TPTP Problem File: SLH0794^1.p
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%------------------------------------------------------------------------------
% 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_00257_013924__6021098_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1445 ( 676 unt; 173 typ; 0 def)
% Number of atoms : 3392 (1749 equ; 0 cnn)
% Maximal formula atoms : 14 ( 2 avg)
% Number of connectives : 11108 ( 327 ~; 56 |; 338 &;9184 @)
% ( 0 <=>;1203 =>; 0 <=; 0 <~>)
% Maximal formula depth : 30 ( 6 avg)
% Number of types : 29 ( 28 usr)
% Number of type conns : 735 ( 735 >; 0 *; 0 +; 0 <<)
% Number of symbols : 148 ( 145 usr; 15 con; 0-3 aty)
% Number of variables : 3841 ( 240 ^;3394 !; 207 ?;3841 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 16:49:00.425
%------------------------------------------------------------------------------
% Could-be-implicit typings (28)
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% Explicit typings (145)
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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__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_001_062_It__Int__Oint_M_Eo_J,type,
ord_less_int_o: ( int > $o ) > ( int > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
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_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Int__Oint_M_Eo_J,type,
ord_less_eq_int_o: ( int > $o ) > ( int > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_Eo_J,type,
ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $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__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_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__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__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_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_cfs,type,
cfs: list_P419547167332244514st_int ).
thf(sy_v_k,type,
k: nat ).
thf(sy_v_x,type,
x: a ).
% Relevant facts (1264)
thf(fact_0_C,axiom,
( ( produc8941309049508465880st_int @ ( nth_Pr8327173284404383691st_int @ cfs @ k ) )
= ( semiri1314217659103216013at_int @ ( size_size_list_instr @ p ) ) ) ).
% C
thf(fact_1_calculation,axiom,
( member_nat
@ ( lattic8721135487736765967in_nat
@ ( collect_nat
@ ^ [K: nat] :
( ( member_nat @ K @ ( set_or4665077453230672383an_nat @ k @ ( size_s229347708945616566st_int @ cfs ) ) )
& ~ ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ p ) ) @ ( produc8941309049508465880st_int @ ( nth_Pr8327173284404383691st_int @ cfs @ K ) ) )
& ( ord_less_int @ ( minus_minus_int @ ( produc8941309049508465880st_int @ ( nth_Pr8327173284404383691st_int @ cfs @ K ) ) @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ p ) ) ) @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ ( p2 @ x ) ) ) ) ) ) ) )
@ ( collect_nat
@ ^ [K: nat] :
( ( member_nat @ K @ ( set_or4665077453230672383an_nat @ k @ ( size_s229347708945616566st_int @ cfs ) ) )
& ~ ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ p ) ) @ ( produc8941309049508465880st_int @ ( nth_Pr8327173284404383691st_int @ cfs @ K ) ) )
& ( ord_less_int @ ( minus_minus_int @ ( produc8941309049508465880st_int @ ( nth_Pr8327173284404383691st_int @ cfs @ K ) ) @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ p ) ) ) @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ ( p2 @ x ) ) ) ) ) ) ) ) ).
% calculation
thf(fact_2_B,axiom,
ord_less_nat @ k @ ( size_s229347708945616566st_int @ cfs ) ).
% B
thf(fact_3_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_4_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_5_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_6_prod_Ocollapse,axiom,
! [Prod: produc4189061985984568957nt_int] :
( ( produc3965054194175396271nt_int @ ( produc811350159336358595nt_int @ Prod ) @ ( produc30867750519413249nt_int @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_7_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_8_prod_Ocollapse,axiom,
! [Prod: produc369741286924889929st_int] :
( ( produc8650753666468850689st_int @ ( produc2155683374303800813st_int @ Prod ) @ ( produc6244060863719409455st_int @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_9_prod_Ocollapse,axiom,
! [Prod: produc4435102495419491129nt_int] :
( ( produc5790713362662368625nt_int @ ( produc9025023058967219549nt_int @ Prod ) @ ( produc179445755590504351nt_int @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_10_prod_Ocollapse,axiom,
! [Prod: produc6425607678544837394st_int] :
( ( produc5086643055186798020st_int @ ( produc8941309049508465880st_int @ Prod ) @ ( produc1296517004818028822st_int @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_11_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_12_of__nat__1,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% of_nat_1
thf(fact_13_of__nat__1,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% of_nat_1
thf(fact_14_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_15_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_16_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_17_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_18_atLeastLessThan__iff,axiom,
! [I: nat,L: nat,U: nat] :
( ( member_nat @ I @ ( set_or4665077453230672383an_nat @ L @ U ) )
= ( ( ord_less_eq_nat @ L @ I )
& ( ord_less_nat @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_19_atLeastLessThan__iff,axiom,
! [I: int,L: int,U: int] :
( ( member_int @ I @ ( set_or4662586982721622107an_int @ L @ U ) )
= ( ( ord_less_eq_int @ L @ I )
& ( ord_less_int @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_20_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_21_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_22_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_23_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_24_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_25_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_26_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_27_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_28_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_29_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_30_Suc__diff__diff,axiom,
! [M: nat,N: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K2 ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K2 ) ) ).
% Suc_diff_diff
thf(fact_31_ivl__subset,axiom,
! [I: nat,J: nat,M: nat,N: nat] :
( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ I @ J ) @ ( set_or4665077453230672383an_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ J @ I )
| ( ( ord_less_eq_nat @ M @ I )
& ( ord_less_eq_nat @ J @ N ) ) ) ) ).
% ivl_subset
thf(fact_32_ivl__subset,axiom,
! [I: int,J: int,M: int,N: int] :
( ( ord_less_eq_set_int @ ( set_or4662586982721622107an_int @ I @ J ) @ ( set_or4662586982721622107an_int @ M @ N ) )
= ( ( ord_less_eq_int @ J @ I )
| ( ( ord_less_eq_int @ M @ I )
& ( ord_less_eq_int @ J @ N ) ) ) ) ).
% ivl_subset
thf(fact_33_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_34_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_35_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_36_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_37_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_38_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_39_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_40_prod_Oinject,axiom,
! [X1: list_list_char,X2: produc4435102495419491129nt_int,Y1: list_list_char,Y2: produc4435102495419491129nt_int] :
( ( ( produc3965054194175396271nt_int @ X1 @ X2 )
= ( produc3965054194175396271nt_int @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_41_prod_Oinject,axiom,
! [X1: int,X2: produc369741286924889929st_int,Y1: int,Y2: produc369741286924889929st_int] :
( ( ( produc5086643055186798020st_int @ X1 @ X2 )
= ( produc5086643055186798020st_int @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_42_prod_Oinject,axiom,
! [X1: int,X2: int,Y1: int,Y2: int] :
( ( ( product_Pair_int_int @ X1 @ X2 )
= ( product_Pair_int_int @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_43_prod_Oinject,axiom,
! [X1: list_char > int,X2: list_int,Y1: list_char > int,Y2: list_int] :
( ( ( produc8650753666468850689st_int @ X1 @ X2 )
= ( produc8650753666468850689st_int @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_44_prod_Oinject,axiom,
! [X1: list_char > int,X2: int,Y1: list_char > int,Y2: int] :
( ( ( produc5790713362662368625nt_int @ X1 @ X2 )
= ( produc5790713362662368625nt_int @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_45_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_46_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_47_nat_Oinject,axiom,
! [X2: nat,Y2: nat] :
( ( ( suc @ X2 )
= ( suc @ Y2 ) )
= ( X2 = Y2 ) ) ).
% nat.inject
thf(fact_48_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_49_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1316708129612266289at_nat @ M )
= ( semiri1316708129612266289at_nat @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_50_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_51_map__ident,axiom,
( ( map_Pr8625093460839654122st_int
@ ^ [X: produc6425607678544837394st_int] : X )
= ( ^ [Xs2: list_P419547167332244514st_int] : Xs2 ) ) ).
% map_ident
thf(fact_52_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_53_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_54_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_55_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_56_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_57_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_58_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_59_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_60_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_61_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_62_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_63_nth__take,axiom,
! [I: nat,N: nat,Xs: list_int] :
( ( ord_less_nat @ I @ N )
=> ( ( nth_int @ ( take_int @ N @ Xs ) @ I )
= ( nth_int @ Xs @ I ) ) ) ).
% nth_take
thf(fact_64_nth__take,axiom,
! [I: nat,N: nat,Xs: list_P419547167332244514st_int] :
( ( ord_less_nat @ I @ N )
=> ( ( nth_Pr8327173284404383691st_int @ ( take_P1004856038796978596st_int @ N @ Xs ) @ I )
= ( nth_Pr8327173284404383691st_int @ Xs @ I ) ) ) ).
% nth_take
thf(fact_65_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_66_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_67_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_68_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_69_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_70_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_71_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_72_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_73_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_74_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_75_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_76_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_77_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M2: nat,N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
& ( M2 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_78_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_79_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_80_less__diff__iff,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_81_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_82_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_83_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_84_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_85_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N3: nat] : ( ord_less_eq_nat @ ( suc @ N3 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_86_mem__Collect__eq,axiom,
! [A: int,P: int > $o] :
( ( member_int @ A @ ( collect_int @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_87_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_88_Collect__mem__eq,axiom,
! [A3: set_int] :
( ( collect_int
@ ^ [X: int] : ( member_int @ X @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_89_Collect__mem__eq,axiom,
! [A3: set_nat] :
( ( collect_nat
@ ^ [X: nat] : ( member_nat @ X @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_90_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_91_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_92_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_93_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_94_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_95_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_96_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_97_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_98_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N3: nat] :
( ( ord_less_nat @ M2 @ N3 )
| ( M2 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_99_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_100_linorder__neqE__nat,axiom,
! [X4: nat,Y: nat] :
( ( X4 != Y )
=> ( ~ ( ord_less_nat @ X4 @ Y )
=> ( ord_less_nat @ Y @ X4 ) ) ) ).
% linorder_neqE_nat
thf(fact_101_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_102_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_103_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_104_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] :
( ( J
= ( suc @ I2 ) )
=> ( P @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( P @ ( suc @ I2 ) )
=> ( P @ I2 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_105_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J2: nat,K3: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ K3 )
=> ( ( P @ I2 @ J2 )
=> ( ( P @ J2 @ K3 )
=> ( P @ I2 @ K3 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_106_less__trans__Suc,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K2 )
=> ( ord_less_nat @ ( suc @ I ) @ K2 ) ) ) ).
% less_trans_Suc
thf(fact_107_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_108_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_109_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M4: nat] :
( ( M
= ( suc @ M4 ) )
& ( ord_less_nat @ N @ M4 ) ) ) ) ).
% Suc_less_eq2
thf(fact_110_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
=> ( P @ I3 ) ) )
= ( ( P @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( P @ I3 ) ) ) ) ).
% All_less_Suc
thf(fact_111_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_112_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_113_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
& ( P @ I3 ) ) )
= ( ( P @ N )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ( P @ I3 ) ) ) ) ).
% Ex_less_Suc
thf(fact_114_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_115_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_116_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_117_Suc__lessE,axiom,
! [I: nat,K2: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K2 )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K2
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_118_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_119_Nat_OlessE,axiom,
! [I: nat,K2: nat] :
( ( ord_less_nat @ I @ K2 )
=> ( ( K2
!= ( suc @ I ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K2
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_120_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X3: nat] : ( R @ X3 @ X3 )
=> ( ! [X3: nat,Y3: nat,Z: nat] :
( ( R @ X3 @ Y3 )
=> ( ( R @ Y3 @ Z )
=> ( R @ X3 @ Z ) ) )
=> ( ! [N2: nat] : ( R @ N2 @ ( suc @ N2 ) )
=> ( R @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_121_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_122_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_123_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_124_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_125_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_126_Suc__le__D,axiom,
! [N: nat,M5: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M5 )
=> ? [M6: nat] :
( M5
= ( suc @ M6 ) ) ) ).
% Suc_le_D
thf(fact_127_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_128_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_129_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_130_length__induct,axiom,
! [P: list_P419547167332244514st_int > $o,Xs: list_P419547167332244514st_int] :
( ! [Xs3: list_P419547167332244514st_int] :
( ! [Ys: list_P419547167332244514st_int] :
( ( ord_less_nat @ ( size_s229347708945616566st_int @ Ys ) @ ( size_s229347708945616566st_int @ Xs3 ) )
=> ( P @ Ys ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_131_length__induct,axiom,
! [P: list_instr > $o,Xs: list_instr] :
( ! [Xs3: list_instr] :
( ! [Ys: list_instr] :
( ( ord_less_nat @ ( size_size_list_instr @ Ys ) @ ( size_size_list_instr @ Xs3 ) )
=> ( P @ Ys ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_132_length__induct,axiom,
! [P: list_list_char > $o,Xs: list_list_char] :
( ! [Xs3: list_list_char] :
( ! [Ys: list_list_char] :
( ( ord_less_nat @ ( size_s356637359517785349t_char @ Ys ) @ ( size_s356637359517785349t_char @ Xs3 ) )
=> ( P @ Ys ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_133_less__imp__diff__less,axiom,
! [J: nat,K2: nat,N: nat] :
( ( ord_less_nat @ J @ K2 )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K2 ) ) ).
% less_imp_diff_less
thf(fact_134_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_135_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_136_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_137_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_138_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_139_Nat_Odiff__diff__eq,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_140_le__diff__iff,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_141_eq__diff__iff,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( ( minus_minus_nat @ M @ K2 )
= ( minus_minus_nat @ N @ K2 ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_142_nth__take__lemma,axiom,
! [K2: nat,Xs: list_int,Ys2: list_int] :
( ( ord_less_eq_nat @ K2 @ ( size_size_list_int @ Xs ) )
=> ( ( ord_less_eq_nat @ K2 @ ( size_size_list_int @ Ys2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K2 )
=> ( ( nth_int @ Xs @ I2 )
= ( nth_int @ Ys2 @ I2 ) ) )
=> ( ( take_int @ K2 @ Xs )
= ( take_int @ K2 @ Ys2 ) ) ) ) ) ).
% nth_take_lemma
thf(fact_143_nth__take__lemma,axiom,
! [K2: nat,Xs: list_P419547167332244514st_int,Ys2: list_P419547167332244514st_int] :
( ( ord_less_eq_nat @ K2 @ ( size_s229347708945616566st_int @ Xs ) )
=> ( ( ord_less_eq_nat @ K2 @ ( size_s229347708945616566st_int @ Ys2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K2 )
=> ( ( nth_Pr8327173284404383691st_int @ Xs @ I2 )
= ( nth_Pr8327173284404383691st_int @ Ys2 @ I2 ) ) )
=> ( ( take_P1004856038796978596st_int @ K2 @ Xs )
= ( take_P1004856038796978596st_int @ K2 @ Ys2 ) ) ) ) ) ).
% nth_take_lemma
thf(fact_144_nth__take__lemma,axiom,
! [K2: nat,Xs: list_instr,Ys2: list_instr] :
( ( ord_less_eq_nat @ K2 @ ( size_size_list_instr @ Xs ) )
=> ( ( ord_less_eq_nat @ K2 @ ( size_size_list_instr @ Ys2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K2 )
=> ( ( nth_instr @ Xs @ I2 )
= ( nth_instr @ Ys2 @ I2 ) ) )
=> ( ( take_instr @ K2 @ Xs )
= ( take_instr @ K2 @ Ys2 ) ) ) ) ) ).
% nth_take_lemma
thf(fact_145_nth__take__lemma,axiom,
! [K2: nat,Xs: list_list_char,Ys2: list_list_char] :
( ( ord_less_eq_nat @ K2 @ ( size_s356637359517785349t_char @ Xs ) )
=> ( ( ord_less_eq_nat @ K2 @ ( size_s356637359517785349t_char @ Ys2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K2 )
=> ( ( nth_list_char @ Xs @ I2 )
= ( nth_list_char @ Ys2 @ I2 ) ) )
=> ( ( take_list_char @ K2 @ Xs )
= ( take_list_char @ K2 @ Ys2 ) ) ) ) ) ).
% nth_take_lemma
thf(fact_146_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_147_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_148_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_149_list__eq__iff__nth__eq,axiom,
( ( ^ [Y4: list_int,Z2: list_int] : ( Y4 = Z2 ) )
= ( ^ [Xs2: list_int,Ys3: list_int] :
( ( ( size_size_list_int @ Xs2 )
= ( size_size_list_int @ Ys3 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs2 ) )
=> ( ( nth_int @ Xs2 @ I3 )
= ( nth_int @ Ys3 @ I3 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_150_list__eq__iff__nth__eq,axiom,
( ( ^ [Y4: list_P419547167332244514st_int,Z2: list_P419547167332244514st_int] : ( Y4 = Z2 ) )
= ( ^ [Xs2: list_P419547167332244514st_int,Ys3: list_P419547167332244514st_int] :
( ( ( size_s229347708945616566st_int @ Xs2 )
= ( size_s229347708945616566st_int @ Ys3 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s229347708945616566st_int @ Xs2 ) )
=> ( ( nth_Pr8327173284404383691st_int @ Xs2 @ I3 )
= ( nth_Pr8327173284404383691st_int @ Ys3 @ I3 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_151_list__eq__iff__nth__eq,axiom,
( ( ^ [Y4: list_instr,Z2: list_instr] : ( Y4 = Z2 ) )
= ( ^ [Xs2: list_instr,Ys3: list_instr] :
( ( ( size_size_list_instr @ Xs2 )
= ( size_size_list_instr @ Ys3 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_instr @ Xs2 ) )
=> ( ( nth_instr @ Xs2 @ I3 )
= ( nth_instr @ Ys3 @ I3 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_152_list__eq__iff__nth__eq,axiom,
( ( ^ [Y4: list_list_char,Z2: list_list_char] : ( Y4 = Z2 ) )
= ( ^ [Xs2: list_list_char,Ys3: list_list_char] :
( ( ( size_s356637359517785349t_char @ Xs2 )
= ( size_s356637359517785349t_char @ Ys3 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s356637359517785349t_char @ Xs2 ) )
=> ( ( nth_list_char @ Xs2 @ I3 )
= ( nth_list_char @ Ys3 @ I3 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_153_Skolem__list__nth,axiom,
! [K2: nat,P: nat > int > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K2 )
=> ? [X5: int] : ( P @ I3 @ X5 ) ) )
= ( ? [Xs2: list_int] :
( ( ( size_size_list_int @ Xs2 )
= K2 )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K2 )
=> ( P @ I3 @ ( nth_int @ Xs2 @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_154_Skolem__list__nth,axiom,
! [K2: nat,P: nat > produc6425607678544837394st_int > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K2 )
=> ? [X5: produc6425607678544837394st_int] : ( P @ I3 @ X5 ) ) )
= ( ? [Xs2: list_P419547167332244514st_int] :
( ( ( size_s229347708945616566st_int @ Xs2 )
= K2 )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K2 )
=> ( P @ I3 @ ( nth_Pr8327173284404383691st_int @ Xs2 @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_155_Skolem__list__nth,axiom,
! [K2: nat,P: nat > instr > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K2 )
=> ? [X5: instr] : ( P @ I3 @ X5 ) ) )
= ( ? [Xs2: list_instr] :
( ( ( size_size_list_instr @ Xs2 )
= K2 )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K2 )
=> ( P @ I3 @ ( nth_instr @ Xs2 @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_156_Skolem__list__nth,axiom,
! [K2: nat,P: nat > list_char > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K2 )
=> ? [X5: list_char] : ( P @ I3 @ X5 ) ) )
= ( ? [Xs2: list_list_char] :
( ( ( size_s356637359517785349t_char @ Xs2 )
= K2 )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K2 )
=> ( P @ I3 @ ( nth_list_char @ Xs2 @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_157_nth__equalityI,axiom,
! [Xs: list_int,Ys2: list_int] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
=> ( ( nth_int @ Xs @ I2 )
= ( nth_int @ Ys2 @ I2 ) ) )
=> ( Xs = Ys2 ) ) ) ).
% nth_equalityI
thf(fact_158_nth__equalityI,axiom,
! [Xs: list_P419547167332244514st_int,Ys2: list_P419547167332244514st_int] :
( ( ( size_s229347708945616566st_int @ Xs )
= ( size_s229347708945616566st_int @ Ys2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s229347708945616566st_int @ Xs ) )
=> ( ( nth_Pr8327173284404383691st_int @ Xs @ I2 )
= ( nth_Pr8327173284404383691st_int @ Ys2 @ I2 ) ) )
=> ( Xs = Ys2 ) ) ) ).
% nth_equalityI
thf(fact_159_nth__equalityI,axiom,
! [Xs: list_instr,Ys2: list_instr] :
( ( ( size_size_list_instr @ Xs )
= ( size_size_list_instr @ Ys2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_instr @ Xs ) )
=> ( ( nth_instr @ Xs @ I2 )
= ( nth_instr @ Ys2 @ I2 ) ) )
=> ( Xs = Ys2 ) ) ) ).
% nth_equalityI
thf(fact_160_nth__equalityI,axiom,
! [Xs: list_list_char,Ys2: list_list_char] :
( ( ( size_s356637359517785349t_char @ Xs )
= ( size_s356637359517785349t_char @ Ys2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s356637359517785349t_char @ Xs ) )
=> ( ( nth_list_char @ Xs @ I2 )
= ( nth_list_char @ Ys2 @ I2 ) ) )
=> ( Xs = Ys2 ) ) ) ).
% nth_equalityI
thf(fact_161_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_162_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_163_map__equality__iff,axiom,
! [F: int > produc6425607678544837394st_int,Xs: list_int,G: produc6425607678544837394st_int > produc6425607678544837394st_int,Ys2: list_P419547167332244514st_int] :
( ( ( map_in8512336359594627332st_int @ F @ Xs )
= ( map_Pr8625093460839654122st_int @ G @ Ys2 ) )
= ( ( ( size_size_list_int @ Xs )
= ( size_s229347708945616566st_int @ Ys2 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s229347708945616566st_int @ Ys2 ) )
=> ( ( F @ ( nth_int @ Xs @ I3 ) )
= ( G @ ( nth_Pr8327173284404383691st_int @ Ys2 @ I3 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_164_map__equality__iff,axiom,
! [F: produc6425607678544837394st_int > produc6425607678544837394st_int,Xs: list_P419547167332244514st_int,G: int > produc6425607678544837394st_int,Ys2: list_int] :
( ( ( map_Pr8625093460839654122st_int @ F @ Xs )
= ( map_in8512336359594627332st_int @ G @ Ys2 ) )
= ( ( ( size_s229347708945616566st_int @ Xs )
= ( size_size_list_int @ Ys2 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_int @ Ys2 ) )
=> ( ( F @ ( nth_Pr8327173284404383691st_int @ Xs @ I3 ) )
= ( G @ ( nth_int @ Ys2 @ I3 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_165_map__equality__iff,axiom,
! [F: produc6425607678544837394st_int > produc6425607678544837394st_int,Xs: list_P419547167332244514st_int,G: produc6425607678544837394st_int > produc6425607678544837394st_int,Ys2: list_P419547167332244514st_int] :
( ( ( map_Pr8625093460839654122st_int @ F @ Xs )
= ( map_Pr8625093460839654122st_int @ G @ Ys2 ) )
= ( ( ( size_s229347708945616566st_int @ Xs )
= ( size_s229347708945616566st_int @ Ys2 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s229347708945616566st_int @ Ys2 ) )
=> ( ( F @ ( nth_Pr8327173284404383691st_int @ Xs @ I3 ) )
= ( G @ ( nth_Pr8327173284404383691st_int @ Ys2 @ I3 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_166_map__equality__iff,axiom,
! [F: produc6425607678544837394st_int > produc6425607678544837394st_int,Xs: list_P419547167332244514st_int,G: instr > produc6425607678544837394st_int,Ys2: list_instr] :
( ( ( map_Pr8625093460839654122st_int @ F @ Xs )
= ( map_in7259965606695656109st_int @ G @ Ys2 ) )
= ( ( ( size_s229347708945616566st_int @ Xs )
= ( size_size_list_instr @ Ys2 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_instr @ Ys2 ) )
=> ( ( F @ ( nth_Pr8327173284404383691st_int @ Xs @ I3 ) )
= ( G @ ( nth_instr @ Ys2 @ I3 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_167_map__equality__iff,axiom,
! [F: produc6425607678544837394st_int > produc6425607678544837394st_int,Xs: list_P419547167332244514st_int,G: list_char > produc6425607678544837394st_int,Ys2: list_list_char] :
( ( ( map_Pr8625093460839654122st_int @ F @ Xs )
= ( map_li6440205260635360185st_int @ G @ Ys2 ) )
= ( ( ( size_s229347708945616566st_int @ Xs )
= ( size_s356637359517785349t_char @ Ys2 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s356637359517785349t_char @ Ys2 ) )
=> ( ( F @ ( nth_Pr8327173284404383691st_int @ Xs @ I3 ) )
= ( G @ ( nth_list_char @ Ys2 @ I3 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_168_map__equality__iff,axiom,
! [F: instr > produc6425607678544837394st_int,Xs: list_instr,G: produc6425607678544837394st_int > produc6425607678544837394st_int,Ys2: list_P419547167332244514st_int] :
( ( ( map_in7259965606695656109st_int @ F @ Xs )
= ( map_Pr8625093460839654122st_int @ G @ Ys2 ) )
= ( ( ( size_size_list_instr @ Xs )
= ( size_s229347708945616566st_int @ Ys2 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s229347708945616566st_int @ Ys2 ) )
=> ( ( F @ ( nth_instr @ Xs @ I3 ) )
= ( G @ ( nth_Pr8327173284404383691st_int @ Ys2 @ I3 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_169_map__equality__iff,axiom,
! [F: list_char > produc6425607678544837394st_int,Xs: list_list_char,G: produc6425607678544837394st_int > produc6425607678544837394st_int,Ys2: list_P419547167332244514st_int] :
( ( ( map_li6440205260635360185st_int @ F @ Xs )
= ( map_Pr8625093460839654122st_int @ G @ Ys2 ) )
= ( ( ( size_s356637359517785349t_char @ Xs )
= ( size_s229347708945616566st_int @ Ys2 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s229347708945616566st_int @ Ys2 ) )
=> ( ( F @ ( nth_list_char @ Xs @ I3 ) )
= ( G @ ( nth_Pr8327173284404383691st_int @ Ys2 @ I3 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_170_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_171_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_172_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_173_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_174_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_175_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_176_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_177_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_178_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_179_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_180_surj__pair,axiom,
! [P2: produc4189061985984568957nt_int] :
? [X3: list_list_char,Y3: produc4435102495419491129nt_int] :
( P2
= ( produc3965054194175396271nt_int @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_181_surj__pair,axiom,
! [P2: produc6425607678544837394st_int] :
? [X3: int,Y3: produc369741286924889929st_int] :
( P2
= ( produc5086643055186798020st_int @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_182_surj__pair,axiom,
! [P2: product_prod_int_int] :
? [X3: int,Y3: int] :
( P2
= ( product_Pair_int_int @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_183_surj__pair,axiom,
! [P2: produc369741286924889929st_int] :
? [X3: list_char > int,Y3: list_int] :
( P2
= ( produc8650753666468850689st_int @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_184_surj__pair,axiom,
! [P2: produc4435102495419491129nt_int] :
? [X3: list_char > int,Y3: int] :
( P2
= ( produc5790713362662368625nt_int @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_185_old_Oprod_Oexhaust,axiom,
! [Y: produc4189061985984568957nt_int] :
~ ! [A4: list_list_char,B3: produc4435102495419491129nt_int] :
( Y
!= ( produc3965054194175396271nt_int @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_186_old_Oprod_Oexhaust,axiom,
! [Y: produc6425607678544837394st_int] :
~ ! [A4: int,B3: produc369741286924889929st_int] :
( Y
!= ( produc5086643055186798020st_int @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_187_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_188_old_Oprod_Oexhaust,axiom,
! [Y: produc369741286924889929st_int] :
~ ! [A4: list_char > int,B3: list_int] :
( Y
!= ( produc8650753666468850689st_int @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_189_old_Oprod_Oexhaust,axiom,
! [Y: produc4435102495419491129nt_int] :
~ ! [A4: list_char > int,B3: int] :
( Y
!= ( produc5790713362662368625nt_int @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_190_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_191_Suc__inject,axiom,
! [X4: nat,Y: nat] :
( ( ( suc @ X4 )
= ( suc @ Y ) )
=> ( X4 = Y ) ) ).
% Suc_inject
thf(fact_192_size__neq__size__imp__neq,axiom,
! [X4: char,Y: char] :
( ( ( size_size_char @ X4 )
!= ( size_size_char @ Y ) )
=> ( X4 != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_193_size__neq__size__imp__neq,axiom,
! [X4: list_P419547167332244514st_int,Y: list_P419547167332244514st_int] :
( ( ( size_s229347708945616566st_int @ X4 )
!= ( size_s229347708945616566st_int @ Y ) )
=> ( X4 != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_194_size__neq__size__imp__neq,axiom,
! [X4: list_instr,Y: list_instr] :
( ( ( size_size_list_instr @ X4 )
!= ( size_size_list_instr @ Y ) )
=> ( X4 != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_195_size__neq__size__imp__neq,axiom,
! [X4: list_list_char,Y: list_list_char] :
( ( ( size_s356637359517785349t_char @ X4 )
!= ( size_s356637359517785349t_char @ Y ) )
=> ( X4 != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_196_size__neq__size__imp__neq,axiom,
! [X4: com,Y: com] :
( ( ( size_size_com @ X4 )
!= ( size_size_com @ Y ) )
=> ( X4 != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_197_neq__if__length__neq,axiom,
! [Xs: list_P419547167332244514st_int,Ys2: list_P419547167332244514st_int] :
( ( ( size_s229347708945616566st_int @ Xs )
!= ( size_s229347708945616566st_int @ Ys2 ) )
=> ( Xs != Ys2 ) ) ).
% neq_if_length_neq
thf(fact_198_neq__if__length__neq,axiom,
! [Xs: list_instr,Ys2: list_instr] :
( ( ( size_size_list_instr @ Xs )
!= ( size_size_list_instr @ Ys2 ) )
=> ( Xs != Ys2 ) ) ).
% neq_if_length_neq
thf(fact_199_neq__if__length__neq,axiom,
! [Xs: list_list_char,Ys2: list_list_char] :
( ( ( size_s356637359517785349t_char @ Xs )
!= ( size_s356637359517785349t_char @ Ys2 ) )
=> ( Xs != Ys2 ) ) ).
% neq_if_length_neq
thf(fact_200_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_P419547167332244514st_int] :
( ( size_s229347708945616566st_int @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_201_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_instr] :
( ( size_size_list_instr @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_202_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_list_char] :
( ( size_s356637359517785349t_char @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_203_diff__commute,axiom,
! [I: nat,J: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K2 )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K2 ) @ J ) ) ).
% diff_commute
thf(fact_204_take__equalityI,axiom,
! [Xs: list_P419547167332244514st_int,Ys2: list_P419547167332244514st_int] :
( ! [I2: nat] :
( ( take_P1004856038796978596st_int @ I2 @ Xs )
= ( take_P1004856038796978596st_int @ I2 @ Ys2 ) )
=> ( Xs = Ys2 ) ) ).
% take_equalityI
thf(fact_205_list_Omap__ident,axiom,
! [T: list_P419547167332244514st_int] :
( ( map_Pr8625093460839654122st_int
@ ^ [X: produc6425607678544837394st_int] : X
@ T )
= T ) ).
% list.map_ident
thf(fact_206_lift__Suc__antimono__le,axiom,
! [F: nat > int,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_int @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_207_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_208_lift__Suc__mono__le,axiom,
! [F: nat > int,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_int @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_209_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_210_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_211_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_212_lift__Suc__mono__less,axiom,
! [F: nat > int,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_less_int @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_213_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_214_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).
% of_nat_mono
thf(fact_215_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).
% of_nat_mono
thf(fact_216_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_217_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_218_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_219_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_220_atLeastLessThan__subset__iff,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ A @ B ) @ ( set_or4665077453230672383an_nat @ C @ D ) )
=> ( ( ord_less_eq_nat @ B @ A )
| ( ( ord_less_eq_nat @ C @ A )
& ( ord_less_eq_nat @ B @ D ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_221_atLeastLessThan__subset__iff,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_set_int @ ( set_or4662586982721622107an_int @ A @ B ) @ ( set_or4662586982721622107an_int @ C @ D ) )
=> ( ( ord_less_eq_int @ B @ A )
| ( ( ord_less_eq_int @ C @ A )
& ( ord_less_eq_int @ B @ D ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_222_atLeastLessThan__eq__iff,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ( set_or4665077453230672383an_nat @ A @ B )
= ( set_or4665077453230672383an_nat @ C @ D ) )
= ( ( A = C )
& ( B = D ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_223_atLeastLessThan__eq__iff,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( ( set_or4662586982721622107an_int @ A @ B )
= ( set_or4662586982721622107an_int @ C @ D ) )
= ( ( A = C )
& ( B = D ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_224_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_225_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_226_atLeastLessThan__inj_I1_J,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ( set_or4665077453230672383an_nat @ A @ B )
= ( set_or4665077453230672383an_nat @ C @ D ) )
=> ( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( A = C ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_227_atLeastLessThan__inj_I1_J,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( set_or4662586982721622107an_int @ A @ B )
= ( set_or4662586982721622107an_int @ C @ D ) )
=> ( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( A = C ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_228_atLeastLessThan__inj_I2_J,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ( set_or4665077453230672383an_nat @ A @ B )
= ( set_or4665077453230672383an_nat @ C @ D ) )
=> ( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( B = D ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_229_atLeastLessThan__inj_I2_J,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( set_or4662586982721622107an_int @ A @ B )
= ( set_or4662586982721622107an_int @ C @ D ) )
=> ( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( B = D ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_230_prod__induct3,axiom,
! [P: produc4189061985984568957nt_int > $o,X4: produc4189061985984568957nt_int] :
( ! [A4: list_list_char,B3: list_char > int,C2: int] : ( P @ ( produc3965054194175396271nt_int @ A4 @ ( produc5790713362662368625nt_int @ B3 @ C2 ) ) )
=> ( P @ X4 ) ) ).
% prod_induct3
thf(fact_231_prod__induct3,axiom,
! [P: produc6425607678544837394st_int > $o,X4: produc6425607678544837394st_int] :
( ! [A4: int,B3: list_char > int,C2: list_int] : ( P @ ( produc5086643055186798020st_int @ A4 @ ( produc8650753666468850689st_int @ B3 @ C2 ) ) )
=> ( P @ X4 ) ) ).
% prod_induct3
thf(fact_232_prod__cases3,axiom,
! [Y: produc4189061985984568957nt_int] :
~ ! [A4: list_list_char,B3: list_char > int,C2: int] :
( Y
!= ( produc3965054194175396271nt_int @ A4 @ ( produc5790713362662368625nt_int @ B3 @ C2 ) ) ) ).
% prod_cases3
thf(fact_233_prod__cases3,axiom,
! [Y: produc6425607678544837394st_int] :
~ ! [A4: int,B3: list_char > int,C2: list_int] :
( Y
!= ( produc5086643055186798020st_int @ A4 @ ( produc8650753666468850689st_int @ B3 @ C2 ) ) ) ).
% prod_cases3
thf(fact_234_zero__induct__lemma,axiom,
! [P: nat > $o,K2: nat,I: nat] :
( ( P @ K2 )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ ( minus_minus_nat @ K2 @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_235_fst__conv,axiom,
! [X1: int,X2: produc369741286924889929st_int] :
( ( produc8941309049508465880st_int @ ( produc5086643055186798020st_int @ X1 @ X2 ) )
= X1 ) ).
% fst_conv
thf(fact_236_fst__conv,axiom,
! [X1: list_char > int,X2: int] :
( ( produc9025023058967219549nt_int @ ( produc5790713362662368625nt_int @ X1 @ X2 ) )
= X1 ) ).
% fst_conv
thf(fact_237_fst__conv,axiom,
! [X1: list_char > int,X2: list_int] :
( ( produc2155683374303800813st_int @ ( produc8650753666468850689st_int @ X1 @ X2 ) )
= X1 ) ).
% fst_conv
thf(fact_238_fst__conv,axiom,
! [X1: int,X2: int] :
( ( product_fst_int_int @ ( product_Pair_int_int @ X1 @ X2 ) )
= X1 ) ).
% fst_conv
thf(fact_239_fst__conv,axiom,
! [X1: list_list_char,X2: produc4435102495419491129nt_int] :
( ( produc811350159336358595nt_int @ ( produc3965054194175396271nt_int @ X1 @ X2 ) )
= X1 ) ).
% fst_conv
thf(fact_240_fst__eqD,axiom,
! [X4: int,Y: produc369741286924889929st_int,A: int] :
( ( ( produc8941309049508465880st_int @ ( produc5086643055186798020st_int @ X4 @ Y ) )
= A )
=> ( X4 = A ) ) ).
% fst_eqD
thf(fact_241_fst__eqD,axiom,
! [X4: list_char > int,Y: int,A: list_char > int] :
( ( ( produc9025023058967219549nt_int @ ( produc5790713362662368625nt_int @ X4 @ Y ) )
= A )
=> ( X4 = A ) ) ).
% fst_eqD
thf(fact_242_fst__eqD,axiom,
! [X4: list_char > int,Y: list_int,A: list_char > int] :
( ( ( produc2155683374303800813st_int @ ( produc8650753666468850689st_int @ X4 @ Y ) )
= A )
=> ( X4 = A ) ) ).
% fst_eqD
thf(fact_243_fst__eqD,axiom,
! [X4: int,Y: int,A: int] :
( ( ( product_fst_int_int @ ( product_Pair_int_int @ X4 @ Y ) )
= A )
=> ( X4 = A ) ) ).
% fst_eqD
thf(fact_244_fst__eqD,axiom,
! [X4: list_list_char,Y: produc4435102495419491129nt_int,A: list_list_char] :
( ( ( produc811350159336358595nt_int @ ( produc3965054194175396271nt_int @ X4 @ Y ) )
= A )
=> ( X4 = A ) ) ).
% fst_eqD
thf(fact_245_snd__conv,axiom,
! [X1: int,X2: produc369741286924889929st_int] :
( ( produc1296517004818028822st_int @ ( produc5086643055186798020st_int @ X1 @ X2 ) )
= X2 ) ).
% snd_conv
thf(fact_246_snd__conv,axiom,
! [X1: list_char > int,X2: int] :
( ( produc179445755590504351nt_int @ ( produc5790713362662368625nt_int @ X1 @ X2 ) )
= X2 ) ).
% snd_conv
thf(fact_247_snd__conv,axiom,
! [X1: list_char > int,X2: list_int] :
( ( produc6244060863719409455st_int @ ( produc8650753666468850689st_int @ X1 @ X2 ) )
= X2 ) ).
% snd_conv
thf(fact_248_snd__conv,axiom,
! [X1: int,X2: int] :
( ( product_snd_int_int @ ( product_Pair_int_int @ X1 @ X2 ) )
= X2 ) ).
% snd_conv
thf(fact_249_snd__conv,axiom,
! [X1: list_list_char,X2: produc4435102495419491129nt_int] :
( ( produc30867750519413249nt_int @ ( produc3965054194175396271nt_int @ X1 @ X2 ) )
= X2 ) ).
% snd_conv
thf(fact_250_snd__eqD,axiom,
! [X4: int,Y: produc369741286924889929st_int,A: produc369741286924889929st_int] :
( ( ( produc1296517004818028822st_int @ ( produc5086643055186798020st_int @ X4 @ Y ) )
= A )
=> ( Y = A ) ) ).
% snd_eqD
thf(fact_251_snd__eqD,axiom,
! [X4: list_char > int,Y: int,A: int] :
( ( ( produc179445755590504351nt_int @ ( produc5790713362662368625nt_int @ X4 @ Y ) )
= A )
=> ( Y = A ) ) ).
% snd_eqD
thf(fact_252_snd__eqD,axiom,
! [X4: list_char > int,Y: list_int,A: list_int] :
( ( ( produc6244060863719409455st_int @ ( produc8650753666468850689st_int @ X4 @ Y ) )
= A )
=> ( Y = A ) ) ).
% snd_eqD
thf(fact_253_snd__eqD,axiom,
! [X4: int,Y: int,A: int] :
( ( ( product_snd_int_int @ ( product_Pair_int_int @ X4 @ Y ) )
= A )
=> ( Y = A ) ) ).
% snd_eqD
thf(fact_254_snd__eqD,axiom,
! [X4: list_list_char,Y: produc4435102495419491129nt_int,A: produc4435102495419491129nt_int] :
( ( ( produc30867750519413249nt_int @ ( produc3965054194175396271nt_int @ X4 @ Y ) )
= A )
=> ( Y = A ) ) ).
% snd_eqD
thf(fact_255_map__eq__imp__length__eq,axiom,
! [F: produc6425607678544837394st_int > produc6425607678544837394st_int,Xs: list_P419547167332244514st_int,G: produc6425607678544837394st_int > produc6425607678544837394st_int,Ys2: list_P419547167332244514st_int] :
( ( ( map_Pr8625093460839654122st_int @ F @ Xs )
= ( map_Pr8625093460839654122st_int @ G @ Ys2 ) )
=> ( ( size_s229347708945616566st_int @ Xs )
= ( size_s229347708945616566st_int @ Ys2 ) ) ) ).
% map_eq_imp_length_eq
thf(fact_256_map__eq__imp__length__eq,axiom,
! [F: produc6425607678544837394st_int > produc6425607678544837394st_int,Xs: list_P419547167332244514st_int,G: instr > produc6425607678544837394st_int,Ys2: list_instr] :
( ( ( map_Pr8625093460839654122st_int @ F @ Xs )
= ( map_in7259965606695656109st_int @ G @ Ys2 ) )
=> ( ( size_s229347708945616566st_int @ Xs )
= ( size_size_list_instr @ Ys2 ) ) ) ).
% map_eq_imp_length_eq
thf(fact_257_map__eq__imp__length__eq,axiom,
! [F: produc6425607678544837394st_int > produc6425607678544837394st_int,Xs: list_P419547167332244514st_int,G: list_char > produc6425607678544837394st_int,Ys2: list_list_char] :
( ( ( map_Pr8625093460839654122st_int @ F @ Xs )
= ( map_li6440205260635360185st_int @ G @ Ys2 ) )
=> ( ( size_s229347708945616566st_int @ Xs )
= ( size_s356637359517785349t_char @ Ys2 ) ) ) ).
% map_eq_imp_length_eq
thf(fact_258_map__eq__imp__length__eq,axiom,
! [F: instr > produc6425607678544837394st_int,Xs: list_instr,G: produc6425607678544837394st_int > produc6425607678544837394st_int,Ys2: list_P419547167332244514st_int] :
( ( ( map_in7259965606695656109st_int @ F @ Xs )
= ( map_Pr8625093460839654122st_int @ G @ Ys2 ) )
=> ( ( size_size_list_instr @ Xs )
= ( size_s229347708945616566st_int @ Ys2 ) ) ) ).
% map_eq_imp_length_eq
thf(fact_259_map__eq__imp__length__eq,axiom,
! [F: list_char > produc6425607678544837394st_int,Xs: list_list_char,G: produc6425607678544837394st_int > produc6425607678544837394st_int,Ys2: list_P419547167332244514st_int] :
( ( ( map_li6440205260635360185st_int @ F @ Xs )
= ( map_Pr8625093460839654122st_int @ G @ Ys2 ) )
=> ( ( size_s356637359517785349t_char @ Xs )
= ( size_s229347708945616566st_int @ Ys2 ) ) ) ).
% map_eq_imp_length_eq
thf(fact_260_prod__eq__iff,axiom,
( ( ^ [Y4: produc6425607678544837394st_int,Z2: produc6425607678544837394st_int] : ( Y4 = Z2 ) )
= ( ^ [S2: produc6425607678544837394st_int,T2: produc6425607678544837394st_int] :
( ( ( produc8941309049508465880st_int @ S2 )
= ( produc8941309049508465880st_int @ T2 ) )
& ( ( produc1296517004818028822st_int @ S2 )
= ( produc1296517004818028822st_int @ T2 ) ) ) ) ) ).
% prod_eq_iff
thf(fact_261_prod__eq__iff,axiom,
( ( ^ [Y4: produc4435102495419491129nt_int,Z2: produc4435102495419491129nt_int] : ( Y4 = Z2 ) )
= ( ^ [S2: produc4435102495419491129nt_int,T2: produc4435102495419491129nt_int] :
( ( ( produc9025023058967219549nt_int @ S2 )
= ( produc9025023058967219549nt_int @ T2 ) )
& ( ( produc179445755590504351nt_int @ S2 )
= ( produc179445755590504351nt_int @ T2 ) ) ) ) ) ).
% prod_eq_iff
thf(fact_262_prod__eq__iff,axiom,
( ( ^ [Y4: produc369741286924889929st_int,Z2: produc369741286924889929st_int] : ( Y4 = Z2 ) )
= ( ^ [S2: produc369741286924889929st_int,T2: produc369741286924889929st_int] :
( ( ( produc2155683374303800813st_int @ S2 )
= ( produc2155683374303800813st_int @ T2 ) )
& ( ( produc6244060863719409455st_int @ S2 )
= ( produc6244060863719409455st_int @ T2 ) ) ) ) ) ).
% prod_eq_iff
thf(fact_263_prod__eq__iff,axiom,
( ( ^ [Y4: product_prod_int_int,Z2: product_prod_int_int] : ( Y4 = Z2 ) )
= ( ^ [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_264_prod__eq__iff,axiom,
( ( ^ [Y4: produc4189061985984568957nt_int,Z2: produc4189061985984568957nt_int] : ( Y4 = Z2 ) )
= ( ^ [S2: produc4189061985984568957nt_int,T2: produc4189061985984568957nt_int] :
( ( ( produc811350159336358595nt_int @ S2 )
= ( produc811350159336358595nt_int @ T2 ) )
& ( ( produc30867750519413249nt_int @ S2 )
= ( produc30867750519413249nt_int @ T2 ) ) ) ) ) ).
% prod_eq_iff
thf(fact_265_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_266_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_267_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_268_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_269_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_270_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_271_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_272_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_273_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_274_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_275_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_276_drop__map,axiom,
! [N: nat,F: list_char > produc6425607678544837394st_int,Xs: list_list_char] :
( ( drop_P7838799151754935884st_int @ N @ ( map_li6440205260635360185st_int @ F @ Xs ) )
= ( map_li6440205260635360185st_int @ F @ ( drop_list_char @ N @ Xs ) ) ) ).
% drop_map
thf(fact_277_drop__map,axiom,
! [N: nat,F: instr > produc6425607678544837394st_int,Xs: list_instr] :
( ( drop_P7838799151754935884st_int @ N @ ( map_in7259965606695656109st_int @ F @ Xs ) )
= ( map_in7259965606695656109st_int @ F @ ( drop_instr @ N @ Xs ) ) ) ).
% drop_map
thf(fact_278_drop__map,axiom,
! [N: nat,F: produc6425607678544837394st_int > list_char,Xs: list_P419547167332244514st_int] :
( ( drop_list_char @ N @ ( map_Pr5338895242362685753t_char @ F @ Xs ) )
= ( map_Pr5338895242362685753t_char @ F @ ( drop_P7838799151754935884st_int @ N @ Xs ) ) ) ).
% drop_map
thf(fact_279_drop__map,axiom,
! [N: nat,F: list_char > list_char,Xs: list_list_char] :
( ( drop_list_char @ N @ ( map_li116305933131242120t_char @ F @ Xs ) )
= ( map_li116305933131242120t_char @ F @ ( drop_list_char @ N @ Xs ) ) ) ).
% drop_map
thf(fact_280_drop__map,axiom,
! [N: nat,F: instr > list_char,Xs: list_instr] :
( ( drop_list_char @ N @ ( map_instr_list_char @ F @ Xs ) )
= ( map_instr_list_char @ F @ ( drop_instr @ N @ Xs ) ) ) ).
% drop_map
thf(fact_281_drop__map,axiom,
! [N: nat,F: produc6425607678544837394st_int > instr,Xs: list_P419547167332244514st_int] :
( ( drop_instr @ N @ ( map_Pr7514435936640407687_instr @ F @ Xs ) )
= ( map_Pr7514435936640407687_instr @ F @ ( drop_P7838799151754935884st_int @ N @ Xs ) ) ) ).
% drop_map
thf(fact_282_drop__map,axiom,
! [N: nat,F: list_char > instr,Xs: list_list_char] :
( ( drop_instr @ N @ ( map_list_char_instr @ F @ Xs ) )
= ( map_list_char_instr @ F @ ( drop_list_char @ N @ Xs ) ) ) ).
% drop_map
thf(fact_283_drop__map,axiom,
! [N: nat,F: instr > instr,Xs: list_instr] :
( ( drop_instr @ N @ ( map_instr_instr @ F @ Xs ) )
= ( map_instr_instr @ F @ ( drop_instr @ N @ Xs ) ) ) ).
% drop_map
thf(fact_284_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_285_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_286_surjective__pairing,axiom,
! [T: produc6425607678544837394st_int] :
( T
= ( produc5086643055186798020st_int @ ( produc8941309049508465880st_int @ T ) @ ( produc1296517004818028822st_int @ T ) ) ) ).
% surjective_pairing
thf(fact_287_surjective__pairing,axiom,
! [T: produc4435102495419491129nt_int] :
( T
= ( produc5790713362662368625nt_int @ ( produc9025023058967219549nt_int @ T ) @ ( produc179445755590504351nt_int @ T ) ) ) ).
% surjective_pairing
thf(fact_288_surjective__pairing,axiom,
! [T: produc369741286924889929st_int] :
( T
= ( produc8650753666468850689st_int @ ( produc2155683374303800813st_int @ T ) @ ( produc6244060863719409455st_int @ T ) ) ) ).
% surjective_pairing
thf(fact_289_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_290_surjective__pairing,axiom,
! [T: produc4189061985984568957nt_int] :
( T
= ( produc3965054194175396271nt_int @ ( produc811350159336358595nt_int @ T ) @ ( produc30867750519413249nt_int @ T ) ) ) ).
% surjective_pairing
thf(fact_291_prod_Oexhaust__sel,axiom,
! [Prod: produc6425607678544837394st_int] :
( Prod
= ( produc5086643055186798020st_int @ ( produc8941309049508465880st_int @ Prod ) @ ( produc1296517004818028822st_int @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_292_prod_Oexhaust__sel,axiom,
! [Prod: produc4435102495419491129nt_int] :
( Prod
= ( produc5790713362662368625nt_int @ ( produc9025023058967219549nt_int @ Prod ) @ ( produc179445755590504351nt_int @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_293_prod_Oexhaust__sel,axiom,
! [Prod: produc369741286924889929st_int] :
( Prod
= ( produc8650753666468850689st_int @ ( produc2155683374303800813st_int @ Prod ) @ ( produc6244060863719409455st_int @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_294_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_295_prod_Oexhaust__sel,axiom,
! [Prod: produc4189061985984568957nt_int] :
( Prod
= ( produc3965054194175396271nt_int @ ( produc811350159336358595nt_int @ Prod ) @ ( produc30867750519413249nt_int @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_296_drop__take,axiom,
! [N: nat,M: nat,Xs: list_list_char] :
( ( drop_list_char @ N @ ( take_list_char @ M @ Xs ) )
= ( take_list_char @ ( minus_minus_nat @ M @ N ) @ ( drop_list_char @ N @ Xs ) ) ) ).
% drop_take
thf(fact_297_drop__take,axiom,
! [N: nat,M: nat,Xs: list_instr] :
( ( drop_instr @ N @ ( take_instr @ M @ Xs ) )
= ( take_instr @ ( minus_minus_nat @ M @ N ) @ ( drop_instr @ N @ Xs ) ) ) ).
% drop_take
thf(fact_298_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_299_pair__list__eqI,axiom,
! [Xs: list_P419547167332244514st_int,Ys2: list_P419547167332244514st_int] :
( ( ( map_Pr3224476576807795204nt_int @ produc8941309049508465880st_int @ Xs )
= ( map_Pr3224476576807795204nt_int @ produc8941309049508465880st_int @ Ys2 ) )
=> ( ( ( map_Pr4101632105909876081st_int @ produc1296517004818028822st_int @ Xs )
= ( map_Pr4101632105909876081st_int @ produc1296517004818028822st_int @ Ys2 ) )
=> ( Xs = Ys2 ) ) ) ).
% pair_list_eqI
thf(fact_300_pair__list__eqI,axiom,
! [Xs: list_P98921565119350719nt_int,Ys2: list_P98921565119350719nt_int] :
( ( ( map_Pr6267657629804605729ar_int @ produc9025023058967219549nt_int @ Xs )
= ( map_Pr6267657629804605729ar_int @ produc9025023058967219549nt_int @ Ys2 ) )
=> ( ( ( map_Pr7025006424454111649nt_int @ produc179445755590504351nt_int @ Xs )
= ( map_Pr7025006424454111649nt_int @ produc179445755590504351nt_int @ Ys2 ) )
=> ( Xs = Ys2 ) ) ) ).
% pair_list_eqI
thf(fact_301_pair__list__eqI,axiom,
! [Xs: list_P8654944379044127311st_int,Ys2: list_P8654944379044127311st_int] :
( ( ( map_Pr4838045029083743153ar_int @ produc2155683374303800813st_int @ Xs )
= ( map_Pr4838045029083743153ar_int @ produc2155683374303800813st_int @ Ys2 ) )
=> ( ( ( map_Pr7594757675026762433st_int @ produc6244060863719409455st_int @ Xs )
= ( map_Pr7594757675026762433st_int @ produc6244060863719409455st_int @ Ys2 ) )
=> ( Xs = Ys2 ) ) ) ).
% pair_list_eqI
thf(fact_302_pair__list__eqI,axiom,
! [Xs: list_P5707943133018811711nt_int,Ys2: list_P5707943133018811711nt_int] :
( ( ( map_Pr6494458893431244577nt_int @ product_fst_int_int @ Xs )
= ( map_Pr6494458893431244577nt_int @ product_fst_int_int @ Ys2 ) )
=> ( ( ( map_Pr6494458893431244577nt_int @ product_snd_int_int @ Xs )
= ( map_Pr6494458893431244577nt_int @ product_snd_int_int @ Ys2 ) )
=> ( Xs = Ys2 ) ) ) ).
% pair_list_eqI
thf(fact_303_pair__list__eqI,axiom,
! [Xs: list_P7813408393781968525nt_int,Ys2: list_P7813408393781968525nt_int] :
( ( ( map_Pr7817743546820071988t_char @ produc811350159336358595nt_int @ Xs )
= ( map_Pr7817743546820071988t_char @ produc811350159336358595nt_int @ Ys2 ) )
=> ( ( ( map_Pr254812380637628598nt_int @ produc30867750519413249nt_int @ Xs )
= ( map_Pr254812380637628598nt_int @ produc30867750519413249nt_int @ Ys2 ) )
=> ( Xs = Ys2 ) ) ) ).
% pair_list_eqI
thf(fact_304_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_305_zle__diff1__eq,axiom,
! [W: int,Z3: int] :
( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z3 @ one_one_int ) )
= ( ord_less_int @ W @ Z3 ) ) ).
% zle_diff1_eq
thf(fact_306_A,axiom,
execl_all @ ( append_instr @ p @ ( append_instr @ ( p2 @ x ) @ p3 ) ) @ cfs ).
% A
thf(fact_307_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_308_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: int > produc369741286924889929st_int > $o,X4: int,Y: produc369741286924889929st_int,A: produc6425607678544837394st_int] :
( ( P @ X4 @ Y )
=> ( ( A
= ( produc5086643055186798020st_int @ X4 @ Y ) )
=> ( P @ ( produc8941309049508465880st_int @ A ) @ ( produc1296517004818028822st_int @ A ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_309_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: ( list_char > int ) > int > $o,X4: list_char > int,Y: int,A: produc4435102495419491129nt_int] :
( ( P @ X4 @ Y )
=> ( ( A
= ( produc5790713362662368625nt_int @ X4 @ Y ) )
=> ( P @ ( produc9025023058967219549nt_int @ A ) @ ( produc179445755590504351nt_int @ A ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_310_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: ( list_char > int ) > list_int > $o,X4: list_char > int,Y: list_int,A: produc369741286924889929st_int] :
( ( P @ X4 @ Y )
=> ( ( A
= ( produc8650753666468850689st_int @ X4 @ Y ) )
=> ( P @ ( produc2155683374303800813st_int @ A ) @ ( produc6244060863719409455st_int @ A ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_311_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: int > int > $o,X4: int,Y: int,A: product_prod_int_int] :
( ( P @ X4 @ Y )
=> ( ( A
= ( product_Pair_int_int @ X4 @ Y ) )
=> ( P @ ( product_fst_int_int @ A ) @ ( product_snd_int_int @ A ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_312_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: list_list_char > produc4435102495419491129nt_int > $o,X4: list_list_char,Y: produc4435102495419491129nt_int,A: produc4189061985984568957nt_int] :
( ( P @ X4 @ Y )
=> ( ( A
= ( produc3965054194175396271nt_int @ X4 @ Y ) )
=> ( P @ ( produc811350159336358595nt_int @ A ) @ ( produc30867750519413249nt_int @ A ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_313_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_314_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_315_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_316_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_317_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_318_exI__realizer,axiom,
! [P: produc369741286924889929st_int > int > $o,Y: produc369741286924889929st_int,X4: int] :
( ( P @ Y @ X4 )
=> ( P @ ( produc1296517004818028822st_int @ ( produc5086643055186798020st_int @ X4 @ Y ) ) @ ( produc8941309049508465880st_int @ ( produc5086643055186798020st_int @ X4 @ Y ) ) ) ) ).
% exI_realizer
thf(fact_319_exI__realizer,axiom,
! [P: int > ( list_char > int ) > $o,Y: int,X4: list_char > int] :
( ( P @ Y @ X4 )
=> ( P @ ( produc179445755590504351nt_int @ ( produc5790713362662368625nt_int @ X4 @ Y ) ) @ ( produc9025023058967219549nt_int @ ( produc5790713362662368625nt_int @ X4 @ Y ) ) ) ) ).
% exI_realizer
thf(fact_320_exI__realizer,axiom,
! [P: list_int > ( list_char > int ) > $o,Y: list_int,X4: list_char > int] :
( ( P @ Y @ X4 )
=> ( P @ ( produc6244060863719409455st_int @ ( produc8650753666468850689st_int @ X4 @ Y ) ) @ ( produc2155683374303800813st_int @ ( produc8650753666468850689st_int @ X4 @ Y ) ) ) ) ).
% exI_realizer
thf(fact_321_exI__realizer,axiom,
! [P: int > int > $o,Y: int,X4: int] :
( ( P @ Y @ X4 )
=> ( P @ ( product_snd_int_int @ ( product_Pair_int_int @ X4 @ Y ) ) @ ( product_fst_int_int @ ( product_Pair_int_int @ X4 @ Y ) ) ) ) ).
% exI_realizer
thf(fact_322_exI__realizer,axiom,
! [P: produc4435102495419491129nt_int > list_list_char > $o,Y: produc4435102495419491129nt_int,X4: list_list_char] :
( ( P @ Y @ X4 )
=> ( P @ ( produc30867750519413249nt_int @ ( produc3965054194175396271nt_int @ X4 @ Y ) ) @ ( produc811350159336358595nt_int @ ( produc3965054194175396271nt_int @ X4 @ Y ) ) ) ) ).
% exI_realizer
thf(fact_323_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_324_execl__sub,axiom,
! [P: list_instr,P3: list_instr,P4: list_instr,Cfs: list_P419547167332244514st_int,M: nat,N: nat] :
( ( execl @ ( append_instr @ P @ ( append_instr @ P3 @ P4 ) ) @ Cfs )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_or4665077453230672383an_nat @ M @ N ) )
=> ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P ) ) @ ( produc8941309049508465880st_int @ ( nth_Pr8327173284404383691st_int @ Cfs @ X3 ) ) )
& ( ord_less_int @ ( minus_minus_int @ ( produc8941309049508465880st_int @ ( nth_Pr8327173284404383691st_int @ Cfs @ X3 ) ) @ ( 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 ) @ Cfs ) ) ) ) ) ) ).
% execl_sub
thf(fact_325_subsetI,axiom,
! [A3: set_nat,B5: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A3 )
=> ( member_nat @ X3 @ B5 ) )
=> ( ord_less_eq_set_nat @ A3 @ B5 ) ) ).
% subsetI
thf(fact_326_subsetI,axiom,
! [A3: set_int,B5: set_int] :
( ! [X3: int] :
( ( member_int @ X3 @ A3 )
=> ( member_int @ X3 @ B5 ) )
=> ( ord_less_eq_set_int @ A3 @ B5 ) ) ).
% subsetI
thf(fact_327_append_Oassoc,axiom,
! [A: list_instr,B: list_instr,C: list_instr] :
( ( append_instr @ ( append_instr @ A @ B ) @ C )
= ( append_instr @ A @ ( append_instr @ B @ C ) ) ) ).
% append.assoc
thf(fact_328_append_Oassoc,axiom,
! [A: list_P419547167332244514st_int,B: list_P419547167332244514st_int,C: list_P419547167332244514st_int] :
( ( append5028292981072010935st_int @ ( append5028292981072010935st_int @ A @ B ) @ C )
= ( append5028292981072010935st_int @ A @ ( append5028292981072010935st_int @ B @ C ) ) ) ).
% append.assoc
thf(fact_329_append_Oassoc,axiom,
! [A: list_int,B: list_int,C: list_int] :
( ( append_int @ ( append_int @ A @ B ) @ C )
= ( append_int @ A @ ( append_int @ B @ C ) ) ) ).
% append.assoc
thf(fact_330_append__assoc,axiom,
! [Xs: list_instr,Ys2: list_instr,Zs: list_instr] :
( ( append_instr @ ( append_instr @ Xs @ Ys2 ) @ Zs )
= ( append_instr @ Xs @ ( append_instr @ Ys2 @ Zs ) ) ) ).
% append_assoc
thf(fact_331_append__assoc,axiom,
! [Xs: list_P419547167332244514st_int,Ys2: list_P419547167332244514st_int,Zs: list_P419547167332244514st_int] :
( ( append5028292981072010935st_int @ ( append5028292981072010935st_int @ Xs @ Ys2 ) @ Zs )
= ( append5028292981072010935st_int @ Xs @ ( append5028292981072010935st_int @ Ys2 @ Zs ) ) ) ).
% append_assoc
thf(fact_332_append__assoc,axiom,
! [Xs: list_int,Ys2: list_int,Zs: list_int] :
( ( append_int @ ( append_int @ Xs @ Ys2 ) @ Zs )
= ( append_int @ Xs @ ( append_int @ Ys2 @ Zs ) ) ) ).
% append_assoc
thf(fact_333_append__same__eq,axiom,
! [Ys2: list_instr,Xs: list_instr,Zs: list_instr] :
( ( ( append_instr @ Ys2 @ Xs )
= ( append_instr @ Zs @ Xs ) )
= ( Ys2 = Zs ) ) ).
% append_same_eq
thf(fact_334_append__same__eq,axiom,
! [Ys2: list_P419547167332244514st_int,Xs: list_P419547167332244514st_int,Zs: list_P419547167332244514st_int] :
( ( ( append5028292981072010935st_int @ Ys2 @ Xs )
= ( append5028292981072010935st_int @ Zs @ Xs ) )
= ( Ys2 = Zs ) ) ).
% append_same_eq
thf(fact_335_append__same__eq,axiom,
! [Ys2: list_int,Xs: list_int,Zs: list_int] :
( ( ( append_int @ Ys2 @ Xs )
= ( append_int @ Zs @ Xs ) )
= ( Ys2 = Zs ) ) ).
% append_same_eq
thf(fact_336_same__append__eq,axiom,
! [Xs: list_instr,Ys2: list_instr,Zs: list_instr] :
( ( ( append_instr @ Xs @ Ys2 )
= ( append_instr @ Xs @ Zs ) )
= ( Ys2 = Zs ) ) ).
% same_append_eq
thf(fact_337_same__append__eq,axiom,
! [Xs: list_P419547167332244514st_int,Ys2: list_P419547167332244514st_int,Zs: list_P419547167332244514st_int] :
( ( ( append5028292981072010935st_int @ Xs @ Ys2 )
= ( append5028292981072010935st_int @ Xs @ Zs ) )
= ( Ys2 = Zs ) ) ).
% same_append_eq
thf(fact_338_same__append__eq,axiom,
! [Xs: list_int,Ys2: list_int,Zs: list_int] :
( ( ( append_int @ Xs @ Ys2 )
= ( append_int @ Xs @ Zs ) )
= ( Ys2 = Zs ) ) ).
% same_append_eq
thf(fact_339_ivl__diff,axiom,
! [I: nat,N: nat,M: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ I @ M ) @ ( set_or4665077453230672383an_nat @ I @ N ) )
= ( set_or4665077453230672383an_nat @ N @ M ) ) ) ).
% ivl_diff
thf(fact_340_ivl__diff,axiom,
! [I: int,N: int,M: int] :
( ( ord_less_eq_int @ I @ N )
=> ( ( minus_minus_set_int @ ( set_or4662586982721622107an_int @ I @ M ) @ ( set_or4662586982721622107an_int @ I @ N ) )
= ( set_or4662586982721622107an_int @ N @ M ) ) ) ).
% ivl_diff
thf(fact_341_append__eq__append__conv,axiom,
! [Xs: list_int,Ys2: list_int,Us: list_int,Vs: list_int] :
( ( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys2 ) )
| ( ( size_size_list_int @ Us )
= ( size_size_list_int @ Vs ) ) )
=> ( ( ( append_int @ Xs @ Us )
= ( append_int @ Ys2 @ Vs ) )
= ( ( Xs = Ys2 )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_342_append__eq__append__conv,axiom,
! [Xs: list_P419547167332244514st_int,Ys2: list_P419547167332244514st_int,Us: list_P419547167332244514st_int,Vs: list_P419547167332244514st_int] :
( ( ( ( size_s229347708945616566st_int @ Xs )
= ( size_s229347708945616566st_int @ Ys2 ) )
| ( ( size_s229347708945616566st_int @ Us )
= ( size_s229347708945616566st_int @ Vs ) ) )
=> ( ( ( append5028292981072010935st_int @ Xs @ Us )
= ( append5028292981072010935st_int @ Ys2 @ Vs ) )
= ( ( Xs = Ys2 )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_343_append__eq__append__conv,axiom,
! [Xs: list_instr,Ys2: list_instr,Us: list_instr,Vs: list_instr] :
( ( ( ( size_size_list_instr @ Xs )
= ( size_size_list_instr @ Ys2 ) )
| ( ( size_size_list_instr @ Us )
= ( size_size_list_instr @ Vs ) ) )
=> ( ( ( append_instr @ Xs @ Us )
= ( append_instr @ Ys2 @ Vs ) )
= ( ( Xs = Ys2 )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_344_append__eq__append__conv,axiom,
! [Xs: list_list_char,Ys2: list_list_char,Us: list_list_char,Vs: list_list_char] :
( ( ( ( size_s356637359517785349t_char @ Xs )
= ( size_s356637359517785349t_char @ Ys2 ) )
| ( ( size_s356637359517785349t_char @ Us )
= ( size_s356637359517785349t_char @ Vs ) ) )
=> ( ( ( append_list_char @ Xs @ Us )
= ( append_list_char @ Ys2 @ Vs ) )
= ( ( Xs = Ys2 )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_345_map__append,axiom,
! [F: instr > instr,Xs: list_instr,Ys2: list_instr] :
( ( map_instr_instr @ F @ ( append_instr @ Xs @ Ys2 ) )
= ( append_instr @ ( map_instr_instr @ F @ Xs ) @ ( map_instr_instr @ F @ Ys2 ) ) ) ).
% map_append
thf(fact_346_map__append,axiom,
! [F: instr > produc6425607678544837394st_int,Xs: list_instr,Ys2: list_instr] :
( ( map_in7259965606695656109st_int @ F @ ( append_instr @ Xs @ Ys2 ) )
= ( append5028292981072010935st_int @ ( map_in7259965606695656109st_int @ F @ Xs ) @ ( map_in7259965606695656109st_int @ F @ Ys2 ) ) ) ).
% map_append
thf(fact_347_map__append,axiom,
! [F: instr > int,Xs: list_instr,Ys2: list_instr] :
( ( map_instr_int @ F @ ( append_instr @ Xs @ Ys2 ) )
= ( append_int @ ( map_instr_int @ F @ Xs ) @ ( map_instr_int @ F @ Ys2 ) ) ) ).
% map_append
thf(fact_348_map__append,axiom,
! [F: produc6425607678544837394st_int > instr,Xs: list_P419547167332244514st_int,Ys2: list_P419547167332244514st_int] :
( ( map_Pr7514435936640407687_instr @ F @ ( append5028292981072010935st_int @ Xs @ Ys2 ) )
= ( append_instr @ ( map_Pr7514435936640407687_instr @ F @ Xs ) @ ( map_Pr7514435936640407687_instr @ F @ Ys2 ) ) ) ).
% map_append
thf(fact_349_map__append,axiom,
! [F: produc6425607678544837394st_int > int,Xs: list_P419547167332244514st_int,Ys2: list_P419547167332244514st_int] :
( ( map_Pr3224476576807795204nt_int @ F @ ( append5028292981072010935st_int @ Xs @ Ys2 ) )
= ( append_int @ ( map_Pr3224476576807795204nt_int @ F @ Xs ) @ ( map_Pr3224476576807795204nt_int @ F @ Ys2 ) ) ) ).
% map_append
thf(fact_350_map__append,axiom,
! [F: int > instr,Xs: list_int,Ys2: list_int] :
( ( map_int_instr @ F @ ( append_int @ Xs @ Ys2 ) )
= ( append_instr @ ( map_int_instr @ F @ Xs ) @ ( map_int_instr @ F @ Ys2 ) ) ) ).
% map_append
thf(fact_351_map__append,axiom,
! [F: int > produc6425607678544837394st_int,Xs: list_int,Ys2: list_int] :
( ( map_in8512336359594627332st_int @ F @ ( append_int @ Xs @ Ys2 ) )
= ( append5028292981072010935st_int @ ( map_in8512336359594627332st_int @ F @ Xs ) @ ( map_in8512336359594627332st_int @ F @ Ys2 ) ) ) ).
% map_append
thf(fact_352_map__append,axiom,
! [F: int > int,Xs: list_int,Ys2: list_int] :
( ( map_int_int @ F @ ( append_int @ Xs @ Ys2 ) )
= ( append_int @ ( map_int_int @ F @ Xs ) @ ( map_int_int @ F @ Ys2 ) ) ) ).
% map_append
thf(fact_353_map__append,axiom,
! [F: produc6425607678544837394st_int > produc6425607678544837394st_int,Xs: list_P419547167332244514st_int,Ys2: list_P419547167332244514st_int] :
( ( map_Pr8625093460839654122st_int @ F @ ( append5028292981072010935st_int @ Xs @ Ys2 ) )
= ( append5028292981072010935st_int @ ( map_Pr8625093460839654122st_int @ F @ Xs ) @ ( map_Pr8625093460839654122st_int @ F @ Ys2 ) ) ) ).
% map_append
thf(fact_354_append__take__drop__id,axiom,
! [N: nat,Xs: list_list_char] :
( ( append_list_char @ ( take_list_char @ N @ Xs ) @ ( drop_list_char @ N @ Xs ) )
= Xs ) ).
% append_take_drop_id
thf(fact_355_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_356_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_357_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_358_take__append,axiom,
! [N: nat,Xs: list_int,Ys2: list_int] :
( ( take_int @ N @ ( append_int @ Xs @ Ys2 ) )
= ( append_int @ ( take_int @ N @ Xs ) @ ( take_int @ ( minus_minus_nat @ N @ ( size_size_list_int @ Xs ) ) @ Ys2 ) ) ) ).
% take_append
thf(fact_359_take__append,axiom,
! [N: nat,Xs: list_P419547167332244514st_int,Ys2: list_P419547167332244514st_int] :
( ( take_P1004856038796978596st_int @ N @ ( append5028292981072010935st_int @ Xs @ Ys2 ) )
= ( append5028292981072010935st_int @ ( take_P1004856038796978596st_int @ N @ Xs ) @ ( take_P1004856038796978596st_int @ ( minus_minus_nat @ N @ ( size_s229347708945616566st_int @ Xs ) ) @ Ys2 ) ) ) ).
% take_append
thf(fact_360_take__append,axiom,
! [N: nat,Xs: list_instr,Ys2: list_instr] :
( ( take_instr @ N @ ( append_instr @ Xs @ Ys2 ) )
= ( append_instr @ ( take_instr @ N @ Xs ) @ ( take_instr @ ( minus_minus_nat @ N @ ( size_size_list_instr @ Xs ) ) @ Ys2 ) ) ) ).
% take_append
thf(fact_361_take__append,axiom,
! [N: nat,Xs: list_list_char,Ys2: list_list_char] :
( ( take_list_char @ N @ ( append_list_char @ Xs @ Ys2 ) )
= ( append_list_char @ ( take_list_char @ N @ Xs ) @ ( take_list_char @ ( minus_minus_nat @ N @ ( size_s356637359517785349t_char @ Xs ) ) @ Ys2 ) ) ) ).
% take_append
thf(fact_362_drop__append,axiom,
! [N: nat,Xs: list_int,Ys2: list_int] :
( ( drop_int @ N @ ( append_int @ Xs @ Ys2 ) )
= ( append_int @ ( drop_int @ N @ Xs ) @ ( drop_int @ ( minus_minus_nat @ N @ ( size_size_list_int @ Xs ) ) @ Ys2 ) ) ) ).
% drop_append
thf(fact_363_drop__append,axiom,
! [N: nat,Xs: list_P419547167332244514st_int,Ys2: list_P419547167332244514st_int] :
( ( drop_P7838799151754935884st_int @ N @ ( append5028292981072010935st_int @ Xs @ Ys2 ) )
= ( append5028292981072010935st_int @ ( drop_P7838799151754935884st_int @ N @ Xs ) @ ( drop_P7838799151754935884st_int @ ( minus_minus_nat @ N @ ( size_s229347708945616566st_int @ Xs ) ) @ Ys2 ) ) ) ).
% drop_append
thf(fact_364_drop__append,axiom,
! [N: nat,Xs: list_instr,Ys2: list_instr] :
( ( drop_instr @ N @ ( append_instr @ Xs @ Ys2 ) )
= ( append_instr @ ( drop_instr @ N @ Xs ) @ ( drop_instr @ ( minus_minus_nat @ N @ ( size_size_list_instr @ Xs ) ) @ Ys2 ) ) ) ).
% drop_append
thf(fact_365_drop__append,axiom,
! [N: nat,Xs: list_list_char,Ys2: list_list_char] :
( ( drop_list_char @ N @ ( append_list_char @ Xs @ Ys2 ) )
= ( append_list_char @ ( drop_list_char @ N @ Xs ) @ ( drop_list_char @ ( minus_minus_nat @ N @ ( size_s356637359517785349t_char @ Xs ) ) @ Ys2 ) ) ) ).
% drop_append
thf(fact_366_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_367_le__trans,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K2 )
=> ( ord_less_eq_nat @ I @ K2 ) ) ) ).
% le_trans
thf(fact_368_psubsetD,axiom,
! [A3: set_nat,B5: set_nat,C: nat] :
( ( ord_less_set_nat @ A3 @ B5 )
=> ( ( member_nat @ C @ A3 )
=> ( member_nat @ C @ B5 ) ) ) ).
% psubsetD
thf(fact_369_psubsetD,axiom,
! [A3: set_int,B5: set_int,C: int] :
( ( ord_less_set_int @ A3 @ B5 )
=> ( ( member_int @ C @ A3 )
=> ( member_int @ C @ B5 ) ) ) ).
% psubsetD
thf(fact_370_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_371_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_372_less__set__def,axiom,
( ord_less_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
( ord_less_nat_o
@ ^ [X: nat] : ( member_nat @ X @ A6 )
@ ^ [X: nat] : ( member_nat @ X @ B6 ) ) ) ) ).
% less_set_def
thf(fact_373_less__set__def,axiom,
( ord_less_set_int
= ( ^ [A6: set_int,B6: set_int] :
( ord_less_int_o
@ ^ [X: int] : ( member_int @ X @ A6 )
@ ^ [X: int] : ( member_int @ X @ B6 ) ) ) ) ).
% less_set_def
thf(fact_374_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_375_append__eq__appendI,axiom,
! [Xs: list_instr,Xs1: list_instr,Zs: list_instr,Ys2: list_instr,Us: list_instr] :
( ( ( append_instr @ Xs @ Xs1 )
= Zs )
=> ( ( Ys2
= ( append_instr @ Xs1 @ Us ) )
=> ( ( append_instr @ Xs @ Ys2 )
= ( append_instr @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_376_append__eq__appendI,axiom,
! [Xs: list_P419547167332244514st_int,Xs1: list_P419547167332244514st_int,Zs: list_P419547167332244514st_int,Ys2: list_P419547167332244514st_int,Us: list_P419547167332244514st_int] :
( ( ( append5028292981072010935st_int @ Xs @ Xs1 )
= Zs )
=> ( ( Ys2
= ( append5028292981072010935st_int @ Xs1 @ Us ) )
=> ( ( append5028292981072010935st_int @ Xs @ Ys2 )
= ( append5028292981072010935st_int @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_377_append__eq__appendI,axiom,
! [Xs: list_int,Xs1: list_int,Zs: list_int,Ys2: list_int,Us: list_int] :
( ( ( append_int @ Xs @ Xs1 )
= Zs )
=> ( ( Ys2
= ( append_int @ Xs1 @ Us ) )
=> ( ( append_int @ Xs @ Ys2 )
= ( append_int @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_378_psubset__imp__ex__mem,axiom,
! [A3: set_nat,B5: set_nat] :
( ( ord_less_set_nat @ A3 @ B5 )
=> ? [B3: nat] : ( member_nat @ B3 @ ( minus_minus_set_nat @ B5 @ A3 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_379_psubset__imp__ex__mem,axiom,
! [A3: set_int,B5: set_int] :
( ( ord_less_set_int @ A3 @ B5 )
=> ? [B3: int] : ( member_int @ B3 @ ( minus_minus_set_int @ B5 @ A3 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_380_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K2: nat,B: nat] :
( ( P @ K2 )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_381_append__eq__append__conv2,axiom,
! [Xs: list_instr,Ys2: list_instr,Zs: list_instr,Ts: list_instr] :
( ( ( append_instr @ Xs @ Ys2 )
= ( append_instr @ Zs @ Ts ) )
= ( ? [Us2: list_instr] :
( ( ( Xs
= ( append_instr @ Zs @ Us2 ) )
& ( ( append_instr @ Us2 @ Ys2 )
= Ts ) )
| ( ( ( append_instr @ Xs @ Us2 )
= Zs )
& ( Ys2
= ( append_instr @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_382_append__eq__append__conv2,axiom,
! [Xs: list_P419547167332244514st_int,Ys2: list_P419547167332244514st_int,Zs: list_P419547167332244514st_int,Ts: list_P419547167332244514st_int] :
( ( ( append5028292981072010935st_int @ Xs @ Ys2 )
= ( append5028292981072010935st_int @ Zs @ Ts ) )
= ( ? [Us2: list_P419547167332244514st_int] :
( ( ( Xs
= ( append5028292981072010935st_int @ Zs @ Us2 ) )
& ( ( append5028292981072010935st_int @ Us2 @ Ys2 )
= Ts ) )
| ( ( ( append5028292981072010935st_int @ Xs @ Us2 )
= Zs )
& ( Ys2
= ( append5028292981072010935st_int @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_383_append__eq__append__conv2,axiom,
! [Xs: list_int,Ys2: list_int,Zs: list_int,Ts: list_int] :
( ( ( append_int @ Xs @ Ys2 )
= ( append_int @ Zs @ Ts ) )
= ( ? [Us2: list_int] :
( ( ( Xs
= ( append_int @ Zs @ Us2 ) )
& ( ( append_int @ Us2 @ Ys2 )
= Ts ) )
| ( ( ( append_int @ Xs @ Us2 )
= Zs )
& ( Ys2
= ( append_int @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_384_bounded__Max__nat,axiom,
! [P: nat > $o,X4: nat,M7: nat] :
( ( P @ X4 )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M7 ) )
=> ~ ! [M6: nat] :
( ( P @ M6 )
=> ~ ! [X6: nat] :
( ( P @ X6 )
=> ( ord_less_eq_nat @ X6 @ M6 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_385_append__eq__map__conv,axiom,
! [Ys2: list_instr,Zs: list_instr,F: instr > instr,Xs: list_instr] :
( ( ( append_instr @ Ys2 @ Zs )
= ( map_instr_instr @ F @ Xs ) )
= ( ? [Us2: list_instr,Vs2: list_instr] :
( ( Xs
= ( append_instr @ Us2 @ Vs2 ) )
& ( Ys2
= ( map_instr_instr @ F @ Us2 ) )
& ( Zs
= ( map_instr_instr @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_386_append__eq__map__conv,axiom,
! [Ys2: list_instr,Zs: list_instr,F: produc6425607678544837394st_int > instr,Xs: list_P419547167332244514st_int] :
( ( ( append_instr @ Ys2 @ Zs )
= ( map_Pr7514435936640407687_instr @ F @ Xs ) )
= ( ? [Us2: list_P419547167332244514st_int,Vs2: list_P419547167332244514st_int] :
( ( Xs
= ( append5028292981072010935st_int @ Us2 @ Vs2 ) )
& ( Ys2
= ( map_Pr7514435936640407687_instr @ F @ Us2 ) )
& ( Zs
= ( map_Pr7514435936640407687_instr @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_387_append__eq__map__conv,axiom,
! [Ys2: list_instr,Zs: list_instr,F: int > instr,Xs: list_int] :
( ( ( append_instr @ Ys2 @ Zs )
= ( map_int_instr @ F @ Xs ) )
= ( ? [Us2: list_int,Vs2: list_int] :
( ( Xs
= ( append_int @ Us2 @ Vs2 ) )
& ( Ys2
= ( map_int_instr @ F @ Us2 ) )
& ( Zs
= ( map_int_instr @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_388_append__eq__map__conv,axiom,
! [Ys2: list_P419547167332244514st_int,Zs: list_P419547167332244514st_int,F: instr > produc6425607678544837394st_int,Xs: list_instr] :
( ( ( append5028292981072010935st_int @ Ys2 @ Zs )
= ( map_in7259965606695656109st_int @ F @ Xs ) )
= ( ? [Us2: list_instr,Vs2: list_instr] :
( ( Xs
= ( append_instr @ Us2 @ Vs2 ) )
& ( Ys2
= ( map_in7259965606695656109st_int @ F @ Us2 ) )
& ( Zs
= ( map_in7259965606695656109st_int @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_389_append__eq__map__conv,axiom,
! [Ys2: list_P419547167332244514st_int,Zs: list_P419547167332244514st_int,F: int > produc6425607678544837394st_int,Xs: list_int] :
( ( ( append5028292981072010935st_int @ Ys2 @ Zs )
= ( map_in8512336359594627332st_int @ F @ Xs ) )
= ( ? [Us2: list_int,Vs2: list_int] :
( ( Xs
= ( append_int @ Us2 @ Vs2 ) )
& ( Ys2
= ( map_in8512336359594627332st_int @ F @ Us2 ) )
& ( Zs
= ( map_in8512336359594627332st_int @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_390_append__eq__map__conv,axiom,
! [Ys2: list_int,Zs: list_int,F: instr > int,Xs: list_instr] :
( ( ( append_int @ Ys2 @ Zs )
= ( map_instr_int @ F @ Xs ) )
= ( ? [Us2: list_instr,Vs2: list_instr] :
( ( Xs
= ( append_instr @ Us2 @ Vs2 ) )
& ( Ys2
= ( map_instr_int @ F @ Us2 ) )
& ( Zs
= ( map_instr_int @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_391_append__eq__map__conv,axiom,
! [Ys2: list_int,Zs: list_int,F: produc6425607678544837394st_int > int,Xs: list_P419547167332244514st_int] :
( ( ( append_int @ Ys2 @ Zs )
= ( map_Pr3224476576807795204nt_int @ F @ Xs ) )
= ( ? [Us2: list_P419547167332244514st_int,Vs2: list_P419547167332244514st_int] :
( ( Xs
= ( append5028292981072010935st_int @ Us2 @ Vs2 ) )
& ( Ys2
= ( map_Pr3224476576807795204nt_int @ F @ Us2 ) )
& ( Zs
= ( map_Pr3224476576807795204nt_int @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_392_append__eq__map__conv,axiom,
! [Ys2: list_int,Zs: list_int,F: int > int,Xs: list_int] :
( ( ( append_int @ Ys2 @ Zs )
= ( map_int_int @ F @ Xs ) )
= ( ? [Us2: list_int,Vs2: list_int] :
( ( Xs
= ( append_int @ Us2 @ Vs2 ) )
& ( Ys2
= ( map_int_int @ F @ Us2 ) )
& ( Zs
= ( map_int_int @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_393_append__eq__map__conv,axiom,
! [Ys2: list_P419547167332244514st_int,Zs: list_P419547167332244514st_int,F: produc6425607678544837394st_int > produc6425607678544837394st_int,Xs: list_P419547167332244514st_int] :
( ( ( append5028292981072010935st_int @ Ys2 @ Zs )
= ( map_Pr8625093460839654122st_int @ F @ Xs ) )
= ( ? [Us2: list_P419547167332244514st_int,Vs2: list_P419547167332244514st_int] :
( ( Xs
= ( append5028292981072010935st_int @ Us2 @ Vs2 ) )
& ( Ys2
= ( map_Pr8625093460839654122st_int @ F @ Us2 ) )
& ( Zs
= ( map_Pr8625093460839654122st_int @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_394_map__eq__append__conv,axiom,
! [F: instr > instr,Xs: list_instr,Ys2: list_instr,Zs: list_instr] :
( ( ( map_instr_instr @ F @ Xs )
= ( append_instr @ Ys2 @ Zs ) )
= ( ? [Us2: list_instr,Vs2: list_instr] :
( ( Xs
= ( append_instr @ Us2 @ Vs2 ) )
& ( Ys2
= ( map_instr_instr @ F @ Us2 ) )
& ( Zs
= ( map_instr_instr @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_395_map__eq__append__conv,axiom,
! [F: produc6425607678544837394st_int > instr,Xs: list_P419547167332244514st_int,Ys2: list_instr,Zs: list_instr] :
( ( ( map_Pr7514435936640407687_instr @ F @ Xs )
= ( append_instr @ Ys2 @ Zs ) )
= ( ? [Us2: list_P419547167332244514st_int,Vs2: list_P419547167332244514st_int] :
( ( Xs
= ( append5028292981072010935st_int @ Us2 @ Vs2 ) )
& ( Ys2
= ( map_Pr7514435936640407687_instr @ F @ Us2 ) )
& ( Zs
= ( map_Pr7514435936640407687_instr @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_396_map__eq__append__conv,axiom,
! [F: int > instr,Xs: list_int,Ys2: list_instr,Zs: list_instr] :
( ( ( map_int_instr @ F @ Xs )
= ( append_instr @ Ys2 @ Zs ) )
= ( ? [Us2: list_int,Vs2: list_int] :
( ( Xs
= ( append_int @ Us2 @ Vs2 ) )
& ( Ys2
= ( map_int_instr @ F @ Us2 ) )
& ( Zs
= ( map_int_instr @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_397_map__eq__append__conv,axiom,
! [F: instr > produc6425607678544837394st_int,Xs: list_instr,Ys2: list_P419547167332244514st_int,Zs: list_P419547167332244514st_int] :
( ( ( map_in7259965606695656109st_int @ F @ Xs )
= ( append5028292981072010935st_int @ Ys2 @ Zs ) )
= ( ? [Us2: list_instr,Vs2: list_instr] :
( ( Xs
= ( append_instr @ Us2 @ Vs2 ) )
& ( Ys2
= ( map_in7259965606695656109st_int @ F @ Us2 ) )
& ( Zs
= ( map_in7259965606695656109st_int @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_398_map__eq__append__conv,axiom,
! [F: int > produc6425607678544837394st_int,Xs: list_int,Ys2: list_P419547167332244514st_int,Zs: list_P419547167332244514st_int] :
( ( ( map_in8512336359594627332st_int @ F @ Xs )
= ( append5028292981072010935st_int @ Ys2 @ Zs ) )
= ( ? [Us2: list_int,Vs2: list_int] :
( ( Xs
= ( append_int @ Us2 @ Vs2 ) )
& ( Ys2
= ( map_in8512336359594627332st_int @ F @ Us2 ) )
& ( Zs
= ( map_in8512336359594627332st_int @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_399_map__eq__append__conv,axiom,
! [F: instr > int,Xs: list_instr,Ys2: list_int,Zs: list_int] :
( ( ( map_instr_int @ F @ Xs )
= ( append_int @ Ys2 @ Zs ) )
= ( ? [Us2: list_instr,Vs2: list_instr] :
( ( Xs
= ( append_instr @ Us2 @ Vs2 ) )
& ( Ys2
= ( map_instr_int @ F @ Us2 ) )
& ( Zs
= ( map_instr_int @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_400_map__eq__append__conv,axiom,
! [F: produc6425607678544837394st_int > int,Xs: list_P419547167332244514st_int,Ys2: list_int,Zs: list_int] :
( ( ( map_Pr3224476576807795204nt_int @ F @ Xs )
= ( append_int @ Ys2 @ Zs ) )
= ( ? [Us2: list_P419547167332244514st_int,Vs2: list_P419547167332244514st_int] :
( ( Xs
= ( append5028292981072010935st_int @ Us2 @ Vs2 ) )
& ( Ys2
= ( map_Pr3224476576807795204nt_int @ F @ Us2 ) )
& ( Zs
= ( map_Pr3224476576807795204nt_int @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_401_map__eq__append__conv,axiom,
! [F: int > int,Xs: list_int,Ys2: list_int,Zs: list_int] :
( ( ( map_int_int @ F @ Xs )
= ( append_int @ Ys2 @ Zs ) )
= ( ? [Us2: list_int,Vs2: list_int] :
( ( Xs
= ( append_int @ Us2 @ Vs2 ) )
& ( Ys2
= ( map_int_int @ F @ Us2 ) )
& ( Zs
= ( map_int_int @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_402_map__eq__append__conv,axiom,
! [F: produc6425607678544837394st_int > produc6425607678544837394st_int,Xs: list_P419547167332244514st_int,Ys2: list_P419547167332244514st_int,Zs: list_P419547167332244514st_int] :
( ( ( map_Pr8625093460839654122st_int @ F @ Xs )
= ( append5028292981072010935st_int @ Ys2 @ Zs ) )
= ( ? [Us2: list_P419547167332244514st_int,Vs2: list_P419547167332244514st_int] :
( ( Xs
= ( append5028292981072010935st_int @ Us2 @ Vs2 ) )
& ( Ys2
= ( map_Pr8625093460839654122st_int @ F @ Us2 ) )
& ( Zs
= ( map_Pr8625093460839654122st_int @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_403_execl__take,axiom,
! [P: list_instr,Cfs: list_P419547167332244514st_int,N: nat] :
( ( execl @ P @ Cfs )
=> ( execl @ P @ ( take_P1004856038796978596st_int @ N @ Cfs ) ) ) ).
% execl_take
thf(fact_404_execl__drop,axiom,
! [P: list_instr,Cfs: list_P419547167332244514st_int,N: nat] :
( ( execl @ P @ Cfs )
=> ( execl @ P @ ( drop_P7838799151754935884st_int @ N @ Cfs ) ) ) ).
% execl_drop
thf(fact_405_verit__comp__simplify1_I2_J,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_406_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_407_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_408_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_409_append__eq__conv__conj,axiom,
! [Xs: list_int,Ys2: list_int,Zs: list_int] :
( ( ( append_int @ Xs @ Ys2 )
= Zs )
= ( ( Xs
= ( take_int @ ( size_size_list_int @ Xs ) @ Zs ) )
& ( Ys2
= ( drop_int @ ( size_size_list_int @ Xs ) @ Zs ) ) ) ) ).
% append_eq_conv_conj
thf(fact_410_append__eq__conv__conj,axiom,
! [Xs: list_P419547167332244514st_int,Ys2: list_P419547167332244514st_int,Zs: list_P419547167332244514st_int] :
( ( ( append5028292981072010935st_int @ Xs @ Ys2 )
= Zs )
= ( ( Xs
= ( take_P1004856038796978596st_int @ ( size_s229347708945616566st_int @ Xs ) @ Zs ) )
& ( Ys2
= ( drop_P7838799151754935884st_int @ ( size_s229347708945616566st_int @ Xs ) @ Zs ) ) ) ) ).
% append_eq_conv_conj
thf(fact_411_append__eq__conv__conj,axiom,
! [Xs: list_instr,Ys2: list_instr,Zs: list_instr] :
( ( ( append_instr @ Xs @ Ys2 )
= Zs )
= ( ( Xs
= ( take_instr @ ( size_size_list_instr @ Xs ) @ Zs ) )
& ( Ys2
= ( drop_instr @ ( size_size_list_instr @ Xs ) @ Zs ) ) ) ) ).
% append_eq_conv_conj
thf(fact_412_append__eq__conv__conj,axiom,
! [Xs: list_list_char,Ys2: list_list_char,Zs: list_list_char] :
( ( ( append_list_char @ Xs @ Ys2 )
= Zs )
= ( ( Xs
= ( take_list_char @ ( size_s356637359517785349t_char @ Xs ) @ Zs ) )
& ( Ys2
= ( drop_list_char @ ( size_s356637359517785349t_char @ Xs ) @ Zs ) ) ) ) ).
% append_eq_conv_conj
thf(fact_413_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_414_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_415_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_416_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_417_verit__comp__simplify1_I1_J,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_418_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_419_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X: nat] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_420_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_421_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
! [T2: nat] :
( ( member_nat @ T2 @ A6 )
=> ( member_nat @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_422_subset__iff,axiom,
( ord_less_eq_set_int
= ( ^ [A6: set_int,B6: set_int] :
! [T2: int] :
( ( member_int @ T2 @ A6 )
=> ( member_int @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_423_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
! [X: nat] :
( ( member_nat @ X @ A6 )
=> ( member_nat @ X @ B6 ) ) ) ) ).
% subset_eq
thf(fact_424_subset__eq,axiom,
( ord_less_eq_set_int
= ( ^ [A6: set_int,B6: set_int] :
! [X: int] :
( ( member_int @ X @ A6 )
=> ( member_int @ X @ B6 ) ) ) ) ).
% subset_eq
thf(fact_425_subsetD,axiom,
! [A3: set_nat,B5: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A3 @ B5 )
=> ( ( member_nat @ C @ A3 )
=> ( member_nat @ C @ B5 ) ) ) ).
% subsetD
thf(fact_426_subsetD,axiom,
! [A3: set_int,B5: set_int,C: int] :
( ( ord_less_eq_set_int @ A3 @ B5 )
=> ( ( member_int @ C @ A3 )
=> ( member_int @ C @ B5 ) ) ) ).
% subsetD
thf(fact_427_in__mono,axiom,
! [A3: set_nat,B5: set_nat,X4: nat] :
( ( ord_less_eq_set_nat @ A3 @ B5 )
=> ( ( member_nat @ X4 @ A3 )
=> ( member_nat @ X4 @ B5 ) ) ) ).
% in_mono
thf(fact_428_in__mono,axiom,
! [A3: set_int,B5: set_int,X4: int] :
( ( ord_less_eq_set_int @ A3 @ B5 )
=> ( ( member_int @ X4 @ A3 )
=> ( member_int @ X4 @ B5 ) ) ) ).
% in_mono
thf(fact_429_verit__la__generic,axiom,
! [A: int,X4: int] :
( ( ord_less_eq_int @ A @ X4 )
| ( A = X4 )
| ( ord_less_eq_int @ X4 @ A ) ) ).
% verit_la_generic
thf(fact_430_nat__int__comparison_I1_J,axiom,
( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
= ( ^ [A5: nat,B4: nat] :
( ( semiri1314217659103216013at_int @ A5 )
= ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_431_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_432_int__int__eq,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% int_int_eq
thf(fact_433_iexec_Ocases,axiom,
! [X4: produc5995290525303592096st_int] :
~ ! [Ins: instr,I2: int,S3: list_char > int,Stk: list_int] :
( X4
!= ( produc1484136438817787418st_int @ Ins @ ( produc5086643055186798020st_int @ I2 @ ( produc8650753666468850689st_int @ S3 @ Stk ) ) ) ) ).
% iexec.cases
thf(fact_434_nth__append,axiom,
! [N: nat,Xs: list_int,Ys2: list_int] :
( ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
=> ( ( nth_int @ ( append_int @ Xs @ Ys2 ) @ N )
= ( nth_int @ Xs @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
=> ( ( nth_int @ ( append_int @ Xs @ Ys2 ) @ N )
= ( nth_int @ Ys2 @ ( minus_minus_nat @ N @ ( size_size_list_int @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_435_nth__append,axiom,
! [N: nat,Xs: list_P419547167332244514st_int,Ys2: list_P419547167332244514st_int] :
( ( ( ord_less_nat @ N @ ( size_s229347708945616566st_int @ Xs ) )
=> ( ( nth_Pr8327173284404383691st_int @ ( append5028292981072010935st_int @ Xs @ Ys2 ) @ N )
= ( nth_Pr8327173284404383691st_int @ Xs @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_s229347708945616566st_int @ Xs ) )
=> ( ( nth_Pr8327173284404383691st_int @ ( append5028292981072010935st_int @ Xs @ Ys2 ) @ N )
= ( nth_Pr8327173284404383691st_int @ Ys2 @ ( minus_minus_nat @ N @ ( size_s229347708945616566st_int @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_436_nth__append,axiom,
! [N: nat,Xs: list_instr,Ys2: list_instr] :
( ( ( ord_less_nat @ N @ ( size_size_list_instr @ Xs ) )
=> ( ( nth_instr @ ( append_instr @ Xs @ Ys2 ) @ N )
= ( nth_instr @ Xs @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_size_list_instr @ Xs ) )
=> ( ( nth_instr @ ( append_instr @ Xs @ Ys2 ) @ N )
= ( nth_instr @ Ys2 @ ( minus_minus_nat @ N @ ( size_size_list_instr @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_437_nth__append,axiom,
! [N: nat,Xs: list_list_char,Ys2: list_list_char] :
( ( ( ord_less_nat @ N @ ( size_s356637359517785349t_char @ Xs ) )
=> ( ( nth_list_char @ ( append_list_char @ Xs @ Ys2 ) @ N )
= ( nth_list_char @ Xs @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_s356637359517785349t_char @ Xs ) )
=> ( ( nth_list_char @ ( append_list_char @ Xs @ Ys2 ) @ N )
= ( nth_list_char @ Ys2 @ ( minus_minus_nat @ N @ ( size_s356637359517785349t_char @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_438_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
@ ^ [X: int] :
( ( member_int @ X @ A3 )
& ( P @ X ) ) ) )
= ( ! [X: int] :
( ( member_int @ X @ B5 )
=> ( P @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_439_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
@ ^ [X: nat] :
( ( member_nat @ X @ A3 )
& ( P @ X ) ) ) )
= ( ! [X: nat] :
( ( member_nat @ X @ B5 )
=> ( P @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_440_subset__CollectI,axiom,
! [B5: set_int,A3: set_int,Q: int > $o,P: int > $o] :
( ( ord_less_eq_set_int @ B5 @ A3 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ B5 )
=> ( ( Q @ X3 )
=> ( P @ X3 ) ) )
=> ( ord_less_eq_set_int
@ ( collect_int
@ ^ [X: int] :
( ( member_int @ X @ B5 )
& ( Q @ X ) ) )
@ ( collect_int
@ ^ [X: int] :
( ( member_int @ X @ A3 )
& ( P @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_441_subset__CollectI,axiom,
! [B5: set_nat,A3: set_nat,Q: nat > $o,P: nat > $o] :
( ( ord_less_eq_set_nat @ B5 @ A3 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B5 )
=> ( ( Q @ X3 )
=> ( P @ X3 ) ) )
=> ( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ B5 )
& ( Q @ X ) ) )
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A3 )
& ( P @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_442_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
( ord_less_eq_nat_o
@ ^ [X: nat] : ( member_nat @ X @ A6 )
@ ^ [X: nat] : ( member_nat @ X @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_443_less__eq__set__def,axiom,
( ord_less_eq_set_int
= ( ^ [A6: set_int,B6: set_int] :
( ord_less_eq_int_o
@ ^ [X: int] : ( member_int @ X @ A6 )
@ ^ [X: int] : ( member_int @ X @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_444_Collect__subset,axiom,
! [A3: set_int,P: int > $o] :
( ord_less_eq_set_int
@ ( collect_int
@ ^ [X: int] :
( ( member_int @ X @ A3 )
& ( P @ X ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_445_Collect__subset,axiom,
! [A3: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A3 )
& ( P @ X ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_446_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_447_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_448_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_449_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_450_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_451_int__le__induct,axiom,
! [I: int,K2: int,P: int > $o] :
( ( ord_less_eq_int @ I @ K2 )
=> ( ( P @ K2 )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ I2 @ K2 )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_le_induct
thf(fact_452_int__diff__cases,axiom,
! [Z3: int] :
~ ! [M6: nat,N2: nat] :
( Z3
!= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M6 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% int_diff_cases
thf(fact_453_int__less__induct,axiom,
! [I: int,K2: int,P: int > $o] :
( ( ord_less_int @ I @ K2 )
=> ( ( P @ ( minus_minus_int @ K2 @ one_one_int ) )
=> ( ! [I2: int] :
( ( ord_less_int @ I2 @ K2 )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_less_induct
thf(fact_454_exE__realizer_H,axiom,
! [P: produc369741286924889929st_int > int > $o,P2: produc6425607678544837394st_int] :
( ( P @ ( produc1296517004818028822st_int @ P2 ) @ ( produc8941309049508465880st_int @ P2 ) )
=> ~ ! [X3: int,Y3: produc369741286924889929st_int] :
~ ( P @ Y3 @ X3 ) ) ).
% exE_realizer'
thf(fact_455_exE__realizer_H,axiom,
! [P: int > ( list_char > int ) > $o,P2: produc4435102495419491129nt_int] :
( ( P @ ( produc179445755590504351nt_int @ P2 ) @ ( produc9025023058967219549nt_int @ P2 ) )
=> ~ ! [X3: list_char > int,Y3: int] :
~ ( P @ Y3 @ X3 ) ) ).
% exE_realizer'
thf(fact_456_exE__realizer_H,axiom,
! [P: list_int > ( list_char > int ) > $o,P2: produc369741286924889929st_int] :
( ( P @ ( produc6244060863719409455st_int @ P2 ) @ ( produc2155683374303800813st_int @ P2 ) )
=> ~ ! [X3: list_char > int,Y3: list_int] :
~ ( P @ Y3 @ X3 ) ) ).
% exE_realizer'
thf(fact_457_exE__realizer_H,axiom,
! [P: int > int > $o,P2: product_prod_int_int] :
( ( P @ ( product_snd_int_int @ P2 ) @ ( product_fst_int_int @ P2 ) )
=> ~ ! [X3: int,Y3: int] :
~ ( P @ Y3 @ X3 ) ) ).
% exE_realizer'
thf(fact_458_exE__realizer_H,axiom,
! [P: produc4435102495419491129nt_int > list_list_char > $o,P2: produc4189061985984568957nt_int] :
( ( P @ ( produc30867750519413249nt_int @ P2 ) @ ( produc811350159336358595nt_int @ P2 ) )
=> ~ ! [X3: list_list_char,Y3: produc4435102495419491129nt_int] :
~ ( P @ Y3 @ X3 ) ) ).
% exE_realizer'
thf(fact_459_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_460_execl__last,axiom,
! [P: list_instr,Cfs: list_P419547167332244514st_int,K2: nat] :
( ( execl @ P @ Cfs )
=> ( ( ord_less_nat @ K2 @ ( size_s229347708945616566st_int @ Cfs ) )
=> ( ~ ( member_int @ ( produc8941309049508465880st_int @ ( nth_Pr8327173284404383691st_int @ Cfs @ K2 ) ) @ ( set_or4662586982721622107an_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P ) ) ) )
=> ( ( minus_minus_nat @ ( size_s229347708945616566st_int @ Cfs ) @ one_one_nat )
= K2 ) ) ) ) ).
% execl_last
thf(fact_461_order__refl,axiom,
! [X4: int] : ( ord_less_eq_int @ X4 @ X4 ) ).
% order_refl
thf(fact_462_order__refl,axiom,
! [X4: nat] : ( ord_less_eq_nat @ X4 @ X4 ) ).
% order_refl
thf(fact_463_dual__order_Orefl,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% dual_order.refl
thf(fact_464_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_465_id__take__nth__drop,axiom,
! [I: nat,Xs: list_int] :
( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
=> ( Xs
= ( append_int @ ( take_int @ I @ Xs ) @ ( cons_int @ ( nth_int @ Xs @ I ) @ ( drop_int @ ( suc @ I ) @ Xs ) ) ) ) ) ).
% id_take_nth_drop
thf(fact_466_id__take__nth__drop,axiom,
! [I: nat,Xs: list_P419547167332244514st_int] :
( ( ord_less_nat @ I @ ( size_s229347708945616566st_int @ Xs ) )
=> ( Xs
= ( append5028292981072010935st_int @ ( take_P1004856038796978596st_int @ I @ Xs ) @ ( cons_P4151473878978182108st_int @ ( nth_Pr8327173284404383691st_int @ Xs @ I ) @ ( drop_P7838799151754935884st_int @ ( suc @ I ) @ Xs ) ) ) ) ) ).
% id_take_nth_drop
thf(fact_467_id__take__nth__drop,axiom,
! [I: nat,Xs: list_instr] :
( ( ord_less_nat @ I @ ( size_size_list_instr @ Xs ) )
=> ( Xs
= ( append_instr @ ( take_instr @ I @ Xs ) @ ( cons_instr @ ( nth_instr @ Xs @ I ) @ ( drop_instr @ ( suc @ I ) @ Xs ) ) ) ) ) ).
% id_take_nth_drop
thf(fact_468_id__take__nth__drop,axiom,
! [I: nat,Xs: list_list_char] :
( ( ord_less_nat @ I @ ( size_s356637359517785349t_char @ Xs ) )
=> ( Xs
= ( append_list_char @ ( take_list_char @ I @ Xs ) @ ( cons_list_char @ ( nth_list_char @ Xs @ I ) @ ( drop_list_char @ ( suc @ I ) @ Xs ) ) ) ) ) ).
% id_take_nth_drop
thf(fact_469_sndI,axiom,
! [X4: produc6425607678544837394st_int,Y: int,Z3: produc369741286924889929st_int] :
( ( X4
= ( produc5086643055186798020st_int @ Y @ Z3 ) )
=> ( ( produc1296517004818028822st_int @ X4 )
= Z3 ) ) ).
% sndI
thf(fact_470_sndI,axiom,
! [X4: produc4435102495419491129nt_int,Y: list_char > int,Z3: int] :
( ( X4
= ( produc5790713362662368625nt_int @ Y @ Z3 ) )
=> ( ( produc179445755590504351nt_int @ X4 )
= Z3 ) ) ).
% sndI
thf(fact_471_sndI,axiom,
! [X4: produc369741286924889929st_int,Y: list_char > int,Z3: list_int] :
( ( X4
= ( produc8650753666468850689st_int @ Y @ Z3 ) )
=> ( ( produc6244060863719409455st_int @ X4 )
= Z3 ) ) ).
% sndI
thf(fact_472_sndI,axiom,
! [X4: product_prod_int_int,Y: int,Z3: int] :
( ( X4
= ( product_Pair_int_int @ Y @ Z3 ) )
=> ( ( product_snd_int_int @ X4 )
= Z3 ) ) ).
% sndI
thf(fact_473_sndI,axiom,
! [X4: produc4189061985984568957nt_int,Y: list_list_char,Z3: produc4435102495419491129nt_int] :
( ( X4
= ( produc3965054194175396271nt_int @ Y @ Z3 ) )
=> ( ( produc30867750519413249nt_int @ X4 )
= Z3 ) ) ).
% sndI
thf(fact_474_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_475_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_476_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_477_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_478_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_479_list_Oinject,axiom,
! [X21: produc6425607678544837394st_int,X22: list_P419547167332244514st_int,Y21: produc6425607678544837394st_int,Y22: list_P419547167332244514st_int] :
( ( ( cons_P4151473878978182108st_int @ X21 @ X22 )
= ( cons_P4151473878978182108st_int @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_480_list_Oinject,axiom,
! [X21: instr,X22: list_instr,Y21: instr,Y22: list_instr] :
( ( ( cons_instr @ X21 @ X22 )
= ( cons_instr @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_481_list_Oinject,axiom,
! [X21: int,X22: list_int,Y21: int,Y22: list_int] :
( ( ( cons_int @ X21 @ X22 )
= ( cons_int @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_482_Diff__iff,axiom,
! [C: nat,A3: set_nat,B5: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B5 ) )
= ( ( member_nat @ C @ A3 )
& ~ ( member_nat @ C @ B5 ) ) ) ).
% Diff_iff
thf(fact_483_Diff__iff,axiom,
! [C: int,A3: set_int,B5: set_int] :
( ( member_int @ C @ ( minus_minus_set_int @ A3 @ B5 ) )
= ( ( member_int @ C @ A3 )
& ~ ( member_int @ C @ B5 ) ) ) ).
% Diff_iff
thf(fact_484_DiffI,axiom,
! [C: nat,A3: set_nat,B5: set_nat] :
( ( member_nat @ C @ A3 )
=> ( ~ ( member_nat @ C @ B5 )
=> ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B5 ) ) ) ) ).
% DiffI
thf(fact_485_DiffI,axiom,
! [C: int,A3: set_int,B5: set_int] :
( ( member_int @ C @ A3 )
=> ( ~ ( member_int @ C @ B5 )
=> ( member_int @ C @ ( minus_minus_set_int @ A3 @ B5 ) ) ) ) ).
% DiffI
thf(fact_486_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_487_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_488_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_489_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_490_diff__zero,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_zero
thf(fact_491_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_492_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_493_diff__0__right,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_0_right
thf(fact_494_diff__self,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% diff_self
thf(fact_495_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_496_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_497_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_498_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_499_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_500_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_501_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_502_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_503_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_504_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_505_nth__Cons__Suc,axiom,
! [X4: produc6425607678544837394st_int,Xs: list_P419547167332244514st_int,N: nat] :
( ( nth_Pr8327173284404383691st_int @ ( cons_P4151473878978182108st_int @ X4 @ Xs ) @ ( suc @ N ) )
= ( nth_Pr8327173284404383691st_int @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_506_nth__Cons__Suc,axiom,
! [X4: instr,Xs: list_instr,N: nat] :
( ( nth_instr @ ( cons_instr @ X4 @ Xs ) @ ( suc @ N ) )
= ( nth_instr @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_507_nth__Cons__Suc,axiom,
! [X4: int,Xs: list_int,N: nat] :
( ( nth_int @ ( cons_int @ X4 @ Xs ) @ ( suc @ N ) )
= ( nth_int @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_508_take__Suc__Cons,axiom,
! [N: nat,X4: produc6425607678544837394st_int,Xs: list_P419547167332244514st_int] :
( ( take_P1004856038796978596st_int @ ( suc @ N ) @ ( cons_P4151473878978182108st_int @ X4 @ Xs ) )
= ( cons_P4151473878978182108st_int @ X4 @ ( take_P1004856038796978596st_int @ N @ Xs ) ) ) ).
% take_Suc_Cons
thf(fact_509_take__Suc__Cons,axiom,
! [N: nat,X4: instr,Xs: list_instr] :
( ( take_instr @ ( suc @ N ) @ ( cons_instr @ X4 @ Xs ) )
= ( cons_instr @ X4 @ ( take_instr @ N @ Xs ) ) ) ).
% take_Suc_Cons
thf(fact_510_take__Suc__Cons,axiom,
! [N: nat,X4: int,Xs: list_int] :
( ( take_int @ ( suc @ N ) @ ( cons_int @ X4 @ Xs ) )
= ( cons_int @ X4 @ ( take_int @ N @ Xs ) ) ) ).
% take_Suc_Cons
thf(fact_511_drop__Suc__Cons,axiom,
! [N: nat,X4: list_char,Xs: list_list_char] :
( ( drop_list_char @ ( suc @ N ) @ ( cons_list_char @ X4 @ Xs ) )
= ( drop_list_char @ N @ Xs ) ) ).
% drop_Suc_Cons
thf(fact_512_drop__Suc__Cons,axiom,
! [N: nat,X4: produc6425607678544837394st_int,Xs: list_P419547167332244514st_int] :
( ( drop_P7838799151754935884st_int @ ( suc @ N ) @ ( cons_P4151473878978182108st_int @ X4 @ Xs ) )
= ( drop_P7838799151754935884st_int @ N @ Xs ) ) ).
% drop_Suc_Cons
thf(fact_513_drop__Suc__Cons,axiom,
! [N: nat,X4: instr,Xs: list_instr] :
( ( drop_instr @ ( suc @ N ) @ ( cons_instr @ X4 @ Xs ) )
= ( drop_instr @ N @ Xs ) ) ).
% drop_Suc_Cons
thf(fact_514_drop__Suc__Cons,axiom,
! [N: nat,X4: int,Xs: list_int] :
( ( drop_int @ ( suc @ N ) @ ( cons_int @ X4 @ Xs ) )
= ( drop_int @ N @ Xs ) ) ).
% drop_Suc_Cons
thf(fact_515_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_516_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_517_nth__append__length,axiom,
! [Xs: list_int,X4: int,Ys2: list_int] :
( ( nth_int @ ( append_int @ Xs @ ( cons_int @ X4 @ Ys2 ) ) @ ( size_size_list_int @ Xs ) )
= X4 ) ).
% nth_append_length
thf(fact_518_nth__append__length,axiom,
! [Xs: list_P419547167332244514st_int,X4: produc6425607678544837394st_int,Ys2: list_P419547167332244514st_int] :
( ( nth_Pr8327173284404383691st_int @ ( append5028292981072010935st_int @ Xs @ ( cons_P4151473878978182108st_int @ X4 @ Ys2 ) ) @ ( size_s229347708945616566st_int @ Xs ) )
= X4 ) ).
% nth_append_length
thf(fact_519_nth__append__length,axiom,
! [Xs: list_instr,X4: instr,Ys2: list_instr] :
( ( nth_instr @ ( append_instr @ Xs @ ( cons_instr @ X4 @ Ys2 ) ) @ ( size_size_list_instr @ Xs ) )
= X4 ) ).
% nth_append_length
thf(fact_520_nth__append__length,axiom,
! [Xs: list_list_char,X4: list_char,Ys2: list_list_char] :
( ( nth_list_char @ ( append_list_char @ Xs @ ( cons_list_char @ X4 @ Ys2 ) ) @ ( size_s356637359517785349t_char @ Xs ) )
= X4 ) ).
% nth_append_length
thf(fact_521_minus__set__def,axiom,
( minus_minus_set_int
= ( ^ [A6: set_int,B6: set_int] :
( collect_int
@ ( minus_minus_int_o
@ ^ [X: int] : ( member_int @ X @ A6 )
@ ^ [X: int] : ( member_int @ X @ B6 ) ) ) ) ) ).
% minus_set_def
thf(fact_522_minus__set__def,axiom,
( minus_minus_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
( collect_nat
@ ( minus_minus_nat_o
@ ^ [X: nat] : ( member_nat @ X @ A6 )
@ ^ [X: nat] : ( member_nat @ X @ B6 ) ) ) ) ) ).
% minus_set_def
thf(fact_523_set__diff__eq,axiom,
( minus_minus_set_int
= ( ^ [A6: set_int,B6: set_int] :
( collect_int
@ ^ [X: int] :
( ( member_int @ X @ A6 )
& ~ ( member_int @ X @ B6 ) ) ) ) ) ).
% set_diff_eq
thf(fact_524_set__diff__eq,axiom,
( minus_minus_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
( collect_nat
@ ^ [X: nat] :
( ( member_nat @ X @ A6 )
& ~ ( member_nat @ X @ B6 ) ) ) ) ) ).
% set_diff_eq
thf(fact_525_DiffD2,axiom,
! [C: nat,A3: set_nat,B5: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B5 ) )
=> ~ ( member_nat @ C @ B5 ) ) ).
% DiffD2
thf(fact_526_DiffD2,axiom,
! [C: int,A3: set_int,B5: set_int] :
( ( member_int @ C @ ( minus_minus_set_int @ A3 @ B5 ) )
=> ~ ( member_int @ C @ B5 ) ) ).
% DiffD2
thf(fact_527_DiffD1,axiom,
! [C: nat,A3: set_nat,B5: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B5 ) )
=> ( member_nat @ C @ A3 ) ) ).
% DiffD1
thf(fact_528_DiffD1,axiom,
! [C: int,A3: set_int,B5: set_int] :
( ( member_int @ C @ ( minus_minus_set_int @ A3 @ B5 ) )
=> ( member_int @ C @ A3 ) ) ).
% DiffD1
thf(fact_529_DiffE,axiom,
! [C: nat,A3: set_nat,B5: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B5 ) )
=> ~ ( ( member_nat @ C @ A3 )
=> ( member_nat @ C @ B5 ) ) ) ).
% DiffE
thf(fact_530_DiffE,axiom,
! [C: int,A3: set_int,B5: set_int] :
( ( member_int @ C @ ( minus_minus_set_int @ A3 @ B5 ) )
=> ~ ( ( member_int @ C @ A3 )
=> ( member_int @ C @ B5 ) ) ) ).
% DiffE
thf(fact_531_zero__le,axiom,
! [X4: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X4 ) ).
% zero_le
thf(fact_532_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_533_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_534_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_535_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_536_eq__iff__diff__eq__0,axiom,
( ( ^ [Y4: int,Z2: int] : ( Y4 = Z2 ) )
= ( ^ [A5: int,B4: int] :
( ( minus_minus_int @ A5 @ B4 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_537_zero__reorient,axiom,
! [X4: int] :
( ( zero_zero_int = X4 )
= ( X4 = zero_zero_int ) ) ).
% zero_reorient
thf(fact_538_zero__reorient,axiom,
! [X4: nat] :
( ( zero_zero_nat = X4 )
= ( X4 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_539_not__Cons__self2,axiom,
! [X4: produc6425607678544837394st_int,Xs: list_P419547167332244514st_int] :
( ( cons_P4151473878978182108st_int @ X4 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_540_not__Cons__self2,axiom,
! [X4: instr,Xs: list_instr] :
( ( cons_instr @ X4 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_541_not__Cons__self2,axiom,
! [X4: int,Xs: list_int] :
( ( cons_int @ X4 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_542_map__eq__Cons__conv,axiom,
! [F: produc6425607678544837394st_int > produc6425607678544837394st_int,Xs: list_P419547167332244514st_int,Y: produc6425607678544837394st_int,Ys2: list_P419547167332244514st_int] :
( ( ( map_Pr8625093460839654122st_int @ F @ Xs )
= ( cons_P4151473878978182108st_int @ Y @ Ys2 ) )
= ( ? [Z4: produc6425607678544837394st_int,Zs2: list_P419547167332244514st_int] :
( ( Xs
= ( cons_P4151473878978182108st_int @ Z4 @ Zs2 ) )
& ( ( F @ Z4 )
= Y )
& ( ( map_Pr8625093460839654122st_int @ F @ Zs2 )
= Ys2 ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_543_map__eq__Cons__conv,axiom,
! [F: instr > produc6425607678544837394st_int,Xs: list_instr,Y: produc6425607678544837394st_int,Ys2: list_P419547167332244514st_int] :
( ( ( map_in7259965606695656109st_int @ F @ Xs )
= ( cons_P4151473878978182108st_int @ Y @ Ys2 ) )
= ( ? [Z4: instr,Zs2: list_instr] :
( ( Xs
= ( cons_instr @ Z4 @ Zs2 ) )
& ( ( F @ Z4 )
= Y )
& ( ( map_in7259965606695656109st_int @ F @ Zs2 )
= Ys2 ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_544_map__eq__Cons__conv,axiom,
! [F: int > produc6425607678544837394st_int,Xs: list_int,Y: produc6425607678544837394st_int,Ys2: list_P419547167332244514st_int] :
( ( ( map_in8512336359594627332st_int @ F @ Xs )
= ( cons_P4151473878978182108st_int @ Y @ Ys2 ) )
= ( ? [Z4: int,Zs2: list_int] :
( ( Xs
= ( cons_int @ Z4 @ Zs2 ) )
& ( ( F @ Z4 )
= Y )
& ( ( map_in8512336359594627332st_int @ F @ Zs2 )
= Ys2 ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_545_map__eq__Cons__conv,axiom,
! [F: produc6425607678544837394st_int > instr,Xs: list_P419547167332244514st_int,Y: instr,Ys2: list_instr] :
( ( ( map_Pr7514435936640407687_instr @ F @ Xs )
= ( cons_instr @ Y @ Ys2 ) )
= ( ? [Z4: produc6425607678544837394st_int,Zs2: list_P419547167332244514st_int] :
( ( Xs
= ( cons_P4151473878978182108st_int @ Z4 @ Zs2 ) )
& ( ( F @ Z4 )
= Y )
& ( ( map_Pr7514435936640407687_instr @ F @ Zs2 )
= Ys2 ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_546_map__eq__Cons__conv,axiom,
! [F: instr > instr,Xs: list_instr,Y: instr,Ys2: list_instr] :
( ( ( map_instr_instr @ F @ Xs )
= ( cons_instr @ Y @ Ys2 ) )
= ( ? [Z4: instr,Zs2: list_instr] :
( ( Xs
= ( cons_instr @ Z4 @ Zs2 ) )
& ( ( F @ Z4 )
= Y )
& ( ( map_instr_instr @ F @ Zs2 )
= Ys2 ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_547_map__eq__Cons__conv,axiom,
! [F: int > instr,Xs: list_int,Y: instr,Ys2: list_instr] :
( ( ( map_int_instr @ F @ Xs )
= ( cons_instr @ Y @ Ys2 ) )
= ( ? [Z4: int,Zs2: list_int] :
( ( Xs
= ( cons_int @ Z4 @ Zs2 ) )
& ( ( F @ Z4 )
= Y )
& ( ( map_int_instr @ F @ Zs2 )
= Ys2 ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_548_map__eq__Cons__conv,axiom,
! [F: produc6425607678544837394st_int > int,Xs: list_P419547167332244514st_int,Y: int,Ys2: list_int] :
( ( ( map_Pr3224476576807795204nt_int @ F @ Xs )
= ( cons_int @ Y @ Ys2 ) )
= ( ? [Z4: produc6425607678544837394st_int,Zs2: list_P419547167332244514st_int] :
( ( Xs
= ( cons_P4151473878978182108st_int @ Z4 @ Zs2 ) )
& ( ( F @ Z4 )
= Y )
& ( ( map_Pr3224476576807795204nt_int @ F @ Zs2 )
= Ys2 ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_549_map__eq__Cons__conv,axiom,
! [F: instr > int,Xs: list_instr,Y: int,Ys2: list_int] :
( ( ( map_instr_int @ F @ Xs )
= ( cons_int @ Y @ Ys2 ) )
= ( ? [Z4: instr,Zs2: list_instr] :
( ( Xs
= ( cons_instr @ Z4 @ Zs2 ) )
& ( ( F @ Z4 )
= Y )
& ( ( map_instr_int @ F @ Zs2 )
= Ys2 ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_550_map__eq__Cons__conv,axiom,
! [F: int > int,Xs: list_int,Y: int,Ys2: list_int] :
( ( ( map_int_int @ F @ Xs )
= ( cons_int @ Y @ Ys2 ) )
= ( ? [Z4: int,Zs2: list_int] :
( ( Xs
= ( cons_int @ Z4 @ Zs2 ) )
& ( ( F @ Z4 )
= Y )
& ( ( map_int_int @ F @ Zs2 )
= Ys2 ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_551_Cons__eq__map__conv,axiom,
! [X4: produc6425607678544837394st_int,Xs: list_P419547167332244514st_int,F: produc6425607678544837394st_int > produc6425607678544837394st_int,Ys2: list_P419547167332244514st_int] :
( ( ( cons_P4151473878978182108st_int @ X4 @ Xs )
= ( map_Pr8625093460839654122st_int @ F @ Ys2 ) )
= ( ? [Z4: produc6425607678544837394st_int,Zs2: list_P419547167332244514st_int] :
( ( Ys2
= ( cons_P4151473878978182108st_int @ Z4 @ Zs2 ) )
& ( X4
= ( F @ Z4 ) )
& ( Xs
= ( map_Pr8625093460839654122st_int @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_552_Cons__eq__map__conv,axiom,
! [X4: produc6425607678544837394st_int,Xs: list_P419547167332244514st_int,F: instr > produc6425607678544837394st_int,Ys2: list_instr] :
( ( ( cons_P4151473878978182108st_int @ X4 @ Xs )
= ( map_in7259965606695656109st_int @ F @ Ys2 ) )
= ( ? [Z4: instr,Zs2: list_instr] :
( ( Ys2
= ( cons_instr @ Z4 @ Zs2 ) )
& ( X4
= ( F @ Z4 ) )
& ( Xs
= ( map_in7259965606695656109st_int @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_553_Cons__eq__map__conv,axiom,
! [X4: produc6425607678544837394st_int,Xs: list_P419547167332244514st_int,F: int > produc6425607678544837394st_int,Ys2: list_int] :
( ( ( cons_P4151473878978182108st_int @ X4 @ Xs )
= ( map_in8512336359594627332st_int @ F @ Ys2 ) )
= ( ? [Z4: int,Zs2: list_int] :
( ( Ys2
= ( cons_int @ Z4 @ Zs2 ) )
& ( X4
= ( F @ Z4 ) )
& ( Xs
= ( map_in8512336359594627332st_int @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_554_Cons__eq__map__conv,axiom,
! [X4: instr,Xs: list_instr,F: produc6425607678544837394st_int > instr,Ys2: list_P419547167332244514st_int] :
( ( ( cons_instr @ X4 @ Xs )
= ( map_Pr7514435936640407687_instr @ F @ Ys2 ) )
= ( ? [Z4: produc6425607678544837394st_int,Zs2: list_P419547167332244514st_int] :
( ( Ys2
= ( cons_P4151473878978182108st_int @ Z4 @ Zs2 ) )
& ( X4
= ( F @ Z4 ) )
& ( Xs
= ( map_Pr7514435936640407687_instr @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_555_Cons__eq__map__conv,axiom,
! [X4: instr,Xs: list_instr,F: instr > instr,Ys2: list_instr] :
( ( ( cons_instr @ X4 @ Xs )
= ( map_instr_instr @ F @ Ys2 ) )
= ( ? [Z4: instr,Zs2: list_instr] :
( ( Ys2
= ( cons_instr @ Z4 @ Zs2 ) )
& ( X4
= ( F @ Z4 ) )
& ( Xs
= ( map_instr_instr @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_556_Cons__eq__map__conv,axiom,
! [X4: instr,Xs: list_instr,F: int > instr,Ys2: list_int] :
( ( ( cons_instr @ X4 @ Xs )
= ( map_int_instr @ F @ Ys2 ) )
= ( ? [Z4: int,Zs2: list_int] :
( ( Ys2
= ( cons_int @ Z4 @ Zs2 ) )
& ( X4
= ( F @ Z4 ) )
& ( Xs
= ( map_int_instr @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_557_Cons__eq__map__conv,axiom,
! [X4: int,Xs: list_int,F: produc6425607678544837394st_int > int,Ys2: list_P419547167332244514st_int] :
( ( ( cons_int @ X4 @ Xs )
= ( map_Pr3224476576807795204nt_int @ F @ Ys2 ) )
= ( ? [Z4: produc6425607678544837394st_int,Zs2: list_P419547167332244514st_int] :
( ( Ys2
= ( cons_P4151473878978182108st_int @ Z4 @ Zs2 ) )
& ( X4
= ( F @ Z4 ) )
& ( Xs
= ( map_Pr3224476576807795204nt_int @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_558_Cons__eq__map__conv,axiom,
! [X4: int,Xs: list_int,F: instr > int,Ys2: list_instr] :
( ( ( cons_int @ X4 @ Xs )
= ( map_instr_int @ F @ Ys2 ) )
= ( ? [Z4: instr,Zs2: list_instr] :
( ( Ys2
= ( cons_instr @ Z4 @ Zs2 ) )
& ( X4
= ( F @ Z4 ) )
& ( Xs
= ( map_instr_int @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_559_Cons__eq__map__conv,axiom,
! [X4: int,Xs: list_int,F: int > int,Ys2: list_int] :
( ( ( cons_int @ X4 @ Xs )
= ( map_int_int @ F @ Ys2 ) )
= ( ? [Z4: int,Zs2: list_int] :
( ( Ys2
= ( cons_int @ Z4 @ Zs2 ) )
& ( X4
= ( F @ Z4 ) )
& ( Xs
= ( map_int_int @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_560_map__eq__Cons__D,axiom,
! [F: produc6425607678544837394st_int > produc6425607678544837394st_int,Xs: list_P419547167332244514st_int,Y: produc6425607678544837394st_int,Ys2: list_P419547167332244514st_int] :
( ( ( map_Pr8625093460839654122st_int @ F @ Xs )
= ( cons_P4151473878978182108st_int @ Y @ Ys2 ) )
=> ? [Z: produc6425607678544837394st_int,Zs3: list_P419547167332244514st_int] :
( ( Xs
= ( cons_P4151473878978182108st_int @ Z @ Zs3 ) )
& ( ( F @ Z )
= Y )
& ( ( map_Pr8625093460839654122st_int @ F @ Zs3 )
= Ys2 ) ) ) ).
% map_eq_Cons_D
thf(fact_561_map__eq__Cons__D,axiom,
! [F: instr > produc6425607678544837394st_int,Xs: list_instr,Y: produc6425607678544837394st_int,Ys2: list_P419547167332244514st_int] :
( ( ( map_in7259965606695656109st_int @ F @ Xs )
= ( cons_P4151473878978182108st_int @ Y @ Ys2 ) )
=> ? [Z: instr,Zs3: list_instr] :
( ( Xs
= ( cons_instr @ Z @ Zs3 ) )
& ( ( F @ Z )
= Y )
& ( ( map_in7259965606695656109st_int @ F @ Zs3 )
= Ys2 ) ) ) ).
% map_eq_Cons_D
thf(fact_562_map__eq__Cons__D,axiom,
! [F: int > produc6425607678544837394st_int,Xs: list_int,Y: produc6425607678544837394st_int,Ys2: list_P419547167332244514st_int] :
( ( ( map_in8512336359594627332st_int @ F @ Xs )
= ( cons_P4151473878978182108st_int @ Y @ Ys2 ) )
=> ? [Z: int,Zs3: list_int] :
( ( Xs
= ( cons_int @ Z @ Zs3 ) )
& ( ( F @ Z )
= Y )
& ( ( map_in8512336359594627332st_int @ F @ Zs3 )
= Ys2 ) ) ) ).
% map_eq_Cons_D
thf(fact_563_map__eq__Cons__D,axiom,
! [F: produc6425607678544837394st_int > instr,Xs: list_P419547167332244514st_int,Y: instr,Ys2: list_instr] :
( ( ( map_Pr7514435936640407687_instr @ F @ Xs )
= ( cons_instr @ Y @ Ys2 ) )
=> ? [Z: produc6425607678544837394st_int,Zs3: list_P419547167332244514st_int] :
( ( Xs
= ( cons_P4151473878978182108st_int @ Z @ Zs3 ) )
& ( ( F @ Z )
= Y )
& ( ( map_Pr7514435936640407687_instr @ F @ Zs3 )
= Ys2 ) ) ) ).
% map_eq_Cons_D
thf(fact_564_map__eq__Cons__D,axiom,
! [F: instr > instr,Xs: list_instr,Y: instr,Ys2: list_instr] :
( ( ( map_instr_instr @ F @ Xs )
= ( cons_instr @ Y @ Ys2 ) )
=> ? [Z: instr,Zs3: list_instr] :
( ( Xs
= ( cons_instr @ Z @ Zs3 ) )
& ( ( F @ Z )
= Y )
& ( ( map_instr_instr @ F @ Zs3 )
= Ys2 ) ) ) ).
% map_eq_Cons_D
thf(fact_565_map__eq__Cons__D,axiom,
! [F: int > instr,Xs: list_int,Y: instr,Ys2: list_instr] :
( ( ( map_int_instr @ F @ Xs )
= ( cons_instr @ Y @ Ys2 ) )
=> ? [Z: int,Zs3: list_int] :
( ( Xs
= ( cons_int @ Z @ Zs3 ) )
& ( ( F @ Z )
= Y )
& ( ( map_int_instr @ F @ Zs3 )
= Ys2 ) ) ) ).
% map_eq_Cons_D
thf(fact_566_map__eq__Cons__D,axiom,
! [F: produc6425607678544837394st_int > int,Xs: list_P419547167332244514st_int,Y: int,Ys2: list_int] :
( ( ( map_Pr3224476576807795204nt_int @ F @ Xs )
= ( cons_int @ Y @ Ys2 ) )
=> ? [Z: produc6425607678544837394st_int,Zs3: list_P419547167332244514st_int] :
( ( Xs
= ( cons_P4151473878978182108st_int @ Z @ Zs3 ) )
& ( ( F @ Z )
= Y )
& ( ( map_Pr3224476576807795204nt_int @ F @ Zs3 )
= Ys2 ) ) ) ).
% map_eq_Cons_D
thf(fact_567_map__eq__Cons__D,axiom,
! [F: instr > int,Xs: list_instr,Y: int,Ys2: list_int] :
( ( ( map_instr_int @ F @ Xs )
= ( cons_int @ Y @ Ys2 ) )
=> ? [Z: instr,Zs3: list_instr] :
( ( Xs
= ( cons_instr @ Z @ Zs3 ) )
& ( ( F @ Z )
= Y )
& ( ( map_instr_int @ F @ Zs3 )
= Ys2 ) ) ) ).
% map_eq_Cons_D
thf(fact_568_map__eq__Cons__D,axiom,
! [F: int > int,Xs: list_int,Y: int,Ys2: list_int] :
( ( ( map_int_int @ F @ Xs )
= ( cons_int @ Y @ Ys2 ) )
=> ? [Z: int,Zs3: list_int] :
( ( Xs
= ( cons_int @ Z @ Zs3 ) )
& ( ( F @ Z )
= Y )
& ( ( map_int_int @ F @ Zs3 )
= Ys2 ) ) ) ).
% map_eq_Cons_D
thf(fact_569_Cons__eq__map__D,axiom,
! [X4: produc6425607678544837394st_int,Xs: list_P419547167332244514st_int,F: produc6425607678544837394st_int > produc6425607678544837394st_int,Ys2: list_P419547167332244514st_int] :
( ( ( cons_P4151473878978182108st_int @ X4 @ Xs )
= ( map_Pr8625093460839654122st_int @ F @ Ys2 ) )
=> ? [Z: produc6425607678544837394st_int,Zs3: list_P419547167332244514st_int] :
( ( Ys2
= ( cons_P4151473878978182108st_int @ Z @ Zs3 ) )
& ( X4
= ( F @ Z ) )
& ( Xs
= ( map_Pr8625093460839654122st_int @ F @ Zs3 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_570_Cons__eq__map__D,axiom,
! [X4: produc6425607678544837394st_int,Xs: list_P419547167332244514st_int,F: instr > produc6425607678544837394st_int,Ys2: list_instr] :
( ( ( cons_P4151473878978182108st_int @ X4 @ Xs )
= ( map_in7259965606695656109st_int @ F @ Ys2 ) )
=> ? [Z: instr,Zs3: list_instr] :
( ( Ys2
= ( cons_instr @ Z @ Zs3 ) )
& ( X4
= ( F @ Z ) )
& ( Xs
= ( map_in7259965606695656109st_int @ F @ Zs3 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_571_Cons__eq__map__D,axiom,
! [X4: produc6425607678544837394st_int,Xs: list_P419547167332244514st_int,F: int > produc6425607678544837394st_int,Ys2: list_int] :
( ( ( cons_P4151473878978182108st_int @ X4 @ Xs )
= ( map_in8512336359594627332st_int @ F @ Ys2 ) )
=> ? [Z: int,Zs3: list_int] :
( ( Ys2
= ( cons_int @ Z @ Zs3 ) )
& ( X4
= ( F @ Z ) )
& ( Xs
= ( map_in8512336359594627332st_int @ F @ Zs3 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_572_Cons__eq__map__D,axiom,
! [X4: instr,Xs: list_instr,F: produc6425607678544837394st_int > instr,Ys2: list_P419547167332244514st_int] :
( ( ( cons_instr @ X4 @ Xs )
= ( map_Pr7514435936640407687_instr @ F @ Ys2 ) )
=> ? [Z: produc6425607678544837394st_int,Zs3: list_P419547167332244514st_int] :
( ( Ys2
= ( cons_P4151473878978182108st_int @ Z @ Zs3 ) )
& ( X4
= ( F @ Z ) )
& ( Xs
= ( map_Pr7514435936640407687_instr @ F @ Zs3 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_573_Cons__eq__map__D,axiom,
! [X4: instr,Xs: list_instr,F: instr > instr,Ys2: list_instr] :
( ( ( cons_instr @ X4 @ Xs )
= ( map_instr_instr @ F @ Ys2 ) )
=> ? [Z: instr,Zs3: list_instr] :
( ( Ys2
= ( cons_instr @ Z @ Zs3 ) )
& ( X4
= ( F @ Z ) )
& ( Xs
= ( map_instr_instr @ F @ Zs3 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_574_Cons__eq__map__D,axiom,
! [X4: instr,Xs: list_instr,F: int > instr,Ys2: list_int] :
( ( ( cons_instr @ X4 @ Xs )
= ( map_int_instr @ F @ Ys2 ) )
=> ? [Z: int,Zs3: list_int] :
( ( Ys2
= ( cons_int @ Z @ Zs3 ) )
& ( X4
= ( F @ Z ) )
& ( Xs
= ( map_int_instr @ F @ Zs3 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_575_Cons__eq__map__D,axiom,
! [X4: int,Xs: list_int,F: produc6425607678544837394st_int > int,Ys2: list_P419547167332244514st_int] :
( ( ( cons_int @ X4 @ Xs )
= ( map_Pr3224476576807795204nt_int @ F @ Ys2 ) )
=> ? [Z: produc6425607678544837394st_int,Zs3: list_P419547167332244514st_int] :
( ( Ys2
= ( cons_P4151473878978182108st_int @ Z @ Zs3 ) )
& ( X4
= ( F @ Z ) )
& ( Xs
= ( map_Pr3224476576807795204nt_int @ F @ Zs3 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_576_Cons__eq__map__D,axiom,
! [X4: int,Xs: list_int,F: instr > int,Ys2: list_instr] :
( ( ( cons_int @ X4 @ Xs )
= ( map_instr_int @ F @ Ys2 ) )
=> ? [Z: instr,Zs3: list_instr] :
( ( Ys2
= ( cons_instr @ Z @ Zs3 ) )
& ( X4
= ( F @ Z ) )
& ( Xs
= ( map_instr_int @ F @ Zs3 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_577_Cons__eq__map__D,axiom,
! [X4: int,Xs: list_int,F: int > int,Ys2: list_int] :
( ( ( cons_int @ X4 @ Xs )
= ( map_int_int @ F @ Ys2 ) )
=> ? [Z: int,Zs3: list_int] :
( ( Ys2
= ( cons_int @ Z @ Zs3 ) )
& ( X4
= ( F @ Z ) )
& ( Xs
= ( map_int_int @ F @ Zs3 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_578_list_Osimps_I9_J,axiom,
! [F: produc6425607678544837394st_int > produc6425607678544837394st_int,X21: produc6425607678544837394st_int,X22: list_P419547167332244514st_int] :
( ( map_Pr8625093460839654122st_int @ F @ ( cons_P4151473878978182108st_int @ X21 @ X22 ) )
= ( cons_P4151473878978182108st_int @ ( F @ X21 ) @ ( map_Pr8625093460839654122st_int @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_579_list_Osimps_I9_J,axiom,
! [F: produc6425607678544837394st_int > instr,X21: produc6425607678544837394st_int,X22: list_P419547167332244514st_int] :
( ( map_Pr7514435936640407687_instr @ F @ ( cons_P4151473878978182108st_int @ X21 @ X22 ) )
= ( cons_instr @ ( F @ X21 ) @ ( map_Pr7514435936640407687_instr @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_580_list_Osimps_I9_J,axiom,
! [F: produc6425607678544837394st_int > int,X21: produc6425607678544837394st_int,X22: list_P419547167332244514st_int] :
( ( map_Pr3224476576807795204nt_int @ F @ ( cons_P4151473878978182108st_int @ X21 @ X22 ) )
= ( cons_int @ ( F @ X21 ) @ ( map_Pr3224476576807795204nt_int @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_581_list_Osimps_I9_J,axiom,
! [F: instr > produc6425607678544837394st_int,X21: instr,X22: list_instr] :
( ( map_in7259965606695656109st_int @ F @ ( cons_instr @ X21 @ X22 ) )
= ( cons_P4151473878978182108st_int @ ( F @ X21 ) @ ( map_in7259965606695656109st_int @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_582_list_Osimps_I9_J,axiom,
! [F: instr > instr,X21: instr,X22: list_instr] :
( ( map_instr_instr @ F @ ( cons_instr @ X21 @ X22 ) )
= ( cons_instr @ ( F @ X21 ) @ ( map_instr_instr @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_583_list_Osimps_I9_J,axiom,
! [F: instr > int,X21: instr,X22: list_instr] :
( ( map_instr_int @ F @ ( cons_instr @ X21 @ X22 ) )
= ( cons_int @ ( F @ X21 ) @ ( map_instr_int @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_584_list_Osimps_I9_J,axiom,
! [F: int > produc6425607678544837394st_int,X21: int,X22: list_int] :
( ( map_in8512336359594627332st_int @ F @ ( cons_int @ X21 @ X22 ) )
= ( cons_P4151473878978182108st_int @ ( F @ X21 ) @ ( map_in8512336359594627332st_int @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_585_list_Osimps_I9_J,axiom,
! [F: int > instr,X21: int,X22: list_int] :
( ( map_int_instr @ F @ ( cons_int @ X21 @ X22 ) )
= ( cons_instr @ ( F @ X21 ) @ ( map_int_instr @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_586_list_Osimps_I9_J,axiom,
! [F: int > int,X21: int,X22: list_int] :
( ( map_int_int @ F @ ( cons_int @ X21 @ X22 ) )
= ( cons_int @ ( F @ X21 ) @ ( map_int_int @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_587_Cons__eq__appendI,axiom,
! [X4: produc6425607678544837394st_int,Xs1: list_P419547167332244514st_int,Ys2: list_P419547167332244514st_int,Xs: list_P419547167332244514st_int,Zs: list_P419547167332244514st_int] :
( ( ( cons_P4151473878978182108st_int @ X4 @ Xs1 )
= Ys2 )
=> ( ( Xs
= ( append5028292981072010935st_int @ Xs1 @ Zs ) )
=> ( ( cons_P4151473878978182108st_int @ X4 @ Xs )
= ( append5028292981072010935st_int @ Ys2 @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_588_Cons__eq__appendI,axiom,
! [X4: instr,Xs1: list_instr,Ys2: list_instr,Xs: list_instr,Zs: list_instr] :
( ( ( cons_instr @ X4 @ Xs1 )
= Ys2 )
=> ( ( Xs
= ( append_instr @ Xs1 @ Zs ) )
=> ( ( cons_instr @ X4 @ Xs )
= ( append_instr @ Ys2 @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_589_Cons__eq__appendI,axiom,
! [X4: int,Xs1: list_int,Ys2: list_int,Xs: list_int,Zs: list_int] :
( ( ( cons_int @ X4 @ Xs1 )
= Ys2 )
=> ( ( Xs
= ( append_int @ Xs1 @ Zs ) )
=> ( ( cons_int @ X4 @ Xs )
= ( append_int @ Ys2 @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_590_append__Cons,axiom,
! [X4: produc6425607678544837394st_int,Xs: list_P419547167332244514st_int,Ys2: list_P419547167332244514st_int] :
( ( append5028292981072010935st_int @ ( cons_P4151473878978182108st_int @ X4 @ Xs ) @ Ys2 )
= ( cons_P4151473878978182108st_int @ X4 @ ( append5028292981072010935st_int @ Xs @ Ys2 ) ) ) ).
% append_Cons
thf(fact_591_append__Cons,axiom,
! [X4: instr,Xs: list_instr,Ys2: list_instr] :
( ( append_instr @ ( cons_instr @ X4 @ Xs ) @ Ys2 )
= ( cons_instr @ X4 @ ( append_instr @ Xs @ Ys2 ) ) ) ).
% append_Cons
thf(fact_592_append__Cons,axiom,
! [X4: int,Xs: list_int,Ys2: list_int] :
( ( append_int @ ( cons_int @ X4 @ Xs ) @ Ys2 )
= ( cons_int @ X4 @ ( append_int @ Xs @ Ys2 ) ) ) ).
% append_Cons
thf(fact_593_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_594_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_595_minus__int__code_I1_J,axiom,
! [K2: int] :
( ( minus_minus_int @ K2 @ zero_zero_int )
= K2 ) ).
% minus_int_code(1)
thf(fact_596_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_597_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_598_length__Suc__conv,axiom,
! [Xs: list_int,N: nat] :
( ( ( size_size_list_int @ Xs )
= ( suc @ N ) )
= ( ? [Y6: int,Ys3: list_int] :
( ( Xs
= ( cons_int @ Y6 @ Ys3 ) )
& ( ( size_size_list_int @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_599_length__Suc__conv,axiom,
! [Xs: list_P419547167332244514st_int,N: nat] :
( ( ( size_s229347708945616566st_int @ Xs )
= ( suc @ N ) )
= ( ? [Y6: produc6425607678544837394st_int,Ys3: list_P419547167332244514st_int] :
( ( Xs
= ( cons_P4151473878978182108st_int @ Y6 @ Ys3 ) )
& ( ( size_s229347708945616566st_int @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_600_length__Suc__conv,axiom,
! [Xs: list_instr,N: nat] :
( ( ( size_size_list_instr @ Xs )
= ( suc @ N ) )
= ( ? [Y6: instr,Ys3: list_instr] :
( ( Xs
= ( cons_instr @ Y6 @ Ys3 ) )
& ( ( size_size_list_instr @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_601_length__Suc__conv,axiom,
! [Xs: list_list_char,N: nat] :
( ( ( size_s356637359517785349t_char @ Xs )
= ( suc @ N ) )
= ( ? [Y6: list_char,Ys3: list_list_char] :
( ( Xs
= ( cons_list_char @ Y6 @ Ys3 ) )
& ( ( size_s356637359517785349t_char @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_602_Suc__length__conv,axiom,
! [N: nat,Xs: list_int] :
( ( ( suc @ N )
= ( size_size_list_int @ Xs ) )
= ( ? [Y6: int,Ys3: list_int] :
( ( Xs
= ( cons_int @ Y6 @ Ys3 ) )
& ( ( size_size_list_int @ Ys3 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_603_Suc__length__conv,axiom,
! [N: nat,Xs: list_P419547167332244514st_int] :
( ( ( suc @ N )
= ( size_s229347708945616566st_int @ Xs ) )
= ( ? [Y6: produc6425607678544837394st_int,Ys3: list_P419547167332244514st_int] :
( ( Xs
= ( cons_P4151473878978182108st_int @ Y6 @ Ys3 ) )
& ( ( size_s229347708945616566st_int @ Ys3 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_604_Suc__length__conv,axiom,
! [N: nat,Xs: list_instr] :
( ( ( suc @ N )
= ( size_size_list_instr @ Xs ) )
= ( ? [Y6: instr,Ys3: list_instr] :
( ( Xs
= ( cons_instr @ Y6 @ Ys3 ) )
& ( ( size_size_list_instr @ Ys3 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_605_Suc__length__conv,axiom,
! [N: nat,Xs: list_list_char] :
( ( ( suc @ N )
= ( size_s356637359517785349t_char @ Xs ) )
= ( ? [Y6: list_char,Ys3: list_list_char] :
( ( Xs
= ( cons_list_char @ Y6 @ Ys3 ) )
& ( ( size_s356637359517785349t_char @ Ys3 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_606_length__Cons,axiom,
! [X4: int,Xs: list_int] :
( ( size_size_list_int @ ( cons_int @ X4 @ Xs ) )
= ( suc @ ( size_size_list_int @ Xs ) ) ) ).
% length_Cons
thf(fact_607_length__Cons,axiom,
! [X4: produc6425607678544837394st_int,Xs: list_P419547167332244514st_int] :
( ( size_s229347708945616566st_int @ ( cons_P4151473878978182108st_int @ X4 @ Xs ) )
= ( suc @ ( size_s229347708945616566st_int @ Xs ) ) ) ).
% length_Cons
thf(fact_608_length__Cons,axiom,
! [X4: instr,Xs: list_instr] :
( ( size_size_list_instr @ ( cons_instr @ X4 @ Xs ) )
= ( suc @ ( size_size_list_instr @ Xs ) ) ) ).
% length_Cons
thf(fact_609_length__Cons,axiom,
! [X4: list_char,Xs: list_list_char] :
( ( size_s356637359517785349t_char @ ( cons_list_char @ X4 @ Xs ) )
= ( suc @ ( size_s356637359517785349t_char @ Xs ) ) ) ).
% length_Cons
thf(fact_610_impossible__Cons,axiom,
! [Xs: list_int,Ys2: list_int,X4: int] :
( ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ ( size_size_list_int @ Ys2 ) )
=> ( Xs
!= ( cons_int @ X4 @ Ys2 ) ) ) ).
% impossible_Cons
thf(fact_611_impossible__Cons,axiom,
! [Xs: list_P419547167332244514st_int,Ys2: list_P419547167332244514st_int,X4: produc6425607678544837394st_int] :
( ( ord_less_eq_nat @ ( size_s229347708945616566st_int @ Xs ) @ ( size_s229347708945616566st_int @ Ys2 ) )
=> ( Xs
!= ( cons_P4151473878978182108st_int @ X4 @ Ys2 ) ) ) ).
% impossible_Cons
thf(fact_612_impossible__Cons,axiom,
! [Xs: list_instr,Ys2: list_instr,X4: instr] :
( ( ord_less_eq_nat @ ( size_size_list_instr @ Xs ) @ ( size_size_list_instr @ Ys2 ) )
=> ( Xs
!= ( cons_instr @ X4 @ Ys2 ) ) ) ).
% impossible_Cons
thf(fact_613_impossible__Cons,axiom,
! [Xs: list_list_char,Ys2: list_list_char,X4: list_char] :
( ( ord_less_eq_nat @ ( size_s356637359517785349t_char @ Xs ) @ ( size_s356637359517785349t_char @ Ys2 ) )
=> ( Xs
!= ( cons_list_char @ X4 @ Ys2 ) ) ) ).
% impossible_Cons
thf(fact_614_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_615_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_616_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_617_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_618_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
!= zero_zero_int ) ).
% of_nat_neq_0
thf(fact_619_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
!= zero_zero_nat ) ).
% of_nat_neq_0
thf(fact_620_nth__via__drop,axiom,
! [N: nat,Xs: list_list_char,Y: list_char,Ys2: list_list_char] :
( ( ( drop_list_char @ N @ Xs )
= ( cons_list_char @ Y @ Ys2 ) )
=> ( ( nth_list_char @ Xs @ N )
= Y ) ) ).
% nth_via_drop
thf(fact_621_nth__via__drop,axiom,
! [N: nat,Xs: list_P419547167332244514st_int,Y: produc6425607678544837394st_int,Ys2: list_P419547167332244514st_int] :
( ( ( drop_P7838799151754935884st_int @ N @ Xs )
= ( cons_P4151473878978182108st_int @ Y @ Ys2 ) )
=> ( ( nth_Pr8327173284404383691st_int @ Xs @ N )
= Y ) ) ).
% nth_via_drop
thf(fact_622_nth__via__drop,axiom,
! [N: nat,Xs: list_instr,Y: instr,Ys2: list_instr] :
( ( ( drop_instr @ N @ Xs )
= ( cons_instr @ Y @ Ys2 ) )
=> ( ( nth_instr @ Xs @ N )
= Y ) ) ).
% nth_via_drop
thf(fact_623_nth__via__drop,axiom,
! [N: nat,Xs: list_int,Y: int,Ys2: list_int] :
( ( ( drop_int @ N @ Xs )
= ( cons_int @ Y @ Ys2 ) )
=> ( ( nth_int @ Xs @ N )
= Y ) ) ).
% nth_via_drop
thf(fact_624_zero__le__imp__eq__int,axiom,
! [K2: int] :
( ( ord_less_eq_int @ zero_zero_int @ K2 )
=> ? [N2: nat] :
( K2
= ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_625_nonneg__int__cases,axiom,
! [K2: int] :
( ( ord_less_eq_int @ zero_zero_int @ K2 )
=> ~ ! [N2: nat] :
( K2
!= ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% nonneg_int_cases
thf(fact_626_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_int] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_int @ Xs ) )
= ( ? [X: int,Ys3: list_int] :
( ( Xs
= ( cons_int @ X @ Ys3 ) )
& ( ord_less_eq_nat @ N @ ( size_size_list_int @ Ys3 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_627_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_P419547167332244514st_int] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_s229347708945616566st_int @ Xs ) )
= ( ? [X: produc6425607678544837394st_int,Ys3: list_P419547167332244514st_int] :
( ( Xs
= ( cons_P4151473878978182108st_int @ X @ Ys3 ) )
& ( ord_less_eq_nat @ N @ ( size_s229347708945616566st_int @ Ys3 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_628_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_instr] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_instr @ Xs ) )
= ( ? [X: instr,Ys3: list_instr] :
( ( Xs
= ( cons_instr @ X @ Ys3 ) )
& ( ord_less_eq_nat @ N @ ( size_size_list_instr @ Ys3 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_629_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_list_char] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_s356637359517785349t_char @ Xs ) )
= ( ? [X: list_char,Ys3: list_list_char] :
( ( Xs
= ( cons_list_char @ X @ Ys3 ) )
& ( ord_less_eq_nat @ N @ ( size_s356637359517785349t_char @ Ys3 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_630_order__antisym__conv,axiom,
! [Y: int,X4: int] :
( ( ord_less_eq_int @ Y @ X4 )
=> ( ( ord_less_eq_int @ X4 @ Y )
= ( X4 = Y ) ) ) ).
% order_antisym_conv
thf(fact_631_order__antisym__conv,axiom,
! [Y: nat,X4: nat] :
( ( ord_less_eq_nat @ Y @ X4 )
=> ( ( ord_less_eq_nat @ X4 @ Y )
= ( X4 = Y ) ) ) ).
% order_antisym_conv
thf(fact_632_linorder__le__cases,axiom,
! [X4: int,Y: int] :
( ~ ( ord_less_eq_int @ X4 @ Y )
=> ( ord_less_eq_int @ Y @ X4 ) ) ).
% linorder_le_cases
thf(fact_633_linorder__le__cases,axiom,
! [X4: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_nat @ Y @ X4 ) ) ).
% linorder_le_cases
thf(fact_634_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_635_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_636_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_637_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_638_ord__eq__le__subst,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_639_ord__eq__le__subst,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_640_ord__eq__le__subst,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_641_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_642_linorder__linear,axiom,
! [X4: int,Y: int] :
( ( ord_less_eq_int @ X4 @ Y )
| ( ord_less_eq_int @ Y @ X4 ) ) ).
% linorder_linear
thf(fact_643_linorder__linear,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
| ( ord_less_eq_nat @ Y @ X4 ) ) ).
% linorder_linear
thf(fact_644_order__eq__refl,axiom,
! [X4: int,Y: int] :
( ( X4 = Y )
=> ( ord_less_eq_int @ X4 @ Y ) ) ).
% order_eq_refl
thf(fact_645_order__eq__refl,axiom,
! [X4: nat,Y: nat] :
( ( X4 = Y )
=> ( ord_less_eq_nat @ X4 @ Y ) ) ).
% order_eq_refl
thf(fact_646_order__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_647_order__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_648_order__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_649_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_650_order__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_651_order__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_652_order__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_653_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_654_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: int,Z2: int] : ( Y4 = Z2 ) )
= ( ^ [A5: int,B4: int] :
( ( ord_less_eq_int @ A5 @ B4 )
& ( ord_less_eq_int @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_655_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ A5 @ B4 )
& ( ord_less_eq_nat @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_656_antisym,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_657_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_658_dual__order_Otrans,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_659_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_660_dual__order_Oantisym,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_661_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_662_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: int,Z2: int] : ( Y4 = Z2 ) )
= ( ^ [A5: int,B4: int] :
( ( ord_less_eq_int @ B4 @ A5 )
& ( ord_less_eq_int @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_663_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ B4 @ A5 )
& ( ord_less_eq_nat @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_664_linorder__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A4: int,B3: int] :
( ( ord_less_eq_int @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: int,B3: int] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_665_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: nat,B3: nat] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_666_order__trans,axiom,
! [X4: int,Y: int,Z3: int] :
( ( ord_less_eq_int @ X4 @ Y )
=> ( ( ord_less_eq_int @ Y @ Z3 )
=> ( ord_less_eq_int @ X4 @ Z3 ) ) ) ).
% order_trans
thf(fact_667_order__trans,axiom,
! [X4: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z3 )
=> ( ord_less_eq_nat @ X4 @ Z3 ) ) ) ).
% order_trans
thf(fact_668_order_Otrans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% order.trans
thf(fact_669_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_670_order__antisym,axiom,
! [X4: int,Y: int] :
( ( ord_less_eq_int @ X4 @ Y )
=> ( ( ord_less_eq_int @ Y @ X4 )
=> ( X4 = Y ) ) ) ).
% order_antisym
thf(fact_671_order__antisym,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ( ord_less_eq_nat @ Y @ X4 )
=> ( X4 = Y ) ) ) ).
% order_antisym
thf(fact_672_ord__le__eq__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_673_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_674_ord__eq__le__trans,axiom,
! [A: int,B: int,C: int] :
( ( A = B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_675_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_676_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: int,Z2: int] : ( Y4 = Z2 ) )
= ( ^ [X: int,Y6: int] :
( ( ord_less_eq_int @ X @ Y6 )
& ( ord_less_eq_int @ Y6 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_677_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
= ( ^ [X: nat,Y6: nat] :
( ( ord_less_eq_nat @ X @ Y6 )
& ( ord_less_eq_nat @ Y6 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_678_le__cases3,axiom,
! [X4: int,Y: int,Z3: int] :
( ( ( ord_less_eq_int @ X4 @ Y )
=> ~ ( ord_less_eq_int @ Y @ Z3 ) )
=> ( ( ( ord_less_eq_int @ Y @ X4 )
=> ~ ( ord_less_eq_int @ X4 @ Z3 ) )
=> ( ( ( ord_less_eq_int @ X4 @ Z3 )
=> ~ ( ord_less_eq_int @ Z3 @ Y ) )
=> ( ( ( ord_less_eq_int @ Z3 @ Y )
=> ~ ( ord_less_eq_int @ Y @ X4 ) )
=> ( ( ( ord_less_eq_int @ Y @ Z3 )
=> ~ ( ord_less_eq_int @ Z3 @ X4 ) )
=> ~ ( ( ord_less_eq_int @ Z3 @ X4 )
=> ~ ( ord_less_eq_int @ X4 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_679_le__cases3,axiom,
! [X4: nat,Y: nat,Z3: nat] :
( ( ( ord_less_eq_nat @ X4 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X4 )
=> ~ ( ord_less_eq_nat @ X4 @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ X4 @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z3 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X4 ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ X4 ) )
=> ~ ( ( ord_less_eq_nat @ Z3 @ X4 )
=> ~ ( ord_less_eq_nat @ X4 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_680_nle__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_eq_int @ A @ B ) )
= ( ( ord_less_eq_int @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_681_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_682_order__less__imp__not__less,axiom,
! [X4: int,Y: int] :
( ( ord_less_int @ X4 @ Y )
=> ~ ( ord_less_int @ Y @ X4 ) ) ).
% order_less_imp_not_less
thf(fact_683_order__less__imp__not__less,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ~ ( ord_less_nat @ Y @ X4 ) ) ).
% order_less_imp_not_less
thf(fact_684_order__less__imp__not__eq2,axiom,
! [X4: int,Y: int] :
( ( ord_less_int @ X4 @ Y )
=> ( Y != X4 ) ) ).
% order_less_imp_not_eq2
thf(fact_685_order__less__imp__not__eq2,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( Y != X4 ) ) ).
% order_less_imp_not_eq2
thf(fact_686_order__less__imp__not__eq,axiom,
! [X4: int,Y: int] :
( ( ord_less_int @ X4 @ Y )
=> ( X4 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_687_order__less__imp__not__eq,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( X4 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_688_linorder__less__linear,axiom,
! [X4: int,Y: int] :
( ( ord_less_int @ X4 @ Y )
| ( X4 = Y )
| ( ord_less_int @ Y @ X4 ) ) ).
% linorder_less_linear
thf(fact_689_linorder__less__linear,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
| ( X4 = Y )
| ( ord_less_nat @ Y @ X4 ) ) ).
% linorder_less_linear
thf(fact_690_order__less__imp__triv,axiom,
! [X4: int,Y: int,P: $o] :
( ( ord_less_int @ X4 @ Y )
=> ( ( ord_less_int @ Y @ X4 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_691_order__less__imp__triv,axiom,
! [X4: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X4 @ Y )
=> ( ( ord_less_nat @ Y @ X4 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_692_order__less__not__sym,axiom,
! [X4: int,Y: int] :
( ( ord_less_int @ X4 @ Y )
=> ~ ( ord_less_int @ Y @ X4 ) ) ).
% order_less_not_sym
thf(fact_693_order__less__not__sym,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ~ ( ord_less_nat @ Y @ X4 ) ) ).
% order_less_not_sym
thf(fact_694_order__less__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_695_order__less__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_696_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_697_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_698_order__less__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_699_order__less__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_700_order__less__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_701_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_702_order__less__irrefl,axiom,
! [X4: int] :
~ ( ord_less_int @ X4 @ X4 ) ).
% order_less_irrefl
thf(fact_703_order__less__irrefl,axiom,
! [X4: nat] :
~ ( ord_less_nat @ X4 @ X4 ) ).
% order_less_irrefl
thf(fact_704_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_705_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_706_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_707_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_708_ord__eq__less__subst,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_709_ord__eq__less__subst,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_710_ord__eq__less__subst,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_711_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_712_order__less__trans,axiom,
! [X4: int,Y: int,Z3: int] :
( ( ord_less_int @ X4 @ Y )
=> ( ( ord_less_int @ Y @ Z3 )
=> ( ord_less_int @ X4 @ Z3 ) ) ) ).
% order_less_trans
thf(fact_713_order__less__trans,axiom,
! [X4: nat,Y: nat,Z3: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ( ord_less_nat @ Y @ Z3 )
=> ( ord_less_nat @ X4 @ Z3 ) ) ) ).
% order_less_trans
thf(fact_714_order__less__asym_H,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order_less_asym'
thf(fact_715_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_716_linorder__neq__iff,axiom,
! [X4: int,Y: int] :
( ( X4 != Y )
= ( ( ord_less_int @ X4 @ Y )
| ( ord_less_int @ Y @ X4 ) ) ) ).
% linorder_neq_iff
thf(fact_717_linorder__neq__iff,axiom,
! [X4: nat,Y: nat] :
( ( X4 != Y )
= ( ( ord_less_nat @ X4 @ Y )
| ( ord_less_nat @ Y @ X4 ) ) ) ).
% linorder_neq_iff
thf(fact_718_order__less__asym,axiom,
! [X4: int,Y: int] :
( ( ord_less_int @ X4 @ Y )
=> ~ ( ord_less_int @ Y @ X4 ) ) ).
% order_less_asym
thf(fact_719_order__less__asym,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ~ ( ord_less_nat @ Y @ X4 ) ) ).
% order_less_asym
thf(fact_720_linorder__neqE,axiom,
! [X4: int,Y: int] :
( ( X4 != Y )
=> ( ~ ( ord_less_int @ X4 @ Y )
=> ( ord_less_int @ Y @ X4 ) ) ) ).
% linorder_neqE
thf(fact_721_linorder__neqE,axiom,
! [X4: nat,Y: nat] :
( ( X4 != Y )
=> ( ~ ( ord_less_nat @ X4 @ Y )
=> ( ord_less_nat @ Y @ X4 ) ) ) ).
% linorder_neqE
thf(fact_722_dual__order_Ostrict__implies__not__eq,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_723_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_724_order_Ostrict__implies__not__eq,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_725_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_726_dual__order_Ostrict__trans,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_727_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_728_not__less__iff__gr__or__eq,axiom,
! [X4: int,Y: int] :
( ( ~ ( ord_less_int @ X4 @ Y ) )
= ( ( ord_less_int @ Y @ X4 )
| ( X4 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_729_not__less__iff__gr__or__eq,axiom,
! [X4: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X4 @ Y ) )
= ( ( ord_less_nat @ Y @ X4 )
| ( X4 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_730_order_Ostrict__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_731_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_732_linorder__less__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A4: int,B3: int] :
( ( ord_less_int @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: int] : ( P @ A4 @ A4 )
=> ( ! [A4: int,B3: int] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_733_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B3: nat] :
( ( ord_less_nat @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: nat] : ( P @ A4 @ A4 )
=> ( ! [A4: nat,B3: nat] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_734_exists__least__iff,axiom,
( ( ^ [P5: nat > $o] :
? [X7: nat] : ( P5 @ X7 ) )
= ( ^ [P6: nat > $o] :
? [N3: nat] :
( ( P6 @ N3 )
& ! [M2: nat] :
( ( ord_less_nat @ M2 @ N3 )
=> ~ ( P6 @ M2 ) ) ) ) ) ).
% exists_least_iff
thf(fact_735_dual__order_Oirrefl,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% dual_order.irrefl
thf(fact_736_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_737_dual__order_Oasym,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ~ ( ord_less_int @ A @ B ) ) ).
% dual_order.asym
thf(fact_738_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_739_linorder__cases,axiom,
! [X4: int,Y: int] :
( ~ ( ord_less_int @ X4 @ Y )
=> ( ( X4 != Y )
=> ( ord_less_int @ Y @ X4 ) ) ) ).
% linorder_cases
thf(fact_740_linorder__cases,axiom,
! [X4: nat,Y: nat] :
( ~ ( ord_less_nat @ X4 @ Y )
=> ( ( X4 != Y )
=> ( ord_less_nat @ Y @ X4 ) ) ) ).
% linorder_cases
thf(fact_741_antisym__conv3,axiom,
! [Y: int,X4: int] :
( ~ ( ord_less_int @ Y @ X4 )
=> ( ( ~ ( ord_less_int @ X4 @ Y ) )
= ( X4 = Y ) ) ) ).
% antisym_conv3
thf(fact_742_antisym__conv3,axiom,
! [Y: nat,X4: nat] :
( ~ ( ord_less_nat @ Y @ X4 )
=> ( ( ~ ( ord_less_nat @ X4 @ Y ) )
= ( X4 = Y ) ) ) ).
% antisym_conv3
thf(fact_743_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X3: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X3 )
=> ( P @ Y5 ) )
=> ( P @ X3 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_744_ord__less__eq__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( B = C )
=> ( ord_less_int @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_745_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_746_ord__eq__less__trans,axiom,
! [A: int,B: int,C: int] :
( ( A = B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_747_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_748_order_Oasym,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order.asym
thf(fact_749_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_750_less__imp__neq,axiom,
! [X4: int,Y: int] :
( ( ord_less_int @ X4 @ Y )
=> ( X4 != Y ) ) ).
% less_imp_neq
thf(fact_751_less__imp__neq,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( X4 != Y ) ) ).
% less_imp_neq
thf(fact_752_gt__ex,axiom,
! [X4: int] :
? [X_1: int] : ( ord_less_int @ X4 @ X_1 ) ).
% gt_ex
thf(fact_753_gt__ex,axiom,
! [X4: nat] :
? [X_1: nat] : ( ord_less_nat @ X4 @ X_1 ) ).
% gt_ex
thf(fact_754_lt__ex,axiom,
! [X4: int] :
? [Y3: int] : ( ord_less_int @ Y3 @ X4 ) ).
% lt_ex
thf(fact_755_one__reorient,axiom,
! [X4: nat] :
( ( one_one_nat = X4 )
= ( X4 = one_one_nat ) ) ).
% one_reorient
thf(fact_756_one__reorient,axiom,
! [X4: int] :
( ( one_one_int = X4 )
= ( X4 = one_one_int ) ) ).
% one_reorient
thf(fact_757_diff__right__commute,axiom,
! [A: int,C: int,B: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B )
= ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% diff_right_commute
thf(fact_758_diff__right__commute,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% diff_right_commute
thf(fact_759_diff__eq__diff__eq,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( A = B )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_760_int__one__le__iff__zero__less,axiom,
! [Z3: int] :
( ( ord_less_eq_int @ one_one_int @ Z3 )
= ( ord_less_int @ zero_zero_int @ Z3 ) ) ).
% int_one_le_iff_zero_less
thf(fact_761_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_762_Cons__nth__drop__Suc,axiom,
! [I: nat,Xs: list_int] :
( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
=> ( ( cons_int @ ( nth_int @ Xs @ I ) @ ( drop_int @ ( suc @ I ) @ Xs ) )
= ( drop_int @ I @ Xs ) ) ) ).
% Cons_nth_drop_Suc
thf(fact_763_Cons__nth__drop__Suc,axiom,
! [I: nat,Xs: list_P419547167332244514st_int] :
( ( ord_less_nat @ I @ ( size_s229347708945616566st_int @ Xs ) )
=> ( ( cons_P4151473878978182108st_int @ ( nth_Pr8327173284404383691st_int @ Xs @ I ) @ ( drop_P7838799151754935884st_int @ ( suc @ I ) @ Xs ) )
= ( drop_P7838799151754935884st_int @ I @ Xs ) ) ) ).
% Cons_nth_drop_Suc
thf(fact_764_Cons__nth__drop__Suc,axiom,
! [I: nat,Xs: list_instr] :
( ( ord_less_nat @ I @ ( size_size_list_instr @ Xs ) )
=> ( ( cons_instr @ ( nth_instr @ Xs @ I ) @ ( drop_instr @ ( suc @ I ) @ Xs ) )
= ( drop_instr @ I @ Xs ) ) ) ).
% Cons_nth_drop_Suc
thf(fact_765_Cons__nth__drop__Suc,axiom,
! [I: nat,Xs: list_list_char] :
( ( ord_less_nat @ I @ ( size_s356637359517785349t_char @ Xs ) )
=> ( ( cons_list_char @ ( nth_list_char @ Xs @ I ) @ ( drop_list_char @ ( suc @ I ) @ Xs ) )
= ( drop_list_char @ I @ Xs ) ) ) ).
% Cons_nth_drop_Suc
thf(fact_766_order__le__imp__less__or__eq,axiom,
! [X4: int,Y: int] :
( ( ord_less_eq_int @ X4 @ Y )
=> ( ( ord_less_int @ X4 @ Y )
| ( X4 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_767_order__le__imp__less__or__eq,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ( ord_less_nat @ X4 @ Y )
| ( X4 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_768_linorder__le__less__linear,axiom,
! [X4: int,Y: int] :
( ( ord_less_eq_int @ X4 @ Y )
| ( ord_less_int @ Y @ X4 ) ) ).
% linorder_le_less_linear
thf(fact_769_linorder__le__less__linear,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
| ( ord_less_nat @ Y @ X4 ) ) ).
% linorder_le_less_linear
thf(fact_770_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_771_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_772_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_773_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_774_order__less__le__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_775_order__less__le__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_776_order__less__le__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_777_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_778_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_779_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_780_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_781_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_782_order__le__less__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_783_order__le__less__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_784_order__le__less__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_785_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_786_order__less__le__trans,axiom,
! [X4: int,Y: int,Z3: int] :
( ( ord_less_int @ X4 @ Y )
=> ( ( ord_less_eq_int @ Y @ Z3 )
=> ( ord_less_int @ X4 @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_787_order__less__le__trans,axiom,
! [X4: nat,Y: nat,Z3: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z3 )
=> ( ord_less_nat @ X4 @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_788_order__le__less__trans,axiom,
! [X4: int,Y: int,Z3: int] :
( ( ord_less_eq_int @ X4 @ Y )
=> ( ( ord_less_int @ Y @ Z3 )
=> ( ord_less_int @ X4 @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_789_order__le__less__trans,axiom,
! [X4: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ( ord_less_nat @ Y @ Z3 )
=> ( ord_less_nat @ X4 @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_790_order__neq__le__trans,axiom,
! [A: int,B: int] :
( ( A != B )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_int @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_791_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_792_order__le__neq__trans,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( A != B )
=> ( ord_less_int @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_793_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_794_order__less__imp__le,axiom,
! [X4: int,Y: int] :
( ( ord_less_int @ X4 @ Y )
=> ( ord_less_eq_int @ X4 @ Y ) ) ).
% order_less_imp_le
thf(fact_795_order__less__imp__le,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_nat @ X4 @ Y )
=> ( ord_less_eq_nat @ X4 @ Y ) ) ).
% order_less_imp_le
thf(fact_796_linorder__not__less,axiom,
! [X4: int,Y: int] :
( ( ~ ( ord_less_int @ X4 @ Y ) )
= ( ord_less_eq_int @ Y @ X4 ) ) ).
% linorder_not_less
thf(fact_797_linorder__not__less,axiom,
! [X4: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X4 @ Y ) )
= ( ord_less_eq_nat @ Y @ X4 ) ) ).
% linorder_not_less
thf(fact_798_linorder__not__le,axiom,
! [X4: int,Y: int] :
( ( ~ ( ord_less_eq_int @ X4 @ Y ) )
= ( ord_less_int @ Y @ X4 ) ) ).
% linorder_not_le
thf(fact_799_linorder__not__le,axiom,
! [X4: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X4 @ Y ) )
= ( ord_less_nat @ Y @ X4 ) ) ).
% linorder_not_le
thf(fact_800_order__less__le,axiom,
( ord_less_int
= ( ^ [X: int,Y6: int] :
( ( ord_less_eq_int @ X @ Y6 )
& ( X != Y6 ) ) ) ) ).
% order_less_le
thf(fact_801_order__less__le,axiom,
( ord_less_nat
= ( ^ [X: nat,Y6: nat] :
( ( ord_less_eq_nat @ X @ Y6 )
& ( X != Y6 ) ) ) ) ).
% order_less_le
thf(fact_802_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X: int,Y6: int] :
( ( ord_less_int @ X @ Y6 )
| ( X = Y6 ) ) ) ) ).
% order_le_less
thf(fact_803_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X: nat,Y6: nat] :
( ( ord_less_nat @ X @ Y6 )
| ( X = Y6 ) ) ) ) ).
% order_le_less
thf(fact_804_dual__order_Ostrict__implies__order,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_eq_int @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_805_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_806_order_Ostrict__implies__order,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_eq_int @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_807_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_808_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B4: int,A5: int] :
( ( ord_less_eq_int @ B4 @ A5 )
& ~ ( ord_less_eq_int @ A5 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_809_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B4: nat,A5: nat] :
( ( ord_less_eq_nat @ B4 @ A5 )
& ~ ( ord_less_eq_nat @ A5 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_810_dual__order_Ostrict__trans2,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_811_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_812_dual__order_Ostrict__trans1,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_813_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_814_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B4: int,A5: int] :
( ( ord_less_eq_int @ B4 @ A5 )
& ( A5 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_815_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B4: nat,A5: nat] :
( ( ord_less_eq_nat @ B4 @ A5 )
& ( A5 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_816_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B4: int,A5: int] :
( ( ord_less_int @ B4 @ A5 )
| ( A5 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_817_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A5: nat] :
( ( ord_less_nat @ B4 @ A5 )
| ( A5 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_818_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A5: int,B4: int] :
( ( ord_less_eq_int @ A5 @ B4 )
& ~ ( ord_less_eq_int @ B4 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_819_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ A5 @ B4 )
& ~ ( ord_less_eq_nat @ B4 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_820_order_Ostrict__trans2,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_821_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_822_order_Ostrict__trans1,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_823_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_824_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A5: int,B4: int] :
( ( ord_less_eq_int @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_825_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_826_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A5: int,B4: int] :
( ( ord_less_int @ A5 @ B4 )
| ( A5 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_827_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_nat @ A5 @ B4 )
| ( A5 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_828_not__le__imp__less,axiom,
! [Y: int,X4: int] :
( ~ ( ord_less_eq_int @ Y @ X4 )
=> ( ord_less_int @ X4 @ Y ) ) ).
% not_le_imp_less
thf(fact_829_not__le__imp__less,axiom,
! [Y: nat,X4: nat] :
( ~ ( ord_less_eq_nat @ Y @ X4 )
=> ( ord_less_nat @ X4 @ Y ) ) ).
% not_le_imp_less
thf(fact_830_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X: int,Y6: int] :
( ( ord_less_eq_int @ X @ Y6 )
& ~ ( ord_less_eq_int @ Y6 @ X ) ) ) ) ).
% less_le_not_le
thf(fact_831_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X: nat,Y6: nat] :
( ( ord_less_eq_nat @ X @ Y6 )
& ~ ( ord_less_eq_nat @ Y6 @ X ) ) ) ) ).
% less_le_not_le
thf(fact_832_antisym__conv2,axiom,
! [X4: int,Y: int] :
( ( ord_less_eq_int @ X4 @ Y )
=> ( ( ~ ( ord_less_int @ X4 @ Y ) )
= ( X4 = Y ) ) ) ).
% antisym_conv2
thf(fact_833_antisym__conv2,axiom,
! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ( ~ ( ord_less_nat @ X4 @ Y ) )
= ( X4 = Y ) ) ) ).
% antisym_conv2
thf(fact_834_antisym__conv1,axiom,
! [X4: int,Y: int] :
( ~ ( ord_less_int @ X4 @ Y )
=> ( ( ord_less_eq_int @ X4 @ Y )
= ( X4 = Y ) ) ) ).
% antisym_conv1
thf(fact_835_antisym__conv1,axiom,
! [X4: nat,Y: nat] :
( ~ ( ord_less_nat @ X4 @ Y )
=> ( ( ord_less_eq_nat @ X4 @ Y )
= ( X4 = Y ) ) ) ).
% antisym_conv1
thf(fact_836_nless__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_int @ A @ B ) )
= ( ~ ( ord_less_eq_int @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_837_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_838_leI,axiom,
! [X4: int,Y: int] :
( ~ ( ord_less_int @ X4 @ Y )
=> ( ord_less_eq_int @ Y @ X4 ) ) ).
% leI
thf(fact_839_leI,axiom,
! [X4: nat,Y: nat] :
( ~ ( ord_less_nat @ X4 @ Y )
=> ( ord_less_eq_nat @ Y @ X4 ) ) ).
% leI
thf(fact_840_leD,axiom,
! [Y: int,X4: int] :
( ( ord_less_eq_int @ Y @ X4 )
=> ~ ( ord_less_int @ X4 @ Y ) ) ).
% leD
thf(fact_841_leD,axiom,
! [Y: nat,X4: nat] :
( ( ord_less_eq_nat @ Y @ X4 )
=> ~ ( ord_less_nat @ X4 @ Y ) ) ).
% leD
thf(fact_842_diff__eq__diff__less__eq,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_eq_int @ A @ B )
= ( ord_less_eq_int @ C @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_843_diff__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_844_diff__left__mono,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_845_diff__mono,axiom,
! [A: int,B: int,D: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ D @ C )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% diff_mono
thf(fact_846_diff__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_847_diff__strict__left__mono,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_848_diff__eq__diff__less,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_int @ A @ B )
= ( ord_less_int @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_849_diff__strict__mono,axiom,
! [A: int,B: int,D: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ D @ C )
=> ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_850_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_851_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_852_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_853_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_854_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_855_fstI,axiom,
! [X4: produc6425607678544837394st_int,Y: int,Z3: produc369741286924889929st_int] :
( ( X4
= ( produc5086643055186798020st_int @ Y @ Z3 ) )
=> ( ( produc8941309049508465880st_int @ X4 )
= Y ) ) ).
% fstI
thf(fact_856_fstI,axiom,
! [X4: produc4435102495419491129nt_int,Y: list_char > int,Z3: int] :
( ( X4
= ( produc5790713362662368625nt_int @ Y @ Z3 ) )
=> ( ( produc9025023058967219549nt_int @ X4 )
= Y ) ) ).
% fstI
thf(fact_857_fstI,axiom,
! [X4: produc369741286924889929st_int,Y: list_char > int,Z3: list_int] :
( ( X4
= ( produc8650753666468850689st_int @ Y @ Z3 ) )
=> ( ( produc2155683374303800813st_int @ X4 )
= Y ) ) ).
% fstI
thf(fact_858_fstI,axiom,
! [X4: product_prod_int_int,Y: int,Z3: int] :
( ( X4
= ( product_Pair_int_int @ Y @ Z3 ) )
=> ( ( product_fst_int_int @ X4 )
= Y ) ) ).
% fstI
thf(fact_859_fstI,axiom,
! [X4: produc4189061985984568957nt_int,Y: list_list_char,Z3: produc4435102495419491129nt_int] :
( ( X4
= ( produc3965054194175396271nt_int @ Y @ Z3 ) )
=> ( ( produc811350159336358595nt_int @ X4 )
= Y ) ) ).
% fstI
thf(fact_860_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_861_zdiff__int__split,axiom,
! [P: int > $o,X4: nat,Y: nat] :
( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X4 @ Y ) ) )
= ( ( ( ord_less_eq_nat @ Y @ X4 )
=> ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X4 ) @ ( semiri1314217659103216013at_int @ Y ) ) ) )
& ( ( ord_less_nat @ X4 @ Y )
=> ( P @ zero_zero_int ) ) ) ) ).
% zdiff_int_split
thf(fact_862_inth__append,axiom,
! [I: int,Xs: list_int,Ys2: list_int] :
( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_size_list_int @ Xs ) ) )
=> ( ( inth_int @ ( append_int @ Xs @ Ys2 ) @ I )
= ( inth_int @ Xs @ I ) ) )
& ( ~ ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_size_list_int @ Xs ) ) )
=> ( ( inth_int @ ( append_int @ Xs @ Ys2 ) @ I )
= ( inth_int @ Ys2 @ ( minus_minus_int @ I @ ( semiri1314217659103216013at_int @ ( size_size_list_int @ Xs ) ) ) ) ) ) ) ) ).
% inth_append
thf(fact_863_inth__append,axiom,
! [I: int,Xs: list_P419547167332244514st_int,Ys2: list_P419547167332244514st_int] :
( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_s229347708945616566st_int @ Xs ) ) )
=> ( ( inth_P8039235741264645271st_int @ ( append5028292981072010935st_int @ Xs @ Ys2 ) @ I )
= ( inth_P8039235741264645271st_int @ Xs @ I ) ) )
& ( ~ ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_s229347708945616566st_int @ Xs ) ) )
=> ( ( inth_P8039235741264645271st_int @ ( append5028292981072010935st_int @ Xs @ Ys2 ) @ I )
= ( inth_P8039235741264645271st_int @ Ys2 @ ( minus_minus_int @ I @ ( semiri1314217659103216013at_int @ ( size_s229347708945616566st_int @ Xs ) ) ) ) ) ) ) ) ).
% inth_append
thf(fact_864_inth__append,axiom,
! [I: int,Xs: list_instr,Ys2: list_instr] :
( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ Xs ) ) )
=> ( ( inth_instr @ ( append_instr @ Xs @ Ys2 ) @ I )
= ( inth_instr @ Xs @ I ) ) )
& ( ~ ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ Xs ) ) )
=> ( ( inth_instr @ ( append_instr @ Xs @ Ys2 ) @ I )
= ( inth_instr @ Ys2 @ ( minus_minus_int @ I @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ Xs ) ) ) ) ) ) ) ) ).
% inth_append
thf(fact_865_inth__append,axiom,
! [I: int,Xs: list_list_char,Ys2: list_list_char] :
( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ Xs ) ) )
=> ( ( inth_list_char @ ( append_list_char @ Xs @ Ys2 ) @ I )
= ( inth_list_char @ Xs @ I ) ) )
& ( ~ ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ Xs ) ) )
=> ( ( inth_list_char @ ( append_list_char @ Xs @ Ys2 ) @ I )
= ( inth_list_char @ Ys2 @ ( minus_minus_int @ I @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ Xs ) ) ) ) ) ) ) ) ).
% inth_append
thf(fact_866_inth__map,axiom,
! [I: int,Xs: list_P419547167332244514st_int,F: produc6425607678544837394st_int > instr] :
( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_s229347708945616566st_int @ Xs ) ) )
=> ( ( inth_instr @ ( map_Pr7514435936640407687_instr @ F @ Xs ) @ I )
= ( F @ ( inth_P8039235741264645271st_int @ Xs @ I ) ) ) ) ) ).
% inth_map
thf(fact_867_inth__map,axiom,
! [I: int,Xs: list_P419547167332244514st_int,F: produc6425607678544837394st_int > list_char] :
( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_s229347708945616566st_int @ Xs ) ) )
=> ( ( inth_list_char @ ( map_Pr5338895242362685753t_char @ F @ Xs ) @ I )
= ( F @ ( inth_P8039235741264645271st_int @ Xs @ I ) ) ) ) ) ).
% inth_map
thf(fact_868_inth__map,axiom,
! [I: int,Xs: list_P419547167332244514st_int,F: produc6425607678544837394st_int > produc6425607678544837394st_int] :
( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_s229347708945616566st_int @ Xs ) ) )
=> ( ( inth_P8039235741264645271st_int @ ( map_Pr8625093460839654122st_int @ F @ Xs ) @ I )
= ( F @ ( inth_P8039235741264645271st_int @ Xs @ I ) ) ) ) ) ).
% inth_map
thf(fact_869_inth__map,axiom,
! [I: int,Xs: list_instr,F: instr > instr] :
( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ Xs ) ) )
=> ( ( inth_instr @ ( map_instr_instr @ F @ Xs ) @ I )
= ( F @ ( inth_instr @ Xs @ I ) ) ) ) ) ).
% inth_map
thf(fact_870_inth__map,axiom,
! [I: int,Xs: list_instr,F: instr > list_char] :
( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ Xs ) ) )
=> ( ( inth_list_char @ ( map_instr_list_char @ F @ Xs ) @ I )
= ( F @ ( inth_instr @ Xs @ I ) ) ) ) ) ).
% inth_map
thf(fact_871_inth__map,axiom,
! [I: int,Xs: list_list_char,F: list_char > instr] :
( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ Xs ) ) )
=> ( ( inth_instr @ ( map_list_char_instr @ F @ Xs ) @ I )
= ( F @ ( inth_list_char @ Xs @ I ) ) ) ) ) ).
% inth_map
thf(fact_872_inth__map,axiom,
! [I: int,Xs: list_list_char,F: list_char > list_char] :
( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ Xs ) ) )
=> ( ( inth_list_char @ ( map_li116305933131242120t_char @ F @ Xs ) @ I )
= ( F @ ( inth_list_char @ Xs @ I ) ) ) ) ) ).
% inth_map
thf(fact_873_execl__next,axiom,
! [P: list_instr,Cfs: list_P419547167332244514st_int,K2: nat] :
( ( execl @ P @ Cfs )
=> ( ( ord_less_nat @ K2 @ ( size_s229347708945616566st_int @ Cfs ) )
=> ( ( K2
!= ( minus_minus_nat @ ( size_s229347708945616566st_int @ Cfs ) @ one_one_nat ) )
=> ( ( iexec_pred @ ( produc1484136438817787418st_int @ ( inth_instr @ P @ ( produc8941309049508465880st_int @ ( nth_Pr8327173284404383691st_int @ Cfs @ K2 ) ) ) @ ( nth_Pr8327173284404383691st_int @ Cfs @ K2 ) ) @ ( nth_Pr8327173284404383691st_int @ Cfs @ ( suc @ K2 ) ) )
& ( ord_less_eq_int @ zero_zero_int @ ( produc8941309049508465880st_int @ ( nth_Pr8327173284404383691st_int @ Cfs @ K2 ) ) )
& ( ord_less_int @ ( produc8941309049508465880st_int @ ( nth_Pr8327173284404383691st_int @ Cfs @ K2 ) ) @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P ) ) ) ) ) ) ) ).
% execl_next
thf(fact_874_execl__sub__aux,axiom,
! [Q: list_instr > produc6425607678544837394st_int > $o,P3: list_instr,Pc: int,S: list_char > int,Stk2: list_int,Cfs: list_P419547167332244514st_int,P: list_instr,F2: list_P419547167332244514st_int > nat > nat > list_P419547167332244514st_int,M: nat,N: nat,Cfs2: list_P419547167332244514st_int] :
( ! [M6: nat,N2: nat] :
( ! [X6: nat] :
( ( member_nat @ X6 @ ( set_or4665077453230672383an_nat @ M6 @ N2 ) )
=> ( Q @ P3 @ ( nth_Pr8327173284404383691st_int @ ( cons_P4151473878978182108st_int @ ( produc5086643055186798020st_int @ Pc @ ( produc8650753666468850689st_int @ S @ Stk2 ) ) @ Cfs ) @ X6 ) ) )
=> ( 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 @ Cfs ) )
@ ^ [O: nat] : ( F2 @ Cfs @ O @ N2 )
@ M6 ) ) ) )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_or4665077453230672383an_nat @ M @ ( plus_plus_nat @ ( plus_plus_nat @ N @ M ) @ ( size_s229347708945616566st_int @ Cfs2 ) ) ) )
=> ( Q @ P3 @ ( nth_Pr8327173284404383691st_int @ ( append5028292981072010935st_int @ Cfs2 @ ( cons_P4151473878978182108st_int @ ( produc5086643055186798020st_int @ Pc @ ( produc8650753666468850689st_int @ S @ Stk2 ) ) @ Cfs ) ) @ ( minus_minus_nat @ X3 @ 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 @ Cfs ) ) ) ) ) ) ).
% execl_sub_aux
thf(fact_875_add__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_876_add__left__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_877_add__right__cancel,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_878_add__right__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_879_add__le__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_880_add__le__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_881_add__le__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_882_add__le__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_883_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_884_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_885_zero__eq__add__iff__both__eq__0,axiom,
! [X4: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X4 @ Y ) )
= ( ( X4 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_886_add__eq__0__iff__both__eq__0,axiom,
! [X4: nat,Y: nat] :
( ( ( plus_plus_nat @ X4 @ Y )
= zero_zero_nat )
= ( ( X4 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_887_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_888_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_889_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_890_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_891_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_892_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_893_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_894_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_895_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_896_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_897_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_898_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_899_add__less__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_900_add__less__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_901_add__less__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_902_add__less__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_903_add__diff__cancel,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_904_diff__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_905_add__diff__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_906_add__diff__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_907_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_908_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_909_add__diff__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_910_add__diff__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_911_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_912_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_913_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_914_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_915_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_916_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_917_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_918_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_919_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_920_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_921_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_922_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_923_nat__add__left__cancel__less,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K2 @ M ) @ ( plus_plus_nat @ K2 @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_924_nat__add__left__cancel__le,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K2 @ M ) @ ( plus_plus_nat @ K2 @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_925_diff__diff__left,axiom,
! [I: nat,J: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K2 )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K2 ) ) ) ).
% diff_diff_left
thf(fact_926_drop0,axiom,
( ( drop_P7838799151754935884st_int @ zero_zero_nat )
= ( ^ [X: list_P419547167332244514st_int] : X ) ) ).
% drop0
thf(fact_927_drop0,axiom,
( ( drop_list_char @ zero_zero_nat )
= ( ^ [X: list_list_char] : X ) ) ).
% drop0
thf(fact_928_drop0,axiom,
( ( drop_instr @ zero_zero_nat )
= ( ^ [X: list_instr] : X ) ) ).
% drop0
thf(fact_929_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_930_drop__drop,axiom,
! [N: nat,M: nat,Xs: list_list_char] :
( ( drop_list_char @ N @ ( drop_list_char @ M @ Xs ) )
= ( drop_list_char @ ( plus_plus_nat @ N @ M ) @ Xs ) ) ).
% drop_drop
thf(fact_931_drop__drop,axiom,
! [N: nat,M: nat,Xs: list_instr] :
( ( drop_instr @ N @ ( drop_instr @ M @ Xs ) )
= ( drop_instr @ ( plus_plus_nat @ N @ M ) @ Xs ) ) ).
% drop_drop
thf(fact_932_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_933_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_934_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_935_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_936_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_937_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_938_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_939_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_940_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_941_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_942_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_943_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_944_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_945_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_946_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_947_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_948_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_949_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_950_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_951_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_952_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_953_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_954_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_955_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_956_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_957_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_958_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_959_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_960_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_961_nth__Cons__0,axiom,
! [X4: produc6425607678544837394st_int,Xs: list_P419547167332244514st_int] :
( ( nth_Pr8327173284404383691st_int @ ( cons_P4151473878978182108st_int @ X4 @ Xs ) @ zero_zero_nat )
= X4 ) ).
% nth_Cons_0
thf(fact_962_nth__Cons__0,axiom,
! [X4: instr,Xs: list_instr] :
( ( nth_instr @ ( cons_instr @ X4 @ Xs ) @ zero_zero_nat )
= X4 ) ).
% nth_Cons_0
thf(fact_963_nth__Cons__0,axiom,
! [X4: int,Xs: list_int] :
( ( nth_int @ ( cons_int @ X4 @ Xs ) @ zero_zero_nat )
= X4 ) ).
% nth_Cons_0
thf(fact_964_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_965_Nat_Odiff__diff__right,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K2 ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_966_Nat_Oadd__diff__assoc2,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K2 ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K2 ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_967_Nat_Oadd__diff__assoc,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K2 ) ) ) ).
% Nat.add_diff_assoc
thf(fact_968_length__append,axiom,
! [Xs: list_int,Ys2: list_int] :
( ( size_size_list_int @ ( append_int @ Xs @ Ys2 ) )
= ( plus_plus_nat @ ( size_size_list_int @ Xs ) @ ( size_size_list_int @ Ys2 ) ) ) ).
% length_append
thf(fact_969_length__append,axiom,
! [Xs: list_P419547167332244514st_int,Ys2: list_P419547167332244514st_int] :
( ( size_s229347708945616566st_int @ ( append5028292981072010935st_int @ Xs @ Ys2 ) )
= ( plus_plus_nat @ ( size_s229347708945616566st_int @ Xs ) @ ( size_s229347708945616566st_int @ Ys2 ) ) ) ).
% length_append
thf(fact_970_length__append,axiom,
! [Xs: list_instr,Ys2: list_instr] :
( ( size_size_list_instr @ ( append_instr @ Xs @ Ys2 ) )
= ( plus_plus_nat @ ( size_size_list_instr @ Xs ) @ ( size_size_list_instr @ Ys2 ) ) ) ).
% length_append
thf(fact_971_length__append,axiom,
! [Xs: list_list_char,Ys2: list_list_char] :
( ( size_s356637359517785349t_char @ ( append_list_char @ Xs @ Ys2 ) )
= ( plus_plus_nat @ ( size_s356637359517785349t_char @ Xs ) @ ( size_s356637359517785349t_char @ Ys2 ) ) ) ).
% length_append
thf(fact_972_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ M ) )
= ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% of_nat_Suc
thf(fact_973_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ M ) )
= ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M ) ) ) ).
% of_nat_Suc
thf(fact_974_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_975_diff__Suc__diff__eq1,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K2 ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K2 ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_976_diff__Suc__diff__eq2,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K2 ) ) @ I )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K2 @ I ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_977_nth__append__length__plus,axiom,
! [Xs: list_int,Ys2: list_int,N: nat] :
( ( nth_int @ ( append_int @ Xs @ Ys2 ) @ ( plus_plus_nat @ ( size_size_list_int @ Xs ) @ N ) )
= ( nth_int @ Ys2 @ N ) ) ).
% nth_append_length_plus
thf(fact_978_nth__append__length__plus,axiom,
! [Xs: list_P419547167332244514st_int,Ys2: list_P419547167332244514st_int,N: nat] :
( ( nth_Pr8327173284404383691st_int @ ( append5028292981072010935st_int @ Xs @ Ys2 ) @ ( plus_plus_nat @ ( size_s229347708945616566st_int @ Xs ) @ N ) )
= ( nth_Pr8327173284404383691st_int @ Ys2 @ N ) ) ).
% nth_append_length_plus
thf(fact_979_nth__append__length__plus,axiom,
! [Xs: list_instr,Ys2: list_instr,N: nat] :
( ( nth_instr @ ( append_instr @ Xs @ Ys2 ) @ ( plus_plus_nat @ ( size_size_list_instr @ Xs ) @ N ) )
= ( nth_instr @ Ys2 @ N ) ) ).
% nth_append_length_plus
thf(fact_980_nth__append__length__plus,axiom,
! [Xs: list_list_char,Ys2: list_list_char,N: nat] :
( ( nth_list_char @ ( append_list_char @ Xs @ Ys2 ) @ ( plus_plus_nat @ ( size_s356637359517785349t_char @ Xs ) @ N ) )
= ( nth_list_char @ Ys2 @ N ) ) ).
% nth_append_length_plus
thf(fact_981_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_982_nth__drop,axiom,
! [N: nat,Xs: list_int,I: nat] :
( ( ord_less_eq_nat @ N @ ( size_size_list_int @ Xs ) )
=> ( ( nth_int @ ( drop_int @ N @ Xs ) @ I )
= ( nth_int @ Xs @ ( plus_plus_nat @ N @ I ) ) ) ) ).
% nth_drop
thf(fact_983_nth__drop,axiom,
! [N: nat,Xs: list_P419547167332244514st_int,I: nat] :
( ( ord_less_eq_nat @ N @ ( size_s229347708945616566st_int @ Xs ) )
=> ( ( nth_Pr8327173284404383691st_int @ ( drop_P7838799151754935884st_int @ N @ Xs ) @ I )
= ( nth_Pr8327173284404383691st_int @ Xs @ ( plus_plus_nat @ N @ I ) ) ) ) ).
% nth_drop
thf(fact_984_nth__drop,axiom,
! [N: nat,Xs: list_instr,I: nat] :
( ( ord_less_eq_nat @ N @ ( size_size_list_instr @ Xs ) )
=> ( ( nth_instr @ ( drop_instr @ N @ Xs ) @ I )
= ( nth_instr @ Xs @ ( plus_plus_nat @ N @ I ) ) ) ) ).
% nth_drop
thf(fact_985_nth__drop,axiom,
! [N: nat,Xs: list_list_char,I: nat] :
( ( ord_less_eq_nat @ N @ ( size_s356637359517785349t_char @ Xs ) )
=> ( ( nth_list_char @ ( drop_list_char @ N @ Xs ) @ I )
= ( nth_list_char @ Xs @ ( plus_plus_nat @ N @ I ) ) ) ) ).
% nth_drop
thf(fact_986_iexec__offset,axiom,
! [Ins2: instr,Pc: int,S: list_char > int,Stk2: list_int,Pc2: int,S4: list_char > int,Stk3: list_int,I: 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 @ I ) @ ( produc8650753666468850689st_int @ S @ Stk2 ) ) ) @ ( produc5086643055186798020st_int @ ( minus_minus_int @ Pc2 @ I ) @ ( produc8650753666468850689st_int @ S4 @ Stk3 ) ) ) ) ).
% iexec_offset
thf(fact_987_nth__Cons__pos,axiom,
! [N: nat,X4: produc6425607678544837394st_int,Xs: list_P419547167332244514st_int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_Pr8327173284404383691st_int @ ( cons_P4151473878978182108st_int @ X4 @ Xs ) @ N )
= ( nth_Pr8327173284404383691st_int @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_988_nth__Cons__pos,axiom,
! [N: nat,X4: instr,Xs: list_instr] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_instr @ ( cons_instr @ X4 @ Xs ) @ N )
= ( nth_instr @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_989_nth__Cons__pos,axiom,
! [N: nat,X4: int,Xs: list_int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_int @ ( cons_int @ X4 @ Xs ) @ N )
= ( nth_int @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_990_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_991_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_992_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ( plus_plus_nat @ I @ K3 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_993_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_994_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_995_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_996_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( I = J )
& ( K2 = L ) )
=> ( ( plus_plus_nat @ I @ K2 )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_997_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: int,J: int,K2: int,L: int] :
( ( ( I = J )
& ( K2 = L ) )
=> ( ( plus_plus_int @ I @ K2 )
= ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_998_group__cancel_Oadd1,axiom,
! [A3: nat,K2: nat,A: nat,B: nat] :
( ( A3
= ( plus_plus_nat @ K2 @ A ) )
=> ( ( plus_plus_nat @ A3 @ B )
= ( plus_plus_nat @ K2 @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_999_group__cancel_Oadd1,axiom,
! [A3: int,K2: int,A: int,B: int] :
( ( A3
= ( plus_plus_int @ K2 @ A ) )
=> ( ( plus_plus_int @ A3 @ B )
= ( plus_plus_int @ K2 @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_1000_group__cancel_Oadd2,axiom,
! [B5: nat,K2: nat,B: nat,A: nat] :
( ( B5
= ( plus_plus_nat @ K2 @ B ) )
=> ( ( plus_plus_nat @ A @ B5 )
= ( plus_plus_nat @ K2 @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_1001_group__cancel_Oadd2,axiom,
! [B5: int,K2: int,B: int,A: int] :
( ( B5
= ( plus_plus_int @ K2 @ B ) )
=> ( ( plus_plus_int @ A @ B5 )
= ( plus_plus_int @ K2 @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_1002_add_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_1003_add_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.assoc
thf(fact_1004_add_Oleft__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_1005_add_Oright__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_1006_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A5: nat,B4: nat] : ( plus_plus_nat @ B4 @ A5 ) ) ) ).
% add.commute
thf(fact_1007_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A5: int,B4: int] : ( plus_plus_int @ B4 @ A5 ) ) ) ).
% add.commute
thf(fact_1008_add_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_1009_add_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C ) )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.left_commute
thf(fact_1010_add__left__imp__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_1011_add__left__imp__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_1012_add__right__imp__eq,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_1013_add__right__imp__eq,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_1014_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_1015_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_1016_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_1017_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_1018_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_1019_is__num__normalize_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_1020_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 )
@ ^ [X: nat] : ( H @ ( F22 @ X ) )
@ Nat ) ) ).
% nat.case_distrib
thf(fact_1021_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 )
@ ^ [X: nat] : ( H @ ( F22 @ X ) )
@ Nat ) ) ).
% nat.case_distrib
thf(fact_1022_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 )
@ ^ [X: nat] : ( H @ ( F22 @ X ) )
@ Nat ) ) ).
% nat.case_distrib
thf(fact_1023_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 )
@ ^ [X: nat] : ( H @ ( F22 @ X ) )
@ Nat ) ) ).
% nat.case_distrib
thf(fact_1024_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 )
@ ^ [X: nat] : ( H @ ( F22 @ X ) )
@ Nat ) ) ).
% nat.case_distrib
thf(fact_1025_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 )
@ ^ [X: nat] : ( H @ ( F22 @ X ) )
@ Nat ) ) ).
% nat.case_distrib
thf(fact_1026_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 )
@ ^ [X: nat] : ( H @ ( F22 @ X ) )
@ Nat ) ) ).
% nat.case_distrib
thf(fact_1027_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 )
@ ^ [X: nat] : ( H @ ( F22 @ X ) )
@ Nat ) ) ).
% nat.case_distrib
thf(fact_1028_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 )
@ ^ [X: nat] : ( H @ ( F22 @ X ) )
@ Nat ) ) ).
% nat.case_distrib
thf(fact_1029_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_1030_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_1031_add__le__imp__le__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_1032_add__le__imp__le__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_1033_add__le__imp__le__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_1034_add__le__imp__le__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_1035_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_1036_add__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_right_mono
thf(fact_1037_add__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_1038_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C2: nat] :
( B
!= ( plus_plus_nat @ A @ C2 ) ) ) ).
% less_eqE
thf(fact_1039_add__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_left_mono
thf(fact_1040_add__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_1041_add__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_mono
thf(fact_1042_add__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_1043_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: int,J: int,K2: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_eq_int @ K2 @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_1044_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K2 @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_1045_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: int,J: int,K2: int,L: int] :
( ( ( I = J )
& ( ord_less_eq_int @ K2 @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_1046_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K2 @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_1047_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: int,J: int,K2: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( K2 = L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_1048_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K2 = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_1049_add_Ogroup__left__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add.group_left_neutral
thf(fact_1050_add_Ocomm__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.comm_neutral
thf(fact_1051_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_1052_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_1053_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_1054_verit__sum__simplify,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% verit_sum_simplify
thf(fact_1055_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_1056_add__less__imp__less__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_1057_add__less__imp__less__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_1058_add__less__imp__less__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_1059_add__less__imp__less__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_1060_add__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_1061_add__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_1062_add__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_1063_add__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_1064_add__strict__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_1065_add__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_1066_add__mono__thms__linordered__field_I1_J,axiom,
! [I: int,J: int,K2: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( K2 = L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_1067_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K2 = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_1068_add__mono__thms__linordered__field_I2_J,axiom,
! [I: int,J: int,K2: int,L: int] :
( ( ( I = J )
& ( ord_less_int @ K2 @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_1069_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( I = J )
& ( ord_less_nat @ K2 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_1070_add__mono__thms__linordered__field_I5_J,axiom,
! [I: int,J: int,K2: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_int @ K2 @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_1071_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_nat @ K2 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_1072_diff__diff__eq,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_1073_diff__diff__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_1074_add__implies__diff,axiom,
! [C: int,B: int,A: int] :
( ( ( plus_plus_int @ C @ B )
= A )
=> ( C
= ( minus_minus_int @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_1075_add__implies__diff,axiom,
! [C: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ C @ B )
= A )
=> ( C
= ( minus_minus_nat @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_1076_diff__add__eq__diff__diff__swap,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) )
= ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_1077_diff__add__eq,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% diff_add_eq
thf(fact_1078_diff__diff__eq2,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% diff_diff_eq2
thf(fact_1079_add__diff__eq,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% add_diff_eq
thf(fact_1080_nat__arith_Osuc1,axiom,
! [A3: nat,K2: nat,A: nat] :
( ( A3
= ( plus_plus_nat @ K2 @ A ) )
=> ( ( suc @ A3 )
= ( plus_plus_nat @ K2 @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_1081_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_1082_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_1083_less__add__eq__less,axiom,
! [K2: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K2 @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K2 @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_1084_trans__less__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_1085_trans__less__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_1086_add__less__mono1,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ K2 ) ) ) ).
% add_less_mono1
thf(fact_1087_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_1088_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_1089_add__less__mono,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K2 @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_1090_add__lessD1,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K2 )
=> ( ord_less_nat @ I @ K2 ) ) ).
% add_lessD1
thf(fact_1091_add__leE,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K2 ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K2 @ N ) ) ) ).
% add_leE
thf(fact_1092_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_1093_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_1094_add__leD1,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K2 ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_1095_add__leD2,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K2 ) @ N )
=> ( ord_less_eq_nat @ K2 @ N ) ) ).
% add_leD2
thf(fact_1096_le__Suc__ex,axiom,
! [K2: nat,L: nat] :
( ( ord_less_eq_nat @ K2 @ L )
=> ? [N2: nat] :
( L
= ( plus_plus_nat @ K2 @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_1097_add__le__mono,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K2 @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_1098_add__le__mono1,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ K2 ) ) ) ).
% add_le_mono1
thf(fact_1099_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_1100_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_1101_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N3: nat] :
? [K4: nat] :
( N3
= ( plus_plus_nat @ M2 @ K4 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1102_Nat_Odiff__cancel,axiom,
! [K2: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K2 @ M ) @ ( plus_plus_nat @ K2 @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_1103_diff__cancel2,axiom,
! [M: nat,K2: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K2 ) @ ( plus_plus_nat @ N @ K2 ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_1104_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_1105_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_1106_nat_Odistinct_I1_J,axiom,
! [X2: nat] :
( zero_zero_nat
!= ( suc @ X2 ) ) ).
% nat.distinct(1)
thf(fact_1107_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_1108_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_1109_nat_OdiscI,axiom,
! [Nat: nat,X2: nat] :
( ( Nat
= ( suc @ X2 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_1110_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_1111_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_1112_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
=> ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
=> ( ! [X3: nat,Y3: nat] :
( ( P @ X3 @ Y3 )
=> ( P @ ( suc @ X3 ) @ ( suc @ Y3 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_1113_zero__induct,axiom,
! [P: nat > $o,K2: nat] :
( ( P @ K2 )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_1114_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_1115_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_1116_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_1117_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M6: nat] :
( N
= ( suc @ M6 ) ) ) ).
% not0_implies_Suc
thf(fact_1118_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_1119_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_1120_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_1121_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_1122_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_1123_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_1124_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_1125_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_1126_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_1127_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_1128_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1129_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_1130_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_1131_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M2: nat,N3: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ N3 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N3 ) ) ) ) ) ).
% add_eq_if
thf(fact_1132_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ? [K3: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K3 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_1133_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M2: nat,N3: nat] :
? [K4: nat] :
( N3
= ( suc @ ( plus_plus_nat @ M2 @ K4 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1134_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
% less_add_Suc2
thf(fact_1135_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
% less_add_Suc1
thf(fact_1136_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_1137_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K2: nat] :
( ! [M6: nat,N2: nat] :
( ( ord_less_nat @ M6 @ N2 )
=> ( ord_less_nat @ ( F @ M6 ) @ ( F @ N2 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K2 ) @ ( F @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1138_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_1139_less__diff__conv,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ J ) ) ).
% less_diff_conv
thf(fact_1140_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K2 )
= ( J
= ( plus_plus_nat @ K2 @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1141_Nat_Odiff__add__assoc2,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K2 )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K2 ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1142_Nat_Odiff__add__assoc,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K2 )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K2 ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1143_Nat_Ole__diff__conv2,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1144_le__diff__conv,axiom,
! [J: nat,K2: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K2 ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K2 ) ) ) ).
% le_diff_conv
thf(fact_1145_Suc__eq__plus1,axiom,
( suc
= ( ^ [N3: nat] : ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_1146_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1147_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_1148_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_1149_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M6: nat] :
( N
= ( suc @ M6 ) ) ) ).
% gr0_implies_Suc
thf(fact_1150_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
=> ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( P @ ( suc @ I3 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_1151_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M2: nat] :
( N
= ( suc @ M2 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1152_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
& ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ( P @ ( suc @ I3 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_1153_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K3 )
=> ~ ( P @ I4 ) )
& ( P @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1154_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_1155_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_1156_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_1157_all__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ! [M2: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( P @ M2 ) ) )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
=> ( P @ X ) ) ) ) ).
% all_nat_less_eq
thf(fact_1158_ex__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ? [M2: nat] :
( ( ord_less_nat @ M2 @ N )
& ( P @ M2 ) ) )
= ( ? [X: nat] :
( ( member_nat @ X @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
& ( P @ X ) ) ) ) ).
% ex_nat_less_eq
thf(fact_1159_less__diff__conv2,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K2 ) @ I )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K2 ) ) ) ) ).
% less_diff_conv2
thf(fact_1160_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_nat @ K3 @ N )
& ! [I4: nat] :
( ( ord_less_eq_nat @ I4 @ K3 )
=> ~ ( P @ I4 ) )
& ( P @ ( suc @ K3 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1161_diff__Suc__less,axiom,
! [N: nat,I: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_1162_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_1163_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_1164_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_1165_pos__int__cases,axiom,
! [K2: int] :
( ( ord_less_int @ zero_zero_int @ K2 )
=> ~ ! [N2: nat] :
( ( K2
= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% pos_int_cases
thf(fact_1166_zero__less__imp__eq__int,axiom,
! [K2: int] :
( ( ord_less_int @ zero_zero_int @ K2 )
=> ? [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
& ( K2
= ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_1167_conj__le__cong,axiom,
! [X4: int,X8: int,P: $o,P3: $o] :
( ( X4 = X8 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X8 )
=> ( P = P3 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X4 )
& P )
= ( ( ord_less_eq_int @ zero_zero_int @ X8 )
& P3 ) ) ) ) ).
% conj_le_cong
thf(fact_1168_imp__le__cong,axiom,
! [X4: int,X8: int,P: $o,P3: $o] :
( ( X4 = X8 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X8 )
=> ( P = P3 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X4 )
=> P )
= ( ( ord_less_eq_int @ zero_zero_int @ X8 )
=> P3 ) ) ) ) ).
% imp_le_cong
thf(fact_1169_execl__all__def,axiom,
( execl_all
= ( ^ [P6: list_instr,Cfs3: list_P419547167332244514st_int] :
( ( execl @ P6 @ 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 @ P6 ) ) ) ) ) ) ) ).
% execl_all_def
thf(fact_1170_zle__add1__eq__le,axiom,
! [W: int,Z3: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z3 @ one_one_int ) )
= ( ord_less_eq_int @ W @ Z3 ) ) ).
% zle_add1_eq_le
thf(fact_1171_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_1172_plus__int__code_I1_J,axiom,
! [K2: int] :
( ( plus_plus_int @ K2 @ zero_zero_int )
= K2 ) ).
% plus_int_code(1)
thf(fact_1173_execl_Ocases,axiom,
! [X4: produc4400946599758736694st_int] :
( ! [P7: list_instr,Cf2: produc6425607678544837394st_int,Cf3: produc6425607678544837394st_int,Cfs4: list_P419547167332244514st_int] :
( X4
!= ( produc7764413618310988720st_int @ P7 @ ( cons_P4151473878978182108st_int @ Cf2 @ ( cons_P4151473878978182108st_int @ Cf3 @ Cfs4 ) ) ) )
=> ( ! [P7: list_instr] :
( X4
!= ( produc7764413618310988720st_int @ P7 @ nil_Pr749310521696113548st_int ) )
=> ~ ! [P7: list_instr,V: produc6425607678544837394st_int] :
( X4
!= ( produc7764413618310988720st_int @ P7 @ ( cons_P4151473878978182108st_int @ V @ nil_Pr749310521696113548st_int ) ) ) ) ) ).
% execl.cases
thf(fact_1174_execl_Osimps_I2_J,axiom,
! [P: list_instr] : ( execl @ P @ nil_Pr749310521696113548st_int ) ).
% execl.simps(2)
thf(fact_1175_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_1176_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_1177_odd__nonzero,axiom,
! [Z3: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z3 ) @ Z3 )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_1178_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_1179_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_1180_zadd__int__left,axiom,
! [M: nat,N: nat,Z3: int] :
( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z3 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) ) @ Z3 ) ) ).
% zadd_int_left
thf(fact_1181_int__ge__induct,axiom,
! [K2: int,I: int,P: int > $o] :
( ( ord_less_eq_int @ K2 @ I )
=> ( ( P @ K2 )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ K2 @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_ge_induct
thf(fact_1182_zle__iff__zadd,axiom,
( ord_less_eq_int
= ( ^ [W2: int,Z4: int] :
? [N3: nat] :
( Z4
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_1183_int__gr__induct,axiom,
! [K2: int,I: int,P: int > $o] :
( ( ord_less_int @ K2 @ I )
=> ( ( P @ ( plus_plus_int @ K2 @ one_one_int ) )
=> ( ! [I2: int] :
( ( ord_less_int @ K2 @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_gr_induct
thf(fact_1184_zless__add1__eq,axiom,
! [W: int,Z3: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z3 @ one_one_int ) )
= ( ( ord_less_int @ W @ Z3 )
| ( W = Z3 ) ) ) ).
% zless_add1_eq
thf(fact_1185_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_1186_int__Suc,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ).
% int_Suc
thf(fact_1187_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_1188_zless__iff__Suc__zadd,axiom,
( ord_less_int
= ( ^ [W2: int,Z4: int] :
? [N3: nat] :
( Z4
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ) ).
% zless_iff_Suc_zadd
thf(fact_1189_odd__less__0__iff,axiom,
! [Z3: int] :
( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z3 ) @ Z3 ) @ zero_zero_int )
= ( ord_less_int @ Z3 @ zero_zero_int ) ) ).
% odd_less_0_iff
thf(fact_1190_zless__imp__add1__zle,axiom,
! [W: int,Z3: int] :
( ( ord_less_int @ W @ Z3 )
=> ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z3 ) ) ).
% zless_imp_add1_zle
thf(fact_1191_add1__zle__eq,axiom,
! [W: int,Z3: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z3 )
= ( ord_less_int @ W @ Z3 ) ) ).
% add1_zle_eq
thf(fact_1192_int__induct,axiom,
! [P: int > $o,K2: int,I: int] :
( ( P @ K2 )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ K2 @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ I2 @ K2 )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_induct
thf(fact_1193_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_1194_le__imp__0__less,axiom,
! [Z3: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z3 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z3 ) ) ) ).
% le_imp_0_less
thf(fact_1195_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_1196_exec1__def,axiom,
( exec1
= ( ^ [P6: list_instr,Cf: produc6425607678544837394st_int,Cf4: produc6425607678544837394st_int] :
( ( iexec_pred @ ( produc1484136438817787418st_int @ ( inth_instr @ P6 @ ( produc8941309049508465880st_int @ Cf ) ) @ Cf ) @ Cf4 )
& ( ord_less_eq_int @ zero_zero_int @ ( produc8941309049508465880st_int @ Cf ) )
& ( ord_less_int @ ( produc8941309049508465880st_int @ Cf ) @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P6 ) ) ) ) ) ) ).
% exec1_def
thf(fact_1197_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_1198_exec1__appendR,axiom,
! [P: list_instr,C: produc6425607678544837394st_int,C4: produc6425607678544837394st_int,P3: list_instr] :
( ( exec1 @ P @ C @ C4 )
=> ( exec1 @ ( append_instr @ P @ P3 ) @ C @ C4 ) ) ).
% exec1_appendR
thf(fact_1199_execl_Oelims_I3_J,axiom,
! [X4: list_instr,Xa: list_P419547167332244514st_int] :
( ~ ( execl @ X4 @ Xa )
=> ~ ! [Cf2: produc6425607678544837394st_int,Cf3: produc6425607678544837394st_int,Cfs4: list_P419547167332244514st_int] :
( ( Xa
= ( cons_P4151473878978182108st_int @ Cf2 @ ( cons_P4151473878978182108st_int @ Cf3 @ Cfs4 ) ) )
=> ( ( exec1 @ X4 @ Cf2 @ Cf3 )
& ( execl @ X4 @ ( cons_P4151473878978182108st_int @ Cf3 @ Cfs4 ) ) ) ) ) ).
% execl.elims(3)
thf(fact_1200_execl_Osimps_I1_J,axiom,
! [P: list_instr,Cf5: produc6425607678544837394st_int,Cf6: produc6425607678544837394st_int,Cfs: list_P419547167332244514st_int] :
( ( execl @ P @ ( cons_P4151473878978182108st_int @ Cf5 @ ( cons_P4151473878978182108st_int @ Cf6 @ Cfs ) ) )
= ( ( exec1 @ P @ Cf5 @ Cf6 )
& ( execl @ P @ ( cons_P4151473878978182108st_int @ Cf6 @ Cfs ) ) ) ) ).
% execl.simps(1)
thf(fact_1201_execl_Oelims_I2_J,axiom,
! [X4: list_instr,Xa: list_P419547167332244514st_int] :
( ( execl @ X4 @ Xa )
=> ( ! [Cf2: produc6425607678544837394st_int,Cf3: produc6425607678544837394st_int,Cfs4: list_P419547167332244514st_int] :
( ( Xa
= ( cons_P4151473878978182108st_int @ Cf2 @ ( cons_P4151473878978182108st_int @ Cf3 @ Cfs4 ) ) )
=> ~ ( ( exec1 @ X4 @ Cf2 @ Cf3 )
& ( execl @ X4 @ ( cons_P4151473878978182108st_int @ Cf3 @ Cfs4 ) ) ) )
=> ( ( Xa != nil_Pr749310521696113548st_int )
=> ~ ! [V: produc6425607678544837394st_int] :
( Xa
!= ( cons_P4151473878978182108st_int @ V @ nil_Pr749310521696113548st_int ) ) ) ) ) ).
% execl.elims(2)
thf(fact_1202_execl_Oelims_I1_J,axiom,
! [X4: list_instr,Xa: list_P419547167332244514st_int,Y: $o] :
( ( ( execl @ X4 @ Xa )
= Y )
=> ( ! [Cf2: produc6425607678544837394st_int,Cf3: produc6425607678544837394st_int,Cfs4: list_P419547167332244514st_int] :
( ( Xa
= ( cons_P4151473878978182108st_int @ Cf2 @ ( cons_P4151473878978182108st_int @ Cf3 @ Cfs4 ) ) )
=> ( Y
= ( ~ ( ( exec1 @ X4 @ Cf2 @ Cf3 )
& ( execl @ X4 @ ( cons_P4151473878978182108st_int @ Cf3 @ Cfs4 ) ) ) ) ) )
=> ( ( ( Xa = nil_Pr749310521696113548st_int )
=> ~ Y )
=> ~ ( ? [V: produc6425607678544837394st_int] :
( Xa
= ( cons_P4151473878978182108st_int @ V @ nil_Pr749310521696113548st_int ) )
=> ~ Y ) ) ) ) ).
% execl.elims(1)
thf(fact_1203_exec1__appendL,axiom,
! [P: list_instr,I: int,S: list_char > int,Stk2: list_int,I5: int,S4: list_char > int,Stk3: list_int,P3: list_instr] :
( ( exec1 @ P @ ( produc5086643055186798020st_int @ I @ ( 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 ) ) @ I ) @ ( 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_1204_to__m__state_Oelims,axiom,
! [X4: list_list_char,Xa: list_char > int,Xb: int,Y: int] :
( ( ( to_m_state @ X4 @ Xa @ Xb )
= Y )
=> ( ( ( ( ord_less_int @ zero_zero_int @ Xb )
& ( ord_less_eq_int @ Xb @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ X4 ) ) ) )
=> ( Y
= ( Xa @ ( inth_list_char @ X4 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ X4 ) ) @ Xb ) ) ) ) )
& ( ~ ( ( ord_less_int @ zero_zero_int @ Xb )
& ( ord_less_eq_int @ Xb @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ X4 ) ) ) )
=> ( Y = zero_zero_int ) ) ) ) ).
% to_m_state.elims
thf(fact_1205_exec1I,axiom,
! [C4: produc6425607678544837394st_int,P: list_instr,I: int,S: list_char > int,Stk2: list_int] :
( ( C4
= ( iexec @ ( inth_instr @ P @ I ) @ ( produc5086643055186798020st_int @ I @ ( produc8650753666468850689st_int @ S @ Stk2 ) ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P ) ) )
=> ( exec1 @ P @ ( produc5086643055186798020st_int @ I @ ( produc8650753666468850689st_int @ S @ Stk2 ) ) @ C4 ) ) ) ) ).
% exec1I
thf(fact_1206_iexec__shift,axiom,
! [N: int,I5: int,S4: list_char > int,Stk3: list_int,Ins2: instr,I: 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 @ I ) @ ( produc8650753666468850689st_int @ S @ Stk2 ) ) ) )
= ( ( produc5086643055186798020st_int @ I5 @ ( produc8650753666468850689st_int @ S4 @ Stk3 ) )
= ( iexec @ Ins2 @ ( produc5086643055186798020st_int @ I @ ( produc8650753666468850689st_int @ S @ Stk2 ) ) ) ) ) ).
% iexec_shift
thf(fact_1207_iexec__pred_Ocases,axiom,
! [A1: produc5995290525303592096st_int,A22: produc6425607678544837394st_int] :
( ( iexec_pred @ A1 @ A22 )
=> ~ ! [Ins: instr,Cf2: produc6425607678544837394st_int] :
( ( A1
= ( produc1484136438817787418st_int @ Ins @ Cf2 ) )
=> ( A22
!= ( iexec @ Ins @ Cf2 ) ) ) ) ).
% iexec_pred.cases
thf(fact_1208_iexec__pred_Osimps,axiom,
( iexec_pred
= ( ^ [A12: produc5995290525303592096st_int,A23: produc6425607678544837394st_int] :
? [Ins3: instr,Cf: produc6425607678544837394st_int] :
( ( A12
= ( produc1484136438817787418st_int @ Ins3 @ Cf ) )
& ( A23
= ( iexec @ Ins3 @ Cf ) ) ) ) ) ).
% iexec_pred.simps
thf(fact_1209_iexec__pred_Ointros,axiom,
! [Ins2: instr,Cf5: produc6425607678544837394st_int] : ( iexec_pred @ ( produc1484136438817787418st_int @ Ins2 @ Cf5 ) @ ( iexec @ Ins2 @ Cf5 ) ) ).
% iexec_pred.intros
thf(fact_1210_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_1211_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_1212_exec__appendL__if,axiom,
! [P3: list_instr,I: 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 ) ) @ I )
=> ( ( star_P707599355569300323st_int @ ( exec1 @ P ) @ ( produc5086643055186798020st_int @ ( minus_minus_int @ I @ ( 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 @ I @ ( produc8650753666468850689st_int @ S @ Stk2 ) ) @ ( produc5086643055186798020st_int @ I5 @ ( produc8650753666468850689st_int @ S4 @ Stk3 ) ) ) ) ) ) ).
% exec_appendL_if
thf(fact_1213_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_1214_exec__appendR,axiom,
! [P: list_instr,C: produc6425607678544837394st_int,C4: produc6425607678544837394st_int,P3: list_instr] :
( ( star_P707599355569300323st_int @ ( exec1 @ P ) @ C @ C4 )
=> ( star_P707599355569300323st_int @ ( exec1 @ ( append_instr @ P @ P3 ) ) @ C @ C4 ) ) ).
% exec_appendR
thf(fact_1215_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_1216_exec__appendL,axiom,
! [P: list_instr,I: 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 @ I @ ( 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 ) ) @ I ) @ ( 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_1217_LoadIE,axiom,
! [I: int,Pc: int,S: list_char > int,Stk2: list_int,Cf5: produc6425607678544837394st_int] :
( ( iexec_pred @ ( produc1484136438817787418st_int @ ( loadi @ I ) @ ( produc5086643055186798020st_int @ Pc @ ( produc8650753666468850689st_int @ S @ Stk2 ) ) ) @ Cf5 )
=> ( Cf5
= ( produc5086643055186798020st_int @ ( plus_plus_int @ Pc @ one_one_int ) @ ( produc8650753666468850689st_int @ S @ ( cons_int @ I @ Stk2 ) ) ) ) ) ).
% LoadIE
thf(fact_1218_LoadE,axiom,
! [X4: list_char,Pc: int,S: list_char > int,Stk2: list_int,Cf5: produc6425607678544837394st_int] :
( ( iexec_pred @ ( produc1484136438817787418st_int @ ( load @ X4 ) @ ( produc5086643055186798020st_int @ Pc @ ( produc8650753666468850689st_int @ S @ Stk2 ) ) ) @ Cf5 )
=> ( Cf5
= ( produc5086643055186798020st_int @ ( plus_plus_int @ Pc @ one_one_int ) @ ( produc8650753666468850689st_int @ S @ ( cons_int @ ( S @ X4 ) @ Stk2 ) ) ) ) ) ).
% LoadE
thf(fact_1219_instr_Oinject_I2_J,axiom,
! [X2: list_char,Y2: list_char] :
( ( ( load @ X2 )
= ( load @ Y2 ) )
= ( X2 = Y2 ) ) ).
% instr.inject(2)
thf(fact_1220_instr_Oinject_I1_J,axiom,
! [X1: int,Y1: int] :
( ( ( loadi @ X1 )
= ( loadi @ Y1 ) )
= ( X1 = Y1 ) ) ).
% instr.inject(1)
thf(fact_1221_instr_Osize_I9_J,axiom,
! [X2: list_char] :
( ( size_size_instr @ ( load @ X2 ) )
= zero_zero_nat ) ).
% instr.size(9)
thf(fact_1222_instr_Osize_I8_J,axiom,
! [X1: int] :
( ( size_size_instr @ ( loadi @ X1 ) )
= zero_zero_nat ) ).
% instr.size(8)
thf(fact_1223_instr_Odistinct_I1_J,axiom,
! [X1: int,X2: list_char] :
( ( loadi @ X1 )
!= ( load @ X2 ) ) ).
% instr.distinct(1)
thf(fact_1224_upto__aux__rec,axiom,
( upto_aux
= ( ^ [I3: int,J3: int,Js: list_int] : ( if_list_int @ ( ord_less_int @ J3 @ I3 ) @ Js @ ( upto_aux @ I3 @ ( minus_minus_int @ J3 @ one_one_int ) @ ( cons_int @ J3 @ Js ) ) ) ) ) ).
% upto_aux_rec
thf(fact_1225_JmpE,axiom,
! [I: int,Pc: int,S: list_char > int,Stk2: list_int,Cf5: produc6425607678544837394st_int] :
( ( iexec_pred @ ( produc1484136438817787418st_int @ ( jmp @ I ) @ ( produc5086643055186798020st_int @ Pc @ ( produc8650753666468850689st_int @ S @ Stk2 ) ) ) @ Cf5 )
=> ( Cf5
= ( produc5086643055186798020st_int @ ( plus_plus_int @ ( plus_plus_int @ Pc @ one_one_int ) @ I ) @ ( produc8650753666468850689st_int @ S @ Stk2 ) ) ) ) ).
% JmpE
thf(fact_1226_instr_Oinject_I4_J,axiom,
! [X52: int,Y52: int] :
( ( ( jmp @ X52 )
= ( jmp @ Y52 ) )
= ( X52 = Y52 ) ) ).
% instr.inject(4)
thf(fact_1227_instr_Odistinct_I7_J,axiom,
! [X1: int,X52: int] :
( ( loadi @ X1 )
!= ( jmp @ X52 ) ) ).
% instr.distinct(7)
thf(fact_1228_instr_Odistinct_I17_J,axiom,
! [X2: list_char,X52: int] :
( ( load @ X2 )
!= ( jmp @ X52 ) ) ).
% instr.distinct(17)
thf(fact_1229_instr_Osize_I12_J,axiom,
! [X52: int] :
( ( size_size_instr @ ( jmp @ X52 ) )
= zero_zero_nat ) ).
% instr.size(12)
thf(fact_1230_size__char__eq__0,axiom,
( size_size_char
= ( ^ [C3: char] : zero_zero_nat ) ) ).
% size_char_eq_0
thf(fact_1231_size_H__char__eq__0,axiom,
( size_char
= ( ^ [C3: char] : zero_zero_nat ) ) ).
% size'_char_eq_0
thf(fact_1232_to__m__state_Opelims,axiom,
! [X4: list_list_char,Xa: list_char > int,Xb: int,Y: int] :
( ( ( to_m_state @ X4 @ Xa @ Xb )
= Y )
=> ( ( accp_P4562858270263085236nt_int @ to_m_state_rel @ ( produc3965054194175396271nt_int @ X4 @ ( produc5790713362662368625nt_int @ Xa @ Xb ) ) )
=> ~ ( ( ( ( ( ord_less_int @ zero_zero_int @ Xb )
& ( ord_less_eq_int @ Xb @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ X4 ) ) ) )
=> ( Y
= ( Xa @ ( inth_list_char @ X4 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ X4 ) ) @ Xb ) ) ) ) )
& ( ~ ( ( ord_less_int @ zero_zero_int @ Xb )
& ( ord_less_eq_int @ Xb @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ X4 ) ) ) )
=> ( Y = zero_zero_int ) ) )
=> ~ ( accp_P4562858270263085236nt_int @ to_m_state_rel @ ( produc3965054194175396271nt_int @ X4 @ ( produc5790713362662368625nt_int @ Xa @ Xb ) ) ) ) ) ) ).
% to_m_state.pelims
thf(fact_1233_pred__def,axiom,
( pred
= ( case_nat_nat @ zero_zero_nat
@ ^ [X23: nat] : X23 ) ) ).
% pred_def
thf(fact_1234_nat0__intermed__int__val,axiom,
! [N: nat,F: nat > int,K2: int] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I2 @ one_one_nat ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K2 )
=> ( ( ord_less_eq_int @ K2 @ ( F @ N ) )
=> ? [I2: nat] :
( ( ord_less_eq_nat @ I2 @ N )
& ( ( F @ I2 )
= K2 ) ) ) ) ) ).
% nat0_intermed_int_val
thf(fact_1235_zabs__less__one__iff,axiom,
! [Z3: int] :
( ( ord_less_int @ ( abs_abs_int @ Z3 ) @ one_one_int )
= ( Z3 = zero_zero_int ) ) ).
% zabs_less_one_iff
thf(fact_1236_nat__intermed__int__val,axiom,
! [M: nat,N: nat,F: nat > int,K2: int] :
( ! [I2: nat] :
( ( ( ord_less_eq_nat @ M @ I2 )
& ( ord_less_nat @ I2 @ N ) )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_int @ ( F @ M ) @ K2 )
=> ( ( ord_less_eq_int @ K2 @ ( F @ N ) )
=> ? [I2: nat] :
( ( ord_less_eq_nat @ M @ I2 )
& ( ord_less_eq_nat @ I2 @ N )
& ( ( F @ I2 )
= K2 ) ) ) ) ) ) ).
% nat_intermed_int_val
thf(fact_1237_nat__ivt__aux,axiom,
! [N: nat,F: nat > int,K2: int] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K2 )
=> ( ( ord_less_eq_int @ K2 @ ( F @ N ) )
=> ? [I2: nat] :
( ( ord_less_eq_nat @ I2 @ N )
& ( ( F @ I2 )
= K2 ) ) ) ) ) ).
% nat_ivt_aux
thf(fact_1238_execl_Opelims_I1_J,axiom,
! [X4: list_instr,Xa: list_P419547167332244514st_int,Y: $o] :
( ( ( execl @ X4 @ Xa )
= Y )
=> ( ( accp_P1482788143903795053st_int @ execl_rel @ ( produc7764413618310988720st_int @ X4 @ Xa ) )
=> ( ! [Cf2: produc6425607678544837394st_int,Cf3: produc6425607678544837394st_int,Cfs4: list_P419547167332244514st_int] :
( ( Xa
= ( cons_P4151473878978182108st_int @ Cf2 @ ( cons_P4151473878978182108st_int @ Cf3 @ Cfs4 ) ) )
=> ( ( Y
= ( ( exec1 @ X4 @ Cf2 @ Cf3 )
& ( execl @ X4 @ ( cons_P4151473878978182108st_int @ Cf3 @ Cfs4 ) ) ) )
=> ~ ( accp_P1482788143903795053st_int @ execl_rel @ ( produc7764413618310988720st_int @ X4 @ ( cons_P4151473878978182108st_int @ Cf2 @ ( cons_P4151473878978182108st_int @ Cf3 @ Cfs4 ) ) ) ) ) )
=> ( ( ( Xa = nil_Pr749310521696113548st_int )
=> ( Y
=> ~ ( accp_P1482788143903795053st_int @ execl_rel @ ( produc7764413618310988720st_int @ X4 @ nil_Pr749310521696113548st_int ) ) ) )
=> ~ ! [V: produc6425607678544837394st_int] :
( ( Xa
= ( cons_P4151473878978182108st_int @ V @ nil_Pr749310521696113548st_int ) )
=> ( Y
=> ~ ( accp_P1482788143903795053st_int @ execl_rel @ ( produc7764413618310988720st_int @ X4 @ ( cons_P4151473878978182108st_int @ V @ nil_Pr749310521696113548st_int ) ) ) ) ) ) ) ) ) ).
% execl.pelims(1)
thf(fact_1239_execl_Opelims_I3_J,axiom,
! [X4: list_instr,Xa: list_P419547167332244514st_int] :
( ~ ( execl @ X4 @ Xa )
=> ( ( accp_P1482788143903795053st_int @ execl_rel @ ( produc7764413618310988720st_int @ X4 @ Xa ) )
=> ~ ! [Cf2: produc6425607678544837394st_int,Cf3: produc6425607678544837394st_int,Cfs4: list_P419547167332244514st_int] :
( ( Xa
= ( cons_P4151473878978182108st_int @ Cf2 @ ( cons_P4151473878978182108st_int @ Cf3 @ Cfs4 ) ) )
=> ( ( accp_P1482788143903795053st_int @ execl_rel @ ( produc7764413618310988720st_int @ X4 @ ( cons_P4151473878978182108st_int @ Cf2 @ ( cons_P4151473878978182108st_int @ Cf3 @ Cfs4 ) ) ) )
=> ( ( exec1 @ X4 @ Cf2 @ Cf3 )
& ( execl @ X4 @ ( cons_P4151473878978182108st_int @ Cf3 @ Cfs4 ) ) ) ) ) ) ) ).
% execl.pelims(3)
thf(fact_1240_execl_Opelims_I2_J,axiom,
! [X4: list_instr,Xa: list_P419547167332244514st_int] :
( ( execl @ X4 @ Xa )
=> ( ( accp_P1482788143903795053st_int @ execl_rel @ ( produc7764413618310988720st_int @ X4 @ Xa ) )
=> ( ! [Cf2: produc6425607678544837394st_int,Cf3: produc6425607678544837394st_int,Cfs4: list_P419547167332244514st_int] :
( ( Xa
= ( cons_P4151473878978182108st_int @ Cf2 @ ( cons_P4151473878978182108st_int @ Cf3 @ Cfs4 ) ) )
=> ( ( accp_P1482788143903795053st_int @ execl_rel @ ( produc7764413618310988720st_int @ X4 @ ( cons_P4151473878978182108st_int @ Cf2 @ ( cons_P4151473878978182108st_int @ Cf3 @ Cfs4 ) ) ) )
=> ~ ( ( exec1 @ X4 @ Cf2 @ Cf3 )
& ( execl @ X4 @ ( cons_P4151473878978182108st_int @ Cf3 @ Cfs4 ) ) ) ) )
=> ( ( ( Xa = nil_Pr749310521696113548st_int )
=> ~ ( accp_P1482788143903795053st_int @ execl_rel @ ( produc7764413618310988720st_int @ X4 @ nil_Pr749310521696113548st_int ) ) )
=> ~ ! [V: produc6425607678544837394st_int] :
( ( Xa
= ( cons_P4151473878978182108st_int @ V @ nil_Pr749310521696113548st_int ) )
=> ~ ( accp_P1482788143903795053st_int @ execl_rel @ ( produc7764413618310988720st_int @ X4 @ ( cons_P4151473878978182108st_int @ V @ nil_Pr749310521696113548st_int ) ) ) ) ) ) ) ) ).
% execl.pelims(2)
thf(fact_1241_execl__all__SKIP,axiom,
! [Cfs: list_P419547167332244514st_int] :
( ( execl_all @ nil_instr @ Cfs )
=> ( ( cpred @ skip @ ( nth_Pr8327173284404383691st_int @ Cfs @ zero_zero_nat ) @ ( nth_Pr8327173284404383691st_int @ Cfs @ ( minus_minus_nat @ ( size_s229347708945616566st_int @ Cfs ) @ ( suc @ zero_zero_nat ) ) ) )
& ( ( size_s229347708945616566st_int @ Cfs )
= ( suc @ zero_zero_nat ) ) ) ) ).
% execl_all_SKIP
thf(fact_1242_com_Osize_I6_J,axiom,
( ( size_size_com @ skip )
= zero_zero_nat ) ).
% com.size(6)
thf(fact_1243_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M: nat] :
( ! [K3: nat] :
( ( ord_less_nat @ N @ K3 )
=> ( P @ K3 ) )
=> ( ! [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
=> ( ! [I4: nat] :
( ( ord_less_nat @ K3 @ I4 )
=> ( P @ I4 ) )
=> ( P @ K3 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_1244_upto_Opinduct,axiom,
! [A0: int,A1: int,P: int > int > $o] :
( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ A0 @ A1 ) )
=> ( ! [I2: int,J2: int] :
( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I2 @ J2 ) )
=> ( ( ( ord_less_eq_int @ I2 @ J2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) @ J2 ) )
=> ( P @ I2 @ J2 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% upto.pinduct
thf(fact_1245_upto_Opsimps,axiom,
! [I: int,J: int] :
( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I @ J ) )
=> ( ( ( ord_less_eq_int @ I @ J )
=> ( ( upto @ I @ J )
= ( cons_int @ I @ ( upto @ ( plus_plus_int @ I @ one_one_int ) @ J ) ) ) )
& ( ~ ( ord_less_eq_int @ I @ J )
=> ( ( upto @ I @ J )
= nil_int ) ) ) ) ).
% upto.psimps
thf(fact_1246_upto_Opelims,axiom,
! [X4: int,Xa: int,Y: list_int] :
( ( ( upto @ X4 @ Xa )
= Y )
=> ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X4 @ Xa ) )
=> ~ ( ( ( ( ord_less_eq_int @ X4 @ Xa )
=> ( Y
= ( cons_int @ X4 @ ( upto @ ( plus_plus_int @ X4 @ one_one_int ) @ Xa ) ) ) )
& ( ~ ( ord_less_eq_int @ X4 @ Xa )
=> ( Y = nil_int ) ) )
=> ~ ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X4 @ Xa ) ) ) ) ) ).
% upto.pelims
thf(fact_1247_upto__Nil,axiom,
! [I: int,J: int] :
( ( ( upto @ I @ J )
= nil_int )
= ( ord_less_int @ J @ I ) ) ).
% upto_Nil
thf(fact_1248_upto__Nil2,axiom,
! [I: int,J: int] :
( ( nil_int
= ( upto @ I @ J ) )
= ( ord_less_int @ J @ I ) ) ).
% upto_Nil2
thf(fact_1249_upto__empty,axiom,
! [J: int,I: int] :
( ( ord_less_int @ J @ I )
=> ( ( upto @ I @ J )
= nil_int ) ) ).
% upto_empty
thf(fact_1250_upto__single,axiom,
! [I: int] :
( ( upto @ I @ I )
= ( cons_int @ I @ nil_int ) ) ).
% upto_single
thf(fact_1251_nth__upto,axiom,
! [I: int,K2: nat,J: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I @ ( semiri1314217659103216013at_int @ K2 ) ) @ J )
=> ( ( nth_int @ ( upto @ I @ J ) @ K2 )
= ( plus_plus_int @ I @ ( semiri1314217659103216013at_int @ K2 ) ) ) ) ).
% nth_upto
thf(fact_1252_upto__code,axiom,
( upto
= ( ^ [I3: int,J3: int] : ( upto_aux @ I3 @ J3 @ nil_int ) ) ) ).
% upto_code
thf(fact_1253_upto__aux__def,axiom,
( upto_aux
= ( ^ [I3: int,J3: int] : ( append_int @ ( upto @ I3 @ J3 ) ) ) ) ).
% upto_aux_def
thf(fact_1254_upto__split2,axiom,
! [I: int,J: int,K2: int] :
( ( ord_less_eq_int @ I @ J )
=> ( ( ord_less_eq_int @ J @ K2 )
=> ( ( upto @ I @ K2 )
= ( append_int @ ( upto @ I @ J ) @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K2 ) ) ) ) ) ).
% upto_split2
thf(fact_1255_upto__split1,axiom,
! [I: int,J: int,K2: int] :
( ( ord_less_eq_int @ I @ J )
=> ( ( ord_less_eq_int @ J @ K2 )
=> ( ( upto @ I @ K2 )
= ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( upto @ J @ K2 ) ) ) ) ) ).
% upto_split1
thf(fact_1256_upto_Oelims,axiom,
! [X4: int,Xa: int,Y: list_int] :
( ( ( upto @ X4 @ Xa )
= Y )
=> ( ( ( ord_less_eq_int @ X4 @ Xa )
=> ( Y
= ( cons_int @ X4 @ ( upto @ ( plus_plus_int @ X4 @ one_one_int ) @ Xa ) ) ) )
& ( ~ ( ord_less_eq_int @ X4 @ Xa )
=> ( Y = nil_int ) ) ) ) ).
% upto.elims
thf(fact_1257_upto_Osimps,axiom,
( upto
= ( ^ [I3: int,J3: int] : ( if_list_int @ ( ord_less_eq_int @ I3 @ J3 ) @ ( cons_int @ I3 @ ( upto @ ( plus_plus_int @ I3 @ one_one_int ) @ J3 ) ) @ nil_int ) ) ) ).
% upto.simps
thf(fact_1258_upto__rec1,axiom,
! [I: int,J: int] :
( ( ord_less_eq_int @ I @ J )
=> ( ( upto @ I @ J )
= ( cons_int @ I @ ( upto @ ( plus_plus_int @ I @ one_one_int ) @ J ) ) ) ) ).
% upto_rec1
thf(fact_1259_upto__rec2,axiom,
! [I: int,J: int] :
( ( ord_less_eq_int @ I @ J )
=> ( ( upto @ I @ J )
= ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ nil_int ) ) ) ) ).
% upto_rec2
thf(fact_1260_upto__split3,axiom,
! [I: int,J: int,K2: int] :
( ( ord_less_eq_int @ I @ J )
=> ( ( ord_less_eq_int @ J @ K2 )
=> ( ( upto @ I @ K2 )
= ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K2 ) ) ) ) ) ) ).
% upto_split3
thf(fact_1261_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_1262_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_1263_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,
! [X4: int,Y: int] :
( ( if_int @ $false @ X4 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X4: int,Y: int] :
( ( if_int @ $true @ X4 @ Y )
= X4 ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X4: nat,Y: nat] :
( ( if_nat @ $false @ X4 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X4: nat,Y: nat] :
( ( if_nat @ $true @ X4 @ Y )
= X4 ) ).
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,
! [X4: list_int,Y: list_int] :
( ( if_list_int @ $false @ X4 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
! [X4: list_int,Y: list_int] :
( ( if_list_int @ $true @ X4 @ Y )
= X4 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( 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
@ ( lattic8721135487736765967in_nat
@ ( collect_nat
@ ^ [K: nat] :
( ( member_nat @ K @ ( set_or4665077453230672383an_nat @ k @ ( size_s229347708945616566st_int @ cfs ) ) )
& ~ ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ p ) ) @ ( produc8941309049508465880st_int @ ( nth_Pr8327173284404383691st_int @ cfs @ K ) ) )
& ( ord_less_int @ ( minus_minus_int @ ( produc8941309049508465880st_int @ ( nth_Pr8327173284404383691st_int @ cfs @ K ) ) @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ p ) ) ) @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ ( p2 @ x ) ) ) ) ) ) ) ) )
@ 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
@ ( lattic8721135487736765967in_nat
@ ( collect_nat
@ ^ [K: nat] :
( ( member_nat @ K @ ( set_or4665077453230672383an_nat @ k @ ( size_s229347708945616566st_int @ cfs ) ) )
& ~ ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ p ) ) @ ( produc8941309049508465880st_int @ ( nth_Pr8327173284404383691st_int @ cfs @ K ) ) )
& ( ord_less_int @ ( minus_minus_int @ ( produc8941309049508465880st_int @ ( nth_Pr8327173284404383691st_int @ cfs @ K ) ) @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ p ) ) ) @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ ( p2 @ x ) ) ) ) ) ) ) ) )
@ cfs ) ) ) )
@ one_one_nat ) )
= ( produc5086643055186798020st_int
@ ( minus_minus_int
@ ( produc8941309049508465880st_int
@ ( nth_Pr8327173284404383691st_int @ cfs
@ ( lattic8721135487736765967in_nat
@ ( collect_nat
@ ^ [K: nat] :
( ( member_nat @ K @ ( set_or4665077453230672383an_nat @ k @ ( size_s229347708945616566st_int @ cfs ) ) )
& ~ ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ p ) ) @ ( produc8941309049508465880st_int @ ( nth_Pr8327173284404383691st_int @ cfs @ K ) ) )
& ( ord_less_int @ ( minus_minus_int @ ( produc8941309049508465880st_int @ ( nth_Pr8327173284404383691st_int @ cfs @ K ) ) @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ p ) ) ) @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ ( p2 @ x ) ) ) ) ) ) ) ) ) )
@ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ p ) ) )
@ ( produc1296517004818028822st_int
@ ( nth_Pr8327173284404383691st_int @ cfs
@ ( lattic8721135487736765967in_nat
@ ( collect_nat
@ ^ [K: nat] :
( ( member_nat @ K @ ( set_or4665077453230672383an_nat @ k @ ( size_s229347708945616566st_int @ cfs ) ) )
& ~ ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ p ) ) @ ( produc8941309049508465880st_int @ ( nth_Pr8327173284404383691st_int @ cfs @ K ) ) )
& ( ord_less_int @ ( minus_minus_int @ ( produc8941309049508465880st_int @ ( nth_Pr8327173284404383691st_int @ cfs @ K ) ) @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ p ) ) ) @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ ( p2 @ x ) ) ) ) ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------