,
! [W: int,Z2: int] :
( ( ord_less_int @ W @ Z2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z2 ) ) ).
% zless_imp_add1_zle
thf(fact_1199_add1__zle__eq,axiom,
! [W: int,Z2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z2 )
= ( ord_less_int @ W @ Z2 ) ) ).
% add1_zle_eq
thf(fact_1200_int__induct,axiom,
! [P: int > $o,K: int,I3: int] :
( ( P @ K )
=> ( ! [I: int] :
( ( ord_less_eq_int @ K @ I )
=> ( ( P @ I )
=> ( P @ ( plus_plus_int @ I @ one_one_int ) ) ) )
=> ( ! [I: int] :
( ( ord_less_eq_int @ I @ K )
=> ( ( P @ I )
=> ( P @ ( minus_minus_int @ I @ one_one_int ) ) ) )
=> ( P @ I3 ) ) ) ) ).
% int_induct
thf(fact_1201_less__eq__nat_Osimps_I2_J,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
= ( case_nat_o @ $false @ ( ord_less_eq_nat @ M ) @ N ) ) ).
% less_eq_nat.simps(2)
thf(fact_1202_le__imp__0__less,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z2 ) ) ) ).
% le_imp_0_less
thf(fact_1203_iexec__simp,axiom,
! [Ins2: instr,Cf5: produc6425607678544837394st_int,Cf6: produc6425607678544837394st_int] :
( ( iexec_pred @ ( produc1484136438817787418st_int @ Ins2 @ Cf5 ) @ Cf6 )
= ( Cf6
= ( iexec @ Ins2 @ Cf5 ) ) ) ).
% iexec_simp
thf(fact_1204_iexec__pred_Ointros,axiom,
! [Ins2: instr,Cf5: produc6425607678544837394st_int] : ( iexec_pred @ ( produc1484136438817787418st_int @ Ins2 @ Cf5 ) @ ( iexec @ Ins2 @ Cf5 ) ) ).
% iexec_pred.intros
thf(fact_1205_iexec__pred_Osimps,axiom,
( iexec_pred
= ( ^ [A1: produc5995290525303592096st_int,A22: produc6425607678544837394st_int] :
? [Ins3: instr,Cf: produc6425607678544837394st_int] :
( ( A1
= ( produc1484136438817787418st_int @ Ins3 @ Cf ) )
& ( A22
= ( iexec @ Ins3 @ Cf ) ) ) ) ) ).
% iexec_pred.simps
thf(fact_1206_iexec__pred_Ocases,axiom,
! [A12: produc5995290525303592096st_int,A23: produc6425607678544837394st_int] :
( ( iexec_pred @ A12 @ A23 )
=> ~ ! [Ins: instr,Cf3: produc6425607678544837394st_int] :
( ( A12
= ( produc1484136438817787418st_int @ Ins @ Cf3 ) )
=> ( A23
!= ( iexec @ Ins @ Cf3 ) ) ) ) ).
% iexec_pred.cases
thf(fact_1207_diff__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ M @ ( suc @ N ) )
= ( case_nat_nat @ zero_zero_nat
@ ^ [K4: nat] : K4
@ ( minus_minus_nat @ M @ N ) ) ) ).
% diff_Suc
thf(fact_1208_exec1__appendL,axiom,
! [P: list_instr,I3: int,S: list_char > int,Stk2: list_int,I5: int,S4: list_char > int,Stk3: list_int,P3: list_instr] :
( ( exec1 @ P @ ( produc5086643055186798020st_int @ I3 @ ( produc8650753666468850689st_int @ S @ Stk2 ) ) @ ( produc5086643055186798020st_int @ I5 @ ( produc8650753666468850689st_int @ S4 @ Stk3 ) ) )
=> ( exec1 @ ( append_instr @ P3 @ P ) @ ( produc5086643055186798020st_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P3 ) ) @ I3 ) @ ( produc8650753666468850689st_int @ S @ Stk2 ) ) @ ( produc5086643055186798020st_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P3 ) ) @ I5 ) @ ( produc8650753666468850689st_int @ S4 @ Stk3 ) ) ) ) ).
% exec1_appendL
thf(fact_1209_exec__append__trans,axiom,
! [P: list_instr,S: list_char > int,Stk2: list_int,I5: int,S4: list_char > int,Stk3: list_int,P3: list_instr,I6: int,S5: list_char > int,Stk4: list_int,J4: int] :
( ( star_P707599355569300323st_int @ ( exec1 @ P ) @ ( produc5086643055186798020st_int @ zero_zero_int @ ( produc8650753666468850689st_int @ S @ Stk2 ) ) @ ( produc5086643055186798020st_int @ I5 @ ( produc8650753666468850689st_int @ S4 @ Stk3 ) ) )
=> ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P ) ) @ I5 )
=> ( ( star_P707599355569300323st_int @ ( exec1 @ P3 ) @ ( produc5086643055186798020st_int @ ( minus_minus_int @ I5 @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P ) ) ) @ ( produc8650753666468850689st_int @ S4 @ Stk3 ) ) @ ( produc5086643055186798020st_int @ I6 @ ( produc8650753666468850689st_int @ S5 @ Stk4 ) ) )
=> ( ( J4
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P ) ) @ I6 ) )
=> ( star_P707599355569300323st_int @ ( exec1 @ ( append_instr @ P @ P3 ) ) @ ( produc5086643055186798020st_int @ zero_zero_int @ ( produc8650753666468850689st_int @ S @ Stk2 ) ) @ ( produc5086643055186798020st_int @ J4 @ ( produc8650753666468850689st_int @ S5 @ Stk4 ) ) ) ) ) ) ) ).
% exec_append_trans
thf(fact_1210_exec__appendL__if,axiom,
! [P3: list_instr,I3: int,P: list_instr,S: list_char > int,Stk2: list_int,J: int,S4: list_char > int,Stk3: list_int,I5: int] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P3 ) ) @ I3 )
=> ( ( star_P707599355569300323st_int @ ( exec1 @ P ) @ ( produc5086643055186798020st_int @ ( minus_minus_int @ I3 @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P3 ) ) ) @ ( produc8650753666468850689st_int @ S @ Stk2 ) ) @ ( produc5086643055186798020st_int @ J @ ( produc8650753666468850689st_int @ S4 @ Stk3 ) ) )
=> ( ( I5
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P3 ) ) @ J ) )
=> ( star_P707599355569300323st_int @ ( exec1 @ ( append_instr @ P3 @ P ) ) @ ( produc5086643055186798020st_int @ I3 @ ( produc8650753666468850689st_int @ S @ Stk2 ) ) @ ( produc5086643055186798020st_int @ I5 @ ( produc8650753666468850689st_int @ S4 @ Stk3 ) ) ) ) ) ) ).
% exec_appendL_if
thf(fact_1211_exec__Cons__1,axiom,
! [P: list_instr,S: list_char > int,Stk2: list_int,J: int,T: list_char > int,Stk3: list_int,Ins2: instr] :
( ( star_P707599355569300323st_int @ ( exec1 @ P ) @ ( produc5086643055186798020st_int @ zero_zero_int @ ( produc8650753666468850689st_int @ S @ Stk2 ) ) @ ( produc5086643055186798020st_int @ J @ ( produc8650753666468850689st_int @ T @ Stk3 ) ) )
=> ( star_P707599355569300323st_int @ ( exec1 @ ( cons_instr @ Ins2 @ P ) ) @ ( produc5086643055186798020st_int @ one_one_int @ ( produc8650753666468850689st_int @ S @ Stk2 ) ) @ ( produc5086643055186798020st_int @ ( plus_plus_int @ one_one_int @ J ) @ ( produc8650753666468850689st_int @ T @ Stk3 ) ) ) ) ).
% exec_Cons_1
thf(fact_1212_exec__appendR,axiom,
! [P: list_instr,C2: produc6425607678544837394st_int,C4: produc6425607678544837394st_int,P3: list_instr] :
( ( star_P707599355569300323st_int @ ( exec1 @ P ) @ C2 @ C4 )
=> ( star_P707599355569300323st_int @ ( exec1 @ ( append_instr @ P @ P3 ) ) @ C2 @ C4 ) ) ).
% exec_appendR
thf(fact_1213_exec__induct,axiom,
! [Pa: list_instr,X1a: int,X1b: list_char > int,X1c: list_int,X2a: int,X2b: list_char > int,X2c: list_int,P: int > ( list_char > int ) > list_int > int > ( list_char > int ) > list_int > $o] :
( ( star_P707599355569300323st_int @ ( exec1 @ Pa ) @ ( produc5086643055186798020st_int @ X1a @ ( produc8650753666468850689st_int @ X1b @ X1c ) ) @ ( produc5086643055186798020st_int @ X2a @ ( produc8650753666468850689st_int @ X2b @ X2c ) ) )
=> ( ! [A4: int,Aa: list_char > int,B3: list_int] : ( P @ A4 @ Aa @ B3 @ A4 @ Aa @ B3 )
=> ( ! [A4: int,Aa: list_char > int,B3: list_int,Ab: int,Ac: list_char > int,Ba: list_int,Ad: int,Ae: list_char > int,Bb: list_int] :
( ( exec1 @ Pa @ ( produc5086643055186798020st_int @ A4 @ ( produc8650753666468850689st_int @ Aa @ B3 ) ) @ ( produc5086643055186798020st_int @ Ab @ ( produc8650753666468850689st_int @ Ac @ Ba ) ) )
=> ( ( star_P707599355569300323st_int @ ( exec1 @ Pa ) @ ( produc5086643055186798020st_int @ Ab @ ( produc8650753666468850689st_int @ Ac @ Ba ) ) @ ( produc5086643055186798020st_int @ Ad @ ( produc8650753666468850689st_int @ Ae @ Bb ) ) )
=> ( ( P @ Ab @ Ac @ Ba @ Ad @ Ae @ Bb )
=> ( P @ A4 @ Aa @ B3 @ Ad @ Ae @ Bb ) ) ) )
=> ( P @ X1a @ X1b @ X1c @ X2a @ X2b @ X2c ) ) ) ) ).
% exec_induct
thf(fact_1214_exec__appendL,axiom,
! [P: list_instr,I3: int,S: list_char > int,Stk2: list_int,I5: int,S4: list_char > int,Stk3: list_int,P3: list_instr] :
( ( star_P707599355569300323st_int @ ( exec1 @ P ) @ ( produc5086643055186798020st_int @ I3 @ ( produc8650753666468850689st_int @ S @ Stk2 ) ) @ ( produc5086643055186798020st_int @ I5 @ ( produc8650753666468850689st_int @ S4 @ Stk3 ) ) )
=> ( star_P707599355569300323st_int @ ( exec1 @ ( append_instr @ P3 @ P ) ) @ ( produc5086643055186798020st_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P3 ) ) @ I3 ) @ ( produc8650753666468850689st_int @ S @ Stk2 ) ) @ ( produc5086643055186798020st_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P3 ) ) @ I5 ) @ ( produc8650753666468850689st_int @ S4 @ Stk3 ) ) ) ) ).
% exec_appendL
thf(fact_1215_LoadIE,axiom,
! [I3: int,Pc: int,S: list_char > int,Stk2: list_int,Cf5: produc6425607678544837394st_int] :
( ( iexec_pred @ ( produc1484136438817787418st_int @ ( loadi @ I3 ) @ ( produc5086643055186798020st_int @ Pc @ ( produc8650753666468850689st_int @ S @ Stk2 ) ) ) @ Cf5 )
=> ( Cf5
= ( produc5086643055186798020st_int @ ( plus_plus_int @ Pc @ one_one_int ) @ ( produc8650753666468850689st_int @ S @ ( cons_int @ I3 @ Stk2 ) ) ) ) ) ).
% LoadIE
thf(fact_1216_instr_Oinject_I1_J,axiom,
! [X1: int,Y1: int] :
( ( ( loadi @ X1 )
= ( loadi @ Y1 ) )
= ( X1 = Y1 ) ) ).
% instr.inject(1)
thf(fact_1217_instr_Osize_I8_J,axiom,
! [X1: int] :
( ( size_size_instr @ ( loadi @ X1 ) )
= zero_zero_nat ) ).
% instr.size(8)
thf(fact_1218_upto__aux__rec,axiom,
( upto_aux
= ( ^ [I2: int,J3: int,Js: list_int] : ( if_list_int @ ( ord_less_int @ J3 @ I2 ) @ Js @ ( upto_aux @ I2 @ ( minus_minus_int @ J3 @ one_one_int ) @ ( cons_int @ J3 @ Js ) ) ) ) ) ).
% upto_aux_rec
thf(fact_1219_LoadE,axiom,
! [X: list_char,Pc: int,S: list_char > int,Stk2: list_int,Cf5: produc6425607678544837394st_int] :
( ( iexec_pred @ ( produc1484136438817787418st_int @ ( load @ X ) @ ( produc5086643055186798020st_int @ Pc @ ( produc8650753666468850689st_int @ S @ Stk2 ) ) ) @ Cf5 )
=> ( Cf5
= ( produc5086643055186798020st_int @ ( plus_plus_int @ Pc @ one_one_int ) @ ( produc8650753666468850689st_int @ S @ ( cons_int @ ( S @ X ) @ Stk2 ) ) ) ) ) ).
% LoadE
thf(fact_1220_JmpE,axiom,
! [I3: int,Pc: int,S: list_char > int,Stk2: list_int,Cf5: produc6425607678544837394st_int] :
( ( iexec_pred @ ( produc1484136438817787418st_int @ ( jmp @ I3 ) @ ( produc5086643055186798020st_int @ Pc @ ( produc8650753666468850689st_int @ S @ Stk2 ) ) ) @ Cf5 )
=> ( Cf5
= ( produc5086643055186798020st_int @ ( plus_plus_int @ ( plus_plus_int @ Pc @ one_one_int ) @ I3 ) @ ( produc8650753666468850689st_int @ S @ Stk2 ) ) ) ) ).
% JmpE
thf(fact_1221_instr_Oinject_I4_J,axiom,
! [X52: int,Y52: int] :
( ( ( jmp @ X52 )
= ( jmp @ Y52 ) )
= ( X52 = Y52 ) ) ).
% instr.inject(4)
thf(fact_1222_instr_Oinject_I2_J,axiom,
! [X22: list_char,Y22: list_char] :
( ( ( load @ X22 )
= ( load @ Y22 ) )
= ( X22 = Y22 ) ) ).
% instr.inject(2)
thf(fact_1223_instr_Odistinct_I1_J,axiom,
! [X1: int,X22: list_char] :
( ( loadi @ X1 )
!= ( load @ X22 ) ) ).
% instr.distinct(1)
thf(fact_1224_instr_Odistinct_I7_J,axiom,
! [X1: int,X52: int] :
( ( loadi @ X1 )
!= ( jmp @ X52 ) ) ).
% instr.distinct(7)
thf(fact_1225_instr_Osize_I12_J,axiom,
! [X52: int] :
( ( size_size_instr @ ( jmp @ X52 ) )
= zero_zero_nat ) ).
% instr.size(12)
thf(fact_1226_instr_Osize_I9_J,axiom,
! [X22: list_char] :
( ( size_size_instr @ ( load @ X22 ) )
= zero_zero_nat ) ).
% instr.size(9)
thf(fact_1227_instr_Odistinct_I17_J,axiom,
! [X22: list_char,X52: int] :
( ( load @ X22 )
!= ( jmp @ X52 ) ) ).
% instr.distinct(17)
thf(fact_1228_to__m__state_Opelims,axiom,
! [X: list_list_char,Xa: list_char > int,Xb: int,Y: int] :
( ( ( to_m_state @ X @ Xa @ Xb )
= Y )
=> ( ( accp_P4562858270263085236nt_int @ to_m_state_rel @ ( produc3965054194175396271nt_int @ X @ ( produc5790713362662368625nt_int @ Xa @ Xb ) ) )
=> ~ ( ( ( ( ( ord_less_int @ zero_zero_int @ Xb )
& ( ord_less_eq_int @ Xb @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ X ) ) ) )
=> ( Y
= ( Xa @ ( inth_list_char @ X @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ X ) ) @ Xb ) ) ) ) )
& ( ~ ( ( ord_less_int @ zero_zero_int @ Xb )
& ( ord_less_eq_int @ Xb @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ X ) ) ) )
=> ( Y = zero_zero_int ) ) )
=> ~ ( accp_P4562858270263085236nt_int @ to_m_state_rel @ ( produc3965054194175396271nt_int @ X @ ( produc5790713362662368625nt_int @ Xa @ Xb ) ) ) ) ) ) ).
% to_m_state.pelims
thf(fact_1229_pred__def,axiom,
( pred
= ( case_nat_nat @ zero_zero_nat
@ ^ [X23: nat] : X23 ) ) ).
% pred_def
thf(fact_1230_execl_Opelims_I1_J,axiom,
! [X: list_instr,Xa: list_P419547167332244514st_int,Y: $o] :
( ( ( execl @ X @ Xa )
= Y )
=> ( ( accp_P1482788143903795053st_int @ execl_rel @ ( produc7764413618310988720st_int @ X @ Xa ) )
=> ( ! [Cf3: produc6425607678544837394st_int,Cf4: produc6425607678544837394st_int,Cfs5: list_P419547167332244514st_int] :
( ( Xa
= ( cons_P4151473878978182108st_int @ Cf3 @ ( cons_P4151473878978182108st_int @ Cf4 @ Cfs5 ) ) )
=> ( ( Y
= ( ( exec1 @ X @ Cf3 @ Cf4 )
& ( execl @ X @ ( cons_P4151473878978182108st_int @ Cf4 @ Cfs5 ) ) ) )
=> ~ ( accp_P1482788143903795053st_int @ execl_rel @ ( produc7764413618310988720st_int @ X @ ( cons_P4151473878978182108st_int @ Cf3 @ ( cons_P4151473878978182108st_int @ Cf4 @ Cfs5 ) ) ) ) ) )
=> ( ( ( Xa = nil_Pr749310521696113548st_int )
=> ( Y
=> ~ ( accp_P1482788143903795053st_int @ execl_rel @ ( produc7764413618310988720st_int @ X @ nil_Pr749310521696113548st_int ) ) ) )
=> ~ ! [V: produc6425607678544837394st_int] :
( ( Xa
= ( cons_P4151473878978182108st_int @ V @ nil_Pr749310521696113548st_int ) )
=> ( Y
=> ~ ( accp_P1482788143903795053st_int @ execl_rel @ ( produc7764413618310988720st_int @ X @ ( cons_P4151473878978182108st_int @ V @ nil_Pr749310521696113548st_int ) ) ) ) ) ) ) ) ) ).
% execl.pelims(1)
thf(fact_1231_execl_Opelims_I3_J,axiom,
! [X: list_instr,Xa: list_P419547167332244514st_int] :
( ~ ( execl @ X @ Xa )
=> ( ( accp_P1482788143903795053st_int @ execl_rel @ ( produc7764413618310988720st_int @ X @ Xa ) )
=> ~ ! [Cf3: produc6425607678544837394st_int,Cf4: produc6425607678544837394st_int,Cfs5: list_P419547167332244514st_int] :
( ( Xa
= ( cons_P4151473878978182108st_int @ Cf3 @ ( cons_P4151473878978182108st_int @ Cf4 @ Cfs5 ) ) )
=> ( ( accp_P1482788143903795053st_int @ execl_rel @ ( produc7764413618310988720st_int @ X @ ( cons_P4151473878978182108st_int @ Cf3 @ ( cons_P4151473878978182108st_int @ Cf4 @ Cfs5 ) ) ) )
=> ( ( exec1 @ X @ Cf3 @ Cf4 )
& ( execl @ X @ ( cons_P4151473878978182108st_int @ Cf4 @ Cfs5 ) ) ) ) ) ) ) ).
% execl.pelims(3)
thf(fact_1232_execl_Opelims_I2_J,axiom,
! [X: list_instr,Xa: list_P419547167332244514st_int] :
( ( execl @ X @ Xa )
=> ( ( accp_P1482788143903795053st_int @ execl_rel @ ( produc7764413618310988720st_int @ X @ Xa ) )
=> ( ! [Cf3: produc6425607678544837394st_int,Cf4: produc6425607678544837394st_int,Cfs5: list_P419547167332244514st_int] :
( ( Xa
= ( cons_P4151473878978182108st_int @ Cf3 @ ( cons_P4151473878978182108st_int @ Cf4 @ Cfs5 ) ) )
=> ( ( accp_P1482788143903795053st_int @ execl_rel @ ( produc7764413618310988720st_int @ X @ ( cons_P4151473878978182108st_int @ Cf3 @ ( cons_P4151473878978182108st_int @ Cf4 @ Cfs5 ) ) ) )
=> ~ ( ( exec1 @ X @ Cf3 @ Cf4 )
& ( execl @ X @ ( cons_P4151473878978182108st_int @ Cf4 @ Cfs5 ) ) ) ) )
=> ( ( ( Xa = nil_Pr749310521696113548st_int )
=> ~ ( accp_P1482788143903795053st_int @ execl_rel @ ( produc7764413618310988720st_int @ X @ nil_Pr749310521696113548st_int ) ) )
=> ~ ! [V: produc6425607678544837394st_int] :
( ( Xa
= ( cons_P4151473878978182108st_int @ V @ nil_Pr749310521696113548st_int ) )
=> ~ ( accp_P1482788143903795053st_int @ execl_rel @ ( produc7764413618310988720st_int @ X @ ( cons_P4151473878978182108st_int @ V @ nil_Pr749310521696113548st_int ) ) ) ) ) ) ) ) ).
% execl.pelims(2)
thf(fact_1233_nat0__intermed__int__val,axiom,
! [N: nat,F: nat > int,K: int] :
( ! [I: nat] :
( ( ord_less_nat @ I @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I @ one_one_nat ) ) @ ( F @ I ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I: nat] :
( ( ord_less_eq_nat @ I @ N )
& ( ( F @ I )
= K ) ) ) ) ) ).
% nat0_intermed_int_val
thf(fact_1234_zabs__less__one__iff,axiom,
! [Z2: int] :
( ( ord_less_int @ ( abs_abs_int @ Z2 ) @ one_one_int )
= ( Z2 = zero_zero_int ) ) ).
% zabs_less_one_iff
thf(fact_1235_nat__intermed__int__val,axiom,
! [M: nat,N: nat,F: nat > int,K: int] :
( ! [I: nat] :
( ( ( ord_less_eq_nat @ M @ I )
& ( ord_less_nat @ I @ N ) )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I ) ) @ ( F @ I ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_int @ ( F @ M ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I: nat] :
( ( ord_less_eq_nat @ M @ I )
& ( ord_less_eq_nat @ I @ N )
& ( ( F @ I )
= K ) ) ) ) ) ) ).
% nat_intermed_int_val
thf(fact_1236_nat__ivt__aux,axiom,
! [N: nat,F: nat > int,K: int] :
( ! [I: nat] :
( ( ord_less_nat @ I @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I ) ) @ ( F @ I ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I: nat] :
( ( ord_less_eq_nat @ I @ N )
& ( ( F @ I )
= K ) ) ) ) ) ).
% nat_ivt_aux
thf(fact_1237_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ! [I4: nat] :
( ( ord_less_nat @ K2 @ I4 )
=> ( P @ I4 ) )
=> ( P @ K2 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_1238_upto_Opinduct,axiom,
! [A0: int,A12: int,P: int > int > $o] :
( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ A0 @ A12 ) )
=> ( ! [I: int,J2: int] :
( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I @ J2 ) )
=> ( ( ( ord_less_eq_int @ I @ J2 )
=> ( P @ ( plus_plus_int @ I @ one_one_int ) @ J2 ) )
=> ( P @ I @ J2 ) ) )
=> ( P @ A0 @ A12 ) ) ) ).
% upto.pinduct
thf(fact_1239_execl__all__SKIP,axiom,
! [Cfs2: list_P419547167332244514st_int] :
( ( execl_all @ nil_instr @ Cfs2 )
=> ( ( cpred @ skip @ ( nth_Pr8327173284404383691st_int @ Cfs2 @ zero_zero_nat ) @ ( nth_Pr8327173284404383691st_int @ Cfs2 @ ( minus_minus_nat @ ( size_s229347708945616566st_int @ Cfs2 ) @ ( suc @ zero_zero_nat ) ) ) )
& ( ( size_s229347708945616566st_int @ Cfs2 )
= ( suc @ zero_zero_nat ) ) ) ) ).
% execl_all_SKIP
thf(fact_1240_upto_Opsimps,axiom,
! [I3: int,J: int] :
( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I3 @ J ) )
=> ( ( ( ord_less_eq_int @ I3 @ J )
=> ( ( upto @ I3 @ J )
= ( cons_int @ I3 @ ( upto @ ( plus_plus_int @ I3 @ one_one_int ) @ J ) ) ) )
& ( ~ ( ord_less_eq_int @ I3 @ J )
=> ( ( upto @ I3 @ J )
= nil_int ) ) ) ) ).
% upto.psimps
thf(fact_1241_upto_Opelims,axiom,
! [X: int,Xa: int,Y: list_int] :
( ( ( upto @ X @ Xa )
= Y )
=> ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X @ Xa ) )
=> ~ ( ( ( ( ord_less_eq_int @ X @ Xa )
=> ( Y
= ( cons_int @ X @ ( upto @ ( plus_plus_int @ X @ one_one_int ) @ Xa ) ) ) )
& ( ~ ( ord_less_eq_int @ X @ Xa )
=> ( Y = nil_int ) ) )
=> ~ ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X @ Xa ) ) ) ) ) ).
% upto.pelims
thf(fact_1242_upto__Nil,axiom,
! [I3: int,J: int] :
( ( ( upto @ I3 @ J )
= nil_int )
= ( ord_less_int @ J @ I3 ) ) ).
% upto_Nil
thf(fact_1243_upto__Nil2,axiom,
! [I3: int,J: int] :
( ( nil_int
= ( upto @ I3 @ J ) )
= ( ord_less_int @ J @ I3 ) ) ).
% upto_Nil2
thf(fact_1244_upto__empty,axiom,
! [J: int,I3: int] :
( ( ord_less_int @ J @ I3 )
=> ( ( upto @ I3 @ J )
= nil_int ) ) ).
% upto_empty
thf(fact_1245_upto__single,axiom,
! [I3: int] :
( ( upto @ I3 @ I3 )
= ( cons_int @ I3 @ nil_int ) ) ).
% upto_single
thf(fact_1246_nth__upto,axiom,
! [I3: int,K: nat,J: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I3 @ ( semiri1314217659103216013at_int @ K ) ) @ J )
=> ( ( nth_int @ ( upto @ I3 @ J ) @ K )
= ( plus_plus_int @ I3 @ ( semiri1314217659103216013at_int @ K ) ) ) ) ).
% nth_upto
thf(fact_1247_upto__code,axiom,
( upto
= ( ^ [I2: int,J3: int] : ( upto_aux @ I2 @ J3 @ nil_int ) ) ) ).
% upto_code
thf(fact_1248_upto__aux__def,axiom,
( upto_aux
= ( ^ [I2: int,J3: int] : ( append_int @ ( upto @ I2 @ J3 ) ) ) ) ).
% upto_aux_def
thf(fact_1249_upto__split2,axiom,
! [I3: int,J: int,K: int] :
( ( ord_less_eq_int @ I3 @ J )
=> ( ( ord_less_eq_int @ J @ K )
=> ( ( upto @ I3 @ K )
= ( append_int @ ( upto @ I3 @ J ) @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ).
% upto_split2
thf(fact_1250_upto__split1,axiom,
! [I3: int,J: int,K: int] :
( ( ord_less_eq_int @ I3 @ J )
=> ( ( ord_less_eq_int @ J @ K )
=> ( ( upto @ I3 @ K )
= ( append_int @ ( upto @ I3 @ ( minus_minus_int @ J @ one_one_int ) ) @ ( upto @ J @ K ) ) ) ) ) ).
% upto_split1
thf(fact_1251_upto_Osimps,axiom,
( upto
= ( ^ [I2: int,J3: int] : ( if_list_int @ ( ord_less_eq_int @ I2 @ J3 ) @ ( cons_int @ I2 @ ( upto @ ( plus_plus_int @ I2 @ one_one_int ) @ J3 ) ) @ nil_int ) ) ) ).
% upto.simps
thf(fact_1252_upto_Oelims,axiom,
! [X: int,Xa: int,Y: list_int] :
( ( ( upto @ X @ Xa )
= Y )
=> ( ( ( ord_less_eq_int @ X @ Xa )
=> ( Y
= ( cons_int @ X @ ( upto @ ( plus_plus_int @ X @ one_one_int ) @ Xa ) ) ) )
& ( ~ ( ord_less_eq_int @ X @ Xa )
=> ( Y = nil_int ) ) ) ) ).
% upto.elims
thf(fact_1253_upto__rec1,axiom,
! [I3: int,J: int] :
( ( ord_less_eq_int @ I3 @ J )
=> ( ( upto @ I3 @ J )
= ( cons_int @ I3 @ ( upto @ ( plus_plus_int @ I3 @ one_one_int ) @ J ) ) ) ) ).
% upto_rec1
thf(fact_1254_upto__rec2,axiom,
! [I3: int,J: int] :
( ( ord_less_eq_int @ I3 @ J )
=> ( ( upto @ I3 @ J )
= ( append_int @ ( upto @ I3 @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ nil_int ) ) ) ) ).
% upto_rec2
thf(fact_1255_upto__split3,axiom,
! [I3: int,J: int,K: int] :
( ( ord_less_eq_int @ I3 @ J )
=> ( ( ord_less_eq_int @ J @ K )
=> ( ( upto @ I3 @ K )
= ( append_int @ ( upto @ I3 @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ) ).
% upto_split3
thf(fact_1256_upto__rec__numeral_I1_J,axiom,
! [M: num,N: num] :
( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
=> ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( numeral_numeral_int @ N ) ) ) ) )
& ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
=> ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= nil_int ) ) ) ).
% upto_rec_numeral(1)
thf(fact_1257_com_Osize_I6_J,axiom,
( ( size_size_com @ skip )
= zero_zero_nat ) ).
% com.size(6)
thf(fact_1258_upto__rec__numeral_I2_J,axiom,
! [M: num,N: num] :
( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
=> ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ) ) )
& ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
=> ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= nil_int ) ) ) ).
% upto_rec_numeral(2)
thf(fact_1259_upto__rec__numeral_I3_J,axiom,
! [M: num,N: num] :
( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
=> ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
= ( cons_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( upto @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) @ ( numeral_numeral_int @ N ) ) ) ) )
& ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
=> ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
= nil_int ) ) ) ).
% upto_rec_numeral(3)
% Helper facts (7)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
! [X: list_int,Y: list_int] :
( ( if_list_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
! [X: list_int,Y: list_int] :
( ( if_list_int @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
? [K5: nat] :
( ( ord_less_nat @ K5 @ ( size_s229347708945616566st_int @ cfs ) )
& ( q @ x @ ( produc5086643055186798020st_int @ ( minus_minus_int @ ( produc8941309049508465880st_int @ ( nth_Pr8327173284404383691st_int @ cfs @ k ) ) @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ p ) ) ) @ ( produc1296517004818028822st_int @ ( nth_Pr8327173284404383691st_int @ cfs @ k ) ) ) @ ( produc5086643055186798020st_int @ ( minus_minus_int @ ( produc8941309049508465880st_int @ ( nth_Pr8327173284404383691st_int @ cfs @ K5 ) ) @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ p ) ) ) @ ( produc1296517004818028822st_int @ ( nth_Pr8327173284404383691st_int @ cfs @ K5 ) ) ) ) ) ).
%------------------------------------------------------------------------------