TPTP Problem File: SLH0327^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : IMP_Compiler_Reuse/0006_Compiler2/prob_00323_016874__6082532_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1485 ( 674 unt; 217 typ; 0 def)
% Number of atoms : 3327 (1688 equ; 0 cnn)
% Maximal formula atoms : 14 ( 2 avg)
% Number of connectives : 10647 ( 437 ~; 70 |; 244 &;8600 @)
% ( 0 <=>;1296 =>; 0 <=; 0 <~>)
% Maximal formula depth : 30 ( 6 avg)
% Number of types : 42 ( 41 usr)
% Number of type conns : 889 ( 889 >; 0 *; 0 +; 0 <<)
% Number of symbols : 179 ( 176 usr; 20 con; 0-4 aty)
% Number of variables : 3721 ( 270 ^;3295 !; 156 ?;3721 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 16:49:18.675
%------------------------------------------------------------------------------
% Could-be-implicit typings (41)
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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_c_member_001t__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__Int__Oint_J,type,
member3599195040465536738nt_int: produc4435102495419491129nt_int > set_Pr5290250543098163609nt_int > $o ).
thf(sy_c_member_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,
member6299553290643319410st_int: produc369741286924889929st_int > set_Pr5580431329626101545st_int > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
member5262025264175285858nt_int: product_prod_int_int > set_Pr958786334691620121nt_int > $o ).
thf(sy_c_member_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,
member9100552558438320297st_int: produc6425607678544837394st_int > set_Pr2116125220558876360st_int > $o ).
thf(sy_c_member_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,
member7220921130490543508nt_int: produc4189061985984568957nt_int > set_Pr4358015327039922227nt_int > $o ).
thf(sy_v_P,type,
p: list_instr ).
thf(sy_v_P_H,type,
p2: a > list_instr ).
thf(sy_v_P_H_H,type,
p3: list_instr ).
thf(sy_v_Q,type,
q: a > produc6425607678544837394st_int > produc6425607678544837394st_int > $o ).
thf(sy_v_cfs,type,
cfs: list_P419547167332244514st_int ).
thf(sy_v_k,type,
k: nat ).
thf(sy_v_thesis____,type,
thesis: $o ).
thf(sy_v_x,type,
x: a ).
% Relevant facts (1259)
thf(fact_0_C,axiom,
( ( produc8941309049508465880st_int @ ( nth_Pr8327173284404383691st_int @ cfs @ k ) )
= ( semiri1314217659103216013at_int @ ( size_size_list_instr @ p ) ) ) ).
% C
thf(fact_1_F,axiom,
! [Cfs: list_P419547167332244514st_int] :
( ( execl_all @ ( p2 @ x ) @ Cfs )
=> ( mpred @ ( p2 @ x ) @ Cfs @ zero_zero_nat @ ( minus_minus_nat @ ( size_s229347708945616566st_int @ Cfs ) @ one_one_nat ) ) ) ).
% F
thf(fact_2_E,axiom,
! [Cfs: list_P419547167332244514st_int] :
( ( execl_all @ ( p2 @ x ) @ Cfs )
=> ( q @ x @ ( nth_Pr8327173284404383691st_int @ Cfs @ zero_zero_nat ) @ ( nth_Pr8327173284404383691st_int @ Cfs @ ( minus_minus_nat @ ( size_s229347708945616566st_int @ Cfs ) @ one_one_nat ) ) ) ) ).
% E
thf(fact_3__092_060open_062_092_060exists_062k_H_092_060in_062_123k_O_O_060length_Acfs_125_O_AP_H_Ax_A_092_060Turnstile_062_Amap_A_Ioff_AP_J_A_Idrop_Ak_A_Itake_A_ISuc_Ak_H_J_Acfs_J_J_092_060box_062_092_060close_062,axiom,
? [X: nat] :
( ( member_nat @ X @ ( set_or4665077453230672383an_nat @ k @ ( size_s229347708945616566st_int @ cfs ) ) )
& ( execl_all @ ( p2 @ x )
@ ( map_Pr8625093460839654122st_int
@ ^ [Cf: produc6425607678544837394st_int] : ( produc5086643055186798020st_int @ ( minus_minus_int @ ( produc8941309049508465880st_int @ Cf ) @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ p ) ) ) @ ( produc1296517004818028822st_int @ Cf ) )
@ ( drop_P7838799151754935884st_int @ k @ ( take_P1004856038796978596st_int @ ( suc @ X ) @ cfs ) ) ) ) ) ).
% \<open>\<exists>k'\<in>{k..<length cfs}. P' x \<Turnstile> map (off P) (drop k (take (Suc k') cfs))\<box>\<close>
thf(fact_4_B,axiom,
ord_less_nat @ k @ ( size_s229347708945616566st_int @ cfs ) ).
% B
thf(fact_5_A,axiom,
execl_all @ ( append_instr @ p @ ( append_instr @ ( p2 @ x ) @ p3 ) ) @ cfs ).
% A
thf(fact_6_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_7_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_8_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_9_length__drop,axiom,
! [N: nat,Xs: list_int] :
( ( size_size_list_int @ ( drop_int @ N @ Xs ) )
= ( minus_minus_nat @ ( size_size_list_int @ Xs ) @ N ) ) ).
% length_drop
thf(fact_10_prod_Ocollapse,axiom,
! [Prod: produc4189061985984568957nt_int] :
( ( produc3965054194175396271nt_int @ ( produc811350159336358595nt_int @ Prod ) @ ( produc30867750519413249nt_int @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_11_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_12_prod_Ocollapse,axiom,
! [Prod: produc4435102495419491129nt_int] :
( ( produc5790713362662368625nt_int @ ( produc9025023058967219549nt_int @ Prod ) @ ( produc179445755590504351nt_int @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_13_prod_Ocollapse,axiom,
! [Prod: produc6425607678544837394st_int] :
( ( produc5086643055186798020st_int @ ( produc8941309049508465880st_int @ Prod ) @ ( produc1296517004818028822st_int @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_14_prod_Ocollapse,axiom,
! [Prod: produc369741286924889929st_int] :
( ( produc8650753666468850689st_int @ ( produc2155683374303800813st_int @ Prod ) @ ( produc6244060863719409455st_int @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_15_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_16_of__nat__1,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% of_nat_1
thf(fact_17_of__nat__1,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% of_nat_1
thf(fact_18_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_19_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_20_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_21_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_22_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_23_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_24_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_25_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_26_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_27_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_28_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_29_drop0,axiom,
( ( drop_P7838799151754935884st_int @ zero_zero_nat )
= ( ^ [X2: list_P419547167332244514st_int] : X2 ) ) ).
% drop0
thf(fact_30_drop0,axiom,
( ( drop_int @ zero_zero_nat )
= ( ^ [X2: list_int] : X2 ) ) ).
% drop0
thf(fact_31_drop0,axiom,
( ( drop_list_char @ zero_zero_nat )
= ( ^ [X2: list_list_char] : X2 ) ) ).
% drop0
thf(fact_32_drop0,axiom,
( ( drop_instr @ zero_zero_nat )
= ( ^ [X2: list_instr] : X2 ) ) ).
% drop0
thf(fact_33_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_34_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_35_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_36_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_37_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_38_prod_Oinject,axiom,
! [X1: list_list_char,X22: produc4435102495419491129nt_int,Y1: list_list_char,Y2: produc4435102495419491129nt_int] :
( ( ( produc3965054194175396271nt_int @ X1 @ X22 )
= ( produc3965054194175396271nt_int @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_39_prod_Oinject,axiom,
! [X1: int,X22: produc369741286924889929st_int,Y1: int,Y2: produc369741286924889929st_int] :
( ( ( produc5086643055186798020st_int @ X1 @ X22 )
= ( produc5086643055186798020st_int @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_40_prod_Oinject,axiom,
! [X1: int,X22: int,Y1: int,Y2: int] :
( ( ( product_Pair_int_int @ X1 @ X22 )
= ( product_Pair_int_int @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_41_prod_Oinject,axiom,
! [X1: list_char > int,X22: list_int,Y1: list_char > int,Y2: list_int] :
( ( ( produc8650753666468850689st_int @ X1 @ X22 )
= ( produc8650753666468850689st_int @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_42_prod_Oinject,axiom,
! [X1: list_char > int,X22: int,Y1: list_char > int,Y2: int] :
( ( ( produc5790713362662368625nt_int @ X1 @ X22 )
= ( produc5790713362662368625nt_int @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_43_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_44_nat_Oinject,axiom,
! [X22: nat,Y2: nat] :
( ( ( suc @ X22 )
= ( suc @ Y2 ) )
= ( X22 = Y2 ) ) ).
% nat.inject
thf(fact_45_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_46_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1316708129612266289at_nat @ M )
= ( semiri1316708129612266289at_nat @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_47_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_48_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_49_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_50_append__assoc,axiom,
! [Xs: list_instr,Ys: list_instr,Zs: list_instr] :
( ( append_instr @ ( append_instr @ Xs @ Ys ) @ Zs )
= ( append_instr @ Xs @ ( append_instr @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_51_append__assoc,axiom,
! [Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int,Zs: list_P419547167332244514st_int] :
( ( append5028292981072010935st_int @ ( append5028292981072010935st_int @ Xs @ Ys ) @ Zs )
= ( append5028292981072010935st_int @ Xs @ ( append5028292981072010935st_int @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_52_append__assoc,axiom,
! [Xs: list_int,Ys: list_int,Zs: list_int] :
( ( append_int @ ( append_int @ Xs @ Ys ) @ Zs )
= ( append_int @ Xs @ ( append_int @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_53_append__same__eq,axiom,
! [Ys: list_instr,Xs: list_instr,Zs: list_instr] :
( ( ( append_instr @ Ys @ Xs )
= ( append_instr @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_54_append__same__eq,axiom,
! [Ys: list_P419547167332244514st_int,Xs: list_P419547167332244514st_int,Zs: list_P419547167332244514st_int] :
( ( ( append5028292981072010935st_int @ Ys @ Xs )
= ( append5028292981072010935st_int @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_55_append__same__eq,axiom,
! [Ys: list_int,Xs: list_int,Zs: list_int] :
( ( ( append_int @ Ys @ Xs )
= ( append_int @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_56_same__append__eq,axiom,
! [Xs: list_instr,Ys: list_instr,Zs: list_instr] :
( ( ( append_instr @ Xs @ Ys )
= ( append_instr @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_57_same__append__eq,axiom,
! [Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int,Zs: list_P419547167332244514st_int] :
( ( ( append5028292981072010935st_int @ Xs @ Ys )
= ( append5028292981072010935st_int @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_58_same__append__eq,axiom,
! [Xs: list_int,Ys: list_int,Zs: list_int] :
( ( ( append_int @ Xs @ Ys )
= ( append_int @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_59_map__ident,axiom,
( ( map_Pr8625093460839654122st_int
@ ^ [X2: produc6425607678544837394st_int] : X2 )
= ( ^ [Xs2: list_P419547167332244514st_int] : Xs2 ) ) ).
% map_ident
thf(fact_60_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_61_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_62_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_63_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_64_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_65_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_66_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_67_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_68_Suc__diff__diff,axiom,
! [M: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_69_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_70_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_71_length__map,axiom,
! [F: int > instr,Xs: list_int] :
( ( size_size_list_instr @ ( map_int_instr @ F @ Xs ) )
= ( size_size_list_int @ Xs ) ) ).
% length_map
thf(fact_72_length__map,axiom,
! [F: instr > int,Xs: list_instr] :
( ( size_size_list_int @ ( map_instr_int @ F @ Xs ) )
= ( size_size_list_instr @ Xs ) ) ).
% length_map
thf(fact_73_length__map,axiom,
! [F: int > int,Xs: list_int] :
( ( size_size_list_int @ ( map_int_int @ F @ Xs ) )
= ( size_size_list_int @ Xs ) ) ).
% length_map
thf(fact_74_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_75_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_76_length__map,axiom,
! [F: int > list_char,Xs: list_int] :
( ( size_s356637359517785349t_char @ ( map_int_list_char @ F @ Xs ) )
= ( size_size_list_int @ Xs ) ) ).
% length_map
thf(fact_77_length__map,axiom,
! [F: list_char > int,Xs: list_list_char] :
( ( size_size_list_int @ ( map_list_char_int @ F @ Xs ) )
= ( size_s356637359517785349t_char @ Xs ) ) ).
% length_map
thf(fact_78_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_79_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_80_append__eq__append__conv,axiom,
! [Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int,Us: list_P419547167332244514st_int,Vs: list_P419547167332244514st_int] :
( ( ( ( size_s229347708945616566st_int @ Xs )
= ( size_s229347708945616566st_int @ Ys ) )
| ( ( size_s229347708945616566st_int @ Us )
= ( size_s229347708945616566st_int @ Vs ) ) )
=> ( ( ( append5028292981072010935st_int @ Xs @ Us )
= ( append5028292981072010935st_int @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_81_append__eq__append__conv,axiom,
! [Xs: list_instr,Ys: list_instr,Us: list_instr,Vs: list_instr] :
( ( ( ( size_size_list_instr @ Xs )
= ( size_size_list_instr @ Ys ) )
| ( ( size_size_list_instr @ Us )
= ( size_size_list_instr @ Vs ) ) )
=> ( ( ( append_instr @ Xs @ Us )
= ( append_instr @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_82_append__eq__append__conv,axiom,
! [Xs: list_list_char,Ys: list_list_char,Us: list_list_char,Vs: list_list_char] :
( ( ( ( size_s356637359517785349t_char @ Xs )
= ( size_s356637359517785349t_char @ Ys ) )
| ( ( size_s356637359517785349t_char @ Us )
= ( size_s356637359517785349t_char @ Vs ) ) )
=> ( ( ( append_list_char @ Xs @ Us )
= ( append_list_char @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_83_append__eq__append__conv,axiom,
! [Xs: list_int,Ys: list_int,Us: list_int,Vs: list_int] :
( ( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys ) )
| ( ( size_size_list_int @ Us )
= ( size_size_list_int @ Vs ) ) )
=> ( ( ( append_int @ Xs @ Us )
= ( append_int @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_84_map__append,axiom,
! [F: instr > instr,Xs: list_instr,Ys: list_instr] :
( ( map_instr_instr @ F @ ( append_instr @ Xs @ Ys ) )
= ( append_instr @ ( map_instr_instr @ F @ Xs ) @ ( map_instr_instr @ F @ Ys ) ) ) ).
% map_append
thf(fact_85_map__append,axiom,
! [F: instr > produc6425607678544837394st_int,Xs: list_instr,Ys: list_instr] :
( ( map_in7259965606695656109st_int @ F @ ( append_instr @ Xs @ Ys ) )
= ( append5028292981072010935st_int @ ( map_in7259965606695656109st_int @ F @ Xs ) @ ( map_in7259965606695656109st_int @ F @ Ys ) ) ) ).
% map_append
thf(fact_86_map__append,axiom,
! [F: instr > int,Xs: list_instr,Ys: list_instr] :
( ( map_instr_int @ F @ ( append_instr @ Xs @ Ys ) )
= ( append_int @ ( map_instr_int @ F @ Xs ) @ ( map_instr_int @ F @ Ys ) ) ) ).
% map_append
thf(fact_87_map__append,axiom,
! [F: produc6425607678544837394st_int > instr,Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int] :
( ( map_Pr7514435936640407687_instr @ F @ ( append5028292981072010935st_int @ Xs @ Ys ) )
= ( append_instr @ ( map_Pr7514435936640407687_instr @ F @ Xs ) @ ( map_Pr7514435936640407687_instr @ F @ Ys ) ) ) ).
% map_append
thf(fact_88_map__append,axiom,
! [F: produc6425607678544837394st_int > int,Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int] :
( ( map_Pr3224476576807795204nt_int @ F @ ( append5028292981072010935st_int @ Xs @ Ys ) )
= ( append_int @ ( map_Pr3224476576807795204nt_int @ F @ Xs ) @ ( map_Pr3224476576807795204nt_int @ F @ Ys ) ) ) ).
% map_append
thf(fact_89_map__append,axiom,
! [F: int > instr,Xs: list_int,Ys: list_int] :
( ( map_int_instr @ F @ ( append_int @ Xs @ Ys ) )
= ( append_instr @ ( map_int_instr @ F @ Xs ) @ ( map_int_instr @ F @ Ys ) ) ) ).
% map_append
thf(fact_90_map__append,axiom,
! [F: int > produc6425607678544837394st_int,Xs: list_int,Ys: list_int] :
( ( map_in8512336359594627332st_int @ F @ ( append_int @ Xs @ Ys ) )
= ( append5028292981072010935st_int @ ( map_in8512336359594627332st_int @ F @ Xs ) @ ( map_in8512336359594627332st_int @ F @ Ys ) ) ) ).
% map_append
thf(fact_91_map__append,axiom,
! [F: int > int,Xs: list_int,Ys: list_int] :
( ( map_int_int @ F @ ( append_int @ Xs @ Ys ) )
= ( append_int @ ( map_int_int @ F @ Xs ) @ ( map_int_int @ F @ Ys ) ) ) ).
% map_append
thf(fact_92_map__append,axiom,
! [F: produc6425607678544837394st_int > produc6425607678544837394st_int,Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int] :
( ( map_Pr8625093460839654122st_int @ F @ ( append5028292981072010935st_int @ Xs @ Ys ) )
= ( append5028292981072010935st_int @ ( map_Pr8625093460839654122st_int @ F @ Xs ) @ ( map_Pr8625093460839654122st_int @ F @ Ys ) ) ) ).
% map_append
thf(fact_93_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_94_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_95_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_96_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_97_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_98_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_99_mem__Collect__eq,axiom,
! [A: int,P: int > $o] :
( ( member_int @ A @ ( collect_int @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_100_Collect__mem__eq,axiom,
! [A3: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_101_Collect__mem__eq,axiom,
! [A3: set_int] :
( ( collect_int
@ ^ [X2: int] : ( member_int @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_102_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_103_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_104_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_105_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_106_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_107_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_108_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_109_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_110_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_111_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_112_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_113_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_114_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_115_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_116_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_117_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_118_take__append,axiom,
! [N: nat,Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int] :
( ( take_P1004856038796978596st_int @ N @ ( append5028292981072010935st_int @ Xs @ Ys ) )
= ( append5028292981072010935st_int @ ( take_P1004856038796978596st_int @ N @ Xs ) @ ( take_P1004856038796978596st_int @ ( minus_minus_nat @ N @ ( size_s229347708945616566st_int @ Xs ) ) @ Ys ) ) ) ).
% take_append
thf(fact_119_take__append,axiom,
! [N: nat,Xs: list_instr,Ys: list_instr] :
( ( take_instr @ N @ ( append_instr @ Xs @ Ys ) )
= ( append_instr @ ( take_instr @ N @ Xs ) @ ( take_instr @ ( minus_minus_nat @ N @ ( size_size_list_instr @ Xs ) ) @ Ys ) ) ) ).
% take_append
thf(fact_120_take__append,axiom,
! [N: nat,Xs: list_list_char,Ys: list_list_char] :
( ( take_list_char @ N @ ( append_list_char @ Xs @ Ys ) )
= ( append_list_char @ ( take_list_char @ N @ Xs ) @ ( take_list_char @ ( minus_minus_nat @ N @ ( size_s356637359517785349t_char @ Xs ) ) @ Ys ) ) ) ).
% take_append
thf(fact_121_take__append,axiom,
! [N: nat,Xs: list_int,Ys: list_int] :
( ( take_int @ N @ ( append_int @ Xs @ Ys ) )
= ( append_int @ ( take_int @ N @ Xs ) @ ( take_int @ ( minus_minus_nat @ N @ ( size_size_list_int @ Xs ) ) @ Ys ) ) ) ).
% take_append
thf(fact_122_drop__append,axiom,
! [N: nat,Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int] :
( ( drop_P7838799151754935884st_int @ N @ ( append5028292981072010935st_int @ Xs @ Ys ) )
= ( append5028292981072010935st_int @ ( drop_P7838799151754935884st_int @ N @ Xs ) @ ( drop_P7838799151754935884st_int @ ( minus_minus_nat @ N @ ( size_s229347708945616566st_int @ Xs ) ) @ Ys ) ) ) ).
% drop_append
thf(fact_123_drop__append,axiom,
! [N: nat,Xs: list_instr,Ys: list_instr] :
( ( drop_instr @ N @ ( append_instr @ Xs @ Ys ) )
= ( append_instr @ ( drop_instr @ N @ Xs ) @ ( drop_instr @ ( minus_minus_nat @ N @ ( size_size_list_instr @ Xs ) ) @ Ys ) ) ) ).
% drop_append
thf(fact_124_drop__append,axiom,
! [N: nat,Xs: list_list_char,Ys: list_list_char] :
( ( drop_list_char @ N @ ( append_list_char @ Xs @ Ys ) )
= ( append_list_char @ ( drop_list_char @ N @ Xs ) @ ( drop_list_char @ ( minus_minus_nat @ N @ ( size_s356637359517785349t_char @ Xs ) ) @ Ys ) ) ) ).
% drop_append
thf(fact_125_drop__append,axiom,
! [N: nat,Xs: list_int,Ys: list_int] :
( ( drop_int @ N @ ( append_int @ Xs @ Ys ) )
= ( append_int @ ( drop_int @ N @ Xs ) @ ( drop_int @ ( minus_minus_nat @ N @ ( size_size_list_int @ Xs ) ) @ Ys ) ) ) ).
% drop_append
thf(fact_126_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_127_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_128_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_129_D,axiom,
ord_less_eq_nat @ ( size_size_list_int @ ( produc6244060863719409455st_int @ ( produc1296517004818028822st_int @ ( nth_Pr8327173284404383691st_int @ cfs @ zero_zero_nat ) ) ) ) @ ( size_size_list_int @ ( produc6244060863719409455st_int @ ( produc1296517004818028822st_int @ ( nth_Pr8327173284404383691st_int @ cfs @ k ) ) ) ) ).
% D
thf(fact_130_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N2 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_131_lift__Suc__mono__less,axiom,
! [F: nat > int,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N2 )
=> ( ord_less_int @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_132_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_133_lift__Suc__mono__less__iff,axiom,
! [F: nat > int,N: nat,M: nat] :
( ! [N3: nat] : ( ord_less_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_int @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_134_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_135_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_136_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_137_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_138_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_139_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_140_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_141_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_142_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_143_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M2: nat] :
( ( ord_less_nat @ M2 @ N3 )
=> ( P @ M2 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_144_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N3 )
& ~ ( P @ M2 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_145_linorder__neqE__nat,axiom,
! [X3: nat,Y: nat] :
( ( X3 != Y )
=> ( ~ ( ord_less_nat @ X3 @ Y )
=> ( ord_less_nat @ Y @ X3 ) ) ) ).
% linorder_neqE_nat
thf(fact_146_append__eq__appendI,axiom,
! [Xs: list_instr,Xs1: list_instr,Zs: list_instr,Ys: list_instr,Us: list_instr] :
( ( ( append_instr @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_instr @ Xs1 @ Us ) )
=> ( ( append_instr @ Xs @ Ys )
= ( append_instr @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_147_append__eq__appendI,axiom,
! [Xs: list_P419547167332244514st_int,Xs1: list_P419547167332244514st_int,Zs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int,Us: list_P419547167332244514st_int] :
( ( ( append5028292981072010935st_int @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append5028292981072010935st_int @ Xs1 @ Us ) )
=> ( ( append5028292981072010935st_int @ Xs @ Ys )
= ( append5028292981072010935st_int @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_148_append__eq__appendI,axiom,
! [Xs: list_int,Xs1: list_int,Zs: list_int,Ys: list_int,Us: list_int] :
( ( ( append_int @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_int @ Xs1 @ Us ) )
=> ( ( append_int @ Xs @ Ys )
= ( append_int @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_149_append__eq__append__conv2,axiom,
! [Xs: list_instr,Ys: list_instr,Zs: list_instr,Ts: list_instr] :
( ( ( append_instr @ Xs @ Ys )
= ( append_instr @ Zs @ Ts ) )
= ( ? [Us2: list_instr] :
( ( ( Xs
= ( append_instr @ Zs @ Us2 ) )
& ( ( append_instr @ Us2 @ Ys )
= Ts ) )
| ( ( ( append_instr @ Xs @ Us2 )
= Zs )
& ( Ys
= ( append_instr @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_150_append__eq__append__conv2,axiom,
! [Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int,Zs: list_P419547167332244514st_int,Ts: list_P419547167332244514st_int] :
( ( ( append5028292981072010935st_int @ Xs @ Ys )
= ( append5028292981072010935st_int @ Zs @ Ts ) )
= ( ? [Us2: list_P419547167332244514st_int] :
( ( ( Xs
= ( append5028292981072010935st_int @ Zs @ Us2 ) )
& ( ( append5028292981072010935st_int @ Us2 @ Ys )
= Ts ) )
| ( ( ( append5028292981072010935st_int @ Xs @ Us2 )
= Zs )
& ( Ys
= ( append5028292981072010935st_int @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_151_append__eq__append__conv2,axiom,
! [Xs: list_int,Ys: list_int,Zs: list_int,Ts: list_int] :
( ( ( append_int @ Xs @ Ys )
= ( append_int @ Zs @ Ts ) )
= ( ? [Us2: list_int] :
( ( ( Xs
= ( append_int @ Zs @ Us2 ) )
& ( ( append_int @ Us2 @ Ys )
= Ts ) )
| ( ( ( append_int @ Xs @ Us2 )
= Zs )
& ( Ys
= ( append_int @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_152_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_153_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_154_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_155_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_156_map__eq__append__conv,axiom,
! [F: instr > instr,Xs: list_instr,Ys: list_instr,Zs: list_instr] :
( ( ( map_instr_instr @ F @ Xs )
= ( append_instr @ Ys @ Zs ) )
= ( ? [Us2: list_instr,Vs2: list_instr] :
( ( Xs
= ( append_instr @ Us2 @ Vs2 ) )
& ( Ys
= ( map_instr_instr @ F @ Us2 ) )
& ( Zs
= ( map_instr_instr @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_157_map__eq__append__conv,axiom,
! [F: produc6425607678544837394st_int > instr,Xs: list_P419547167332244514st_int,Ys: list_instr,Zs: list_instr] :
( ( ( map_Pr7514435936640407687_instr @ F @ Xs )
= ( append_instr @ Ys @ Zs ) )
= ( ? [Us2: list_P419547167332244514st_int,Vs2: list_P419547167332244514st_int] :
( ( Xs
= ( append5028292981072010935st_int @ Us2 @ Vs2 ) )
& ( Ys
= ( map_Pr7514435936640407687_instr @ F @ Us2 ) )
& ( Zs
= ( map_Pr7514435936640407687_instr @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_158_map__eq__append__conv,axiom,
! [F: int > instr,Xs: list_int,Ys: list_instr,Zs: list_instr] :
( ( ( map_int_instr @ F @ Xs )
= ( append_instr @ Ys @ Zs ) )
= ( ? [Us2: list_int,Vs2: list_int] :
( ( Xs
= ( append_int @ Us2 @ Vs2 ) )
& ( Ys
= ( map_int_instr @ F @ Us2 ) )
& ( Zs
= ( map_int_instr @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_159_map__eq__append__conv,axiom,
! [F: instr > produc6425607678544837394st_int,Xs: list_instr,Ys: list_P419547167332244514st_int,Zs: list_P419547167332244514st_int] :
( ( ( map_in7259965606695656109st_int @ F @ Xs )
= ( append5028292981072010935st_int @ Ys @ Zs ) )
= ( ? [Us2: list_instr,Vs2: list_instr] :
( ( Xs
= ( append_instr @ Us2 @ Vs2 ) )
& ( Ys
= ( map_in7259965606695656109st_int @ F @ Us2 ) )
& ( Zs
= ( map_in7259965606695656109st_int @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_160_map__eq__append__conv,axiom,
! [F: int > produc6425607678544837394st_int,Xs: list_int,Ys: list_P419547167332244514st_int,Zs: list_P419547167332244514st_int] :
( ( ( map_in8512336359594627332st_int @ F @ Xs )
= ( append5028292981072010935st_int @ Ys @ Zs ) )
= ( ? [Us2: list_int,Vs2: list_int] :
( ( Xs
= ( append_int @ Us2 @ Vs2 ) )
& ( Ys
= ( map_in8512336359594627332st_int @ F @ Us2 ) )
& ( Zs
= ( map_in8512336359594627332st_int @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_161_map__eq__append__conv,axiom,
! [F: instr > int,Xs: list_instr,Ys: list_int,Zs: list_int] :
( ( ( map_instr_int @ F @ Xs )
= ( append_int @ Ys @ Zs ) )
= ( ? [Us2: list_instr,Vs2: list_instr] :
( ( Xs
= ( append_instr @ Us2 @ Vs2 ) )
& ( Ys
= ( map_instr_int @ F @ Us2 ) )
& ( Zs
= ( map_instr_int @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_162_map__eq__append__conv,axiom,
! [F: produc6425607678544837394st_int > int,Xs: list_P419547167332244514st_int,Ys: list_int,Zs: list_int] :
( ( ( map_Pr3224476576807795204nt_int @ F @ Xs )
= ( append_int @ Ys @ Zs ) )
= ( ? [Us2: list_P419547167332244514st_int,Vs2: list_P419547167332244514st_int] :
( ( Xs
= ( append5028292981072010935st_int @ Us2 @ Vs2 ) )
& ( Ys
= ( map_Pr3224476576807795204nt_int @ F @ Us2 ) )
& ( Zs
= ( map_Pr3224476576807795204nt_int @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_163_map__eq__append__conv,axiom,
! [F: int > int,Xs: list_int,Ys: list_int,Zs: list_int] :
( ( ( map_int_int @ F @ Xs )
= ( append_int @ Ys @ Zs ) )
= ( ? [Us2: list_int,Vs2: list_int] :
( ( Xs
= ( append_int @ Us2 @ Vs2 ) )
& ( Ys
= ( map_int_int @ F @ Us2 ) )
& ( Zs
= ( map_int_int @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_164_map__eq__append__conv,axiom,
! [F: produc6425607678544837394st_int > produc6425607678544837394st_int,Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int,Zs: list_P419547167332244514st_int] :
( ( ( map_Pr8625093460839654122st_int @ F @ Xs )
= ( append5028292981072010935st_int @ Ys @ Zs ) )
= ( ? [Us2: list_P419547167332244514st_int,Vs2: list_P419547167332244514st_int] :
( ( Xs
= ( append5028292981072010935st_int @ Us2 @ Vs2 ) )
& ( Ys
= ( map_Pr8625093460839654122st_int @ F @ Us2 ) )
& ( Zs
= ( map_Pr8625093460839654122st_int @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_165_append__eq__map__conv,axiom,
! [Ys: list_instr,Zs: list_instr,F: instr > instr,Xs: list_instr] :
( ( ( append_instr @ Ys @ Zs )
= ( map_instr_instr @ F @ Xs ) )
= ( ? [Us2: list_instr,Vs2: list_instr] :
( ( Xs
= ( append_instr @ Us2 @ Vs2 ) )
& ( Ys
= ( map_instr_instr @ F @ Us2 ) )
& ( Zs
= ( map_instr_instr @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_166_append__eq__map__conv,axiom,
! [Ys: list_instr,Zs: list_instr,F: produc6425607678544837394st_int > instr,Xs: list_P419547167332244514st_int] :
( ( ( append_instr @ Ys @ Zs )
= ( map_Pr7514435936640407687_instr @ F @ Xs ) )
= ( ? [Us2: list_P419547167332244514st_int,Vs2: list_P419547167332244514st_int] :
( ( Xs
= ( append5028292981072010935st_int @ Us2 @ Vs2 ) )
& ( Ys
= ( map_Pr7514435936640407687_instr @ F @ Us2 ) )
& ( Zs
= ( map_Pr7514435936640407687_instr @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_167_append__eq__map__conv,axiom,
! [Ys: list_instr,Zs: list_instr,F: int > instr,Xs: list_int] :
( ( ( append_instr @ Ys @ Zs )
= ( map_int_instr @ F @ Xs ) )
= ( ? [Us2: list_int,Vs2: list_int] :
( ( Xs
= ( append_int @ Us2 @ Vs2 ) )
& ( Ys
= ( map_int_instr @ F @ Us2 ) )
& ( Zs
= ( map_int_instr @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_168_append__eq__map__conv,axiom,
! [Ys: list_P419547167332244514st_int,Zs: list_P419547167332244514st_int,F: instr > produc6425607678544837394st_int,Xs: list_instr] :
( ( ( append5028292981072010935st_int @ Ys @ Zs )
= ( map_in7259965606695656109st_int @ F @ Xs ) )
= ( ? [Us2: list_instr,Vs2: list_instr] :
( ( Xs
= ( append_instr @ Us2 @ Vs2 ) )
& ( Ys
= ( map_in7259965606695656109st_int @ F @ Us2 ) )
& ( Zs
= ( map_in7259965606695656109st_int @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_169_append__eq__map__conv,axiom,
! [Ys: list_P419547167332244514st_int,Zs: list_P419547167332244514st_int,F: int > produc6425607678544837394st_int,Xs: list_int] :
( ( ( append5028292981072010935st_int @ Ys @ Zs )
= ( map_in8512336359594627332st_int @ F @ Xs ) )
= ( ? [Us2: list_int,Vs2: list_int] :
( ( Xs
= ( append_int @ Us2 @ Vs2 ) )
& ( Ys
= ( map_in8512336359594627332st_int @ F @ Us2 ) )
& ( Zs
= ( map_in8512336359594627332st_int @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_170_append__eq__map__conv,axiom,
! [Ys: list_int,Zs: list_int,F: instr > int,Xs: list_instr] :
( ( ( append_int @ Ys @ Zs )
= ( map_instr_int @ F @ Xs ) )
= ( ? [Us2: list_instr,Vs2: list_instr] :
( ( Xs
= ( append_instr @ Us2 @ Vs2 ) )
& ( Ys
= ( map_instr_int @ F @ Us2 ) )
& ( Zs
= ( map_instr_int @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_171_append__eq__map__conv,axiom,
! [Ys: list_int,Zs: list_int,F: produc6425607678544837394st_int > int,Xs: list_P419547167332244514st_int] :
( ( ( append_int @ Ys @ Zs )
= ( map_Pr3224476576807795204nt_int @ F @ Xs ) )
= ( ? [Us2: list_P419547167332244514st_int,Vs2: list_P419547167332244514st_int] :
( ( Xs
= ( append5028292981072010935st_int @ Us2 @ Vs2 ) )
& ( Ys
= ( map_Pr3224476576807795204nt_int @ F @ Us2 ) )
& ( Zs
= ( map_Pr3224476576807795204nt_int @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_172_append__eq__map__conv,axiom,
! [Ys: list_int,Zs: list_int,F: int > int,Xs: list_int] :
( ( ( append_int @ Ys @ Zs )
= ( map_int_int @ F @ Xs ) )
= ( ? [Us2: list_int,Vs2: list_int] :
( ( Xs
= ( append_int @ Us2 @ Vs2 ) )
& ( Ys
= ( map_int_int @ F @ Us2 ) )
& ( Zs
= ( map_int_int @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_173_append__eq__map__conv,axiom,
! [Ys: list_P419547167332244514st_int,Zs: list_P419547167332244514st_int,F: produc6425607678544837394st_int > produc6425607678544837394st_int,Xs: list_P419547167332244514st_int] :
( ( ( append5028292981072010935st_int @ Ys @ Zs )
= ( map_Pr8625093460839654122st_int @ F @ Xs ) )
= ( ? [Us2: list_P419547167332244514st_int,Vs2: list_P419547167332244514st_int] :
( ( Xs
= ( append5028292981072010935st_int @ Us2 @ Vs2 ) )
& ( Ys
= ( map_Pr8625093460839654122st_int @ F @ Us2 ) )
& ( Zs
= ( map_Pr8625093460839654122st_int @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_174_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N3 )
& ~ ( P @ M2 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_175_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_176_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_177_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_178_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_179_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_180_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_181_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_182_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_183_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,K2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ K2 )
=> ( ( P @ I2 @ J2 )
=> ( ( P @ J2 @ K2 )
=> ( P @ I2 @ K2 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_184_less__trans__Suc,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_185_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_186_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_187_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M3: nat] :
( ( M
= ( suc @ M3 ) )
& ( ord_less_nat @ N @ M3 ) ) ) ) ).
% Suc_less_eq2
thf(fact_188_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_189_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_190_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_191_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_192_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_193_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_194_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_195_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_196_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_197_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_198_length__induct,axiom,
! [P: list_P419547167332244514st_int > $o,Xs: list_P419547167332244514st_int] :
( ! [Xs3: list_P419547167332244514st_int] :
( ! [Ys2: list_P419547167332244514st_int] :
( ( ord_less_nat @ ( size_s229347708945616566st_int @ Ys2 ) @ ( size_s229347708945616566st_int @ Xs3 ) )
=> ( P @ Ys2 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_199_length__induct,axiom,
! [P: list_instr > $o,Xs: list_instr] :
( ! [Xs3: list_instr] :
( ! [Ys2: list_instr] :
( ( ord_less_nat @ ( size_size_list_instr @ Ys2 ) @ ( size_size_list_instr @ Xs3 ) )
=> ( P @ Ys2 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_200_length__induct,axiom,
! [P: list_list_char > $o,Xs: list_list_char] :
( ! [Xs3: list_list_char] :
( ! [Ys2: list_list_char] :
( ( ord_less_nat @ ( size_s356637359517785349t_char @ Ys2 ) @ ( size_s356637359517785349t_char @ Xs3 ) )
=> ( P @ Ys2 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_201_length__induct,axiom,
! [P: list_int > $o,Xs: list_int] :
( ! [Xs3: list_int] :
( ! [Ys2: list_int] :
( ( ord_less_nat @ ( size_size_list_int @ Ys2 ) @ ( size_size_list_int @ Xs3 ) )
=> ( P @ Ys2 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_202_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_203_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_204_nth__append,axiom,
! [N: nat,Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int] :
( ( ( ord_less_nat @ N @ ( size_s229347708945616566st_int @ Xs ) )
=> ( ( nth_Pr8327173284404383691st_int @ ( append5028292981072010935st_int @ Xs @ Ys ) @ N )
= ( nth_Pr8327173284404383691st_int @ Xs @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_s229347708945616566st_int @ Xs ) )
=> ( ( nth_Pr8327173284404383691st_int @ ( append5028292981072010935st_int @ Xs @ Ys ) @ N )
= ( nth_Pr8327173284404383691st_int @ Ys @ ( minus_minus_nat @ N @ ( size_s229347708945616566st_int @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_205_nth__append,axiom,
! [N: nat,Xs: list_instr,Ys: list_instr] :
( ( ( ord_less_nat @ N @ ( size_size_list_instr @ Xs ) )
=> ( ( nth_instr @ ( append_instr @ Xs @ Ys ) @ N )
= ( nth_instr @ Xs @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_size_list_instr @ Xs ) )
=> ( ( nth_instr @ ( append_instr @ Xs @ Ys ) @ N )
= ( nth_instr @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_instr @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_206_nth__append,axiom,
! [N: nat,Xs: list_list_char,Ys: list_list_char] :
( ( ( ord_less_nat @ N @ ( size_s356637359517785349t_char @ Xs ) )
=> ( ( nth_list_char @ ( append_list_char @ Xs @ Ys ) @ N )
= ( nth_list_char @ Xs @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_s356637359517785349t_char @ Xs ) )
=> ( ( nth_list_char @ ( append_list_char @ Xs @ Ys ) @ N )
= ( nth_list_char @ Ys @ ( minus_minus_nat @ N @ ( size_s356637359517785349t_char @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_207_nth__append,axiom,
! [N: nat,Xs: list_int,Ys: list_int] :
( ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
=> ( ( nth_int @ ( append_int @ Xs @ Ys ) @ N )
= ( nth_int @ Xs @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
=> ( ( nth_int @ ( append_int @ Xs @ Ys ) @ N )
= ( nth_int @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_int @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_208_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_209_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_210_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_211_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_212_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_213_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% gr0_implies_Suc
thf(fact_214_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_215_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_216_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_217_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_218_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_219_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_220_all__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ! [M5: nat] :
( ( ord_less_nat @ M5 @ N )
=> ( P @ M5 ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
=> ( P @ X2 ) ) ) ) ).
% all_nat_less_eq
thf(fact_221_ex__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ? [M5: nat] :
( ( ord_less_nat @ M5 @ N )
& ( P @ M5 ) ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
& ( P @ X2 ) ) ) ) ).
% ex_nat_less_eq
thf(fact_222_list__eq__iff__nth__eq,axiom,
( ( ^ [Y3: list_P419547167332244514st_int,Z: list_P419547167332244514st_int] : ( Y3 = Z ) )
= ( ^ [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_223_list__eq__iff__nth__eq,axiom,
( ( ^ [Y3: list_instr,Z: list_instr] : ( Y3 = Z ) )
= ( ^ [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_224_list__eq__iff__nth__eq,axiom,
( ( ^ [Y3: list_list_char,Z: list_list_char] : ( Y3 = Z ) )
= ( ^ [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_225_list__eq__iff__nth__eq,axiom,
( ( ^ [Y3: list_int,Z: list_int] : ( Y3 = Z ) )
= ( ^ [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_226_Skolem__list__nth,axiom,
! [K: nat,P: nat > produc6425607678544837394st_int > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ? [X4: produc6425607678544837394st_int] : ( P @ I3 @ X4 ) ) )
= ( ? [Xs2: list_P419547167332244514st_int] :
( ( ( size_s229347708945616566st_int @ Xs2 )
= K )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( P @ I3 @ ( nth_Pr8327173284404383691st_int @ Xs2 @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_227_Skolem__list__nth,axiom,
! [K: nat,P: nat > instr > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ? [X4: instr] : ( P @ I3 @ X4 ) ) )
= ( ? [Xs2: list_instr] :
( ( ( size_size_list_instr @ Xs2 )
= K )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( P @ I3 @ ( nth_instr @ Xs2 @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_228_Skolem__list__nth,axiom,
! [K: nat,P: nat > list_char > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ? [X4: list_char] : ( P @ I3 @ X4 ) ) )
= ( ? [Xs2: list_list_char] :
( ( ( size_s356637359517785349t_char @ Xs2 )
= K )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( P @ I3 @ ( nth_list_char @ Xs2 @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_229_Skolem__list__nth,axiom,
! [K: nat,P: nat > int > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ? [X4: int] : ( P @ I3 @ X4 ) ) )
= ( ? [Xs2: list_int] :
( ( ( size_size_list_int @ Xs2 )
= K )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( P @ I3 @ ( nth_int @ Xs2 @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_230_nth__equalityI,axiom,
! [Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int] :
( ( ( size_s229347708945616566st_int @ Xs )
= ( size_s229347708945616566st_int @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s229347708945616566st_int @ Xs ) )
=> ( ( nth_Pr8327173284404383691st_int @ Xs @ I2 )
= ( nth_Pr8327173284404383691st_int @ Ys @ I2 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_231_nth__equalityI,axiom,
! [Xs: list_instr,Ys: list_instr] :
( ( ( size_size_list_instr @ Xs )
= ( size_size_list_instr @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_instr @ Xs ) )
=> ( ( nth_instr @ Xs @ I2 )
= ( nth_instr @ Ys @ I2 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_232_nth__equalityI,axiom,
! [Xs: list_list_char,Ys: list_list_char] :
( ( ( size_s356637359517785349t_char @ Xs )
= ( size_s356637359517785349t_char @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s356637359517785349t_char @ Xs ) )
=> ( ( nth_list_char @ Xs @ I2 )
= ( nth_list_char @ Ys @ I2 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_233_nth__equalityI,axiom,
! [Xs: list_int,Ys: list_int] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
=> ( ( nth_int @ Xs @ I2 )
= ( nth_int @ Ys @ I2 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_234_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_235_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_236_append__eq__conv__conj,axiom,
! [Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int,Zs: list_P419547167332244514st_int] :
( ( ( append5028292981072010935st_int @ Xs @ Ys )
= Zs )
= ( ( Xs
= ( take_P1004856038796978596st_int @ ( size_s229347708945616566st_int @ Xs ) @ Zs ) )
& ( Ys
= ( drop_P7838799151754935884st_int @ ( size_s229347708945616566st_int @ Xs ) @ Zs ) ) ) ) ).
% append_eq_conv_conj
thf(fact_237_append__eq__conv__conj,axiom,
! [Xs: list_instr,Ys: list_instr,Zs: list_instr] :
( ( ( append_instr @ Xs @ Ys )
= Zs )
= ( ( Xs
= ( take_instr @ ( size_size_list_instr @ Xs ) @ Zs ) )
& ( Ys
= ( drop_instr @ ( size_size_list_instr @ Xs ) @ Zs ) ) ) ) ).
% append_eq_conv_conj
thf(fact_238_append__eq__conv__conj,axiom,
! [Xs: list_list_char,Ys: list_list_char,Zs: list_list_char] :
( ( ( append_list_char @ Xs @ Ys )
= Zs )
= ( ( Xs
= ( take_list_char @ ( size_s356637359517785349t_char @ Xs ) @ Zs ) )
& ( Ys
= ( drop_list_char @ ( size_s356637359517785349t_char @ Xs ) @ Zs ) ) ) ) ).
% append_eq_conv_conj
thf(fact_239_append__eq__conv__conj,axiom,
! [Xs: list_int,Ys: list_int,Zs: list_int] :
( ( ( append_int @ Xs @ Ys )
= Zs )
= ( ( Xs
= ( take_int @ ( size_size_list_int @ Xs ) @ Zs ) )
& ( Ys
= ( drop_int @ ( size_size_list_int @ Xs ) @ Zs ) ) ) ) ).
% append_eq_conv_conj
thf(fact_240_map__equality__iff,axiom,
! [F: produc6425607678544837394st_int > produc6425607678544837394st_int,Xs: list_P419547167332244514st_int,G: produc6425607678544837394st_int > produc6425607678544837394st_int,Ys: list_P419547167332244514st_int] :
( ( ( map_Pr8625093460839654122st_int @ F @ Xs )
= ( map_Pr8625093460839654122st_int @ G @ Ys ) )
= ( ( ( size_s229347708945616566st_int @ Xs )
= ( size_s229347708945616566st_int @ Ys ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s229347708945616566st_int @ Ys ) )
=> ( ( F @ ( nth_Pr8327173284404383691st_int @ Xs @ I3 ) )
= ( G @ ( nth_Pr8327173284404383691st_int @ Ys @ I3 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_241_map__equality__iff,axiom,
! [F: produc6425607678544837394st_int > produc6425607678544837394st_int,Xs: list_P419547167332244514st_int,G: instr > produc6425607678544837394st_int,Ys: list_instr] :
( ( ( map_Pr8625093460839654122st_int @ F @ Xs )
= ( map_in7259965606695656109st_int @ G @ Ys ) )
= ( ( ( size_s229347708945616566st_int @ Xs )
= ( size_size_list_instr @ Ys ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_instr @ Ys ) )
=> ( ( F @ ( nth_Pr8327173284404383691st_int @ Xs @ I3 ) )
= ( G @ ( nth_instr @ Ys @ I3 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_242_map__equality__iff,axiom,
! [F: produc6425607678544837394st_int > produc6425607678544837394st_int,Xs: list_P419547167332244514st_int,G: list_char > produc6425607678544837394st_int,Ys: list_list_char] :
( ( ( map_Pr8625093460839654122st_int @ F @ Xs )
= ( map_li6440205260635360185st_int @ G @ Ys ) )
= ( ( ( size_s229347708945616566st_int @ Xs )
= ( size_s356637359517785349t_char @ Ys ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s356637359517785349t_char @ Ys ) )
=> ( ( F @ ( nth_Pr8327173284404383691st_int @ Xs @ I3 ) )
= ( G @ ( nth_list_char @ Ys @ I3 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_243_map__equality__iff,axiom,
! [F: produc6425607678544837394st_int > produc6425607678544837394st_int,Xs: list_P419547167332244514st_int,G: int > produc6425607678544837394st_int,Ys: list_int] :
( ( ( map_Pr8625093460839654122st_int @ F @ Xs )
= ( map_in8512336359594627332st_int @ G @ Ys ) )
= ( ( ( size_s229347708945616566st_int @ Xs )
= ( size_size_list_int @ Ys ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_int @ Ys ) )
=> ( ( F @ ( nth_Pr8327173284404383691st_int @ Xs @ I3 ) )
= ( G @ ( nth_int @ Ys @ I3 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_244_map__equality__iff,axiom,
! [F: instr > produc6425607678544837394st_int,Xs: list_instr,G: produc6425607678544837394st_int > produc6425607678544837394st_int,Ys: list_P419547167332244514st_int] :
( ( ( map_in7259965606695656109st_int @ F @ Xs )
= ( map_Pr8625093460839654122st_int @ G @ Ys ) )
= ( ( ( size_size_list_instr @ Xs )
= ( size_s229347708945616566st_int @ Ys ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s229347708945616566st_int @ Ys ) )
=> ( ( F @ ( nth_instr @ Xs @ I3 ) )
= ( G @ ( nth_Pr8327173284404383691st_int @ Ys @ I3 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_245_map__equality__iff,axiom,
! [F: list_char > produc6425607678544837394st_int,Xs: list_list_char,G: produc6425607678544837394st_int > produc6425607678544837394st_int,Ys: list_P419547167332244514st_int] :
( ( ( map_li6440205260635360185st_int @ F @ Xs )
= ( map_Pr8625093460839654122st_int @ G @ Ys ) )
= ( ( ( size_s356637359517785349t_char @ Xs )
= ( size_s229347708945616566st_int @ Ys ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s229347708945616566st_int @ Ys ) )
=> ( ( F @ ( nth_list_char @ Xs @ I3 ) )
= ( G @ ( nth_Pr8327173284404383691st_int @ Ys @ I3 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_246_map__equality__iff,axiom,
! [F: int > produc6425607678544837394st_int,Xs: list_int,G: produc6425607678544837394st_int > produc6425607678544837394st_int,Ys: list_P419547167332244514st_int] :
( ( ( map_in8512336359594627332st_int @ F @ Xs )
= ( map_Pr8625093460839654122st_int @ G @ Ys ) )
= ( ( ( size_size_list_int @ Xs )
= ( size_s229347708945616566st_int @ Ys ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s229347708945616566st_int @ Ys ) )
=> ( ( F @ ( nth_int @ Xs @ I3 ) )
= ( G @ ( nth_Pr8327173284404383691st_int @ Ys @ I3 ) ) ) ) ) ) ).
% map_equality_iff
thf(fact_247_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_248_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_249_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_250_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_251_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_252_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_253_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_254_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_255_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_256_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_257_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_258_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_259_surj__pair,axiom,
! [P2: produc4189061985984568957nt_int] :
? [X: list_list_char,Y4: produc4435102495419491129nt_int] :
( P2
= ( produc3965054194175396271nt_int @ X @ Y4 ) ) ).
% surj_pair
thf(fact_260_surj__pair,axiom,
! [P2: produc6425607678544837394st_int] :
? [X: int,Y4: produc369741286924889929st_int] :
( P2
= ( produc5086643055186798020st_int @ X @ Y4 ) ) ).
% surj_pair
thf(fact_261_surj__pair,axiom,
! [P2: product_prod_int_int] :
? [X: int,Y4: int] :
( P2
= ( product_Pair_int_int @ X @ Y4 ) ) ).
% surj_pair
thf(fact_262_surj__pair,axiom,
! [P2: produc369741286924889929st_int] :
? [X: list_char > int,Y4: list_int] :
( P2
= ( produc8650753666468850689st_int @ X @ Y4 ) ) ).
% surj_pair
thf(fact_263_surj__pair,axiom,
! [P2: produc4435102495419491129nt_int] :
? [X: list_char > int,Y4: int] :
( P2
= ( produc5790713362662368625nt_int @ X @ Y4 ) ) ).
% surj_pair
thf(fact_264_old_Oprod_Oexhaust,axiom,
! [Y: produc4189061985984568957nt_int] :
~ ! [A4: list_list_char,B3: produc4435102495419491129nt_int] :
( Y
!= ( produc3965054194175396271nt_int @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_265_old_Oprod_Oexhaust,axiom,
! [Y: produc6425607678544837394st_int] :
~ ! [A4: int,B3: produc369741286924889929st_int] :
( Y
!= ( produc5086643055186798020st_int @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_266_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_267_old_Oprod_Oexhaust,axiom,
! [Y: produc369741286924889929st_int] :
~ ! [A4: list_char > int,B3: list_int] :
( Y
!= ( produc8650753666468850689st_int @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_268_old_Oprod_Oexhaust,axiom,
! [Y: produc4435102495419491129nt_int] :
~ ! [A4: list_char > int,B3: int] :
( Y
!= ( produc5790713362662368625nt_int @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_269_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_270_Suc__inject,axiom,
! [X3: nat,Y: nat] :
( ( ( suc @ X3 )
= ( suc @ Y ) )
=> ( X3 = Y ) ) ).
% Suc_inject
thf(fact_271_size__neq__size__imp__neq,axiom,
! [X3: list_P419547167332244514st_int,Y: list_P419547167332244514st_int] :
( ( ( size_s229347708945616566st_int @ X3 )
!= ( size_s229347708945616566st_int @ Y ) )
=> ( X3 != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_272_size__neq__size__imp__neq,axiom,
! [X3: list_instr,Y: list_instr] :
( ( ( size_size_list_instr @ X3 )
!= ( size_size_list_instr @ Y ) )
=> ( X3 != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_273_size__neq__size__imp__neq,axiom,
! [X3: list_list_char,Y: list_list_char] :
( ( ( size_s356637359517785349t_char @ X3 )
!= ( size_s356637359517785349t_char @ Y ) )
=> ( X3 != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_274_size__neq__size__imp__neq,axiom,
! [X3: list_int,Y: list_int] :
( ( ( size_size_list_int @ X3 )
!= ( size_size_list_int @ Y ) )
=> ( X3 != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_275_size__neq__size__imp__neq,axiom,
! [X3: com,Y: com] :
( ( ( size_size_com @ X3 )
!= ( size_size_com @ Y ) )
=> ( X3 != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_276_neq__if__length__neq,axiom,
! [Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int] :
( ( ( size_s229347708945616566st_int @ Xs )
!= ( size_s229347708945616566st_int @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_277_neq__if__length__neq,axiom,
! [Xs: list_instr,Ys: list_instr] :
( ( ( size_size_list_instr @ Xs )
!= ( size_size_list_instr @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_278_neq__if__length__neq,axiom,
! [Xs: list_list_char,Ys: list_list_char] :
( ( ( size_s356637359517785349t_char @ Xs )
!= ( size_s356637359517785349t_char @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_279_neq__if__length__neq,axiom,
! [Xs: list_int,Ys: list_int] :
( ( ( size_size_list_int @ Xs )
!= ( size_size_list_int @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_280_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_P419547167332244514st_int] :
( ( size_s229347708945616566st_int @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_281_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_instr] :
( ( size_size_list_instr @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_282_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_list_char] :
( ( size_s356637359517785349t_char @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_283_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_int] :
( ( size_size_list_int @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_284_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_285_take__equalityI,axiom,
! [Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int] :
( ! [I2: nat] :
( ( take_P1004856038796978596st_int @ I2 @ Xs )
= ( take_P1004856038796978596st_int @ I2 @ Ys ) )
=> ( Xs = Ys ) ) ).
% take_equalityI
thf(fact_286_list_Omap__ident,axiom,
! [T: list_P419547167332244514st_int] :
( ( map_Pr8625093460839654122st_int
@ ^ [X2: produc6425607678544837394st_int] : X2
@ T )
= T ) ).
% list.map_ident
thf(fact_287_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% not0_implies_Suc
thf(fact_288_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_289_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_290_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_291_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_292_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X: nat] : ( P @ X @ zero_zero_nat )
=> ( ! [Y4: nat] : ( P @ zero_zero_nat @ ( suc @ Y4 ) )
=> ( ! [X: nat,Y4: nat] :
( ( P @ X @ Y4 )
=> ( P @ ( suc @ X ) @ ( suc @ Y4 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_293_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_294_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_295_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_296_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_297_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_298_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_299_prod__induct3,axiom,
! [P: produc4189061985984568957nt_int > $o,X3: produc4189061985984568957nt_int] :
( ! [A4: list_list_char,B3: list_char > int,C2: int] : ( P @ ( produc3965054194175396271nt_int @ A4 @ ( produc5790713362662368625nt_int @ B3 @ C2 ) ) )
=> ( P @ X3 ) ) ).
% prod_induct3
thf(fact_300_prod__induct3,axiom,
! [P: produc6425607678544837394st_int > $o,X3: produc6425607678544837394st_int] :
( ! [A4: int,B3: list_char > int,C2: list_int] : ( P @ ( produc5086643055186798020st_int @ A4 @ ( produc8650753666468850689st_int @ B3 @ C2 ) ) )
=> ( P @ X3 ) ) ).
% prod_induct3
thf(fact_301_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_302_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_303_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_304_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_305_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_306_fst__conv,axiom,
! [X1: int,X22: produc369741286924889929st_int] :
( ( produc8941309049508465880st_int @ ( produc5086643055186798020st_int @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_307_fst__conv,axiom,
! [X1: list_char > int,X22: list_int] :
( ( produc2155683374303800813st_int @ ( produc8650753666468850689st_int @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_308_fst__conv,axiom,
! [X1: list_char > int,X22: int] :
( ( produc9025023058967219549nt_int @ ( produc5790713362662368625nt_int @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_309_fst__conv,axiom,
! [X1: int,X22: int] :
( ( product_fst_int_int @ ( product_Pair_int_int @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_310_fst__conv,axiom,
! [X1: list_list_char,X22: produc4435102495419491129nt_int] :
( ( produc811350159336358595nt_int @ ( produc3965054194175396271nt_int @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_311_fst__eqD,axiom,
! [X3: int,Y: produc369741286924889929st_int,A: int] :
( ( ( produc8941309049508465880st_int @ ( produc5086643055186798020st_int @ X3 @ Y ) )
= A )
=> ( X3 = A ) ) ).
% fst_eqD
thf(fact_312_fst__eqD,axiom,
! [X3: list_char > int,Y: list_int,A: list_char > int] :
( ( ( produc2155683374303800813st_int @ ( produc8650753666468850689st_int @ X3 @ Y ) )
= A )
=> ( X3 = A ) ) ).
% fst_eqD
thf(fact_313_fst__eqD,axiom,
! [X3: list_char > int,Y: int,A: list_char > int] :
( ( ( produc9025023058967219549nt_int @ ( produc5790713362662368625nt_int @ X3 @ Y ) )
= A )
=> ( X3 = A ) ) ).
% fst_eqD
thf(fact_314_fst__eqD,axiom,
! [X3: int,Y: int,A: int] :
( ( ( product_fst_int_int @ ( product_Pair_int_int @ X3 @ Y ) )
= A )
=> ( X3 = A ) ) ).
% fst_eqD
thf(fact_315_fst__eqD,axiom,
! [X3: list_list_char,Y: produc4435102495419491129nt_int,A: list_list_char] :
( ( ( produc811350159336358595nt_int @ ( produc3965054194175396271nt_int @ X3 @ Y ) )
= A )
=> ( X3 = A ) ) ).
% fst_eqD
thf(fact_316_snd__conv,axiom,
! [X1: int,X22: produc369741286924889929st_int] :
( ( produc1296517004818028822st_int @ ( produc5086643055186798020st_int @ X1 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_317_snd__conv,axiom,
! [X1: list_char > int,X22: list_int] :
( ( produc6244060863719409455st_int @ ( produc8650753666468850689st_int @ X1 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_318_snd__conv,axiom,
! [X1: list_char > int,X22: int] :
( ( produc179445755590504351nt_int @ ( produc5790713362662368625nt_int @ X1 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_319_snd__conv,axiom,
! [X1: int,X22: int] :
( ( product_snd_int_int @ ( product_Pair_int_int @ X1 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_320_snd__conv,axiom,
! [X1: list_list_char,X22: produc4435102495419491129nt_int] :
( ( produc30867750519413249nt_int @ ( produc3965054194175396271nt_int @ X1 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_321_snd__eqD,axiom,
! [X3: int,Y: produc369741286924889929st_int,A: produc369741286924889929st_int] :
( ( ( produc1296517004818028822st_int @ ( produc5086643055186798020st_int @ X3 @ Y ) )
= A )
=> ( Y = A ) ) ).
% snd_eqD
thf(fact_322_snd__eqD,axiom,
! [X3: list_char > int,Y: list_int,A: list_int] :
( ( ( produc6244060863719409455st_int @ ( produc8650753666468850689st_int @ X3 @ Y ) )
= A )
=> ( Y = A ) ) ).
% snd_eqD
thf(fact_323_snd__eqD,axiom,
! [X3: list_char > int,Y: int,A: int] :
( ( ( produc179445755590504351nt_int @ ( produc5790713362662368625nt_int @ X3 @ Y ) )
= A )
=> ( Y = A ) ) ).
% snd_eqD
thf(fact_324_snd__eqD,axiom,
! [X3: int,Y: int,A: int] :
( ( ( product_snd_int_int @ ( product_Pair_int_int @ X3 @ Y ) )
= A )
=> ( Y = A ) ) ).
% snd_eqD
thf(fact_325_snd__eqD,axiom,
! [X3: list_list_char,Y: produc4435102495419491129nt_int,A: produc4435102495419491129nt_int] :
( ( ( produc30867750519413249nt_int @ ( produc3965054194175396271nt_int @ X3 @ Y ) )
= A )
=> ( Y = A ) ) ).
% snd_eqD
thf(fact_326_map__eq__imp__length__eq,axiom,
! [F: produc6425607678544837394st_int > produc6425607678544837394st_int,Xs: list_P419547167332244514st_int,G: produc6425607678544837394st_int > produc6425607678544837394st_int,Ys: list_P419547167332244514st_int] :
( ( ( map_Pr8625093460839654122st_int @ F @ Xs )
= ( map_Pr8625093460839654122st_int @ G @ Ys ) )
=> ( ( size_s229347708945616566st_int @ Xs )
= ( size_s229347708945616566st_int @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_327_map__eq__imp__length__eq,axiom,
! [F: produc6425607678544837394st_int > produc6425607678544837394st_int,Xs: list_P419547167332244514st_int,G: instr > produc6425607678544837394st_int,Ys: list_instr] :
( ( ( map_Pr8625093460839654122st_int @ F @ Xs )
= ( map_in7259965606695656109st_int @ G @ Ys ) )
=> ( ( size_s229347708945616566st_int @ Xs )
= ( size_size_list_instr @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_328_map__eq__imp__length__eq,axiom,
! [F: produc6425607678544837394st_int > produc6425607678544837394st_int,Xs: list_P419547167332244514st_int,G: list_char > produc6425607678544837394st_int,Ys: list_list_char] :
( ( ( map_Pr8625093460839654122st_int @ F @ Xs )
= ( map_li6440205260635360185st_int @ G @ Ys ) )
=> ( ( size_s229347708945616566st_int @ Xs )
= ( size_s356637359517785349t_char @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_329_map__eq__imp__length__eq,axiom,
! [F: produc6425607678544837394st_int > produc6425607678544837394st_int,Xs: list_P419547167332244514st_int,G: int > produc6425607678544837394st_int,Ys: list_int] :
( ( ( map_Pr8625093460839654122st_int @ F @ Xs )
= ( map_in8512336359594627332st_int @ G @ Ys ) )
=> ( ( size_s229347708945616566st_int @ Xs )
= ( size_size_list_int @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_330_map__eq__imp__length__eq,axiom,
! [F: instr > produc6425607678544837394st_int,Xs: list_instr,G: produc6425607678544837394st_int > produc6425607678544837394st_int,Ys: list_P419547167332244514st_int] :
( ( ( map_in7259965606695656109st_int @ F @ Xs )
= ( map_Pr8625093460839654122st_int @ G @ Ys ) )
=> ( ( size_size_list_instr @ Xs )
= ( size_s229347708945616566st_int @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_331_map__eq__imp__length__eq,axiom,
! [F: list_char > produc6425607678544837394st_int,Xs: list_list_char,G: produc6425607678544837394st_int > produc6425607678544837394st_int,Ys: list_P419547167332244514st_int] :
( ( ( map_li6440205260635360185st_int @ F @ Xs )
= ( map_Pr8625093460839654122st_int @ G @ Ys ) )
=> ( ( size_s356637359517785349t_char @ Xs )
= ( size_s229347708945616566st_int @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_332_map__eq__imp__length__eq,axiom,
! [F: int > produc6425607678544837394st_int,Xs: list_int,G: produc6425607678544837394st_int > produc6425607678544837394st_int,Ys: list_P419547167332244514st_int] :
( ( ( map_in8512336359594627332st_int @ F @ Xs )
= ( map_Pr8625093460839654122st_int @ G @ Ys ) )
=> ( ( size_size_list_int @ Xs )
= ( size_s229347708945616566st_int @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_333_drop__0,axiom,
! [Xs: list_P419547167332244514st_int] :
( ( drop_P7838799151754935884st_int @ zero_zero_nat @ Xs )
= Xs ) ).
% drop_0
thf(fact_334_drop__0,axiom,
! [Xs: list_int] :
( ( drop_int @ zero_zero_nat @ Xs )
= Xs ) ).
% drop_0
thf(fact_335_drop__0,axiom,
! [Xs: list_list_char] :
( ( drop_list_char @ zero_zero_nat @ Xs )
= Xs ) ).
% drop_0
thf(fact_336_drop__0,axiom,
! [Xs: list_instr] :
( ( drop_instr @ zero_zero_nat @ Xs )
= Xs ) ).
% drop_0
thf(fact_337_prod__eq__iff,axiom,
( ( ^ [Y3: produc6425607678544837394st_int,Z: produc6425607678544837394st_int] : ( Y3 = Z ) )
= ( ^ [S2: produc6425607678544837394st_int,T2: produc6425607678544837394st_int] :
( ( ( produc8941309049508465880st_int @ S2 )
= ( produc8941309049508465880st_int @ T2 ) )
& ( ( produc1296517004818028822st_int @ S2 )
= ( produc1296517004818028822st_int @ T2 ) ) ) ) ) ).
% prod_eq_iff
thf(fact_338_prod__eq__iff,axiom,
( ( ^ [Y3: produc369741286924889929st_int,Z: produc369741286924889929st_int] : ( Y3 = Z ) )
= ( ^ [S2: produc369741286924889929st_int,T2: produc369741286924889929st_int] :
( ( ( produc2155683374303800813st_int @ S2 )
= ( produc2155683374303800813st_int @ T2 ) )
& ( ( produc6244060863719409455st_int @ S2 )
= ( produc6244060863719409455st_int @ T2 ) ) ) ) ) ).
% prod_eq_iff
thf(fact_339_prod__eq__iff,axiom,
( ( ^ [Y3: produc4435102495419491129nt_int,Z: produc4435102495419491129nt_int] : ( Y3 = Z ) )
= ( ^ [S2: produc4435102495419491129nt_int,T2: produc4435102495419491129nt_int] :
( ( ( produc9025023058967219549nt_int @ S2 )
= ( produc9025023058967219549nt_int @ T2 ) )
& ( ( produc179445755590504351nt_int @ S2 )
= ( produc179445755590504351nt_int @ T2 ) ) ) ) ) ).
% prod_eq_iff
thf(fact_340_prod__eq__iff,axiom,
( ( ^ [Y3: product_prod_int_int,Z: product_prod_int_int] : ( Y3 = Z ) )
= ( ^ [S2: product_prod_int_int,T2: product_prod_int_int] :
( ( ( product_fst_int_int @ S2 )
= ( product_fst_int_int @ T2 ) )
& ( ( product_snd_int_int @ S2 )
= ( product_snd_int_int @ T2 ) ) ) ) ) ).
% prod_eq_iff
thf(fact_341_prod__eq__iff,axiom,
( ( ^ [Y3: produc4189061985984568957nt_int,Z: produc4189061985984568957nt_int] : ( Y3 = Z ) )
= ( ^ [S2: produc4189061985984568957nt_int,T2: produc4189061985984568957nt_int] :
( ( ( produc811350159336358595nt_int @ S2 )
= ( produc811350159336358595nt_int @ T2 ) )
& ( ( produc30867750519413249nt_int @ S2 )
= ( produc30867750519413249nt_int @ T2 ) ) ) ) ) ).
% prod_eq_iff
thf(fact_342_prod__eqI,axiom,
! [P2: produc6425607678544837394st_int,Q: produc6425607678544837394st_int] :
( ( ( produc8941309049508465880st_int @ P2 )
= ( produc8941309049508465880st_int @ Q ) )
=> ( ( ( produc1296517004818028822st_int @ P2 )
= ( produc1296517004818028822st_int @ Q ) )
=> ( P2 = Q ) ) ) ).
% prod_eqI
thf(fact_343_prod__eqI,axiom,
! [P2: produc369741286924889929st_int,Q: produc369741286924889929st_int] :
( ( ( produc2155683374303800813st_int @ P2 )
= ( produc2155683374303800813st_int @ Q ) )
=> ( ( ( produc6244060863719409455st_int @ P2 )
= ( produc6244060863719409455st_int @ Q ) )
=> ( P2 = Q ) ) ) ).
% prod_eqI
thf(fact_344_prod__eqI,axiom,
! [P2: produc4435102495419491129nt_int,Q: produc4435102495419491129nt_int] :
( ( ( produc9025023058967219549nt_int @ P2 )
= ( produc9025023058967219549nt_int @ Q ) )
=> ( ( ( produc179445755590504351nt_int @ P2 )
= ( produc179445755590504351nt_int @ Q ) )
=> ( P2 = Q ) ) ) ).
% prod_eqI
thf(fact_345_prod__eqI,axiom,
! [P2: product_prod_int_int,Q: product_prod_int_int] :
( ( ( product_fst_int_int @ P2 )
= ( product_fst_int_int @ Q ) )
=> ( ( ( product_snd_int_int @ P2 )
= ( product_snd_int_int @ Q ) )
=> ( P2 = Q ) ) ) ).
% prod_eqI
thf(fact_346_prod__eqI,axiom,
! [P2: produc4189061985984568957nt_int,Q: produc4189061985984568957nt_int] :
( ( ( produc811350159336358595nt_int @ P2 )
= ( produc811350159336358595nt_int @ Q ) )
=> ( ( ( produc30867750519413249nt_int @ P2 )
= ( produc30867750519413249nt_int @ Q ) )
=> ( P2 = Q ) ) ) ).
% prod_eqI
thf(fact_347_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_348_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_349_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_350_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_351_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_352_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_353_drop__map,axiom,
! [N: nat,F: int > int,Xs: list_int] :
( ( drop_int @ N @ ( map_int_int @ F @ Xs ) )
= ( map_int_int @ F @ ( drop_int @ N @ Xs ) ) ) ).
% drop_map
thf(fact_354_drop__map,axiom,
! [N: nat,F: instr > int,Xs: list_instr] :
( ( drop_int @ N @ ( map_instr_int @ F @ Xs ) )
= ( map_instr_int @ F @ ( drop_instr @ N @ Xs ) ) ) ).
% drop_map
thf(fact_355_drop__map,axiom,
! [N: nat,F: int > instr,Xs: list_int] :
( ( drop_instr @ N @ ( map_int_instr @ F @ Xs ) )
= ( map_int_instr @ F @ ( drop_int @ N @ Xs ) ) ) ).
% drop_map
thf(fact_356_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_357_drop__map,axiom,
! [N: nat,F: list_char > int,Xs: list_list_char] :
( ( drop_int @ N @ ( map_list_char_int @ F @ Xs ) )
= ( map_list_char_int @ F @ ( drop_list_char @ N @ Xs ) ) ) ).
% drop_map
thf(fact_358_drop__map,axiom,
! [N: nat,F: int > list_char,Xs: list_int] :
( ( drop_list_char @ N @ ( map_int_list_char @ F @ Xs ) )
= ( map_int_list_char @ F @ ( drop_int @ N @ Xs ) ) ) ).
% drop_map
thf(fact_359_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_360_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_361_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_362_drop__map,axiom,
! [N: nat,F: int > produc6425607678544837394st_int,Xs: list_int] :
( ( drop_P7838799151754935884st_int @ N @ ( map_in8512336359594627332st_int @ F @ Xs ) )
= ( map_in8512336359594627332st_int @ F @ ( drop_int @ N @ Xs ) ) ) ).
% drop_map
thf(fact_363_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
!= zero_zero_int ) ).
% of_nat_neq_0
thf(fact_364_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
!= zero_zero_nat ) ).
% of_nat_neq_0
thf(fact_365_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_366_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_367_surjective__pairing,axiom,
! [T: produc6425607678544837394st_int] :
( T
= ( produc5086643055186798020st_int @ ( produc8941309049508465880st_int @ T ) @ ( produc1296517004818028822st_int @ T ) ) ) ).
% surjective_pairing
thf(fact_368_surjective__pairing,axiom,
! [T: produc369741286924889929st_int] :
( T
= ( produc8650753666468850689st_int @ ( produc2155683374303800813st_int @ T ) @ ( produc6244060863719409455st_int @ T ) ) ) ).
% surjective_pairing
thf(fact_369_surjective__pairing,axiom,
! [T: produc4435102495419491129nt_int] :
( T
= ( produc5790713362662368625nt_int @ ( produc9025023058967219549nt_int @ T ) @ ( produc179445755590504351nt_int @ T ) ) ) ).
% surjective_pairing
thf(fact_370_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_371_surjective__pairing,axiom,
! [T: produc4189061985984568957nt_int] :
( T
= ( produc3965054194175396271nt_int @ ( produc811350159336358595nt_int @ T ) @ ( produc30867750519413249nt_int @ T ) ) ) ).
% surjective_pairing
thf(fact_372_prod_Oexhaust__sel,axiom,
! [Prod: produc6425607678544837394st_int] :
( Prod
= ( produc5086643055186798020st_int @ ( produc8941309049508465880st_int @ Prod ) @ ( produc1296517004818028822st_int @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_373_prod_Oexhaust__sel,axiom,
! [Prod: produc369741286924889929st_int] :
( Prod
= ( produc8650753666468850689st_int @ ( produc2155683374303800813st_int @ Prod ) @ ( produc6244060863719409455st_int @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_374_prod_Oexhaust__sel,axiom,
! [Prod: produc4435102495419491129nt_int] :
( Prod
= ( produc5790713362662368625nt_int @ ( produc9025023058967219549nt_int @ Prod ) @ ( produc179445755590504351nt_int @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_375_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_376_prod_Oexhaust__sel,axiom,
! [Prod: produc4189061985984568957nt_int] :
( Prod
= ( produc3965054194175396271nt_int @ ( produc811350159336358595nt_int @ Prod ) @ ( produc30867750519413249nt_int @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_377_drop__take,axiom,
! [N: nat,M: nat,Xs: list_int] :
( ( drop_int @ N @ ( take_int @ M @ Xs ) )
= ( take_int @ ( minus_minus_nat @ M @ N ) @ ( drop_int @ N @ Xs ) ) ) ).
% drop_take
thf(fact_378_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_379_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_380_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_381_pair__list__eqI,axiom,
! [Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int] :
( ( ( map_Pr3224476576807795204nt_int @ produc8941309049508465880st_int @ Xs )
= ( map_Pr3224476576807795204nt_int @ produc8941309049508465880st_int @ Ys ) )
=> ( ( ( map_Pr4101632105909876081st_int @ produc1296517004818028822st_int @ Xs )
= ( map_Pr4101632105909876081st_int @ produc1296517004818028822st_int @ Ys ) )
=> ( Xs = Ys ) ) ) ).
% pair_list_eqI
thf(fact_382_pair__list__eqI,axiom,
! [Xs: list_P8654944379044127311st_int,Ys: list_P8654944379044127311st_int] :
( ( ( map_Pr4838045029083743153ar_int @ produc2155683374303800813st_int @ Xs )
= ( map_Pr4838045029083743153ar_int @ produc2155683374303800813st_int @ Ys ) )
=> ( ( ( map_Pr7594757675026762433st_int @ produc6244060863719409455st_int @ Xs )
= ( map_Pr7594757675026762433st_int @ produc6244060863719409455st_int @ Ys ) )
=> ( Xs = Ys ) ) ) ).
% pair_list_eqI
thf(fact_383_pair__list__eqI,axiom,
! [Xs: list_P98921565119350719nt_int,Ys: list_P98921565119350719nt_int] :
( ( ( map_Pr6267657629804605729ar_int @ produc9025023058967219549nt_int @ Xs )
= ( map_Pr6267657629804605729ar_int @ produc9025023058967219549nt_int @ Ys ) )
=> ( ( ( map_Pr7025006424454111649nt_int @ produc179445755590504351nt_int @ Xs )
= ( map_Pr7025006424454111649nt_int @ produc179445755590504351nt_int @ Ys ) )
=> ( Xs = Ys ) ) ) ).
% pair_list_eqI
thf(fact_384_pair__list__eqI,axiom,
! [Xs: list_P5707943133018811711nt_int,Ys: list_P5707943133018811711nt_int] :
( ( ( map_Pr6494458893431244577nt_int @ product_fst_int_int @ Xs )
= ( map_Pr6494458893431244577nt_int @ product_fst_int_int @ Ys ) )
=> ( ( ( map_Pr6494458893431244577nt_int @ product_snd_int_int @ Xs )
= ( map_Pr6494458893431244577nt_int @ product_snd_int_int @ Ys ) )
=> ( Xs = Ys ) ) ) ).
% pair_list_eqI
thf(fact_385_pair__list__eqI,axiom,
! [Xs: list_P7813408393781968525nt_int,Ys: list_P7813408393781968525nt_int] :
( ( ( map_Pr7817743546820071988t_char @ produc811350159336358595nt_int @ Xs )
= ( map_Pr7817743546820071988t_char @ produc811350159336358595nt_int @ Ys ) )
=> ( ( ( map_Pr254812380637628598nt_int @ produc30867750519413249nt_int @ Xs )
= ( map_Pr254812380637628598nt_int @ produc30867750519413249nt_int @ Ys ) )
=> ( Xs = Ys ) ) ) ).
% pair_list_eqI
thf(fact_386_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_387_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_388_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_389_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_390_diff__zero,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_zero
thf(fact_391_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_392_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_393_diff__0__right,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_0_right
thf(fact_394_diff__self,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% diff_self
thf(fact_395_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_396_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: int > produc369741286924889929st_int > $o,X3: int,Y: produc369741286924889929st_int,A: produc6425607678544837394st_int] :
( ( P @ X3 @ Y )
=> ( ( A
= ( produc5086643055186798020st_int @ X3 @ Y ) )
=> ( P @ ( produc8941309049508465880st_int @ A ) @ ( produc1296517004818028822st_int @ A ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_397_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: ( list_char > int ) > list_int > $o,X3: list_char > int,Y: list_int,A: produc369741286924889929st_int] :
( ( P @ X3 @ Y )
=> ( ( A
= ( produc8650753666468850689st_int @ X3 @ Y ) )
=> ( P @ ( produc2155683374303800813st_int @ A ) @ ( produc6244060863719409455st_int @ A ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_398_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: ( list_char > int ) > int > $o,X3: list_char > int,Y: int,A: produc4435102495419491129nt_int] :
( ( P @ X3 @ Y )
=> ( ( A
= ( produc5790713362662368625nt_int @ X3 @ Y ) )
=> ( P @ ( produc9025023058967219549nt_int @ A ) @ ( produc179445755590504351nt_int @ A ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_399_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: int > int > $o,X3: int,Y: int,A: product_prod_int_int] :
( ( P @ X3 @ Y )
=> ( ( A
= ( product_Pair_int_int @ X3 @ Y ) )
=> ( P @ ( product_fst_int_int @ A ) @ ( product_snd_int_int @ A ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_400_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: list_list_char > produc4435102495419491129nt_int > $o,X3: list_list_char,Y: produc4435102495419491129nt_int,A: produc4189061985984568957nt_int] :
( ( P @ X3 @ Y )
=> ( ( A
= ( produc3965054194175396271nt_int @ X3 @ Y ) )
=> ( P @ ( produc811350159336358595nt_int @ A ) @ ( produc30867750519413249nt_int @ A ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_401_conjI__realizer,axiom,
! [P: int > $o,P2: int,Q2: produc369741286924889929st_int > $o,Q: produc369741286924889929st_int] :
( ( P @ P2 )
=> ( ( Q2 @ Q )
=> ( ( P @ ( produc8941309049508465880st_int @ ( produc5086643055186798020st_int @ P2 @ Q ) ) )
& ( Q2 @ ( produc1296517004818028822st_int @ ( produc5086643055186798020st_int @ P2 @ Q ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_402_conjI__realizer,axiom,
! [P: ( list_char > int ) > $o,P2: list_char > int,Q2: list_int > $o,Q: list_int] :
( ( P @ P2 )
=> ( ( Q2 @ Q )
=> ( ( P @ ( produc2155683374303800813st_int @ ( produc8650753666468850689st_int @ P2 @ Q ) ) )
& ( Q2 @ ( produc6244060863719409455st_int @ ( produc8650753666468850689st_int @ P2 @ Q ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_403_conjI__realizer,axiom,
! [P: ( list_char > int ) > $o,P2: list_char > int,Q2: int > $o,Q: int] :
( ( P @ P2 )
=> ( ( Q2 @ Q )
=> ( ( P @ ( produc9025023058967219549nt_int @ ( produc5790713362662368625nt_int @ P2 @ Q ) ) )
& ( Q2 @ ( produc179445755590504351nt_int @ ( produc5790713362662368625nt_int @ P2 @ Q ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_404_conjI__realizer,axiom,
! [P: int > $o,P2: int,Q2: int > $o,Q: int] :
( ( P @ P2 )
=> ( ( Q2 @ Q )
=> ( ( P @ ( product_fst_int_int @ ( product_Pair_int_int @ P2 @ Q ) ) )
& ( Q2 @ ( product_snd_int_int @ ( product_Pair_int_int @ P2 @ Q ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_405_conjI__realizer,axiom,
! [P: list_list_char > $o,P2: list_list_char,Q2: produc4435102495419491129nt_int > $o,Q: produc4435102495419491129nt_int] :
( ( P @ P2 )
=> ( ( Q2 @ Q )
=> ( ( P @ ( produc811350159336358595nt_int @ ( produc3965054194175396271nt_int @ P2 @ Q ) ) )
& ( Q2 @ ( produc30867750519413249nt_int @ ( produc3965054194175396271nt_int @ P2 @ Q ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_406_exI__realizer,axiom,
! [P: produc369741286924889929st_int > int > $o,Y: produc369741286924889929st_int,X3: int] :
( ( P @ Y @ X3 )
=> ( P @ ( produc1296517004818028822st_int @ ( produc5086643055186798020st_int @ X3 @ Y ) ) @ ( produc8941309049508465880st_int @ ( produc5086643055186798020st_int @ X3 @ Y ) ) ) ) ).
% exI_realizer
thf(fact_407_exI__realizer,axiom,
! [P: list_int > ( list_char > int ) > $o,Y: list_int,X3: list_char > int] :
( ( P @ Y @ X3 )
=> ( P @ ( produc6244060863719409455st_int @ ( produc8650753666468850689st_int @ X3 @ Y ) ) @ ( produc2155683374303800813st_int @ ( produc8650753666468850689st_int @ X3 @ Y ) ) ) ) ).
% exI_realizer
thf(fact_408_exI__realizer,axiom,
! [P: int > ( list_char > int ) > $o,Y: int,X3: list_char > int] :
( ( P @ Y @ X3 )
=> ( P @ ( produc179445755590504351nt_int @ ( produc5790713362662368625nt_int @ X3 @ Y ) ) @ ( produc9025023058967219549nt_int @ ( produc5790713362662368625nt_int @ X3 @ Y ) ) ) ) ).
% exI_realizer
thf(fact_409_exI__realizer,axiom,
! [P: int > int > $o,Y: int,X3: int] :
( ( P @ Y @ X3 )
=> ( P @ ( product_snd_int_int @ ( product_Pair_int_int @ X3 @ Y ) ) @ ( product_fst_int_int @ ( product_Pair_int_int @ X3 @ Y ) ) ) ) ).
% exI_realizer
thf(fact_410_exI__realizer,axiom,
! [P: produc4435102495419491129nt_int > list_list_char > $o,Y: produc4435102495419491129nt_int,X3: list_list_char] :
( ( P @ Y @ X3 )
=> ( P @ ( produc30867750519413249nt_int @ ( produc3965054194175396271nt_int @ X3 @ Y ) ) @ ( produc811350159336358595nt_int @ ( produc3965054194175396271nt_int @ X3 @ Y ) ) ) ) ).
% exI_realizer
thf(fact_411_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_412_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_413_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_414_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_415_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_416_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_417_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_418_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_419_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_420_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_421_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_422_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_423_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_424_take__all,axiom,
! [Xs: list_int,N: nat] :
( ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ N )
=> ( ( take_int @ N @ Xs )
= Xs ) ) ).
% take_all
thf(fact_425_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_426_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_427_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_428_take__all__iff,axiom,
! [N: nat,Xs: list_int] :
( ( ( take_int @ N @ Xs )
= Xs )
= ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ N ) ) ).
% take_all_iff
thf(fact_429_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_430_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_431_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_432_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_433_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_434_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_435_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_436_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ B ) )
=> ? [X: nat] :
( ( P @ X )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_437_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_438_lift__Suc__antimono__le,axiom,
! [F: nat > int,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_less_eq_int @ ( F @ N2 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_439_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_440_lift__Suc__mono__le,axiom,
! [F: nat > int,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_less_eq_int @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_441_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_442_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_443_zero__le,axiom,
! [X3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X3 ) ).
% zero_le
thf(fact_444_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_445_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_446_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_447_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_448_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_449_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_450_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_451_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_452_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_453_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_454_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_455_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_456_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_457_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_458_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_459_Suc__le__D,axiom,
! [N: nat,M6: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M6 )
=> ? [M4: nat] :
( M6
= ( suc @ M4 ) ) ) ).
% Suc_le_D
thf(fact_460_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_461_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_462_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_463_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M2: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N3 )
=> ( P @ M2 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_464_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_465_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X: nat] : ( R @ X @ X )
=> ( ! [X: nat,Y4: nat,Z2: nat] :
( ( R @ X @ Y4 )
=> ( ( R @ Y4 @ Z2 )
=> ( R @ X @ Z2 ) ) )
=> ( ! [N3: nat] : ( R @ N3 @ ( suc @ N3 ) )
=> ( R @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_466_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_467_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_468_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_469_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M5: nat,N4: nat] :
( ( ord_less_nat @ M5 @ N4 )
| ( M5 = N4 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_470_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_471_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M5: nat,N4: nat] :
( ( ord_less_eq_nat @ M5 @ N4 )
& ( M5 != N4 ) ) ) ) ).
% nat_less_le
thf(fact_472_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_473_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_474_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_475_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_476_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_477_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_478_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_479_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_480_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_481_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_482_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_483_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_484_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N4: nat] : ( ord_less_eq_nat @ ( suc @ N4 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_485_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_486_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_487_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_488_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_489_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_490_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_491_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_492_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_493_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_494_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_495_zero__reorient,axiom,
! [X3: nat] :
( ( zero_zero_nat = X3 )
= ( X3 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_496_zero__reorient,axiom,
! [X3: int] :
( ( zero_zero_int = X3 )
= ( X3 = zero_zero_int ) ) ).
% zero_reorient
thf(fact_497_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N )
& ! [I4: nat] :
( ( ord_less_eq_nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_498_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_499_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_500_one__reorient,axiom,
! [X3: nat] :
( ( one_one_nat = X3 )
= ( X3 = one_one_nat ) ) ).
% one_reorient
thf(fact_501_one__reorient,axiom,
! [X3: int] :
( ( one_one_int = X3 )
= ( X3 = one_one_int ) ) ).
% one_reorient
thf(fact_502_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_503_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_504_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_505_iexec_Ocases,axiom,
! [X3: produc5995290525303592096st_int] :
~ ! [Ins: instr,I2: int,S3: list_char > int,Stk: list_int] :
( X3
!= ( produc1484136438817787418st_int @ Ins @ ( produc5086643055186798020st_int @ I2 @ ( produc8650753666468850689st_int @ S3 @ Stk ) ) ) ) ).
% iexec.cases
thf(fact_506_nth__take__lemma,axiom,
! [K: nat,Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int] :
( ( ord_less_eq_nat @ K @ ( size_s229347708945616566st_int @ Xs ) )
=> ( ( ord_less_eq_nat @ K @ ( size_s229347708945616566st_int @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( ( nth_Pr8327173284404383691st_int @ Xs @ I2 )
= ( nth_Pr8327173284404383691st_int @ Ys @ I2 ) ) )
=> ( ( take_P1004856038796978596st_int @ K @ Xs )
= ( take_P1004856038796978596st_int @ K @ Ys ) ) ) ) ) ).
% nth_take_lemma
thf(fact_507_nth__take__lemma,axiom,
! [K: nat,Xs: list_instr,Ys: list_instr] :
( ( ord_less_eq_nat @ K @ ( size_size_list_instr @ Xs ) )
=> ( ( ord_less_eq_nat @ K @ ( size_size_list_instr @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( ( nth_instr @ Xs @ I2 )
= ( nth_instr @ Ys @ I2 ) ) )
=> ( ( take_instr @ K @ Xs )
= ( take_instr @ K @ Ys ) ) ) ) ) ).
% nth_take_lemma
thf(fact_508_nth__take__lemma,axiom,
! [K: nat,Xs: list_list_char,Ys: list_list_char] :
( ( ord_less_eq_nat @ K @ ( size_s356637359517785349t_char @ Xs ) )
=> ( ( ord_less_eq_nat @ K @ ( size_s356637359517785349t_char @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( ( nth_list_char @ Xs @ I2 )
= ( nth_list_char @ Ys @ I2 ) ) )
=> ( ( take_list_char @ K @ Xs )
= ( take_list_char @ K @ Ys ) ) ) ) ) ).
% nth_take_lemma
thf(fact_509_nth__take__lemma,axiom,
! [K: nat,Xs: list_int,Ys: list_int] :
( ( ord_less_eq_nat @ K @ ( size_size_list_int @ Xs ) )
=> ( ( ord_less_eq_nat @ K @ ( size_size_list_int @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( ( nth_int @ Xs @ I2 )
= ( nth_int @ Ys @ I2 ) ) )
=> ( ( take_int @ K @ Xs )
= ( take_int @ K @ Ys ) ) ) ) ) ).
% nth_take_lemma
thf(fact_510_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_511_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_512_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_513_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_514_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_515_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_516_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_517_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_518_eq__iff__diff__eq__0,axiom,
( ( ^ [Y3: int,Z: int] : ( Y3 = Z ) )
= ( ^ [A5: int,B4: int] :
( ( minus_minus_int @ A5 @ B4 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_519_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_520_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_521_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_522_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_523_exE__realizer_H,axiom,
! [P: produc369741286924889929st_int > int > $o,P2: produc6425607678544837394st_int] :
( ( P @ ( produc1296517004818028822st_int @ P2 ) @ ( produc8941309049508465880st_int @ P2 ) )
=> ~ ! [X: int,Y4: produc369741286924889929st_int] :
~ ( P @ Y4 @ X ) ) ).
% exE_realizer'
thf(fact_524_exE__realizer_H,axiom,
! [P: list_int > ( list_char > int ) > $o,P2: produc369741286924889929st_int] :
( ( P @ ( produc6244060863719409455st_int @ P2 ) @ ( produc2155683374303800813st_int @ P2 ) )
=> ~ ! [X: list_char > int,Y4: list_int] :
~ ( P @ Y4 @ X ) ) ).
% exE_realizer'
thf(fact_525_exE__realizer_H,axiom,
! [P: int > ( list_char > int ) > $o,P2: produc4435102495419491129nt_int] :
( ( P @ ( produc179445755590504351nt_int @ P2 ) @ ( produc9025023058967219549nt_int @ P2 ) )
=> ~ ! [X: list_char > int,Y4: int] :
~ ( P @ Y4 @ X ) ) ).
% exE_realizer'
thf(fact_526_exE__realizer_H,axiom,
! [P: int > int > $o,P2: product_prod_int_int] :
( ( P @ ( product_snd_int_int @ P2 ) @ ( product_fst_int_int @ P2 ) )
=> ~ ! [X: int,Y4: int] :
~ ( P @ Y4 @ X ) ) ).
% exE_realizer'
thf(fact_527_exE__realizer_H,axiom,
! [P: produc4435102495419491129nt_int > list_list_char > $o,P2: produc4189061985984568957nt_int] :
( ( P @ ( produc30867750519413249nt_int @ P2 ) @ ( produc811350159336358595nt_int @ P2 ) )
=> ~ ! [X: list_list_char,Y4: produc4435102495419491129nt_int] :
~ ( P @ Y4 @ X ) ) ).
% exE_realizer'
thf(fact_528_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_529_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_530_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_531_zdiff__int__split,axiom,
! [P: int > $o,X3: nat,Y: nat] :
( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X3 @ Y ) ) )
= ( ( ( ord_less_eq_nat @ Y @ X3 )
=> ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X3 ) @ ( semiri1314217659103216013at_int @ Y ) ) ) )
& ( ( ord_less_nat @ X3 @ Y )
=> ( P @ zero_zero_int ) ) ) ) ).
% zdiff_int_split
thf(fact_532_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_533_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_534_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_535_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_536_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_537_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N3: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% pos_int_cases
thf(fact_538_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
& ( K
= ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_539_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_540_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_541_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_542_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_543_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_544_imp__le__cong,axiom,
! [X3: int,X5: int,P: $o,P3: $o] :
( ( X3 = X5 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X5 )
=> ( P = P3 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X3 )
=> P )
= ( ( ord_less_eq_int @ zero_zero_int @ X5 )
=> P3 ) ) ) ) ).
% imp_le_cong
thf(fact_545_conj__le__cong,axiom,
! [X3: int,X5: int,P: $o,P3: $o] :
( ( X3 = X5 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X5 )
=> ( P = P3 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X3 )
& P )
= ( ( ord_less_eq_int @ zero_zero_int @ X5 )
& P3 ) ) ) ) ).
% conj_le_cong
thf(fact_546_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_547_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
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A3 )
& ( P @ X2 ) ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ B5 )
=> ( P @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_548_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
@ ^ [X2: int] :
( ( member_int @ X2 @ A3 )
& ( P @ X2 ) ) ) )
= ( ! [X2: int] :
( ( member_int @ X2 @ B5 )
=> ( P @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_549_subset__CollectI,axiom,
! [B5: set_nat,A3: set_nat,Q2: nat > $o,P: nat > $o] :
( ( ord_less_eq_set_nat @ B5 @ A3 )
=> ( ! [X: nat] :
( ( member_nat @ X @ B5 )
=> ( ( Q2 @ X )
=> ( P @ X ) ) )
=> ( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ B5 )
& ( Q2 @ X2 ) ) )
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A3 )
& ( P @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_550_subset__CollectI,axiom,
! [B5: set_int,A3: set_int,Q2: int > $o,P: int > $o] :
( ( ord_less_eq_set_int @ B5 @ A3 )
=> ( ! [X: int] :
( ( member_int @ X @ B5 )
=> ( ( Q2 @ X )
=> ( P @ X ) ) )
=> ( ord_less_eq_set_int
@ ( collect_int
@ ^ [X2: int] :
( ( member_int @ X2 @ B5 )
& ( Q2 @ X2 ) ) )
@ ( collect_int
@ ^ [X2: int] :
( ( member_int @ X2 @ A3 )
& ( P @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_551_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ? [N3: nat] :
( K
= ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_552_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ~ ! [N3: nat] :
( K
!= ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% nonneg_int_cases
thf(fact_553_int__le__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_eq_int @ I @ K )
=> ( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_le_induct
thf(fact_554_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_555_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_556_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_557_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_558_verit__comp__simplify1_I2_J,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_559_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_560_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_561_minf_I7_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z2 )
=> ~ ( ord_less_nat @ T @ X6 ) ) ).
% minf(7)
thf(fact_562_minf_I7_J,axiom,
! [T: int] :
? [Z2: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z2 )
=> ~ ( ord_less_int @ T @ X6 ) ) ).
% minf(7)
thf(fact_563_minf_I5_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z2 )
=> ( ord_less_nat @ X6 @ T ) ) ).
% minf(5)
thf(fact_564_minf_I5_J,axiom,
! [T: int] :
? [Z2: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z2 )
=> ( ord_less_int @ X6 @ T ) ) ).
% minf(5)
thf(fact_565_minf_I4_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z2 )
=> ( X6 != T ) ) ).
% minf(4)
thf(fact_566_minf_I4_J,axiom,
! [T: int] :
? [Z2: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z2 )
=> ( X6 != T ) ) ).
% minf(4)
thf(fact_567_minf_I3_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z2 )
=> ( X6 != T ) ) ).
% minf(3)
thf(fact_568_minf_I3_J,axiom,
! [T: int] :
? [Z2: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z2 )
=> ( X6 != T ) ) ).
% minf(3)
thf(fact_569_minf_I2_J,axiom,
! [P: nat > $o,P3: nat > $o,Q2: nat > $o,Q3: nat > $o] :
( ? [Z4: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z4 )
=> ( ( P @ X )
= ( P3 @ X ) ) )
=> ( ? [Z4: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z4 )
=> ( ( Q2 @ X )
= ( Q3 @ X ) ) )
=> ? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z2 )
=> ( ( ( P @ X6 )
| ( Q2 @ X6 ) )
= ( ( P3 @ X6 )
| ( Q3 @ X6 ) ) ) ) ) ) ).
% minf(2)
thf(fact_570_minf_I2_J,axiom,
! [P: int > $o,P3: int > $o,Q2: int > $o,Q3: int > $o] :
( ? [Z4: int] :
! [X: int] :
( ( ord_less_int @ X @ Z4 )
=> ( ( P @ X )
= ( P3 @ X ) ) )
=> ( ? [Z4: int] :
! [X: int] :
( ( ord_less_int @ X @ Z4 )
=> ( ( Q2 @ X )
= ( Q3 @ X ) ) )
=> ? [Z2: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z2 )
=> ( ( ( P @ X6 )
| ( Q2 @ X6 ) )
= ( ( P3 @ X6 )
| ( Q3 @ X6 ) ) ) ) ) ) ).
% minf(2)
thf(fact_571_minf_I1_J,axiom,
! [P: nat > $o,P3: nat > $o,Q2: nat > $o,Q3: nat > $o] :
( ? [Z4: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z4 )
=> ( ( P @ X )
= ( P3 @ X ) ) )
=> ( ? [Z4: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z4 )
=> ( ( Q2 @ X )
= ( Q3 @ X ) ) )
=> ? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z2 )
=> ( ( ( P @ X6 )
& ( Q2 @ X6 ) )
= ( ( P3 @ X6 )
& ( Q3 @ X6 ) ) ) ) ) ) ).
% minf(1)
thf(fact_572_minf_I1_J,axiom,
! [P: int > $o,P3: int > $o,Q2: int > $o,Q3: int > $o] :
( ? [Z4: int] :
! [X: int] :
( ( ord_less_int @ X @ Z4 )
=> ( ( P @ X )
= ( P3 @ X ) ) )
=> ( ? [Z4: int] :
! [X: int] :
( ( ord_less_int @ X @ Z4 )
=> ( ( Q2 @ X )
= ( Q3 @ X ) ) )
=> ? [Z2: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z2 )
=> ( ( ( P @ X6 )
& ( Q2 @ X6 ) )
= ( ( P3 @ X6 )
& ( Q3 @ X6 ) ) ) ) ) ) ).
% minf(1)
thf(fact_573_pinf_I7_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z2 @ X6 )
=> ( ord_less_nat @ T @ X6 ) ) ).
% pinf(7)
thf(fact_574_pinf_I7_J,axiom,
! [T: int] :
? [Z2: int] :
! [X6: int] :
( ( ord_less_int @ Z2 @ X6 )
=> ( ord_less_int @ T @ X6 ) ) ).
% pinf(7)
thf(fact_575_pinf_I5_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z2 @ X6 )
=> ~ ( ord_less_nat @ X6 @ T ) ) ).
% pinf(5)
thf(fact_576_pinf_I5_J,axiom,
! [T: int] :
? [Z2: int] :
! [X6: int] :
( ( ord_less_int @ Z2 @ X6 )
=> ~ ( ord_less_int @ X6 @ T ) ) ).
% pinf(5)
thf(fact_577_pinf_I4_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z2 @ X6 )
=> ( X6 != T ) ) ).
% pinf(4)
thf(fact_578_pinf_I4_J,axiom,
! [T: int] :
? [Z2: int] :
! [X6: int] :
( ( ord_less_int @ Z2 @ X6 )
=> ( X6 != T ) ) ).
% pinf(4)
thf(fact_579_pinf_I3_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z2 @ X6 )
=> ( X6 != T ) ) ).
% pinf(3)
thf(fact_580_pinf_I3_J,axiom,
! [T: int] :
? [Z2: int] :
! [X6: int] :
( ( ord_less_int @ Z2 @ X6 )
=> ( X6 != T ) ) ).
% pinf(3)
thf(fact_581_pinf_I2_J,axiom,
! [P: nat > $o,P3: nat > $o,Q2: nat > $o,Q3: nat > $o] :
( ? [Z4: nat] :
! [X: nat] :
( ( ord_less_nat @ Z4 @ X )
=> ( ( P @ X )
= ( P3 @ X ) ) )
=> ( ? [Z4: nat] :
! [X: nat] :
( ( ord_less_nat @ Z4 @ X )
=> ( ( Q2 @ X )
= ( Q3 @ X ) ) )
=> ? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z2 @ X6 )
=> ( ( ( P @ X6 )
| ( Q2 @ X6 ) )
= ( ( P3 @ X6 )
| ( Q3 @ X6 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_582_pinf_I2_J,axiom,
! [P: int > $o,P3: int > $o,Q2: int > $o,Q3: int > $o] :
( ? [Z4: int] :
! [X: int] :
( ( ord_less_int @ Z4 @ X )
=> ( ( P @ X )
= ( P3 @ X ) ) )
=> ( ? [Z4: int] :
! [X: int] :
( ( ord_less_int @ Z4 @ X )
=> ( ( Q2 @ X )
= ( Q3 @ X ) ) )
=> ? [Z2: int] :
! [X6: int] :
( ( ord_less_int @ Z2 @ X6 )
=> ( ( ( P @ X6 )
| ( Q2 @ X6 ) )
= ( ( P3 @ X6 )
| ( Q3 @ X6 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_583_pinf_I1_J,axiom,
! [P: nat > $o,P3: nat > $o,Q2: nat > $o,Q3: nat > $o] :
( ? [Z4: nat] :
! [X: nat] :
( ( ord_less_nat @ Z4 @ X )
=> ( ( P @ X )
= ( P3 @ X ) ) )
=> ( ? [Z4: nat] :
! [X: nat] :
( ( ord_less_nat @ Z4 @ X )
=> ( ( Q2 @ X )
= ( Q3 @ X ) ) )
=> ? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z2 @ X6 )
=> ( ( ( P @ X6 )
& ( Q2 @ X6 ) )
= ( ( P3 @ X6 )
& ( Q3 @ X6 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_584_pinf_I1_J,axiom,
! [P: int > $o,P3: int > $o,Q2: int > $o,Q3: int > $o] :
( ? [Z4: int] :
! [X: int] :
( ( ord_less_int @ Z4 @ X )
=> ( ( P @ X )
= ( P3 @ X ) ) )
=> ( ? [Z4: int] :
! [X: int] :
( ( ord_less_int @ Z4 @ X )
=> ( ( Q2 @ X )
= ( Q3 @ X ) ) )
=> ? [Z2: int] :
! [X6: int] :
( ( ord_less_int @ Z2 @ X6 )
=> ( ( ( P @ X6 )
& ( Q2 @ X6 ) )
= ( ( P3 @ X6 )
& ( Q3 @ X6 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_585_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_586_verit__comp__simplify1_I1_J,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_587_linorder__neqE__linordered__idom,axiom,
! [X3: int,Y: int] :
( ( X3 != Y )
=> ( ~ ( ord_less_int @ X3 @ Y )
=> ( ord_less_int @ Y @ X3 ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_588_bounded__Max__nat,axiom,
! [P: nat > $o,X3: nat,M7: nat] :
( ( P @ X3 )
=> ( ! [X: nat] :
( ( P @ X )
=> ( ord_less_eq_nat @ X @ M7 ) )
=> ~ ! [M4: nat] :
( ( P @ M4 )
=> ~ ! [X6: nat] :
( ( P @ X6 )
=> ( ord_less_eq_nat @ X6 @ M4 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_589_int__int__eq,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% int_int_eq
thf(fact_590_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_591_nat__int__comparison_I1_J,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [A5: nat,B4: nat] :
( ( semiri1314217659103216013at_int @ A5 )
= ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_592_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus_int @ K @ zero_zero_int )
= K ) ).
% minus_int_code(1)
thf(fact_593_int__less__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_int @ I @ K )
=> ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
=> ( ! [I2: int] :
( ( ord_less_int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_less_induct
thf(fact_594_minf_I8_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z2 )
=> ~ ( ord_less_eq_nat @ T @ X6 ) ) ).
% minf(8)
thf(fact_595_minf_I8_J,axiom,
! [T: int] :
? [Z2: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z2 )
=> ~ ( ord_less_eq_int @ T @ X6 ) ) ).
% minf(8)
thf(fact_596_minf_I6_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z2 )
=> ( ord_less_eq_nat @ X6 @ T ) ) ).
% minf(6)
thf(fact_597_minf_I6_J,axiom,
! [T: int] :
? [Z2: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z2 )
=> ( ord_less_eq_int @ X6 @ T ) ) ).
% minf(6)
thf(fact_598_pinf_I8_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z2 @ X6 )
=> ( ord_less_eq_nat @ T @ X6 ) ) ).
% pinf(8)
thf(fact_599_pinf_I8_J,axiom,
! [T: int] :
? [Z2: int] :
! [X6: int] :
( ( ord_less_int @ Z2 @ X6 )
=> ( ord_less_eq_int @ T @ X6 ) ) ).
% pinf(8)
thf(fact_600_pinf_I6_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z2 @ X6 )
=> ~ ( ord_less_eq_nat @ X6 @ T ) ) ).
% pinf(6)
thf(fact_601_pinf_I6_J,axiom,
! [T: int] :
? [Z2: int] :
! [X6: int] :
( ( ord_less_int @ Z2 @ X6 )
=> ~ ( ord_less_eq_int @ X6 @ T ) ) ).
% pinf(6)
thf(fact_602_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_603_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_604_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_605_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_606_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_607_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_608_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_609_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_610_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_611_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_612_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_613_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_614_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_615_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_616_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_617_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_618_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_619_int__diff__cases,axiom,
! [Z3: int] :
~ ! [M4: nat,N3: nat] :
( Z3
!= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M4 ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% int_diff_cases
thf(fact_620_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_621_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_622_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_623_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_624_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_625_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_626_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% zero_less_one_class.zero_le_one
thf(fact_627_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_628_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_629_execl__sub,axiom,
! [P: list_instr,P3: list_instr,P4: list_instr,Cfs2: list_P419547167332244514st_int,M: nat,N: nat] :
( ( execl @ ( append_instr @ P @ ( append_instr @ P3 @ P4 ) ) @ Cfs2 )
=> ( ! [X: nat] :
( ( member_nat @ X @ ( set_or4665077453230672383an_nat @ M @ N ) )
=> ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P ) ) @ ( produc8941309049508465880st_int @ ( nth_Pr8327173284404383691st_int @ Cfs2 @ X ) ) )
& ( ord_less_int @ ( minus_minus_int @ ( produc8941309049508465880st_int @ ( nth_Pr8327173284404383691st_int @ Cfs2 @ X ) ) @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P ) ) ) @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P3 ) ) ) ) )
=> ( execl @ P3
@ ( map_Pr8625093460839654122st_int
@ ^ [Cf: produc6425607678544837394st_int] : ( produc5086643055186798020st_int @ ( minus_minus_int @ ( produc8941309049508465880st_int @ Cf ) @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P ) ) ) @ ( produc1296517004818028822st_int @ Cf ) )
@ ( drop_P7838799151754935884st_int @ M @ ( take_P1004856038796978596st_int @ ( suc @ N ) @ Cfs2 ) ) ) ) ) ) ).
% execl_sub
thf(fact_630_inth__append,axiom,
! [I: int,Xs: list_P419547167332244514st_int,Ys: 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 @ Ys ) @ I )
= ( inth_P8039235741264645271st_int @ Xs @ I ) ) )
& ( ~ ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_s229347708945616566st_int @ Xs ) ) )
=> ( ( inth_P8039235741264645271st_int @ ( append5028292981072010935st_int @ Xs @ Ys ) @ I )
= ( inth_P8039235741264645271st_int @ Ys @ ( minus_minus_int @ I @ ( semiri1314217659103216013at_int @ ( size_s229347708945616566st_int @ Xs ) ) ) ) ) ) ) ) ).
% inth_append
thf(fact_631_inth__append,axiom,
! [I: int,Xs: list_instr,Ys: 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 @ Ys ) @ I )
= ( inth_instr @ Xs @ I ) ) )
& ( ~ ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ Xs ) ) )
=> ( ( inth_instr @ ( append_instr @ Xs @ Ys ) @ I )
= ( inth_instr @ Ys @ ( minus_minus_int @ I @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ Xs ) ) ) ) ) ) ) ) ).
% inth_append
thf(fact_632_inth__append,axiom,
! [I: int,Xs: list_list_char,Ys: 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 @ Ys ) @ I )
= ( inth_list_char @ Xs @ I ) ) )
& ( ~ ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ Xs ) ) )
=> ( ( inth_list_char @ ( append_list_char @ Xs @ Ys ) @ I )
= ( inth_list_char @ Ys @ ( minus_minus_int @ I @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ Xs ) ) ) ) ) ) ) ) ).
% inth_append
thf(fact_633_inth__append,axiom,
! [I: int,Xs: list_int,Ys: 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 @ Ys ) @ I )
= ( inth_int @ Xs @ I ) ) )
& ( ~ ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_size_list_int @ Xs ) ) )
=> ( ( inth_int @ ( append_int @ Xs @ Ys ) @ I )
= ( inth_int @ Ys @ ( minus_minus_int @ I @ ( semiri1314217659103216013at_int @ ( size_size_list_int @ Xs ) ) ) ) ) ) ) ) ).
% inth_append
thf(fact_634_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_635_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_636_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_637_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_638_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_639_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_640_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_641_inth__map,axiom,
! [I: int,Xs: list_int,F: int > instr] :
( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_size_list_int @ Xs ) ) )
=> ( ( inth_instr @ ( map_int_instr @ F @ Xs ) @ I )
= ( F @ ( inth_int @ Xs @ I ) ) ) ) ) ).
% inth_map
thf(fact_642_inth__map,axiom,
! [I: int,Xs: list_int,F: int > list_char] :
( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_size_list_int @ Xs ) ) )
=> ( ( inth_list_char @ ( map_int_list_char @ F @ Xs ) @ I )
= ( F @ ( inth_int @ Xs @ I ) ) ) ) ) ).
% inth_map
thf(fact_643_execl__last,axiom,
! [P: list_instr,Cfs2: list_P419547167332244514st_int,K: nat] :
( ( execl @ P @ Cfs2 )
=> ( ( ord_less_nat @ K @ ( size_s229347708945616566st_int @ Cfs2 ) )
=> ( ~ ( member_int @ ( produc8941309049508465880st_int @ ( nth_Pr8327173284404383691st_int @ Cfs2 @ K ) ) @ ( set_or4662586982721622107an_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P ) ) ) )
=> ( ( minus_minus_nat @ ( size_s229347708945616566st_int @ Cfs2 ) @ one_one_nat )
= K ) ) ) ) ).
% execl_last
thf(fact_644_subsetI,axiom,
! [A3: set_nat,B5: set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A3 )
=> ( member_nat @ X @ B5 ) )
=> ( ord_less_eq_set_nat @ A3 @ B5 ) ) ).
% subsetI
thf(fact_645_subsetI,axiom,
! [A3: set_int,B5: set_int] :
( ! [X: int] :
( ( member_int @ X @ A3 )
=> ( member_int @ X @ B5 ) )
=> ( ord_less_eq_set_int @ A3 @ B5 ) ) ).
% subsetI
thf(fact_646_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_647_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_648_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_649_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_650_verit__la__generic,axiom,
! [A: int,X3: int] :
( ( ord_less_eq_int @ A @ X3 )
| ( A = X3 )
| ( ord_less_eq_int @ X3 @ A ) ) ).
% verit_la_generic
thf(fact_651_execl__take,axiom,
! [P: list_instr,Cfs2: list_P419547167332244514st_int,N: nat] :
( ( execl @ P @ Cfs2 )
=> ( execl @ P @ ( take_P1004856038796978596st_int @ N @ Cfs2 ) ) ) ).
% execl_take
thf(fact_652_execl__drop,axiom,
! [P: list_instr,Cfs2: list_P419547167332244514st_int,N: nat] :
( ( execl @ P @ Cfs2 )
=> ( execl @ P @ ( drop_P7838799151754935884st_int @ N @ Cfs2 ) ) ) ).
% execl_drop
thf(fact_653_in__mono,axiom,
! [A3: set_nat,B5: set_nat,X3: nat] :
( ( ord_less_eq_set_nat @ A3 @ B5 )
=> ( ( member_nat @ X3 @ A3 )
=> ( member_nat @ X3 @ B5 ) ) ) ).
% in_mono
thf(fact_654_in__mono,axiom,
! [A3: set_int,B5: set_int,X3: int] :
( ( ord_less_eq_set_int @ A3 @ B5 )
=> ( ( member_int @ X3 @ A3 )
=> ( member_int @ X3 @ B5 ) ) ) ).
% in_mono
thf(fact_655_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_656_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_657_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
! [X2: nat] :
( ( member_nat @ X2 @ A6 )
=> ( member_nat @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_658_subset__eq,axiom,
( ord_less_eq_set_int
= ( ^ [A6: set_int,B6: set_int] :
! [X2: int] :
( ( member_int @ X2 @ A6 )
=> ( member_int @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_659_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_660_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_661_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_662_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_663_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_664_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_665_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_666_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_667_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_668_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_669_Collect__subset,axiom,
! [A3: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A3 )
& ( P @ X2 ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_670_Collect__subset,axiom,
! [A3: set_int,P: int > $o] :
( ord_less_eq_set_int
@ ( collect_int
@ ^ [X2: int] :
( ( member_int @ X2 @ A3 )
& ( P @ X2 ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_671_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
( ord_less_eq_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ A6 )
@ ^ [X2: nat] : ( member_nat @ X2 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_672_less__eq__set__def,axiom,
( ord_less_eq_set_int
= ( ^ [A6: set_int,B6: set_int] :
( ord_less_eq_int_o
@ ^ [X2: int] : ( member_int @ X2 @ A6 )
@ ^ [X2: int] : ( member_int @ X2 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_673_minus__set__def,axiom,
( minus_minus_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
( collect_nat
@ ( minus_minus_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ A6 )
@ ^ [X2: nat] : ( member_nat @ X2 @ B6 ) ) ) ) ) ).
% minus_set_def
thf(fact_674_minus__set__def,axiom,
( minus_minus_set_int
= ( ^ [A6: set_int,B6: set_int] :
( collect_int
@ ( minus_minus_int_o
@ ^ [X2: int] : ( member_int @ X2 @ A6 )
@ ^ [X2: int] : ( member_int @ X2 @ B6 ) ) ) ) ) ).
% minus_set_def
thf(fact_675_set__diff__eq,axiom,
( minus_minus_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A6 )
& ~ ( member_nat @ X2 @ B6 ) ) ) ) ) ).
% set_diff_eq
thf(fact_676_set__diff__eq,axiom,
( minus_minus_set_int
= ( ^ [A6: set_int,B6: set_int] :
( collect_int
@ ^ [X2: int] :
( ( member_int @ X2 @ A6 )
& ~ ( member_int @ X2 @ B6 ) ) ) ) ) ).
% set_diff_eq
thf(fact_677_execl__next,axiom,
! [P: list_instr,Cfs2: list_P419547167332244514st_int,K: nat] :
( ( execl @ P @ Cfs2 )
=> ( ( ord_less_nat @ K @ ( size_s229347708945616566st_int @ Cfs2 ) )
=> ( ( K
!= ( minus_minus_nat @ ( size_s229347708945616566st_int @ Cfs2 ) @ one_one_nat ) )
=> ( ( iexec_pred @ ( produc1484136438817787418st_int @ ( inth_instr @ P @ ( produc8941309049508465880st_int @ ( nth_Pr8327173284404383691st_int @ Cfs2 @ K ) ) ) @ ( nth_Pr8327173284404383691st_int @ Cfs2 @ K ) ) @ ( nth_Pr8327173284404383691st_int @ Cfs2 @ ( suc @ K ) ) )
& ( ord_less_eq_int @ zero_zero_int @ ( produc8941309049508465880st_int @ ( nth_Pr8327173284404383691st_int @ Cfs2 @ K ) ) )
& ( ord_less_int @ ( produc8941309049508465880st_int @ ( nth_Pr8327173284404383691st_int @ Cfs2 @ K ) ) @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P ) ) ) ) ) ) ) ).
% execl_next
thf(fact_678_execl__all__def,axiom,
( execl_all
= ( ^ [P5: list_instr,Cfs3: list_P419547167332244514st_int] :
( ( execl @ P5 @ Cfs3 )
& ( Cfs3 != nil_Pr749310521696113548st_int )
& ( ( produc8941309049508465880st_int @ ( nth_Pr8327173284404383691st_int @ Cfs3 @ zero_zero_nat ) )
= zero_zero_int )
& ~ ( member_int @ ( produc8941309049508465880st_int @ ( nth_Pr8327173284404383691st_int @ Cfs3 @ ( minus_minus_nat @ ( size_s229347708945616566st_int @ Cfs3 ) @ one_one_nat ) ) ) @ ( set_or4662586982721622107an_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P5 ) ) ) ) ) ) ) ).
% execl_all_def
thf(fact_679_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_680_to__m__state_Oelims,axiom,
! [X3: list_list_char,Xa: list_char > int,Xb: int,Y: int] :
( ( ( to_m_state @ X3 @ Xa @ Xb )
= Y )
=> ( ( ( ( ord_less_int @ zero_zero_int @ Xb )
& ( ord_less_eq_int @ Xb @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ X3 ) ) ) )
=> ( Y
= ( Xa @ ( inth_list_char @ X3 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ X3 ) ) @ Xb ) ) ) ) )
& ( ~ ( ( ord_less_int @ zero_zero_int @ Xb )
& ( ord_less_eq_int @ Xb @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ X3 ) ) ) )
=> ( Y = zero_zero_int ) ) ) ) ).
% to_m_state.elims
thf(fact_681_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_682_map__is__Nil__conv,axiom,
! [F: instr > produc6425607678544837394st_int,Xs: list_instr] :
( ( ( map_in7259965606695656109st_int @ F @ Xs )
= nil_Pr749310521696113548st_int )
= ( Xs = nil_instr ) ) ).
% map_is_Nil_conv
thf(fact_683_map__is__Nil__conv,axiom,
! [F: int > produc6425607678544837394st_int,Xs: list_int] :
( ( ( map_in8512336359594627332st_int @ F @ Xs )
= nil_Pr749310521696113548st_int )
= ( Xs = nil_int ) ) ).
% map_is_Nil_conv
thf(fact_684_map__is__Nil__conv,axiom,
! [F: produc6425607678544837394st_int > instr,Xs: list_P419547167332244514st_int] :
( ( ( map_Pr7514435936640407687_instr @ F @ Xs )
= nil_instr )
= ( Xs = nil_Pr749310521696113548st_int ) ) ).
% map_is_Nil_conv
thf(fact_685_map__is__Nil__conv,axiom,
! [F: instr > instr,Xs: list_instr] :
( ( ( map_instr_instr @ F @ Xs )
= nil_instr )
= ( Xs = nil_instr ) ) ).
% map_is_Nil_conv
thf(fact_686_map__is__Nil__conv,axiom,
! [F: int > instr,Xs: list_int] :
( ( ( map_int_instr @ F @ Xs )
= nil_instr )
= ( Xs = nil_int ) ) ).
% map_is_Nil_conv
thf(fact_687_map__is__Nil__conv,axiom,
! [F: produc6425607678544837394st_int > int,Xs: list_P419547167332244514st_int] :
( ( ( map_Pr3224476576807795204nt_int @ F @ Xs )
= nil_int )
= ( Xs = nil_Pr749310521696113548st_int ) ) ).
% map_is_Nil_conv
thf(fact_688_map__is__Nil__conv,axiom,
! [F: instr > int,Xs: list_instr] :
( ( ( map_instr_int @ F @ Xs )
= nil_int )
= ( Xs = nil_instr ) ) ).
% map_is_Nil_conv
thf(fact_689_map__is__Nil__conv,axiom,
! [F: int > int,Xs: list_int] :
( ( ( map_int_int @ F @ Xs )
= nil_int )
= ( Xs = nil_int ) ) ).
% map_is_Nil_conv
thf(fact_690_map__is__Nil__conv,axiom,
! [F: produc6425607678544837394st_int > produc6425607678544837394st_int,Xs: list_P419547167332244514st_int] :
( ( ( map_Pr8625093460839654122st_int @ F @ Xs )
= nil_Pr749310521696113548st_int )
= ( Xs = nil_Pr749310521696113548st_int ) ) ).
% map_is_Nil_conv
thf(fact_691_Nil__is__map__conv,axiom,
! [F: instr > produc6425607678544837394st_int,Xs: list_instr] :
( ( nil_Pr749310521696113548st_int
= ( map_in7259965606695656109st_int @ F @ Xs ) )
= ( Xs = nil_instr ) ) ).
% Nil_is_map_conv
thf(fact_692_Nil__is__map__conv,axiom,
! [F: int > produc6425607678544837394st_int,Xs: list_int] :
( ( nil_Pr749310521696113548st_int
= ( map_in8512336359594627332st_int @ F @ Xs ) )
= ( Xs = nil_int ) ) ).
% Nil_is_map_conv
thf(fact_693_Nil__is__map__conv,axiom,
! [F: produc6425607678544837394st_int > instr,Xs: list_P419547167332244514st_int] :
( ( nil_instr
= ( map_Pr7514435936640407687_instr @ F @ Xs ) )
= ( Xs = nil_Pr749310521696113548st_int ) ) ).
% Nil_is_map_conv
thf(fact_694_Nil__is__map__conv,axiom,
! [F: instr > instr,Xs: list_instr] :
( ( nil_instr
= ( map_instr_instr @ F @ Xs ) )
= ( Xs = nil_instr ) ) ).
% Nil_is_map_conv
thf(fact_695_Nil__is__map__conv,axiom,
! [F: int > instr,Xs: list_int] :
( ( nil_instr
= ( map_int_instr @ F @ Xs ) )
= ( Xs = nil_int ) ) ).
% Nil_is_map_conv
thf(fact_696_Nil__is__map__conv,axiom,
! [F: produc6425607678544837394st_int > int,Xs: list_P419547167332244514st_int] :
( ( nil_int
= ( map_Pr3224476576807795204nt_int @ F @ Xs ) )
= ( Xs = nil_Pr749310521696113548st_int ) ) ).
% Nil_is_map_conv
thf(fact_697_Nil__is__map__conv,axiom,
! [F: instr > int,Xs: list_instr] :
( ( nil_int
= ( map_instr_int @ F @ Xs ) )
= ( Xs = nil_instr ) ) ).
% Nil_is_map_conv
thf(fact_698_Nil__is__map__conv,axiom,
! [F: int > int,Xs: list_int] :
( ( nil_int
= ( map_int_int @ F @ Xs ) )
= ( Xs = nil_int ) ) ).
% Nil_is_map_conv
thf(fact_699_Nil__is__map__conv,axiom,
! [F: produc6425607678544837394st_int > produc6425607678544837394st_int,Xs: list_P419547167332244514st_int] :
( ( nil_Pr749310521696113548st_int
= ( map_Pr8625093460839654122st_int @ F @ Xs ) )
= ( Xs = nil_Pr749310521696113548st_int ) ) ).
% Nil_is_map_conv
thf(fact_700_list_Omap__disc__iff,axiom,
! [F: instr > produc6425607678544837394st_int,A: list_instr] :
( ( ( map_in7259965606695656109st_int @ F @ A )
= nil_Pr749310521696113548st_int )
= ( A = nil_instr ) ) ).
% list.map_disc_iff
thf(fact_701_list_Omap__disc__iff,axiom,
! [F: int > produc6425607678544837394st_int,A: list_int] :
( ( ( map_in8512336359594627332st_int @ F @ A )
= nil_Pr749310521696113548st_int )
= ( A = nil_int ) ) ).
% list.map_disc_iff
thf(fact_702_list_Omap__disc__iff,axiom,
! [F: produc6425607678544837394st_int > instr,A: list_P419547167332244514st_int] :
( ( ( map_Pr7514435936640407687_instr @ F @ A )
= nil_instr )
= ( A = nil_Pr749310521696113548st_int ) ) ).
% list.map_disc_iff
thf(fact_703_list_Omap__disc__iff,axiom,
! [F: instr > instr,A: list_instr] :
( ( ( map_instr_instr @ F @ A )
= nil_instr )
= ( A = nil_instr ) ) ).
% list.map_disc_iff
thf(fact_704_list_Omap__disc__iff,axiom,
! [F: int > instr,A: list_int] :
( ( ( map_int_instr @ F @ A )
= nil_instr )
= ( A = nil_int ) ) ).
% list.map_disc_iff
thf(fact_705_list_Omap__disc__iff,axiom,
! [F: produc6425607678544837394st_int > int,A: list_P419547167332244514st_int] :
( ( ( map_Pr3224476576807795204nt_int @ F @ A )
= nil_int )
= ( A = nil_Pr749310521696113548st_int ) ) ).
% list.map_disc_iff
thf(fact_706_list_Omap__disc__iff,axiom,
! [F: instr > int,A: list_instr] :
( ( ( map_instr_int @ F @ A )
= nil_int )
= ( A = nil_instr ) ) ).
% list.map_disc_iff
thf(fact_707_list_Omap__disc__iff,axiom,
! [F: int > int,A: list_int] :
( ( ( map_int_int @ F @ A )
= nil_int )
= ( A = nil_int ) ) ).
% list.map_disc_iff
thf(fact_708_list_Omap__disc__iff,axiom,
! [F: produc6425607678544837394st_int > produc6425607678544837394st_int,A: list_P419547167332244514st_int] :
( ( ( map_Pr8625093460839654122st_int @ F @ A )
= nil_Pr749310521696113548st_int )
= ( A = nil_Pr749310521696113548st_int ) ) ).
% list.map_disc_iff
thf(fact_709_append_Oright__neutral,axiom,
! [A: list_P419547167332244514st_int] :
( ( append5028292981072010935st_int @ A @ nil_Pr749310521696113548st_int )
= A ) ).
% append.right_neutral
thf(fact_710_append_Oright__neutral,axiom,
! [A: list_instr] :
( ( append_instr @ A @ nil_instr )
= A ) ).
% append.right_neutral
thf(fact_711_append_Oright__neutral,axiom,
! [A: list_int] :
( ( append_int @ A @ nil_int )
= A ) ).
% append.right_neutral
thf(fact_712_append__Nil2,axiom,
! [Xs: list_P419547167332244514st_int] :
( ( append5028292981072010935st_int @ Xs @ nil_Pr749310521696113548st_int )
= Xs ) ).
% append_Nil2
thf(fact_713_append__Nil2,axiom,
! [Xs: list_instr] :
( ( append_instr @ Xs @ nil_instr )
= Xs ) ).
% append_Nil2
thf(fact_714_append__Nil2,axiom,
! [Xs: list_int] :
( ( append_int @ Xs @ nil_int )
= Xs ) ).
% append_Nil2
thf(fact_715_append__self__conv,axiom,
! [Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int] :
( ( ( append5028292981072010935st_int @ Xs @ Ys )
= Xs )
= ( Ys = nil_Pr749310521696113548st_int ) ) ).
% append_self_conv
thf(fact_716_append__self__conv,axiom,
! [Xs: list_instr,Ys: list_instr] :
( ( ( append_instr @ Xs @ Ys )
= Xs )
= ( Ys = nil_instr ) ) ).
% append_self_conv
thf(fact_717_append__self__conv,axiom,
! [Xs: list_int,Ys: list_int] :
( ( ( append_int @ Xs @ Ys )
= Xs )
= ( Ys = nil_int ) ) ).
% append_self_conv
thf(fact_718_self__append__conv,axiom,
! [Y: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int] :
( ( Y
= ( append5028292981072010935st_int @ Y @ Ys ) )
= ( Ys = nil_Pr749310521696113548st_int ) ) ).
% self_append_conv
thf(fact_719_self__append__conv,axiom,
! [Y: list_instr,Ys: list_instr] :
( ( Y
= ( append_instr @ Y @ Ys ) )
= ( Ys = nil_instr ) ) ).
% self_append_conv
thf(fact_720_self__append__conv,axiom,
! [Y: list_int,Ys: list_int] :
( ( Y
= ( append_int @ Y @ Ys ) )
= ( Ys = nil_int ) ) ).
% self_append_conv
thf(fact_721_append__self__conv2,axiom,
! [Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int] :
( ( ( append5028292981072010935st_int @ Xs @ Ys )
= Ys )
= ( Xs = nil_Pr749310521696113548st_int ) ) ).
% append_self_conv2
thf(fact_722_append__self__conv2,axiom,
! [Xs: list_instr,Ys: list_instr] :
( ( ( append_instr @ Xs @ Ys )
= Ys )
= ( Xs = nil_instr ) ) ).
% append_self_conv2
thf(fact_723_append__self__conv2,axiom,
! [Xs: list_int,Ys: list_int] :
( ( ( append_int @ Xs @ Ys )
= Ys )
= ( Xs = nil_int ) ) ).
% append_self_conv2
thf(fact_724_self__append__conv2,axiom,
! [Y: list_P419547167332244514st_int,Xs: list_P419547167332244514st_int] :
( ( Y
= ( append5028292981072010935st_int @ Xs @ Y ) )
= ( Xs = nil_Pr749310521696113548st_int ) ) ).
% self_append_conv2
thf(fact_725_self__append__conv2,axiom,
! [Y: list_instr,Xs: list_instr] :
( ( Y
= ( append_instr @ Xs @ Y ) )
= ( Xs = nil_instr ) ) ).
% self_append_conv2
thf(fact_726_self__append__conv2,axiom,
! [Y: list_int,Xs: list_int] :
( ( Y
= ( append_int @ Xs @ Y ) )
= ( Xs = nil_int ) ) ).
% self_append_conv2
thf(fact_727_Nil__is__append__conv,axiom,
! [Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int] :
( ( nil_Pr749310521696113548st_int
= ( append5028292981072010935st_int @ Xs @ Ys ) )
= ( ( Xs = nil_Pr749310521696113548st_int )
& ( Ys = nil_Pr749310521696113548st_int ) ) ) ).
% Nil_is_append_conv
thf(fact_728_Nil__is__append__conv,axiom,
! [Xs: list_instr,Ys: list_instr] :
( ( nil_instr
= ( append_instr @ Xs @ Ys ) )
= ( ( Xs = nil_instr )
& ( Ys = nil_instr ) ) ) ).
% Nil_is_append_conv
thf(fact_729_Nil__is__append__conv,axiom,
! [Xs: list_int,Ys: list_int] :
( ( nil_int
= ( append_int @ Xs @ Ys ) )
= ( ( Xs = nil_int )
& ( Ys = nil_int ) ) ) ).
% Nil_is_append_conv
thf(fact_730_append__is__Nil__conv,axiom,
! [Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int] :
( ( ( append5028292981072010935st_int @ Xs @ Ys )
= nil_Pr749310521696113548st_int )
= ( ( Xs = nil_Pr749310521696113548st_int )
& ( Ys = nil_Pr749310521696113548st_int ) ) ) ).
% append_is_Nil_conv
thf(fact_731_append__is__Nil__conv,axiom,
! [Xs: list_instr,Ys: list_instr] :
( ( ( append_instr @ Xs @ Ys )
= nil_instr )
= ( ( Xs = nil_instr )
& ( Ys = nil_instr ) ) ) ).
% append_is_Nil_conv
thf(fact_732_append__is__Nil__conv,axiom,
! [Xs: list_int,Ys: list_int] :
( ( ( append_int @ Xs @ Ys )
= nil_int )
= ( ( Xs = nil_int )
& ( Ys = nil_int ) ) ) ).
% append_is_Nil_conv
thf(fact_733_length__0__conv,axiom,
! [Xs: list_P419547167332244514st_int] :
( ( ( size_s229347708945616566st_int @ Xs )
= zero_zero_nat )
= ( Xs = nil_Pr749310521696113548st_int ) ) ).
% length_0_conv
thf(fact_734_length__0__conv,axiom,
! [Xs: list_instr] :
( ( ( size_size_list_instr @ Xs )
= zero_zero_nat )
= ( Xs = nil_instr ) ) ).
% length_0_conv
thf(fact_735_length__0__conv,axiom,
! [Xs: list_list_char] :
( ( ( size_s356637359517785349t_char @ Xs )
= zero_zero_nat )
= ( Xs = nil_list_char ) ) ).
% length_0_conv
thf(fact_736_length__0__conv,axiom,
! [Xs: list_int] :
( ( ( size_size_list_int @ Xs )
= zero_zero_nat )
= ( Xs = nil_int ) ) ).
% length_0_conv
thf(fact_737_take__eq__Nil2,axiom,
! [N: nat,Xs: list_P419547167332244514st_int] :
( ( nil_Pr749310521696113548st_int
= ( take_P1004856038796978596st_int @ N @ Xs ) )
= ( ( N = zero_zero_nat )
| ( Xs = nil_Pr749310521696113548st_int ) ) ) ).
% take_eq_Nil2
thf(fact_738_take__eq__Nil2,axiom,
! [N: nat,Xs: list_instr] :
( ( nil_instr
= ( take_instr @ N @ Xs ) )
= ( ( N = zero_zero_nat )
| ( Xs = nil_instr ) ) ) ).
% take_eq_Nil2
thf(fact_739_take__eq__Nil2,axiom,
! [N: nat,Xs: list_int] :
( ( nil_int
= ( take_int @ N @ Xs ) )
= ( ( N = zero_zero_nat )
| ( Xs = nil_int ) ) ) ).
% take_eq_Nil2
thf(fact_740_take__eq__Nil,axiom,
! [N: nat,Xs: list_P419547167332244514st_int] :
( ( ( take_P1004856038796978596st_int @ N @ Xs )
= nil_Pr749310521696113548st_int )
= ( ( N = zero_zero_nat )
| ( Xs = nil_Pr749310521696113548st_int ) ) ) ).
% take_eq_Nil
thf(fact_741_take__eq__Nil,axiom,
! [N: nat,Xs: list_instr] :
( ( ( take_instr @ N @ Xs )
= nil_instr )
= ( ( N = zero_zero_nat )
| ( Xs = nil_instr ) ) ) ).
% take_eq_Nil
thf(fact_742_take__eq__Nil,axiom,
! [N: nat,Xs: list_int] :
( ( ( take_int @ N @ Xs )
= nil_int )
= ( ( N = zero_zero_nat )
| ( Xs = nil_int ) ) ) ).
% take_eq_Nil
thf(fact_743_take0,axiom,
( ( take_P1004856038796978596st_int @ zero_zero_nat )
= ( ^ [Xs2: list_P419547167332244514st_int] : nil_Pr749310521696113548st_int ) ) ).
% take0
thf(fact_744_take0,axiom,
( ( take_instr @ zero_zero_nat )
= ( ^ [Xs2: list_instr] : nil_instr ) ) ).
% take0
thf(fact_745_take0,axiom,
( ( take_int @ zero_zero_nat )
= ( ^ [Xs2: list_int] : nil_int ) ) ).
% take0
thf(fact_746_length__greater__0__conv,axiom,
! [Xs: list_P419547167332244514st_int] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s229347708945616566st_int @ Xs ) )
= ( Xs != nil_Pr749310521696113548st_int ) ) ).
% length_greater_0_conv
thf(fact_747_length__greater__0__conv,axiom,
! [Xs: list_instr] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_instr @ Xs ) )
= ( Xs != nil_instr ) ) ).
% length_greater_0_conv
thf(fact_748_length__greater__0__conv,axiom,
! [Xs: list_list_char] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s356637359517785349t_char @ Xs ) )
= ( Xs != nil_list_char ) ) ).
% length_greater_0_conv
thf(fact_749_length__greater__0__conv,axiom,
! [Xs: list_int] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_int @ Xs ) )
= ( Xs != nil_int ) ) ).
% length_greater_0_conv
thf(fact_750_drop__eq__Nil2,axiom,
! [N: nat,Xs: list_P419547167332244514st_int] :
( ( nil_Pr749310521696113548st_int
= ( drop_P7838799151754935884st_int @ N @ Xs ) )
= ( ord_less_eq_nat @ ( size_s229347708945616566st_int @ Xs ) @ N ) ) ).
% drop_eq_Nil2
thf(fact_751_drop__eq__Nil2,axiom,
! [N: nat,Xs: list_instr] :
( ( nil_instr
= ( drop_instr @ N @ Xs ) )
= ( ord_less_eq_nat @ ( size_size_list_instr @ Xs ) @ N ) ) ).
% drop_eq_Nil2
thf(fact_752_drop__eq__Nil2,axiom,
! [N: nat,Xs: list_list_char] :
( ( nil_list_char
= ( drop_list_char @ N @ Xs ) )
= ( ord_less_eq_nat @ ( size_s356637359517785349t_char @ Xs ) @ N ) ) ).
% drop_eq_Nil2
thf(fact_753_drop__eq__Nil2,axiom,
! [N: nat,Xs: list_int] :
( ( nil_int
= ( drop_int @ N @ Xs ) )
= ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ N ) ) ).
% drop_eq_Nil2
thf(fact_754_drop__eq__Nil,axiom,
! [N: nat,Xs: list_P419547167332244514st_int] :
( ( ( drop_P7838799151754935884st_int @ N @ Xs )
= nil_Pr749310521696113548st_int )
= ( ord_less_eq_nat @ ( size_s229347708945616566st_int @ Xs ) @ N ) ) ).
% drop_eq_Nil
thf(fact_755_drop__eq__Nil,axiom,
! [N: nat,Xs: list_instr] :
( ( ( drop_instr @ N @ Xs )
= nil_instr )
= ( ord_less_eq_nat @ ( size_size_list_instr @ Xs ) @ N ) ) ).
% drop_eq_Nil
thf(fact_756_drop__eq__Nil,axiom,
! [N: nat,Xs: list_list_char] :
( ( ( drop_list_char @ N @ Xs )
= nil_list_char )
= ( ord_less_eq_nat @ ( size_s356637359517785349t_char @ Xs ) @ N ) ) ).
% drop_eq_Nil
thf(fact_757_drop__eq__Nil,axiom,
! [N: nat,Xs: list_int] :
( ( ( drop_int @ N @ Xs )
= nil_int )
= ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ N ) ) ).
% drop_eq_Nil
thf(fact_758_drop__all,axiom,
! [Xs: list_P419547167332244514st_int,N: nat] :
( ( ord_less_eq_nat @ ( size_s229347708945616566st_int @ Xs ) @ N )
=> ( ( drop_P7838799151754935884st_int @ N @ Xs )
= nil_Pr749310521696113548st_int ) ) ).
% drop_all
thf(fact_759_drop__all,axiom,
! [Xs: list_instr,N: nat] :
( ( ord_less_eq_nat @ ( size_size_list_instr @ Xs ) @ N )
=> ( ( drop_instr @ N @ Xs )
= nil_instr ) ) ).
% drop_all
thf(fact_760_drop__all,axiom,
! [Xs: list_list_char,N: nat] :
( ( ord_less_eq_nat @ ( size_s356637359517785349t_char @ Xs ) @ N )
=> ( ( drop_list_char @ N @ Xs )
= nil_list_char ) ) ).
% drop_all
thf(fact_761_drop__all,axiom,
! [Xs: list_int,N: nat] :
( ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ N )
=> ( ( drop_int @ N @ Xs )
= nil_int ) ) ).
% drop_all
thf(fact_762_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_763_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_764_less__set__def,axiom,
( ord_less_set_nat
= ( ^ [A6: set_nat,B6: set_nat] :
( ord_less_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ A6 )
@ ^ [X2: nat] : ( member_nat @ X2 @ B6 ) ) ) ) ).
% less_set_def
thf(fact_765_less__set__def,axiom,
( ord_less_set_int
= ( ^ [A6: set_int,B6: set_int] :
( ord_less_int_o
@ ^ [X2: int] : ( member_int @ X2 @ A6 )
@ ^ [X2: int] : ( member_int @ X2 @ B6 ) ) ) ) ).
% less_set_def
thf(fact_766_list_Osimps_I8_J,axiom,
! [F: produc6425607678544837394st_int > instr] :
( ( map_Pr7514435936640407687_instr @ F @ nil_Pr749310521696113548st_int )
= nil_instr ) ).
% list.simps(8)
thf(fact_767_list_Osimps_I8_J,axiom,
! [F: produc6425607678544837394st_int > int] :
( ( map_Pr3224476576807795204nt_int @ F @ nil_Pr749310521696113548st_int )
= nil_int ) ).
% list.simps(8)
thf(fact_768_list_Osimps_I8_J,axiom,
! [F: instr > produc6425607678544837394st_int] :
( ( map_in7259965606695656109st_int @ F @ nil_instr )
= nil_Pr749310521696113548st_int ) ).
% list.simps(8)
thf(fact_769_list_Osimps_I8_J,axiom,
! [F: instr > instr] :
( ( map_instr_instr @ F @ nil_instr )
= nil_instr ) ).
% list.simps(8)
thf(fact_770_list_Osimps_I8_J,axiom,
! [F: instr > int] :
( ( map_instr_int @ F @ nil_instr )
= nil_int ) ).
% list.simps(8)
thf(fact_771_list_Osimps_I8_J,axiom,
! [F: int > produc6425607678544837394st_int] :
( ( map_in8512336359594627332st_int @ F @ nil_int )
= nil_Pr749310521696113548st_int ) ).
% list.simps(8)
thf(fact_772_list_Osimps_I8_J,axiom,
! [F: int > instr] :
( ( map_int_instr @ F @ nil_int )
= nil_instr ) ).
% list.simps(8)
thf(fact_773_list_Osimps_I8_J,axiom,
! [F: int > int] :
( ( map_int_int @ F @ nil_int )
= nil_int ) ).
% list.simps(8)
thf(fact_774_list_Osimps_I8_J,axiom,
! [F: produc6425607678544837394st_int > produc6425607678544837394st_int] :
( ( map_Pr8625093460839654122st_int @ F @ nil_Pr749310521696113548st_int )
= nil_Pr749310521696113548st_int ) ).
% list.simps(8)
thf(fact_775_append__Nil,axiom,
! [Ys: list_P419547167332244514st_int] :
( ( append5028292981072010935st_int @ nil_Pr749310521696113548st_int @ Ys )
= Ys ) ).
% append_Nil
thf(fact_776_append__Nil,axiom,
! [Ys: list_instr] :
( ( append_instr @ nil_instr @ Ys )
= Ys ) ).
% append_Nil
thf(fact_777_append__Nil,axiom,
! [Ys: list_int] :
( ( append_int @ nil_int @ Ys )
= Ys ) ).
% append_Nil
thf(fact_778_append_Oleft__neutral,axiom,
! [A: list_P419547167332244514st_int] :
( ( append5028292981072010935st_int @ nil_Pr749310521696113548st_int @ A )
= A ) ).
% append.left_neutral
thf(fact_779_append_Oleft__neutral,axiom,
! [A: list_instr] :
( ( append_instr @ nil_instr @ A )
= A ) ).
% append.left_neutral
thf(fact_780_append_Oleft__neutral,axiom,
! [A: list_int] :
( ( append_int @ nil_int @ A )
= A ) ).
% append.left_neutral
thf(fact_781_eq__Nil__appendI,axiom,
! [Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int] :
( ( Xs = Ys )
=> ( Xs
= ( append5028292981072010935st_int @ nil_Pr749310521696113548st_int @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_782_eq__Nil__appendI,axiom,
! [Xs: list_instr,Ys: list_instr] :
( ( Xs = Ys )
=> ( Xs
= ( append_instr @ nil_instr @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_783_eq__Nil__appendI,axiom,
! [Xs: list_int,Ys: list_int] :
( ( Xs = Ys )
=> ( Xs
= ( append_int @ nil_int @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_784_take__Nil,axiom,
! [N: nat] :
( ( take_P1004856038796978596st_int @ N @ nil_Pr749310521696113548st_int )
= nil_Pr749310521696113548st_int ) ).
% take_Nil
thf(fact_785_take__Nil,axiom,
! [N: nat] :
( ( take_instr @ N @ nil_instr )
= nil_instr ) ).
% take_Nil
thf(fact_786_take__Nil,axiom,
! [N: nat] :
( ( take_int @ N @ nil_int )
= nil_int ) ).
% take_Nil
thf(fact_787_drop__Nil,axiom,
! [N: nat] :
( ( drop_list_char @ N @ nil_list_char )
= nil_list_char ) ).
% drop_Nil
thf(fact_788_drop__Nil,axiom,
! [N: nat] :
( ( drop_P7838799151754935884st_int @ N @ nil_Pr749310521696113548st_int )
= nil_Pr749310521696113548st_int ) ).
% drop_Nil
thf(fact_789_drop__Nil,axiom,
! [N: nat] :
( ( drop_instr @ N @ nil_instr )
= nil_instr ) ).
% drop_Nil
thf(fact_790_drop__Nil,axiom,
! [N: nat] :
( ( drop_int @ N @ nil_int )
= nil_int ) ).
% drop_Nil
thf(fact_791_execl_Osimps_I2_J,axiom,
! [P: list_instr] : ( execl @ P @ nil_Pr749310521696113548st_int ) ).
% execl.simps(2)
thf(fact_792_list_Osize_I3_J,axiom,
( ( size_s229347708945616566st_int @ nil_Pr749310521696113548st_int )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_793_list_Osize_I3_J,axiom,
( ( size_size_list_instr @ nil_instr )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_794_list_Osize_I3_J,axiom,
( ( size_s356637359517785349t_char @ nil_list_char )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_795_list_Osize_I3_J,axiom,
( ( size_size_list_int @ nil_int )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_796_take__0,axiom,
! [Xs: list_P419547167332244514st_int] :
( ( take_P1004856038796978596st_int @ zero_zero_nat @ Xs )
= nil_Pr749310521696113548st_int ) ).
% take_0
thf(fact_797_take__0,axiom,
! [Xs: list_instr] :
( ( take_instr @ zero_zero_nat @ Xs )
= nil_instr ) ).
% take_0
thf(fact_798_take__0,axiom,
! [Xs: list_int] :
( ( take_int @ zero_zero_nat @ Xs )
= nil_int ) ).
% take_0
thf(fact_799_exec1__def,axiom,
( exec1
= ( ^ [P5: list_instr,Cf: produc6425607678544837394st_int,Cf2: produc6425607678544837394st_int] :
( ( iexec_pred @ ( produc1484136438817787418st_int @ ( inth_instr @ P5 @ ( produc8941309049508465880st_int @ Cf ) ) @ Cf ) @ Cf2 )
& ( ord_less_eq_int @ zero_zero_int @ ( produc8941309049508465880st_int @ Cf ) )
& ( ord_less_int @ ( produc8941309049508465880st_int @ Cf ) @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P5 ) ) ) ) ) ) ).
% exec1_def
thf(fact_800_pred__subset__eq,axiom,
! [R: set_nat,S5: set_nat] :
( ( ord_less_eq_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ R )
@ ^ [X2: nat] : ( member_nat @ X2 @ S5 ) )
= ( ord_less_eq_set_nat @ R @ S5 ) ) ).
% pred_subset_eq
thf(fact_801_pred__subset__eq,axiom,
! [R: set_int,S5: set_int] :
( ( ord_less_eq_int_o
@ ^ [X2: int] : ( member_int @ X2 @ R )
@ ^ [X2: int] : ( member_int @ X2 @ S5 ) )
= ( ord_less_eq_set_int @ R @ S5 ) ) ).
% pred_subset_eq
thf(fact_802_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_803_dual__order_Orefl,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% dual_order.refl
thf(fact_804_order__refl,axiom,
! [X3: nat] : ( ord_less_eq_nat @ X3 @ X3 ) ).
% order_refl
thf(fact_805_order__refl,axiom,
! [X3: int] : ( ord_less_eq_int @ X3 @ X3 ) ).
% order_refl
thf(fact_806_exec1__appendR,axiom,
! [P: list_instr,C: produc6425607678544837394st_int,C3: produc6425607678544837394st_int,P3: list_instr] :
( ( exec1 @ P @ C @ C3 )
=> ( exec1 @ ( append_instr @ P @ P3 ) @ C @ C3 ) ) ).
% exec1_appendR
thf(fact_807_order__antisym__conv,axiom,
! [Y: nat,X3: nat] :
( ( ord_less_eq_nat @ Y @ X3 )
=> ( ( ord_less_eq_nat @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% order_antisym_conv
thf(fact_808_order__antisym__conv,axiom,
! [Y: int,X3: int] :
( ( ord_less_eq_int @ Y @ X3 )
=> ( ( ord_less_eq_int @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% order_antisym_conv
thf(fact_809_linorder__le__cases,axiom,
! [X3: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ Y @ X3 ) ) ).
% linorder_le_cases
thf(fact_810_linorder__le__cases,axiom,
! [X3: int,Y: int] :
( ~ ( ord_less_eq_int @ X3 @ Y )
=> ( ord_less_eq_int @ Y @ X3 ) ) ).
% linorder_le_cases
thf(fact_811_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_812_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_813_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_814_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ Y4 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_815_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_816_ord__eq__le__subst,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_817_ord__eq__le__subst,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_818_ord__eq__le__subst,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ Y4 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_819_linorder__linear,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
| ( ord_less_eq_nat @ Y @ X3 ) ) ).
% linorder_linear
thf(fact_820_linorder__linear,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_int @ X3 @ Y )
| ( ord_less_eq_int @ Y @ X3 ) ) ).
% linorder_linear
thf(fact_821_order__eq__refl,axiom,
! [X3: nat,Y: nat] :
( ( X3 = Y )
=> ( ord_less_eq_nat @ X3 @ Y ) ) ).
% order_eq_refl
thf(fact_822_order__eq__refl,axiom,
! [X3: int,Y: int] :
( ( X3 = Y )
=> ( ord_less_eq_int @ X3 @ Y ) ) ).
% order_eq_refl
thf(fact_823_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 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_824_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 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_825_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 )
=> ( ! [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_826_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 )
=> ( ! [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ Y4 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_827_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 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_828_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 )
=> ( ! [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_829_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 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_830_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 )
=> ( ! [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ Y4 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_831_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ A5 @ B4 )
& ( ord_less_eq_nat @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_832_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: int,Z: int] : ( Y3 = Z ) )
= ( ^ [A5: int,B4: int] :
( ( ord_less_eq_int @ A5 @ B4 )
& ( ord_less_eq_int @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_833_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_834_antisym,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_835_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_836_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_837_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_838_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_839_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ B4 @ A5 )
& ( ord_less_eq_nat @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_840_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: int,Z: int] : ( Y3 = Z ) )
= ( ^ [A5: int,B4: int] :
( ( ord_less_eq_int @ B4 @ A5 )
& ( ord_less_eq_int @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_841_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_842_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_843_order__trans,axiom,
! [X3: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z3 )
=> ( ord_less_eq_nat @ X3 @ Z3 ) ) ) ).
% order_trans
thf(fact_844_order__trans,axiom,
! [X3: int,Y: int,Z3: int] :
( ( ord_less_eq_int @ X3 @ Y )
=> ( ( ord_less_eq_int @ Y @ Z3 )
=> ( ord_less_eq_int @ X3 @ Z3 ) ) ) ).
% order_trans
thf(fact_845_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_846_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_847_order__antisym,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( ord_less_eq_nat @ Y @ X3 )
=> ( X3 = Y ) ) ) ).
% order_antisym
thf(fact_848_order__antisym,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_int @ X3 @ Y )
=> ( ( ord_less_eq_int @ Y @ X3 )
=> ( X3 = Y ) ) ) ).
% order_antisym
thf(fact_849_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_850_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_851_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_852_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_853_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [X2: nat,Y6: nat] :
( ( ord_less_eq_nat @ X2 @ Y6 )
& ( ord_less_eq_nat @ Y6 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_854_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: int,Z: int] : ( Y3 = Z ) )
= ( ^ [X2: int,Y6: int] :
( ( ord_less_eq_int @ X2 @ Y6 )
& ( ord_less_eq_int @ Y6 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_855_le__cases3,axiom,
! [X3: nat,Y: nat,Z3: nat] :
( ( ( ord_less_eq_nat @ X3 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ X3 @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z3 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X3 ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ X3 ) )
=> ~ ( ( ord_less_eq_nat @ Z3 @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_856_le__cases3,axiom,
! [X3: int,Y: int,Z3: int] :
( ( ( ord_less_eq_int @ X3 @ Y )
=> ~ ( ord_less_eq_int @ Y @ Z3 ) )
=> ( ( ( ord_less_eq_int @ Y @ X3 )
=> ~ ( ord_less_eq_int @ X3 @ Z3 ) )
=> ( ( ( ord_less_eq_int @ X3 @ Z3 )
=> ~ ( ord_less_eq_int @ Z3 @ Y ) )
=> ( ( ( ord_less_eq_int @ Z3 @ Y )
=> ~ ( ord_less_eq_int @ Y @ X3 ) )
=> ( ( ( ord_less_eq_int @ Y @ Z3 )
=> ~ ( ord_less_eq_int @ Z3 @ X3 ) )
=> ~ ( ( ord_less_eq_int @ Z3 @ X3 )
=> ~ ( ord_less_eq_int @ X3 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_857_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_858_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_859_order__less__imp__not__less,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ~ ( ord_less_nat @ Y @ X3 ) ) ).
% order_less_imp_not_less
thf(fact_860_order__less__imp__not__less,axiom,
! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
=> ~ ( ord_less_int @ Y @ X3 ) ) ).
% order_less_imp_not_less
thf(fact_861_order__less__imp__not__eq2,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( Y != X3 ) ) ).
% order_less_imp_not_eq2
thf(fact_862_order__less__imp__not__eq2,axiom,
! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
=> ( Y != X3 ) ) ).
% order_less_imp_not_eq2
thf(fact_863_order__less__imp__not__eq,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( X3 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_864_order__less__imp__not__eq,axiom,
! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
=> ( X3 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_865_linorder__less__linear,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
| ( X3 = Y )
| ( ord_less_nat @ Y @ X3 ) ) ).
% linorder_less_linear
thf(fact_866_linorder__less__linear,axiom,
! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
| ( X3 = Y )
| ( ord_less_int @ Y @ X3 ) ) ).
% linorder_less_linear
thf(fact_867_order__less__imp__triv,axiom,
! [X3: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X3 @ Y )
=> ( ( ord_less_nat @ Y @ X3 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_868_order__less__imp__triv,axiom,
! [X3: int,Y: int,P: $o] :
( ( ord_less_int @ X3 @ Y )
=> ( ( ord_less_int @ Y @ X3 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_869_order__less__not__sym,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ~ ( ord_less_nat @ Y @ X3 ) ) ).
% order_less_not_sym
thf(fact_870_order__less__not__sym,axiom,
! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
=> ~ ( ord_less_int @ Y @ X3 ) ) ).
% order_less_not_sym
thf(fact_871_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_872_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_873_order__less__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_int @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_874_order__less__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_int @ X @ Y4 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_875_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_876_order__less__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_int @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_877_order__less__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_878_order__less__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_int @ X @ Y4 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_879_order__less__irrefl,axiom,
! [X3: nat] :
~ ( ord_less_nat @ X3 @ X3 ) ).
% order_less_irrefl
thf(fact_880_order__less__irrefl,axiom,
! [X3: int] :
~ ( ord_less_int @ X3 @ X3 ) ).
% order_less_irrefl
thf(fact_881_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_882_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_883_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_int @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_884_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_int @ X @ Y4 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_885_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_886_ord__eq__less__subst,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_887_ord__eq__less__subst,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_int @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_888_ord__eq__less__subst,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X: int,Y4: int] :
( ( ord_less_int @ X @ Y4 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_889_order__less__trans,axiom,
! [X3: nat,Y: nat,Z3: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ( ord_less_nat @ Y @ Z3 )
=> ( ord_less_nat @ X3 @ Z3 ) ) ) ).
% order_less_trans
thf(fact_890_order__less__trans,axiom,
! [X3: int,Y: int,Z3: int] :
( ( ord_less_int @ X3 @ Y )
=> ( ( ord_less_int @ Y @ Z3 )
=> ( ord_less_int @ X3 @ Z3 ) ) ) ).
% order_less_trans
thf(fact_891_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_892_order__less__asym_H,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order_less_asym'
thf(fact_893_linorder__neq__iff,axiom,
! [X3: nat,Y: nat] :
( ( X3 != Y )
= ( ( ord_less_nat @ X3 @ Y )
| ( ord_less_nat @ Y @ X3 ) ) ) ).
% linorder_neq_iff
thf(fact_894_linorder__neq__iff,axiom,
! [X3: int,Y: int] :
( ( X3 != Y )
= ( ( ord_less_int @ X3 @ Y )
| ( ord_less_int @ Y @ X3 ) ) ) ).
% linorder_neq_iff
thf(fact_895_order__less__asym,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ~ ( ord_less_nat @ Y @ X3 ) ) ).
% order_less_asym
thf(fact_896_order__less__asym,axiom,
! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
=> ~ ( ord_less_int @ Y @ X3 ) ) ).
% order_less_asym
thf(fact_897_linorder__neqE,axiom,
! [X3: nat,Y: nat] :
( ( X3 != Y )
=> ( ~ ( ord_less_nat @ X3 @ Y )
=> ( ord_less_nat @ Y @ X3 ) ) ) ).
% linorder_neqE
thf(fact_898_linorder__neqE,axiom,
! [X3: int,Y: int] :
( ( X3 != Y )
=> ( ~ ( ord_less_int @ X3 @ Y )
=> ( ord_less_int @ Y @ X3 ) ) ) ).
% linorder_neqE
thf(fact_899_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_900_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_901_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_902_order_Ostrict__implies__not__eq,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_903_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_904_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_905_not__less__iff__gr__or__eq,axiom,
! [X3: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X3 @ Y ) )
= ( ( ord_less_nat @ Y @ X3 )
| ( X3 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_906_not__less__iff__gr__or__eq,axiom,
! [X3: int,Y: int] :
( ( ~ ( ord_less_int @ X3 @ Y ) )
= ( ( ord_less_int @ Y @ X3 )
| ( X3 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_907_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_908_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_909_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_910_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_911_exists__least__iff,axiom,
( ( ^ [P6: nat > $o] :
? [X7: nat] : ( P6 @ X7 ) )
= ( ^ [P5: nat > $o] :
? [N4: nat] :
( ( P5 @ N4 )
& ! [M5: nat] :
( ( ord_less_nat @ M5 @ N4 )
=> ~ ( P5 @ M5 ) ) ) ) ) ).
% exists_least_iff
thf(fact_912_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_913_dual__order_Oirrefl,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% dual_order.irrefl
thf(fact_914_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_915_dual__order_Oasym,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ~ ( ord_less_int @ A @ B ) ) ).
% dual_order.asym
thf(fact_916_linorder__cases,axiom,
! [X3: nat,Y: nat] :
( ~ ( ord_less_nat @ X3 @ Y )
=> ( ( X3 != Y )
=> ( ord_less_nat @ Y @ X3 ) ) ) ).
% linorder_cases
thf(fact_917_linorder__cases,axiom,
! [X3: int,Y: int] :
( ~ ( ord_less_int @ X3 @ Y )
=> ( ( X3 != Y )
=> ( ord_less_int @ Y @ X3 ) ) ) ).
% linorder_cases
thf(fact_918_antisym__conv3,axiom,
! [Y: nat,X3: nat] :
( ~ ( ord_less_nat @ Y @ X3 )
=> ( ( ~ ( ord_less_nat @ X3 @ Y ) )
= ( X3 = Y ) ) ) ).
% antisym_conv3
thf(fact_919_antisym__conv3,axiom,
! [Y: int,X3: int] :
( ~ ( ord_less_int @ Y @ X3 )
=> ( ( ~ ( ord_less_int @ X3 @ Y ) )
= ( X3 = Y ) ) ) ).
% antisym_conv3
thf(fact_920_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X )
=> ( P @ Y5 ) )
=> ( P @ X ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_921_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_922_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_923_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_924_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_925_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_926_order_Oasym,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order.asym
thf(fact_927_less__imp__neq,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( X3 != Y ) ) ).
% less_imp_neq
thf(fact_928_less__imp__neq,axiom,
! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
=> ( X3 != Y ) ) ).
% less_imp_neq
thf(fact_929_gt__ex,axiom,
! [X3: nat] :
? [X_1: nat] : ( ord_less_nat @ X3 @ X_1 ) ).
% gt_ex
thf(fact_930_gt__ex,axiom,
! [X3: int] :
? [X_1: int] : ( ord_less_int @ X3 @ X_1 ) ).
% gt_ex
thf(fact_931_lt__ex,axiom,
! [X3: int] :
? [Y4: int] : ( ord_less_int @ Y4 @ X3 ) ).
% lt_ex
thf(fact_932_subrelI,axiom,
! [R2: set_Pr4358015327039922227nt_int,S: set_Pr4358015327039922227nt_int] :
( ! [X: list_list_char,Y4: produc4435102495419491129nt_int] :
( ( member7220921130490543508nt_int @ ( produc3965054194175396271nt_int @ X @ Y4 ) @ R2 )
=> ( member7220921130490543508nt_int @ ( produc3965054194175396271nt_int @ X @ Y4 ) @ S ) )
=> ( ord_le5912571515218079891nt_int @ R2 @ S ) ) ).
% subrelI
thf(fact_933_subrelI,axiom,
! [R2: set_Pr2116125220558876360st_int,S: set_Pr2116125220558876360st_int] :
( ! [X: int,Y4: produc369741286924889929st_int] :
( ( member9100552558438320297st_int @ ( produc5086643055186798020st_int @ X @ Y4 ) @ R2 )
=> ( member9100552558438320297st_int @ ( produc5086643055186798020st_int @ X @ Y4 ) @ S ) )
=> ( ord_le8444298136753458984st_int @ R2 @ S ) ) ).
% subrelI
thf(fact_934_subrelI,axiom,
! [R2: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int] :
( ! [X: int,Y4: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y4 ) @ R2 )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y4 ) @ S ) )
=> ( ord_le2843351958646193337nt_int @ R2 @ S ) ) ).
% subrelI
thf(fact_935_subrelI,axiom,
! [R2: set_Pr5580431329626101545st_int,S: set_Pr5580431329626101545st_int] :
( ! [X: list_char > int,Y4: list_int] :
( ( member6299553290643319410st_int @ ( produc8650753666468850689st_int @ X @ Y4 ) @ R2 )
=> ( member6299553290643319410st_int @ ( produc8650753666468850689st_int @ X @ Y4 ) @ S ) )
=> ( ord_le5738032636065246921st_int @ R2 @ S ) ) ).
% subrelI
thf(fact_936_subrelI,axiom,
! [R2: set_Pr5290250543098163609nt_int,S: set_Pr5290250543098163609nt_int] :
( ! [X: list_char > int,Y4: int] :
( ( member3599195040465536738nt_int @ ( produc5790713362662368625nt_int @ X @ Y4 ) @ R2 )
=> ( member3599195040465536738nt_int @ ( produc5790713362662368625nt_int @ X @ Y4 ) @ S ) )
=> ( ord_le5108646760900237625nt_int @ R2 @ S ) ) ).
% subrelI
thf(fact_937_pred__equals__eq2,axiom,
! [R: set_Pr4358015327039922227nt_int,S5: set_Pr4358015327039922227nt_int] :
( ( ( ^ [X2: list_list_char,Y6: produc4435102495419491129nt_int] : ( member7220921130490543508nt_int @ ( produc3965054194175396271nt_int @ X2 @ Y6 ) @ R ) )
= ( ^ [X2: list_list_char,Y6: produc4435102495419491129nt_int] : ( member7220921130490543508nt_int @ ( produc3965054194175396271nt_int @ X2 @ Y6 ) @ S5 ) ) )
= ( R = S5 ) ) ).
% pred_equals_eq2
thf(fact_938_pred__equals__eq2,axiom,
! [R: set_Pr2116125220558876360st_int,S5: set_Pr2116125220558876360st_int] :
( ( ( ^ [X2: int,Y6: produc369741286924889929st_int] : ( member9100552558438320297st_int @ ( produc5086643055186798020st_int @ X2 @ Y6 ) @ R ) )
= ( ^ [X2: int,Y6: produc369741286924889929st_int] : ( member9100552558438320297st_int @ ( produc5086643055186798020st_int @ X2 @ Y6 ) @ S5 ) ) )
= ( R = S5 ) ) ).
% pred_equals_eq2
thf(fact_939_pred__equals__eq2,axiom,
! [R: set_Pr958786334691620121nt_int,S5: set_Pr958786334691620121nt_int] :
( ( ( ^ [X2: int,Y6: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y6 ) @ R ) )
= ( ^ [X2: int,Y6: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y6 ) @ S5 ) ) )
= ( R = S5 ) ) ).
% pred_equals_eq2
thf(fact_940_pred__equals__eq2,axiom,
! [R: set_Pr5580431329626101545st_int,S5: set_Pr5580431329626101545st_int] :
( ( ( ^ [X2: list_char > int,Y6: list_int] : ( member6299553290643319410st_int @ ( produc8650753666468850689st_int @ X2 @ Y6 ) @ R ) )
= ( ^ [X2: list_char > int,Y6: list_int] : ( member6299553290643319410st_int @ ( produc8650753666468850689st_int @ X2 @ Y6 ) @ S5 ) ) )
= ( R = S5 ) ) ).
% pred_equals_eq2
thf(fact_941_pred__equals__eq2,axiom,
! [R: set_Pr5290250543098163609nt_int,S5: set_Pr5290250543098163609nt_int] :
( ( ( ^ [X2: list_char > int,Y6: int] : ( member3599195040465536738nt_int @ ( produc5790713362662368625nt_int @ X2 @ Y6 ) @ R ) )
= ( ^ [X2: list_char > int,Y6: int] : ( member3599195040465536738nt_int @ ( produc5790713362662368625nt_int @ X2 @ Y6 ) @ S5 ) ) )
= ( R = S5 ) ) ).
% pred_equals_eq2
thf(fact_942_pred__subset__eq2,axiom,
! [R: set_Pr4358015327039922227nt_int,S5: set_Pr4358015327039922227nt_int] :
( ( ord_le8337283501185079284_int_o
@ ^ [X2: list_list_char,Y6: produc4435102495419491129nt_int] : ( member7220921130490543508nt_int @ ( produc3965054194175396271nt_int @ X2 @ Y6 ) @ R )
@ ^ [X2: list_list_char,Y6: produc4435102495419491129nt_int] : ( member7220921130490543508nt_int @ ( produc3965054194175396271nt_int @ X2 @ Y6 ) @ S5 ) )
= ( ord_le5912571515218079891nt_int @ R @ S5 ) ) ).
% pred_subset_eq2
thf(fact_943_pred__subset__eq2,axiom,
! [R: set_Pr2116125220558876360st_int,S5: set_Pr2116125220558876360st_int] :
( ( ord_le8490654098287043871_int_o
@ ^ [X2: int,Y6: produc369741286924889929st_int] : ( member9100552558438320297st_int @ ( produc5086643055186798020st_int @ X2 @ Y6 ) @ R )
@ ^ [X2: int,Y6: produc369741286924889929st_int] : ( member9100552558438320297st_int @ ( produc5086643055186798020st_int @ X2 @ Y6 ) @ S5 ) )
= ( ord_le8444298136753458984st_int @ R @ S5 ) ) ).
% pred_subset_eq2
thf(fact_944_pred__subset__eq2,axiom,
! [R: set_Pr958786334691620121nt_int,S5: set_Pr958786334691620121nt_int] :
( ( ord_le6741204236512500942_int_o
@ ^ [X2: int,Y6: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y6 ) @ R )
@ ^ [X2: int,Y6: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y6 ) @ S5 ) )
= ( ord_le2843351958646193337nt_int @ R @ S5 ) ) ).
% pred_subset_eq2
thf(fact_945_pred__subset__eq2,axiom,
! [R: set_Pr5580431329626101545st_int,S5: set_Pr5580431329626101545st_int] :
( ( ord_le4781006829409452478_int_o
@ ^ [X2: list_char > int,Y6: list_int] : ( member6299553290643319410st_int @ ( produc8650753666468850689st_int @ X2 @ Y6 ) @ R )
@ ^ [X2: list_char > int,Y6: list_int] : ( member6299553290643319410st_int @ ( produc8650753666468850689st_int @ X2 @ Y6 ) @ S5 ) )
= ( ord_le5738032636065246921st_int @ R @ S5 ) ) ).
% pred_subset_eq2
thf(fact_946_pred__subset__eq2,axiom,
! [R: set_Pr5290250543098163609nt_int,S5: set_Pr5290250543098163609nt_int] :
( ( ord_le8207994782381187918_int_o
@ ^ [X2: list_char > int,Y6: int] : ( member3599195040465536738nt_int @ ( produc5790713362662368625nt_int @ X2 @ Y6 ) @ R )
@ ^ [X2: list_char > int,Y6: int] : ( member3599195040465536738nt_int @ ( produc5790713362662368625nt_int @ X2 @ Y6 ) @ S5 ) )
= ( ord_le5108646760900237625nt_int @ R @ S5 ) ) ).
% pred_subset_eq2
thf(fact_947_order__le__imp__less__or__eq,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( ord_less_nat @ X3 @ Y )
| ( X3 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_948_order__le__imp__less__or__eq,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_int @ X3 @ Y )
=> ( ( ord_less_int @ X3 @ Y )
| ( X3 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_949_linorder__le__less__linear,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
| ( ord_less_nat @ Y @ X3 ) ) ).
% linorder_le_less_linear
thf(fact_950_linorder__le__less__linear,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_int @ X3 @ Y )
| ( ord_less_int @ Y @ X3 ) ) ).
% linorder_le_less_linear
thf(fact_951_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 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_952_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 )
=> ( ! [X: int,Y4: int] :
( ( ord_less_int @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_953_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 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_954_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 )
=> ( ! [X: int,Y4: int] :
( ( ord_less_int @ X @ Y4 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_955_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 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_956_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 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_957_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 )
=> ( ! [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_958_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 )
=> ( ! [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ Y4 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_959_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 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_960_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 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_961_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 )
=> ( ! [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_962_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 )
=> ( ! [X: int,Y4: int] :
( ( ord_less_eq_int @ X @ Y4 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_963_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 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_964_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 )
=> ( ! [X: int,Y4: int] :
( ( ord_less_int @ X @ Y4 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_965_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 )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_nat @ X @ Y4 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_966_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 )
=> ( ! [X: int,Y4: int] :
( ( ord_less_int @ X @ Y4 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_967_order__less__le__trans,axiom,
! [X3: nat,Y: nat,Z3: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z3 )
=> ( ord_less_nat @ X3 @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_968_order__less__le__trans,axiom,
! [X3: int,Y: int,Z3: int] :
( ( ord_less_int @ X3 @ Y )
=> ( ( ord_less_eq_int @ Y @ Z3 )
=> ( ord_less_int @ X3 @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_969_order__le__less__trans,axiom,
! [X3: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( ord_less_nat @ Y @ Z3 )
=> ( ord_less_nat @ X3 @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_970_order__le__less__trans,axiom,
! [X3: int,Y: int,Z3: int] :
( ( ord_less_eq_int @ X3 @ Y )
=> ( ( ord_less_int @ Y @ Z3 )
=> ( ord_less_int @ X3 @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_971_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_972_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_973_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_974_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_975_order__less__imp__le,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ X3 @ Y ) ) ).
% order_less_imp_le
thf(fact_976_order__less__imp__le,axiom,
! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
=> ( ord_less_eq_int @ X3 @ Y ) ) ).
% order_less_imp_le
thf(fact_977_linorder__not__less,axiom,
! [X3: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X3 @ Y ) )
= ( ord_less_eq_nat @ Y @ X3 ) ) ).
% linorder_not_less
thf(fact_978_linorder__not__less,axiom,
! [X3: int,Y: int] :
( ( ~ ( ord_less_int @ X3 @ Y ) )
= ( ord_less_eq_int @ Y @ X3 ) ) ).
% linorder_not_less
thf(fact_979_linorder__not__le,axiom,
! [X3: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X3 @ Y ) )
= ( ord_less_nat @ Y @ X3 ) ) ).
% linorder_not_le
thf(fact_980_linorder__not__le,axiom,
! [X3: int,Y: int] :
( ( ~ ( ord_less_eq_int @ X3 @ Y ) )
= ( ord_less_int @ Y @ X3 ) ) ).
% linorder_not_le
thf(fact_981_order__less__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y6: nat] :
( ( ord_less_eq_nat @ X2 @ Y6 )
& ( X2 != Y6 ) ) ) ) ).
% order_less_le
thf(fact_982_order__less__le,axiom,
( ord_less_int
= ( ^ [X2: int,Y6: int] :
( ( ord_less_eq_int @ X2 @ Y6 )
& ( X2 != Y6 ) ) ) ) ).
% order_less_le
thf(fact_983_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y6: nat] :
( ( ord_less_nat @ X2 @ Y6 )
| ( X2 = Y6 ) ) ) ) ).
% order_le_less
thf(fact_984_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X2: int,Y6: int] :
( ( ord_less_int @ X2 @ Y6 )
| ( X2 = Y6 ) ) ) ) ).
% order_le_less
thf(fact_985_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_986_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_987_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_988_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_989_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_990_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_991_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_992_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_993_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_994_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_995_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_996_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_997_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_998_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_999_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_1000_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_1001_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_1002_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_1003_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_1004_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_1005_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_1006_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_1007_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_1008_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_1009_not__le__imp__less,axiom,
! [Y: nat,X3: nat] :
( ~ ( ord_less_eq_nat @ Y @ X3 )
=> ( ord_less_nat @ X3 @ Y ) ) ).
% not_le_imp_less
thf(fact_1010_not__le__imp__less,axiom,
! [Y: int,X3: int] :
( ~ ( ord_less_eq_int @ Y @ X3 )
=> ( ord_less_int @ X3 @ Y ) ) ).
% not_le_imp_less
thf(fact_1011_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y6: nat] :
( ( ord_less_eq_nat @ X2 @ Y6 )
& ~ ( ord_less_eq_nat @ Y6 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_1012_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X2: int,Y6: int] :
( ( ord_less_eq_int @ X2 @ Y6 )
& ~ ( ord_less_eq_int @ Y6 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_1013_antisym__conv2,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( ~ ( ord_less_nat @ X3 @ Y ) )
= ( X3 = Y ) ) ) ).
% antisym_conv2
thf(fact_1014_antisym__conv2,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_int @ X3 @ Y )
=> ( ( ~ ( ord_less_int @ X3 @ Y ) )
= ( X3 = Y ) ) ) ).
% antisym_conv2
thf(fact_1015_antisym__conv1,axiom,
! [X3: nat,Y: nat] :
( ~ ( ord_less_nat @ X3 @ Y )
=> ( ( ord_less_eq_nat @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% antisym_conv1
thf(fact_1016_antisym__conv1,axiom,
! [X3: int,Y: int] :
( ~ ( ord_less_int @ X3 @ Y )
=> ( ( ord_less_eq_int @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% antisym_conv1
thf(fact_1017_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_1018_nless__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_int @ A @ B ) )
= ( ~ ( ord_less_eq_int @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_1019_leI,axiom,
! [X3: nat,Y: nat] :
( ~ ( ord_less_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ Y @ X3 ) ) ).
% leI
thf(fact_1020_leI,axiom,
! [X3: int,Y: int] :
( ~ ( ord_less_int @ X3 @ Y )
=> ( ord_less_eq_int @ Y @ X3 ) ) ).
% leI
thf(fact_1021_leD,axiom,
! [Y: nat,X3: nat] :
( ( ord_less_eq_nat @ Y @ X3 )
=> ~ ( ord_less_nat @ X3 @ Y ) ) ).
% leD
thf(fact_1022_leD,axiom,
! [Y: int,X3: int] :
( ( ord_less_eq_int @ Y @ X3 )
=> ~ ( ord_less_int @ X3 @ Y ) ) ).
% leD
thf(fact_1023_exec1I,axiom,
! [C3: produc6425607678544837394st_int,P: list_instr,I: int,S: list_char > int,Stk2: list_int] :
( ( C3
= ( 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 ) ) @ C3 ) ) ) ) ).
% exec1I
thf(fact_1024_size__char__eq__0,axiom,
( size_size_char
= ( ^ [C4: char] : zero_zero_nat ) ) ).
% size_char_eq_0
thf(fact_1025_to__m__state_Opelims,axiom,
! [X3: list_list_char,Xa: list_char > int,Xb: int,Y: int] :
( ( ( to_m_state @ X3 @ Xa @ Xb )
= Y )
=> ( ( accp_P4562858270263085236nt_int @ to_m_state_rel @ ( produc3965054194175396271nt_int @ X3 @ ( produc5790713362662368625nt_int @ Xa @ Xb ) ) )
=> ~ ( ( ( ( ( ord_less_int @ zero_zero_int @ Xb )
& ( ord_less_eq_int @ Xb @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ X3 ) ) ) )
=> ( Y
= ( Xa @ ( inth_list_char @ X3 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ X3 ) ) @ Xb ) ) ) ) )
& ( ~ ( ( ord_less_int @ zero_zero_int @ Xb )
& ( ord_less_eq_int @ Xb @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ X3 ) ) ) )
=> ( Y = zero_zero_int ) ) )
=> ~ ( accp_P4562858270263085236nt_int @ to_m_state_rel @ ( produc3965054194175396271nt_int @ X3 @ ( produc5790713362662368625nt_int @ Xa @ Xb ) ) ) ) ) ) ).
% to_m_state.pelims
thf(fact_1026_execl__all__SKIP,axiom,
! [Cfs2: list_P419547167332244514st_int] :
( ( execl_all @ nil_instr @ Cfs2 )
=> ( ( cpred @ skip @ ( nth_Pr8327173284404383691st_int @ Cfs2 @ zero_zero_nat ) @ ( nth_Pr8327173284404383691st_int @ Cfs2 @ ( minus_minus_nat @ ( size_s229347708945616566st_int @ Cfs2 ) @ ( suc @ zero_zero_nat ) ) ) )
& ( ( size_s229347708945616566st_int @ Cfs2 )
= ( suc @ zero_zero_nat ) ) ) ) ).
% execl_all_SKIP
thf(fact_1027_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_1028_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_1029_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_1030_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_1031_list_Oinject,axiom,
! [X21: produc6425607678544837394st_int,X222: list_P419547167332244514st_int,Y21: produc6425607678544837394st_int,Y22: list_P419547167332244514st_int] :
( ( ( cons_P4151473878978182108st_int @ X21 @ X222 )
= ( cons_P4151473878978182108st_int @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X222 = Y22 ) ) ) ).
% list.inject
thf(fact_1032_list_Oinject,axiom,
! [X21: int,X222: list_int,Y21: int,Y22: list_int] :
( ( ( cons_int @ X21 @ X222 )
= ( cons_int @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X222 = Y22 ) ) ) ).
% list.inject
thf(fact_1033_list_Oinject,axiom,
! [X21: instr,X222: list_instr,Y21: instr,Y22: list_instr] :
( ( ( cons_instr @ X21 @ X222 )
= ( cons_instr @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X222 = Y22 ) ) ) ).
% list.inject
thf(fact_1034_append1__eq__conv,axiom,
! [Xs: list_P419547167332244514st_int,X3: produc6425607678544837394st_int,Ys: list_P419547167332244514st_int,Y: produc6425607678544837394st_int] :
( ( ( append5028292981072010935st_int @ Xs @ ( cons_P4151473878978182108st_int @ X3 @ nil_Pr749310521696113548st_int ) )
= ( append5028292981072010935st_int @ Ys @ ( cons_P4151473878978182108st_int @ Y @ nil_Pr749310521696113548st_int ) ) )
= ( ( Xs = Ys )
& ( X3 = Y ) ) ) ).
% append1_eq_conv
thf(fact_1035_append1__eq__conv,axiom,
! [Xs: list_int,X3: int,Ys: list_int,Y: int] :
( ( ( append_int @ Xs @ ( cons_int @ X3 @ nil_int ) )
= ( append_int @ Ys @ ( cons_int @ Y @ nil_int ) ) )
= ( ( Xs = Ys )
& ( X3 = Y ) ) ) ).
% append1_eq_conv
thf(fact_1036_append1__eq__conv,axiom,
! [Xs: list_instr,X3: instr,Ys: list_instr,Y: instr] :
( ( ( append_instr @ Xs @ ( cons_instr @ X3 @ nil_instr ) )
= ( append_instr @ Ys @ ( cons_instr @ Y @ nil_instr ) ) )
= ( ( Xs = Ys )
& ( X3 = Y ) ) ) ).
% append1_eq_conv
thf(fact_1037_nth__Cons__0,axiom,
! [X3: produc6425607678544837394st_int,Xs: list_P419547167332244514st_int] :
( ( nth_Pr8327173284404383691st_int @ ( cons_P4151473878978182108st_int @ X3 @ Xs ) @ zero_zero_nat )
= X3 ) ).
% nth_Cons_0
thf(fact_1038_nth__Cons__0,axiom,
! [X3: int,Xs: list_int] :
( ( nth_int @ ( cons_int @ X3 @ Xs ) @ zero_zero_nat )
= X3 ) ).
% nth_Cons_0
thf(fact_1039_nth__Cons__0,axiom,
! [X3: instr,Xs: list_instr] :
( ( nth_instr @ ( cons_instr @ X3 @ Xs ) @ zero_zero_nat )
= X3 ) ).
% nth_Cons_0
thf(fact_1040_nth__Cons__Suc,axiom,
! [X3: produc6425607678544837394st_int,Xs: list_P419547167332244514st_int,N: nat] :
( ( nth_Pr8327173284404383691st_int @ ( cons_P4151473878978182108st_int @ X3 @ Xs ) @ ( suc @ N ) )
= ( nth_Pr8327173284404383691st_int @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_1041_nth__Cons__Suc,axiom,
! [X3: int,Xs: list_int,N: nat] :
( ( nth_int @ ( cons_int @ X3 @ Xs ) @ ( suc @ N ) )
= ( nth_int @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_1042_nth__Cons__Suc,axiom,
! [X3: instr,Xs: list_instr,N: nat] :
( ( nth_instr @ ( cons_instr @ X3 @ Xs ) @ ( suc @ N ) )
= ( nth_instr @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_1043_take__Suc__Cons,axiom,
! [N: nat,X3: produc6425607678544837394st_int,Xs: list_P419547167332244514st_int] :
( ( take_P1004856038796978596st_int @ ( suc @ N ) @ ( cons_P4151473878978182108st_int @ X3 @ Xs ) )
= ( cons_P4151473878978182108st_int @ X3 @ ( take_P1004856038796978596st_int @ N @ Xs ) ) ) ).
% take_Suc_Cons
thf(fact_1044_take__Suc__Cons,axiom,
! [N: nat,X3: int,Xs: list_int] :
( ( take_int @ ( suc @ N ) @ ( cons_int @ X3 @ Xs ) )
= ( cons_int @ X3 @ ( take_int @ N @ Xs ) ) ) ).
% take_Suc_Cons
thf(fact_1045_take__Suc__Cons,axiom,
! [N: nat,X3: instr,Xs: list_instr] :
( ( take_instr @ ( suc @ N ) @ ( cons_instr @ X3 @ Xs ) )
= ( cons_instr @ X3 @ ( take_instr @ N @ Xs ) ) ) ).
% take_Suc_Cons
thf(fact_1046_drop__Suc__Cons,axiom,
! [N: nat,X3: list_char,Xs: list_list_char] :
( ( drop_list_char @ ( suc @ N ) @ ( cons_list_char @ X3 @ Xs ) )
= ( drop_list_char @ N @ Xs ) ) ).
% drop_Suc_Cons
thf(fact_1047_drop__Suc__Cons,axiom,
! [N: nat,X3: produc6425607678544837394st_int,Xs: list_P419547167332244514st_int] :
( ( drop_P7838799151754935884st_int @ ( suc @ N ) @ ( cons_P4151473878978182108st_int @ X3 @ Xs ) )
= ( drop_P7838799151754935884st_int @ N @ Xs ) ) ).
% drop_Suc_Cons
thf(fact_1048_drop__Suc__Cons,axiom,
! [N: nat,X3: int,Xs: list_int] :
( ( drop_int @ ( suc @ N ) @ ( cons_int @ X3 @ Xs ) )
= ( drop_int @ N @ Xs ) ) ).
% drop_Suc_Cons
thf(fact_1049_drop__Suc__Cons,axiom,
! [N: nat,X3: instr,Xs: list_instr] :
( ( drop_instr @ ( suc @ N ) @ ( cons_instr @ X3 @ Xs ) )
= ( drop_instr @ N @ Xs ) ) ).
% drop_Suc_Cons
thf(fact_1050_nth__append__length,axiom,
! [Xs: list_P419547167332244514st_int,X3: produc6425607678544837394st_int,Ys: list_P419547167332244514st_int] :
( ( nth_Pr8327173284404383691st_int @ ( append5028292981072010935st_int @ Xs @ ( cons_P4151473878978182108st_int @ X3 @ Ys ) ) @ ( size_s229347708945616566st_int @ Xs ) )
= X3 ) ).
% nth_append_length
thf(fact_1051_nth__append__length,axiom,
! [Xs: list_instr,X3: instr,Ys: list_instr] :
( ( nth_instr @ ( append_instr @ Xs @ ( cons_instr @ X3 @ Ys ) ) @ ( size_size_list_instr @ Xs ) )
= X3 ) ).
% nth_append_length
thf(fact_1052_nth__append__length,axiom,
! [Xs: list_list_char,X3: list_char,Ys: list_list_char] :
( ( nth_list_char @ ( append_list_char @ Xs @ ( cons_list_char @ X3 @ Ys ) ) @ ( size_s356637359517785349t_char @ Xs ) )
= X3 ) ).
% nth_append_length
thf(fact_1053_nth__append__length,axiom,
! [Xs: list_int,X3: int,Ys: list_int] :
( ( nth_int @ ( append_int @ Xs @ ( cons_int @ X3 @ Ys ) ) @ ( size_size_list_int @ Xs ) )
= X3 ) ).
% nth_append_length
thf(fact_1054_nth__Cons__pos,axiom,
! [N: nat,X3: produc6425607678544837394st_int,Xs: list_P419547167332244514st_int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_Pr8327173284404383691st_int @ ( cons_P4151473878978182108st_int @ X3 @ Xs ) @ N )
= ( nth_Pr8327173284404383691st_int @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_1055_nth__Cons__pos,axiom,
! [N: nat,X3: int,Xs: list_int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_int @ ( cons_int @ X3 @ Xs ) @ N )
= ( nth_int @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_1056_nth__Cons__pos,axiom,
! [N: nat,X3: instr,Xs: list_instr] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_instr @ ( cons_instr @ X3 @ Xs ) @ N )
= ( nth_instr @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_1057_list__nonempty__induct,axiom,
! [Xs: list_P419547167332244514st_int,P: list_P419547167332244514st_int > $o] :
( ( Xs != nil_Pr749310521696113548st_int )
=> ( ! [X: produc6425607678544837394st_int] : ( P @ ( cons_P4151473878978182108st_int @ X @ nil_Pr749310521696113548st_int ) )
=> ( ! [X: produc6425607678544837394st_int,Xs3: list_P419547167332244514st_int] :
( ( Xs3 != nil_Pr749310521696113548st_int )
=> ( ( P @ Xs3 )
=> ( P @ ( cons_P4151473878978182108st_int @ X @ Xs3 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_1058_list__nonempty__induct,axiom,
! [Xs: list_int,P: list_int > $o] :
( ( Xs != nil_int )
=> ( ! [X: int] : ( P @ ( cons_int @ X @ nil_int ) )
=> ( ! [X: int,Xs3: list_int] :
( ( Xs3 != nil_int )
=> ( ( P @ Xs3 )
=> ( P @ ( cons_int @ X @ Xs3 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_1059_list__nonempty__induct,axiom,
! [Xs: list_instr,P: list_instr > $o] :
( ( Xs != nil_instr )
=> ( ! [X: instr] : ( P @ ( cons_instr @ X @ nil_instr ) )
=> ( ! [X: instr,Xs3: list_instr] :
( ( Xs3 != nil_instr )
=> ( ( P @ Xs3 )
=> ( P @ ( cons_instr @ X @ Xs3 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_1060_list__induct2_H,axiom,
! [P: list_P419547167332244514st_int > list_P419547167332244514st_int > $o,Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int] :
( ( P @ nil_Pr749310521696113548st_int @ nil_Pr749310521696113548st_int )
=> ( ! [X: produc6425607678544837394st_int,Xs3: list_P419547167332244514st_int] : ( P @ ( cons_P4151473878978182108st_int @ X @ Xs3 ) @ nil_Pr749310521696113548st_int )
=> ( ! [Y4: produc6425607678544837394st_int,Ys4: list_P419547167332244514st_int] : ( P @ nil_Pr749310521696113548st_int @ ( cons_P4151473878978182108st_int @ Y4 @ Ys4 ) )
=> ( ! [X: produc6425607678544837394st_int,Xs3: list_P419547167332244514st_int,Y4: produc6425607678544837394st_int,Ys4: list_P419547167332244514st_int] :
( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons_P4151473878978182108st_int @ X @ Xs3 ) @ ( cons_P4151473878978182108st_int @ Y4 @ Ys4 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_1061_list__induct2_H,axiom,
! [P: list_P419547167332244514st_int > list_int > $o,Xs: list_P419547167332244514st_int,Ys: list_int] :
( ( P @ nil_Pr749310521696113548st_int @ nil_int )
=> ( ! [X: produc6425607678544837394st_int,Xs3: list_P419547167332244514st_int] : ( P @ ( cons_P4151473878978182108st_int @ X @ Xs3 ) @ nil_int )
=> ( ! [Y4: int,Ys4: list_int] : ( P @ nil_Pr749310521696113548st_int @ ( cons_int @ Y4 @ Ys4 ) )
=> ( ! [X: produc6425607678544837394st_int,Xs3: list_P419547167332244514st_int,Y4: int,Ys4: list_int] :
( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons_P4151473878978182108st_int @ X @ Xs3 ) @ ( cons_int @ Y4 @ Ys4 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_1062_list__induct2_H,axiom,
! [P: list_P419547167332244514st_int > list_instr > $o,Xs: list_P419547167332244514st_int,Ys: list_instr] :
( ( P @ nil_Pr749310521696113548st_int @ nil_instr )
=> ( ! [X: produc6425607678544837394st_int,Xs3: list_P419547167332244514st_int] : ( P @ ( cons_P4151473878978182108st_int @ X @ Xs3 ) @ nil_instr )
=> ( ! [Y4: instr,Ys4: list_instr] : ( P @ nil_Pr749310521696113548st_int @ ( cons_instr @ Y4 @ Ys4 ) )
=> ( ! [X: produc6425607678544837394st_int,Xs3: list_P419547167332244514st_int,Y4: instr,Ys4: list_instr] :
( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons_P4151473878978182108st_int @ X @ Xs3 ) @ ( cons_instr @ Y4 @ Ys4 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_1063_list__induct2_H,axiom,
! [P: list_int > list_P419547167332244514st_int > $o,Xs: list_int,Ys: list_P419547167332244514st_int] :
( ( P @ nil_int @ nil_Pr749310521696113548st_int )
=> ( ! [X: int,Xs3: list_int] : ( P @ ( cons_int @ X @ Xs3 ) @ nil_Pr749310521696113548st_int )
=> ( ! [Y4: produc6425607678544837394st_int,Ys4: list_P419547167332244514st_int] : ( P @ nil_int @ ( cons_P4151473878978182108st_int @ Y4 @ Ys4 ) )
=> ( ! [X: int,Xs3: list_int,Y4: produc6425607678544837394st_int,Ys4: list_P419547167332244514st_int] :
( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons_int @ X @ Xs3 ) @ ( cons_P4151473878978182108st_int @ Y4 @ Ys4 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_1064_list__induct2_H,axiom,
! [P: list_int > list_int > $o,Xs: list_int,Ys: list_int] :
( ( P @ nil_int @ nil_int )
=> ( ! [X: int,Xs3: list_int] : ( P @ ( cons_int @ X @ Xs3 ) @ nil_int )
=> ( ! [Y4: int,Ys4: list_int] : ( P @ nil_int @ ( cons_int @ Y4 @ Ys4 ) )
=> ( ! [X: int,Xs3: list_int,Y4: int,Ys4: list_int] :
( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons_int @ X @ Xs3 ) @ ( cons_int @ Y4 @ Ys4 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_1065_list__induct2_H,axiom,
! [P: list_int > list_instr > $o,Xs: list_int,Ys: list_instr] :
( ( P @ nil_int @ nil_instr )
=> ( ! [X: int,Xs3: list_int] : ( P @ ( cons_int @ X @ Xs3 ) @ nil_instr )
=> ( ! [Y4: instr,Ys4: list_instr] : ( P @ nil_int @ ( cons_instr @ Y4 @ Ys4 ) )
=> ( ! [X: int,Xs3: list_int,Y4: instr,Ys4: list_instr] :
( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons_int @ X @ Xs3 ) @ ( cons_instr @ Y4 @ Ys4 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_1066_list__induct2_H,axiom,
! [P: list_instr > list_P419547167332244514st_int > $o,Xs: list_instr,Ys: list_P419547167332244514st_int] :
( ( P @ nil_instr @ nil_Pr749310521696113548st_int )
=> ( ! [X: instr,Xs3: list_instr] : ( P @ ( cons_instr @ X @ Xs3 ) @ nil_Pr749310521696113548st_int )
=> ( ! [Y4: produc6425607678544837394st_int,Ys4: list_P419547167332244514st_int] : ( P @ nil_instr @ ( cons_P4151473878978182108st_int @ Y4 @ Ys4 ) )
=> ( ! [X: instr,Xs3: list_instr,Y4: produc6425607678544837394st_int,Ys4: list_P419547167332244514st_int] :
( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons_instr @ X @ Xs3 ) @ ( cons_P4151473878978182108st_int @ Y4 @ Ys4 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_1067_list__induct2_H,axiom,
! [P: list_instr > list_int > $o,Xs: list_instr,Ys: list_int] :
( ( P @ nil_instr @ nil_int )
=> ( ! [X: instr,Xs3: list_instr] : ( P @ ( cons_instr @ X @ Xs3 ) @ nil_int )
=> ( ! [Y4: int,Ys4: list_int] : ( P @ nil_instr @ ( cons_int @ Y4 @ Ys4 ) )
=> ( ! [X: instr,Xs3: list_instr,Y4: int,Ys4: list_int] :
( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons_instr @ X @ Xs3 ) @ ( cons_int @ Y4 @ Ys4 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_1068_list__induct2_H,axiom,
! [P: list_instr > list_instr > $o,Xs: list_instr,Ys: list_instr] :
( ( P @ nil_instr @ nil_instr )
=> ( ! [X: instr,Xs3: list_instr] : ( P @ ( cons_instr @ X @ Xs3 ) @ nil_instr )
=> ( ! [Y4: instr,Ys4: list_instr] : ( P @ nil_instr @ ( cons_instr @ Y4 @ Ys4 ) )
=> ( ! [X: instr,Xs3: list_instr,Y4: instr,Ys4: list_instr] :
( ( P @ Xs3 @ Ys4 )
=> ( P @ ( cons_instr @ X @ Xs3 ) @ ( cons_instr @ Y4 @ Ys4 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_1069_neq__Nil__conv,axiom,
! [Xs: list_P419547167332244514st_int] :
( ( Xs != nil_Pr749310521696113548st_int )
= ( ? [Y6: produc6425607678544837394st_int,Ys3: list_P419547167332244514st_int] :
( Xs
= ( cons_P4151473878978182108st_int @ Y6 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_1070_neq__Nil__conv,axiom,
! [Xs: list_int] :
( ( Xs != nil_int )
= ( ? [Y6: int,Ys3: list_int] :
( Xs
= ( cons_int @ Y6 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_1071_neq__Nil__conv,axiom,
! [Xs: list_instr] :
( ( Xs != nil_instr )
= ( ? [Y6: instr,Ys3: list_instr] :
( Xs
= ( cons_instr @ Y6 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_1072_remdups__adj_Ocases,axiom,
! [X3: list_P419547167332244514st_int] :
( ( X3 != nil_Pr749310521696113548st_int )
=> ( ! [X: produc6425607678544837394st_int] :
( X3
!= ( cons_P4151473878978182108st_int @ X @ nil_Pr749310521696113548st_int ) )
=> ~ ! [X: produc6425607678544837394st_int,Y4: produc6425607678544837394st_int,Xs3: list_P419547167332244514st_int] :
( X3
!= ( cons_P4151473878978182108st_int @ X @ ( cons_P4151473878978182108st_int @ Y4 @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_1073_remdups__adj_Ocases,axiom,
! [X3: list_int] :
( ( X3 != nil_int )
=> ( ! [X: int] :
( X3
!= ( cons_int @ X @ nil_int ) )
=> ~ ! [X: int,Y4: int,Xs3: list_int] :
( X3
!= ( cons_int @ X @ ( cons_int @ Y4 @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_1074_remdups__adj_Ocases,axiom,
! [X3: list_instr] :
( ( X3 != nil_instr )
=> ( ! [X: instr] :
( X3
!= ( cons_instr @ X @ nil_instr ) )
=> ~ ! [X: instr,Y4: instr,Xs3: list_instr] :
( X3
!= ( cons_instr @ X @ ( cons_instr @ Y4 @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_1075_transpose_Ocases,axiom,
! [X3: list_l6422821129637969586st_int] :
( ( X3 != nil_li4597314861645524892st_int )
=> ( ! [Xss: list_l6422821129637969586st_int] :
( X3
!= ( cons_l8648290059330449388st_int @ nil_Pr749310521696113548st_int @ Xss ) )
=> ~ ! [X: produc6425607678544837394st_int,Xs3: list_P419547167332244514st_int,Xss: list_l6422821129637969586st_int] :
( X3
!= ( cons_l8648290059330449388st_int @ ( cons_P4151473878978182108st_int @ X @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_1076_transpose_Ocases,axiom,
! [X3: list_list_int] :
( ( X3 != nil_list_int )
=> ( ! [Xss: list_list_int] :
( X3
!= ( cons_list_int @ nil_int @ Xss ) )
=> ~ ! [X: int,Xs3: list_int,Xss: list_list_int] :
( X3
!= ( cons_list_int @ ( cons_int @ X @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_1077_transpose_Ocases,axiom,
! [X3: list_list_instr] :
( ( X3 != nil_list_instr )
=> ( ! [Xss: list_list_instr] :
( X3
!= ( cons_list_instr @ nil_instr @ Xss ) )
=> ~ ! [X: instr,Xs3: list_instr,Xss: list_list_instr] :
( X3
!= ( cons_list_instr @ ( cons_instr @ X @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_1078_min__list_Ocases,axiom,
! [X3: list_int] :
( ! [X: int,Xs3: list_int] :
( X3
!= ( cons_int @ X @ Xs3 ) )
=> ( X3 = nil_int ) ) ).
% min_list.cases
thf(fact_1079_list_Oexhaust,axiom,
! [Y: list_P419547167332244514st_int] :
( ( Y != nil_Pr749310521696113548st_int )
=> ~ ! [X212: produc6425607678544837394st_int,X223: list_P419547167332244514st_int] :
( Y
!= ( cons_P4151473878978182108st_int @ X212 @ X223 ) ) ) ).
% list.exhaust
thf(fact_1080_list_Oexhaust,axiom,
! [Y: list_int] :
( ( Y != nil_int )
=> ~ ! [X212: int,X223: list_int] :
( Y
!= ( cons_int @ X212 @ X223 ) ) ) ).
% list.exhaust
thf(fact_1081_list_Oexhaust,axiom,
! [Y: list_instr] :
( ( Y != nil_instr )
=> ~ ! [X212: instr,X223: list_instr] :
( Y
!= ( cons_instr @ X212 @ X223 ) ) ) ).
% list.exhaust
thf(fact_1082_list_OdiscI,axiom,
! [List: list_P419547167332244514st_int,X21: produc6425607678544837394st_int,X222: list_P419547167332244514st_int] :
( ( List
= ( cons_P4151473878978182108st_int @ X21 @ X222 ) )
=> ( List != nil_Pr749310521696113548st_int ) ) ).
% list.discI
thf(fact_1083_list_OdiscI,axiom,
! [List: list_int,X21: int,X222: list_int] :
( ( List
= ( cons_int @ X21 @ X222 ) )
=> ( List != nil_int ) ) ).
% list.discI
thf(fact_1084_list_OdiscI,axiom,
! [List: list_instr,X21: instr,X222: list_instr] :
( ( List
= ( cons_instr @ X21 @ X222 ) )
=> ( List != nil_instr ) ) ).
% list.discI
thf(fact_1085_list_Odistinct_I1_J,axiom,
! [X21: produc6425607678544837394st_int,X222: list_P419547167332244514st_int] :
( nil_Pr749310521696113548st_int
!= ( cons_P4151473878978182108st_int @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_1086_list_Odistinct_I1_J,axiom,
! [X21: int,X222: list_int] :
( nil_int
!= ( cons_int @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_1087_list_Odistinct_I1_J,axiom,
! [X21: instr,X222: list_instr] :
( nil_instr
!= ( cons_instr @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_1088_successively_Ocases,axiom,
! [X3: produc2468426419292222131st_int] :
( ! [P7: produc6425607678544837394st_int > produc6425607678544837394st_int > $o] :
( X3
!= ( produc554038597315947693st_int @ P7 @ nil_Pr749310521696113548st_int ) )
=> ( ! [P7: produc6425607678544837394st_int > produc6425607678544837394st_int > $o,X: produc6425607678544837394st_int] :
( X3
!= ( produc554038597315947693st_int @ P7 @ ( cons_P4151473878978182108st_int @ X @ nil_Pr749310521696113548st_int ) ) )
=> ~ ! [P7: produc6425607678544837394st_int > produc6425607678544837394st_int > $o,X: produc6425607678544837394st_int,Y4: produc6425607678544837394st_int,Xs3: list_P419547167332244514st_int] :
( X3
!= ( produc554038597315947693st_int @ P7 @ ( cons_P4151473878978182108st_int @ X @ ( cons_P4151473878978182108st_int @ Y4 @ Xs3 ) ) ) ) ) ) ).
% successively.cases
thf(fact_1089_successively_Ocases,axiom,
! [X3: produc5834231552977413017st_int] :
( ! [P7: int > int > $o] :
( X3
!= ( produc8618682346314911123st_int @ P7 @ nil_int ) )
=> ( ! [P7: int > int > $o,X: int] :
( X3
!= ( produc8618682346314911123st_int @ P7 @ ( cons_int @ X @ nil_int ) ) )
=> ~ ! [P7: int > int > $o,X: int,Y4: int,Xs3: list_int] :
( X3
!= ( produc8618682346314911123st_int @ P7 @ ( cons_int @ X @ ( cons_int @ Y4 @ Xs3 ) ) ) ) ) ) ).
% successively.cases
thf(fact_1090_successively_Ocases,axiom,
! [X3: produc7133927873007960030_instr] :
( ! [P7: instr > instr > $o] :
( X3
!= ( produc7818391707689388878_instr @ P7 @ nil_instr ) )
=> ( ! [P7: instr > instr > $o,X: instr] :
( X3
!= ( produc7818391707689388878_instr @ P7 @ ( cons_instr @ X @ nil_instr ) ) )
=> ~ ! [P7: instr > instr > $o,X: instr,Y4: instr,Xs3: list_instr] :
( X3
!= ( produc7818391707689388878_instr @ P7 @ ( cons_instr @ X @ ( cons_instr @ Y4 @ Xs3 ) ) ) ) ) ) ).
% successively.cases
thf(fact_1091_sorted__wrt_Ocases,axiom,
! [X3: produc2468426419292222131st_int] :
( ! [P7: produc6425607678544837394st_int > produc6425607678544837394st_int > $o] :
( X3
!= ( produc554038597315947693st_int @ P7 @ nil_Pr749310521696113548st_int ) )
=> ~ ! [P7: produc6425607678544837394st_int > produc6425607678544837394st_int > $o,X: produc6425607678544837394st_int,Ys4: list_P419547167332244514st_int] :
( X3
!= ( produc554038597315947693st_int @ P7 @ ( cons_P4151473878978182108st_int @ X @ Ys4 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_1092_sorted__wrt_Ocases,axiom,
! [X3: produc5834231552977413017st_int] :
( ! [P7: int > int > $o] :
( X3
!= ( produc8618682346314911123st_int @ P7 @ nil_int ) )
=> ~ ! [P7: int > int > $o,X: int,Ys4: list_int] :
( X3
!= ( produc8618682346314911123st_int @ P7 @ ( cons_int @ X @ Ys4 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_1093_sorted__wrt_Ocases,axiom,
! [X3: produc7133927873007960030_instr] :
( ! [P7: instr > instr > $o] :
( X3
!= ( produc7818391707689388878_instr @ P7 @ nil_instr ) )
=> ~ ! [P7: instr > instr > $o,X: instr,Ys4: list_instr] :
( X3
!= ( produc7818391707689388878_instr @ P7 @ ( cons_instr @ X @ Ys4 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_1094_shuffles_Ocases,axiom,
! [X3: produc5848496107505245989st_int] :
( ! [Ys4: list_P419547167332244514st_int] :
( X3
!= ( produc2639222612625641053st_int @ nil_Pr749310521696113548st_int @ Ys4 ) )
=> ( ! [Xs3: list_P419547167332244514st_int] :
( X3
!= ( produc2639222612625641053st_int @ Xs3 @ nil_Pr749310521696113548st_int ) )
=> ~ ! [X: produc6425607678544837394st_int,Xs3: list_P419547167332244514st_int,Y4: produc6425607678544837394st_int,Ys4: list_P419547167332244514st_int] :
( X3
!= ( produc2639222612625641053st_int @ ( cons_P4151473878978182108st_int @ X @ Xs3 ) @ ( cons_P4151473878978182108st_int @ Y4 @ Ys4 ) ) ) ) ) ).
% shuffles.cases
thf(fact_1095_shuffles_Ocases,axiom,
! [X3: produc1186641810826059865st_int] :
( ! [Ys4: list_int] :
( X3
!= ( produc364263696895485585st_int @ nil_int @ Ys4 ) )
=> ( ! [Xs3: list_int] :
( X3
!= ( produc364263696895485585st_int @ Xs3 @ nil_int ) )
=> ~ ! [X: int,Xs3: list_int,Y4: int,Ys4: list_int] :
( X3
!= ( produc364263696895485585st_int @ ( cons_int @ X @ Xs3 ) @ ( cons_int @ Y4 @ Ys4 ) ) ) ) ) ).
% shuffles.cases
thf(fact_1096_shuffles_Ocases,axiom,
! [X3: produc8891168062546637159_instr] :
( ! [Ys4: list_instr] :
( X3
!= ( produc7750586281064920791_instr @ nil_instr @ Ys4 ) )
=> ( ! [Xs3: list_instr] :
( X3
!= ( produc7750586281064920791_instr @ Xs3 @ nil_instr ) )
=> ~ ! [X: instr,Xs3: list_instr,Y4: instr,Ys4: list_instr] :
( X3
!= ( produc7750586281064920791_instr @ ( cons_instr @ X @ Xs3 ) @ ( cons_instr @ Y4 @ Ys4 ) ) ) ) ) ).
% shuffles.cases
thf(fact_1097_splice_Ocases,axiom,
! [X3: produc5848496107505245989st_int] :
( ! [Ys4: list_P419547167332244514st_int] :
( X3
!= ( produc2639222612625641053st_int @ nil_Pr749310521696113548st_int @ Ys4 ) )
=> ~ ! [X: produc6425607678544837394st_int,Xs3: list_P419547167332244514st_int,Ys4: list_P419547167332244514st_int] :
( X3
!= ( produc2639222612625641053st_int @ ( cons_P4151473878978182108st_int @ X @ Xs3 ) @ Ys4 ) ) ) ).
% splice.cases
thf(fact_1098_splice_Ocases,axiom,
! [X3: produc1186641810826059865st_int] :
( ! [Ys4: list_int] :
( X3
!= ( produc364263696895485585st_int @ nil_int @ Ys4 ) )
=> ~ ! [X: int,Xs3: list_int,Ys4: list_int] :
( X3
!= ( produc364263696895485585st_int @ ( cons_int @ X @ Xs3 ) @ Ys4 ) ) ) ).
% splice.cases
thf(fact_1099_splice_Ocases,axiom,
! [X3: produc8891168062546637159_instr] :
( ! [Ys4: list_instr] :
( X3
!= ( produc7750586281064920791_instr @ nil_instr @ Ys4 ) )
=> ~ ! [X: instr,Xs3: list_instr,Ys4: list_instr] :
( X3
!= ( produc7750586281064920791_instr @ ( cons_instr @ X @ Xs3 ) @ Ys4 ) ) ) ).
% splice.cases
thf(fact_1100_list_Osimps_I9_J,axiom,
! [F: produc6425607678544837394st_int > produc6425607678544837394st_int,X21: produc6425607678544837394st_int,X222: list_P419547167332244514st_int] :
( ( map_Pr8625093460839654122st_int @ F @ ( cons_P4151473878978182108st_int @ X21 @ X222 ) )
= ( cons_P4151473878978182108st_int @ ( F @ X21 ) @ ( map_Pr8625093460839654122st_int @ F @ X222 ) ) ) ).
% list.simps(9)
thf(fact_1101_list_Osimps_I9_J,axiom,
! [F: produc6425607678544837394st_int > int,X21: produc6425607678544837394st_int,X222: list_P419547167332244514st_int] :
( ( map_Pr3224476576807795204nt_int @ F @ ( cons_P4151473878978182108st_int @ X21 @ X222 ) )
= ( cons_int @ ( F @ X21 ) @ ( map_Pr3224476576807795204nt_int @ F @ X222 ) ) ) ).
% list.simps(9)
thf(fact_1102_list_Osimps_I9_J,axiom,
! [F: produc6425607678544837394st_int > instr,X21: produc6425607678544837394st_int,X222: list_P419547167332244514st_int] :
( ( map_Pr7514435936640407687_instr @ F @ ( cons_P4151473878978182108st_int @ X21 @ X222 ) )
= ( cons_instr @ ( F @ X21 ) @ ( map_Pr7514435936640407687_instr @ F @ X222 ) ) ) ).
% list.simps(9)
thf(fact_1103_list_Osimps_I9_J,axiom,
! [F: int > produc6425607678544837394st_int,X21: int,X222: list_int] :
( ( map_in8512336359594627332st_int @ F @ ( cons_int @ X21 @ X222 ) )
= ( cons_P4151473878978182108st_int @ ( F @ X21 ) @ ( map_in8512336359594627332st_int @ F @ X222 ) ) ) ).
% list.simps(9)
thf(fact_1104_list_Osimps_I9_J,axiom,
! [F: int > int,X21: int,X222: list_int] :
( ( map_int_int @ F @ ( cons_int @ X21 @ X222 ) )
= ( cons_int @ ( F @ X21 ) @ ( map_int_int @ F @ X222 ) ) ) ).
% list.simps(9)
thf(fact_1105_list_Osimps_I9_J,axiom,
! [F: int > instr,X21: int,X222: list_int] :
( ( map_int_instr @ F @ ( cons_int @ X21 @ X222 ) )
= ( cons_instr @ ( F @ X21 ) @ ( map_int_instr @ F @ X222 ) ) ) ).
% list.simps(9)
thf(fact_1106_list_Osimps_I9_J,axiom,
! [F: instr > produc6425607678544837394st_int,X21: instr,X222: list_instr] :
( ( map_in7259965606695656109st_int @ F @ ( cons_instr @ X21 @ X222 ) )
= ( cons_P4151473878978182108st_int @ ( F @ X21 ) @ ( map_in7259965606695656109st_int @ F @ X222 ) ) ) ).
% list.simps(9)
thf(fact_1107_list_Osimps_I9_J,axiom,
! [F: instr > int,X21: instr,X222: list_instr] :
( ( map_instr_int @ F @ ( cons_instr @ X21 @ X222 ) )
= ( cons_int @ ( F @ X21 ) @ ( map_instr_int @ F @ X222 ) ) ) ).
% list.simps(9)
thf(fact_1108_list_Osimps_I9_J,axiom,
! [F: instr > instr,X21: instr,X222: list_instr] :
( ( map_instr_instr @ F @ ( cons_instr @ X21 @ X222 ) )
= ( cons_instr @ ( F @ X21 ) @ ( map_instr_instr @ F @ X222 ) ) ) ).
% list.simps(9)
thf(fact_1109_execl_Ocases,axiom,
! [X3: produc4400946599758736694st_int] :
( ! [P7: list_instr,Cf3: produc6425607678544837394st_int,Cf4: produc6425607678544837394st_int,Cfs4: list_P419547167332244514st_int] :
( X3
!= ( produc7764413618310988720st_int @ P7 @ ( cons_P4151473878978182108st_int @ Cf3 @ ( cons_P4151473878978182108st_int @ Cf4 @ Cfs4 ) ) ) )
=> ( ! [P7: list_instr] :
( X3
!= ( produc7764413618310988720st_int @ P7 @ nil_Pr749310521696113548st_int ) )
=> ~ ! [P7: list_instr,V: produc6425607678544837394st_int] :
( X3
!= ( produc7764413618310988720st_int @ P7 @ ( cons_P4151473878978182108st_int @ V @ nil_Pr749310521696113548st_int ) ) ) ) ) ).
% execl.cases
thf(fact_1110_execl_Osimps_I1_J,axiom,
! [P: list_instr,Cf5: produc6425607678544837394st_int,Cf6: produc6425607678544837394st_int,Cfs2: list_P419547167332244514st_int] :
( ( execl @ P @ ( cons_P4151473878978182108st_int @ Cf5 @ ( cons_P4151473878978182108st_int @ Cf6 @ Cfs2 ) ) )
= ( ( exec1 @ P @ Cf5 @ Cf6 )
& ( execl @ P @ ( cons_P4151473878978182108st_int @ Cf6 @ Cfs2 ) ) ) ) ).
% execl.simps(1)
thf(fact_1111_execl_Oelims_I3_J,axiom,
! [X3: list_instr,Xa: list_P419547167332244514st_int] :
( ~ ( execl @ X3 @ Xa )
=> ~ ! [Cf3: produc6425607678544837394st_int,Cf4: produc6425607678544837394st_int,Cfs4: list_P419547167332244514st_int] :
( ( Xa
= ( cons_P4151473878978182108st_int @ Cf3 @ ( cons_P4151473878978182108st_int @ Cf4 @ Cfs4 ) ) )
=> ( ( exec1 @ X3 @ Cf3 @ Cf4 )
& ( execl @ X3 @ ( cons_P4151473878978182108st_int @ Cf4 @ Cfs4 ) ) ) ) ) ).
% execl.elims(3)
thf(fact_1112_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_1113_execl_Oelims_I1_J,axiom,
! [X3: list_instr,Xa: list_P419547167332244514st_int,Y: $o] :
( ( ( execl @ X3 @ Xa )
= Y )
=> ( ! [Cf3: produc6425607678544837394st_int,Cf4: produc6425607678544837394st_int,Cfs4: list_P419547167332244514st_int] :
( ( Xa
= ( cons_P4151473878978182108st_int @ Cf3 @ ( cons_P4151473878978182108st_int @ Cf4 @ Cfs4 ) ) )
=> ( Y
= ( ~ ( ( exec1 @ X3 @ Cf3 @ Cf4 )
& ( execl @ X3 @ ( cons_P4151473878978182108st_int @ Cf4 @ Cfs4 ) ) ) ) ) )
=> ( ( ( Xa = nil_Pr749310521696113548st_int )
=> ~ Y )
=> ~ ( ? [V: produc6425607678544837394st_int] :
( Xa
= ( cons_P4151473878978182108st_int @ V @ nil_Pr749310521696113548st_int ) )
=> ~ Y ) ) ) ) ).
% execl.elims(1)
thf(fact_1114_execl_Oelims_I2_J,axiom,
! [X3: list_instr,Xa: list_P419547167332244514st_int] :
( ( execl @ X3 @ Xa )
=> ( ! [Cf3: produc6425607678544837394st_int,Cf4: produc6425607678544837394st_int,Cfs4: list_P419547167332244514st_int] :
( ( Xa
= ( cons_P4151473878978182108st_int @ Cf3 @ ( cons_P4151473878978182108st_int @ Cf4 @ Cfs4 ) ) )
=> ~ ( ( exec1 @ X3 @ Cf3 @ Cf4 )
& ( execl @ X3 @ ( cons_P4151473878978182108st_int @ Cf4 @ Cfs4 ) ) ) )
=> ( ( Xa != nil_Pr749310521696113548st_int )
=> ~ ! [V: produc6425607678544837394st_int] :
( Xa
!= ( cons_P4151473878978182108st_int @ V @ nil_Pr749310521696113548st_int ) ) ) ) ) ).
% execl.elims(2)
thf(fact_1115_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_1116_iexec__pred_Ointros,axiom,
! [Ins2: instr,Cf5: produc6425607678544837394st_int] : ( iexec_pred @ ( produc1484136438817787418st_int @ Ins2 @ Cf5 ) @ ( iexec @ Ins2 @ Cf5 ) ) ).
% iexec_pred.intros
thf(fact_1117_iexec__pred_Osimps,axiom,
( iexec_pred
= ( ^ [A1: produc5995290525303592096st_int,A22: produc6425607678544837394st_int] :
? [Ins3: instr,Cf: produc6425607678544837394st_int] :
( ( A1
= ( produc1484136438817787418st_int @ Ins3 @ Cf ) )
& ( A22
= ( iexec @ Ins3 @ Cf ) ) ) ) ) ).
% iexec_pred.simps
thf(fact_1118_iexec__pred_Ocases,axiom,
! [A12: produc5995290525303592096st_int,A23: produc6425607678544837394st_int] :
( ( iexec_pred @ A12 @ A23 )
=> ~ ! [Ins: instr,Cf3: produc6425607678544837394st_int] :
( ( A12
= ( produc1484136438817787418st_int @ Ins @ Cf3 ) )
=> ( A23
!= ( iexec @ Ins @ Cf3 ) ) ) ) ).
% iexec_pred.cases
thf(fact_1119_size_H__char__eq__0,axiom,
( size_char
= ( ^ [C4: char] : zero_zero_nat ) ) ).
% size'_char_eq_0
thf(fact_1120_execl__sub__aux,axiom,
! [Q2: list_instr > produc6425607678544837394st_int > $o,P3: list_instr,Pc: int,S: list_char > int,Stk2: list_int,Cfs2: list_P419547167332244514st_int,P: list_instr,F2: list_P419547167332244514st_int > nat > nat > list_P419547167332244514st_int,M: nat,N: nat,Cfs5: list_P419547167332244514st_int] :
( ! [M4: nat,N3: nat] :
( ! [X6: nat] :
( ( member_nat @ X6 @ ( set_or4665077453230672383an_nat @ M4 @ N3 ) )
=> ( Q2 @ P3 @ ( nth_Pr8327173284404383691st_int @ ( cons_P4151473878978182108st_int @ ( produc5086643055186798020st_int @ Pc @ ( produc8650753666468850689st_int @ S @ Stk2 ) ) @ Cfs2 ) @ 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 @ N3 @ Cfs2 ) )
@ ^ [O: nat] : ( F2 @ Cfs2 @ O @ N3 )
@ M4 ) ) ) )
=> ( ! [X: nat] :
( ( member_nat @ X @ ( set_or4665077453230672383an_nat @ M @ ( plus_plus_nat @ ( plus_plus_nat @ N @ M ) @ ( size_s229347708945616566st_int @ Cfs5 ) ) ) )
=> ( Q2 @ P3 @ ( nth_Pr8327173284404383691st_int @ ( append5028292981072010935st_int @ Cfs5 @ ( cons_P4151473878978182108st_int @ ( produc5086643055186798020st_int @ Pc @ ( produc8650753666468850689st_int @ S @ Stk2 ) ) @ Cfs2 ) ) @ ( minus_minus_nat @ X @ M ) ) ) )
=> ( execl @ P3
@ ( cons_P4151473878978182108st_int @ ( produc5086643055186798020st_int @ ( minus_minus_int @ Pc @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P ) ) ) @ ( produc8650753666468850689st_int @ S @ Stk2 ) )
@ ( map_Pr8625093460839654122st_int
@ ^ [Cf: produc6425607678544837394st_int] : ( produc5086643055186798020st_int @ ( minus_minus_int @ ( produc8941309049508465880st_int @ Cf ) @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P ) ) ) @ ( produc1296517004818028822st_int @ Cf ) )
@ ( take_P1004856038796978596st_int @ N @ Cfs2 ) ) ) ) ) ) ).
% execl_sub_aux
thf(fact_1121_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_1122_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_1123_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_1124_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_1125_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_1126_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_1127_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_1128_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_1129_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1130_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1131_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1132_diff__Suc__diff__eq1,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_1133_diff__Suc__diff__eq2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_1134_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_1135_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_1136_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_1137_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_1138_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_1139_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_1140_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_1141_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_1142_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_1143_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_1144_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_1145_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_1146_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_1147_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N3: nat] :
( L
= ( plus_plus_nat @ K @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_1148_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_1149_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_1150_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_1151_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_1152_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M5: nat,N4: nat] :
? [K3: nat] :
( N4
= ( plus_plus_nat @ M5 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1153_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_1154_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_1155_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_1156_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_1157_nat__arith_Osuc1,axiom,
! [A3: nat,K: nat,A: nat] :
( ( A3
= ( plus_plus_nat @ K @ A ) )
=> ( ( suc @ A3 )
= ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_1158_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_1159_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_1160_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_1161_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_1162_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_1163_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_1164_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I @ K2 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1165_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ? [K2: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_1166_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M5: nat,N4: nat] :
? [K3: nat] :
( N4
= ( suc @ ( plus_plus_nat @ M5 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1167_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
% less_add_Suc2
thf(fact_1168_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
% less_add_Suc1
thf(fact_1169_less__natE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ! [Q4: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M @ Q4 ) ) ) ) ).
% less_natE
thf(fact_1170_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M4: nat,N3: nat] :
( ( ord_less_nat @ M4 @ N3 )
=> ( ord_less_nat @ ( F @ M4 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1171_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_1172_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_1173_less__diff__conv,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_1174_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K )
= ( J
= ( plus_plus_nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1175_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1176_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1177_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1178_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_1179_Suc__eq__plus1,axiom,
( suc
= ( ^ [N4: nat] : ( plus_plus_nat @ N4 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_1180_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1181_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_1182_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_1183_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_1184_less__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_1185_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M5: nat,N4: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ N4 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M5 @ one_one_nat ) @ N4 ) ) ) ) ) ).
% add_eq_if
thf(fact_1186_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_1187_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_1188_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus_int @ K @ zero_zero_int )
= K ) ).
% plus_int_code(1)
thf(fact_1189_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_1190_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_1191_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_1192_odd__nonzero,axiom,
! [Z3: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z3 ) @ Z3 )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_1193_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_1194_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_1195_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_1196_int__ge__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_eq_int @ K @ I )
=> ( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_ge_induct
thf(fact_1197_zle__iff__zadd,axiom,
( ord_less_eq_int
= ( ^ [W2: int,Z5: int] :
? [N4: nat] :
( Z5
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ N4 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_1198_int__gr__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_int @ K @ I )
=> ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
=> ( ! [I2: int] :
( ( ord_less_int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_gr_induct
thf(fact_1199_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_1200_int__Suc,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ).
% int_Suc
thf(fact_1201_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_1202_zless__iff__Suc__zadd,axiom,
( ord_less_int
= ( ^ [W2: int,Z5: int] :
? [N4: nat] :
( Z5
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ ( suc @ N4 ) ) ) ) ) ) ).
% zless_iff_Suc_zadd
thf(fact_1203_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_1204_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_1205_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_1206_int__induct,axiom,
! [P: int > $o,K: int,I: int] :
( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_induct
thf(fact_1207_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_1208_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_1209_diff__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ M @ ( suc @ N ) )
= ( case_nat_nat @ zero_zero_nat
@ ^ [K3: nat] : K3
@ ( minus_minus_nat @ M @ N ) ) ) ).
% diff_Suc
thf(fact_1210_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_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,S6: 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 @ S6 @ 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 @ S6 @ Stk4 ) ) ) ) ) ) ) ).
% exec_append_trans
thf(fact_1212_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_1213_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_1214_exec__appendR,axiom,
! [P: list_instr,C: produc6425607678544837394st_int,C3: produc6425607678544837394st_int,P3: list_instr] :
( ( star_P707599355569300323st_int @ ( exec1 @ P ) @ C @ C3 )
=> ( star_P707599355569300323st_int @ ( exec1 @ ( append_instr @ P @ P3 ) ) @ C @ C3 ) ) ).
% 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,
! [X3: list_char,Pc: int,S: list_char > int,Stk2: list_int,Cf5: produc6425607678544837394st_int] :
( ( iexec_pred @ ( produc1484136438817787418st_int @ ( load @ X3 ) @ ( produc5086643055186798020st_int @ Pc @ ( produc8650753666468850689st_int @ S @ Stk2 ) ) ) @ Cf5 )
=> ( Cf5
= ( produc5086643055186798020st_int @ ( plus_plus_int @ Pc @ one_one_int ) @ ( produc8650753666468850689st_int @ S @ ( cons_int @ ( S @ X3 ) @ Stk2 ) ) ) ) ) ).
% LoadE
thf(fact_1219_instr_Oinject_I2_J,axiom,
! [X22: list_char,Y2: list_char] :
( ( ( load @ X22 )
= ( load @ Y2 ) )
= ( X22 = 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,
! [X22: list_char] :
( ( size_size_instr @ ( load @ X22 ) )
= 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,X22: list_char] :
( ( loadi @ X1 )
!= ( load @ X22 ) ) ).
% instr.distinct(1)
thf(fact_1224_execl_Opelims_I2_J,axiom,
! [X3: list_instr,Xa: list_P419547167332244514st_int] :
( ( execl @ X3 @ Xa )
=> ( ( accp_P1482788143903795053st_int @ execl_rel @ ( produc7764413618310988720st_int @ X3 @ Xa ) )
=> ( ! [Cf3: produc6425607678544837394st_int,Cf4: produc6425607678544837394st_int,Cfs4: list_P419547167332244514st_int] :
( ( Xa
= ( cons_P4151473878978182108st_int @ Cf3 @ ( cons_P4151473878978182108st_int @ Cf4 @ Cfs4 ) ) )
=> ( ( accp_P1482788143903795053st_int @ execl_rel @ ( produc7764413618310988720st_int @ X3 @ ( cons_P4151473878978182108st_int @ Cf3 @ ( cons_P4151473878978182108st_int @ Cf4 @ Cfs4 ) ) ) )
=> ~ ( ( exec1 @ X3 @ Cf3 @ Cf4 )
& ( execl @ X3 @ ( cons_P4151473878978182108st_int @ Cf4 @ Cfs4 ) ) ) ) )
=> ( ( ( Xa = nil_Pr749310521696113548st_int )
=> ~ ( accp_P1482788143903795053st_int @ execl_rel @ ( produc7764413618310988720st_int @ X3 @ nil_Pr749310521696113548st_int ) ) )
=> ~ ! [V: produc6425607678544837394st_int] :
( ( Xa
= ( cons_P4151473878978182108st_int @ V @ nil_Pr749310521696113548st_int ) )
=> ~ ( accp_P1482788143903795053st_int @ execl_rel @ ( produc7764413618310988720st_int @ X3 @ ( cons_P4151473878978182108st_int @ V @ nil_Pr749310521696113548st_int ) ) ) ) ) ) ) ) ).
% execl.pelims(2)
thf(fact_1225_execl_Opelims_I3_J,axiom,
! [X3: list_instr,Xa: list_P419547167332244514st_int] :
( ~ ( execl @ X3 @ Xa )
=> ( ( accp_P1482788143903795053st_int @ execl_rel @ ( produc7764413618310988720st_int @ X3 @ Xa ) )
=> ~ ! [Cf3: produc6425607678544837394st_int,Cf4: produc6425607678544837394st_int,Cfs4: list_P419547167332244514st_int] :
( ( Xa
= ( cons_P4151473878978182108st_int @ Cf3 @ ( cons_P4151473878978182108st_int @ Cf4 @ Cfs4 ) ) )
=> ( ( accp_P1482788143903795053st_int @ execl_rel @ ( produc7764413618310988720st_int @ X3 @ ( cons_P4151473878978182108st_int @ Cf3 @ ( cons_P4151473878978182108st_int @ Cf4 @ Cfs4 ) ) ) )
=> ( ( exec1 @ X3 @ Cf3 @ Cf4 )
& ( execl @ X3 @ ( cons_P4151473878978182108st_int @ Cf4 @ Cfs4 ) ) ) ) ) ) ) ).
% execl.pelims(3)
thf(fact_1226_execl_Opelims_I1_J,axiom,
! [X3: list_instr,Xa: list_P419547167332244514st_int,Y: $o] :
( ( ( execl @ X3 @ Xa )
= Y )
=> ( ( accp_P1482788143903795053st_int @ execl_rel @ ( produc7764413618310988720st_int @ X3 @ Xa ) )
=> ( ! [Cf3: produc6425607678544837394st_int,Cf4: produc6425607678544837394st_int,Cfs4: list_P419547167332244514st_int] :
( ( Xa
= ( cons_P4151473878978182108st_int @ Cf3 @ ( cons_P4151473878978182108st_int @ Cf4 @ Cfs4 ) ) )
=> ( ( Y
= ( ( exec1 @ X3 @ Cf3 @ Cf4 )
& ( execl @ X3 @ ( cons_P4151473878978182108st_int @ Cf4 @ Cfs4 ) ) ) )
=> ~ ( accp_P1482788143903795053st_int @ execl_rel @ ( produc7764413618310988720st_int @ X3 @ ( cons_P4151473878978182108st_int @ Cf3 @ ( cons_P4151473878978182108st_int @ Cf4 @ Cfs4 ) ) ) ) ) )
=> ( ( ( Xa = nil_Pr749310521696113548st_int )
=> ( Y
=> ~ ( accp_P1482788143903795053st_int @ execl_rel @ ( produc7764413618310988720st_int @ X3 @ nil_Pr749310521696113548st_int ) ) ) )
=> ~ ! [V: produc6425607678544837394st_int] :
( ( Xa
= ( cons_P4151473878978182108st_int @ V @ nil_Pr749310521696113548st_int ) )
=> ( Y
=> ~ ( accp_P1482788143903795053st_int @ execl_rel @ ( produc7764413618310988720st_int @ X3 @ ( cons_P4151473878978182108st_int @ V @ nil_Pr749310521696113548st_int ) ) ) ) ) ) ) ) ) ).
% execl.pelims(1)
thf(fact_1227_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_1228_instr_Oinject_I4_J,axiom,
! [X52: int,Y52: int] :
( ( ( jmp @ X52 )
= ( jmp @ Y52 ) )
= ( X52 = Y52 ) ) ).
% instr.inject(4)
thf(fact_1229_instr_Odistinct_I7_J,axiom,
! [X1: int,X52: int] :
( ( loadi @ X1 )
!= ( jmp @ X52 ) ) ).
% instr.distinct(7)
thf(fact_1230_instr_Odistinct_I17_J,axiom,
! [X22: list_char,X52: int] :
( ( load @ X22 )
!= ( jmp @ X52 ) ) ).
% instr.distinct(17)
thf(fact_1231_instr_Osize_I12_J,axiom,
! [X52: int] :
( ( size_size_instr @ ( jmp @ X52 ) )
= zero_zero_nat ) ).
% instr.size(12)
thf(fact_1232_com_Osize_I6_J,axiom,
( ( size_size_com @ skip )
= zero_zero_nat ) ).
% com.size(6)
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,K: 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 ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I2: nat] :
( ( ord_less_eq_nat @ I2 @ N )
& ( ( F @ I2 )
= K ) ) ) ) ) ).
% 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,K: 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 ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I2: nat] :
( ( ord_less_eq_nat @ M @ I2 )
& ( ord_less_eq_nat @ I2 @ N )
& ( ( F @ I2 )
= K ) ) ) ) ) ) ).
% nat_intermed_int_val
thf(fact_1237_nat__ivt__aux,axiom,
! [N: nat,F: nat > int,K: 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 ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I2: nat] :
( ( ord_less_eq_nat @ I2 @ N )
& ( ( F @ I2 )
= K ) ) ) ) ) ).
% nat_ivt_aux
thf(fact_1238_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ! [I4: nat] :
( ( ord_less_nat @ K2 @ I4 )
=> ( P @ I4 ) )
=> ( P @ K2 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_1239_upto_Opinduct,axiom,
! [A0: int,A12: int,P: int > int > $o] :
( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ A0 @ A12 ) )
=> ( ! [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 @ A12 ) ) ) ).
% upto.pinduct
thf(fact_1240_upto_Opelims,axiom,
! [X3: int,Xa: int,Y: list_int] :
( ( ( upto @ X3 @ Xa )
= Y )
=> ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X3 @ Xa ) )
=> ~ ( ( ( ( ord_less_eq_int @ X3 @ Xa )
=> ( Y
= ( cons_int @ X3 @ ( upto @ ( plus_plus_int @ X3 @ one_one_int ) @ Xa ) ) ) )
& ( ~ ( ord_less_eq_int @ X3 @ Xa )
=> ( Y = nil_int ) ) )
=> ~ ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X3 @ Xa ) ) ) ) ) ).
% upto.pelims
thf(fact_1241_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_1242_upto__Nil,axiom,
! [I: int,J: int] :
( ( ( upto @ I @ J )
= nil_int )
= ( ord_less_int @ J @ I ) ) ).
% upto_Nil
thf(fact_1243_upto__Nil2,axiom,
! [I: int,J: int] :
( ( nil_int
= ( upto @ I @ J ) )
= ( ord_less_int @ J @ I ) ) ).
% upto_Nil2
thf(fact_1244_upto__empty,axiom,
! [J: int,I: int] :
( ( ord_less_int @ J @ I )
=> ( ( upto @ I @ J )
= nil_int ) ) ).
% upto_empty
thf(fact_1245_upto__single,axiom,
! [I: int] :
( ( upto @ I @ I )
= ( cons_int @ I @ nil_int ) ) ).
% upto_single
thf(fact_1246_nth__upto,axiom,
! [I: int,K: nat,J: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I @ ( semiri1314217659103216013at_int @ K ) ) @ J )
=> ( ( nth_int @ ( upto @ I @ J ) @ K )
= ( plus_plus_int @ I @ ( semiri1314217659103216013at_int @ K ) ) ) ) ).
% nth_upto
thf(fact_1247_upto__aux__def,axiom,
( upto_aux
= ( ^ [I3: int,J3: int] : ( append_int @ ( upto @ I3 @ J3 ) ) ) ) ).
% upto_aux_def
thf(fact_1248_upto__code,axiom,
( upto
= ( ^ [I3: int,J3: int] : ( upto_aux @ I3 @ J3 @ nil_int ) ) ) ).
% upto_code
thf(fact_1249_upto__split2,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_eq_int @ I @ J )
=> ( ( ord_less_eq_int @ J @ K )
=> ( ( upto @ I @ K )
= ( append_int @ ( upto @ I @ J ) @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ).
% upto_split2
thf(fact_1250_upto__split1,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_eq_int @ I @ J )
=> ( ( ord_less_eq_int @ J @ K )
=> ( ( upto @ I @ K )
= ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( upto @ J @ K ) ) ) ) ) ).
% upto_split1
thf(fact_1251_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_1252_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_1253_upto_Oelims,axiom,
! [X3: int,Xa: int,Y: list_int] :
( ( ( upto @ X3 @ Xa )
= Y )
=> ( ( ( ord_less_eq_int @ X3 @ Xa )
=> ( Y
= ( cons_int @ X3 @ ( upto @ ( plus_plus_int @ X3 @ one_one_int ) @ Xa ) ) ) )
& ( ~ ( ord_less_eq_int @ X3 @ Xa )
=> ( Y = nil_int ) ) ) ) ).
% upto.elims
thf(fact_1254_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_1255_upto__split3,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_eq_int @ I @ J )
=> ( ( ord_less_eq_int @ J @ K )
=> ( ( upto @ I @ K )
= ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ) ).
% upto_split3
thf(fact_1256_upto__rec__numeral_I1_J,axiom,
! [M: num,N: num] :
( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
=> ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( numeral_numeral_int @ N ) ) ) ) )
& ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
=> ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= nil_int ) ) ) ).
% upto_rec_numeral(1)
thf(fact_1257_upto__rec__numeral_I4_J,axiom,
! [M: num,N: num] :
( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
=> ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( 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 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ) ) )
& ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
=> ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= nil_int ) ) ) ).
% upto_rec_numeral(4)
thf(fact_1258_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,
! [X3: int,Y: int] :
( ( if_int @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X3: int,Y: int] :
( ( if_int @ $true @ X3 @ Y )
= X3 ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X3: nat,Y: nat] :
( ( if_nat @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X3: nat,Y: nat] :
( ( if_nat @ $true @ X3 @ Y )
= X3 ) ).
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,
! [X3: list_int,Y: list_int] :
( ( if_list_int @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
! [X3: list_int,Y: list_int] :
( ( if_list_int @ $true @ X3 @ Y )
= X3 ) ).
% Conjectures (2)
thf(conj_0,hypothesis,
! [K4: nat] :
( ( member_nat @ K4 @ ( set_or4665077453230672383an_nat @ k @ ( size_s229347708945616566st_int @ cfs ) ) )
=> ( ( execl_all @ ( p2 @ x )
@ ( map_Pr8625093460839654122st_int
@ ^ [Cf: produc6425607678544837394st_int] : ( produc5086643055186798020st_int @ ( minus_minus_int @ ( produc8941309049508465880st_int @ Cf ) @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ p ) ) ) @ ( produc1296517004818028822st_int @ Cf ) )
@ ( drop_P7838799151754935884st_int @ k @ ( take_P1004856038796978596st_int @ ( suc @ K4 ) @ cfs ) ) ) )
=> ( ( q @ x
@ ( nth_Pr8327173284404383691st_int
@ ( map_Pr8625093460839654122st_int
@ ^ [Cf: produc6425607678544837394st_int] : ( produc5086643055186798020st_int @ ( minus_minus_int @ ( produc8941309049508465880st_int @ Cf ) @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ p ) ) ) @ ( produc1296517004818028822st_int @ Cf ) )
@ ( drop_P7838799151754935884st_int @ k @ ( take_P1004856038796978596st_int @ ( suc @ K4 ) @ cfs ) ) )
@ zero_zero_nat )
@ ( nth_Pr8327173284404383691st_int
@ ( map_Pr8625093460839654122st_int
@ ^ [Cf: produc6425607678544837394st_int] : ( produc5086643055186798020st_int @ ( minus_minus_int @ ( produc8941309049508465880st_int @ Cf ) @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ p ) ) ) @ ( produc1296517004818028822st_int @ Cf ) )
@ ( drop_P7838799151754935884st_int @ k @ ( take_P1004856038796978596st_int @ ( suc @ K4 ) @ 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 @ K4 ) @ cfs ) ) ) )
@ one_one_nat ) ) )
=> thesis ) ) ) ).
thf(conj_1,conjecture,
thesis ).
%------------------------------------------------------------------------------