TPTP Problem File: SLH0787^1.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_00616_031391__6290322_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1580 ( 732 unt; 312 typ; 0 def)
% Number of atoms : 3033 (1878 equ; 0 cnn)
% Maximal formula atoms : 11 ( 2 avg)
% Number of connectives : 10722 ( 445 ~; 55 |; 200 &;8898 @)
% ( 0 <=>;1124 =>; 0 <=; 0 <~>)
% Maximal formula depth : 30 ( 6 avg)
% Number of types : 70 ( 69 usr)
% Number of type conns : 766 ( 766 >; 0 *; 0 +; 0 <<)
% Number of symbols : 246 ( 243 usr; 35 con; 0-8 aty)
% Number of variables : 3574 ( 89 ^;3367 !; 118 ?;3574 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 16:59:09.769
%------------------------------------------------------------------------------
% Could-be-implicit typings (69)
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thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Compiler__Oinstr_J,type,
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thf(ty_n_t__List__Olist_It__List__Olist_It__String__Ochar_J_J,type,
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thf(ty_n_t__List__Olist_I_062_It__Int__Oint_Mt__Int__Oint_J_J,type,
list_int_int: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__String__Ochar_J_J,type,
set_list_char: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Int__Oint_J_J,type,
list_list_int: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J,type,
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thf(ty_n_t__List__Olist_It__Compiler__Oinstr_J,type,
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thf(ty_n_t__List__Olist_It__String__Ochar_J,type,
list_char: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__List__Olist_It__Int__Oint_J,type,
list_int: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
set_int: $tType ).
thf(ty_n_t__Compiler__Oinstr,type,
instr: $tType ).
thf(ty_n_t__String__Ochar,type,
char: $tType ).
thf(ty_n_t__BExp__Obexp,type,
bexp: $tType ).
thf(ty_n_t__AExp__Oaexp,type,
aexp: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
thf(ty_n_t__Com__Ocom,type,
com: $tType ).
% Explicit typings (243)
thf(sy_c_AExp_Oaval,type,
aval: aexp > ( list_char > int ) > int ).
thf(sy_c_BExp_Obexp_OAnd,type,
and: bexp > bexp > bexp ).
thf(sy_c_BExp_Obexp_OBc,type,
bc: $o > bexp ).
thf(sy_c_BExp_Obexp_Osize__bexp,type,
size_bexp: bexp > nat ).
thf(sy_c_BExp_Obval,type,
bval: bexp > ( list_char > int ) > $o ).
thf(sy_c_Big__Step_Obig__step,type,
big_big_step: produc2931317944591925149ar_int > ( list_char > int ) > $o ).
thf(sy_c_Com_Ocom_OAssign,type,
assign: list_char > aexp > com ).
thf(sy_c_Com_Ocom_OIf,type,
if: bexp > com > com > com ).
thf(sy_c_Com_Ocom_OSKIP,type,
skip: com ).
thf(sy_c_Com_Ocom_OSeq,type,
seq: com > com > com ).
thf(sy_c_Com_Ocom_OWhile,type,
while: bexp > com > com ).
thf(sy_c_Com_Ocom_Osize__com,type,
size_com: com > nat ).
thf(sy_c_Compiler2_Oapred,type,
apred: aexp > produc6425607678544837394st_int > produc6425607678544837394st_int > $o ).
thf(sy_c_Compiler2_Obpred,type,
bpred: produc1897111610453708512_o_int > produc6425607678544837394st_int > produc6425607678544837394st_int > $o ).
thf(sy_c_Compiler2_Ocpred,type,
cpred: com > produc6425607678544837394st_int > produc6425607678544837394st_int > $o ).
thf(sy_c_Compiler2_Oexecl,type,
execl: list_instr > list_P419547167332244514st_int > $o ).
thf(sy_c_Compiler2_Oexecl__all,type,
execl_all: list_instr > list_P419547167332244514st_int > $o ).
thf(sy_c_Compiler2_Ompred,type,
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thf(sy_c_Compiler_Oacomp,type,
acomp: aexp > list_instr ).
thf(sy_c_Compiler_Oadd__stack,type,
add_stack: nat > ( int > int ) > list_int ).
thf(sy_c_Compiler_Oaddr__of,type,
addr_of: list_list_char > list_char > int ).
thf(sy_c_Compiler_Obcomp,type,
bcomp: produc1897111610453708512_o_int > list_instr ).
thf(sy_c_Compiler_Occomp,type,
ccomp: com > list_instr ).
thf(sy_c_Compiler_Oexec1,type,
exec1: list_instr > produc6425607678544837394st_int > produc6425607678544837394st_int > $o ).
thf(sy_c_Compiler_Oiexec,type,
iexec: instr > produc6425607678544837394st_int > produc6425607678544837394st_int ).
thf(sy_c_Compiler_Oiexec__pred,type,
iexec_pred: produc5995290525303592096st_int > produc6425607678544837394st_int > $o ).
thf(sy_c_Compiler_Oinstr_OJMP,type,
jmp: int > instr ).
thf(sy_c_Compiler_Oinstr_OLOAD,type,
load: list_char > instr ).
thf(sy_c_Compiler_Oinstr_OLOADI,type,
loadi: int > instr ).
thf(sy_c_Compiler_Oinstr_OSTORE,type,
store: list_char > instr ).
thf(sy_c_Compiler_Oinstr_Ocase__instr_001t__List__Olist_It__List__Olist_It__String__Ochar_J_J,type,
case_i7427087243572719837t_char: ( int > list_list_char ) > ( list_char > list_list_char ) > list_list_char > ( list_char > list_list_char ) > ( int > list_list_char ) > ( int > list_list_char ) > ( int > list_list_char ) > instr > list_list_char ).
thf(sy_c_Compiler_Ointh_001t__Compiler__Oinstr,type,
inth_instr: list_instr > int > instr ).
thf(sy_c_Compiler_Ointh_001t__Int__Oint,type,
inth_int: list_int > int > int ).
thf(sy_c_Compiler_Ointh_001t__List__Olist_It__String__Ochar_J,type,
inth_list_char: list_list_char > int > list_char ).
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inth_P8039235741264645271st_int: list_P419547167332244514st_int > int > produc6425607678544837394st_int ).
thf(sy_c_Compiler_Om__config__equiv,type,
m_config_equiv: produc1019184382078353393nt_nat > produc1019184382078353393nt_nat > $o ).
thf(sy_c_Compiler_Oto__m__state,type,
to_m_state: list_list_char > ( list_char > int ) > int > int ).
thf(sy_c_Compiler_Oto__m__state__rel,type,
to_m_state_rel: produc4189061985984568957nt_int > produc4189061985984568957nt_int > $o ).
thf(sy_c_Compiler_Ovars,type,
vars: list_instr > list_list_char ).
thf(sy_c_Groups_Oabs__class_Oabs_001t__Int__Oint,type,
abs_abs_int: int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
minus_minus_int: int > int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
plus_plus_int: int > int > int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
uminus_uminus_int: int > int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_If_001t__Int__Oint,type,
if_int: $o > int > int > int ).
thf(sy_c_If_001t__List__Olist_It__Int__Oint_J,type,
if_list_int: $o > list_int > list_int > list_int ).
thf(sy_c_If_001t__List__Olist_It__List__Olist_It__String__Ochar_J_J,type,
if_list_list_char: $o > list_list_char > list_list_char > list_list_char ).
thf(sy_c_If_001t__Nat__Onat,type,
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member_int: int > set_int > $o ).
thf(sy_c_member_001t__List__Olist_It__String__Ochar_J,type,
member_list_char: list_char > set_list_char > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_v_b,type,
b: bexp ).
thf(sy_v_c,type,
c: com ).
thf(sy_v_cfsa____,type,
cfsa: list_P419547167332244514st_int ).
thf(sy_v_k____,type,
k: nat ).
thf(sy_v_s_H____,type,
s: list_char > int ).
thf(sy_v_s____,type,
s2: list_char > int ).
thf(sy_v_stk____,type,
stk: list_int ).
% Relevant facts (1258)
thf(fact_0__092_060open_062bval_Ab_As_092_060close_062,axiom,
bval @ b @ s2 ).
% \<open>bval b s\<close>
thf(fact_1__092_060open_062_Ic_M_As_J_A_092_060Rightarrow_062_As_H_092_060close_062,axiom,
big_big_step @ ( produc5595214716300948949ar_int @ c @ s2 ) @ s ).
% \<open>(c, s) \<Rightarrow> s'\<close>
thf(fact_2_D,axiom,
( ( nth_Pr8327173284404383691st_int @ cfsa @ ( suc @ k ) )
= ( produc5086643055186798020st_int @ zero_zero_int @ ( produc8650753666468850689st_int @ s @ stk ) ) ) ).
% D
thf(fact_3__092_060open_062cpred_A_IWHILE_Ab_ADO_Ac_J_A_Icfs_A_B_ASuc_Ak_J_A_Icfs_A_B_A_Ilength_Acfs_A_N_A1_J_J_092_060close_062,axiom,
cpred @ ( while @ b @ c ) @ ( nth_Pr8327173284404383691st_int @ cfsa @ ( suc @ k ) ) @ ( nth_Pr8327173284404383691st_int @ cfsa @ ( minus_minus_nat @ ( size_s229347708945616566st_int @ cfsa ) @ one_one_nat ) ) ).
% \<open>cpred (WHILE b DO c) (cfs ! Suc k) (cfs ! (length cfs - 1))\<close>
thf(fact_4_C,axiom,
( k
!= ( minus_minus_nat @ ( size_s229347708945616566st_int @ cfsa ) @ one_one_nat ) ) ).
% C
thf(fact_5_B,axiom,
ord_less_nat @ k @ ( size_s229347708945616566st_int @ cfsa ) ).
% B
thf(fact_6_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_7_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_8_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_9_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_10_diff__self,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% diff_self
thf(fact_11_diff__0__right,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_0_right
thf(fact_12_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_13_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_14_diff__zero,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_zero
thf(fact_15_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_16_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_17_com_Oinject_I4_J,axiom,
! [X51: bexp,X52: com,Y51: bexp,Y52: com] :
( ( ( while @ X51 @ X52 )
= ( while @ Y51 @ Y52 ) )
= ( ( X51 = Y51 )
& ( X52 = Y52 ) ) ) ).
% com.inject(4)
thf(fact_18_nat_Oinject,axiom,
! [X2: nat,Y2: nat] :
( ( ( suc @ X2 )
= ( suc @ Y2 ) )
= ( X2 = Y2 ) ) ).
% nat.inject
thf(fact_19_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_20_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_21_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_22_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_23_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_24_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_25_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_26_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_27_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_28_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_29_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_30_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_31_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_32_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_33_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_34_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_35_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_36_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_37_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_38_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_39_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_40_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_41_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_42_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_43_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_44_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
= ( X = one_one_int ) ) ).
% one_reorient
thf(fact_45_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_46_mem__Collect__eq,axiom,
! [A: int,P: int > $o] :
( ( member_int @ A @ ( collect_int @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_47_mem__Collect__eq,axiom,
! [A: list_char,P: list_char > $o] :
( ( member_list_char @ A @ ( collect_list_char @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_48_Collect__mem__eq,axiom,
! [A2: set_int] :
( ( collect_int
@ ^ [X3: int] : ( member_int @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_49_Collect__mem__eq,axiom,
! [A2: set_list_char] :
( ( collect_list_char
@ ^ [X3: list_char] : ( member_list_char @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_50_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_51_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_52_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_53_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_54_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_55_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_56_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_57_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_58_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_59_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_60_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_61_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_62_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_63_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_64_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_65_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_66_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_67_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_68_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_69_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_70_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_71_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_72_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_73_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_74_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_75_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_76_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_77_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_78_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_79_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_80_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_81_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_82_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_83_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_84_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_85_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_86_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_87_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_88_less__iff__diff__less__0,axiom,
( ord_less_int
= ( ^ [A3: int,B2: int] : ( ord_less_int @ ( minus_minus_int @ A3 @ B2 ) @ zero_zero_int ) ) ) ).
% less_iff_diff_less_0
thf(fact_89_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_90_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% gr0_implies_Suc
thf(fact_91_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_92_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_93_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_94_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_95_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_96_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_97_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_98_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_99_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_100_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_101_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_102_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_103_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_104_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_105_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_106_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_107_size__neq__size__imp__neq,axiom,
! [X: list_instr,Y: list_instr] :
( ( ( size_size_list_instr @ X )
!= ( size_size_list_instr @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_108_size__neq__size__imp__neq,axiom,
! [X: list_list_char,Y: list_list_char] :
( ( ( size_s356637359517785349t_char @ X )
!= ( size_s356637359517785349t_char @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_109_size__neq__size__imp__neq,axiom,
! [X: list_int,Y: list_int] :
( ( ( size_size_list_int @ X )
!= ( size_size_list_int @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_110_size__neq__size__imp__neq,axiom,
! [X: bexp,Y: bexp] :
( ( ( size_size_bexp @ X )
!= ( size_size_bexp @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_111_size__neq__size__imp__neq,axiom,
! [X: com,Y: com] :
( ( ( size_size_com @ X )
!= ( size_size_com @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_112_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_113_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_114_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_115_eq__iff__diff__eq__0,axiom,
( ( ^ [Y3: int,Z: int] : ( Y3 = Z ) )
= ( ^ [A3: int,B2: int] :
( ( minus_minus_int @ A3 @ B2 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_116_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% not0_implies_Suc
thf(fact_117_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_118_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_119_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_120_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_121_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X4: nat] : ( P @ X4 @ zero_zero_nat )
=> ( ! [Y4: nat] : ( P @ zero_zero_nat @ ( suc @ Y4 ) )
=> ( ! [X4: nat,Y4: nat] :
( ( P @ X4 @ Y4 )
=> ( P @ ( suc @ X4 ) @ ( suc @ Y4 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_122_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_123_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_124_nat_OdiscI,axiom,
! [Nat: nat,X2: nat] :
( ( Nat
= ( suc @ X2 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_125_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_126_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_127_nat_Odistinct_I1_J,axiom,
! [X2: nat] :
( zero_zero_nat
!= ( suc @ X2 ) ) ).
% nat.distinct(1)
thf(fact_128_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_129_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_130_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_131_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_132_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_133_WhileE,axiom,
! [B: bexp,C: com,S: list_char > int,T: list_char > int] :
( ( big_big_step @ ( produc5595214716300948949ar_int @ ( while @ B @ C ) @ S ) @ T )
=> ( ( ( T = S )
=> ( bval @ B @ S ) )
=> ~ ( ( bval @ B @ S )
=> ! [S_2: list_char > int] :
( ( big_big_step @ ( produc5595214716300948949ar_int @ C @ S ) @ S_2 )
=> ~ ( big_big_step @ ( produc5595214716300948949ar_int @ ( while @ B @ C ) @ S_2 ) @ T ) ) ) ) ) ).
% WhileE
thf(fact_134_WhileFalse,axiom,
! [B: bexp,S: list_char > int,C: com] :
( ~ ( bval @ B @ S )
=> ( big_big_step @ ( produc5595214716300948949ar_int @ ( while @ B @ C ) @ S ) @ S ) ) ).
% WhileFalse
thf(fact_135_WhileTrue,axiom,
! [B: bexp,S_1: list_char > int,C: com,S_22: list_char > int,S_3: list_char > int] :
( ( bval @ B @ S_1 )
=> ( ( big_big_step @ ( produc5595214716300948949ar_int @ C @ S_1 ) @ S_22 )
=> ( ( big_big_step @ ( produc5595214716300948949ar_int @ ( while @ B @ C ) @ S_22 ) @ S_3 )
=> ( big_big_step @ ( produc5595214716300948949ar_int @ ( while @ B @ C ) @ S_1 ) @ S_3 ) ) ) ) ).
% WhileTrue
thf(fact_136_sim__while__cong__aux,axiom,
! [B: bexp,C: com,S: list_char > int,T: list_char > int,C2: com] :
( ( big_big_step @ ( produc5595214716300948949ar_int @ ( while @ B @ C ) @ S ) @ T )
=> ( ! [S2: list_char > int,T2: list_char > int] :
( ( big_big_step @ ( produc5595214716300948949ar_int @ C @ S2 ) @ T2 )
= ( big_big_step @ ( produc5595214716300948949ar_int @ C2 @ S2 ) @ T2 ) )
=> ( big_big_step @ ( produc5595214716300948949ar_int @ ( while @ B @ C2 ) @ S ) @ T ) ) ) ).
% sim_while_cong_aux
thf(fact_137_sim__while__cong,axiom,
! [C: com,C2: com,B: bexp] :
( ! [S2: list_char > int,T2: list_char > int] :
( ( big_big_step @ ( produc5595214716300948949ar_int @ C @ S2 ) @ T2 )
= ( big_big_step @ ( produc5595214716300948949ar_int @ C2 @ S2 ) @ T2 ) )
=> ! [S3: list_char > int,T3: list_char > int] :
( ( big_big_step @ ( produc5595214716300948949ar_int @ ( while @ B @ C ) @ S3 ) @ T3 )
= ( big_big_step @ ( produc5595214716300948949ar_int @ ( while @ B @ C2 ) @ S3 ) @ T3 ) ) ) ).
% sim_while_cong
thf(fact_138_list__eq__iff__nth__eq,axiom,
( ( ^ [Y3: list_P419547167332244514st_int,Z: list_P419547167332244514st_int] : ( Y3 = Z ) )
= ( ^ [Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int] :
( ( ( size_s229347708945616566st_int @ Xs )
= ( size_s229347708945616566st_int @ Ys ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s229347708945616566st_int @ Xs ) )
=> ( ( nth_Pr8327173284404383691st_int @ Xs @ I3 )
= ( nth_Pr8327173284404383691st_int @ Ys @ I3 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_139_list__eq__iff__nth__eq,axiom,
( ( ^ [Y3: list_instr,Z: list_instr] : ( Y3 = Z ) )
= ( ^ [Xs: list_instr,Ys: list_instr] :
( ( ( size_size_list_instr @ Xs )
= ( size_size_list_instr @ Ys ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_instr @ Xs ) )
=> ( ( nth_instr @ Xs @ I3 )
= ( nth_instr @ Ys @ I3 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_140_list__eq__iff__nth__eq,axiom,
( ( ^ [Y3: list_list_char,Z: list_list_char] : ( Y3 = Z ) )
= ( ^ [Xs: list_list_char,Ys: list_list_char] :
( ( ( size_s356637359517785349t_char @ Xs )
= ( size_s356637359517785349t_char @ Ys ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s356637359517785349t_char @ Xs ) )
=> ( ( nth_list_char @ Xs @ I3 )
= ( nth_list_char @ Ys @ I3 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_141_list__eq__iff__nth__eq,axiom,
( ( ^ [Y3: list_int,Z: list_int] : ( Y3 = Z ) )
= ( ^ [Xs: list_int,Ys: list_int] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs ) )
=> ( ( nth_int @ Xs @ I3 )
= ( nth_int @ Ys @ I3 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_142_Skolem__list__nth,axiom,
! [K: nat,P: nat > produc6425607678544837394st_int > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ? [X5: produc6425607678544837394st_int] : ( P @ I3 @ X5 ) ) )
= ( ? [Xs: list_P419547167332244514st_int] :
( ( ( size_s229347708945616566st_int @ Xs )
= K )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( P @ I3 @ ( nth_Pr8327173284404383691st_int @ Xs @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_143_Skolem__list__nth,axiom,
! [K: nat,P: nat > instr > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ? [X5: instr] : ( P @ I3 @ X5 ) ) )
= ( ? [Xs: list_instr] :
( ( ( size_size_list_instr @ Xs )
= K )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( P @ I3 @ ( nth_instr @ Xs @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_144_Skolem__list__nth,axiom,
! [K: nat,P: nat > list_char > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ? [X5: list_char] : ( P @ I3 @ X5 ) ) )
= ( ? [Xs: list_list_char] :
( ( ( size_s356637359517785349t_char @ Xs )
= K )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( P @ I3 @ ( nth_list_char @ Xs @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_145_Skolem__list__nth,axiom,
! [K: nat,P: nat > int > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ? [X5: int] : ( P @ I3 @ X5 ) ) )
= ( ? [Xs: list_int] :
( ( ( size_size_list_int @ Xs )
= K )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( P @ I3 @ ( nth_int @ Xs @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_146_nth__equalityI,axiom,
! [Xs2: list_P419547167332244514st_int,Ys2: list_P419547167332244514st_int] :
( ( ( size_s229347708945616566st_int @ Xs2 )
= ( size_s229347708945616566st_int @ Ys2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s229347708945616566st_int @ Xs2 ) )
=> ( ( nth_Pr8327173284404383691st_int @ Xs2 @ I2 )
= ( nth_Pr8327173284404383691st_int @ Ys2 @ I2 ) ) )
=> ( Xs2 = Ys2 ) ) ) ).
% nth_equalityI
thf(fact_147_nth__equalityI,axiom,
! [Xs2: list_instr,Ys2: list_instr] :
( ( ( size_size_list_instr @ Xs2 )
= ( size_size_list_instr @ Ys2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_instr @ Xs2 ) )
=> ( ( nth_instr @ Xs2 @ I2 )
= ( nth_instr @ Ys2 @ I2 ) ) )
=> ( Xs2 = Ys2 ) ) ) ).
% nth_equalityI
thf(fact_148_nth__equalityI,axiom,
! [Xs2: list_list_char,Ys2: list_list_char] :
( ( ( size_s356637359517785349t_char @ Xs2 )
= ( size_s356637359517785349t_char @ Ys2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s356637359517785349t_char @ Xs2 ) )
=> ( ( nth_list_char @ Xs2 @ I2 )
= ( nth_list_char @ Ys2 @ I2 ) ) )
=> ( Xs2 = Ys2 ) ) ) ).
% nth_equalityI
thf(fact_149_nth__equalityI,axiom,
! [Xs2: list_int,Ys2: list_int] :
( ( ( size_size_list_int @ Xs2 )
= ( size_size_list_int @ Ys2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs2 ) )
=> ( ( nth_int @ Xs2 @ I2 )
= ( nth_int @ Ys2 @ I2 ) ) )
=> ( Xs2 = Ys2 ) ) ) ).
% nth_equalityI
thf(fact_150_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_151_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_152_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_153_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_154_linorder__neqE__linordered__idom,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_155_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_instr] :
( ( size_size_list_instr @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_156_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_list_char] :
( ( size_s356637359517785349t_char @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_157_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs3: list_int] :
( ( size_size_list_int @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_158_neq__if__length__neq,axiom,
! [Xs2: list_instr,Ys2: list_instr] :
( ( ( size_size_list_instr @ Xs2 )
!= ( size_size_list_instr @ Ys2 ) )
=> ( Xs2 != Ys2 ) ) ).
% neq_if_length_neq
thf(fact_159_neq__if__length__neq,axiom,
! [Xs2: list_list_char,Ys2: list_list_char] :
( ( ( size_s356637359517785349t_char @ Xs2 )
!= ( size_s356637359517785349t_char @ Ys2 ) )
=> ( Xs2 != Ys2 ) ) ).
% neq_if_length_neq
thf(fact_160_neq__if__length__neq,axiom,
! [Xs2: list_int,Ys2: list_int] :
( ( ( size_size_list_int @ Xs2 )
!= ( size_size_list_int @ Ys2 ) )
=> ( Xs2 != Ys2 ) ) ).
% neq_if_length_neq
thf(fact_161_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_162_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_163_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_164_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_165_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_166_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_167_length__induct,axiom,
! [P: list_instr > $o,Xs2: list_instr] :
( ! [Xs3: list_instr] :
( ! [Ys3: list_instr] :
( ( ord_less_nat @ ( size_size_list_instr @ Ys3 ) @ ( size_size_list_instr @ Xs3 ) )
=> ( P @ Ys3 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs2 ) ) ).
% length_induct
thf(fact_168_length__induct,axiom,
! [P: list_list_char > $o,Xs2: list_list_char] :
( ! [Xs3: list_list_char] :
( ! [Ys3: list_list_char] :
( ( ord_less_nat @ ( size_s356637359517785349t_char @ Ys3 ) @ ( size_s356637359517785349t_char @ Xs3 ) )
=> ( P @ Ys3 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs2 ) ) ).
% length_induct
thf(fact_169_length__induct,axiom,
! [P: list_int > $o,Xs2: list_int] :
( ! [Xs3: list_int] :
( ! [Ys3: list_int] :
( ( ord_less_nat @ ( size_size_list_int @ Ys3 ) @ ( size_size_list_int @ Xs3 ) )
=> ( P @ Ys3 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs2 ) ) ).
% length_induct
thf(fact_170_big__step__determ,axiom,
! [C: com,S: list_char > int,T: list_char > int,U: list_char > int] :
( ( big_big_step @ ( produc5595214716300948949ar_int @ C @ S ) @ T )
=> ( ( big_big_step @ ( produc5595214716300948949ar_int @ C @ S ) @ U )
=> ( U = T ) ) ) ).
% big_step_determ
thf(fact_171_sim__trans,axiom,
! [C: com,C2: com,C3: com] :
( ! [S2: list_char > int,T2: list_char > int] :
( ( big_big_step @ ( produc5595214716300948949ar_int @ C @ S2 ) @ T2 )
= ( big_big_step @ ( produc5595214716300948949ar_int @ C2 @ S2 ) @ T2 ) )
=> ( ! [S2: list_char > int,T2: list_char > int] :
( ( big_big_step @ ( produc5595214716300948949ar_int @ C2 @ S2 ) @ T2 )
= ( big_big_step @ ( produc5595214716300948949ar_int @ C3 @ S2 ) @ T2 ) )
=> ! [S3: list_char > int,T3: list_char > int] :
( ( big_big_step @ ( produc5595214716300948949ar_int @ C @ S3 ) @ T3 )
= ( big_big_step @ ( produc5595214716300948949ar_int @ C3 @ S3 ) @ T3 ) ) ) ) ).
% sim_trans
thf(fact_172_sim__refl,axiom,
! [C: com,S3: list_char > int,T3: list_char > int] :
( ( big_big_step @ ( produc5595214716300948949ar_int @ C @ S3 ) @ T3 )
= ( big_big_step @ ( produc5595214716300948949ar_int @ C @ S3 ) @ T3 ) ) ).
% sim_refl
thf(fact_173_sim__sym,axiom,
! [C: com,C2: com] :
( ( ! [S4: list_char > int,T4: list_char > int] :
( ( big_big_step @ ( produc5595214716300948949ar_int @ C @ S4 ) @ T4 )
= ( big_big_step @ ( produc5595214716300948949ar_int @ C2 @ S4 ) @ T4 ) ) )
= ( ! [S4: list_char > int,T4: list_char > int] :
( ( big_big_step @ ( produc5595214716300948949ar_int @ C2 @ S4 ) @ T4 )
= ( big_big_step @ ( produc5595214716300948949ar_int @ C @ S4 ) @ T4 ) ) ) ) ).
% sim_sym
thf(fact_174_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_175_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_176_size__char__eq__0,axiom,
( size_size_char
= ( ^ [C4: char] : zero_zero_nat ) ) ).
% size_char_eq_0
thf(fact_177_prod_Oinject,axiom,
! [X1: int,X2: produc1534919314292393896nt_nat,Y1: int,Y2: produc1534919314292393896nt_nat] :
( ( ( produc8342123264692915235nt_nat @ X1 @ X2 )
= ( produc8342123264692915235nt_nat @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_178_prod_Oinject,axiom,
! [X1: com,X2: list_char > int,Y1: com,Y2: list_char > int] :
( ( ( produc5595214716300948949ar_int @ X1 @ X2 )
= ( produc5595214716300948949ar_int @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_179_prod_Oinject,axiom,
! [X1: list_char > int,X2: list_int,Y1: list_char > int,Y2: list_int] :
( ( ( produc8650753666468850689st_int @ X1 @ X2 )
= ( produc8650753666468850689st_int @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_180_prod_Oinject,axiom,
! [X1: list_char > int,X2: int,Y1: list_char > int,Y2: int] :
( ( ( produc5790713362662368625nt_int @ X1 @ X2 )
= ( produc5790713362662368625nt_int @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_181_prod_Oinject,axiom,
! [X1: int > int,X2: nat,Y1: int > int,Y2: nat] :
( ( ( produc2630796075472406112nt_nat @ X1 @ X2 )
= ( produc2630796075472406112nt_nat @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_182_old_Oprod_Oinject,axiom,
! [A: int,B: produc1534919314292393896nt_nat,A4: int,B3: produc1534919314292393896nt_nat] :
( ( ( produc8342123264692915235nt_nat @ A @ B )
= ( produc8342123264692915235nt_nat @ A4 @ B3 ) )
= ( ( A = A4 )
& ( B = B3 ) ) ) ).
% old.prod.inject
thf(fact_183_old_Oprod_Oinject,axiom,
! [A: com,B: list_char > int,A4: com,B3: list_char > int] :
( ( ( produc5595214716300948949ar_int @ A @ B )
= ( produc5595214716300948949ar_int @ A4 @ B3 ) )
= ( ( A = A4 )
& ( B = B3 ) ) ) ).
% old.prod.inject
thf(fact_184_old_Oprod_Oinject,axiom,
! [A: list_char > int,B: list_int,A4: list_char > int,B3: list_int] :
( ( ( produc8650753666468850689st_int @ A @ B )
= ( produc8650753666468850689st_int @ A4 @ B3 ) )
= ( ( A = A4 )
& ( B = B3 ) ) ) ).
% old.prod.inject
thf(fact_185_old_Oprod_Oinject,axiom,
! [A: list_char > int,B: int,A4: list_char > int,B3: int] :
( ( ( produc5790713362662368625nt_int @ A @ B )
= ( produc5790713362662368625nt_int @ A4 @ B3 ) )
= ( ( A = A4 )
& ( B = B3 ) ) ) ).
% old.prod.inject
thf(fact_186_old_Oprod_Oinject,axiom,
! [A: int > int,B: nat,A4: int > int,B3: nat] :
( ( ( produc2630796075472406112nt_nat @ A @ B )
= ( produc2630796075472406112nt_nat @ A4 @ B3 ) )
= ( ( A = A4 )
& ( B = B3 ) ) ) ).
% old.prod.inject
thf(fact_187_prod__cases3,axiom,
! [Y: produc1019184382078353393nt_nat] :
~ ! [A5: int,B4: int > int,C5: nat] :
( Y
!= ( produc8342123264692915235nt_nat @ A5 @ ( produc2630796075472406112nt_nat @ B4 @ C5 ) ) ) ).
% prod_cases3
thf(fact_188_iexec_Ocases,axiom,
! [X: produc5995290525303592096st_int] :
~ ! [Ins: instr,I2: int,S2: list_char > int,Stk: list_int] :
( X
!= ( produc1484136438817787418st_int @ Ins @ ( produc5086643055186798020st_int @ I2 @ ( produc8650753666468850689st_int @ S2 @ Stk ) ) ) ) ).
% iexec.cases
thf(fact_189_Pair__inject,axiom,
! [A: int,B: produc1534919314292393896nt_nat,A4: int,B3: produc1534919314292393896nt_nat] :
( ( ( produc8342123264692915235nt_nat @ A @ B )
= ( produc8342123264692915235nt_nat @ A4 @ B3 ) )
=> ~ ( ( A = A4 )
=> ( B != B3 ) ) ) ).
% Pair_inject
thf(fact_190_Pair__inject,axiom,
! [A: com,B: list_char > int,A4: com,B3: list_char > int] :
( ( ( produc5595214716300948949ar_int @ A @ B )
= ( produc5595214716300948949ar_int @ A4 @ B3 ) )
=> ~ ( ( A = A4 )
=> ( B != B3 ) ) ) ).
% Pair_inject
thf(fact_191_Pair__inject,axiom,
! [A: list_char > int,B: list_int,A4: list_char > int,B3: list_int] :
( ( ( produc8650753666468850689st_int @ A @ B )
= ( produc8650753666468850689st_int @ A4 @ B3 ) )
=> ~ ( ( A = A4 )
=> ( B != B3 ) ) ) ).
% Pair_inject
thf(fact_192_Pair__inject,axiom,
! [A: list_char > int,B: int,A4: list_char > int,B3: int] :
( ( ( produc5790713362662368625nt_int @ A @ B )
= ( produc5790713362662368625nt_int @ A4 @ B3 ) )
=> ~ ( ( A = A4 )
=> ( B != B3 ) ) ) ).
% Pair_inject
thf(fact_193_Pair__inject,axiom,
! [A: int > int,B: nat,A4: int > int,B3: nat] :
( ( ( produc2630796075472406112nt_nat @ A @ B )
= ( produc2630796075472406112nt_nat @ A4 @ B3 ) )
=> ~ ( ( A = A4 )
=> ( B != B3 ) ) ) ).
% Pair_inject
thf(fact_194_prod__cases,axiom,
! [P: produc1019184382078353393nt_nat > $o,P2: produc1019184382078353393nt_nat] :
( ! [A5: int,B4: produc1534919314292393896nt_nat] : ( P @ ( produc8342123264692915235nt_nat @ A5 @ B4 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_195_prod__cases,axiom,
! [P: produc2931317944591925149ar_int > $o,P2: produc2931317944591925149ar_int] :
( ! [A5: com,B4: list_char > int] : ( P @ ( produc5595214716300948949ar_int @ A5 @ B4 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_196_prod__cases,axiom,
! [P: produc369741286924889929st_int > $o,P2: produc369741286924889929st_int] :
( ! [A5: list_char > int,B4: list_int] : ( P @ ( produc8650753666468850689st_int @ A5 @ B4 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_197_prod__cases,axiom,
! [P: produc4435102495419491129nt_int > $o,P2: produc4435102495419491129nt_int] :
( ! [A5: list_char > int,B4: int] : ( P @ ( produc5790713362662368625nt_int @ A5 @ B4 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_198_prod__cases,axiom,
! [P: produc1534919314292393896nt_nat > $o,P2: produc1534919314292393896nt_nat] :
( ! [A5: int > int,B4: nat] : ( P @ ( produc2630796075472406112nt_nat @ A5 @ B4 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_199_surj__pair,axiom,
! [P2: produc1019184382078353393nt_nat] :
? [X4: int,Y4: produc1534919314292393896nt_nat] :
( P2
= ( produc8342123264692915235nt_nat @ X4 @ Y4 ) ) ).
% surj_pair
thf(fact_200_surj__pair,axiom,
! [P2: produc2931317944591925149ar_int] :
? [X4: com,Y4: list_char > int] :
( P2
= ( produc5595214716300948949ar_int @ X4 @ Y4 ) ) ).
% surj_pair
thf(fact_201_surj__pair,axiom,
! [P2: produc369741286924889929st_int] :
? [X4: list_char > int,Y4: list_int] :
( P2
= ( produc8650753666468850689st_int @ X4 @ Y4 ) ) ).
% surj_pair
thf(fact_202_surj__pair,axiom,
! [P2: produc4435102495419491129nt_int] :
? [X4: list_char > int,Y4: int] :
( P2
= ( produc5790713362662368625nt_int @ X4 @ Y4 ) ) ).
% surj_pair
thf(fact_203_surj__pair,axiom,
! [P2: produc1534919314292393896nt_nat] :
? [X4: int > int,Y4: nat] :
( P2
= ( produc2630796075472406112nt_nat @ X4 @ Y4 ) ) ).
% surj_pair
thf(fact_204_old_Oprod_Oexhaust,axiom,
! [Y: produc1019184382078353393nt_nat] :
~ ! [A5: int,B4: produc1534919314292393896nt_nat] :
( Y
!= ( produc8342123264692915235nt_nat @ A5 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_205_old_Oprod_Oexhaust,axiom,
! [Y: produc2931317944591925149ar_int] :
~ ! [A5: com,B4: list_char > int] :
( Y
!= ( produc5595214716300948949ar_int @ A5 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_206_old_Oprod_Oexhaust,axiom,
! [Y: produc369741286924889929st_int] :
~ ! [A5: list_char > int,B4: list_int] :
( Y
!= ( produc8650753666468850689st_int @ A5 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_207_old_Oprod_Oexhaust,axiom,
! [Y: produc4435102495419491129nt_int] :
~ ! [A5: list_char > int,B4: int] :
( Y
!= ( produc5790713362662368625nt_int @ A5 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_208_old_Oprod_Oexhaust,axiom,
! [Y: produc1534919314292393896nt_nat] :
~ ! [A5: int > int,B4: nat] :
( Y
!= ( produc2630796075472406112nt_nat @ A5 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_209_prod__induct3,axiom,
! [P: produc1019184382078353393nt_nat > $o,X: produc1019184382078353393nt_nat] :
( ! [A5: int,B4: int > int,C5: nat] : ( P @ ( produc8342123264692915235nt_nat @ A5 @ ( produc2630796075472406112nt_nat @ B4 @ C5 ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_210_size_H__char__eq__0,axiom,
( size_char
= ( ^ [C4: char] : zero_zero_nat ) ) ).
% size'_char_eq_0
thf(fact_211__092_060open_062cfs_A_B_Ak_A_061_A_Isize_A_Ibcomp_A_Ib_M_AFalse_M_Asize_A_Iccomp_Ac_J_A_L_A1_J_J_A_L_Asize_A_Iccomp_Ac_J_M_As_H_M_Astk_J_092_060close_062,axiom,
( ( nth_Pr8327173284404383691st_int @ cfsa @ k )
= ( produc5086643055186798020st_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ ( bcomp @ ( produc4047900504771817624_o_int @ b @ ( product_Pair_o_int @ $false @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ ( ccomp @ c ) ) ) @ one_one_int ) ) ) ) ) ) @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ ( ccomp @ c ) ) ) ) @ ( produc8650753666468850689st_int @ s @ stk ) ) ) ).
% \<open>cfs ! k = (size (bcomp (b, False, size (ccomp c) + 1)) + size (ccomp c), s', stk)\<close>
thf(fact_212_nth__Cons__pos,axiom,
! [N: nat,X: int,Xs2: list_int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_int @ ( cons_int @ X @ Xs2 ) @ N )
= ( nth_int @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_213_nth__Cons__pos,axiom,
! [N: nat,X: instr,Xs2: list_instr] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_instr @ ( cons_instr @ X @ Xs2 ) @ N )
= ( nth_instr @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_214_nth__Cons__pos,axiom,
! [N: nat,X: produc6425607678544837394st_int,Xs2: list_P419547167332244514st_int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_Pr8327173284404383691st_int @ ( cons_P4151473878978182108st_int @ X @ Xs2 ) @ N )
= ( nth_Pr8327173284404383691st_int @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_215_nth__Cons__pos,axiom,
! [N: nat,X: list_char,Xs2: list_list_char] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_list_char @ ( cons_list_char @ X @ Xs2 ) @ N )
= ( nth_list_char @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_216_dbl__inc__simps_I2_J,axiom,
( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
= one_one_int ) ).
% dbl_inc_simps(2)
thf(fact_217_E,axiom,
( ( plus_plus_nat @ ( suc @ k ) @ ( minus_minus_nat @ ( size_s229347708945616566st_int @ ( drop_P7838799151754935884st_int @ ( suc @ k ) @ cfsa ) ) @ one_one_nat ) )
= ( minus_minus_nat @ ( size_s229347708945616566st_int @ cfsa ) @ one_one_nat ) ) ).
% E
thf(fact_218_nth__zip,axiom,
! [I: nat,Xs2: list_instr,Ys2: list_instr] :
( ( ord_less_nat @ I @ ( size_size_list_instr @ Xs2 ) )
=> ( ( ord_less_nat @ I @ ( size_size_list_instr @ Ys2 ) )
=> ( ( nth_Pr2944702272814216302_instr @ ( zip_instr_instr @ Xs2 @ Ys2 ) @ I )
= ( produc7105755196911784279_instr @ ( nth_instr @ Xs2 @ I ) @ ( nth_instr @ Ys2 @ I ) ) ) ) ) ).
% nth_zip
thf(fact_219_nth__zip,axiom,
! [I: nat,Xs2: list_instr,Ys2: list_int] :
( ( ord_less_nat @ I @ ( size_size_list_instr @ Xs2 ) )
=> ( ( ord_less_nat @ I @ ( size_size_list_int @ Ys2 ) )
=> ( ( nth_Pr7669067734896111347tr_int @ ( zip_instr_int @ Xs2 @ Ys2 ) @ I )
= ( produc1181339232725390644tr_int @ ( nth_instr @ Xs2 @ I ) @ ( nth_int @ Ys2 @ I ) ) ) ) ) ).
% nth_zip
thf(fact_220_nth__zip,axiom,
! [I: nat,Xs2: list_int,Ys2: list_instr] :
( ( ord_less_nat @ I @ ( size_size_list_int @ Xs2 ) )
=> ( ( ord_less_nat @ I @ ( size_size_list_instr @ Ys2 ) )
=> ( ( nth_Pr5016122532423702369_instr @ ( zip_int_instr @ Xs2 @ Ys2 ) @ I )
= ( produc3816574238604233306_instr @ ( nth_int @ Xs2 @ I ) @ ( nth_instr @ Ys2 @ I ) ) ) ) ) ).
% nth_zip
thf(fact_221_nth__zip,axiom,
! [I: nat,Xs2: list_int,Ys2: list_int] :
( ( ord_less_nat @ I @ ( size_size_list_int @ Xs2 ) )
=> ( ( ord_less_nat @ I @ ( size_size_list_int @ Ys2 ) )
=> ( ( nth_Pr4439495888332055232nt_int @ ( zip_int_int @ Xs2 @ Ys2 ) @ I )
= ( product_Pair_int_int @ ( nth_int @ Xs2 @ I ) @ ( nth_int @ Ys2 @ I ) ) ) ) ) ).
% nth_zip
thf(fact_222_nth__zip,axiom,
! [I: nat,Xs2: list_instr,Ys2: list_list_char] :
( ( ord_less_nat @ I @ ( size_size_list_instr @ Xs2 ) )
=> ( ( ord_less_nat @ I @ ( size_s356637359517785349t_char @ Ys2 ) )
=> ( ( nth_Pr2000311911037713832t_char @ ( zip_instr_list_char @ Xs2 @ Ys2 ) @ I )
= ( produc97871184544046185t_char @ ( nth_instr @ Xs2 @ I ) @ ( nth_list_char @ Ys2 @ I ) ) ) ) ) ).
% nth_zip
thf(fact_223_nth__zip,axiom,
! [I: nat,Xs2: list_list_char,Ys2: list_instr] :
( ( ord_less_nat @ I @ ( size_s356637359517785349t_char @ Xs2 ) )
=> ( ( ord_less_nat @ I @ ( size_size_list_instr @ Ys2 ) )
=> ( ( nth_Pr7517844949280720108_instr @ ( zip_list_char_instr @ Xs2 @ Ys2 ) @ I )
= ( produc8288463027051613157_instr @ ( nth_list_char @ Xs2 @ I ) @ ( nth_instr @ Ys2 @ I ) ) ) ) ) ).
% nth_zip
thf(fact_224_nth__zip,axiom,
! [I: nat,Xs2: list_list_char,Ys2: list_int] :
( ( ord_less_nat @ I @ ( size_s356637359517785349t_char @ Xs2 ) )
=> ( ( ord_less_nat @ I @ ( size_size_list_int @ Ys2 ) )
=> ( ( nth_Pr181807947379245557ar_int @ ( zip_list_char_int @ Xs2 @ Ys2 ) @ I )
= ( produc8699855389908574758ar_int @ ( nth_list_char @ Xs2 @ I ) @ ( nth_int @ Ys2 @ I ) ) ) ) ) ).
% nth_zip
thf(fact_225_nth__zip,axiom,
! [I: nat,Xs2: list_int,Ys2: list_list_char] :
( ( ord_less_nat @ I @ ( size_size_list_int @ Xs2 ) )
=> ( ( ord_less_nat @ I @ ( size_s356637359517785349t_char @ Ys2 ) )
=> ( ( nth_Pr337767877128616437t_char @ ( zip_int_list_char @ Xs2 @ Ys2 ) @ I )
= ( produc2247564686889860774t_char @ ( nth_int @ Xs2 @ I ) @ ( nth_list_char @ Ys2 @ I ) ) ) ) ) ).
% nth_zip
thf(fact_226_nth__zip,axiom,
! [I: nat,Xs2: list_int_int,Ys2: list_nat] :
( ( ord_less_nat @ I @ ( size_s7905348490449908891nt_int @ Xs2 ) )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys2 ) )
=> ( ( nth_Pr7511481218797999151nt_nat @ ( zip_int_int_nat @ Xs2 @ Ys2 ) @ I )
= ( produc2630796075472406112nt_nat @ ( nth_int_int @ Xs2 @ I ) @ ( nth_nat @ Ys2 @ I ) ) ) ) ) ).
% nth_zip
thf(fact_227_nth__zip,axiom,
! [I: nat,Xs2: list_list_char,Ys2: list_list_char] :
( ( ord_less_nat @ I @ ( size_s356637359517785349t_char @ Xs2 ) )
=> ( ( ord_less_nat @ I @ ( size_s356637359517785349t_char @ Ys2 ) )
=> ( ( nth_Pr7491354432233842090t_char @ ( zip_li5382857322596760955t_char @ Xs2 @ Ys2 ) @ I )
= ( produc1606739095562851931t_char @ ( nth_list_char @ Xs2 @ I ) @ ( nth_list_char @ Ys2 @ I ) ) ) ) ) ).
% nth_zip
thf(fact_228_add__left__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_229_add__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_230_add__right__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_231_add__right__cancel,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_232_list_Oinject,axiom,
! [X21: int,X22: list_int,Y21: int,Y22: list_int] :
( ( ( cons_int @ X21 @ X22 )
= ( cons_int @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_233_list_Oinject,axiom,
! [X21: instr,X22: list_instr,Y21: instr,Y22: list_instr] :
( ( ( cons_instr @ X21 @ X22 )
= ( cons_instr @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_234_list_Oinject,axiom,
! [X21: produc6425607678544837394st_int,X22: list_P419547167332244514st_int,Y21: produc6425607678544837394st_int,Y22: list_P419547167332244514st_int] :
( ( ( cons_P4151473878978182108st_int @ X21 @ X22 )
= ( cons_P4151473878978182108st_int @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_235_list_Oinject,axiom,
! [X21: list_char,X22: list_list_char,Y21: list_char,Y22: list_list_char] :
( ( ( cons_list_char @ X21 @ X22 )
= ( cons_list_char @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_236_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_237_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_238_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_239_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_240_add__cancel__left__left,axiom,
! [B: int,A: int] :
( ( ( plus_plus_int @ B @ A )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_left
thf(fact_241_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_242_add__cancel__left__right,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_right
thf(fact_243_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_244_add__cancel__right__left,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ B @ A ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_left
thf(fact_245_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_246_add__cancel__right__right,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ A @ B ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_right
thf(fact_247_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_248_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_249_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y ) )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_250_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_251_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_252_add__less__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_253_add__less__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_254_add__less__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_255_add__less__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_256_add__diff__cancel,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_257_diff__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_258_add__diff__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_259_add__diff__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_260_add__diff__cancel__left_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_261_add__diff__cancel__left_H,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_262_add__diff__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_263_add__diff__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_264_add__diff__cancel__right_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_265_add__diff__cancel__right_H,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_266_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_267_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_268_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_269_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_270_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_271_drop0,axiom,
( ( drop_P7838799151754935884st_int @ zero_zero_nat )
= ( ^ [X3: list_P419547167332244514st_int] : X3 ) ) ).
% drop0
thf(fact_272_drop__drop,axiom,
! [N: nat,M: nat,Xs2: list_P419547167332244514st_int] :
( ( drop_P7838799151754935884st_int @ N @ ( drop_P7838799151754935884st_int @ M @ Xs2 ) )
= ( drop_P7838799151754935884st_int @ ( plus_plus_nat @ N @ M ) @ Xs2 ) ) ).
% drop_drop
thf(fact_273_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_274_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_275_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_276_less__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel2
thf(fact_277_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_278_less__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel1
thf(fact_279_add__less__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_280_add__less__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel2
thf(fact_281_add__less__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_282_add__less__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel1
thf(fact_283_diff__add__zero,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_284_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_285_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_286_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_287_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_288_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_289_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_290_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_291_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_292_of__nat__add,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_add
thf(fact_293_of__nat__add,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_add
thf(fact_294_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_295_nth__Cons__0,axiom,
! [X: int,Xs2: list_int] :
( ( nth_int @ ( cons_int @ X @ Xs2 ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_296_nth__Cons__0,axiom,
! [X: instr,Xs2: list_instr] :
( ( nth_instr @ ( cons_instr @ X @ Xs2 ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_297_nth__Cons__0,axiom,
! [X: produc6425607678544837394st_int,Xs2: list_P419547167332244514st_int] :
( ( nth_Pr8327173284404383691st_int @ ( cons_P4151473878978182108st_int @ X @ Xs2 ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_298_nth__Cons__0,axiom,
! [X: list_char,Xs2: list_list_char] :
( ( nth_list_char @ ( cons_list_char @ X @ Xs2 ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_299_nth__Cons__Suc,axiom,
! [X: int,Xs2: list_int,N: nat] :
( ( nth_int @ ( cons_int @ X @ Xs2 ) @ ( suc @ N ) )
= ( nth_int @ Xs2 @ N ) ) ).
% nth_Cons_Suc
thf(fact_300_nth__Cons__Suc,axiom,
! [X: instr,Xs2: list_instr,N: nat] :
( ( nth_instr @ ( cons_instr @ X @ Xs2 ) @ ( suc @ N ) )
= ( nth_instr @ Xs2 @ N ) ) ).
% nth_Cons_Suc
thf(fact_301_nth__Cons__Suc,axiom,
! [X: produc6425607678544837394st_int,Xs2: list_P419547167332244514st_int,N: nat] :
( ( nth_Pr8327173284404383691st_int @ ( cons_P4151473878978182108st_int @ X @ Xs2 ) @ ( suc @ N ) )
= ( nth_Pr8327173284404383691st_int @ Xs2 @ N ) ) ).
% nth_Cons_Suc
thf(fact_302_nth__Cons__Suc,axiom,
! [X: list_char,Xs2: list_list_char,N: nat] :
( ( nth_list_char @ ( cons_list_char @ X @ Xs2 ) @ ( suc @ N ) )
= ( nth_list_char @ Xs2 @ N ) ) ).
% nth_Cons_Suc
thf(fact_303_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_304_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_305_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_306_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_307_of__nat__1,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% of_nat_1
thf(fact_308_of__nat__1,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% of_nat_1
thf(fact_309_zip__Cons__Cons,axiom,
! [X: int,Xs2: list_int,Y: int,Ys2: list_int] :
( ( zip_int_int @ ( cons_int @ X @ Xs2 ) @ ( cons_int @ Y @ Ys2 ) )
= ( cons_P3334398858971670639nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( zip_int_int @ Xs2 @ Ys2 ) ) ) ).
% zip_Cons_Cons
thf(fact_310_zip__Cons__Cons,axiom,
! [X: int,Xs2: list_int,Y: instr,Ys2: list_instr] :
( ( zip_int_instr @ ( cons_int @ X @ Xs2 ) @ ( cons_instr @ Y @ Ys2 ) )
= ( cons_P5330466886524163058_instr @ ( produc3816574238604233306_instr @ X @ Y ) @ ( zip_int_instr @ Xs2 @ Ys2 ) ) ) ).
% zip_Cons_Cons
thf(fact_311_zip__Cons__Cons,axiom,
! [X: instr,Xs2: list_instr,Y: int,Ys2: list_int] :
( ( zip_instr_int @ ( cons_instr @ X @ Xs2 ) @ ( cons_int @ Y @ Ys2 ) )
= ( cons_P7983412088996572036tr_int @ ( produc1181339232725390644tr_int @ X @ Y ) @ ( zip_instr_int @ Xs2 @ Ys2 ) ) ) ).
% zip_Cons_Cons
thf(fact_312_zip__Cons__Cons,axiom,
! [X: instr,Xs2: list_instr,Y: instr,Ys2: list_instr] :
( ( zip_instr_instr @ ( cons_instr @ X @ Xs2 ) @ ( cons_instr @ Y @ Ys2 ) )
= ( cons_P7619363388684002717_instr @ ( produc7105755196911784279_instr @ X @ Y ) @ ( zip_instr_instr @ Xs2 @ Ys2 ) ) ) ).
% zip_Cons_Cons
thf(fact_313_zip__Cons__Cons,axiom,
! [X: int,Xs2: list_int,Y: list_char,Ys2: list_list_char] :
( ( zip_int_list_char @ ( cons_int @ X @ Xs2 ) @ ( cons_list_char @ Y @ Ys2 ) )
= ( cons_P8827863618816167204t_char @ ( produc2247564686889860774t_char @ X @ Y ) @ ( zip_int_list_char @ Xs2 @ Ys2 ) ) ) ).
% zip_Cons_Cons
thf(fact_314_zip__Cons__Cons,axiom,
! [X: instr,Xs2: list_instr,Y: list_char,Ys2: list_list_char] :
( ( zip_instr_list_char @ ( cons_instr @ X @ Xs2 ) @ ( cons_list_char @ Y @ Ys2 ) )
= ( cons_P4723849340467793081t_char @ ( produc97871184544046185t_char @ X @ Y ) @ ( zip_instr_list_char @ Xs2 @ Ys2 ) ) ) ).
% zip_Cons_Cons
thf(fact_315_zip__Cons__Cons,axiom,
! [X: list_char,Xs2: list_list_char,Y: int,Ys2: list_int] :
( ( zip_list_char_int @ ( cons_list_char @ X @ Xs2 ) @ ( cons_int @ Y @ Ys2 ) )
= ( cons_P8671903689066796324ar_int @ ( produc8699855389908574758ar_int @ X @ Y ) @ ( zip_list_char_int @ Xs2 @ Ys2 ) ) ) ).
% zip_Cons_Cons
thf(fact_316_zip__Cons__Cons,axiom,
! [X: list_char,Xs2: list_list_char,Y: instr,Ys2: list_instr] :
( ( zip_list_char_instr @ ( cons_list_char @ X @ Xs2 ) @ ( cons_instr @ Y @ Ys2 ) )
= ( cons_P1018010341856023549_instr @ ( produc8288463027051613157_instr @ X @ Y ) @ ( zip_list_char_instr @ Xs2 @ Ys2 ) ) ) ).
% zip_Cons_Cons
thf(fact_317_zip__Cons__Cons,axiom,
! [X: list_char,Xs2: list_list_char,Y: list_char,Ys2: list_list_char] :
( ( zip_li5382857322596760955t_char @ ( cons_list_char @ X @ Xs2 ) @ ( cons_list_char @ Y @ Ys2 ) )
= ( cons_P922464210127128153t_char @ ( produc1606739095562851931t_char @ X @ Y ) @ ( zip_li5382857322596760955t_char @ Xs2 @ Ys2 ) ) ) ).
% zip_Cons_Cons
thf(fact_318_zip__Cons__Cons,axiom,
! [X: int > int,Xs2: list_int_int,Y: nat,Ys2: list_nat] :
( ( zip_int_int_nat @ ( cons_int_int @ X @ Xs2 ) @ ( cons_nat @ Y @ Ys2 ) )
= ( cons_P6778204923630774110nt_nat @ ( produc2630796075472406112nt_nat @ X @ Y ) @ ( zip_int_int_nat @ Xs2 @ Ys2 ) ) ) ).
% zip_Cons_Cons
thf(fact_319_drop__Suc__Cons,axiom,
! [N: nat,X: int,Xs2: list_int] :
( ( drop_int @ ( suc @ N ) @ ( cons_int @ X @ Xs2 ) )
= ( drop_int @ N @ Xs2 ) ) ).
% drop_Suc_Cons
thf(fact_320_drop__Suc__Cons,axiom,
! [N: nat,X: instr,Xs2: list_instr] :
( ( drop_instr @ ( suc @ N ) @ ( cons_instr @ X @ Xs2 ) )
= ( drop_instr @ N @ Xs2 ) ) ).
% drop_Suc_Cons
thf(fact_321_drop__Suc__Cons,axiom,
! [N: nat,X: produc6425607678544837394st_int,Xs2: list_P419547167332244514st_int] :
( ( drop_P7838799151754935884st_int @ ( suc @ N ) @ ( cons_P4151473878978182108st_int @ X @ Xs2 ) )
= ( drop_P7838799151754935884st_int @ N @ Xs2 ) ) ).
% drop_Suc_Cons
thf(fact_322_drop__Suc__Cons,axiom,
! [N: nat,X: list_char,Xs2: list_list_char] :
( ( drop_list_char @ ( suc @ N ) @ ( cons_list_char @ X @ Xs2 ) )
= ( drop_list_char @ N @ Xs2 ) ) ).
% drop_Suc_Cons
thf(fact_323_length__drop,axiom,
! [N: nat,Xs2: list_P419547167332244514st_int] :
( ( size_s229347708945616566st_int @ ( drop_P7838799151754935884st_int @ N @ Xs2 ) )
= ( minus_minus_nat @ ( size_s229347708945616566st_int @ Xs2 ) @ N ) ) ).
% length_drop
thf(fact_324_length__drop,axiom,
! [N: nat,Xs2: list_instr] :
( ( size_size_list_instr @ ( drop_instr @ N @ Xs2 ) )
= ( minus_minus_nat @ ( size_size_list_instr @ Xs2 ) @ N ) ) ).
% length_drop
thf(fact_325_length__drop,axiom,
! [N: nat,Xs2: list_list_char] :
( ( size_s356637359517785349t_char @ ( drop_list_char @ N @ Xs2 ) )
= ( minus_minus_nat @ ( size_s356637359517785349t_char @ Xs2 ) @ N ) ) ).
% length_drop
thf(fact_326_length__drop,axiom,
! [N: nat,Xs2: list_int] :
( ( size_size_list_int @ ( drop_int @ N @ Xs2 ) )
= ( minus_minus_nat @ ( size_size_list_int @ Xs2 ) @ N ) ) ).
% length_drop
thf(fact_327_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ M ) )
= ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M ) ) ) ).
% of_nat_Suc
thf(fact_328_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ M ) )
= ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% of_nat_Suc
thf(fact_329_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_330_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_331_zip__eq__ConsE,axiom,
! [Xs2: list_int,Ys2: list_int,Xy: product_prod_int_int,Xys: list_P5707943133018811711nt_int] :
( ( ( zip_int_int @ Xs2 @ Ys2 )
= ( cons_P3334398858971670639nt_int @ Xy @ Xys ) )
=> ~ ! [X4: int,Xs4: list_int] :
( ( Xs2
= ( cons_int @ X4 @ Xs4 ) )
=> ! [Y4: int,Ys4: list_int] :
( ( Ys2
= ( cons_int @ Y4 @ Ys4 ) )
=> ( ( Xy
= ( product_Pair_int_int @ X4 @ Y4 ) )
=> ( Xys
!= ( zip_int_int @ Xs4 @ Ys4 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_332_zip__eq__ConsE,axiom,
! [Xs2: list_int,Ys2: list_instr,Xy: produc2765154750163803688_instr,Xys: list_P8574115254301748920_instr] :
( ( ( zip_int_instr @ Xs2 @ Ys2 )
= ( cons_P5330466886524163058_instr @ Xy @ Xys ) )
=> ~ ! [X4: int,Xs4: list_int] :
( ( Xs2
= ( cons_int @ X4 @ Xs4 ) )
=> ! [Y4: instr,Ys4: list_instr] :
( ( Ys2
= ( cons_instr @ Y4 @ Ys4 ) )
=> ( ( Xy
= ( produc3816574238604233306_instr @ X4 @ Y4 ) )
=> ( Xys
!= ( zip_int_instr @ Xs4 @ Ys4 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_333_zip__eq__ConsE,axiom,
! [Xs2: list_instr,Ys2: list_int,Xy: produc5418099952636212666tr_int,Xys: list_P6195009360020970826tr_int] :
( ( ( zip_instr_int @ Xs2 @ Ys2 )
= ( cons_P7983412088996572036tr_int @ Xy @ Xys ) )
=> ~ ! [X4: instr,Xs4: list_instr] :
( ( Xs2
= ( cons_instr @ X4 @ Xs4 ) )
=> ! [Y4: int,Ys4: list_int] :
( ( Ys2
= ( cons_int @ Y4 @ Ys4 ) )
=> ( ( Xy
= ( produc1181339232725390644tr_int @ X4 @ Y4 ) )
=> ( Xys
!= ( zip_instr_int @ Xs4 @ Ys4 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_334_zip__eq__ConsE,axiom,
! [Xs2: list_instr,Ys2: list_instr,Xy: produc6572865875912645095_instr,Xys: list_P492468431389647341_instr] :
( ( ( zip_instr_instr @ Xs2 @ Ys2 )
= ( cons_P7619363388684002717_instr @ Xy @ Xys ) )
=> ~ ! [X4: instr,Xs4: list_instr] :
( ( Xs2
= ( cons_instr @ X4 @ Xs4 ) )
=> ! [Y4: instr,Ys4: list_instr] :
( ( Ys2
= ( cons_instr @ Y4 @ Ys4 ) )
=> ( ( Xy
= ( produc7105755196911784279_instr @ X4 @ Y4 ) )
=> ( Xys
!= ( zip_instr_instr @ Xs4 @ Ys4 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_335_zip__eq__ConsE,axiom,
! [Xs2: list_int,Ys2: list_list_char,Xy: produc3584578009477786990t_char,Xys: list_P4190774984311663604t_char] :
( ( ( zip_int_list_char @ Xs2 @ Ys2 )
= ( cons_P8827863618816167204t_char @ Xy @ Xys ) )
=> ~ ! [X4: int,Xs4: list_int] :
( ( Xs2
= ( cons_int @ X4 @ Xs4 ) )
=> ! [Y4: list_char,Ys4: list_list_char] :
( ( Ys2
= ( cons_list_char @ Y4 @ Ys4 ) )
=> ( ( Xy
= ( produc2247564686889860774t_char @ X4 @ Y4 ) )
=> ( Xys
!= ( zip_int_list_char @ Xs4 @ Ys4 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_336_zip__eq__ConsE,axiom,
! [Xs2: list_instr,Ys2: list_list_char,Xy: produc2120242088828816879t_char,Xys: list_P1193574006058364799t_char] :
( ( ( zip_instr_list_char @ Xs2 @ Ys2 )
= ( cons_P4723849340467793081t_char @ Xy @ Xys ) )
=> ~ ! [X4: instr,Xs4: list_instr] :
( ( Xs2
= ( cons_instr @ X4 @ Xs4 ) )
=> ! [Y4: list_char,Ys4: list_list_char] :
( ( Ys2
= ( cons_list_char @ Y4 @ Ys4 ) )
=> ( ( Xy
= ( produc97871184544046185t_char @ X4 @ Y4 ) )
=> ( Xys
!= ( zip_instr_list_char @ Xs4 @ Ys4 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_337_zip__eq__ConsE,axiom,
! [Xs2: list_list_char,Ys2: list_int,Xy: produc3428618079728416110ar_int,Xys: list_P2585337669149094900ar_int] :
( ( ( zip_list_char_int @ Xs2 @ Ys2 )
= ( cons_P8671903689066796324ar_int @ Xy @ Xys ) )
=> ~ ! [X4: list_char,Xs4: list_list_char] :
( ( Xs2
= ( cons_list_char @ X4 @ Xs4 ) )
=> ! [Y4: int,Ys4: list_int] :
( ( Ys2
= ( cons_int @ Y4 @ Ys4 ) )
=> ( ( Xy
= ( produc8699855389908574758ar_int @ X4 @ Y4 ) )
=> ( Xys
!= ( zip_list_char_int @ Xs4 @ Ys4 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_338_zip__eq__ConsE,axiom,
! [Xs2: list_list_char,Ys2: list_instr,Xy: produc7637775127071823155_instr,Xys: list_P9124871753606960323_instr] :
( ( ( zip_list_char_instr @ Xs2 @ Ys2 )
= ( cons_P1018010341856023549_instr @ Xy @ Xys ) )
=> ~ ! [X4: list_char,Xs4: list_list_char] :
( ( Xs2
= ( cons_list_char @ X4 @ Xs4 ) )
=> ! [Y4: instr,Ys4: list_instr] :
( ( Ys2
= ( cons_instr @ Y4 @ Ys4 ) )
=> ( ( Xy
= ( produc8288463027051613157_instr @ X4 @ Y4 ) )
=> ( Xys
!= ( zip_list_char_instr @ Xs4 @ Ys4 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_339_zip__eq__ConsE,axiom,
! [Xs2: list_list_char,Ys2: list_list_char,Xy: produc6859378861088059299t_char,Xys: list_P8512142354091196201t_char] :
( ( ( zip_li5382857322596760955t_char @ Xs2 @ Ys2 )
= ( cons_P922464210127128153t_char @ Xy @ Xys ) )
=> ~ ! [X4: list_char,Xs4: list_list_char] :
( ( Xs2
= ( cons_list_char @ X4 @ Xs4 ) )
=> ! [Y4: list_char,Ys4: list_list_char] :
( ( Ys2
= ( cons_list_char @ Y4 @ Ys4 ) )
=> ( ( Xy
= ( produc1606739095562851931t_char @ X4 @ Y4 ) )
=> ( Xys
!= ( zip_li5382857322596760955t_char @ Xs4 @ Ys4 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_340_zip__eq__ConsE,axiom,
! [Xs2: list_int_int,Ys2: list_nat,Xy: produc1534919314292393896nt_nat,Xys: list_P5370272772617341230nt_nat] :
( ( ( zip_int_int_nat @ Xs2 @ Ys2 )
= ( cons_P6778204923630774110nt_nat @ Xy @ Xys ) )
=> ~ ! [X4: int > int,Xs4: list_int_int] :
( ( Xs2
= ( cons_int_int @ X4 @ Xs4 ) )
=> ! [Y4: nat,Ys4: list_nat] :
( ( Ys2
= ( cons_nat @ Y4 @ Ys4 ) )
=> ( ( Xy
= ( produc2630796075472406112nt_nat @ X4 @ Y4 ) )
=> ( Xys
!= ( zip_int_int_nat @ Xs4 @ Ys4 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_341_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_342_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_343_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_int @ I @ K )
= ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_344_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_345_group__cancel_Oadd1,axiom,
! [A2: int,K: int,A: int,B: int] :
( ( A2
= ( plus_plus_int @ K @ A ) )
=> ( ( plus_plus_int @ A2 @ B )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_346_group__cancel_Oadd1,axiom,
! [A2: nat,K: nat,A: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_347_group__cancel_Oadd2,axiom,
! [B5: int,K: int,B: int,A: int] :
( ( B5
= ( plus_plus_int @ K @ B ) )
=> ( ( plus_plus_int @ A @ B5 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_348_group__cancel_Oadd2,axiom,
! [B5: nat,K: nat,B: nat,A: nat] :
( ( B5
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B5 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_349_add_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.assoc
thf(fact_350_add_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_351_add_Oleft__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_352_add_Oright__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_353_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A3: int,B2: int] : ( plus_plus_int @ B2 @ A3 ) ) ) ).
% add.commute
thf(fact_354_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A3: nat,B2: nat] : ( plus_plus_nat @ B2 @ A3 ) ) ) ).
% add.commute
thf(fact_355_add_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C ) )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.left_commute
thf(fact_356_add_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_357_add__left__imp__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_358_add__left__imp__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_359_add__right__imp__eq,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_360_add__right__imp__eq,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_361_is__num__normalize_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_362_drop__zip,axiom,
! [N: nat,Xs2: list_P419547167332244514st_int,Ys2: list_P419547167332244514st_int] :
( ( drop_P4239379478136815563st_int @ N @ ( zip_Pr2166390094044666077st_int @ Xs2 @ Ys2 ) )
= ( zip_Pr2166390094044666077st_int @ ( drop_P7838799151754935884st_int @ N @ Xs2 ) @ ( drop_P7838799151754935884st_int @ N @ Ys2 ) ) ) ).
% drop_zip
thf(fact_363_drop__zip,axiom,
! [N: nat,Xs2: list_int,Ys2: list_P8654944379044127311st_int] :
( ( drop_P7838799151754935884st_int @ N @ ( zip_in2431647089413498532st_int @ Xs2 @ Ys2 ) )
= ( zip_in2431647089413498532st_int @ ( drop_int @ N @ Xs2 ) @ ( drop_P3377237446922189967st_int @ N @ Ys2 ) ) ) ).
% drop_zip
thf(fact_364_nth__via__drop,axiom,
! [N: nat,Xs2: list_int,Y: int,Ys2: list_int] :
( ( ( drop_int @ N @ Xs2 )
= ( cons_int @ Y @ Ys2 ) )
=> ( ( nth_int @ Xs2 @ N )
= Y ) ) ).
% nth_via_drop
thf(fact_365_nth__via__drop,axiom,
! [N: nat,Xs2: list_instr,Y: instr,Ys2: list_instr] :
( ( ( drop_instr @ N @ Xs2 )
= ( cons_instr @ Y @ Ys2 ) )
=> ( ( nth_instr @ Xs2 @ N )
= Y ) ) ).
% nth_via_drop
thf(fact_366_nth__via__drop,axiom,
! [N: nat,Xs2: list_P419547167332244514st_int,Y: produc6425607678544837394st_int,Ys2: list_P419547167332244514st_int] :
( ( ( drop_P7838799151754935884st_int @ N @ Xs2 )
= ( cons_P4151473878978182108st_int @ Y @ Ys2 ) )
=> ( ( nth_Pr8327173284404383691st_int @ Xs2 @ N )
= Y ) ) ).
% nth_via_drop
thf(fact_367_nth__via__drop,axiom,
! [N: nat,Xs2: list_list_char,Y: list_char,Ys2: list_list_char] :
( ( ( drop_list_char @ N @ Xs2 )
= ( cons_list_char @ Y @ Ys2 ) )
=> ( ( nth_list_char @ Xs2 @ N )
= Y ) ) ).
% nth_via_drop
thf(fact_368_not__Cons__self2,axiom,
! [X: int,Xs2: list_int] :
( ( cons_int @ X @ Xs2 )
!= Xs2 ) ).
% not_Cons_self2
thf(fact_369_not__Cons__self2,axiom,
! [X: instr,Xs2: list_instr] :
( ( cons_instr @ X @ Xs2 )
!= Xs2 ) ).
% not_Cons_self2
thf(fact_370_not__Cons__self2,axiom,
! [X: produc6425607678544837394st_int,Xs2: list_P419547167332244514st_int] :
( ( cons_P4151473878978182108st_int @ X @ Xs2 )
!= Xs2 ) ).
% not_Cons_self2
thf(fact_371_not__Cons__self2,axiom,
! [X: list_char,Xs2: list_list_char] :
( ( cons_list_char @ X @ Xs2 )
!= Xs2 ) ).
% not_Cons_self2
thf(fact_372_dbl__inc__def,axiom,
( neg_nu5851722552734809277nc_int
= ( ^ [X3: int] : ( plus_plus_int @ ( plus_plus_int @ X3 @ X3 ) @ one_one_int ) ) ) ).
% dbl_inc_def
thf(fact_373_drop__0,axiom,
! [Xs2: list_P419547167332244514st_int] :
( ( drop_P7838799151754935884st_int @ zero_zero_nat @ Xs2 )
= Xs2 ) ).
% drop_0
thf(fact_374_comm__monoid__add__class_Oadd__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_375_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_376_add_Ocomm__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.comm_neutral
thf(fact_377_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_378_add_Ogroup__left__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add.group_left_neutral
thf(fact_379_add__less__imp__less__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_380_add__less__imp__less__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_381_add__less__imp__less__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_382_add__less__imp__less__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_383_add__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_384_add__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_385_add__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_386_add__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_387_add__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_388_add__strict__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_389_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_390_add__mono__thms__linordered__field_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( K = L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_391_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_392_add__mono__thms__linordered__field_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_393_add__mono__thms__linordered__field_I5_J,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_mono_thms_linordered_field(5)
thf(fact_394_add__mono__thms__linordered__field_I5_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_395_group__cancel_Osub1,axiom,
! [A2: int,K: int,A: int,B: int] :
( ( A2
= ( plus_plus_int @ K @ A ) )
=> ( ( minus_minus_int @ A2 @ B )
= ( plus_plus_int @ K @ ( minus_minus_int @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_396_diff__eq__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( minus_minus_int @ A @ B )
= C )
= ( A
= ( plus_plus_int @ C @ B ) ) ) ).
% diff_eq_eq
thf(fact_397_eq__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( A
= ( minus_minus_int @ C @ B ) )
= ( ( plus_plus_int @ A @ B )
= C ) ) ).
% eq_diff_eq
thf(fact_398_add__diff__eq,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% add_diff_eq
thf(fact_399_diff__diff__eq2,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% diff_diff_eq2
thf(fact_400_diff__add__eq,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% diff_add_eq
thf(fact_401_diff__add__eq__diff__diff__swap,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) )
= ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_402_add__implies__diff,axiom,
! [C: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ C @ B )
= A )
=> ( C
= ( minus_minus_nat @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_403_add__implies__diff,axiom,
! [C: int,B: int,A: int] :
( ( ( plus_plus_int @ C @ B )
= A )
=> ( C
= ( minus_minus_int @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_404_diff__diff__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_405_diff__diff__eq,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_406_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_407_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_408_nat__arith_Osuc1,axiom,
! [A2: nat,K: nat,A: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( suc @ A2 )
= ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_409_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_410_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_411_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_412_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_413_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_414_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_415_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_416_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_417_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_418_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_419_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_420_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_421_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_422_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_423_drop__Cons_H,axiom,
! [N: nat,X: int,Xs2: list_int] :
( ( ( N = zero_zero_nat )
=> ( ( drop_int @ N @ ( cons_int @ X @ Xs2 ) )
= ( cons_int @ X @ Xs2 ) ) )
& ( ( N != zero_zero_nat )
=> ( ( drop_int @ N @ ( cons_int @ X @ Xs2 ) )
= ( drop_int @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs2 ) ) ) ) ).
% drop_Cons'
thf(fact_424_drop__Cons_H,axiom,
! [N: nat,X: instr,Xs2: list_instr] :
( ( ( N = zero_zero_nat )
=> ( ( drop_instr @ N @ ( cons_instr @ X @ Xs2 ) )
= ( cons_instr @ X @ Xs2 ) ) )
& ( ( N != zero_zero_nat )
=> ( ( drop_instr @ N @ ( cons_instr @ X @ Xs2 ) )
= ( drop_instr @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs2 ) ) ) ) ).
% drop_Cons'
thf(fact_425_drop__Cons_H,axiom,
! [N: nat,X: produc6425607678544837394st_int,Xs2: list_P419547167332244514st_int] :
( ( ( N = zero_zero_nat )
=> ( ( drop_P7838799151754935884st_int @ N @ ( cons_P4151473878978182108st_int @ X @ Xs2 ) )
= ( cons_P4151473878978182108st_int @ X @ Xs2 ) ) )
& ( ( N != zero_zero_nat )
=> ( ( drop_P7838799151754935884st_int @ N @ ( cons_P4151473878978182108st_int @ X @ Xs2 ) )
= ( drop_P7838799151754935884st_int @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs2 ) ) ) ) ).
% drop_Cons'
thf(fact_426_drop__Cons_H,axiom,
! [N: nat,X: list_char,Xs2: list_list_char] :
( ( ( N = zero_zero_nat )
=> ( ( drop_list_char @ N @ ( cons_list_char @ X @ Xs2 ) )
= ( cons_list_char @ X @ Xs2 ) ) )
& ( ( N != zero_zero_nat )
=> ( ( drop_list_char @ N @ ( cons_list_char @ X @ Xs2 ) )
= ( drop_list_char @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs2 ) ) ) ) ).
% drop_Cons'
thf(fact_427_list_Osize_I4_J,axiom,
! [X21: produc6425607678544837394st_int,X22: list_P419547167332244514st_int] :
( ( size_s229347708945616566st_int @ ( cons_P4151473878978182108st_int @ X21 @ X22 ) )
= ( plus_plus_nat @ ( size_s229347708945616566st_int @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% list.size(4)
thf(fact_428_list_Osize_I4_J,axiom,
! [X21: instr,X22: list_instr] :
( ( size_size_list_instr @ ( cons_instr @ X21 @ X22 ) )
= ( plus_plus_nat @ ( size_size_list_instr @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% list.size(4)
thf(fact_429_list_Osize_I4_J,axiom,
! [X21: list_char,X22: list_list_char] :
( ( size_s356637359517785349t_char @ ( cons_list_char @ X21 @ X22 ) )
= ( plus_plus_nat @ ( size_s356637359517785349t_char @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% list.size(4)
thf(fact_430_list_Osize_I4_J,axiom,
! [X21: int,X22: list_int] :
( ( size_size_list_int @ ( cons_int @ X21 @ X22 ) )
= ( plus_plus_nat @ ( size_size_list_int @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% list.size(4)
thf(fact_431_Cons__nth__drop__Suc,axiom,
! [I: nat,Xs2: list_P419547167332244514st_int] :
( ( ord_less_nat @ I @ ( size_s229347708945616566st_int @ Xs2 ) )
=> ( ( cons_P4151473878978182108st_int @ ( nth_Pr8327173284404383691st_int @ Xs2 @ I ) @ ( drop_P7838799151754935884st_int @ ( suc @ I ) @ Xs2 ) )
= ( drop_P7838799151754935884st_int @ I @ Xs2 ) ) ) ).
% Cons_nth_drop_Suc
thf(fact_432_Cons__nth__drop__Suc,axiom,
! [I: nat,Xs2: list_instr] :
( ( ord_less_nat @ I @ ( size_size_list_instr @ Xs2 ) )
=> ( ( cons_instr @ ( nth_instr @ Xs2 @ I ) @ ( drop_instr @ ( suc @ I ) @ Xs2 ) )
= ( drop_instr @ I @ Xs2 ) ) ) ).
% Cons_nth_drop_Suc
thf(fact_433_Cons__nth__drop__Suc,axiom,
! [I: nat,Xs2: list_list_char] :
( ( ord_less_nat @ I @ ( size_s356637359517785349t_char @ Xs2 ) )
=> ( ( cons_list_char @ ( nth_list_char @ Xs2 @ I ) @ ( drop_list_char @ ( suc @ I ) @ Xs2 ) )
= ( drop_list_char @ I @ Xs2 ) ) ) ).
% Cons_nth_drop_Suc
thf(fact_434_Cons__nth__drop__Suc,axiom,
! [I: nat,Xs2: list_int] :
( ( ord_less_nat @ I @ ( size_size_list_int @ Xs2 ) )
=> ( ( cons_int @ ( nth_int @ Xs2 @ I ) @ ( drop_int @ ( suc @ I ) @ Xs2 ) )
= ( drop_int @ I @ Xs2 ) ) ) ).
% Cons_nth_drop_Suc
thf(fact_435_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_436_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_437_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
!= zero_zero_nat ) ).
% of_nat_neq_0
thf(fact_438_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
!= zero_zero_int ) ).
% of_nat_neq_0
thf(fact_439_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_440_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_441_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_442_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_443_add__less__zeroD,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ ( plus_plus_int @ X @ Y ) @ zero_zero_int )
=> ( ( ord_less_int @ X @ zero_zero_int )
| ( ord_less_int @ Y @ zero_zero_int ) ) ) ).
% add_less_zeroD
thf(fact_444_pos__add__strict,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_445_pos__add__strict,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_446_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C5: nat] :
( ( B
= ( plus_plus_nat @ A @ C5 ) )
=> ( C5 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_447_add__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_448_add__pos__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_449_add__neg__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_450_add__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_neg_neg
thf(fact_451_add__mono1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_452_add__mono1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ one_one_int ) @ ( plus_plus_int @ B @ one_one_int ) ) ) ).
% add_mono1
thf(fact_453_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_454_less__add__one,axiom,
! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).
% less_add_one
thf(fact_455_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ~ ( ord_less_nat @ A @ B )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_456_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: int,B: int] :
( ~ ( ord_less_int @ A @ B )
=> ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_457_less__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ A @ ( minus_minus_int @ C @ B ) )
= ( ord_less_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% less_diff_eq
thf(fact_458_diff__less__eq,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ ( minus_minus_int @ A @ B ) @ C )
= ( ord_less_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% diff_less_eq
thf(fact_459_Suc__length__conv,axiom,
! [N: nat,Xs2: list_P419547167332244514st_int] :
( ( ( suc @ N )
= ( size_s229347708945616566st_int @ Xs2 ) )
= ( ? [Y5: produc6425607678544837394st_int,Ys: list_P419547167332244514st_int] :
( ( Xs2
= ( cons_P4151473878978182108st_int @ Y5 @ Ys ) )
& ( ( size_s229347708945616566st_int @ Ys )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_460_Suc__length__conv,axiom,
! [N: nat,Xs2: list_instr] :
( ( ( suc @ N )
= ( size_size_list_instr @ Xs2 ) )
= ( ? [Y5: instr,Ys: list_instr] :
( ( Xs2
= ( cons_instr @ Y5 @ Ys ) )
& ( ( size_size_list_instr @ Ys )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_461_Suc__length__conv,axiom,
! [N: nat,Xs2: list_list_char] :
( ( ( suc @ N )
= ( size_s356637359517785349t_char @ Xs2 ) )
= ( ? [Y5: list_char,Ys: list_list_char] :
( ( Xs2
= ( cons_list_char @ Y5 @ Ys ) )
& ( ( size_s356637359517785349t_char @ Ys )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_462_Suc__length__conv,axiom,
! [N: nat,Xs2: list_int] :
( ( ( suc @ N )
= ( size_size_list_int @ Xs2 ) )
= ( ? [Y5: int,Ys: list_int] :
( ( Xs2
= ( cons_int @ Y5 @ Ys ) )
& ( ( size_size_list_int @ Ys )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_463_length__Suc__conv,axiom,
! [Xs2: list_P419547167332244514st_int,N: nat] :
( ( ( size_s229347708945616566st_int @ Xs2 )
= ( suc @ N ) )
= ( ? [Y5: produc6425607678544837394st_int,Ys: list_P419547167332244514st_int] :
( ( Xs2
= ( cons_P4151473878978182108st_int @ Y5 @ Ys ) )
& ( ( size_s229347708945616566st_int @ Ys )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_464_length__Suc__conv,axiom,
! [Xs2: list_instr,N: nat] :
( ( ( size_size_list_instr @ Xs2 )
= ( suc @ N ) )
= ( ? [Y5: instr,Ys: list_instr] :
( ( Xs2
= ( cons_instr @ Y5 @ Ys ) )
& ( ( size_size_list_instr @ Ys )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_465_length__Suc__conv,axiom,
! [Xs2: list_list_char,N: nat] :
( ( ( size_s356637359517785349t_char @ Xs2 )
= ( suc @ N ) )
= ( ? [Y5: list_char,Ys: list_list_char] :
( ( Xs2
= ( cons_list_char @ Y5 @ Ys ) )
& ( ( size_s356637359517785349t_char @ Ys )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_466_length__Suc__conv,axiom,
! [Xs2: list_int,N: nat] :
( ( ( size_size_list_int @ Xs2 )
= ( suc @ N ) )
= ( ? [Y5: int,Ys: list_int] :
( ( Xs2
= ( cons_int @ Y5 @ Ys ) )
& ( ( size_size_list_int @ Ys )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_467_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_468_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_469_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_470_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_471_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_472_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
% less_add_Suc2
thf(fact_473_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
% less_add_Suc1
thf(fact_474_less__natE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ! [Q: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M @ Q ) ) ) ) ).
% less_natE
thf(fact_475_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_476_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_477_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_478_Suc__eq__plus1,axiom,
( suc
= ( ^ [N4: nat] : ( plus_plus_nat @ N4 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_479_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_480_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_481_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_482_zero__less__two,axiom,
ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% zero_less_two
thf(fact_483_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_484_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_485_com_Osize_I10_J,axiom,
! [X51: bexp,X52: com] :
( ( size_size_com @ ( while @ X51 @ X52 ) )
= ( plus_plus_nat @ ( size_size_com @ X52 ) @ ( suc @ zero_zero_nat ) ) ) ).
% com.size(10)
thf(fact_486_nth__Cons_H,axiom,
! [N: nat,X: int,Xs2: list_int] :
( ( ( N = zero_zero_nat )
=> ( ( nth_int @ ( cons_int @ X @ Xs2 ) @ N )
= X ) )
& ( ( N != zero_zero_nat )
=> ( ( nth_int @ ( cons_int @ X @ Xs2 ) @ N )
= ( nth_int @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_487_nth__Cons_H,axiom,
! [N: nat,X: instr,Xs2: list_instr] :
( ( ( N = zero_zero_nat )
=> ( ( nth_instr @ ( cons_instr @ X @ Xs2 ) @ N )
= X ) )
& ( ( N != zero_zero_nat )
=> ( ( nth_instr @ ( cons_instr @ X @ Xs2 ) @ N )
= ( nth_instr @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_488_nth__Cons_H,axiom,
! [N: nat,X: produc6425607678544837394st_int,Xs2: list_P419547167332244514st_int] :
( ( ( N = zero_zero_nat )
=> ( ( nth_Pr8327173284404383691st_int @ ( cons_P4151473878978182108st_int @ X @ Xs2 ) @ N )
= X ) )
& ( ( N != zero_zero_nat )
=> ( ( nth_Pr8327173284404383691st_int @ ( cons_P4151473878978182108st_int @ X @ Xs2 ) @ N )
= ( nth_Pr8327173284404383691st_int @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_489_nth__Cons_H,axiom,
! [N: nat,X: list_char,Xs2: list_list_char] :
( ( ( N = zero_zero_nat )
=> ( ( nth_list_char @ ( cons_list_char @ X @ Xs2 ) @ N )
= X ) )
& ( ( N != zero_zero_nat )
=> ( ( nth_list_char @ ( cons_list_char @ X @ Xs2 ) @ N )
= ( nth_list_char @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_490_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_491_nth__non__equal__first__eq,axiom,
! [X: int,Y: int,Xs2: list_int,N: nat] :
( ( X != Y )
=> ( ( ( nth_int @ ( cons_int @ X @ Xs2 ) @ N )
= Y )
= ( ( ( nth_int @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) )
= Y )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_492_nth__non__equal__first__eq,axiom,
! [X: instr,Y: instr,Xs2: list_instr,N: nat] :
( ( X != Y )
=> ( ( ( nth_instr @ ( cons_instr @ X @ Xs2 ) @ N )
= Y )
= ( ( ( nth_instr @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) )
= Y )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_493_nth__non__equal__first__eq,axiom,
! [X: produc6425607678544837394st_int,Y: produc6425607678544837394st_int,Xs2: list_P419547167332244514st_int,N: nat] :
( ( X != Y )
=> ( ( ( nth_Pr8327173284404383691st_int @ ( cons_P4151473878978182108st_int @ X @ Xs2 ) @ N )
= Y )
= ( ( ( nth_Pr8327173284404383691st_int @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) )
= Y )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_494_nth__non__equal__first__eq,axiom,
! [X: list_char,Y: list_char,Xs2: list_list_char,N: nat] :
( ( X != Y )
=> ( ( ( nth_list_char @ ( cons_list_char @ X @ Xs2 ) @ N )
= Y )
= ( ( ( nth_list_char @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) )
= Y )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_495_double__eq__0__iff,axiom,
! [A: int] :
( ( ( plus_plus_int @ A @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% double_eq_0_iff
thf(fact_496_int__Suc,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ).
% int_Suc
thf(fact_497_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_498_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_499_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_500_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_501_zadd__int__left,axiom,
! [M: nat,N: nat,Z2: int] :
( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z2 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) ) @ Z2 ) ) ).
% zadd_int_left
thf(fact_502_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_503_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_504_verit__comp__simplify1_I1_J,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_505_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus_int @ K @ zero_zero_int )
= K ) ).
% minus_int_code(1)
thf(fact_506_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_507_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_508_nat__int__comparison_I1_J,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [A3: nat,B2: nat] :
( ( semiri1314217659103216013at_int @ A3 )
= ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_509_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_510_int__int__eq,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% int_int_eq
thf(fact_511_int__diff__cases,axiom,
! [Z2: int] :
~ ! [M4: nat,N3: nat] :
( Z2
!= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M4 ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% int_diff_cases
thf(fact_512_verit__sum__simplify,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% verit_sum_simplify
thf(fact_513_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_514_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A3: nat,B2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_515_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_516_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus_int @ K @ zero_zero_int )
= K ) ).
% plus_int_code(1)
thf(fact_517_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_518_zless__add1__eq,axiom,
! [W: int,Z2: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z2 @ one_one_int ) )
= ( ( ord_less_int @ W @ Z2 )
| ( W = Z2 ) ) ) ).
% zless_add1_eq
thf(fact_519_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_520_zless__iff__Suc__zadd,axiom,
( ord_less_int
= ( ^ [W2: int,Z3: int] :
? [N4: nat] :
( Z3
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ ( suc @ N4 ) ) ) ) ) ) ).
% zless_iff_Suc_zadd
thf(fact_521_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_522_odd__nonzero,axiom,
! [Z2: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_523_odd__less__0__iff,axiom,
! [Z2: int] :
( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 ) @ zero_zero_int )
= ( ord_less_int @ Z2 @ zero_zero_int ) ) ).
% odd_less_0_iff
thf(fact_524_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_525_length__Cons,axiom,
! [X: produc6425607678544837394st_int,Xs2: list_P419547167332244514st_int] :
( ( size_s229347708945616566st_int @ ( cons_P4151473878978182108st_int @ X @ Xs2 ) )
= ( suc @ ( size_s229347708945616566st_int @ Xs2 ) ) ) ).
% length_Cons
thf(fact_526_length__Cons,axiom,
! [X: instr,Xs2: list_instr] :
( ( size_size_list_instr @ ( cons_instr @ X @ Xs2 ) )
= ( suc @ ( size_size_list_instr @ Xs2 ) ) ) ).
% length_Cons
thf(fact_527_length__Cons,axiom,
! [X: list_char,Xs2: list_list_char] :
( ( size_s356637359517785349t_char @ ( cons_list_char @ X @ Xs2 ) )
= ( suc @ ( size_s356637359517785349t_char @ Xs2 ) ) ) ).
% length_Cons
thf(fact_528_length__Cons,axiom,
! [X: int,Xs2: list_int] :
( ( size_size_list_int @ ( cons_int @ X @ Xs2 ) )
= ( suc @ ( size_size_list_int @ Xs2 ) ) ) ).
% length_Cons
thf(fact_529_iexec__shift,axiom,
! [N: int,I4: int,S5: list_char > int,Stk2: list_int,Ins2: instr,I: int,S: list_char > int,Stk3: list_int] :
( ( ( produc5086643055186798020st_int @ ( plus_plus_int @ N @ I4 ) @ ( produc8650753666468850689st_int @ S5 @ Stk2 ) )
= ( iexec @ Ins2 @ ( produc5086643055186798020st_int @ ( plus_plus_int @ N @ I ) @ ( produc8650753666468850689st_int @ S @ Stk3 ) ) ) )
= ( ( produc5086643055186798020st_int @ I4 @ ( produc8650753666468850689st_int @ S5 @ Stk2 ) )
= ( iexec @ Ins2 @ ( produc5086643055186798020st_int @ I @ ( produc8650753666468850689st_int @ S @ Stk3 ) ) ) ) ) ).
% iexec_shift
thf(fact_530_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_531_iexec__offset,axiom,
! [Ins2: instr,Pc: int,S: list_char > int,Stk3: list_int,Pc2: int,S5: list_char > int,Stk2: list_int,I: int] :
( ( iexec_pred @ ( produc1484136438817787418st_int @ Ins2 @ ( produc5086643055186798020st_int @ Pc @ ( produc8650753666468850689st_int @ S @ Stk3 ) ) ) @ ( produc5086643055186798020st_int @ Pc2 @ ( produc8650753666468850689st_int @ S5 @ Stk2 ) ) )
=> ( iexec_pred @ ( produc1484136438817787418st_int @ Ins2 @ ( produc5086643055186798020st_int @ ( minus_minus_int @ Pc @ I ) @ ( produc8650753666468850689st_int @ S @ Stk3 ) ) ) @ ( produc5086643055186798020st_int @ ( minus_minus_int @ Pc2 @ I ) @ ( produc8650753666468850689st_int @ S5 @ Stk2 ) ) ) ) ).
% iexec_offset
thf(fact_532_inth_Osimps,axiom,
! [I: int,X: int,Xs2: list_int] :
( ( ( I = zero_zero_int )
=> ( ( inth_int @ ( cons_int @ X @ Xs2 ) @ I )
= X ) )
& ( ( I != zero_zero_int )
=> ( ( inth_int @ ( cons_int @ X @ Xs2 ) @ I )
= ( inth_int @ Xs2 @ ( minus_minus_int @ I @ one_one_int ) ) ) ) ) ).
% inth.simps
thf(fact_533_inth_Osimps,axiom,
! [I: int,X: instr,Xs2: list_instr] :
( ( ( I = zero_zero_int )
=> ( ( inth_instr @ ( cons_instr @ X @ Xs2 ) @ I )
= X ) )
& ( ( I != zero_zero_int )
=> ( ( inth_instr @ ( cons_instr @ X @ Xs2 ) @ I )
= ( inth_instr @ Xs2 @ ( minus_minus_int @ I @ one_one_int ) ) ) ) ) ).
% inth.simps
thf(fact_534_inth_Osimps,axiom,
! [I: int,X: produc6425607678544837394st_int,Xs2: list_P419547167332244514st_int] :
( ( ( I = zero_zero_int )
=> ( ( inth_P8039235741264645271st_int @ ( cons_P4151473878978182108st_int @ X @ Xs2 ) @ I )
= X ) )
& ( ( I != zero_zero_int )
=> ( ( inth_P8039235741264645271st_int @ ( cons_P4151473878978182108st_int @ X @ Xs2 ) @ I )
= ( inth_P8039235741264645271st_int @ Xs2 @ ( minus_minus_int @ I @ one_one_int ) ) ) ) ) ).
% inth.simps
thf(fact_535_inth_Osimps,axiom,
! [I: int,X: list_char,Xs2: list_list_char] :
( ( ( I = zero_zero_int )
=> ( ( inth_list_char @ ( cons_list_char @ X @ Xs2 ) @ I )
= X ) )
& ( ( I != zero_zero_int )
=> ( ( inth_list_char @ ( cons_list_char @ X @ Xs2 ) @ I )
= ( inth_list_char @ Xs2 @ ( minus_minus_int @ I @ one_one_int ) ) ) ) ) ).
% inth.simps
thf(fact_536_com_Osize__gen_I5_J,axiom,
! [X51: bexp,X52: com] :
( ( size_com @ ( while @ X51 @ X52 ) )
= ( plus_plus_nat @ ( size_com @ X52 ) @ ( suc @ zero_zero_nat ) ) ) ).
% com.size_gen(5)
thf(fact_537_iexec__simp,axiom,
! [Ins2: instr,Cf: produc6425607678544837394st_int,Cf2: produc6425607678544837394st_int] :
( ( iexec_pred @ ( produc1484136438817787418st_int @ Ins2 @ Cf ) @ Cf2 )
= ( Cf2
= ( iexec @ Ins2 @ Cf ) ) ) ).
% iexec_simp
thf(fact_538_iexec__pred_Ointros,axiom,
! [Ins2: instr,Cf: produc6425607678544837394st_int] : ( iexec_pred @ ( produc1484136438817787418st_int @ Ins2 @ Cf ) @ ( iexec @ Ins2 @ Cf ) ) ).
% iexec_pred.intros
thf(fact_539_iexec__pred_Osimps,axiom,
( iexec_pred
= ( ^ [A1: produc5995290525303592096st_int,A22: produc6425607678544837394st_int] :
? [Ins3: instr,Cf3: produc6425607678544837394st_int] :
( ( A1
= ( produc1484136438817787418st_int @ Ins3 @ Cf3 ) )
& ( A22
= ( iexec @ Ins3 @ Cf3 ) ) ) ) ) ).
% iexec_pred.simps
thf(fact_540_iexec__pred_Ocases,axiom,
! [A12: produc5995290525303592096st_int,A23: produc6425607678544837394st_int] :
( ( iexec_pred @ A12 @ A23 )
=> ~ ! [Ins: instr,Cf4: produc6425607678544837394st_int] :
( ( A12
= ( produc1484136438817787418st_int @ Ins @ Cf4 ) )
=> ( A23
!= ( iexec @ Ins @ Cf4 ) ) ) ) ).
% iexec_pred.cases
thf(fact_541_LoadIE,axiom,
! [I: int,Pc: int,S: list_char > int,Stk3: list_int,Cf: produc6425607678544837394st_int] :
( ( iexec_pred @ ( produc1484136438817787418st_int @ ( loadi @ I ) @ ( produc5086643055186798020st_int @ Pc @ ( produc8650753666468850689st_int @ S @ Stk3 ) ) ) @ Cf )
=> ( Cf
= ( produc5086643055186798020st_int @ ( plus_plus_int @ Pc @ one_one_int ) @ ( produc8650753666468850689st_int @ S @ ( cons_int @ I @ Stk3 ) ) ) ) ) ).
% LoadIE
thf(fact_542_LoadE,axiom,
! [X: list_char,Pc: int,S: list_char > int,Stk3: list_int,Cf: produc6425607678544837394st_int] :
( ( iexec_pred @ ( produc1484136438817787418st_int @ ( load @ X ) @ ( produc5086643055186798020st_int @ Pc @ ( produc8650753666468850689st_int @ S @ Stk3 ) ) ) @ Cf )
=> ( Cf
= ( produc5086643055186798020st_int @ ( plus_plus_int @ Pc @ one_one_int ) @ ( produc8650753666468850689st_int @ S @ ( cons_int @ ( S @ X ) @ Stk3 ) ) ) ) ) ).
% LoadE
thf(fact_543_JmpE,axiom,
! [I: int,Pc: int,S: list_char > int,Stk3: list_int,Cf: produc6425607678544837394st_int] :
( ( iexec_pred @ ( produc1484136438817787418st_int @ ( jmp @ I ) @ ( produc5086643055186798020st_int @ Pc @ ( produc8650753666468850689st_int @ S @ Stk3 ) ) ) @ Cf )
=> ( Cf
= ( produc5086643055186798020st_int @ ( plus_plus_int @ ( plus_plus_int @ Pc @ one_one_int ) @ I ) @ ( produc8650753666468850689st_int @ S @ Stk3 ) ) ) ) ).
% JmpE
thf(fact_544_Euclid__induct,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A5: nat,B4: nat] :
( ( P @ A5 @ B4 )
= ( P @ B4 @ A5 ) )
=> ( ! [A5: nat] : ( P @ A5 @ zero_zero_nat )
=> ( ! [A5: nat,B4: nat] :
( ( P @ A5 @ B4 )
=> ( P @ A5 @ ( plus_plus_nat @ A5 @ B4 ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Euclid_induct
thf(fact_545_add__0__iff,axiom,
! [B: int,A: int] :
( ( B
= ( plus_plus_int @ B @ A ) )
= ( A = zero_zero_int ) ) ).
% add_0_iff
thf(fact_546_add__0__iff,axiom,
! [B: nat,A: nat] :
( ( B
= ( plus_plus_nat @ B @ A ) )
= ( A = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_547_instr_Oinject_I4_J,axiom,
! [X53: int,Y53: int] :
( ( ( jmp @ X53 )
= ( jmp @ Y53 ) )
= ( X53 = Y53 ) ) ).
% instr.inject(4)
thf(fact_548_instr_Oinject_I2_J,axiom,
! [X2: list_char,Y2: list_char] :
( ( ( load @ X2 )
= ( load @ Y2 ) )
= ( X2 = Y2 ) ) ).
% instr.inject(2)
thf(fact_549_instr_Oinject_I1_J,axiom,
! [X1: int,Y1: int] :
( ( ( loadi @ X1 )
= ( loadi @ Y1 ) )
= ( X1 = Y1 ) ) ).
% instr.inject(1)
thf(fact_550_instr_Odistinct_I17_J,axiom,
! [X2: list_char,X53: int] :
( ( load @ X2 )
!= ( jmp @ X53 ) ) ).
% instr.distinct(17)
thf(fact_551_instr_Odistinct_I7_J,axiom,
! [X1: int,X53: int] :
( ( loadi @ X1 )
!= ( jmp @ X53 ) ) ).
% instr.distinct(7)
thf(fact_552_instr_Odistinct_I1_J,axiom,
! [X1: int,X2: list_char] :
( ( loadi @ X1 )
!= ( load @ X2 ) ) ).
% instr.distinct(1)
thf(fact_553_instr_Osize_I12_J,axiom,
! [X53: int] :
( ( size_size_instr @ ( jmp @ X53 ) )
= zero_zero_nat ) ).
% instr.size(12)
thf(fact_554_instr_Osize_I9_J,axiom,
! [X2: list_char] :
( ( size_size_instr @ ( load @ X2 ) )
= zero_zero_nat ) ).
% instr.size(9)
thf(fact_555_instr_Osize_I8_J,axiom,
! [X1: int] :
( ( size_size_instr @ ( loadi @ X1 ) )
= zero_zero_nat ) ).
% instr.size(8)
thf(fact_556__092_060open_062bcomp_A_Ib_M_AFalse_M_Asize_A_Iccomp_Ac_J_A_L_A1_J_A_064_Accomp_Ac_A_064_A_091JMP_A_I_N_Asize_A_Ibcomp_A_Ib_M_AFalse_M_Asize_A_Iccomp_Ac_J_A_L_A1_J_J_A_N_Asize_A_Iccomp_Ac_J_A_N_A1_J_093_A_092_060Turnstile_062_Adrop_A_ISuc_Ak_J_Acfs_092_060box_062_092_060close_062,axiom,
execl_all @ ( append_instr @ ( bcomp @ ( produc4047900504771817624_o_int @ b @ ( product_Pair_o_int @ $false @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ ( ccomp @ c ) ) ) @ one_one_int ) ) ) ) @ ( append_instr @ ( ccomp @ c ) @ ( cons_instr @ ( jmp @ ( minus_minus_int @ ( minus_minus_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ ( bcomp @ ( produc4047900504771817624_o_int @ b @ ( product_Pair_o_int @ $false @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ ( ccomp @ c ) ) ) @ one_one_int ) ) ) ) ) ) ) @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ ( ccomp @ c ) ) ) ) @ one_one_int ) ) @ nil_instr ) ) ) @ ( drop_P7838799151754935884st_int @ ( suc @ k ) @ cfsa ) ).
% \<open>bcomp (b, False, size (ccomp c) + 1) @ ccomp c @ [JMP (- size (bcomp (b, False, size (ccomp c) + 1)) - size (ccomp c) - 1)] \<Turnstile> drop (Suc k) cfs\<box>\<close>
thf(fact_557_A,axiom,
execl_all @ ( append_instr @ ( bcomp @ ( produc4047900504771817624_o_int @ b @ ( product_Pair_o_int @ $false @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ ( ccomp @ c ) ) ) @ one_one_int ) ) ) ) @ ( append_instr @ ( ccomp @ c ) @ ( cons_instr @ ( jmp @ ( minus_minus_int @ ( minus_minus_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ ( bcomp @ ( produc4047900504771817624_o_int @ b @ ( product_Pair_o_int @ $false @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ ( ccomp @ c ) ) ) @ one_one_int ) ) ) ) ) ) ) @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ ( ccomp @ c ) ) ) ) @ one_one_int ) ) @ nil_instr ) ) ) @ cfsa ).
% A
thf(fact_558_com_Osize_I8_J,axiom,
! [X31: com,X32: com] :
( ( size_size_com @ ( seq @ X31 @ X32 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( size_size_com @ X31 ) @ ( size_size_com @ X32 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% com.size(8)
thf(fact_559_com_Osize__gen_I3_J,axiom,
! [X31: com,X32: com] :
( ( size_com @ ( seq @ X31 @ X32 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( size_com @ X31 ) @ ( size_com @ X32 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% com.size_gen(3)
thf(fact_560_nth__enumerate__eq,axiom,
! [M: nat,Xs2: list_P419547167332244514st_int,N: nat] :
( ( ord_less_nat @ M @ ( size_s229347708945616566st_int @ Xs2 ) )
=> ( ( nth_Pr6118945435929909578st_int @ ( enumer1800257932645307117st_int @ N @ Xs2 ) @ M )
= ( produc2664891610604065915st_int @ ( plus_plus_nat @ N @ M ) @ ( nth_Pr8327173284404383691st_int @ Xs2 @ M ) ) ) ) ).
% nth_enumerate_eq
thf(fact_561_nth__enumerate__eq,axiom,
! [M: nat,Xs2: list_instr,N: nat] :
( ( ord_less_nat @ M @ ( size_size_list_instr @ Xs2 ) )
=> ( ( nth_Pr4329546499383151805_instr @ ( enumerate_instr @ N @ Xs2 ) @ M )
= ( produc3129998205563682742_instr @ ( plus_plus_nat @ N @ M ) @ ( nth_instr @ Xs2 @ M ) ) ) ) ).
% nth_enumerate_eq
thf(fact_562_nth__enumerate__eq,axiom,
! [M: nat,Xs2: list_list_char,N: nat] :
( ( ord_less_nat @ M @ ( size_s356637359517785349t_char @ Xs2 ) )
=> ( ( nth_Pr7433773183981329817t_char @ ( enumerate_list_char @ N @ Xs2 ) @ M )
= ( produc120197956887798346t_char @ ( plus_plus_nat @ N @ M ) @ ( nth_list_char @ Xs2 @ M ) ) ) ) ).
% nth_enumerate_eq
thf(fact_563_nth__enumerate__eq,axiom,
! [M: nat,Xs2: list_int,N: nat] :
( ( ord_less_nat @ M @ ( size_size_list_int @ Xs2 ) )
=> ( ( nth_Pr3440142176431000676at_int @ ( enumerate_int @ N @ Xs2 ) @ M )
= ( product_Pair_nat_int @ ( plus_plus_nat @ N @ M ) @ ( nth_int @ Xs2 @ M ) ) ) ) ).
% nth_enumerate_eq
thf(fact_564_neg__equal__iff__equal,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= ( uminus_uminus_int @ B ) )
= ( A = B ) ) ).
% neg_equal_iff_equal
thf(fact_565_add_Oinverse__inverse,axiom,
! [A: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_566_verit__minus__simplify_I4_J,axiom,
! [B: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_567_same__append__eq,axiom,
! [Xs2: list_instr,Ys2: list_instr,Zs: list_instr] :
( ( ( append_instr @ Xs2 @ Ys2 )
= ( append_instr @ Xs2 @ Zs ) )
= ( Ys2 = Zs ) ) ).
% same_append_eq
thf(fact_568_same__append__eq,axiom,
! [Xs2: list_list_char,Ys2: list_list_char,Zs: list_list_char] :
( ( ( append_list_char @ Xs2 @ Ys2 )
= ( append_list_char @ Xs2 @ Zs ) )
= ( Ys2 = Zs ) ) ).
% same_append_eq
thf(fact_569_append__same__eq,axiom,
! [Ys2: list_instr,Xs2: list_instr,Zs: list_instr] :
( ( ( append_instr @ Ys2 @ Xs2 )
= ( append_instr @ Zs @ Xs2 ) )
= ( Ys2 = Zs ) ) ).
% append_same_eq
thf(fact_570_append__same__eq,axiom,
! [Ys2: list_list_char,Xs2: list_list_char,Zs: list_list_char] :
( ( ( append_list_char @ Ys2 @ Xs2 )
= ( append_list_char @ Zs @ Xs2 ) )
= ( Ys2 = Zs ) ) ).
% append_same_eq
thf(fact_571_append__assoc,axiom,
! [Xs2: list_instr,Ys2: list_instr,Zs: list_instr] :
( ( append_instr @ ( append_instr @ Xs2 @ Ys2 ) @ Zs )
= ( append_instr @ Xs2 @ ( append_instr @ Ys2 @ Zs ) ) ) ).
% append_assoc
thf(fact_572_append__assoc,axiom,
! [Xs2: list_list_char,Ys2: list_list_char,Zs: list_list_char] :
( ( append_list_char @ ( append_list_char @ Xs2 @ Ys2 ) @ Zs )
= ( append_list_char @ Xs2 @ ( append_list_char @ Ys2 @ Zs ) ) ) ).
% append_assoc
thf(fact_573_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_574_append_Oassoc,axiom,
! [A: list_list_char,B: list_list_char,C: list_list_char] :
( ( append_list_char @ ( append_list_char @ A @ B ) @ C )
= ( append_list_char @ A @ ( append_list_char @ B @ C ) ) ) ).
% append.assoc
thf(fact_575_com_Oinject_I2_J,axiom,
! [X31: com,X32: com,Y31: com,Y32: com] :
( ( ( seq @ X31 @ X32 )
= ( seq @ Y31 @ Y32 ) )
= ( ( X31 = Y31 )
& ( X32 = Y32 ) ) ) ).
% com.inject(2)
thf(fact_576_neg__equal__zero,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= A )
= ( A = zero_zero_int ) ) ).
% neg_equal_zero
thf(fact_577_equal__neg__zero,axiom,
! [A: int] :
( ( A
= ( uminus_uminus_int @ A ) )
= ( A = zero_zero_int ) ) ).
% equal_neg_zero
thf(fact_578_neg__equal__0__iff__equal,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% neg_equal_0_iff_equal
thf(fact_579_neg__0__equal__iff__equal,axiom,
! [A: int] :
( ( zero_zero_int
= ( uminus_uminus_int @ A ) )
= ( zero_zero_int = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_580_add_Oinverse__neutral,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% add.inverse_neutral
thf(fact_581_neg__less__iff__less,axiom,
! [B: int,A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ B ) ) ).
% neg_less_iff_less
thf(fact_582_minus__add__distrib,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ) ).
% minus_add_distrib
thf(fact_583_minus__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( plus_plus_int @ A @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_584_add__minus__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ A @ ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_585_minus__diff__eq,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) )
= ( minus_minus_int @ B @ A ) ) ).
% minus_diff_eq
thf(fact_586_append__is__Nil__conv,axiom,
! [Xs2: list_instr,Ys2: list_instr] :
( ( ( append_instr @ Xs2 @ Ys2 )
= nil_instr )
= ( ( Xs2 = nil_instr )
& ( Ys2 = nil_instr ) ) ) ).
% append_is_Nil_conv
thf(fact_587_append__is__Nil__conv,axiom,
! [Xs2: list_P419547167332244514st_int,Ys2: list_P419547167332244514st_int] :
( ( ( append5028292981072010935st_int @ Xs2 @ Ys2 )
= nil_Pr749310521696113548st_int )
= ( ( Xs2 = nil_Pr749310521696113548st_int )
& ( Ys2 = nil_Pr749310521696113548st_int ) ) ) ).
% append_is_Nil_conv
thf(fact_588_append__is__Nil__conv,axiom,
! [Xs2: list_int,Ys2: list_int] :
( ( ( append_int @ Xs2 @ Ys2 )
= nil_int )
= ( ( Xs2 = nil_int )
& ( Ys2 = nil_int ) ) ) ).
% append_is_Nil_conv
thf(fact_589_append__is__Nil__conv,axiom,
! [Xs2: list_list_char,Ys2: list_list_char] :
( ( ( append_list_char @ Xs2 @ Ys2 )
= nil_list_char )
= ( ( Xs2 = nil_list_char )
& ( Ys2 = nil_list_char ) ) ) ).
% append_is_Nil_conv
thf(fact_590_Nil__is__append__conv,axiom,
! [Xs2: list_instr,Ys2: list_instr] :
( ( nil_instr
= ( append_instr @ Xs2 @ Ys2 ) )
= ( ( Xs2 = nil_instr )
& ( Ys2 = nil_instr ) ) ) ).
% Nil_is_append_conv
thf(fact_591_Nil__is__append__conv,axiom,
! [Xs2: list_P419547167332244514st_int,Ys2: list_P419547167332244514st_int] :
( ( nil_Pr749310521696113548st_int
= ( append5028292981072010935st_int @ Xs2 @ Ys2 ) )
= ( ( Xs2 = nil_Pr749310521696113548st_int )
& ( Ys2 = nil_Pr749310521696113548st_int ) ) ) ).
% Nil_is_append_conv
thf(fact_592_Nil__is__append__conv,axiom,
! [Xs2: list_int,Ys2: list_int] :
( ( nil_int
= ( append_int @ Xs2 @ Ys2 ) )
= ( ( Xs2 = nil_int )
& ( Ys2 = nil_int ) ) ) ).
% Nil_is_append_conv
thf(fact_593_Nil__is__append__conv,axiom,
! [Xs2: list_list_char,Ys2: list_list_char] :
( ( nil_list_char
= ( append_list_char @ Xs2 @ Ys2 ) )
= ( ( Xs2 = nil_list_char )
& ( Ys2 = nil_list_char ) ) ) ).
% Nil_is_append_conv
thf(fact_594_self__append__conv2,axiom,
! [Y: list_instr,Xs2: list_instr] :
( ( Y
= ( append_instr @ Xs2 @ Y ) )
= ( Xs2 = nil_instr ) ) ).
% self_append_conv2
thf(fact_595_self__append__conv2,axiom,
! [Y: list_P419547167332244514st_int,Xs2: list_P419547167332244514st_int] :
( ( Y
= ( append5028292981072010935st_int @ Xs2 @ Y ) )
= ( Xs2 = nil_Pr749310521696113548st_int ) ) ).
% self_append_conv2
thf(fact_596_self__append__conv2,axiom,
! [Y: list_int,Xs2: list_int] :
( ( Y
= ( append_int @ Xs2 @ Y ) )
= ( Xs2 = nil_int ) ) ).
% self_append_conv2
thf(fact_597_self__append__conv2,axiom,
! [Y: list_list_char,Xs2: list_list_char] :
( ( Y
= ( append_list_char @ Xs2 @ Y ) )
= ( Xs2 = nil_list_char ) ) ).
% self_append_conv2
thf(fact_598_append__self__conv2,axiom,
! [Xs2: list_instr,Ys2: list_instr] :
( ( ( append_instr @ Xs2 @ Ys2 )
= Ys2 )
= ( Xs2 = nil_instr ) ) ).
% append_self_conv2
thf(fact_599_append__self__conv2,axiom,
! [Xs2: list_P419547167332244514st_int,Ys2: list_P419547167332244514st_int] :
( ( ( append5028292981072010935st_int @ Xs2 @ Ys2 )
= Ys2 )
= ( Xs2 = nil_Pr749310521696113548st_int ) ) ).
% append_self_conv2
thf(fact_600_append__self__conv2,axiom,
! [Xs2: list_int,Ys2: list_int] :
( ( ( append_int @ Xs2 @ Ys2 )
= Ys2 )
= ( Xs2 = nil_int ) ) ).
% append_self_conv2
thf(fact_601_append__self__conv2,axiom,
! [Xs2: list_list_char,Ys2: list_list_char] :
( ( ( append_list_char @ Xs2 @ Ys2 )
= Ys2 )
= ( Xs2 = nil_list_char ) ) ).
% append_self_conv2
thf(fact_602_self__append__conv,axiom,
! [Y: list_instr,Ys2: list_instr] :
( ( Y
= ( append_instr @ Y @ Ys2 ) )
= ( Ys2 = nil_instr ) ) ).
% self_append_conv
thf(fact_603_self__append__conv,axiom,
! [Y: list_P419547167332244514st_int,Ys2: list_P419547167332244514st_int] :
( ( Y
= ( append5028292981072010935st_int @ Y @ Ys2 ) )
= ( Ys2 = nil_Pr749310521696113548st_int ) ) ).
% self_append_conv
thf(fact_604_self__append__conv,axiom,
! [Y: list_int,Ys2: list_int] :
( ( Y
= ( append_int @ Y @ Ys2 ) )
= ( Ys2 = nil_int ) ) ).
% self_append_conv
thf(fact_605_self__append__conv,axiom,
! [Y: list_list_char,Ys2: list_list_char] :
( ( Y
= ( append_list_char @ Y @ Ys2 ) )
= ( Ys2 = nil_list_char ) ) ).
% self_append_conv
thf(fact_606_append__self__conv,axiom,
! [Xs2: list_instr,Ys2: list_instr] :
( ( ( append_instr @ Xs2 @ Ys2 )
= Xs2 )
= ( Ys2 = nil_instr ) ) ).
% append_self_conv
thf(fact_607_append__self__conv,axiom,
! [Xs2: list_P419547167332244514st_int,Ys2: list_P419547167332244514st_int] :
( ( ( append5028292981072010935st_int @ Xs2 @ Ys2 )
= Xs2 )
= ( Ys2 = nil_Pr749310521696113548st_int ) ) ).
% append_self_conv
thf(fact_608_append__self__conv,axiom,
! [Xs2: list_int,Ys2: list_int] :
( ( ( append_int @ Xs2 @ Ys2 )
= Xs2 )
= ( Ys2 = nil_int ) ) ).
% append_self_conv
thf(fact_609_append__self__conv,axiom,
! [Xs2: list_list_char,Ys2: list_list_char] :
( ( ( append_list_char @ Xs2 @ Ys2 )
= Xs2 )
= ( Ys2 = nil_list_char ) ) ).
% append_self_conv
thf(fact_610_append__Nil2,axiom,
! [Xs2: list_instr] :
( ( append_instr @ Xs2 @ nil_instr )
= Xs2 ) ).
% append_Nil2
thf(fact_611_append__Nil2,axiom,
! [Xs2: list_P419547167332244514st_int] :
( ( append5028292981072010935st_int @ Xs2 @ nil_Pr749310521696113548st_int )
= Xs2 ) ).
% append_Nil2
thf(fact_612_append__Nil2,axiom,
! [Xs2: list_int] :
( ( append_int @ Xs2 @ nil_int )
= Xs2 ) ).
% append_Nil2
thf(fact_613_append__Nil2,axiom,
! [Xs2: list_list_char] :
( ( append_list_char @ Xs2 @ nil_list_char )
= Xs2 ) ).
% append_Nil2
thf(fact_614_append_Oright__neutral,axiom,
! [A: list_instr] :
( ( append_instr @ A @ nil_instr )
= A ) ).
% append.right_neutral
thf(fact_615_append_Oright__neutral,axiom,
! [A: list_P419547167332244514st_int] :
( ( append5028292981072010935st_int @ A @ nil_Pr749310521696113548st_int )
= A ) ).
% append.right_neutral
thf(fact_616_append_Oright__neutral,axiom,
! [A: list_int] :
( ( append_int @ A @ nil_int )
= A ) ).
% append.right_neutral
thf(fact_617_append_Oright__neutral,axiom,
! [A: list_list_char] :
( ( append_list_char @ A @ nil_list_char )
= A ) ).
% append.right_neutral
thf(fact_618_append__eq__append__conv,axiom,
! [Xs2: list_instr,Ys2: list_instr,Us: list_instr,Vs: list_instr] :
( ( ( ( size_size_list_instr @ Xs2 )
= ( size_size_list_instr @ Ys2 ) )
| ( ( size_size_list_instr @ Us )
= ( size_size_list_instr @ Vs ) ) )
=> ( ( ( append_instr @ Xs2 @ Us )
= ( append_instr @ Ys2 @ Vs ) )
= ( ( Xs2 = Ys2 )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_619_append__eq__append__conv,axiom,
! [Xs2: list_list_char,Ys2: list_list_char,Us: list_list_char,Vs: list_list_char] :
( ( ( ( size_s356637359517785349t_char @ Xs2 )
= ( size_s356637359517785349t_char @ Ys2 ) )
| ( ( size_s356637359517785349t_char @ Us )
= ( size_s356637359517785349t_char @ Vs ) ) )
=> ( ( ( append_list_char @ Xs2 @ Us )
= ( append_list_char @ Ys2 @ Vs ) )
= ( ( Xs2 = Ys2 )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_620_append__eq__append__conv,axiom,
! [Xs2: list_int,Ys2: list_int,Us: list_int,Vs: list_int] :
( ( ( ( size_size_list_int @ Xs2 )
= ( size_size_list_int @ Ys2 ) )
| ( ( size_size_list_int @ Us )
= ( size_size_list_int @ Vs ) ) )
=> ( ( ( append_int @ Xs2 @ Us )
= ( append_int @ Ys2 @ Vs ) )
= ( ( Xs2 = Ys2 )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_621_zip__Nil,axiom,
! [Ys2: list_P8654944379044127311st_int] :
( ( zip_in2431647089413498532st_int @ nil_int @ Ys2 )
= nil_Pr749310521696113548st_int ) ).
% zip_Nil
thf(fact_622_Nil__eq__zip__iff,axiom,
! [Xs2: list_instr,Ys2: list_instr] :
( ( nil_Pr2203008264400890349_instr
= ( zip_instr_instr @ Xs2 @ Ys2 ) )
= ( ( Xs2 = nil_instr )
| ( Ys2 = nil_instr ) ) ) ).
% Nil_eq_zip_iff
thf(fact_623_Nil__eq__zip__iff,axiom,
! [Xs2: list_instr,Ys2: list_int] :
( ( nil_Pr74588867288800052tr_int
= ( zip_instr_int @ Xs2 @ Ys2 ) )
= ( ( Xs2 = nil_instr )
| ( Ys2 = nil_int ) ) ) ).
% Nil_eq_zip_iff
thf(fact_624_Nil__eq__zip__iff,axiom,
! [Xs2: list_int,Ys2: list_instr] :
( ( nil_Pr6645015701671166882_instr
= ( zip_int_instr @ Xs2 @ Ys2 ) )
= ( ( Xs2 = nil_int )
| ( Ys2 = nil_instr ) ) ) ).
% Nil_eq_zip_iff
thf(fact_625_Nil__eq__zip__iff,axiom,
! [Xs2: list_int,Ys2: list_int] :
( ( nil_Pr2300489316682597567nt_int
= ( zip_int_int @ Xs2 @ Ys2 ) )
= ( ( Xs2 = nil_int )
| ( Ys2 = nil_int ) ) ) ).
% Nil_eq_zip_iff
thf(fact_626_Nil__eq__zip__iff,axiom,
! [Xs2: list_instr,Ys2: list_list_char] :
( ( nil_Pr8046467687620146281t_char
= ( zip_instr_list_char @ Xs2 @ Ys2 ) )
= ( ( Xs2 = nil_instr )
| ( Ys2 = nil_list_char ) ) ) ).
% Nil_eq_zip_iff
thf(fact_627_Nil__eq__zip__iff,axiom,
! [Xs2: list_int,Ys2: list_list_char] :
( ( nil_Pr6831627861760291188t_char
= ( zip_int_list_char @ Xs2 @ Ys2 ) )
= ( ( Xs2 = nil_int )
| ( Ys2 = nil_list_char ) ) ) ).
% Nil_eq_zip_iff
thf(fact_628_Nil__eq__zip__iff,axiom,
! [Xs2: list_list_char,Ys2: list_instr] :
( ( nil_Pr4340628689008376749_instr
= ( zip_list_char_instr @ Xs2 @ Ys2 ) )
= ( ( Xs2 = nil_list_char )
| ( Ys2 = nil_instr ) ) ) ).
% Nil_eq_zip_iff
thf(fact_629_Nil__eq__zip__iff,axiom,
! [Xs2: list_list_char,Ys2: list_int] :
( ( nil_Pr6675667932010920308ar_int
= ( zip_list_char_int @ Xs2 @ Ys2 ) )
= ( ( Xs2 = nil_list_char )
| ( Ys2 = nil_int ) ) ) ).
% Nil_eq_zip_iff
thf(fact_630_Nil__eq__zip__iff,axiom,
! [Xs2: list_list_char,Ys2: list_list_char] :
( ( nil_Pr3678812991771325097t_char
= ( zip_li5382857322596760955t_char @ Xs2 @ Ys2 ) )
= ( ( Xs2 = nil_list_char )
| ( Ys2 = nil_list_char ) ) ) ).
% Nil_eq_zip_iff
thf(fact_631_Nil__eq__zip__iff,axiom,
! [Xs2: list_int,Ys2: list_P8654944379044127311st_int] :
( ( nil_Pr749310521696113548st_int
= ( zip_in2431647089413498532st_int @ Xs2 @ Ys2 ) )
= ( ( Xs2 = nil_int )
| ( Ys2 = nil_Pr9161362577040144207st_int ) ) ) ).
% Nil_eq_zip_iff
thf(fact_632_zip__eq__Nil__iff,axiom,
! [Xs2: list_instr,Ys2: list_instr] :
( ( ( zip_instr_instr @ Xs2 @ Ys2 )
= nil_Pr2203008264400890349_instr )
= ( ( Xs2 = nil_instr )
| ( Ys2 = nil_instr ) ) ) ).
% zip_eq_Nil_iff
thf(fact_633_zip__eq__Nil__iff,axiom,
! [Xs2: list_instr,Ys2: list_int] :
( ( ( zip_instr_int @ Xs2 @ Ys2 )
= nil_Pr74588867288800052tr_int )
= ( ( Xs2 = nil_instr )
| ( Ys2 = nil_int ) ) ) ).
% zip_eq_Nil_iff
thf(fact_634_zip__eq__Nil__iff,axiom,
! [Xs2: list_int,Ys2: list_instr] :
( ( ( zip_int_instr @ Xs2 @ Ys2 )
= nil_Pr6645015701671166882_instr )
= ( ( Xs2 = nil_int )
| ( Ys2 = nil_instr ) ) ) ).
% zip_eq_Nil_iff
thf(fact_635_zip__eq__Nil__iff,axiom,
! [Xs2: list_int,Ys2: list_int] :
( ( ( zip_int_int @ Xs2 @ Ys2 )
= nil_Pr2300489316682597567nt_int )
= ( ( Xs2 = nil_int )
| ( Ys2 = nil_int ) ) ) ).
% zip_eq_Nil_iff
thf(fact_636_zip__eq__Nil__iff,axiom,
! [Xs2: list_instr,Ys2: list_list_char] :
( ( ( zip_instr_list_char @ Xs2 @ Ys2 )
= nil_Pr8046467687620146281t_char )
= ( ( Xs2 = nil_instr )
| ( Ys2 = nil_list_char ) ) ) ).
% zip_eq_Nil_iff
thf(fact_637_zip__eq__Nil__iff,axiom,
! [Xs2: list_int,Ys2: list_list_char] :
( ( ( zip_int_list_char @ Xs2 @ Ys2 )
= nil_Pr6831627861760291188t_char )
= ( ( Xs2 = nil_int )
| ( Ys2 = nil_list_char ) ) ) ).
% zip_eq_Nil_iff
thf(fact_638_zip__eq__Nil__iff,axiom,
! [Xs2: list_list_char,Ys2: list_instr] :
( ( ( zip_list_char_instr @ Xs2 @ Ys2 )
= nil_Pr4340628689008376749_instr )
= ( ( Xs2 = nil_list_char )
| ( Ys2 = nil_instr ) ) ) ).
% zip_eq_Nil_iff
thf(fact_639_zip__eq__Nil__iff,axiom,
! [Xs2: list_list_char,Ys2: list_int] :
( ( ( zip_list_char_int @ Xs2 @ Ys2 )
= nil_Pr6675667932010920308ar_int )
= ( ( Xs2 = nil_list_char )
| ( Ys2 = nil_int ) ) ) ).
% zip_eq_Nil_iff
thf(fact_640_zip__eq__Nil__iff,axiom,
! [Xs2: list_list_char,Ys2: list_list_char] :
( ( ( zip_li5382857322596760955t_char @ Xs2 @ Ys2 )
= nil_Pr3678812991771325097t_char )
= ( ( Xs2 = nil_list_char )
| ( Ys2 = nil_list_char ) ) ) ).
% zip_eq_Nil_iff
thf(fact_641_zip__eq__Nil__iff,axiom,
! [Xs2: list_int,Ys2: list_P8654944379044127311st_int] :
( ( ( zip_in2431647089413498532st_int @ Xs2 @ Ys2 )
= nil_Pr749310521696113548st_int )
= ( ( Xs2 = nil_int )
| ( Ys2 = nil_Pr9161362577040144207st_int ) ) ) ).
% zip_eq_Nil_iff
thf(fact_642_enumerate__simps_I1_J,axiom,
! [N: nat] :
( ( enumerate_instr @ N @ nil_instr )
= nil_Pr5958439668630616318_instr ) ).
% enumerate_simps(1)
thf(fact_643_enumerate__simps_I1_J,axiom,
! [N: nat] :
( ( enumer1800257932645307117st_int @ N @ nil_Pr749310521696113548st_int )
= nil_Pr215578427632615497st_int ) ).
% enumerate_simps(1)
thf(fact_644_enumerate__simps_I1_J,axiom,
! [N: nat] :
( ( enumerate_int @ N @ nil_int )
= nil_Pr1301135604781543011at_int ) ).
% enumerate_simps(1)
thf(fact_645_enumerate__simps_I1_J,axiom,
! [N: nat] :
( ( enumerate_list_char @ N @ nil_list_char )
= nil_Pr4704261131758228760t_char ) ).
% enumerate_simps(1)
thf(fact_646_length__enumerate,axiom,
! [N: nat,Xs2: list_instr] :
( ( size_s142077898034696872_instr @ ( enumerate_instr @ N @ Xs2 ) )
= ( size_size_list_instr @ Xs2 ) ) ).
% length_enumerate
thf(fact_647_length__enumerate,axiom,
! [N: nat,Xs2: list_list_char] :
( ( size_s3402568042334145540t_char @ ( enumerate_list_char @ N @ Xs2 ) )
= ( size_s356637359517785349t_char @ Xs2 ) ) ).
% length_enumerate
thf(fact_648_length__enumerate,axiom,
! [N: nat,Xs2: list_int] :
( ( size_s2970893825323803983at_int @ ( enumerate_int @ N @ Xs2 ) )
= ( size_size_list_int @ Xs2 ) ) ).
% length_enumerate
thf(fact_649_less__neg__neg,axiom,
! [A: int] :
( ( ord_less_int @ A @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% less_neg_neg
thf(fact_650_neg__less__pos,axiom,
! [A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ A )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% neg_less_pos
thf(fact_651_neg__0__less__iff__less,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% neg_0_less_iff_less
thf(fact_652_neg__less__0__iff__less,axiom,
! [A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% neg_less_0_iff_less
thf(fact_653_ab__left__minus,axiom,
! [A: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
= zero_zero_int ) ).
% ab_left_minus
thf(fact_654_add_Oright__inverse,axiom,
! [A: int] :
( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
= zero_zero_int ) ).
% add.right_inverse
thf(fact_655_verit__minus__simplify_I3_J,axiom,
! [B: int] :
( ( minus_minus_int @ zero_zero_int @ B )
= ( uminus_uminus_int @ B ) ) ).
% verit_minus_simplify(3)
thf(fact_656_diff__0,axiom,
! [A: int] :
( ( minus_minus_int @ zero_zero_int @ A )
= ( uminus_uminus_int @ A ) ) ).
% diff_0
thf(fact_657_diff__minus__eq__add,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ A @ ( uminus_uminus_int @ B ) )
= ( plus_plus_int @ A @ B ) ) ).
% diff_minus_eq_add
thf(fact_658_uminus__add__conv__diff,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B )
= ( minus_minus_int @ B @ A ) ) ).
% uminus_add_conv_diff
thf(fact_659_length__0__conv,axiom,
! [Xs2: list_P419547167332244514st_int] :
( ( ( size_s229347708945616566st_int @ Xs2 )
= zero_zero_nat )
= ( Xs2 = nil_Pr749310521696113548st_int ) ) ).
% length_0_conv
thf(fact_660_length__0__conv,axiom,
! [Xs2: list_instr] :
( ( ( size_size_list_instr @ Xs2 )
= zero_zero_nat )
= ( Xs2 = nil_instr ) ) ).
% length_0_conv
thf(fact_661_length__0__conv,axiom,
! [Xs2: list_list_char] :
( ( ( size_s356637359517785349t_char @ Xs2 )
= zero_zero_nat )
= ( Xs2 = nil_list_char ) ) ).
% length_0_conv
thf(fact_662_length__0__conv,axiom,
! [Xs2: list_int] :
( ( ( size_size_list_int @ Xs2 )
= zero_zero_nat )
= ( Xs2 = nil_int ) ) ).
% length_0_conv
thf(fact_663_append1__eq__conv,axiom,
! [Xs2: list_int,X: int,Ys2: list_int,Y: int] :
( ( ( append_int @ Xs2 @ ( cons_int @ X @ nil_int ) )
= ( append_int @ Ys2 @ ( cons_int @ Y @ nil_int ) ) )
= ( ( Xs2 = Ys2 )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_664_append1__eq__conv,axiom,
! [Xs2: list_instr,X: instr,Ys2: list_instr,Y: instr] :
( ( ( append_instr @ Xs2 @ ( cons_instr @ X @ nil_instr ) )
= ( append_instr @ Ys2 @ ( cons_instr @ Y @ nil_instr ) ) )
= ( ( Xs2 = Ys2 )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_665_append1__eq__conv,axiom,
! [Xs2: list_P419547167332244514st_int,X: produc6425607678544837394st_int,Ys2: list_P419547167332244514st_int,Y: produc6425607678544837394st_int] :
( ( ( append5028292981072010935st_int @ Xs2 @ ( cons_P4151473878978182108st_int @ X @ nil_Pr749310521696113548st_int ) )
= ( append5028292981072010935st_int @ Ys2 @ ( cons_P4151473878978182108st_int @ Y @ nil_Pr749310521696113548st_int ) ) )
= ( ( Xs2 = Ys2 )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_666_append1__eq__conv,axiom,
! [Xs2: list_list_char,X: list_char,Ys2: list_list_char,Y: list_char] :
( ( ( append_list_char @ Xs2 @ ( cons_list_char @ X @ nil_list_char ) )
= ( append_list_char @ Ys2 @ ( cons_list_char @ Y @ nil_list_char ) ) )
= ( ( Xs2 = Ys2 )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_667_length__append,axiom,
! [Xs2: list_instr,Ys2: list_instr] :
( ( size_size_list_instr @ ( append_instr @ Xs2 @ Ys2 ) )
= ( plus_plus_nat @ ( size_size_list_instr @ Xs2 ) @ ( size_size_list_instr @ Ys2 ) ) ) ).
% length_append
thf(fact_668_length__append,axiom,
! [Xs2: list_list_char,Ys2: list_list_char] :
( ( size_s356637359517785349t_char @ ( append_list_char @ Xs2 @ Ys2 ) )
= ( plus_plus_nat @ ( size_s356637359517785349t_char @ Xs2 ) @ ( size_s356637359517785349t_char @ Ys2 ) ) ) ).
% length_append
thf(fact_669_length__append,axiom,
! [Xs2: list_int,Ys2: list_int] :
( ( size_size_list_int @ ( append_int @ Xs2 @ Ys2 ) )
= ( plus_plus_nat @ ( size_size_list_int @ Xs2 ) @ ( size_size_list_int @ Ys2 ) ) ) ).
% length_append
thf(fact_670_negative__eq__positive,axiom,
! [N: nat,M: nat] :
( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ M ) )
= ( ( N = zero_zero_nat )
& ( M = zero_zero_nat ) ) ) ).
% negative_eq_positive
thf(fact_671_zip__append,axiom,
! [Xs2: list_instr,Us: list_instr,Ys2: list_instr,Vs: list_instr] :
( ( ( size_size_list_instr @ Xs2 )
= ( size_size_list_instr @ Us ) )
=> ( ( zip_instr_instr @ ( append_instr @ Xs2 @ Ys2 ) @ ( append_instr @ Us @ Vs ) )
= ( append3578712275673033858_instr @ ( zip_instr_instr @ Xs2 @ Us ) @ ( zip_instr_instr @ Ys2 @ Vs ) ) ) ) ).
% zip_append
thf(fact_672_zip__append,axiom,
! [Xs2: list_instr,Us: list_list_char,Ys2: list_instr,Vs: list_list_char] :
( ( ( size_size_list_instr @ Xs2 )
= ( size_s356637359517785349t_char @ Us ) )
=> ( ( zip_instr_list_char @ ( append_instr @ Xs2 @ Ys2 ) @ ( append_list_char @ Us @ Vs ) )
= ( append3243301028656582804t_char @ ( zip_instr_list_char @ Xs2 @ Us ) @ ( zip_instr_list_char @ Ys2 @ Vs ) ) ) ) ).
% zip_append
thf(fact_673_zip__append,axiom,
! [Xs2: list_instr,Us: list_int,Ys2: list_instr,Vs: list_int] :
( ( ( size_size_list_instr @ Xs2 )
= ( size_size_list_int @ Us ) )
=> ( ( zip_instr_int @ ( append_instr @ Xs2 @ Ys2 ) @ ( append_int @ Us @ Vs ) )
= ( append3567499337439220703tr_int @ ( zip_instr_int @ Xs2 @ Us ) @ ( zip_instr_int @ Ys2 @ Vs ) ) ) ) ).
% zip_append
thf(fact_674_zip__append,axiom,
! [Xs2: list_list_char,Us: list_instr,Ys2: list_list_char,Vs: list_instr] :
( ( ( size_s356637359517785349t_char @ Xs2 )
= ( size_size_list_instr @ Us ) )
=> ( ( zip_list_char_instr @ ( append_list_char @ Xs2 @ Ys2 ) @ ( append_instr @ Us @ Vs ) )
= ( append8760834066899589080_instr @ ( zip_list_char_instr @ Xs2 @ Us ) @ ( zip_list_char_instr @ Ys2 @ Vs ) ) ) ) ).
% zip_append
thf(fact_675_zip__append,axiom,
! [Xs2: list_list_char,Us: list_list_char,Ys2: list_list_char,Vs: list_list_char] :
( ( ( size_s356637359517785349t_char @ Xs2 )
= ( size_s356637359517785349t_char @ Us ) )
=> ( ( zip_li5382857322596760955t_char @ ( append_list_char @ Xs2 @ Ys2 ) @ ( append_list_char @ Us @ Vs ) )
= ( append655500003500722110t_char @ ( zip_li5382857322596760955t_char @ Xs2 @ Us ) @ ( zip_li5382857322596760955t_char @ Ys2 @ Vs ) ) ) ) ).
% zip_append
thf(fact_676_zip__append,axiom,
! [Xs2: list_list_char,Us: list_int,Ys2: list_list_char,Vs: list_int] :
( ( ( size_s356637359517785349t_char @ Xs2 )
= ( size_size_list_int @ Us ) )
=> ( ( zip_list_char_int @ ( append_list_char @ Xs2 @ Ys2 ) @ ( append_int @ Us @ Vs ) )
= ( append4109703874471170569ar_int @ ( zip_list_char_int @ Xs2 @ Us ) @ ( zip_list_char_int @ Ys2 @ Vs ) ) ) ) ).
% zip_append
thf(fact_677_zip__append,axiom,
! [Xs2: list_int,Us: list_instr,Ys2: list_int,Vs: list_instr] :
( ( ( size_size_list_int @ Xs2 )
= ( size_size_list_instr @ Us ) )
=> ( ( zip_int_instr @ ( append_int @ Xs2 @ Ys2 ) @ ( append_instr @ Us @ Vs ) )
= ( append914554134966811725_instr @ ( zip_int_instr @ Xs2 @ Us ) @ ( zip_int_instr @ Ys2 @ Vs ) ) ) ) ).
% zip_append
thf(fact_678_zip__append,axiom,
! [Xs2: list_int,Us: list_list_char,Ys2: list_int,Vs: list_list_char] :
( ( ( size_size_list_int @ Xs2 )
= ( size_s356637359517785349t_char @ Us ) )
=> ( ( zip_int_list_char @ ( append_int @ Xs2 @ Ys2 ) @ ( append_list_char @ Us @ Vs ) )
= ( append4265663804220541449t_char @ ( zip_int_list_char @ Xs2 @ Us ) @ ( zip_int_list_char @ Ys2 @ Vs ) ) ) ) ).
% zip_append
thf(fact_679_zip__append,axiom,
! [Xs2: list_int,Us: list_int,Ys2: list_int,Vs: list_int] :
( ( ( size_size_list_int @ Xs2 )
= ( size_size_list_int @ Us ) )
=> ( ( zip_int_int @ ( append_int @ Xs2 @ Ys2 ) @ ( append_int @ Us @ Vs ) )
= ( append7030698103840186580nt_int @ ( zip_int_int @ Xs2 @ Us ) @ ( zip_int_int @ Ys2 @ Vs ) ) ) ) ).
% zip_append
thf(fact_680_dbl__inc__simps_I4_J,axiom,
( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% dbl_inc_simps(4)
thf(fact_681_add__neg__numeral__special_I8_J,axiom,
( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
= zero_zero_int ) ).
% add_neg_numeral_special(8)
thf(fact_682_add__neg__numeral__special_I7_J,axiom,
( ( plus_plus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
= zero_zero_int ) ).
% add_neg_numeral_special(7)
thf(fact_683_diff__numeral__special_I12_J,axiom,
( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
= zero_zero_int ) ).
% diff_numeral_special(12)
thf(fact_684_length__greater__0__conv,axiom,
! [Xs2: list_P419547167332244514st_int] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s229347708945616566st_int @ Xs2 ) )
= ( Xs2 != nil_Pr749310521696113548st_int ) ) ).
% length_greater_0_conv
thf(fact_685_length__greater__0__conv,axiom,
! [Xs2: list_instr] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_instr @ Xs2 ) )
= ( Xs2 != nil_instr ) ) ).
% length_greater_0_conv
thf(fact_686_length__greater__0__conv,axiom,
! [Xs2: list_list_char] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s356637359517785349t_char @ Xs2 ) )
= ( Xs2 != nil_list_char ) ) ).
% length_greater_0_conv
thf(fact_687_length__greater__0__conv,axiom,
! [Xs2: list_int] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_int @ Xs2 ) )
= ( Xs2 != nil_int ) ) ).
% length_greater_0_conv
thf(fact_688_nth__append__length,axiom,
! [Xs2: list_P419547167332244514st_int,X: produc6425607678544837394st_int,Ys2: list_P419547167332244514st_int] :
( ( nth_Pr8327173284404383691st_int @ ( append5028292981072010935st_int @ Xs2 @ ( cons_P4151473878978182108st_int @ X @ Ys2 ) ) @ ( size_s229347708945616566st_int @ Xs2 ) )
= X ) ).
% nth_append_length
thf(fact_689_nth__append__length,axiom,
! [Xs2: list_instr,X: instr,Ys2: list_instr] :
( ( nth_instr @ ( append_instr @ Xs2 @ ( cons_instr @ X @ Ys2 ) ) @ ( size_size_list_instr @ Xs2 ) )
= X ) ).
% nth_append_length
thf(fact_690_nth__append__length,axiom,
! [Xs2: list_list_char,X: list_char,Ys2: list_list_char] :
( ( nth_list_char @ ( append_list_char @ Xs2 @ ( cons_list_char @ X @ Ys2 ) ) @ ( size_s356637359517785349t_char @ Xs2 ) )
= X ) ).
% nth_append_length
thf(fact_691_nth__append__length,axiom,
! [Xs2: list_int,X: int,Ys2: list_int] :
( ( nth_int @ ( append_int @ Xs2 @ ( cons_int @ X @ Ys2 ) ) @ ( size_size_list_int @ Xs2 ) )
= X ) ).
% nth_append_length
thf(fact_692_nth__append__length__plus,axiom,
! [Xs2: list_P419547167332244514st_int,Ys2: list_P419547167332244514st_int,N: nat] :
( ( nth_Pr8327173284404383691st_int @ ( append5028292981072010935st_int @ Xs2 @ Ys2 ) @ ( plus_plus_nat @ ( size_s229347708945616566st_int @ Xs2 ) @ N ) )
= ( nth_Pr8327173284404383691st_int @ Ys2 @ N ) ) ).
% nth_append_length_plus
thf(fact_693_nth__append__length__plus,axiom,
! [Xs2: list_instr,Ys2: list_instr,N: nat] :
( ( nth_instr @ ( append_instr @ Xs2 @ Ys2 ) @ ( plus_plus_nat @ ( size_size_list_instr @ Xs2 ) @ N ) )
= ( nth_instr @ Ys2 @ N ) ) ).
% nth_append_length_plus
thf(fact_694_nth__append__length__plus,axiom,
! [Xs2: list_list_char,Ys2: list_list_char,N: nat] :
( ( nth_list_char @ ( append_list_char @ Xs2 @ Ys2 ) @ ( plus_plus_nat @ ( size_s356637359517785349t_char @ Xs2 ) @ N ) )
= ( nth_list_char @ Ys2 @ N ) ) ).
% nth_append_length_plus
thf(fact_695_nth__append__length__plus,axiom,
! [Xs2: list_int,Ys2: list_int,N: nat] :
( ( nth_int @ ( append_int @ Xs2 @ Ys2 ) @ ( plus_plus_nat @ ( size_size_list_int @ Xs2 ) @ N ) )
= ( nth_int @ Ys2 @ N ) ) ).
% nth_append_length_plus
thf(fact_696_negative__zless,axiom,
! [N: nat,M: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% negative_zless
thf(fact_697_drop__append,axiom,
! [N: nat,Xs2: list_P419547167332244514st_int,Ys2: list_P419547167332244514st_int] :
( ( drop_P7838799151754935884st_int @ N @ ( append5028292981072010935st_int @ Xs2 @ Ys2 ) )
= ( append5028292981072010935st_int @ ( drop_P7838799151754935884st_int @ N @ Xs2 ) @ ( drop_P7838799151754935884st_int @ ( minus_minus_nat @ N @ ( size_s229347708945616566st_int @ Xs2 ) ) @ Ys2 ) ) ) ).
% drop_append
thf(fact_698_drop__append,axiom,
! [N: nat,Xs2: list_instr,Ys2: list_instr] :
( ( drop_instr @ N @ ( append_instr @ Xs2 @ Ys2 ) )
= ( append_instr @ ( drop_instr @ N @ Xs2 ) @ ( drop_instr @ ( minus_minus_nat @ N @ ( size_size_list_instr @ Xs2 ) ) @ Ys2 ) ) ) ).
% drop_append
thf(fact_699_drop__append,axiom,
! [N: nat,Xs2: list_list_char,Ys2: list_list_char] :
( ( drop_list_char @ N @ ( append_list_char @ Xs2 @ Ys2 ) )
= ( append_list_char @ ( drop_list_char @ N @ Xs2 ) @ ( drop_list_char @ ( minus_minus_nat @ N @ ( size_s356637359517785349t_char @ Xs2 ) ) @ Ys2 ) ) ) ).
% drop_append
thf(fact_700_drop__append,axiom,
! [N: nat,Xs2: list_int,Ys2: list_int] :
( ( drop_int @ N @ ( append_int @ Xs2 @ Ys2 ) )
= ( append_int @ ( drop_int @ N @ Xs2 ) @ ( drop_int @ ( minus_minus_nat @ N @ ( size_size_list_int @ Xs2 ) ) @ Ys2 ) ) ) ).
% drop_append
thf(fact_701_enumerate__simps_I2_J,axiom,
! [N: nat,X: int,Xs2: list_int] :
( ( enumerate_int @ N @ ( cons_int @ X @ Xs2 ) )
= ( cons_P2335045147070616083at_int @ ( product_Pair_nat_int @ N @ X ) @ ( enumerate_int @ ( suc @ N ) @ Xs2 ) ) ) ).
% enumerate_simps(2)
thf(fact_702_enumerate__simps_I2_J,axiom,
! [N: nat,X: instr,Xs2: list_instr] :
( ( enumerate_instr @ N @ ( cons_instr @ X @ Xs2 ) )
= ( cons_P4643890853483612494_instr @ ( produc3129998205563682742_instr @ N @ X ) @ ( enumerate_instr @ ( suc @ N ) @ Xs2 ) ) ) ).
% enumerate_simps(2)
thf(fact_703_enumerate__simps_I2_J,axiom,
! [N: nat,X: produc6425607678544837394st_int,Xs2: list_P419547167332244514st_int] :
( ( enumer1800257932645307117st_int @ N @ ( cons_P4151473878978182108st_int @ X @ Xs2 ) )
= ( cons_P6123078848854619129st_int @ ( produc2664891610604065915st_int @ N @ X ) @ ( enumer1800257932645307117st_int @ ( suc @ N ) @ Xs2 ) ) ) ).
% enumerate_simps(2)
thf(fact_704_enumerate__simps_I2_J,axiom,
! [N: nat,X: list_char,Xs2: list_list_char] :
( ( enumerate_list_char @ N @ ( cons_list_char @ X @ Xs2 ) )
= ( cons_P6700496888814104776t_char @ ( produc120197956887798346t_char @ N @ X ) @ ( enumerate_list_char @ ( suc @ N ) @ Xs2 ) ) ) ).
% enumerate_simps(2)
thf(fact_705_zip_Osimps_I1_J,axiom,
! [Xs2: list_int] :
( ( zip_in2431647089413498532st_int @ Xs2 @ nil_Pr9161362577040144207st_int )
= nil_Pr749310521696113548st_int ) ).
% zip.simps(1)
thf(fact_706_append__eq__append__conv2,axiom,
! [Xs2: list_instr,Ys2: list_instr,Zs: list_instr,Ts: list_instr] :
( ( ( append_instr @ Xs2 @ Ys2 )
= ( append_instr @ Zs @ Ts ) )
= ( ? [Us2: list_instr] :
( ( ( Xs2
= ( append_instr @ Zs @ Us2 ) )
& ( ( append_instr @ Us2 @ Ys2 )
= Ts ) )
| ( ( ( append_instr @ Xs2 @ Us2 )
= Zs )
& ( Ys2
= ( append_instr @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_707_append__eq__append__conv2,axiom,
! [Xs2: list_list_char,Ys2: list_list_char,Zs: list_list_char,Ts: list_list_char] :
( ( ( append_list_char @ Xs2 @ Ys2 )
= ( append_list_char @ Zs @ Ts ) )
= ( ? [Us2: list_list_char] :
( ( ( Xs2
= ( append_list_char @ Zs @ Us2 ) )
& ( ( append_list_char @ Us2 @ Ys2 )
= Ts ) )
| ( ( ( append_list_char @ Xs2 @ Us2 )
= Zs )
& ( Ys2
= ( append_list_char @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_708_append__eq__appendI,axiom,
! [Xs2: list_instr,Xs1: list_instr,Zs: list_instr,Ys2: list_instr,Us: list_instr] :
( ( ( append_instr @ Xs2 @ Xs1 )
= Zs )
=> ( ( Ys2
= ( append_instr @ Xs1 @ Us ) )
=> ( ( append_instr @ Xs2 @ Ys2 )
= ( append_instr @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_709_append__eq__appendI,axiom,
! [Xs2: list_list_char,Xs1: list_list_char,Zs: list_list_char,Ys2: list_list_char,Us: list_list_char] :
( ( ( append_list_char @ Xs2 @ Xs1 )
= Zs )
=> ( ( Ys2
= ( append_list_char @ Xs1 @ Us ) )
=> ( ( append_list_char @ Xs2 @ Ys2 )
= ( append_list_char @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_710_eq__Nil__appendI,axiom,
! [Xs2: list_instr,Ys2: list_instr] :
( ( Xs2 = Ys2 )
=> ( Xs2
= ( append_instr @ nil_instr @ Ys2 ) ) ) ).
% eq_Nil_appendI
thf(fact_711_eq__Nil__appendI,axiom,
! [Xs2: list_P419547167332244514st_int,Ys2: list_P419547167332244514st_int] :
( ( Xs2 = Ys2 )
=> ( Xs2
= ( append5028292981072010935st_int @ nil_Pr749310521696113548st_int @ Ys2 ) ) ) ).
% eq_Nil_appendI
thf(fact_712_eq__Nil__appendI,axiom,
! [Xs2: list_int,Ys2: list_int] :
( ( Xs2 = Ys2 )
=> ( Xs2
= ( append_int @ nil_int @ Ys2 ) ) ) ).
% eq_Nil_appendI
thf(fact_713_eq__Nil__appendI,axiom,
! [Xs2: list_list_char,Ys2: list_list_char] :
( ( Xs2 = Ys2 )
=> ( Xs2
= ( append_list_char @ nil_list_char @ Ys2 ) ) ) ).
% eq_Nil_appendI
thf(fact_714_append_Oleft__neutral,axiom,
! [A: list_instr] :
( ( append_instr @ nil_instr @ A )
= A ) ).
% append.left_neutral
thf(fact_715_append_Oleft__neutral,axiom,
! [A: list_P419547167332244514st_int] :
( ( append5028292981072010935st_int @ nil_Pr749310521696113548st_int @ A )
= A ) ).
% append.left_neutral
thf(fact_716_append_Oleft__neutral,axiom,
! [A: list_int] :
( ( append_int @ nil_int @ A )
= A ) ).
% append.left_neutral
thf(fact_717_append_Oleft__neutral,axiom,
! [A: list_list_char] :
( ( append_list_char @ nil_list_char @ A )
= A ) ).
% append.left_neutral
thf(fact_718_append__Nil,axiom,
! [Ys2: list_instr] :
( ( append_instr @ nil_instr @ Ys2 )
= Ys2 ) ).
% append_Nil
thf(fact_719_append__Nil,axiom,
! [Ys2: list_P419547167332244514st_int] :
( ( append5028292981072010935st_int @ nil_Pr749310521696113548st_int @ Ys2 )
= Ys2 ) ).
% append_Nil
thf(fact_720_append__Nil,axiom,
! [Ys2: list_int] :
( ( append_int @ nil_int @ Ys2 )
= Ys2 ) ).
% append_Nil
thf(fact_721_append__Nil,axiom,
! [Ys2: list_list_char] :
( ( append_list_char @ nil_list_char @ Ys2 )
= Ys2 ) ).
% append_Nil
thf(fact_722_minus__equation__iff,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= B )
= ( ( uminus_uminus_int @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_723_equation__minus__iff,axiom,
! [A: int,B: int] :
( ( A
= ( uminus_uminus_int @ B ) )
= ( B
= ( uminus_uminus_int @ A ) ) ) ).
% equation_minus_iff
thf(fact_724_ccomp_Osimps_I3_J,axiom,
! [C_1: com,C_2: com] :
( ( ccomp @ ( seq @ C_1 @ C_2 ) )
= ( append_instr @ ( ccomp @ C_1 ) @ ( ccomp @ C_2 ) ) ) ).
% ccomp.simps(3)
thf(fact_725_verit__negate__coefficient_I3_J,axiom,
! [A: int,B: int] :
( ( A = B )
=> ( ( uminus_uminus_int @ A )
= ( uminus_uminus_int @ B ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_726_enumerate__append__eq,axiom,
! [N: nat,Xs2: list_instr,Ys2: list_instr] :
( ( enumerate_instr @ N @ ( append_instr @ Xs2 @ Ys2 ) )
= ( append227978101926261161_instr @ ( enumerate_instr @ N @ Xs2 ) @ ( enumerate_instr @ ( plus_plus_nat @ N @ ( size_size_list_instr @ Xs2 ) ) @ Ys2 ) ) ) ).
% enumerate_append_eq
thf(fact_727_enumerate__append__eq,axiom,
! [N: nat,Xs2: list_list_char,Ys2: list_list_char] :
( ( enumerate_list_char @ N @ ( append_list_char @ Xs2 @ Ys2 ) )
= ( append2138297074218479021t_char @ ( enumerate_list_char @ N @ Xs2 ) @ ( enumerate_list_char @ ( plus_plus_nat @ N @ ( size_s356637359517785349t_char @ Xs2 ) ) @ Ys2 ) ) ) ).
% enumerate_append_eq
thf(fact_728_enumerate__append__eq,axiom,
! [N: nat,Xs2: list_int,Ys2: list_int] :
( ( enumerate_int @ N @ ( append_int @ Xs2 @ Ys2 ) )
= ( append6031344391939132024at_int @ ( enumerate_int @ N @ Xs2 ) @ ( enumerate_int @ ( plus_plus_nat @ N @ ( size_size_list_int @ Xs2 ) ) @ Ys2 ) ) ) ).
% enumerate_append_eq
thf(fact_729_transpose_Ocases,axiom,
! [X: list_list_int] :
( ( X != nil_list_int )
=> ( ! [Xss: list_list_int] :
( X
!= ( cons_list_int @ nil_int @ Xss ) )
=> ~ ! [X4: int,Xs3: list_int,Xss: list_list_int] :
( X
!= ( cons_list_int @ ( cons_int @ X4 @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_730_transpose_Ocases,axiom,
! [X: list_list_instr] :
( ( X != nil_list_instr )
=> ( ! [Xss: list_list_instr] :
( X
!= ( cons_list_instr @ nil_instr @ Xss ) )
=> ~ ! [X4: instr,Xs3: list_instr,Xss: list_list_instr] :
( X
!= ( cons_list_instr @ ( cons_instr @ X4 @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_731_transpose_Ocases,axiom,
! [X: list_l6422821129637969586st_int] :
( ( X != nil_li4597314861645524892st_int )
=> ( ! [Xss: list_l6422821129637969586st_int] :
( X
!= ( cons_l8648290059330449388st_int @ nil_Pr749310521696113548st_int @ Xss ) )
=> ~ ! [X4: produc6425607678544837394st_int,Xs3: list_P419547167332244514st_int,Xss: list_l6422821129637969586st_int] :
( X
!= ( cons_l8648290059330449388st_int @ ( cons_P4151473878978182108st_int @ X4 @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_732_transpose_Ocases,axiom,
! [X: list_list_list_char] :
( ( X != nil_list_list_char )
=> ( ! [Xss: list_list_list_char] :
( X
!= ( cons_list_list_char @ nil_list_char @ Xss ) )
=> ~ ! [X4: list_char,Xs3: list_list_char,Xss: list_list_list_char] :
( X
!= ( cons_list_list_char @ ( cons_list_char @ X4 @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_733_transpose_Ocases,axiom,
! [X: list_list_char] :
( ( X != nil_list_char )
=> ( ! [Xss: list_list_char] :
( X
!= ( cons_list_char @ nil_char @ Xss ) )
=> ~ ! [X4: char,Xs3: list_char,Xss: list_list_char] :
( X
!= ( cons_list_char @ ( cons_char @ X4 @ Xs3 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_734_rev__induct,axiom,
! [P: list_int > $o,Xs2: list_int] :
( ( P @ nil_int )
=> ( ! [X4: int,Xs3: list_int] :
( ( P @ Xs3 )
=> ( P @ ( append_int @ Xs3 @ ( cons_int @ X4 @ nil_int ) ) ) )
=> ( P @ Xs2 ) ) ) ).
% rev_induct
thf(fact_735_rev__induct,axiom,
! [P: list_instr > $o,Xs2: list_instr] :
( ( P @ nil_instr )
=> ( ! [X4: instr,Xs3: list_instr] :
( ( P @ Xs3 )
=> ( P @ ( append_instr @ Xs3 @ ( cons_instr @ X4 @ nil_instr ) ) ) )
=> ( P @ Xs2 ) ) ) ).
% rev_induct
thf(fact_736_rev__induct,axiom,
! [P: list_P419547167332244514st_int > $o,Xs2: list_P419547167332244514st_int] :
( ( P @ nil_Pr749310521696113548st_int )
=> ( ! [X4: produc6425607678544837394st_int,Xs3: list_P419547167332244514st_int] :
( ( P @ Xs3 )
=> ( P @ ( append5028292981072010935st_int @ Xs3 @ ( cons_P4151473878978182108st_int @ X4 @ nil_Pr749310521696113548st_int ) ) ) )
=> ( P @ Xs2 ) ) ) ).
% rev_induct
thf(fact_737_rev__induct,axiom,
! [P: list_list_char > $o,Xs2: list_list_char] :
( ( P @ nil_list_char )
=> ( ! [X4: list_char,Xs3: list_list_char] :
( ( P @ Xs3 )
=> ( P @ ( append_list_char @ Xs3 @ ( cons_list_char @ X4 @ nil_list_char ) ) ) )
=> ( P @ Xs2 ) ) ) ).
% rev_induct
thf(fact_738_rev__exhaust,axiom,
! [Xs2: list_int] :
( ( Xs2 != nil_int )
=> ~ ! [Ys5: list_int,Y4: int] :
( Xs2
!= ( append_int @ Ys5 @ ( cons_int @ Y4 @ nil_int ) ) ) ) ).
% rev_exhaust
thf(fact_739_rev__exhaust,axiom,
! [Xs2: list_instr] :
( ( Xs2 != nil_instr )
=> ~ ! [Ys5: list_instr,Y4: instr] :
( Xs2
!= ( append_instr @ Ys5 @ ( cons_instr @ Y4 @ nil_instr ) ) ) ) ).
% rev_exhaust
thf(fact_740_rev__exhaust,axiom,
! [Xs2: list_P419547167332244514st_int] :
( ( Xs2 != nil_Pr749310521696113548st_int )
=> ~ ! [Ys5: list_P419547167332244514st_int,Y4: produc6425607678544837394st_int] :
( Xs2
!= ( append5028292981072010935st_int @ Ys5 @ ( cons_P4151473878978182108st_int @ Y4 @ nil_Pr749310521696113548st_int ) ) ) ) ).
% rev_exhaust
thf(fact_741_rev__exhaust,axiom,
! [Xs2: list_list_char] :
( ( Xs2 != nil_list_char )
=> ~ ! [Ys5: list_list_char,Y4: list_char] :
( Xs2
!= ( append_list_char @ Ys5 @ ( cons_list_char @ Y4 @ nil_list_char ) ) ) ) ).
% rev_exhaust
thf(fact_742_Cons__eq__append__conv,axiom,
! [X: int,Xs2: list_int,Ys2: list_int,Zs: list_int] :
( ( ( cons_int @ X @ Xs2 )
= ( append_int @ Ys2 @ Zs ) )
= ( ( ( Ys2 = nil_int )
& ( ( cons_int @ X @ Xs2 )
= Zs ) )
| ? [Ys6: list_int] :
( ( ( cons_int @ X @ Ys6 )
= Ys2 )
& ( Xs2
= ( append_int @ Ys6 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_743_Cons__eq__append__conv,axiom,
! [X: instr,Xs2: list_instr,Ys2: list_instr,Zs: list_instr] :
( ( ( cons_instr @ X @ Xs2 )
= ( append_instr @ Ys2 @ Zs ) )
= ( ( ( Ys2 = nil_instr )
& ( ( cons_instr @ X @ Xs2 )
= Zs ) )
| ? [Ys6: list_instr] :
( ( ( cons_instr @ X @ Ys6 )
= Ys2 )
& ( Xs2
= ( append_instr @ Ys6 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_744_Cons__eq__append__conv,axiom,
! [X: produc6425607678544837394st_int,Xs2: list_P419547167332244514st_int,Ys2: list_P419547167332244514st_int,Zs: list_P419547167332244514st_int] :
( ( ( cons_P4151473878978182108st_int @ X @ Xs2 )
= ( append5028292981072010935st_int @ Ys2 @ Zs ) )
= ( ( ( Ys2 = nil_Pr749310521696113548st_int )
& ( ( cons_P4151473878978182108st_int @ X @ Xs2 )
= Zs ) )
| ? [Ys6: list_P419547167332244514st_int] :
( ( ( cons_P4151473878978182108st_int @ X @ Ys6 )
= Ys2 )
& ( Xs2
= ( append5028292981072010935st_int @ Ys6 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_745_Cons__eq__append__conv,axiom,
! [X: list_char,Xs2: list_list_char,Ys2: list_list_char,Zs: list_list_char] :
( ( ( cons_list_char @ X @ Xs2 )
= ( append_list_char @ Ys2 @ Zs ) )
= ( ( ( Ys2 = nil_list_char )
& ( ( cons_list_char @ X @ Xs2 )
= Zs ) )
| ? [Ys6: list_list_char] :
( ( ( cons_list_char @ X @ Ys6 )
= Ys2 )
& ( Xs2
= ( append_list_char @ Ys6 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_746_append__eq__Cons__conv,axiom,
! [Ys2: list_int,Zs: list_int,X: int,Xs2: list_int] :
( ( ( append_int @ Ys2 @ Zs )
= ( cons_int @ X @ Xs2 ) )
= ( ( ( Ys2 = nil_int )
& ( Zs
= ( cons_int @ X @ Xs2 ) ) )
| ? [Ys6: list_int] :
( ( Ys2
= ( cons_int @ X @ Ys6 ) )
& ( ( append_int @ Ys6 @ Zs )
= Xs2 ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_747_append__eq__Cons__conv,axiom,
! [Ys2: list_instr,Zs: list_instr,X: instr,Xs2: list_instr] :
( ( ( append_instr @ Ys2 @ Zs )
= ( cons_instr @ X @ Xs2 ) )
= ( ( ( Ys2 = nil_instr )
& ( Zs
= ( cons_instr @ X @ Xs2 ) ) )
| ? [Ys6: list_instr] :
( ( Ys2
= ( cons_instr @ X @ Ys6 ) )
& ( ( append_instr @ Ys6 @ Zs )
= Xs2 ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_748_append__eq__Cons__conv,axiom,
! [Ys2: list_P419547167332244514st_int,Zs: list_P419547167332244514st_int,X: produc6425607678544837394st_int,Xs2: list_P419547167332244514st_int] :
( ( ( append5028292981072010935st_int @ Ys2 @ Zs )
= ( cons_P4151473878978182108st_int @ X @ Xs2 ) )
= ( ( ( Ys2 = nil_Pr749310521696113548st_int )
& ( Zs
= ( cons_P4151473878978182108st_int @ X @ Xs2 ) ) )
| ? [Ys6: list_P419547167332244514st_int] :
( ( Ys2
= ( cons_P4151473878978182108st_int @ X @ Ys6 ) )
& ( ( append5028292981072010935st_int @ Ys6 @ Zs )
= Xs2 ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_749_append__eq__Cons__conv,axiom,
! [Ys2: list_list_char,Zs: list_list_char,X: list_char,Xs2: list_list_char] :
( ( ( append_list_char @ Ys2 @ Zs )
= ( cons_list_char @ X @ Xs2 ) )
= ( ( ( Ys2 = nil_list_char )
& ( Zs
= ( cons_list_char @ X @ Xs2 ) ) )
| ? [Ys6: list_list_char] :
( ( Ys2
= ( cons_list_char @ X @ Ys6 ) )
& ( ( append_list_char @ Ys6 @ Zs )
= Xs2 ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_750_rev__nonempty__induct,axiom,
! [Xs2: list_int,P: list_int > $o] :
( ( Xs2 != nil_int )
=> ( ! [X4: int] : ( P @ ( cons_int @ X4 @ nil_int ) )
=> ( ! [X4: int,Xs3: list_int] :
( ( Xs3 != nil_int )
=> ( ( P @ Xs3 )
=> ( P @ ( append_int @ Xs3 @ ( cons_int @ X4 @ nil_int ) ) ) ) )
=> ( P @ Xs2 ) ) ) ) ).
% rev_nonempty_induct
thf(fact_751_rev__nonempty__induct,axiom,
! [Xs2: list_instr,P: list_instr > $o] :
( ( Xs2 != nil_instr )
=> ( ! [X4: instr] : ( P @ ( cons_instr @ X4 @ nil_instr ) )
=> ( ! [X4: instr,Xs3: list_instr] :
( ( Xs3 != nil_instr )
=> ( ( P @ Xs3 )
=> ( P @ ( append_instr @ Xs3 @ ( cons_instr @ X4 @ nil_instr ) ) ) ) )
=> ( P @ Xs2 ) ) ) ) ).
% rev_nonempty_induct
thf(fact_752_rev__nonempty__induct,axiom,
! [Xs2: list_P419547167332244514st_int,P: list_P419547167332244514st_int > $o] :
( ( Xs2 != nil_Pr749310521696113548st_int )
=> ( ! [X4: produc6425607678544837394st_int] : ( P @ ( cons_P4151473878978182108st_int @ X4 @ nil_Pr749310521696113548st_int ) )
=> ( ! [X4: produc6425607678544837394st_int,Xs3: list_P419547167332244514st_int] :
( ( Xs3 != nil_Pr749310521696113548st_int )
=> ( ( P @ Xs3 )
=> ( P @ ( append5028292981072010935st_int @ Xs3 @ ( cons_P4151473878978182108st_int @ X4 @ nil_Pr749310521696113548st_int ) ) ) ) )
=> ( P @ Xs2 ) ) ) ) ).
% rev_nonempty_induct
thf(fact_753_rev__nonempty__induct,axiom,
! [Xs2: list_list_char,P: list_list_char > $o] :
( ( Xs2 != nil_list_char )
=> ( ! [X4: list_char] : ( P @ ( cons_list_char @ X4 @ nil_list_char ) )
=> ( ! [X4: list_char,Xs3: list_list_char] :
( ( Xs3 != nil_list_char )
=> ( ( P @ Xs3 )
=> ( P @ ( append_list_char @ Xs3 @ ( cons_list_char @ X4 @ nil_list_char ) ) ) ) )
=> ( P @ Xs2 ) ) ) ) ).
% rev_nonempty_induct
thf(fact_754_same__length__different,axiom,
! [Xs2: list_P419547167332244514st_int,Ys2: list_P419547167332244514st_int] :
( ( Xs2 != Ys2 )
=> ( ( ( size_s229347708945616566st_int @ Xs2 )
= ( size_s229347708945616566st_int @ Ys2 ) )
=> ? [Pre: list_P419547167332244514st_int,X4: produc6425607678544837394st_int,Xs4: list_P419547167332244514st_int,Y4: produc6425607678544837394st_int,Ys4: list_P419547167332244514st_int] :
( ( X4 != Y4 )
& ( Xs2
= ( append5028292981072010935st_int @ Pre @ ( append5028292981072010935st_int @ ( cons_P4151473878978182108st_int @ X4 @ nil_Pr749310521696113548st_int ) @ Xs4 ) ) )
& ( Ys2
= ( append5028292981072010935st_int @ Pre @ ( append5028292981072010935st_int @ ( cons_P4151473878978182108st_int @ Y4 @ nil_Pr749310521696113548st_int ) @ Ys4 ) ) ) ) ) ) ).
% same_length_different
thf(fact_755_same__length__different,axiom,
! [Xs2: list_instr,Ys2: list_instr] :
( ( Xs2 != Ys2 )
=> ( ( ( size_size_list_instr @ Xs2 )
= ( size_size_list_instr @ Ys2 ) )
=> ? [Pre: list_instr,X4: instr,Xs4: list_instr,Y4: instr,Ys4: list_instr] :
( ( X4 != Y4 )
& ( Xs2
= ( append_instr @ Pre @ ( append_instr @ ( cons_instr @ X4 @ nil_instr ) @ Xs4 ) ) )
& ( Ys2
= ( append_instr @ Pre @ ( append_instr @ ( cons_instr @ Y4 @ nil_instr ) @ Ys4 ) ) ) ) ) ) ).
% same_length_different
thf(fact_756_same__length__different,axiom,
! [Xs2: list_list_char,Ys2: list_list_char] :
( ( Xs2 != Ys2 )
=> ( ( ( size_s356637359517785349t_char @ Xs2 )
= ( size_s356637359517785349t_char @ Ys2 ) )
=> ? [Pre: list_list_char,X4: list_char,Xs4: list_list_char,Y4: list_char,Ys4: list_list_char] :
( ( X4 != Y4 )
& ( Xs2
= ( append_list_char @ Pre @ ( append_list_char @ ( cons_list_char @ X4 @ nil_list_char ) @ Xs4 ) ) )
& ( Ys2
= ( append_list_char @ Pre @ ( append_list_char @ ( cons_list_char @ Y4 @ nil_list_char ) @ Ys4 ) ) ) ) ) ) ).
% same_length_different
thf(fact_757_same__length__different,axiom,
! [Xs2: list_int,Ys2: list_int] :
( ( Xs2 != Ys2 )
=> ( ( ( size_size_list_int @ Xs2 )
= ( size_size_list_int @ Ys2 ) )
=> ? [Pre: list_int,X4: int,Xs4: list_int,Y4: int,Ys4: list_int] :
( ( X4 != Y4 )
& ( Xs2
= ( append_int @ Pre @ ( append_int @ ( cons_int @ X4 @ nil_int ) @ Xs4 ) ) )
& ( Ys2
= ( append_int @ Pre @ ( append_int @ ( cons_int @ Y4 @ nil_int ) @ Ys4 ) ) ) ) ) ) ).
% same_length_different
thf(fact_758_Cons__eq__appendI,axiom,
! [X: int,Xs1: list_int,Ys2: list_int,Xs2: list_int,Zs: list_int] :
( ( ( cons_int @ X @ Xs1 )
= Ys2 )
=> ( ( Xs2
= ( append_int @ Xs1 @ Zs ) )
=> ( ( cons_int @ X @ Xs2 )
= ( append_int @ Ys2 @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_759_Cons__eq__appendI,axiom,
! [X: instr,Xs1: list_instr,Ys2: list_instr,Xs2: list_instr,Zs: list_instr] :
( ( ( cons_instr @ X @ Xs1 )
= Ys2 )
=> ( ( Xs2
= ( append_instr @ Xs1 @ Zs ) )
=> ( ( cons_instr @ X @ Xs2 )
= ( append_instr @ Ys2 @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_760_Cons__eq__appendI,axiom,
! [X: produc6425607678544837394st_int,Xs1: list_P419547167332244514st_int,Ys2: list_P419547167332244514st_int,Xs2: list_P419547167332244514st_int,Zs: list_P419547167332244514st_int] :
( ( ( cons_P4151473878978182108st_int @ X @ Xs1 )
= Ys2 )
=> ( ( Xs2
= ( append5028292981072010935st_int @ Xs1 @ Zs ) )
=> ( ( cons_P4151473878978182108st_int @ X @ Xs2 )
= ( append5028292981072010935st_int @ Ys2 @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_761_Cons__eq__appendI,axiom,
! [X: list_char,Xs1: list_list_char,Ys2: list_list_char,Xs2: list_list_char,Zs: list_list_char] :
( ( ( cons_list_char @ X @ Xs1 )
= Ys2 )
=> ( ( Xs2
= ( append_list_char @ Xs1 @ Zs ) )
=> ( ( cons_list_char @ X @ Xs2 )
= ( append_list_char @ Ys2 @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_762_append__Cons,axiom,
! [X: int,Xs2: list_int,Ys2: list_int] :
( ( append_int @ ( cons_int @ X @ Xs2 ) @ Ys2 )
= ( cons_int @ X @ ( append_int @ Xs2 @ Ys2 ) ) ) ).
% append_Cons
thf(fact_763_append__Cons,axiom,
! [X: instr,Xs2: list_instr,Ys2: list_instr] :
( ( append_instr @ ( cons_instr @ X @ Xs2 ) @ Ys2 )
= ( cons_instr @ X @ ( append_instr @ Xs2 @ Ys2 ) ) ) ).
% append_Cons
thf(fact_764_append__Cons,axiom,
! [X: produc6425607678544837394st_int,Xs2: list_P419547167332244514st_int,Ys2: list_P419547167332244514st_int] :
( ( append5028292981072010935st_int @ ( cons_P4151473878978182108st_int @ X @ Xs2 ) @ Ys2 )
= ( cons_P4151473878978182108st_int @ X @ ( append5028292981072010935st_int @ Xs2 @ Ys2 ) ) ) ).
% append_Cons
thf(fact_765_append__Cons,axiom,
! [X: list_char,Xs2: list_list_char,Ys2: list_list_char] :
( ( append_list_char @ ( cons_list_char @ X @ Xs2 ) @ Ys2 )
= ( cons_list_char @ X @ ( append_list_char @ Xs2 @ Ys2 ) ) ) ).
% append_Cons
thf(fact_766_verit__negate__coefficient_I2_J,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_767_less__minus__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( uminus_uminus_int @ B ) )
= ( ord_less_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% less_minus_iff
thf(fact_768_minus__less__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ B )
= ( ord_less_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% minus_less_iff
thf(fact_769_is__num__normalize_I8_J,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% is_num_normalize(8)
thf(fact_770_add_Oinverse__distrib__swap,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% add.inverse_distrib_swap
thf(fact_771_group__cancel_Oneg1,axiom,
! [A2: int,K: int,A: int] :
( ( A2
= ( plus_plus_int @ K @ A ) )
=> ( ( uminus_uminus_int @ A2 )
= ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A ) ) ) ) ).
% group_cancel.neg1
thf(fact_772_one__neq__neg__one,axiom,
( one_one_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% one_neq_neg_one
thf(fact_773_minus__diff__commute,axiom,
! [B: int,A: int] :
( ( minus_minus_int @ ( uminus_uminus_int @ B ) @ A )
= ( minus_minus_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% minus_diff_commute
thf(fact_774_uminus__int__code_I1_J,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% uminus_int_code(1)
thf(fact_775_int__cases2,axiom,
! [Z2: int] :
( ! [N3: nat] :
( Z2
!= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ! [N3: nat] :
( Z2
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% int_cases2
thf(fact_776_list__nonempty__induct,axiom,
! [Xs2: list_int,P: list_int > $o] :
( ( Xs2 != nil_int )
=> ( ! [X4: int] : ( P @ ( cons_int @ X4 @ nil_int ) )
=> ( ! [X4: int,Xs3: list_int] :
( ( Xs3 != nil_int )
=> ( ( P @ Xs3 )
=> ( P @ ( cons_int @ X4 @ Xs3 ) ) ) )
=> ( P @ Xs2 ) ) ) ) ).
% list_nonempty_induct
thf(fact_777_list__nonempty__induct,axiom,
! [Xs2: list_instr,P: list_instr > $o] :
( ( Xs2 != nil_instr )
=> ( ! [X4: instr] : ( P @ ( cons_instr @ X4 @ nil_instr ) )
=> ( ! [X4: instr,Xs3: list_instr] :
( ( Xs3 != nil_instr )
=> ( ( P @ Xs3 )
=> ( P @ ( cons_instr @ X4 @ Xs3 ) ) ) )
=> ( P @ Xs2 ) ) ) ) ).
% list_nonempty_induct
thf(fact_778_list__nonempty__induct,axiom,
! [Xs2: list_P419547167332244514st_int,P: list_P419547167332244514st_int > $o] :
( ( Xs2 != nil_Pr749310521696113548st_int )
=> ( ! [X4: produc6425607678544837394st_int] : ( P @ ( cons_P4151473878978182108st_int @ X4 @ nil_Pr749310521696113548st_int ) )
=> ( ! [X4: produc6425607678544837394st_int,Xs3: list_P419547167332244514st_int] :
( ( Xs3 != nil_Pr749310521696113548st_int )
=> ( ( P @ Xs3 )
=> ( P @ ( cons_P4151473878978182108st_int @ X4 @ Xs3 ) ) ) )
=> ( P @ Xs2 ) ) ) ) ).
% list_nonempty_induct
thf(fact_779_list__nonempty__induct,axiom,
! [Xs2: list_list_char,P: list_list_char > $o] :
( ( Xs2 != nil_list_char )
=> ( ! [X4: list_char] : ( P @ ( cons_list_char @ X4 @ nil_list_char ) )
=> ( ! [X4: list_char,Xs3: list_list_char] :
( ( Xs3 != nil_list_char )
=> ( ( P @ Xs3 )
=> ( P @ ( cons_list_char @ X4 @ Xs3 ) ) ) )
=> ( P @ Xs2 ) ) ) ) ).
% list_nonempty_induct
thf(fact_780_list__induct2_H,axiom,
! [P: list_int > list_int > $o,Xs2: list_int,Ys2: list_int] :
( ( P @ nil_int @ nil_int )
=> ( ! [X4: int,Xs3: list_int] : ( P @ ( cons_int @ X4 @ Xs3 ) @ nil_int )
=> ( ! [Y4: int,Ys5: list_int] : ( P @ nil_int @ ( cons_int @ Y4 @ Ys5 ) )
=> ( ! [X4: int,Xs3: list_int,Y4: int,Ys5: list_int] :
( ( P @ Xs3 @ Ys5 )
=> ( P @ ( cons_int @ X4 @ Xs3 ) @ ( cons_int @ Y4 @ Ys5 ) ) )
=> ( P @ Xs2 @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_781_list__induct2_H,axiom,
! [P: list_int > list_instr > $o,Xs2: list_int,Ys2: list_instr] :
( ( P @ nil_int @ nil_instr )
=> ( ! [X4: int,Xs3: list_int] : ( P @ ( cons_int @ X4 @ Xs3 ) @ nil_instr )
=> ( ! [Y4: instr,Ys5: list_instr] : ( P @ nil_int @ ( cons_instr @ Y4 @ Ys5 ) )
=> ( ! [X4: int,Xs3: list_int,Y4: instr,Ys5: list_instr] :
( ( P @ Xs3 @ Ys5 )
=> ( P @ ( cons_int @ X4 @ Xs3 ) @ ( cons_instr @ Y4 @ Ys5 ) ) )
=> ( P @ Xs2 @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_782_list__induct2_H,axiom,
! [P: list_instr > list_int > $o,Xs2: list_instr,Ys2: list_int] :
( ( P @ nil_instr @ nil_int )
=> ( ! [X4: instr,Xs3: list_instr] : ( P @ ( cons_instr @ X4 @ Xs3 ) @ nil_int )
=> ( ! [Y4: int,Ys5: list_int] : ( P @ nil_instr @ ( cons_int @ Y4 @ Ys5 ) )
=> ( ! [X4: instr,Xs3: list_instr,Y4: int,Ys5: list_int] :
( ( P @ Xs3 @ Ys5 )
=> ( P @ ( cons_instr @ X4 @ Xs3 ) @ ( cons_int @ Y4 @ Ys5 ) ) )
=> ( P @ Xs2 @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_783_list__induct2_H,axiom,
! [P: list_instr > list_instr > $o,Xs2: list_instr,Ys2: list_instr] :
( ( P @ nil_instr @ nil_instr )
=> ( ! [X4: instr,Xs3: list_instr] : ( P @ ( cons_instr @ X4 @ Xs3 ) @ nil_instr )
=> ( ! [Y4: instr,Ys5: list_instr] : ( P @ nil_instr @ ( cons_instr @ Y4 @ Ys5 ) )
=> ( ! [X4: instr,Xs3: list_instr,Y4: instr,Ys5: list_instr] :
( ( P @ Xs3 @ Ys5 )
=> ( P @ ( cons_instr @ X4 @ Xs3 ) @ ( cons_instr @ Y4 @ Ys5 ) ) )
=> ( P @ Xs2 @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_784_list__induct2_H,axiom,
! [P: list_int > list_list_char > $o,Xs2: list_int,Ys2: list_list_char] :
( ( P @ nil_int @ nil_list_char )
=> ( ! [X4: int,Xs3: list_int] : ( P @ ( cons_int @ X4 @ Xs3 ) @ nil_list_char )
=> ( ! [Y4: list_char,Ys5: list_list_char] : ( P @ nil_int @ ( cons_list_char @ Y4 @ Ys5 ) )
=> ( ! [X4: int,Xs3: list_int,Y4: list_char,Ys5: list_list_char] :
( ( P @ Xs3 @ Ys5 )
=> ( P @ ( cons_int @ X4 @ Xs3 ) @ ( cons_list_char @ Y4 @ Ys5 ) ) )
=> ( P @ Xs2 @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_785_list__induct2_H,axiom,
! [P: list_instr > list_list_char > $o,Xs2: list_instr,Ys2: list_list_char] :
( ( P @ nil_instr @ nil_list_char )
=> ( ! [X4: instr,Xs3: list_instr] : ( P @ ( cons_instr @ X4 @ Xs3 ) @ nil_list_char )
=> ( ! [Y4: list_char,Ys5: list_list_char] : ( P @ nil_instr @ ( cons_list_char @ Y4 @ Ys5 ) )
=> ( ! [X4: instr,Xs3: list_instr,Y4: list_char,Ys5: list_list_char] :
( ( P @ Xs3 @ Ys5 )
=> ( P @ ( cons_instr @ X4 @ Xs3 ) @ ( cons_list_char @ Y4 @ Ys5 ) ) )
=> ( P @ Xs2 @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_786_list__induct2_H,axiom,
! [P: list_list_char > list_int > $o,Xs2: list_list_char,Ys2: list_int] :
( ( P @ nil_list_char @ nil_int )
=> ( ! [X4: list_char,Xs3: list_list_char] : ( P @ ( cons_list_char @ X4 @ Xs3 ) @ nil_int )
=> ( ! [Y4: int,Ys5: list_int] : ( P @ nil_list_char @ ( cons_int @ Y4 @ Ys5 ) )
=> ( ! [X4: list_char,Xs3: list_list_char,Y4: int,Ys5: list_int] :
( ( P @ Xs3 @ Ys5 )
=> ( P @ ( cons_list_char @ X4 @ Xs3 ) @ ( cons_int @ Y4 @ Ys5 ) ) )
=> ( P @ Xs2 @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_787_list__induct2_H,axiom,
! [P: list_list_char > list_instr > $o,Xs2: list_list_char,Ys2: list_instr] :
( ( P @ nil_list_char @ nil_instr )
=> ( ! [X4: list_char,Xs3: list_list_char] : ( P @ ( cons_list_char @ X4 @ Xs3 ) @ nil_instr )
=> ( ! [Y4: instr,Ys5: list_instr] : ( P @ nil_list_char @ ( cons_instr @ Y4 @ Ys5 ) )
=> ( ! [X4: list_char,Xs3: list_list_char,Y4: instr,Ys5: list_instr] :
( ( P @ Xs3 @ Ys5 )
=> ( P @ ( cons_list_char @ X4 @ Xs3 ) @ ( cons_instr @ Y4 @ Ys5 ) ) )
=> ( P @ Xs2 @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_788_list__induct2_H,axiom,
! [P: list_list_char > list_list_char > $o,Xs2: list_list_char,Ys2: list_list_char] :
( ( P @ nil_list_char @ nil_list_char )
=> ( ! [X4: list_char,Xs3: list_list_char] : ( P @ ( cons_list_char @ X4 @ Xs3 ) @ nil_list_char )
=> ( ! [Y4: list_char,Ys5: list_list_char] : ( P @ nil_list_char @ ( cons_list_char @ Y4 @ Ys5 ) )
=> ( ! [X4: list_char,Xs3: list_list_char,Y4: list_char,Ys5: list_list_char] :
( ( P @ Xs3 @ Ys5 )
=> ( P @ ( cons_list_char @ X4 @ Xs3 ) @ ( cons_list_char @ Y4 @ Ys5 ) ) )
=> ( P @ Xs2 @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_789_list__induct2_H,axiom,
! [P: list_int > list_P419547167332244514st_int > $o,Xs2: list_int,Ys2: list_P419547167332244514st_int] :
( ( P @ nil_int @ nil_Pr749310521696113548st_int )
=> ( ! [X4: int,Xs3: list_int] : ( P @ ( cons_int @ X4 @ Xs3 ) @ nil_Pr749310521696113548st_int )
=> ( ! [Y4: produc6425607678544837394st_int,Ys5: list_P419547167332244514st_int] : ( P @ nil_int @ ( cons_P4151473878978182108st_int @ Y4 @ Ys5 ) )
=> ( ! [X4: int,Xs3: list_int,Y4: produc6425607678544837394st_int,Ys5: list_P419547167332244514st_int] :
( ( P @ Xs3 @ Ys5 )
=> ( P @ ( cons_int @ X4 @ Xs3 ) @ ( cons_P4151473878978182108st_int @ Y4 @ Ys5 ) ) )
=> ( P @ Xs2 @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_790_neq__Nil__conv,axiom,
! [Xs2: list_int] :
( ( Xs2 != nil_int )
= ( ? [Y5: int,Ys: list_int] :
( Xs2
= ( cons_int @ Y5 @ Ys ) ) ) ) ).
% neq_Nil_conv
thf(fact_791_neq__Nil__conv,axiom,
! [Xs2: list_instr] :
( ( Xs2 != nil_instr )
= ( ? [Y5: instr,Ys: list_instr] :
( Xs2
= ( cons_instr @ Y5 @ Ys ) ) ) ) ).
% neq_Nil_conv
thf(fact_792_neq__Nil__conv,axiom,
! [Xs2: list_P419547167332244514st_int] :
( ( Xs2 != nil_Pr749310521696113548st_int )
= ( ? [Y5: produc6425607678544837394st_int,Ys: list_P419547167332244514st_int] :
( Xs2
= ( cons_P4151473878978182108st_int @ Y5 @ Ys ) ) ) ) ).
% neq_Nil_conv
thf(fact_793_neq__Nil__conv,axiom,
! [Xs2: list_list_char] :
( ( Xs2 != nil_list_char )
= ( ? [Y5: list_char,Ys: list_list_char] :
( Xs2
= ( cons_list_char @ Y5 @ Ys ) ) ) ) ).
% neq_Nil_conv
thf(fact_794_remdups__adj_Ocases,axiom,
! [X: list_int] :
( ( X != nil_int )
=> ( ! [X4: int] :
( X
!= ( cons_int @ X4 @ nil_int ) )
=> ~ ! [X4: int,Y4: int,Xs3: list_int] :
( X
!= ( cons_int @ X4 @ ( cons_int @ Y4 @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_795_remdups__adj_Ocases,axiom,
! [X: list_instr] :
( ( X != nil_instr )
=> ( ! [X4: instr] :
( X
!= ( cons_instr @ X4 @ nil_instr ) )
=> ~ ! [X4: instr,Y4: instr,Xs3: list_instr] :
( X
!= ( cons_instr @ X4 @ ( cons_instr @ Y4 @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_796_remdups__adj_Ocases,axiom,
! [X: list_P419547167332244514st_int] :
( ( X != nil_Pr749310521696113548st_int )
=> ( ! [X4: produc6425607678544837394st_int] :
( X
!= ( cons_P4151473878978182108st_int @ X4 @ nil_Pr749310521696113548st_int ) )
=> ~ ! [X4: produc6425607678544837394st_int,Y4: produc6425607678544837394st_int,Xs3: list_P419547167332244514st_int] :
( X
!= ( cons_P4151473878978182108st_int @ X4 @ ( cons_P4151473878978182108st_int @ Y4 @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_797_remdups__adj_Ocases,axiom,
! [X: list_list_char] :
( ( X != nil_list_char )
=> ( ! [X4: list_char] :
( X
!= ( cons_list_char @ X4 @ nil_list_char ) )
=> ~ ! [X4: list_char,Y4: list_char,Xs3: list_list_char] :
( X
!= ( cons_list_char @ X4 @ ( cons_list_char @ Y4 @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_798_min__list_Ocases,axiom,
! [X: list_int] :
( ! [X4: int,Xs3: list_int] :
( X
!= ( cons_int @ X4 @ Xs3 ) )
=> ( X = nil_int ) ) ).
% min_list.cases
thf(fact_799_list_Oexhaust,axiom,
! [Y: list_int] :
( ( Y != nil_int )
=> ~ ! [X212: int,X222: list_int] :
( Y
!= ( cons_int @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_800_list_Oexhaust,axiom,
! [Y: list_instr] :
( ( Y != nil_instr )
=> ~ ! [X212: instr,X222: list_instr] :
( Y
!= ( cons_instr @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_801_list_Oexhaust,axiom,
! [Y: list_P419547167332244514st_int] :
( ( Y != nil_Pr749310521696113548st_int )
=> ~ ! [X212: produc6425607678544837394st_int,X222: list_P419547167332244514st_int] :
( Y
!= ( cons_P4151473878978182108st_int @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_802_list_Oexhaust,axiom,
! [Y: list_list_char] :
( ( Y != nil_list_char )
=> ~ ! [X212: list_char,X222: list_list_char] :
( Y
!= ( cons_list_char @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_803_list_OdiscI,axiom,
! [List: list_int,X21: int,X22: list_int] :
( ( List
= ( cons_int @ X21 @ X22 ) )
=> ( List != nil_int ) ) ).
% list.discI
thf(fact_804_list_OdiscI,axiom,
! [List: list_instr,X21: instr,X22: list_instr] :
( ( List
= ( cons_instr @ X21 @ X22 ) )
=> ( List != nil_instr ) ) ).
% list.discI
thf(fact_805_list_OdiscI,axiom,
! [List: list_P419547167332244514st_int,X21: produc6425607678544837394st_int,X22: list_P419547167332244514st_int] :
( ( List
= ( cons_P4151473878978182108st_int @ X21 @ X22 ) )
=> ( List != nil_Pr749310521696113548st_int ) ) ).
% list.discI
thf(fact_806_list_OdiscI,axiom,
! [List: list_list_char,X21: list_char,X22: list_list_char] :
( ( List
= ( cons_list_char @ X21 @ X22 ) )
=> ( List != nil_list_char ) ) ).
% list.discI
thf(fact_807_list_Odistinct_I1_J,axiom,
! [X21: int,X22: list_int] :
( nil_int
!= ( cons_int @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_808_list_Odistinct_I1_J,axiom,
! [X21: instr,X22: list_instr] :
( nil_instr
!= ( cons_instr @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_809_list_Odistinct_I1_J,axiom,
! [X21: produc6425607678544837394st_int,X22: list_P419547167332244514st_int] :
( nil_Pr749310521696113548st_int
!= ( cons_P4151473878978182108st_int @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_810_list_Odistinct_I1_J,axiom,
! [X21: list_char,X22: list_list_char] :
( nil_list_char
!= ( cons_list_char @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_811_successively_Ocases,axiom,
! [X: produc5834231552977413017st_int] :
( ! [P3: int > int > $o] :
( X
!= ( produc8618682346314911123st_int @ P3 @ nil_int ) )
=> ( ! [P3: int > int > $o,X4: int] :
( X
!= ( produc8618682346314911123st_int @ P3 @ ( cons_int @ X4 @ nil_int ) ) )
=> ~ ! [P3: int > int > $o,X4: int,Y4: int,Xs3: list_int] :
( X
!= ( produc8618682346314911123st_int @ P3 @ ( cons_int @ X4 @ ( cons_int @ Y4 @ Xs3 ) ) ) ) ) ) ).
% successively.cases
thf(fact_812_successively_Ocases,axiom,
! [X: produc7133927873007960030_instr] :
( ! [P3: instr > instr > $o] :
( X
!= ( produc7818391707689388878_instr @ P3 @ nil_instr ) )
=> ( ! [P3: instr > instr > $o,X4: instr] :
( X
!= ( produc7818391707689388878_instr @ P3 @ ( cons_instr @ X4 @ nil_instr ) ) )
=> ~ ! [P3: instr > instr > $o,X4: instr,Y4: instr,Xs3: list_instr] :
( X
!= ( produc7818391707689388878_instr @ P3 @ ( cons_instr @ X4 @ ( cons_instr @ Y4 @ Xs3 ) ) ) ) ) ) ).
% successively.cases
thf(fact_813_successively_Ocases,axiom,
! [X: produc2468426419292222131st_int] :
( ! [P3: produc6425607678544837394st_int > produc6425607678544837394st_int > $o] :
( X
!= ( produc554038597315947693st_int @ P3 @ nil_Pr749310521696113548st_int ) )
=> ( ! [P3: produc6425607678544837394st_int > produc6425607678544837394st_int > $o,X4: produc6425607678544837394st_int] :
( X
!= ( produc554038597315947693st_int @ P3 @ ( cons_P4151473878978182108st_int @ X4 @ nil_Pr749310521696113548st_int ) ) )
=> ~ ! [P3: produc6425607678544837394st_int > produc6425607678544837394st_int > $o,X4: produc6425607678544837394st_int,Y4: produc6425607678544837394st_int,Xs3: list_P419547167332244514st_int] :
( X
!= ( produc554038597315947693st_int @ P3 @ ( cons_P4151473878978182108st_int @ X4 @ ( cons_P4151473878978182108st_int @ Y4 @ Xs3 ) ) ) ) ) ) ).
% successively.cases
thf(fact_814_successively_Ocases,axiom,
! [X: produc8794813868974875620t_char] :
( ! [P3: list_char > list_char > $o] :
( X
!= ( produc1436111165245395422t_char @ P3 @ nil_list_char ) )
=> ( ! [P3: list_char > list_char > $o,X4: list_char] :
( X
!= ( produc1436111165245395422t_char @ P3 @ ( cons_list_char @ X4 @ nil_list_char ) ) )
=> ~ ! [P3: list_char > list_char > $o,X4: list_char,Y4: list_char,Xs3: list_list_char] :
( X
!= ( produc1436111165245395422t_char @ P3 @ ( cons_list_char @ X4 @ ( cons_list_char @ Y4 @ Xs3 ) ) ) ) ) ) ).
% successively.cases
thf(fact_815_sorted__wrt_Ocases,axiom,
! [X: produc5834231552977413017st_int] :
( ! [P3: int > int > $o] :
( X
!= ( produc8618682346314911123st_int @ P3 @ nil_int ) )
=> ~ ! [P3: int > int > $o,X4: int,Ys5: list_int] :
( X
!= ( produc8618682346314911123st_int @ P3 @ ( cons_int @ X4 @ Ys5 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_816_sorted__wrt_Ocases,axiom,
! [X: produc7133927873007960030_instr] :
( ! [P3: instr > instr > $o] :
( X
!= ( produc7818391707689388878_instr @ P3 @ nil_instr ) )
=> ~ ! [P3: instr > instr > $o,X4: instr,Ys5: list_instr] :
( X
!= ( produc7818391707689388878_instr @ P3 @ ( cons_instr @ X4 @ Ys5 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_817_sorted__wrt_Ocases,axiom,
! [X: produc2468426419292222131st_int] :
( ! [P3: produc6425607678544837394st_int > produc6425607678544837394st_int > $o] :
( X
!= ( produc554038597315947693st_int @ P3 @ nil_Pr749310521696113548st_int ) )
=> ~ ! [P3: produc6425607678544837394st_int > produc6425607678544837394st_int > $o,X4: produc6425607678544837394st_int,Ys5: list_P419547167332244514st_int] :
( X
!= ( produc554038597315947693st_int @ P3 @ ( cons_P4151473878978182108st_int @ X4 @ Ys5 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_818_sorted__wrt_Ocases,axiom,
! [X: produc8794813868974875620t_char] :
( ! [P3: list_char > list_char > $o] :
( X
!= ( produc1436111165245395422t_char @ P3 @ nil_list_char ) )
=> ~ ! [P3: list_char > list_char > $o,X4: list_char,Ys5: list_list_char] :
( X
!= ( produc1436111165245395422t_char @ P3 @ ( cons_list_char @ X4 @ Ys5 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_819_shuffles_Ocases,axiom,
! [X: produc1186641810826059865st_int] :
( ! [Ys5: list_int] :
( X
!= ( produc364263696895485585st_int @ nil_int @ Ys5 ) )
=> ( ! [Xs3: list_int] :
( X
!= ( produc364263696895485585st_int @ Xs3 @ nil_int ) )
=> ~ ! [X4: int,Xs3: list_int,Y4: int,Ys5: list_int] :
( X
!= ( produc364263696895485585st_int @ ( cons_int @ X4 @ Xs3 ) @ ( cons_int @ Y4 @ Ys5 ) ) ) ) ) ).
% shuffles.cases
thf(fact_820_shuffles_Ocases,axiom,
! [X: produc8891168062546637159_instr] :
( ! [Ys5: list_instr] :
( X
!= ( produc7750586281064920791_instr @ nil_instr @ Ys5 ) )
=> ( ! [Xs3: list_instr] :
( X
!= ( produc7750586281064920791_instr @ Xs3 @ nil_instr ) )
=> ~ ! [X4: instr,Xs3: list_instr,Y4: instr,Ys5: list_instr] :
( X
!= ( produc7750586281064920791_instr @ ( cons_instr @ X4 @ Xs3 ) @ ( cons_instr @ Y4 @ Ys5 ) ) ) ) ) ).
% shuffles.cases
thf(fact_821_shuffles_Ocases,axiom,
! [X: produc5848496107505245989st_int] :
( ! [Ys5: list_P419547167332244514st_int] :
( X
!= ( produc2639222612625641053st_int @ nil_Pr749310521696113548st_int @ Ys5 ) )
=> ( ! [Xs3: list_P419547167332244514st_int] :
( X
!= ( produc2639222612625641053st_int @ Xs3 @ nil_Pr749310521696113548st_int ) )
=> ~ ! [X4: produc6425607678544837394st_int,Xs3: list_P419547167332244514st_int,Y4: produc6425607678544837394st_int,Ys5: list_P419547167332244514st_int] :
( X
!= ( produc2639222612625641053st_int @ ( cons_P4151473878978182108st_int @ X4 @ Xs3 ) @ ( cons_P4151473878978182108st_int @ Y4 @ Ys5 ) ) ) ) ) ).
% shuffles.cases
thf(fact_822_shuffles_Ocases,axiom,
! [X: produc525958773753613763t_char] :
( ! [Ys5: list_list_char] :
( X
!= ( produc9134825680149865339t_char @ nil_list_char @ Ys5 ) )
=> ( ! [Xs3: list_list_char] :
( X
!= ( produc9134825680149865339t_char @ Xs3 @ nil_list_char ) )
=> ~ ! [X4: list_char,Xs3: list_list_char,Y4: list_char,Ys5: list_list_char] :
( X
!= ( produc9134825680149865339t_char @ ( cons_list_char @ X4 @ Xs3 ) @ ( cons_list_char @ Y4 @ Ys5 ) ) ) ) ) ).
% shuffles.cases
thf(fact_823_splice_Ocases,axiom,
! [X: produc1186641810826059865st_int] :
( ! [Ys5: list_int] :
( X
!= ( produc364263696895485585st_int @ nil_int @ Ys5 ) )
=> ~ ! [X4: int,Xs3: list_int,Ys5: list_int] :
( X
!= ( produc364263696895485585st_int @ ( cons_int @ X4 @ Xs3 ) @ Ys5 ) ) ) ).
% splice.cases
thf(fact_824_splice_Ocases,axiom,
! [X: produc8891168062546637159_instr] :
( ! [Ys5: list_instr] :
( X
!= ( produc7750586281064920791_instr @ nil_instr @ Ys5 ) )
=> ~ ! [X4: instr,Xs3: list_instr,Ys5: list_instr] :
( X
!= ( produc7750586281064920791_instr @ ( cons_instr @ X4 @ Xs3 ) @ Ys5 ) ) ) ).
% splice.cases
thf(fact_825_splice_Ocases,axiom,
! [X: produc5848496107505245989st_int] :
( ! [Ys5: list_P419547167332244514st_int] :
( X
!= ( produc2639222612625641053st_int @ nil_Pr749310521696113548st_int @ Ys5 ) )
=> ~ ! [X4: produc6425607678544837394st_int,Xs3: list_P419547167332244514st_int,Ys5: list_P419547167332244514st_int] :
( X
!= ( produc2639222612625641053st_int @ ( cons_P4151473878978182108st_int @ X4 @ Xs3 ) @ Ys5 ) ) ) ).
% splice.cases
thf(fact_826_splice_Ocases,axiom,
! [X: produc525958773753613763t_char] :
( ! [Ys5: list_list_char] :
( X
!= ( produc9134825680149865339t_char @ nil_list_char @ Ys5 ) )
=> ~ ! [X4: list_char,Xs3: list_list_char,Ys5: list_list_char] :
( X
!= ( produc9134825680149865339t_char @ ( cons_list_char @ X4 @ Xs3 ) @ Ys5 ) ) ) ).
% splice.cases
thf(fact_827_drop__Nil,axiom,
! [N: nat] :
( ( drop_instr @ N @ nil_instr )
= nil_instr ) ).
% drop_Nil
thf(fact_828_drop__Nil,axiom,
! [N: nat] :
( ( drop_P7838799151754935884st_int @ N @ nil_Pr749310521696113548st_int )
= nil_Pr749310521696113548st_int ) ).
% drop_Nil
thf(fact_829_drop__Nil,axiom,
! [N: nat] :
( ( drop_int @ N @ nil_int )
= nil_int ) ).
% drop_Nil
thf(fact_830_drop__Nil,axiom,
! [N: nat] :
( ( drop_list_char @ N @ nil_list_char )
= nil_list_char ) ).
% drop_Nil
thf(fact_831_length__Suc__conv__rev,axiom,
! [Xs2: list_P419547167332244514st_int,N: nat] :
( ( ( size_s229347708945616566st_int @ Xs2 )
= ( suc @ N ) )
= ( ? [Y5: produc6425607678544837394st_int,Ys: list_P419547167332244514st_int] :
( ( Xs2
= ( append5028292981072010935st_int @ Ys @ ( cons_P4151473878978182108st_int @ Y5 @ nil_Pr749310521696113548st_int ) ) )
& ( ( size_s229347708945616566st_int @ Ys )
= N ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_832_length__Suc__conv__rev,axiom,
! [Xs2: list_instr,N: nat] :
( ( ( size_size_list_instr @ Xs2 )
= ( suc @ N ) )
= ( ? [Y5: instr,Ys: list_instr] :
( ( Xs2
= ( append_instr @ Ys @ ( cons_instr @ Y5 @ nil_instr ) ) )
& ( ( size_size_list_instr @ Ys )
= N ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_833_length__Suc__conv__rev,axiom,
! [Xs2: list_list_char,N: nat] :
( ( ( size_s356637359517785349t_char @ Xs2 )
= ( suc @ N ) )
= ( ? [Y5: list_char,Ys: list_list_char] :
( ( Xs2
= ( append_list_char @ Ys @ ( cons_list_char @ Y5 @ nil_list_char ) ) )
& ( ( size_s356637359517785349t_char @ Ys )
= N ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_834_length__Suc__conv__rev,axiom,
! [Xs2: list_int,N: nat] :
( ( ( size_size_list_int @ Xs2 )
= ( suc @ N ) )
= ( ? [Y5: int,Ys: list_int] :
( ( Xs2
= ( append_int @ Ys @ ( cons_int @ Y5 @ nil_int ) ) )
& ( ( size_size_list_int @ Ys )
= N ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_835_length__append__singleton,axiom,
! [Xs2: list_P419547167332244514st_int,X: produc6425607678544837394st_int] :
( ( size_s229347708945616566st_int @ ( append5028292981072010935st_int @ Xs2 @ ( cons_P4151473878978182108st_int @ X @ nil_Pr749310521696113548st_int ) ) )
= ( suc @ ( size_s229347708945616566st_int @ Xs2 ) ) ) ).
% length_append_singleton
thf(fact_836_length__append__singleton,axiom,
! [Xs2: list_instr,X: instr] :
( ( size_size_list_instr @ ( append_instr @ Xs2 @ ( cons_instr @ X @ nil_instr ) ) )
= ( suc @ ( size_size_list_instr @ Xs2 ) ) ) ).
% length_append_singleton
thf(fact_837_length__append__singleton,axiom,
! [Xs2: list_list_char,X: list_char] :
( ( size_s356637359517785349t_char @ ( append_list_char @ Xs2 @ ( cons_list_char @ X @ nil_list_char ) ) )
= ( suc @ ( size_s356637359517785349t_char @ Xs2 ) ) ) ).
% length_append_singleton
thf(fact_838_length__append__singleton,axiom,
! [Xs2: list_int,X: int] :
( ( size_size_list_int @ ( append_int @ Xs2 @ ( cons_int @ X @ nil_int ) ) )
= ( suc @ ( size_size_list_int @ Xs2 ) ) ) ).
% length_append_singleton
thf(fact_839_com_Odistinct_I17_J,axiom,
! [X31: com,X32: com,X51: bexp,X52: com] :
( ( seq @ X31 @ X32 )
!= ( while @ X51 @ X52 ) ) ).
% com.distinct(17)
thf(fact_840_neg__eq__iff__add__eq__0,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= B )
= ( ( plus_plus_int @ A @ B )
= zero_zero_int ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_841_eq__neg__iff__add__eq__0,axiom,
! [A: int,B: int] :
( ( A
= ( uminus_uminus_int @ B ) )
= ( ( plus_plus_int @ A @ B )
= zero_zero_int ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_842_add_Oinverse__unique,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= zero_zero_int )
=> ( ( uminus_uminus_int @ A )
= B ) ) ).
% add.inverse_unique
thf(fact_843_ab__group__add__class_Oab__left__minus,axiom,
! [A: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
= zero_zero_int ) ).
% ab_group_add_class.ab_left_minus
thf(fact_844_add__eq__0__iff,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= zero_zero_int )
= ( B
= ( uminus_uminus_int @ A ) ) ) ).
% add_eq_0_iff
thf(fact_845_zero__neq__neg__one,axiom,
( zero_zero_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% zero_neq_neg_one
thf(fact_846_less__minus__one__simps_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% less_minus_one_simps(4)
thf(fact_847_less__minus__one__simps_I2_J,axiom,
ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% less_minus_one_simps(2)
thf(fact_848_group__cancel_Osub2,axiom,
! [B5: int,K: int,B: int,A: int] :
( ( B5
= ( plus_plus_int @ K @ B ) )
=> ( ( minus_minus_int @ A @ B5 )
= ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A @ B ) ) ) ) ).
% group_cancel.sub2
thf(fact_849_diff__conv__add__uminus,axiom,
( minus_minus_int
= ( ^ [A3: int,B2: int] : ( plus_plus_int @ A3 @ ( uminus_uminus_int @ B2 ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_850_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_minus_int
= ( ^ [A3: int,B2: int] : ( plus_plus_int @ A3 @ ( uminus_uminus_int @ B2 ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_851_int__cases,axiom,
! [Z2: int] :
( ! [N3: nat] :
( Z2
!= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ! [N3: nat] :
( Z2
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ).
% int_cases
thf(fact_852_int__of__nat__induct,axiom,
! [P: int > $o,Z2: int] :
( ! [N3: nat] : ( P @ ( semiri1314217659103216013at_int @ N3 ) )
=> ( ! [N3: nat] : ( P @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) )
=> ( P @ Z2 ) ) ) ).
% int_of_nat_induct
thf(fact_853_not__int__zless__negative,axiom,
! [N: nat,M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% not_int_zless_negative
thf(fact_854_minus__int__code_I2_J,axiom,
! [L: int] :
( ( minus_minus_int @ zero_zero_int @ L )
= ( uminus_uminus_int @ L ) ) ).
% minus_int_code(2)
thf(fact_855_list_Osize_I3_J,axiom,
( ( size_s229347708945616566st_int @ nil_Pr749310521696113548st_int )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_856_list_Osize_I3_J,axiom,
( ( size_size_list_instr @ nil_instr )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_857_list_Osize_I3_J,axiom,
( ( size_s356637359517785349t_char @ nil_list_char )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_858_list_Osize_I3_J,axiom,
( ( size_size_list_int @ nil_int )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_859_list__induct2,axiom,
! [Xs2: list_instr,Ys2: list_instr,P: list_instr > list_instr > $o] :
( ( ( size_size_list_instr @ Xs2 )
= ( size_size_list_instr @ Ys2 ) )
=> ( ( P @ nil_instr @ nil_instr )
=> ( ! [X4: instr,Xs3: list_instr,Y4: instr,Ys5: list_instr] :
( ( ( size_size_list_instr @ Xs3 )
= ( size_size_list_instr @ Ys5 ) )
=> ( ( P @ Xs3 @ Ys5 )
=> ( P @ ( cons_instr @ X4 @ Xs3 ) @ ( cons_instr @ Y4 @ Ys5 ) ) ) )
=> ( P @ Xs2 @ Ys2 ) ) ) ) ).
% list_induct2
thf(fact_860_list__induct2,axiom,
! [Xs2: list_instr,Ys2: list_int,P: list_instr > list_int > $o] :
( ( ( size_size_list_instr @ Xs2 )
= ( size_size_list_int @ Ys2 ) )
=> ( ( P @ nil_instr @ nil_int )
=> ( ! [X4: instr,Xs3: list_instr,Y4: int,Ys5: list_int] :
( ( ( size_size_list_instr @ Xs3 )
= ( size_size_list_int @ Ys5 ) )
=> ( ( P @ Xs3 @ Ys5 )
=> ( P @ ( cons_instr @ X4 @ Xs3 ) @ ( cons_int @ Y4 @ Ys5 ) ) ) )
=> ( P @ Xs2 @ Ys2 ) ) ) ) ).
% list_induct2
thf(fact_861_list__induct2,axiom,
! [Xs2: list_int,Ys2: list_instr,P: list_int > list_instr > $o] :
( ( ( size_size_list_int @ Xs2 )
= ( size_size_list_instr @ Ys2 ) )
=> ( ( P @ nil_int @ nil_instr )
=> ( ! [X4: int,Xs3: list_int,Y4: instr,Ys5: list_instr] :
( ( ( size_size_list_int @ Xs3 )
= ( size_size_list_instr @ Ys5 ) )
=> ( ( P @ Xs3 @ Ys5 )
=> ( P @ ( cons_int @ X4 @ Xs3 ) @ ( cons_instr @ Y4 @ Ys5 ) ) ) )
=> ( P @ Xs2 @ Ys2 ) ) ) ) ).
% list_induct2
thf(fact_862_list__induct2,axiom,
! [Xs2: list_int,Ys2: list_int,P: list_int > list_int > $o] :
( ( ( size_size_list_int @ Xs2 )
= ( size_size_list_int @ Ys2 ) )
=> ( ( P @ nil_int @ nil_int )
=> ( ! [X4: int,Xs3: list_int,Y4: int,Ys5: list_int] :
( ( ( size_size_list_int @ Xs3 )
= ( size_size_list_int @ Ys5 ) )
=> ( ( P @ Xs3 @ Ys5 )
=> ( P @ ( cons_int @ X4 @ Xs3 ) @ ( cons_int @ Y4 @ Ys5 ) ) ) )
=> ( P @ Xs2 @ Ys2 ) ) ) ) ).
% list_induct2
thf(fact_863_list__induct2,axiom,
! [Xs2: list_instr,Ys2: list_list_char,P: list_instr > list_list_char > $o] :
( ( ( size_size_list_instr @ Xs2 )
= ( size_s356637359517785349t_char @ Ys2 ) )
=> ( ( P @ nil_instr @ nil_list_char )
=> ( ! [X4: instr,Xs3: list_instr,Y4: list_char,Ys5: list_list_char] :
( ( ( size_size_list_instr @ Xs3 )
= ( size_s356637359517785349t_char @ Ys5 ) )
=> ( ( P @ Xs3 @ Ys5 )
=> ( P @ ( cons_instr @ X4 @ Xs3 ) @ ( cons_list_char @ Y4 @ Ys5 ) ) ) )
=> ( P @ Xs2 @ Ys2 ) ) ) ) ).
% list_induct2
thf(fact_864_list__induct2,axiom,
! [Xs2: list_list_char,Ys2: list_instr,P: list_list_char > list_instr > $o] :
( ( ( size_s356637359517785349t_char @ Xs2 )
= ( size_size_list_instr @ Ys2 ) )
=> ( ( P @ nil_list_char @ nil_instr )
=> ( ! [X4: list_char,Xs3: list_list_char,Y4: instr,Ys5: list_instr] :
( ( ( size_s356637359517785349t_char @ Xs3 )
= ( size_size_list_instr @ Ys5 ) )
=> ( ( P @ Xs3 @ Ys5 )
=> ( P @ ( cons_list_char @ X4 @ Xs3 ) @ ( cons_instr @ Y4 @ Ys5 ) ) ) )
=> ( P @ Xs2 @ Ys2 ) ) ) ) ).
% list_induct2
thf(fact_865_list__induct2,axiom,
! [Xs2: list_list_char,Ys2: list_int,P: list_list_char > list_int > $o] :
( ( ( size_s356637359517785349t_char @ Xs2 )
= ( size_size_list_int @ Ys2 ) )
=> ( ( P @ nil_list_char @ nil_int )
=> ( ! [X4: list_char,Xs3: list_list_char,Y4: int,Ys5: list_int] :
( ( ( size_s356637359517785349t_char @ Xs3 )
= ( size_size_list_int @ Ys5 ) )
=> ( ( P @ Xs3 @ Ys5 )
=> ( P @ ( cons_list_char @ X4 @ Xs3 ) @ ( cons_int @ Y4 @ Ys5 ) ) ) )
=> ( P @ Xs2 @ Ys2 ) ) ) ) ).
% list_induct2
thf(fact_866_list__induct2,axiom,
! [Xs2: list_int,Ys2: list_list_char,P: list_int > list_list_char > $o] :
( ( ( size_size_list_int @ Xs2 )
= ( size_s356637359517785349t_char @ Ys2 ) )
=> ( ( P @ nil_int @ nil_list_char )
=> ( ! [X4: int,Xs3: list_int,Y4: list_char,Ys5: list_list_char] :
( ( ( size_size_list_int @ Xs3 )
= ( size_s356637359517785349t_char @ Ys5 ) )
=> ( ( P @ Xs3 @ Ys5 )
=> ( P @ ( cons_int @ X4 @ Xs3 ) @ ( cons_list_char @ Y4 @ Ys5 ) ) ) )
=> ( P @ Xs2 @ Ys2 ) ) ) ) ).
% list_induct2
thf(fact_867_list__induct2,axiom,
! [Xs2: list_list_char,Ys2: list_list_char,P: list_list_char > list_list_char > $o] :
( ( ( size_s356637359517785349t_char @ Xs2 )
= ( size_s356637359517785349t_char @ Ys2 ) )
=> ( ( P @ nil_list_char @ nil_list_char )
=> ( ! [X4: list_char,Xs3: list_list_char,Y4: list_char,Ys5: list_list_char] :
( ( ( size_s356637359517785349t_char @ Xs3 )
= ( size_s356637359517785349t_char @ Ys5 ) )
=> ( ( P @ Xs3 @ Ys5 )
=> ( P @ ( cons_list_char @ X4 @ Xs3 ) @ ( cons_list_char @ Y4 @ Ys5 ) ) ) )
=> ( P @ Xs2 @ Ys2 ) ) ) ) ).
% list_induct2
thf(fact_868_list__induct2,axiom,
! [Xs2: list_P419547167332244514st_int,Ys2: list_instr,P: list_P419547167332244514st_int > list_instr > $o] :
( ( ( size_s229347708945616566st_int @ Xs2 )
= ( size_size_list_instr @ Ys2 ) )
=> ( ( P @ nil_Pr749310521696113548st_int @ nil_instr )
=> ( ! [X4: produc6425607678544837394st_int,Xs3: list_P419547167332244514st_int,Y4: instr,Ys5: list_instr] :
( ( ( size_s229347708945616566st_int @ Xs3 )
= ( size_size_list_instr @ Ys5 ) )
=> ( ( P @ Xs3 @ Ys5 )
=> ( P @ ( cons_P4151473878978182108st_int @ X4 @ Xs3 ) @ ( cons_instr @ Y4 @ Ys5 ) ) ) )
=> ( P @ Xs2 @ Ys2 ) ) ) ) ).
% list_induct2
thf(fact_869_list__induct3,axiom,
! [Xs2: list_instr,Ys2: list_instr,Zs: list_instr,P: list_instr > list_instr > list_instr > $o] :
( ( ( size_size_list_instr @ Xs2 )
= ( size_size_list_instr @ Ys2 ) )
=> ( ( ( size_size_list_instr @ Ys2 )
= ( size_size_list_instr @ Zs ) )
=> ( ( P @ nil_instr @ nil_instr @ nil_instr )
=> ( ! [X4: instr,Xs3: list_instr,Y4: instr,Ys5: list_instr,Z4: instr,Zs2: list_instr] :
( ( ( size_size_list_instr @ Xs3 )
= ( size_size_list_instr @ Ys5 ) )
=> ( ( ( size_size_list_instr @ Ys5 )
= ( size_size_list_instr @ Zs2 ) )
=> ( ( P @ Xs3 @ Ys5 @ Zs2 )
=> ( P @ ( cons_instr @ X4 @ Xs3 ) @ ( cons_instr @ Y4 @ Ys5 ) @ ( cons_instr @ Z4 @ Zs2 ) ) ) ) )
=> ( P @ Xs2 @ Ys2 @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_870_list__induct3,axiom,
! [Xs2: list_instr,Ys2: list_instr,Zs: list_int,P: list_instr > list_instr > list_int > $o] :
( ( ( size_size_list_instr @ Xs2 )
= ( size_size_list_instr @ Ys2 ) )
=> ( ( ( size_size_list_instr @ Ys2 )
= ( size_size_list_int @ Zs ) )
=> ( ( P @ nil_instr @ nil_instr @ nil_int )
=> ( ! [X4: instr,Xs3: list_instr,Y4: instr,Ys5: list_instr,Z4: int,Zs2: list_int] :
( ( ( size_size_list_instr @ Xs3 )
= ( size_size_list_instr @ Ys5 ) )
=> ( ( ( size_size_list_instr @ Ys5 )
= ( size_size_list_int @ Zs2 ) )
=> ( ( P @ Xs3 @ Ys5 @ Zs2 )
=> ( P @ ( cons_instr @ X4 @ Xs3 ) @ ( cons_instr @ Y4 @ Ys5 ) @ ( cons_int @ Z4 @ Zs2 ) ) ) ) )
=> ( P @ Xs2 @ Ys2 @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_871_list__induct3,axiom,
! [Xs2: list_instr,Ys2: list_int,Zs: list_instr,P: list_instr > list_int > list_instr > $o] :
( ( ( size_size_list_instr @ Xs2 )
= ( size_size_list_int @ Ys2 ) )
=> ( ( ( size_size_list_int @ Ys2 )
= ( size_size_list_instr @ Zs ) )
=> ( ( P @ nil_instr @ nil_int @ nil_instr )
=> ( ! [X4: instr,Xs3: list_instr,Y4: int,Ys5: list_int,Z4: instr,Zs2: list_instr] :
( ( ( size_size_list_instr @ Xs3 )
= ( size_size_list_int @ Ys5 ) )
=> ( ( ( size_size_list_int @ Ys5 )
= ( size_size_list_instr @ Zs2 ) )
=> ( ( P @ Xs3 @ Ys5 @ Zs2 )
=> ( P @ ( cons_instr @ X4 @ Xs3 ) @ ( cons_int @ Y4 @ Ys5 ) @ ( cons_instr @ Z4 @ Zs2 ) ) ) ) )
=> ( P @ Xs2 @ Ys2 @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_872_list__induct3,axiom,
! [Xs2: list_instr,Ys2: list_int,Zs: list_int,P: list_instr > list_int > list_int > $o] :
( ( ( size_size_list_instr @ Xs2 )
= ( size_size_list_int @ Ys2 ) )
=> ( ( ( size_size_list_int @ Ys2 )
= ( size_size_list_int @ Zs ) )
=> ( ( P @ nil_instr @ nil_int @ nil_int )
=> ( ! [X4: instr,Xs3: list_instr,Y4: int,Ys5: list_int,Z4: int,Zs2: list_int] :
( ( ( size_size_list_instr @ Xs3 )
= ( size_size_list_int @ Ys5 ) )
=> ( ( ( size_size_list_int @ Ys5 )
= ( size_size_list_int @ Zs2 ) )
=> ( ( P @ Xs3 @ Ys5 @ Zs2 )
=> ( P @ ( cons_instr @ X4 @ Xs3 ) @ ( cons_int @ Y4 @ Ys5 ) @ ( cons_int @ Z4 @ Zs2 ) ) ) ) )
=> ( P @ Xs2 @ Ys2 @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_873_list__induct3,axiom,
! [Xs2: list_int,Ys2: list_instr,Zs: list_instr,P: list_int > list_instr > list_instr > $o] :
( ( ( size_size_list_int @ Xs2 )
= ( size_size_list_instr @ Ys2 ) )
=> ( ( ( size_size_list_instr @ Ys2 )
= ( size_size_list_instr @ Zs ) )
=> ( ( P @ nil_int @ nil_instr @ nil_instr )
=> ( ! [X4: int,Xs3: list_int,Y4: instr,Ys5: list_instr,Z4: instr,Zs2: list_instr] :
( ( ( size_size_list_int @ Xs3 )
= ( size_size_list_instr @ Ys5 ) )
=> ( ( ( size_size_list_instr @ Ys5 )
= ( size_size_list_instr @ Zs2 ) )
=> ( ( P @ Xs3 @ Ys5 @ Zs2 )
=> ( P @ ( cons_int @ X4 @ Xs3 ) @ ( cons_instr @ Y4 @ Ys5 ) @ ( cons_instr @ Z4 @ Zs2 ) ) ) ) )
=> ( P @ Xs2 @ Ys2 @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_874_list__induct3,axiom,
! [Xs2: list_int,Ys2: list_instr,Zs: list_int,P: list_int > list_instr > list_int > $o] :
( ( ( size_size_list_int @ Xs2 )
= ( size_size_list_instr @ Ys2 ) )
=> ( ( ( size_size_list_instr @ Ys2 )
= ( size_size_list_int @ Zs ) )
=> ( ( P @ nil_int @ nil_instr @ nil_int )
=> ( ! [X4: int,Xs3: list_int,Y4: instr,Ys5: list_instr,Z4: int,Zs2: list_int] :
( ( ( size_size_list_int @ Xs3 )
= ( size_size_list_instr @ Ys5 ) )
=> ( ( ( size_size_list_instr @ Ys5 )
= ( size_size_list_int @ Zs2 ) )
=> ( ( P @ Xs3 @ Ys5 @ Zs2 )
=> ( P @ ( cons_int @ X4 @ Xs3 ) @ ( cons_instr @ Y4 @ Ys5 ) @ ( cons_int @ Z4 @ Zs2 ) ) ) ) )
=> ( P @ Xs2 @ Ys2 @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_875_list__induct3,axiom,
! [Xs2: list_int,Ys2: list_int,Zs: list_instr,P: list_int > list_int > list_instr > $o] :
( ( ( size_size_list_int @ Xs2 )
= ( size_size_list_int @ Ys2 ) )
=> ( ( ( size_size_list_int @ Ys2 )
= ( size_size_list_instr @ Zs ) )
=> ( ( P @ nil_int @ nil_int @ nil_instr )
=> ( ! [X4: int,Xs3: list_int,Y4: int,Ys5: list_int,Z4: instr,Zs2: list_instr] :
( ( ( size_size_list_int @ Xs3 )
= ( size_size_list_int @ Ys5 ) )
=> ( ( ( size_size_list_int @ Ys5 )
= ( size_size_list_instr @ Zs2 ) )
=> ( ( P @ Xs3 @ Ys5 @ Zs2 )
=> ( P @ ( cons_int @ X4 @ Xs3 ) @ ( cons_int @ Y4 @ Ys5 ) @ ( cons_instr @ Z4 @ Zs2 ) ) ) ) )
=> ( P @ Xs2 @ Ys2 @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_876_list__induct3,axiom,
! [Xs2: list_int,Ys2: list_int,Zs: list_int,P: list_int > list_int > list_int > $o] :
( ( ( size_size_list_int @ Xs2 )
= ( size_size_list_int @ Ys2 ) )
=> ( ( ( size_size_list_int @ Ys2 )
= ( size_size_list_int @ Zs ) )
=> ( ( P @ nil_int @ nil_int @ nil_int )
=> ( ! [X4: int,Xs3: list_int,Y4: int,Ys5: list_int,Z4: int,Zs2: list_int] :
( ( ( size_size_list_int @ Xs3 )
= ( size_size_list_int @ Ys5 ) )
=> ( ( ( size_size_list_int @ Ys5 )
= ( size_size_list_int @ Zs2 ) )
=> ( ( P @ Xs3 @ Ys5 @ Zs2 )
=> ( P @ ( cons_int @ X4 @ Xs3 ) @ ( cons_int @ Y4 @ Ys5 ) @ ( cons_int @ Z4 @ Zs2 ) ) ) ) )
=> ( P @ Xs2 @ Ys2 @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_877_list__induct3,axiom,
! [Xs2: list_instr,Ys2: list_instr,Zs: list_list_char,P: list_instr > list_instr > list_list_char > $o] :
( ( ( size_size_list_instr @ Xs2 )
= ( size_size_list_instr @ Ys2 ) )
=> ( ( ( size_size_list_instr @ Ys2 )
= ( size_s356637359517785349t_char @ Zs ) )
=> ( ( P @ nil_instr @ nil_instr @ nil_list_char )
=> ( ! [X4: instr,Xs3: list_instr,Y4: instr,Ys5: list_instr,Z4: list_char,Zs2: list_list_char] :
( ( ( size_size_list_instr @ Xs3 )
= ( size_size_list_instr @ Ys5 ) )
=> ( ( ( size_size_list_instr @ Ys5 )
= ( size_s356637359517785349t_char @ Zs2 ) )
=> ( ( P @ Xs3 @ Ys5 @ Zs2 )
=> ( P @ ( cons_instr @ X4 @ Xs3 ) @ ( cons_instr @ Y4 @ Ys5 ) @ ( cons_list_char @ Z4 @ Zs2 ) ) ) ) )
=> ( P @ Xs2 @ Ys2 @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_878_list__induct3,axiom,
! [Xs2: list_instr,Ys2: list_list_char,Zs: list_instr,P: list_instr > list_list_char > list_instr > $o] :
( ( ( size_size_list_instr @ Xs2 )
= ( size_s356637359517785349t_char @ Ys2 ) )
=> ( ( ( size_s356637359517785349t_char @ Ys2 )
= ( size_size_list_instr @ Zs ) )
=> ( ( P @ nil_instr @ nil_list_char @ nil_instr )
=> ( ! [X4: instr,Xs3: list_instr,Y4: list_char,Ys5: list_list_char,Z4: instr,Zs2: list_instr] :
( ( ( size_size_list_instr @ Xs3 )
= ( size_s356637359517785349t_char @ Ys5 ) )
=> ( ( ( size_s356637359517785349t_char @ Ys5 )
= ( size_size_list_instr @ Zs2 ) )
=> ( ( P @ Xs3 @ Ys5 @ Zs2 )
=> ( P @ ( cons_instr @ X4 @ Xs3 ) @ ( cons_list_char @ Y4 @ Ys5 ) @ ( cons_instr @ Z4 @ Zs2 ) ) ) ) )
=> ( P @ Xs2 @ Ys2 @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_879_less__minus__one__simps_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% less_minus_one_simps(3)
thf(fact_880_less__minus__one__simps_I1_J,axiom,
ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% less_minus_one_simps(1)
thf(fact_881_int__cases4,axiom,
! [M: int] :
( ! [N3: nat] :
( M
!= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( M
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% int_cases4
thf(fact_882_Seq,axiom,
! [C_1: com,S_1: list_char > int,S_22: list_char > int,C_2: com,S_3: list_char > int] :
( ( big_big_step @ ( produc5595214716300948949ar_int @ C_1 @ S_1 ) @ S_22 )
=> ( ( big_big_step @ ( produc5595214716300948949ar_int @ C_2 @ S_22 ) @ S_3 )
=> ( big_big_step @ ( produc5595214716300948949ar_int @ ( seq @ C_1 @ C_2 ) @ S_1 ) @ S_3 ) ) ) ).
% Seq
thf(fact_883_SeqE,axiom,
! [C1: com,C22: com,S1: list_char > int,S32: list_char > int] :
( ( big_big_step @ ( produc5595214716300948949ar_int @ ( seq @ C1 @ C22 ) @ S1 ) @ S32 )
=> ~ ! [S_2: list_char > int] :
( ( big_big_step @ ( produc5595214716300948949ar_int @ C1 @ S1 ) @ S_2 )
=> ~ ( big_big_step @ ( produc5595214716300948949ar_int @ C22 @ S_2 ) @ S32 ) ) ) ).
% SeqE
thf(fact_884_Seq__assoc,axiom,
! [C1: com,C22: com,C32: com,S: list_char > int,S5: list_char > int] :
( ( big_big_step @ ( produc5595214716300948949ar_int @ ( seq @ ( seq @ C1 @ C22 ) @ C32 ) @ S ) @ S5 )
= ( big_big_step @ ( produc5595214716300948949ar_int @ ( seq @ C1 @ ( seq @ C22 @ C32 ) ) @ S ) @ S5 ) ) ).
% Seq_assoc
thf(fact_885_nth__append,axiom,
! [N: nat,Xs2: list_P419547167332244514st_int,Ys2: list_P419547167332244514st_int] :
( ( ( ord_less_nat @ N @ ( size_s229347708945616566st_int @ Xs2 ) )
=> ( ( nth_Pr8327173284404383691st_int @ ( append5028292981072010935st_int @ Xs2 @ Ys2 ) @ N )
= ( nth_Pr8327173284404383691st_int @ Xs2 @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_s229347708945616566st_int @ Xs2 ) )
=> ( ( nth_Pr8327173284404383691st_int @ ( append5028292981072010935st_int @ Xs2 @ Ys2 ) @ N )
= ( nth_Pr8327173284404383691st_int @ Ys2 @ ( minus_minus_nat @ N @ ( size_s229347708945616566st_int @ Xs2 ) ) ) ) ) ) ).
% nth_append
thf(fact_886_nth__append,axiom,
! [N: nat,Xs2: list_instr,Ys2: list_instr] :
( ( ( ord_less_nat @ N @ ( size_size_list_instr @ Xs2 ) )
=> ( ( nth_instr @ ( append_instr @ Xs2 @ Ys2 ) @ N )
= ( nth_instr @ Xs2 @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_size_list_instr @ Xs2 ) )
=> ( ( nth_instr @ ( append_instr @ Xs2 @ Ys2 ) @ N )
= ( nth_instr @ Ys2 @ ( minus_minus_nat @ N @ ( size_size_list_instr @ Xs2 ) ) ) ) ) ) ).
% nth_append
thf(fact_887_nth__append,axiom,
! [N: nat,Xs2: list_list_char,Ys2: list_list_char] :
( ( ( ord_less_nat @ N @ ( size_s356637359517785349t_char @ Xs2 ) )
=> ( ( nth_list_char @ ( append_list_char @ Xs2 @ Ys2 ) @ N )
= ( nth_list_char @ Xs2 @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_s356637359517785349t_char @ Xs2 ) )
=> ( ( nth_list_char @ ( append_list_char @ Xs2 @ Ys2 ) @ N )
= ( nth_list_char @ Ys2 @ ( minus_minus_nat @ N @ ( size_s356637359517785349t_char @ Xs2 ) ) ) ) ) ) ).
% nth_append
thf(fact_888_nth__append,axiom,
! [N: nat,Xs2: list_int,Ys2: list_int] :
( ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs2 ) )
=> ( ( nth_int @ ( append_int @ Xs2 @ Ys2 ) @ N )
= ( nth_int @ Xs2 @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_size_list_int @ Xs2 ) )
=> ( ( nth_int @ ( append_int @ Xs2 @ Ys2 ) @ N )
= ( nth_int @ Ys2 @ ( minus_minus_nat @ N @ ( size_size_list_int @ Xs2 ) ) ) ) ) ) ).
% nth_append
thf(fact_889_int__cases3,axiom,
! [K: int] :
( ( K != zero_zero_int )
=> ( ! [N3: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) )
=> ~ ! [N3: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ) ).
% int_cases3
thf(fact_890_negative__zless__0,axiom,
! [N: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ zero_zero_int ) ).
% negative_zless_0
thf(fact_891_negD,axiom,
! [X: int] :
( ( ord_less_int @ X @ zero_zero_int )
=> ? [N3: nat] :
( X
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ).
% negD
thf(fact_892_neg__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ K @ zero_zero_int )
=> ~ ! [N3: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% neg_int_cases
thf(fact_893_execl__all__SKIP,axiom,
! [Cfs: list_P419547167332244514st_int] :
( ( execl_all @ nil_instr @ Cfs )
=> ( ( cpred @ skip @ ( nth_Pr8327173284404383691st_int @ Cfs @ zero_zero_nat ) @ ( nth_Pr8327173284404383691st_int @ Cfs @ ( minus_minus_nat @ ( size_s229347708945616566st_int @ Cfs ) @ ( suc @ zero_zero_nat ) ) ) )
& ( ( size_s229347708945616566st_int @ Cfs )
= ( suc @ zero_zero_nat ) ) ) ) ).
% execl_all_SKIP
thf(fact_894_inth__append,axiom,
! [I: int,Xs2: list_instr,Ys2: list_instr] :
( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ Xs2 ) ) )
=> ( ( inth_instr @ ( append_instr @ Xs2 @ Ys2 ) @ I )
= ( inth_instr @ Xs2 @ I ) ) )
& ( ~ ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ Xs2 ) ) )
=> ( ( inth_instr @ ( append_instr @ Xs2 @ Ys2 ) @ I )
= ( inth_instr @ Ys2 @ ( minus_minus_int @ I @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ Xs2 ) ) ) ) ) ) ) ) ).
% inth_append
thf(fact_895_inth__append,axiom,
! [I: int,Xs2: list_list_char,Ys2: list_list_char] :
( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ Xs2 ) ) )
=> ( ( inth_list_char @ ( append_list_char @ Xs2 @ Ys2 ) @ I )
= ( inth_list_char @ Xs2 @ I ) ) )
& ( ~ ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ Xs2 ) ) )
=> ( ( inth_list_char @ ( append_list_char @ Xs2 @ Ys2 ) @ I )
= ( inth_list_char @ Ys2 @ ( minus_minus_int @ I @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ Xs2 ) ) ) ) ) ) ) ) ).
% inth_append
thf(fact_896_inth__append,axiom,
! [I: int,Xs2: list_int,Ys2: list_int] :
( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_size_list_int @ Xs2 ) ) )
=> ( ( inth_int @ ( append_int @ Xs2 @ Ys2 ) @ I )
= ( inth_int @ Xs2 @ I ) ) )
& ( ~ ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_size_list_int @ Xs2 ) ) )
=> ( ( inth_int @ ( append_int @ Xs2 @ Ys2 ) @ I )
= ( inth_int @ Ys2 @ ( minus_minus_int @ I @ ( semiri1314217659103216013at_int @ ( size_size_list_int @ Xs2 ) ) ) ) ) ) ) ) ).
% inth_append
thf(fact_897_dbl__dec__simps_I2_J,axiom,
( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
= ( uminus_uminus_int @ one_one_int ) ) ).
% dbl_dec_simps(2)
thf(fact_898_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_899_add__le__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_900_add__le__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_901_add__le__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_902_add__le__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_903_neg__le__iff__le,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% neg_le_iff_le
thf(fact_904_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_905_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_906_dbl__dec__simps_I3_J,axiom,
( ( neg_nu3811975205180677377ec_int @ one_one_int )
= one_one_int ) ).
% dbl_dec_simps(3)
thf(fact_907_add__le__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel1
thf(fact_908_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_909_add__le__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel2
thf(fact_910_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_911_le__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel1
thf(fact_912_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_913_le__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel2
thf(fact_914_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_915_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_916_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_917_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_918_neg__0__le__iff__le,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% neg_0_le_iff_le
thf(fact_919_neg__le__0__iff__le,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% neg_le_0_iff_le
thf(fact_920_less__eq__neg__nonpos,axiom,
! [A: int] :
( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% less_eq_neg_nonpos
thf(fact_921_neg__less__eq__nonneg,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ A )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% neg_less_eq_nonneg
thf(fact_922_le__add__diff__inverse,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_923_le__add__diff__inverse,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_924_le__add__diff__inverse2,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_925_le__add__diff__inverse2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_926_negative__zle,axiom,
! [N: nat,M: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% negative_zle
thf(fact_927_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_928_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_929_zle__add1__eq__le,axiom,
! [W: int,Z2: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z2 @ one_one_int ) )
= ( ord_less_eq_int @ W @ Z2 ) ) ).
% zle_add1_eq_le
thf(fact_930_zle__diff1__eq,axiom,
! [W: int,Z2: int] :
( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z2 @ one_one_int ) )
= ( ord_less_int @ W @ Z2 ) ) ).
% zle_diff1_eq
thf(fact_931_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_932_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_933_verit__comp__simplify1_I2_J,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_934_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_935_verit__la__generic,axiom,
! [A: int,X: int] :
( ( ord_less_eq_int @ A @ X )
| ( A = X )
| ( ord_less_eq_int @ X @ A ) ) ).
% verit_la_generic
thf(fact_936_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_937_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_938_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_939_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_940_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_941_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_942_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_943_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_944_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_945_verit__comp__simplify1_I3_J,axiom,
! [B3: int,A4: int] :
( ( ~ ( ord_less_eq_int @ B3 @ A4 ) )
= ( ord_less_int @ A4 @ B3 ) ) ).
% verit_comp_simplify1(3)
thf(fact_946_verit__comp__simplify1_I3_J,axiom,
! [B3: nat,A4: nat] :
( ( ~ ( ord_less_eq_nat @ B3 @ A4 ) )
= ( ord_less_nat @ A4 @ B3 ) ) ).
% verit_comp_simplify1(3)
thf(fact_947_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_948_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_949_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_950_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_951_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_952_add__mono__thms__linordered__semiring_I1_J,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_mono_thms_linordered_semiring(1)
thf(fact_953_add__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_mono
thf(fact_954_add__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_955_add__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_left_mono
thf(fact_956_add__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_957_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C5: nat] :
( B
!= ( plus_plus_nat @ A @ C5 ) ) ) ).
% less_eqE
thf(fact_958_add__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_right_mono
thf(fact_959_add__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_960_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B2: nat] :
? [C4: nat] :
( B2
= ( plus_plus_nat @ A3 @ C4 ) ) ) ) ).
% le_iff_add
thf(fact_961_add__le__imp__le__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_962_add__le__imp__le__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_963_add__le__imp__le__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_964_add__le__imp__le__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_965_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_966_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_967_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_968_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_969_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_970_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_971_le__minus__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ B ) )
= ( ord_less_eq_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% le_minus_iff
thf(fact_972_minus__le__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
= ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% minus_le_iff
thf(fact_973_le__imp__neg__le,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% le_imp_neg_le
thf(fact_974_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_975_add__decreasing,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_976_add__decreasing,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_977_add__increasing,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_978_add__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_979_add__decreasing2,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_980_add__decreasing2,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_981_add__increasing2,axiom,
! [C: int,B: int,A: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_982_add__increasing2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_983_add__nonneg__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_984_add__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_985_add__nonpos__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_nonpos
thf(fact_986_add__nonpos__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_987_add__nonneg__eq__0__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ( plus_plus_int @ X @ Y )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_988_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_989_add__nonpos__eq__0__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ zero_zero_int )
=> ( ( ord_less_eq_int @ Y @ zero_zero_int )
=> ( ( ( plus_plus_int @ X @ Y )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_990_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_991_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_992_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_993_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_994_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_995_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_996_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_997_add__mono__thms__linordered__field_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_998_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_999_add__mono__thms__linordered__field_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_1000_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_1001_add__le__less__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_1002_add__le__less__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_1003_add__less__le__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_1004_add__less__le__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_1005_le__iff__diff__le__0,axiom,
( ord_less_eq_int
= ( ^ [A3: int,B2: int] : ( ord_less_eq_int @ ( minus_minus_int @ A3 @ B2 ) @ zero_zero_int ) ) ) ).
% le_iff_diff_le_0
thf(fact_1006_add__le__imp__le__diff,axiom,
! [I: int,K: int,N: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
=> ( ord_less_eq_int @ I @ ( minus_minus_int @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_1007_add__le__imp__le__diff,axiom,
! [I: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_1008_add__le__add__imp__diff__le,axiom,
! [I: int,K: int,N: int,J: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
=> ( ord_less_eq_int @ ( minus_minus_int @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_1009_add__le__add__imp__diff__le,axiom,
! [I: nat,K: nat,N: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_1010_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ( ( minus_minus_nat @ B @ A )
= C )
= ( B
= ( plus_plus_nat @ C @ A ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_1011_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_1012_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_1013_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
= ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_1014_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_1015_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_1016_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_1017_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_1018_le__add__diff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% le_add_diff
thf(fact_1019_diff__add,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
= B ) ) ).
% diff_add
thf(fact_1020_le__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ A @ ( minus_minus_int @ C @ B ) )
= ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% le_diff_eq
thf(fact_1021_diff__le__eq,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ ( minus_minus_int @ A @ B ) @ C )
= ( ord_less_eq_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% diff_le_eq
thf(fact_1022_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_1023_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_1024_le__minus__one__simps_I4_J,axiom,
~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% le_minus_one_simps(4)
thf(fact_1025_le__minus__one__simps_I2_J,axiom,
ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% le_minus_one_simps(2)
thf(fact_1026_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_1027_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_1028_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_1029_zle__iff__zadd,axiom,
( ord_less_eq_int
= ( ^ [W2: int,Z3: int] :
? [N4: nat] :
( Z3
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ N4 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_1030_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_1031_com_Odistinct_I7_J,axiom,
! [X51: bexp,X52: com] :
( skip
!= ( while @ X51 @ X52 ) ) ).
% com.distinct(7)
thf(fact_1032_com_Odistinct_I3_J,axiom,
! [X31: com,X32: com] :
( skip
!= ( seq @ X31 @ X32 ) ) ).
% com.distinct(3)
thf(fact_1033_add__strict__increasing2,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_1034_add__strict__increasing2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_1035_add__strict__increasing,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_1036_add__strict__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_1037_add__pos__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_1038_add__pos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_1039_add__nonpos__neg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_neg
thf(fact_1040_add__nonpos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_1041_add__nonneg__pos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_1042_add__nonneg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_1043_add__neg__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_neg_nonpos
thf(fact_1044_add__neg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_1045_le__minus__one__simps_I3_J,axiom,
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% le_minus_one_simps(3)
thf(fact_1046_le__minus__one__simps_I1_J,axiom,
ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% le_minus_one_simps(1)
thf(fact_1047_int__zle__neg,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) )
= ( ( N = zero_zero_nat )
& ( M = zero_zero_nat ) ) ) ).
% int_zle_neg
thf(fact_1048_int__one__le__iff__zero__less,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ one_one_int @ Z2 )
= ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% int_one_le_iff_zero_less
thf(fact_1049_nonpos__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ~ ! [N3: nat] :
( K
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% nonpos_int_cases
thf(fact_1050_negative__zle__0,axiom,
! [N: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ zero_zero_int ) ).
% negative_zle_0
thf(fact_1051_zless__imp__add1__zle,axiom,
! [W: int,Z2: int] :
( ( ord_less_int @ W @ Z2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z2 ) ) ).
% zless_imp_add1_zle
thf(fact_1052_add1__zle__eq,axiom,
! [W: int,Z2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z2 )
= ( ord_less_int @ W @ Z2 ) ) ).
% add1_zle_eq
thf(fact_1053_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_1054_SkipE,axiom,
! [S: list_char > int,T: list_char > int] :
( ( big_big_step @ ( produc5595214716300948949ar_int @ skip @ S ) @ T )
=> ( T = S ) ) ).
% SkipE
thf(fact_1055_Skip,axiom,
! [S: list_char > int] : ( big_big_step @ ( produc5595214716300948949ar_int @ skip @ S ) @ S ) ).
% Skip
thf(fact_1056_com_Osize_I6_J,axiom,
( ( size_size_com @ skip )
= zero_zero_nat ) ).
% com.size(6)
thf(fact_1057_com_Osize__gen_I1_J,axiom,
( ( size_com @ skip )
= zero_zero_nat ) ).
% com.size_gen(1)
thf(fact_1058_ccomp_Osimps_I1_J,axiom,
( ( ccomp @ skip )
= nil_instr ) ).
% ccomp.simps(1)
thf(fact_1059_not__zle__0__negative,axiom,
! [N: nat] :
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ).
% not_zle_0_negative
thf(fact_1060_le__imp__0__less,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z2 ) ) ) ).
% le_imp_0_less
thf(fact_1061_dbl__dec__def,axiom,
( neg_nu3811975205180677377ec_int
= ( ^ [X3: int] : ( minus_minus_int @ ( plus_plus_int @ X3 @ X3 ) @ one_one_int ) ) ) ).
% dbl_dec_def
thf(fact_1062_n__lists__Nil,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( n_lists_instr @ N @ nil_instr )
= ( cons_list_instr @ nil_instr @ nil_list_instr ) ) )
& ( ( N != zero_zero_nat )
=> ( ( n_lists_instr @ N @ nil_instr )
= nil_list_instr ) ) ) ).
% n_lists_Nil
thf(fact_1063_n__lists__Nil,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( n_list7311978343830957444st_int @ N @ nil_Pr749310521696113548st_int )
= ( cons_l8648290059330449388st_int @ nil_Pr749310521696113548st_int @ nil_li4597314861645524892st_int ) ) )
& ( ( N != zero_zero_nat )
=> ( ( n_list7311978343830957444st_int @ N @ nil_Pr749310521696113548st_int )
= nil_li4597314861645524892st_int ) ) ) ).
% n_lists_Nil
thf(fact_1064_n__lists__Nil,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( n_lists_int @ N @ nil_int )
= ( cons_list_int @ nil_int @ nil_list_int ) ) )
& ( ( N != zero_zero_nat )
=> ( ( n_lists_int @ N @ nil_int )
= nil_list_int ) ) ) ).
% n_lists_Nil
thf(fact_1065_n__lists__Nil,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( n_lists_list_char @ N @ nil_list_char )
= ( cons_list_list_char @ nil_list_char @ nil_list_list_char ) ) )
& ( ( N != zero_zero_nat )
=> ( ( n_lists_list_char @ N @ nil_list_char )
= nil_list_list_char ) ) ) ).
% n_lists_Nil
thf(fact_1066_n__lists__Nil,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( n_lists_char @ N @ nil_char )
= ( cons_list_char @ nil_char @ nil_list_char ) ) )
& ( ( N != zero_zero_nat )
=> ( ( n_lists_char @ N @ nil_char )
= nil_list_char ) ) ) ).
% n_lists_Nil
thf(fact_1067_exec1I,axiom,
! [C2: produc6425607678544837394st_int,P: list_instr,I: int,S: list_char > int,Stk3: list_int] :
( ( C2
= ( iexec @ ( inth_instr @ P @ I ) @ ( produc5086643055186798020st_int @ I @ ( produc8650753666468850689st_int @ S @ Stk3 ) ) ) )
=> ( ( 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 @ Stk3 ) ) @ C2 ) ) ) ) ).
% exec1I
thf(fact_1068_m__config__equiv_Osimps,axiom,
! [I: int,Ms: int > int,Sp: nat,I4: int,Ms2: int > int,Sp2: nat] :
( ( m_config_equiv @ ( produc8342123264692915235nt_nat @ I @ ( produc2630796075472406112nt_nat @ Ms @ Sp ) ) @ ( produc8342123264692915235nt_nat @ I4 @ ( produc2630796075472406112nt_nat @ Ms2 @ Sp2 ) ) )
= ( ( I = I4 )
& ( Sp = Sp2 )
& ! [A3: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ Sp ) ) @ A3 )
=> ( ( Ms @ A3 )
= ( Ms2 @ A3 ) ) ) ) ) ).
% m_config_equiv.simps
thf(fact_1069_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_1070_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_1071_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_1072_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_1073_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_1074_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_1075_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_1076_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_1077_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_1078_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_1079_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_1080_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_1081_drop__all,axiom,
! [Xs2: list_P419547167332244514st_int,N: nat] :
( ( ord_less_eq_nat @ ( size_s229347708945616566st_int @ Xs2 ) @ N )
=> ( ( drop_P7838799151754935884st_int @ N @ Xs2 )
= nil_Pr749310521696113548st_int ) ) ).
% drop_all
thf(fact_1082_drop__all,axiom,
! [Xs2: list_instr,N: nat] :
( ( ord_less_eq_nat @ ( size_size_list_instr @ Xs2 ) @ N )
=> ( ( drop_instr @ N @ Xs2 )
= nil_instr ) ) ).
% drop_all
thf(fact_1083_drop__all,axiom,
! [Xs2: list_list_char,N: nat] :
( ( ord_less_eq_nat @ ( size_s356637359517785349t_char @ Xs2 ) @ N )
=> ( ( drop_list_char @ N @ Xs2 )
= nil_list_char ) ) ).
% drop_all
thf(fact_1084_drop__all,axiom,
! [Xs2: list_int,N: nat] :
( ( ord_less_eq_nat @ ( size_size_list_int @ Xs2 ) @ N )
=> ( ( drop_int @ N @ Xs2 )
= nil_int ) ) ).
% drop_all
thf(fact_1085_drop__eq__Nil,axiom,
! [N: nat,Xs2: list_P419547167332244514st_int] :
( ( ( drop_P7838799151754935884st_int @ N @ Xs2 )
= nil_Pr749310521696113548st_int )
= ( ord_less_eq_nat @ ( size_s229347708945616566st_int @ Xs2 ) @ N ) ) ).
% drop_eq_Nil
thf(fact_1086_drop__eq__Nil,axiom,
! [N: nat,Xs2: list_instr] :
( ( ( drop_instr @ N @ Xs2 )
= nil_instr )
= ( ord_less_eq_nat @ ( size_size_list_instr @ Xs2 ) @ N ) ) ).
% drop_eq_Nil
thf(fact_1087_drop__eq__Nil,axiom,
! [N: nat,Xs2: list_list_char] :
( ( ( drop_list_char @ N @ Xs2 )
= nil_list_char )
= ( ord_less_eq_nat @ ( size_s356637359517785349t_char @ Xs2 ) @ N ) ) ).
% drop_eq_Nil
thf(fact_1088_drop__eq__Nil,axiom,
! [N: nat,Xs2: list_int] :
( ( ( drop_int @ N @ Xs2 )
= nil_int )
= ( ord_less_eq_nat @ ( size_size_list_int @ Xs2 ) @ N ) ) ).
% drop_eq_Nil
thf(fact_1089_drop__eq__Nil2,axiom,
! [N: nat,Xs2: list_instr] :
( ( nil_instr
= ( drop_instr @ N @ Xs2 ) )
= ( ord_less_eq_nat @ ( size_size_list_instr @ Xs2 ) @ N ) ) ).
% drop_eq_Nil2
thf(fact_1090_drop__eq__Nil2,axiom,
! [N: nat,Xs2: list_list_char] :
( ( nil_list_char
= ( drop_list_char @ N @ Xs2 ) )
= ( ord_less_eq_nat @ ( size_s356637359517785349t_char @ Xs2 ) @ N ) ) ).
% drop_eq_Nil2
thf(fact_1091_drop__eq__Nil2,axiom,
! [N: nat,Xs2: list_int] :
( ( nil_int
= ( drop_int @ N @ Xs2 ) )
= ( ord_less_eq_nat @ ( size_size_list_int @ Xs2 ) @ N ) ) ).
% drop_eq_Nil2
thf(fact_1092_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_1093_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_1094_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_1095_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_1096_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_1097_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 ) )
=> ? [X4: nat] :
( ( P @ X4 )
& ! [Y6: nat] :
( ( P @ Y6 )
=> ( ord_less_eq_nat @ Y6 @ X4 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_1098_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_1099_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_1100_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_1101_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1102_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_1103_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_1104_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_1105_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_1106_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_1107_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_1108_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_1109_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_1110_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_1111_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X4: nat] : ( R @ X4 @ X4 )
=> ( ! [X4: nat,Y4: nat,Z4: nat] :
( ( R @ X4 @ Y4 )
=> ( ( R @ Y4 @ Z4 )
=> ( R @ X4 @ Z4 ) ) )
=> ( ! [N3: nat] : ( R @ N3 @ ( suc @ N3 ) )
=> ( R @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1112_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_1113_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_1114_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_1115_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_1116_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_1117_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M5: nat,N4: nat] :
( ( ord_less_eq_nat @ M5 @ N4 )
& ( M5 != N4 ) ) ) ) ).
% nat_less_le
thf(fact_1118_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_1119_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_1120_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_1121_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_1122_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_1123_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_1124_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_1125_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_1126_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_1127_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_1128_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_1129_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_1130_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_1131_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_1132_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_1133_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_1134_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_1135_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_1136_exec1__appendR,axiom,
! [P: list_instr,C: produc6425607678544837394st_int,C2: produc6425607678544837394st_int,P4: list_instr] :
( ( exec1 @ P @ C @ C2 )
=> ( exec1 @ ( append_instr @ P @ P4 ) @ C @ C2 ) ) ).
% exec1_appendR
thf(fact_1137_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ K2 )
=> ~ ( P @ I5 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1138_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_1139_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_1140_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_1141_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_1142_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_1143_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_1144_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_1145_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N4: nat] : ( ord_less_eq_nat @ ( suc @ N4 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1146_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_1147_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_1148_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_1149_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_1150_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_1151_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_1152_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_1153_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_1154_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_1155_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_1156_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B2: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_1157_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_1158_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N )
& ! [I5: nat] :
( ( ord_less_eq_nat @ I5 @ K2 )
=> ~ ( P @ I5 ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1159_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_1160_exec1__appendL,axiom,
! [P: list_instr,I: int,S: list_char > int,Stk3: list_int,I4: int,S5: list_char > int,Stk2: list_int,P4: list_instr] :
( ( exec1 @ P @ ( produc5086643055186798020st_int @ I @ ( produc8650753666468850689st_int @ S @ Stk3 ) ) @ ( produc5086643055186798020st_int @ I4 @ ( produc8650753666468850689st_int @ S5 @ Stk2 ) ) )
=> ( exec1 @ ( append_instr @ P4 @ P ) @ ( produc5086643055186798020st_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P4 ) ) @ I ) @ ( produc8650753666468850689st_int @ S @ Stk3 ) ) @ ( produc5086643055186798020st_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P4 ) ) @ I4 ) @ ( produc8650753666468850689st_int @ S5 @ Stk2 ) ) ) ) ).
% exec1_appendL
thf(fact_1161_zdiff__int__split,axiom,
! [P: int > $o,X: nat,Y: nat] :
( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X @ Y ) ) )
= ( ( ( ord_less_eq_nat @ Y @ X )
=> ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y ) ) ) )
& ( ( ord_less_nat @ X @ Y )
=> ( P @ zero_zero_int ) ) ) ) ).
% zdiff_int_split
thf(fact_1162_Compiler_Obcomp__correct,axiom,
! [I: int,B: bexp,F: $o,S: list_char > int,Stk3: list_int] :
( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( star_P707599355569300323st_int @ ( exec1 @ ( bcomp @ ( produc4047900504771817624_o_int @ B @ ( product_Pair_o_int @ F @ I ) ) ) ) @ ( produc5086643055186798020st_int @ zero_zero_int @ ( produc8650753666468850689st_int @ S @ Stk3 ) )
@ ( produc5086643055186798020st_int
@ ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ ( bcomp @ ( produc4047900504771817624_o_int @ B @ ( product_Pair_o_int @ F @ I ) ) ) ) )
@ ( if_int
@ ( F
= ( bval @ B @ S ) )
@ I
@ zero_zero_int ) )
@ ( produc8650753666468850689st_int @ S @ Stk3 ) ) ) ) ).
% Compiler.bcomp_correct
thf(fact_1163_exec__append__trans,axiom,
! [P: list_instr,S: list_char > int,Stk3: list_int,I4: int,S5: list_char > int,Stk2: list_int,P4: 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 @ Stk3 ) ) @ ( produc5086643055186798020st_int @ I4 @ ( produc8650753666468850689st_int @ S5 @ Stk2 ) ) )
=> ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P ) ) @ I4 )
=> ( ( star_P707599355569300323st_int @ ( exec1 @ P4 ) @ ( produc5086643055186798020st_int @ ( minus_minus_int @ I4 @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P ) ) ) @ ( produc8650753666468850689st_int @ S5 @ Stk2 ) ) @ ( 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 @ P4 ) ) @ ( produc5086643055186798020st_int @ zero_zero_int @ ( produc8650753666468850689st_int @ S @ Stk3 ) ) @ ( produc5086643055186798020st_int @ J4 @ ( produc8650753666468850689st_int @ S6 @ Stk4 ) ) ) ) ) ) ) ).
% exec_append_trans
thf(fact_1164_exec__Cons__1,axiom,
! [P: list_instr,S: list_char > int,Stk3: list_int,J: int,T: list_char > int,Stk2: list_int,Ins2: instr] :
( ( star_P707599355569300323st_int @ ( exec1 @ P ) @ ( produc5086643055186798020st_int @ zero_zero_int @ ( produc8650753666468850689st_int @ S @ Stk3 ) ) @ ( produc5086643055186798020st_int @ J @ ( produc8650753666468850689st_int @ T @ Stk2 ) ) )
=> ( star_P707599355569300323st_int @ ( exec1 @ ( cons_instr @ Ins2 @ P ) ) @ ( produc5086643055186798020st_int @ one_one_int @ ( produc8650753666468850689st_int @ S @ Stk3 ) ) @ ( produc5086643055186798020st_int @ ( plus_plus_int @ one_one_int @ J ) @ ( produc8650753666468850689st_int @ T @ Stk2 ) ) ) ) ).
% exec_Cons_1
thf(fact_1165_exec__appendL__if,axiom,
! [P4: list_instr,I: int,P: list_instr,S: list_char > int,Stk3: list_int,J: int,S5: list_char > int,Stk2: list_int,I4: int] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P4 ) ) @ I )
=> ( ( star_P707599355569300323st_int @ ( exec1 @ P ) @ ( produc5086643055186798020st_int @ ( minus_minus_int @ I @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P4 ) ) ) @ ( produc8650753666468850689st_int @ S @ Stk3 ) ) @ ( produc5086643055186798020st_int @ J @ ( produc8650753666468850689st_int @ S5 @ Stk2 ) ) )
=> ( ( I4
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P4 ) ) @ J ) )
=> ( star_P707599355569300323st_int @ ( exec1 @ ( append_instr @ P4 @ P ) ) @ ( produc5086643055186798020st_int @ I @ ( produc8650753666468850689st_int @ S @ Stk3 ) ) @ ( produc5086643055186798020st_int @ I4 @ ( produc8650753666468850689st_int @ S5 @ Stk2 ) ) ) ) ) ) ).
% exec_appendL_if
thf(fact_1166_exec__appendR,axiom,
! [P: list_instr,C: produc6425607678544837394st_int,C2: produc6425607678544837394st_int,P4: list_instr] :
( ( star_P707599355569300323st_int @ ( exec1 @ P ) @ C @ C2 )
=> ( star_P707599355569300323st_int @ ( exec1 @ ( append_instr @ P @ P4 ) ) @ C @ C2 ) ) ).
% exec_appendR
thf(fact_1167_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 ) ) )
=> ( ! [A5: int,Aa: list_char > int,B4: list_int] : ( P @ A5 @ Aa @ B4 @ A5 @ Aa @ B4 )
=> ( ! [A5: int,Aa: list_char > int,B4: list_int,Ab: int,Ac: list_char > int,Ba: list_int,Ad: int,Ae: list_char > int,Bb: list_int] :
( ( exec1 @ Pa @ ( produc5086643055186798020st_int @ A5 @ ( produc8650753666468850689st_int @ Aa @ B4 ) ) @ ( 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 @ A5 @ Aa @ B4 @ Ad @ Ae @ Bb ) ) ) )
=> ( P @ X1a @ X1b @ X1c @ X2a @ X2b @ X2c ) ) ) ) ).
% exec_induct
thf(fact_1168_conj__le__cong,axiom,
! [X: int,X6: int,P: $o,P4: $o] :
( ( X = X6 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> ( P = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
& P )
= ( ( ord_less_eq_int @ zero_zero_int @ X6 )
& P4 ) ) ) ) ).
% conj_le_cong
thf(fact_1169_imp__le__cong,axiom,
! [X: int,X6: int,P: $o,P4: $o] :
( ( X = X6 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> ( P = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> P )
= ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> P4 ) ) ) ) ).
% imp_le_cong
thf(fact_1170_exec__appendL,axiom,
! [P: list_instr,I: int,S: list_char > int,Stk3: list_int,I4: int,S5: list_char > int,Stk2: list_int,P4: list_instr] :
( ( star_P707599355569300323st_int @ ( exec1 @ P ) @ ( produc5086643055186798020st_int @ I @ ( produc8650753666468850689st_int @ S @ Stk3 ) ) @ ( produc5086643055186798020st_int @ I4 @ ( produc8650753666468850689st_int @ S5 @ Stk2 ) ) )
=> ( star_P707599355569300323st_int @ ( exec1 @ ( append_instr @ P4 @ P ) ) @ ( produc5086643055186798020st_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P4 ) ) @ I ) @ ( produc8650753666468850689st_int @ S @ Stk3 ) ) @ ( produc5086643055186798020st_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P4 ) ) @ I4 ) @ ( produc8650753666468850689st_int @ S5 @ Stk2 ) ) ) ) ).
% exec_appendL
thf(fact_1171_ccomp__bigstep,axiom,
! [C: com,S: list_char > int,T: list_char > int,Stk3: list_int] :
( ( big_big_step @ ( produc5595214716300948949ar_int @ C @ S ) @ T )
=> ( star_P707599355569300323st_int @ ( exec1 @ ( ccomp @ C ) ) @ ( produc5086643055186798020st_int @ zero_zero_int @ ( produc8650753666468850689st_int @ S @ Stk3 ) ) @ ( produc5086643055186798020st_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ ( ccomp @ C ) ) ) @ ( produc8650753666468850689st_int @ T @ Stk3 ) ) ) ) ).
% ccomp_bigstep
thf(fact_1172_to__m__state_Oelims,axiom,
! [X: list_list_char,Xa: list_char > int,Xb: int,Y: int] :
( ( ( to_m_state @ X @ Xa @ Xb )
= Y )
=> ( ( ( ( ord_less_int @ zero_zero_int @ Xb )
& ( ord_less_eq_int @ Xb @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ X ) ) ) )
=> ( Y
= ( Xa @ ( inth_list_char @ X @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ X ) ) @ Xb ) ) ) ) )
& ( ~ ( ( ord_less_int @ zero_zero_int @ Xb )
& ( ord_less_eq_int @ Xb @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ X ) ) ) )
=> ( Y = zero_zero_int ) ) ) ) ).
% to_m_state.elims
thf(fact_1173_to__m__state_Osimps,axiom,
( to_m_state
= ( ^ [Xs: list_list_char,S4: list_char > int,A3: int] :
( if_int
@ ( ( ord_less_int @ zero_zero_int @ A3 )
& ( ord_less_eq_int @ A3 @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ Xs ) ) ) )
@ ( S4 @ ( inth_list_char @ Xs @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ Xs ) ) @ A3 ) ) )
@ zero_zero_int ) ) ) ).
% to_m_state.simps
thf(fact_1174_exec1__def,axiom,
( exec1
= ( ^ [P5: list_instr,Cf3: produc6425607678544837394st_int,Cf5: produc6425607678544837394st_int] :
( ( iexec_pred @ ( produc1484136438817787418st_int @ ( inth_instr @ P5 @ ( produc8941309049508465880st_int @ Cf3 ) ) @ Cf3 ) @ Cf5 )
& ( ord_less_eq_int @ zero_zero_int @ ( produc8941309049508465880st_int @ Cf3 ) )
& ( ord_less_int @ ( produc8941309049508465880st_int @ Cf3 ) @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P5 ) ) ) ) ) ) ).
% exec1_def
thf(fact_1175_while__unfold,axiom,
! [B: bexp,C: com,S3: list_char > int,T3: list_char > int] :
( ( big_big_step @ ( produc5595214716300948949ar_int @ ( while @ B @ C ) @ S3 ) @ T3 )
= ( big_big_step @ ( produc5595214716300948949ar_int @ ( if @ B @ ( seq @ C @ ( while @ B @ C ) ) @ skip ) @ S3 ) @ T3 ) ) ).
% while_unfold
thf(fact_1176_com_Oinject_I3_J,axiom,
! [X41: bexp,X42: com,X43: com,Y41: bexp,Y42: com,Y43: com] :
( ( ( if @ X41 @ X42 @ X43 )
= ( if @ Y41 @ Y42 @ Y43 ) )
= ( ( X41 = Y41 )
& ( X42 = Y42 )
& ( X43 = Y43 ) ) ) ).
% com.inject(3)
thf(fact_1177_com_Odistinct_I19_J,axiom,
! [X41: bexp,X42: com,X43: com,X51: bexp,X52: com] :
( ( if @ X41 @ X42 @ X43 )
!= ( while @ X51 @ X52 ) ) ).
% com.distinct(19)
thf(fact_1178_com_Odistinct_I15_J,axiom,
! [X31: com,X32: com,X41: bexp,X42: com,X43: com] :
( ( seq @ X31 @ X32 )
!= ( if @ X41 @ X42 @ X43 ) ) ).
% com.distinct(15)
thf(fact_1179_com_Odistinct_I5_J,axiom,
! [X41: bexp,X42: com,X43: com] :
( skip
!= ( if @ X41 @ X42 @ X43 ) ) ).
% com.distinct(5)
thf(fact_1180_commute__if,axiom,
! [B1: bexp,B22: bexp,C11: com,C12: com,C22: com,S3: list_char > int,T3: list_char > int] :
( ( big_big_step @ ( produc5595214716300948949ar_int @ ( if @ B1 @ ( if @ B22 @ C11 @ C12 ) @ C22 ) @ S3 ) @ T3 )
= ( big_big_step @ ( produc5595214716300948949ar_int @ ( if @ B22 @ ( if @ B1 @ C11 @ C22 ) @ ( if @ B1 @ C12 @ C22 ) ) @ S3 ) @ T3 ) ) ).
% commute_if
thf(fact_1181_triv__if,axiom,
! [B: bexp,C: com,S3: list_char > int,T3: list_char > int] :
( ( big_big_step @ ( produc5595214716300948949ar_int @ ( if @ B @ C @ C ) @ S3 ) @ T3 )
= ( big_big_step @ ( produc5595214716300948949ar_int @ C @ S3 ) @ T3 ) ) ).
% triv_if
thf(fact_1182_IfE,axiom,
! [B: bexp,C1: com,C22: com,S: list_char > int,T: list_char > int] :
( ( big_big_step @ ( produc5595214716300948949ar_int @ ( if @ B @ C1 @ C22 ) @ S ) @ T )
=> ( ( ( bval @ B @ S )
=> ~ ( big_big_step @ ( produc5595214716300948949ar_int @ C1 @ S ) @ T ) )
=> ~ ( ~ ( bval @ B @ S )
=> ~ ( big_big_step @ ( produc5595214716300948949ar_int @ C22 @ S ) @ T ) ) ) ) ).
% IfE
thf(fact_1183_IfFalse,axiom,
! [B: bexp,S: list_char > int,C_2: com,T: list_char > int,C_1: com] :
( ~ ( bval @ B @ S )
=> ( ( big_big_step @ ( produc5595214716300948949ar_int @ C_2 @ S ) @ T )
=> ( big_big_step @ ( produc5595214716300948949ar_int @ ( if @ B @ C_1 @ C_2 ) @ S ) @ T ) ) ) ).
% IfFalse
thf(fact_1184_IfTrue,axiom,
! [B: bexp,S: list_char > int,C_1: com,T: list_char > int,C_2: com] :
( ( bval @ B @ S )
=> ( ( big_big_step @ ( produc5595214716300948949ar_int @ C_1 @ S ) @ T )
=> ( big_big_step @ ( produc5595214716300948949ar_int @ ( if @ B @ C_1 @ C_2 ) @ S ) @ T ) ) ) ).
% IfTrue
thf(fact_1185_com_Osize_I9_J,axiom,
! [X41: bexp,X42: com,X43: com] :
( ( size_size_com @ ( if @ X41 @ X42 @ X43 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( size_size_com @ X42 ) @ ( size_size_com @ X43 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% com.size(9)
thf(fact_1186_com_Osize__gen_I4_J,axiom,
! [X41: bexp,X42: com,X43: com] :
( ( size_com @ ( if @ X41 @ X42 @ X43 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( size_com @ X42 ) @ ( size_com @ X43 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% com.size_gen(4)
thf(fact_1187_execl__next,axiom,
! [P: list_instr,Cfs: list_P419547167332244514st_int,K: nat] :
( ( execl @ P @ Cfs )
=> ( ( ord_less_nat @ K @ ( size_s229347708945616566st_int @ Cfs ) )
=> ( ( K
!= ( minus_minus_nat @ ( size_s229347708945616566st_int @ Cfs ) @ one_one_nat ) )
=> ( ( iexec_pred @ ( produc1484136438817787418st_int @ ( inth_instr @ P @ ( produc8941309049508465880st_int @ ( nth_Pr8327173284404383691st_int @ Cfs @ K ) ) ) @ ( nth_Pr8327173284404383691st_int @ Cfs @ K ) ) @ ( nth_Pr8327173284404383691st_int @ Cfs @ ( suc @ K ) ) )
& ( ord_less_eq_int @ zero_zero_int @ ( produc8941309049508465880st_int @ ( nth_Pr8327173284404383691st_int @ Cfs @ K ) ) )
& ( ord_less_int @ ( produc8941309049508465880st_int @ ( nth_Pr8327173284404383691st_int @ Cfs @ K ) ) @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P ) ) ) ) ) ) ) ).
% execl_next
thf(fact_1188_execl__drop,axiom,
! [P: list_instr,Cfs: list_P419547167332244514st_int,N: nat] :
( ( execl @ P @ Cfs )
=> ( execl @ P @ ( drop_P7838799151754935884st_int @ N @ Cfs ) ) ) ).
% execl_drop
thf(fact_1189_execl_Osimps_I2_J,axiom,
! [P: list_instr] : ( execl @ P @ nil_Pr749310521696113548st_int ) ).
% execl.simps(2)
thf(fact_1190_execl_Osimps_I3_J,axiom,
! [P: list_instr,V: produc6425607678544837394st_int] : ( execl @ P @ ( cons_P4151473878978182108st_int @ V @ nil_Pr749310521696113548st_int ) ) ).
% execl.simps(3)
thf(fact_1191_execl_Oelims_I3_J,axiom,
! [X: list_instr,Xa: list_P419547167332244514st_int] :
( ~ ( execl @ X @ Xa )
=> ~ ! [Cf4: produc6425607678544837394st_int,Cf6: produc6425607678544837394st_int,Cfs2: list_P419547167332244514st_int] :
( ( Xa
= ( cons_P4151473878978182108st_int @ Cf4 @ ( cons_P4151473878978182108st_int @ Cf6 @ Cfs2 ) ) )
=> ( ( exec1 @ X @ Cf4 @ Cf6 )
& ( execl @ X @ ( cons_P4151473878978182108st_int @ Cf6 @ Cfs2 ) ) ) ) ) ).
% execl.elims(3)
thf(fact_1192_execl_Oelims_I2_J,axiom,
! [X: list_instr,Xa: list_P419547167332244514st_int] :
( ( execl @ X @ Xa )
=> ( ! [Cf4: produc6425607678544837394st_int,Cf6: produc6425607678544837394st_int,Cfs2: list_P419547167332244514st_int] :
( ( Xa
= ( cons_P4151473878978182108st_int @ Cf4 @ ( cons_P4151473878978182108st_int @ Cf6 @ Cfs2 ) ) )
=> ~ ( ( exec1 @ X @ Cf4 @ Cf6 )
& ( execl @ X @ ( cons_P4151473878978182108st_int @ Cf6 @ Cfs2 ) ) ) )
=> ( ( Xa != nil_Pr749310521696113548st_int )
=> ~ ! [V2: produc6425607678544837394st_int] :
( Xa
!= ( cons_P4151473878978182108st_int @ V2 @ nil_Pr749310521696113548st_int ) ) ) ) ) ).
% execl.elims(2)
thf(fact_1193_execl_Oelims_I1_J,axiom,
! [X: list_instr,Xa: list_P419547167332244514st_int,Y: $o] :
( ( ( execl @ X @ Xa )
= Y )
=> ( ! [Cf4: produc6425607678544837394st_int,Cf6: produc6425607678544837394st_int,Cfs2: list_P419547167332244514st_int] :
( ( Xa
= ( cons_P4151473878978182108st_int @ Cf4 @ ( cons_P4151473878978182108st_int @ Cf6 @ Cfs2 ) ) )
=> ( Y
= ( ~ ( ( exec1 @ X @ Cf4 @ Cf6 )
& ( execl @ X @ ( cons_P4151473878978182108st_int @ Cf6 @ Cfs2 ) ) ) ) ) )
=> ( ( ( Xa = nil_Pr749310521696113548st_int )
=> ~ Y )
=> ~ ( ? [V2: produc6425607678544837394st_int] :
( Xa
= ( cons_P4151473878978182108st_int @ V2 @ nil_Pr749310521696113548st_int ) )
=> ~ Y ) ) ) ) ).
% execl.elims(1)
thf(fact_1194_execl_Osimps_I1_J,axiom,
! [P: list_instr,Cf: produc6425607678544837394st_int,Cf2: produc6425607678544837394st_int,Cfs: list_P419547167332244514st_int] :
( ( execl @ P @ ( cons_P4151473878978182108st_int @ Cf @ ( cons_P4151473878978182108st_int @ Cf2 @ Cfs ) ) )
= ( ( exec1 @ P @ Cf @ Cf2 )
& ( execl @ P @ ( cons_P4151473878978182108st_int @ Cf2 @ Cfs ) ) ) ) ).
% execl.simps(1)
thf(fact_1195_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_1196_execl_Ocases,axiom,
! [X: produc4400946599758736694st_int] :
( ! [P3: list_instr,Cf4: produc6425607678544837394st_int,Cf6: produc6425607678544837394st_int,Cfs2: list_P419547167332244514st_int] :
( X
!= ( produc7764413618310988720st_int @ P3 @ ( cons_P4151473878978182108st_int @ Cf4 @ ( cons_P4151473878978182108st_int @ Cf6 @ Cfs2 ) ) ) )
=> ( ! [P3: list_instr] :
( X
!= ( produc7764413618310988720st_int @ P3 @ nil_Pr749310521696113548st_int ) )
=> ~ ! [P3: list_instr,V2: produc6425607678544837394st_int] :
( X
!= ( produc7764413618310988720st_int @ P3 @ ( cons_P4151473878978182108st_int @ V2 @ nil_Pr749310521696113548st_int ) ) ) ) ) ).
% execl.cases
thf(fact_1197_execl__last,axiom,
! [P: list_instr,Cfs: list_P419547167332244514st_int,K: nat] :
( ( execl @ P @ Cfs )
=> ( ( ord_less_nat @ K @ ( size_s229347708945616566st_int @ Cfs ) )
=> ( ~ ( member_int @ ( produc8941309049508465880st_int @ ( nth_Pr8327173284404383691st_int @ Cfs @ K ) ) @ ( set_or4662586982721622107an_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P ) ) ) )
=> ( ( minus_minus_nat @ ( size_s229347708945616566st_int @ Cfs ) @ one_one_nat )
= K ) ) ) ) ).
% execl_last
thf(fact_1198_Compiler2_Obcomp__correct,axiom,
! [X: produc1897111610453708512_o_int,Cfs: list_P419547167332244514st_int] :
( ( execl_all @ ( bcomp @ X ) @ Cfs )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( product_snd_o_int @ ( produc7219675232234870470_o_int @ X ) ) )
=> ( bpred @ X @ ( nth_Pr8327173284404383691st_int @ Cfs @ zero_zero_nat ) @ ( nth_Pr8327173284404383691st_int @ Cfs @ ( minus_minus_nat @ ( size_s229347708945616566st_int @ Cfs ) @ ( suc @ zero_zero_nat ) ) ) ) ) ) ).
% Compiler2.bcomp_correct
thf(fact_1199_ex__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ? [M5: nat] :
( ( ord_less_nat @ M5 @ N )
& ( P @ M5 ) ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
& ( P @ X3 ) ) ) ) ).
% ex_nat_less_eq
thf(fact_1200_all__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ! [M5: nat] :
( ( ord_less_nat @ M5 @ N )
=> ( P @ M5 ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
=> ( P @ X3 ) ) ) ) ).
% all_nat_less_eq
thf(fact_1201_mpred__merge,axiom,
! [P: list_instr,Cfs: list_P419547167332244514st_int,K: nat,M: nat,N: nat] :
( ( mpred @ P @ Cfs @ K @ M )
=> ( ( mpred @ P @ Cfs @ M @ N )
=> ( mpred @ P @ Cfs @ K @ N ) ) ) ).
% mpred_merge
thf(fact_1202_mpred__drop,axiom,
! [K: nat,Cfs: list_P419547167332244514st_int,P: list_instr,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ ( size_s229347708945616566st_int @ Cfs ) )
=> ( ( 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 ) ) ) ) )
=> ( ( mpred @ P @ ( drop_P7838799151754935884st_int @ K @ Cfs ) @ M @ N )
=> ( mpred @ P @ Cfs @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) ) ) ) ) ).
% mpred_drop
thf(fact_1203_add__stack__eq2,axiom,
! [N: nat,Ms2: int > int,I: int,Ms: int > int] :
( ! [X4: int] :
( ( member_int @ X4 @ ( set_or4662586982721622107an_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ zero_zero_int ) )
=> ( ( ( X4
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) )
=> ( ( Ms2 @ X4 )
= I ) )
& ( ( X4
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) )
=> ( ( Ms2 @ X4 )
= ( Ms @ X4 ) ) ) ) )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( add_stack @ N @ Ms2 )
= ( cons_int @ I @ ( add_stack @ ( minus_minus_nat @ N @ one_one_nat ) @ Ms ) ) ) ) ) ).
% add_stack_eq2
thf(fact_1204_to__m__state_Opelims,axiom,
! [X: list_list_char,Xa: list_char > int,Xb: int,Y: int] :
( ( ( to_m_state @ X @ Xa @ Xb )
= Y )
=> ( ( accp_P4562858270263085236nt_int @ to_m_state_rel @ ( produc3965054194175396271nt_int @ X @ ( produc5790713362662368625nt_int @ Xa @ Xb ) ) )
=> ~ ( ( ( ( ( ord_less_int @ zero_zero_int @ Xb )
& ( ord_less_eq_int @ Xb @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ X ) ) ) )
=> ( Y
= ( Xa @ ( inth_list_char @ X @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ X ) ) @ Xb ) ) ) ) )
& ( ~ ( ( ord_less_int @ zero_zero_int @ Xb )
& ( ord_less_eq_int @ Xb @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ X ) ) ) )
=> ( Y = zero_zero_int ) ) )
=> ~ ( accp_P4562858270263085236nt_int @ to_m_state_rel @ ( produc3965054194175396271nt_int @ X @ ( produc5790713362662368625nt_int @ Xa @ Xb ) ) ) ) ) ) ).
% to_m_state.pelims
thf(fact_1205_add__stack_Osimps_I1_J,axiom,
! [Uu: int > int] :
( ( add_stack @ zero_zero_nat @ Uu )
= nil_int ) ).
% add_stack.simps(1)
thf(fact_1206_add__stack__nnil,axiom,
! [N: nat,Ms: int > int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( add_stack @ N @ Ms )
!= nil_int ) ) ).
% add_stack_nnil
thf(fact_1207_add__stack_Osimps_I2_J,axiom,
! [N: nat,Ms: int > int] :
( ( add_stack @ ( suc @ N ) @ Ms )
= ( cons_int @ ( Ms @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) @ ( add_stack @ N @ Ms ) ) ) ).
% add_stack.simps(2)
thf(fact_1208_add__stack__eq,axiom,
! [M: nat,Ms2: int > int,Ms: int > int,N: nat] :
( ! [X4: int] :
( ( member_int @ X4 @ ( set_or4662586982721622107an_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) @ zero_zero_int ) )
=> ( ( Ms2 @ X4 )
= ( Ms @ X4 ) ) )
=> ( ( M = N )
=> ( ( add_stack @ M @ Ms2 )
= ( add_stack @ N @ Ms ) ) ) ) ).
% add_stack_eq
thf(fact_1209_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_1210_zabs__less__one__iff,axiom,
! [Z2: int] :
( ( ord_less_int @ ( abs_abs_int @ Z2 ) @ one_one_int )
= ( Z2 = zero_zero_int ) ) ).
% zabs_less_one_iff
thf(fact_1211_zabs__def,axiom,
( abs_abs_int
= ( ^ [I3: int] : ( if_int @ ( ord_less_int @ I3 @ zero_zero_int ) @ ( uminus_uminus_int @ I3 ) @ I3 ) ) ) ).
% zabs_def
thf(fact_1212_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_1213_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_1214_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 )
=> ( ! [I5: nat] :
( ( ord_less_nat @ K2 @ I5 )
=> ( P @ I5 ) )
=> ( P @ K2 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_1215_com_Oexhaust,axiom,
! [Y: com] :
( ( Y != skip )
=> ( ! [X212: list_char,X222: aexp] :
( Y
!= ( assign @ X212 @ X222 ) )
=> ( ! [X312: com,X322: com] :
( Y
!= ( seq @ X312 @ X322 ) )
=> ( ! [X412: bexp,X422: com,X432: com] :
( Y
!= ( if @ X412 @ X422 @ X432 ) )
=> ~ ! [X512: bexp,X522: com] :
( Y
!= ( while @ X512 @ X522 ) ) ) ) ) ) ).
% com.exhaust
thf(fact_1216_com_Oinject_I1_J,axiom,
! [X21: list_char,X22: aexp,Y21: list_char,Y22: aexp] :
( ( ( assign @ X21 @ X22 )
= ( assign @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% com.inject(1)
thf(fact_1217_com_Odistinct_I1_J,axiom,
! [X21: list_char,X22: aexp] :
( skip
!= ( assign @ X21 @ X22 ) ) ).
% com.distinct(1)
thf(fact_1218_com_Odistinct_I9_J,axiom,
! [X21: list_char,X22: aexp,X31: com,X32: com] :
( ( assign @ X21 @ X22 )
!= ( seq @ X31 @ X32 ) ) ).
% com.distinct(9)
thf(fact_1219_com_Odistinct_I13_J,axiom,
! [X21: list_char,X22: aexp,X51: bexp,X52: com] :
( ( assign @ X21 @ X22 )
!= ( while @ X51 @ X52 ) ) ).
% com.distinct(13)
thf(fact_1220_com_Odistinct_I11_J,axiom,
! [X21: list_char,X22: aexp,X41: bexp,X42: com,X43: com] :
( ( assign @ X21 @ X22 )
!= ( if @ X41 @ X42 @ X43 ) ) ).
% com.distinct(11)
thf(fact_1221_com_Osize_I7_J,axiom,
! [X21: list_char,X22: aexp] :
( ( size_size_com @ ( assign @ X21 @ X22 ) )
= zero_zero_nat ) ).
% com.size(7)
thf(fact_1222_com_Osize__gen_I2_J,axiom,
! [X21: list_char,X22: aexp] :
( ( size_com @ ( assign @ X21 @ X22 ) )
= zero_zero_nat ) ).
% com.size_gen(2)
thf(fact_1223_ccomp_Osimps_I5_J,axiom,
! [B: bexp,C: com] :
( ( ccomp @ ( while @ B @ C ) )
= ( append_instr @ ( bcomp @ ( produc4047900504771817624_o_int @ B @ ( product_Pair_o_int @ $false @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ ( ccomp @ C ) ) ) @ one_one_int ) ) ) ) @ ( append_instr @ ( ccomp @ C ) @ ( cons_instr @ ( jmp @ ( uminus_uminus_int @ ( plus_plus_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ ( bcomp @ ( produc4047900504771817624_o_int @ B @ ( product_Pair_o_int @ $false @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ ( ccomp @ C ) ) ) @ one_one_int ) ) ) ) ) ) @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ ( ccomp @ C ) ) ) ) @ one_one_int ) ) ) @ nil_instr ) ) ) ) ).
% ccomp.simps(5)
thf(fact_1224_nat__less__as__int,axiom,
( ord_less_nat
= ( ^ [A3: nat,B2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_less_as_int
thf(fact_1225_nat__leq__as__int,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B2: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% nat_leq_as_int
thf(fact_1226_ccomp_Osimps_I4_J,axiom,
! [B: bexp,C_1: com,C_2: com] :
( ( ccomp @ ( if @ B @ C_1 @ C_2 ) )
= ( append_instr @ ( bcomp @ ( produc4047900504771817624_o_int @ B @ ( product_Pair_o_int @ $false @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ ( ccomp @ C_1 ) ) ) @ one_one_int ) ) ) ) @ ( append_instr @ ( ccomp @ C_1 ) @ ( cons_instr @ ( jmp @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ ( ccomp @ C_2 ) ) ) ) @ ( ccomp @ C_2 ) ) ) ) ) ).
% ccomp.simps(4)
thf(fact_1227_bcomp_Osimps_I3_J,axiom,
! [B_1: bexp,B_2: bexp,F: $o,I: int] :
( ( bcomp @ ( produc4047900504771817624_o_int @ ( and @ B_1 @ B_2 ) @ ( product_Pair_o_int @ F @ I ) ) )
= ( append_instr @ ( bcomp @ ( produc4047900504771817624_o_int @ B_1 @ ( product_Pair_o_int @ $false @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ ( bcomp @ ( produc4047900504771817624_o_int @ B_2 @ ( product_Pair_o_int @ F @ I ) ) ) ) ) @ ( if_int @ F @ zero_zero_int @ I ) ) ) ) ) @ ( bcomp @ ( produc4047900504771817624_o_int @ B_2 @ ( product_Pair_o_int @ F @ I ) ) ) ) ) ).
% bcomp.simps(3)
thf(fact_1228_bexp_Osize_I7_J,axiom,
! [X31: bexp,X32: bexp] :
( ( size_size_bexp @ ( and @ X31 @ X32 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( size_size_bexp @ X31 ) @ ( size_size_bexp @ X32 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% bexp.size(7)
thf(fact_1229_vars__load,axiom,
! [I: int,P: list_instr,X: list_char] :
( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P ) ) )
=> ( ( ( inth_instr @ P @ I )
= ( load @ X ) )
=> ( member_list_char @ X @ ( set_list_char2 @ ( vars @ P ) ) ) ) ) ) ).
% vars_load
thf(fact_1230_vars_Osimps_I1_J,axiom,
( ( vars @ nil_instr )
= nil_list_char ) ).
% vars.simps(1)
thf(fact_1231_bval_Osimps_I3_J,axiom,
! [B_1: bexp,B_2: bexp,S: list_char > int] :
( ( bval @ ( and @ B_1 @ B_2 ) @ S )
= ( ( bval @ B_1 @ S )
& ( bval @ B_2 @ S ) ) ) ).
% bval.simps(3)
thf(fact_1232_vars__store,axiom,
! [I: int,P: list_instr,X: list_char] :
( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P ) ) )
=> ( ( ( inth_instr @ P @ I )
= ( store @ X ) )
=> ( member_list_char @ X @ ( set_list_char2 @ ( vars @ P ) ) ) ) ) ) ).
% vars_store
thf(fact_1233_bexp_Osize__gen_I3_J,axiom,
! [X31: bexp,X32: bexp] :
( ( size_bexp @ ( and @ X31 @ X32 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( size_bexp @ X31 ) @ ( size_bexp @ X32 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% bexp.size_gen(3)
thf(fact_1234_instr_Oinject_I3_J,axiom,
! [X44: list_char,Y44: list_char] :
( ( ( store @ X44 )
= ( store @ Y44 ) )
= ( X44 = Y44 ) ) ).
% instr.inject(3)
thf(fact_1235_instr_Odistinct_I5_J,axiom,
! [X1: int,X44: list_char] :
( ( loadi @ X1 )
!= ( store @ X44 ) ) ).
% instr.distinct(5)
thf(fact_1236_instr_Odistinct_I15_J,axiom,
! [X2: list_char,X44: list_char] :
( ( load @ X2 )
!= ( store @ X44 ) ) ).
% instr.distinct(15)
thf(fact_1237_instr_Odistinct_I31_J,axiom,
! [X44: list_char,X53: int] :
( ( store @ X44 )
!= ( jmp @ X53 ) ) ).
% instr.distinct(31)
thf(fact_1238_instr_Osize_I11_J,axiom,
! [X44: list_char] :
( ( size_size_instr @ ( store @ X44 ) )
= zero_zero_nat ) ).
% instr.size(11)
thf(fact_1239_ccomp_Osimps_I2_J,axiom,
! [X: list_char,A: aexp] :
( ( ccomp @ ( assign @ X @ A ) )
= ( append_instr @ ( acomp @ A ) @ ( cons_instr @ ( store @ X ) @ nil_instr ) ) ) ).
% ccomp.simps(2)
thf(fact_1240_vars_Osimps_I2_J,axiom,
! [Ins2: instr,P: list_instr] :
( ( vars @ ( cons_instr @ Ins2 @ P ) )
= ( append_list_char
@ ( case_i7427087243572719837t_char
@ ^ [Int: int] : nil_list_char
@ ^ [X3: list_char] : ( if_list_list_char @ ( member_list_char @ X3 @ ( set_list_char2 @ ( vars @ P ) ) ) @ nil_list_char @ ( cons_list_char @ X3 @ nil_list_char ) )
@ nil_list_char
@ ^ [X3: list_char] : ( if_list_list_char @ ( member_list_char @ X3 @ ( set_list_char2 @ ( vars @ P ) ) ) @ nil_list_char @ ( cons_list_char @ X3 @ nil_list_char ) )
@ ^ [Int: int] : nil_list_char
@ ^ [Int: int] : nil_list_char
@ ^ [Int: int] : nil_list_char
@ Ins2 )
@ ( vars @ P ) ) ) ).
% vars.simps(2)
thf(fact_1241_Compiler2_Oacomp__correct,axiom,
! [A: aexp,Cfs: list_P419547167332244514st_int] :
( ( execl_all @ ( acomp @ A ) @ Cfs )
=> ( apred @ A @ ( nth_Pr8327173284404383691st_int @ Cfs @ zero_zero_nat ) @ ( nth_Pr8327173284404383691st_int @ Cfs @ ( minus_minus_nat @ ( size_s229347708945616566st_int @ Cfs ) @ ( suc @ zero_zero_nat ) ) ) ) ) ).
% Compiler2.acomp_correct
thf(fact_1242_Compiler_Oacomp__correct,axiom,
! [A: aexp,S: list_char > int,Stk3: list_int] : ( star_P707599355569300323st_int @ ( exec1 @ ( acomp @ A ) ) @ ( produc5086643055186798020st_int @ zero_zero_int @ ( produc8650753666468850689st_int @ S @ Stk3 ) ) @ ( produc5086643055186798020st_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ ( acomp @ A ) ) ) @ ( produc8650753666468850689st_int @ S @ ( cons_int @ ( aval @ A @ S ) @ Stk3 ) ) ) ) ).
% Compiler.acomp_correct
thf(fact_1243_addr__of__correct,axiom,
! [X: list_char,Xs2: list_list_char] :
( ( member_list_char @ X @ ( set_list_char2 @ Xs2 ) )
=> ( ( inth_list_char @ Xs2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ Xs2 ) ) @ ( addr_of @ Xs2 @ X ) ) )
= X ) ) ).
% addr_of_correct
thf(fact_1244_addr__of__nneg,axiom,
! [Xs2: list_list_char,X: list_char] : ( ord_less_eq_int @ zero_zero_int @ ( addr_of @ Xs2 @ X ) ) ).
% addr_of_nneg
thf(fact_1245_addr__of__neq2,axiom,
! [X: list_char,Xs2: list_list_char,X6: list_char] :
( ( member_list_char @ X @ ( set_list_char2 @ Xs2 ) )
=> ( ( X6 != X )
=> ( ( addr_of @ Xs2 @ X6 )
!= ( addr_of @ Xs2 @ X ) ) ) ) ).
% addr_of_neq2
thf(fact_1246_addr__of_Osimps_I1_J,axiom,
! [Uu: list_char] :
( ( addr_of @ nil_list_char @ Uu )
= zero_zero_int ) ).
% addr_of.simps(1)
thf(fact_1247_addr__of__nset,axiom,
! [X: list_char,Xs2: list_list_char] :
( ~ ( member_list_char @ X @ ( set_list_char2 @ Xs2 ) )
=> ( ( addr_of @ Xs2 @ X )
= zero_zero_int ) ) ).
% addr_of_nset
thf(fact_1248_addr__of__max,axiom,
! [Xs2: list_list_char,X: list_char] : ( ord_less_eq_int @ ( addr_of @ Xs2 @ X ) @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ Xs2 ) ) ) ).
% addr_of_max
thf(fact_1249_addr__of__set,axiom,
! [X: list_char,Xs2: list_list_char] :
( ( member_list_char @ X @ ( set_list_char2 @ Xs2 ) )
=> ( ord_less_int @ zero_zero_int @ ( addr_of @ Xs2 @ X ) ) ) ).
% addr_of_set
thf(fact_1250_addr__of__neq,axiom,
! [Xs2: list_list_char,X: list_char] :
( ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ Xs2 ) ) )
!= ( addr_of @ Xs2 @ X ) ) ).
% addr_of_neq
thf(fact_1251_addr__of_Osimps_I2_J,axiom,
! [X: list_char,Y: list_char,Xs2: list_list_char] :
( ( ( X = Y )
=> ( ( addr_of @ ( cons_list_char @ X @ Xs2 ) @ Y )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ Xs2 ) ) @ one_one_int ) ) )
& ( ( X != Y )
=> ( ( addr_of @ ( cons_list_char @ X @ Xs2 ) @ Y )
= ( addr_of @ Xs2 @ Y ) ) ) ) ).
% addr_of.simps(2)
thf(fact_1252_addr__of__unique,axiom,
! [Xs2: list_list_char,A: int] :
( ( distinct_list_char @ Xs2 )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ A @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ Xs2 ) ) )
=> ( ( addr_of @ Xs2 @ ( inth_list_char @ Xs2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ Xs2 ) ) @ A ) ) )
= A ) ) ) ) ).
% addr_of_unique
thf(fact_1253_vars__dist,axiom,
! [P: list_instr] : ( distinct_list_char @ ( vars @ P ) ) ).
% vars_dist
thf(fact_1254_bcomp_Osimps_I1_J,axiom,
! [V: $o,F: $o,I: int] :
( ( ( V = F )
=> ( ( bcomp @ ( produc4047900504771817624_o_int @ ( bc @ V ) @ ( product_Pair_o_int @ F @ I ) ) )
= ( cons_instr @ ( jmp @ I ) @ nil_instr ) ) )
& ( ( V = ~ F )
=> ( ( bcomp @ ( produc4047900504771817624_o_int @ ( bc @ V ) @ ( product_Pair_o_int @ F @ I ) ) )
= nil_instr ) ) ) ).
% bcomp.simps(1)
thf(fact_1255_bval_Osimps_I1_J,axiom,
! [V: $o,S: list_char > int] :
( ( bval @ ( bc @ V ) @ S )
= V ) ).
% bval.simps(1)
thf(fact_1256_bexp_Osize_I5_J,axiom,
! [X1: $o] :
( ( size_size_bexp @ ( bc @ X1 ) )
= zero_zero_nat ) ).
% bexp.size(5)
thf(fact_1257_bexp_Osize__gen_I1_J,axiom,
! [X1: $o] :
( ( size_bexp @ ( bc @ X1 ) )
= zero_zero_nat ) ).
% bexp.size_gen(1)
% Helper facts (9)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
! [X: list_int,Y: list_int] :
( ( if_list_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
! [X: list_int,Y: list_int] :
( ( if_list_int @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__List__Olist_It__List__Olist_It__String__Ochar_J_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_It__List__Olist_It__String__Ochar_J_J_T,axiom,
! [X: list_list_char,Y: list_list_char] :
( ( if_list_list_char @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__List__Olist_It__String__Ochar_J_J_T,axiom,
! [X: list_list_char,Y: list_list_char] :
( ( if_list_list_char @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
cpred @ ( while @ b @ c ) @ ( produc5086643055186798020st_int @ zero_zero_int @ ( produc8650753666468850689st_int @ s2 @ stk ) ) @ ( nth_Pr8327173284404383691st_int @ cfsa @ ( minus_minus_nat @ ( size_s229347708945616566st_int @ cfsa ) @ ( suc @ zero_zero_nat ) ) ) ).
%------------------------------------------------------------------------------