TPTP Problem File: SLH0793^1.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_00652_033164__6329812_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1454 ( 656 unt; 185 typ; 0 def)
% Number of atoms : 3063 (1748 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 10284 ( 422 ~; 59 |; 239 &;8314 @)
% ( 0 <=>;1250 =>; 0 <=; 0 <~>)
% Maximal formula depth : 30 ( 6 avg)
% Number of types : 36 ( 35 usr)
% Number of type conns : 664 ( 664 >; 0 *; 0 +; 0 <<)
% Number of symbols : 153 ( 150 usr; 18 con; 0-4 aty)
% Number of variables : 3597 ( 108 ^;3295 !; 194 ?;3597 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 16:59:15.988
%------------------------------------------------------------------------------
% Could-be-implicit typings (35)
thf(ty_n_t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J_J_M_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J_J_M_Eo_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J_J_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Compiler__Oinstr_J_Mt__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J_J_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_I_062_It__Compiler__Oinstr_M_062_It__Compiler__Oinstr_M_Eo_J_J_Mt__List__Olist_It__Compiler__Oinstr_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Compiler__Oinstr_J_Mt__List__Olist_It__Compiler__Oinstr_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Com__Ocom_M_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__Int__Oint_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__BExp__Obexp_Mt__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(ty_n_t__List__Olist_It__List__Olist_It__Compiler__Oinstr_J_J,type,
list_list_instr: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__String__Ochar_J_J,type,
list_list_char: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Int__Oint_J_J,type,
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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,
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thf(ty_n_t__List__Olist_It__Int__Oint_J,type,
list_int: $tType ).
thf(ty_n_t__Set__Oset_It__Num__Onum_J,type,
set_num: $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__Num__Onum,type,
num: $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 (150)
thf(sy_c_AExp_Oaexp_ON,type,
n: int > aexp ).
thf(sy_c_AExp_Oaexp_OV,type,
v: list_char > aexp ).
thf(sy_c_AExp_Oaval,type,
aval: aexp > ( list_char > int ) > int ).
thf(sy_c_BExp_Obexp_OBc,type,
bc: $o > bexp ).
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_OSKIP,type,
skip: com ).
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,
mpred: list_instr > list_P419547167332244514st_int > nat > nat > $o ).
thf(sy_c_Compiler_Oacomp,type,
acomp: aexp > list_instr ).
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_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,
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inth_P8039235741264645271st_int: list_P419547167332244514st_int > int > produc6425607678544837394st_int ).
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_GCD_Obezw,type,
bezw: nat > nat > product_prod_int_int ).
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_Ominus__class_Ominus_001t__Set__Oset_It__Int__Oint_J,type,
minus_minus_set_int: set_int > set_int > set_int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Num__Onum_J,type,
minus_minus_set_num: set_num > set_num > set_num ).
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_Oplus__class_Oplus_001t__Num__Onum,type,
plus_plus_num: num > num > num ).
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__Compiler__Oinstr_J,type,
if_list_instr: $o > list_instr > list_instr > list_instr ).
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__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_List_Oappend_001t__Compiler__Oinstr,type,
append_instr: list_instr > list_instr > list_instr ).
thf(sy_c_List_Oappend_001t__Int__Oint,type,
append_int: list_int > list_int > list_int ).
thf(sy_c_List_Oappend_001t__List__Olist_It__String__Ochar_J,type,
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thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__Int__Oint_Mt__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J_J,type,
append5028292981072010935st_int: list_P419547167332244514st_int > list_P419547167332244514st_int > list_P419547167332244514st_int ).
thf(sy_c_List_Odrop_001t__Compiler__Oinstr,type,
drop_instr: nat > list_instr > list_instr ).
thf(sy_c_List_Odrop_001t__Int__Oint,type,
drop_int: nat > list_int > list_int ).
thf(sy_c_List_Odrop_001t__Product____Type__Oprod_It__Int__Oint_Mt__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J_J,type,
drop_P7838799151754935884st_int: nat > list_P419547167332244514st_int > list_P419547167332244514st_int ).
thf(sy_c_List_Olist_OCons_001t__Compiler__Oinstr,type,
cons_instr: instr > list_instr > list_instr ).
thf(sy_c_List_Olist_OCons_001t__Int__Oint,type,
cons_int: int > list_int > list_int ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__Compiler__Oinstr_J,type,
cons_list_instr: list_instr > list_list_instr > list_list_instr ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__Int__Oint_J,type,
cons_list_int: list_int > list_list_int > list_list_int ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J_J_J,type,
cons_l8648290059330449388st_int: list_P419547167332244514st_int > list_l6422821129637969586st_int > list_l6422821129637969586st_int ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__String__Ochar_J,type,
cons_list_char: list_char > list_list_char > list_list_char ).
thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__Int__Oint_Mt__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J_J,type,
cons_P4151473878978182108st_int: produc6425607678544837394st_int > list_P419547167332244514st_int > list_P419547167332244514st_int ).
thf(sy_c_List_Olist_ONil_001t__Compiler__Oinstr,type,
nil_instr: list_instr ).
thf(sy_c_List_Olist_ONil_001t__Int__Oint,type,
nil_int: list_int ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__Compiler__Oinstr_J,type,
nil_list_instr: list_list_instr ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__Int__Oint_J,type,
nil_list_int: list_list_int ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J_J_J,type,
nil_li4597314861645524892st_int: list_l6422821129637969586st_int ).
thf(sy_c_List_Olist_ONil_001t__Product____Type__Oprod_It__Int__Oint_Mt__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J_J,type,
nil_Pr749310521696113548st_int: list_P419547167332244514st_int ).
thf(sy_c_List_On__lists_001t__Compiler__Oinstr,type,
n_lists_instr: nat > list_instr > list_list_instr ).
thf(sy_c_List_On__lists_001t__Int__Oint,type,
n_lists_int: nat > list_int > list_list_int ).
thf(sy_c_List_On__lists_001t__Product____Type__Oprod_It__Int__Oint_Mt__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J_J,type,
n_list7311978343830957444st_int: nat > list_P419547167332244514st_int > list_l6422821129637969586st_int ).
thf(sy_c_List_Onth_001t__Compiler__Oinstr,type,
nth_instr: list_instr > nat > instr ).
thf(sy_c_List_Onth_001t__Int__Oint,type,
nth_int: list_int > nat > int ).
thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Int__Oint_Mt__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J_J,type,
nth_Pr8327173284404383691st_int: list_P419547167332244514st_int > nat > produc6425607678544837394st_int ).
thf(sy_c_List_Oupto__aux,type,
upto_aux: int > int > list_int > list_int ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
semiri1316708129612266289at_nat: nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__AExp__Oaexp,type,
size_size_aexp: aexp > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__BExp__Obexp,type,
size_size_bexp: bexp > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Com__Ocom,type,
size_size_com: com > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Compiler__Oinstr,type,
size_size_instr: instr > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Compiler__Oinstr_J,type,
size_size_list_instr: list_instr > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Int__Oint_J,type,
size_size_list_int: list_int > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__String__Ochar_J_J,type,
size_s356637359517785349t_char: list_list_char > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J_J_J,type,
size_s229347708945616566st_int: list_P419547167332244514st_int > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Num__Onum,type,
size_size_num: num > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__String__Ochar,type,
size_size_char: char > nat ).
thf(sy_c_Num_OBitM,type,
bitM: num > num ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Int__Oint,type,
neg_nu5851722552734809277nc_int: int > int ).
thf(sy_c_Num_Onum_OBit0,type,
bit0: num > num ).
thf(sy_c_Num_Onum_OBit1,type,
bit1: num > num ).
thf(sy_c_Num_Onum_OOne,type,
one: num ).
thf(sy_c_Num_Onum_Osize__num,type,
size_num: num > nat ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Int__Oint,type,
numeral_numeral_int: num > int ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
numeral_numeral_nat: num > nat ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Num__Onum,type,
ord_less_num: num > num > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Num__Onum,type,
ord_less_eq_num: num > num > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_eq_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Num__Onum_J,type,
ord_less_eq_set_num: set_num > set_num > $o ).
thf(sy_c_Parity_Oadjust__mod,type,
adjust_mod: num > int > int ).
thf(sy_c_Parity_Ounique__euclidean__semiring__with__nat__division__class_Odivmod_001t__Int__Oint,type,
unique5403075989570733136od_int: num > num > product_prod_int_int ).
thf(sy_c_Product__Type_OPair_001_062_It__Compiler__Oinstr_M_062_It__Compiler__Oinstr_M_Eo_J_J_001t__List__Olist_It__Compiler__Oinstr_J,type,
produc7818391707689388878_instr: ( instr > instr > $o ) > list_instr > produc7133927873007960030_instr ).
thf(sy_c_Product__Type_OPair_001_062_It__Int__Oint_M_062_It__Int__Oint_M_Eo_J_J_001t__List__Olist_It__Int__Oint_J,type,
produc8618682346314911123st_int: ( int > int > $o ) > list_int > produc5834231552977413017st_int ).
thf(sy_c_Product__Type_OPair_001_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_001t__Int__Oint,type,
produc5790713362662368625nt_int: ( list_char > int ) > int > produc4435102495419491129nt_int ).
thf(sy_c_Product__Type_OPair_001_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_001t__List__Olist_It__Int__Oint_J,type,
produc8650753666468850689st_int: ( list_char > int ) > list_int > produc369741286924889929st_int ).
thf(sy_c_Product__Type_OPair_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J_J_M_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J_J_M_Eo_J_J_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J_J_J,type,
produc554038597315947693st_int: ( produc6425607678544837394st_int > produc6425607678544837394st_int > $o ) > list_P419547167332244514st_int > produc2468426419292222131st_int ).
thf(sy_c_Product__Type_OPair_001_Eo_001t__Int__Oint,type,
product_Pair_o_int: $o > int > product_prod_o_int ).
thf(sy_c_Product__Type_OPair_001t__BExp__Obexp_001t__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J,type,
produc4047900504771817624_o_int: bexp > product_prod_o_int > produc1897111610453708512_o_int ).
thf(sy_c_Product__Type_OPair_001t__Com__Ocom_001_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J,type,
produc5595214716300948949ar_int: com > ( list_char > int ) > produc2931317944591925149ar_int ).
thf(sy_c_Product__Type_OPair_001t__Compiler__Oinstr_001t__Product____Type__Oprod_It__Int__Oint_Mt__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J_J,type,
produc1484136438817787418st_int: instr > produc6425607678544837394st_int > produc5995290525303592096st_int ).
thf(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Int__Oint,type,
product_Pair_int_int: int > int > product_prod_int_int ).
thf(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J,type,
produc5086643055186798020st_int: int > produc369741286924889929st_int > produc6425607678544837394st_int ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Compiler__Oinstr_J_001t__List__Olist_It__Compiler__Oinstr_J,type,
produc7750586281064920791_instr: list_instr > list_instr > produc8891168062546637159_instr ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Compiler__Oinstr_J_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J_J_J,type,
produc7764413618310988720st_int: list_instr > list_P419547167332244514st_int > produc4400946599758736694st_int ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Int__Oint_J_001t__List__Olist_It__Int__Oint_J,type,
produc364263696895485585st_int: list_int > list_int > produc1186641810826059865st_int ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__List__Olist_It__String__Ochar_J_J_001t__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__Int__Oint_J,type,
produc3965054194175396271nt_int: list_list_char > produc4435102495419491129nt_int > produc4189061985984568957nt_int ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J_J_J_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J_J_J,type,
produc2639222612625641053st_int: list_P419547167332244514st_int > list_P419547167332244514st_int > produc5848496107505245989st_int ).
thf(sy_c_Product__Type_Oprod_Ofst_001_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_001t__Int__Oint,type,
produc9025023058967219549nt_int: produc4435102495419491129nt_int > list_char > int ).
thf(sy_c_Product__Type_Oprod_Ofst_001_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_001t__List__Olist_It__Int__Oint_J,type,
produc2155683374303800813st_int: produc369741286924889929st_int > list_char > int ).
thf(sy_c_Product__Type_Oprod_Ofst_001_Eo_001t__Int__Oint,type,
product_fst_o_int: product_prod_o_int > $o ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__BExp__Obexp_001t__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J,type,
produc2736798884409471108_o_int: produc1897111610453708512_o_int > bexp ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Com__Ocom_001_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J,type,
produc8829524412605799873ar_int: produc2931317944591925149ar_int > com ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Int__Oint_001t__Int__Oint,type,
product_fst_int_int: product_prod_int_int > int ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Int__Oint_001t__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J,type,
produc8941309049508465880st_int: produc6425607678544837394st_int > int ).
thf(sy_c_Product__Type_Oprod_Osnd_001_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_001t__Int__Oint,type,
produc179445755590504351nt_int: produc4435102495419491129nt_int > int ).
thf(sy_c_Product__Type_Oprod_Osnd_001_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_001t__List__Olist_It__Int__Oint_J,type,
produc6244060863719409455st_int: produc369741286924889929st_int > list_int ).
thf(sy_c_Product__Type_Oprod_Osnd_001_Eo_001t__Int__Oint,type,
product_snd_o_int: product_prod_o_int > int ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__BExp__Obexp_001t__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J,type,
produc7219675232234870470_o_int: produc1897111610453708512_o_int > product_prod_o_int ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Com__Ocom_001_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J,type,
produc9207319146083860483ar_int: produc2931317944591925149ar_int > list_char > int ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Int__Oint_001t__Int__Oint,type,
product_snd_int_int: product_prod_int_int > int ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Int__Oint_001t__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J,type,
produc1296517004818028822st_int: produc6425607678544837394st_int > produc369741286924889929st_int ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat,type,
dvd_dvd_nat: nat > nat > $o ).
thf(sy_c_Set_OCollect_001t__Int__Oint,type,
collect_int: ( int > $o ) > set_int ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Int__Oint,type,
set_or4662586982721622107an_int: int > int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
set_or4665077453230672383an_nat: nat > nat > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Num__Onum,type,
set_or1222409239386451017an_num: num > num > set_num ).
thf(sy_c_Star_Ostar_001t__Product____Type__Oprod_It__Int__Oint_Mt__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J_J,type,
star_P707599355569300323st_int: ( produc6425607678544837394st_int > produc6425607678544837394st_int > $o ) > produc6425607678544837394st_int > produc6425607678544837394st_int > $o ).
thf(sy_c_String_Ochar_Osize__char,type,
size_char: char > nat ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__String__Ochar_J_J_Mt__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__Int__Oint_J_J,type,
accp_P4562858270263085236nt_int: ( produc4189061985984568957nt_int > produc4189061985984568957nt_int > $o ) > produc4189061985984568957nt_int > $o ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Num__Onum,type,
member_num: num > set_num > $o ).
thf(sy_v_b____,type,
b: bexp ).
thf(sy_v_c_092_060_094sub_0621____,type,
c_1: com ).
thf(sy_v_c_092_060_094sub_0622____,type,
c_2: com ).
thf(sy_v_cfsa____,type,
cfsa: list_P419547167332244514st_int ).
% Relevant facts (1259)
thf(fact_0_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_1_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_2_length__greater__0__conv,axiom,
! [Xs: list_instr] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_instr @ Xs ) )
= ( Xs != nil_instr ) ) ).
% length_greater_0_conv
thf(fact_3_length__greater__0__conv,axiom,
! [Xs: list_P419547167332244514st_int] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s229347708945616566st_int @ Xs ) )
= ( Xs != nil_Pr749310521696113548st_int ) ) ).
% length_greater_0_conv
thf(fact_4_length__greater__0__conv,axiom,
! [Xs: list_int] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_int @ Xs ) )
= ( Xs != nil_int ) ) ).
% length_greater_0_conv
thf(fact_5_prod_Ocollapse,axiom,
! [Prod: product_prod_int_int] :
( ( product_Pair_int_int @ ( product_fst_int_int @ Prod ) @ ( product_snd_int_int @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_6_prod_Ocollapse,axiom,
! [Prod: produc2931317944591925149ar_int] :
( ( produc5595214716300948949ar_int @ ( produc8829524412605799873ar_int @ Prod ) @ ( produc9207319146083860483ar_int @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_7_prod_Ocollapse,axiom,
! [Prod: produc4435102495419491129nt_int] :
( ( produc5790713362662368625nt_int @ ( produc9025023058967219549nt_int @ Prod ) @ ( produc179445755590504351nt_int @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_8_prod_Ocollapse,axiom,
! [Prod: produc6425607678544837394st_int] :
( ( produc5086643055186798020st_int @ ( produc8941309049508465880st_int @ Prod ) @ ( produc1296517004818028822st_int @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_9_prod_Ocollapse,axiom,
! [Prod: produc369741286924889929st_int] :
( ( produc8650753666468850689st_int @ ( produc2155683374303800813st_int @ Prod ) @ ( produc6244060863719409455st_int @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_10_prod_Ocollapse,axiom,
! [Prod: product_prod_o_int] :
( ( product_Pair_o_int @ ( product_fst_o_int @ Prod ) @ ( product_snd_o_int @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_11_prod_Ocollapse,axiom,
! [Prod: produc1897111610453708512_o_int] :
( ( produc4047900504771817624_o_int @ ( produc2736798884409471108_o_int @ Prod ) @ ( produc7219675232234870470_o_int @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_12_length__0__conv,axiom,
! [Xs: list_instr] :
( ( ( size_size_list_instr @ Xs )
= zero_zero_nat )
= ( Xs = nil_instr ) ) ).
% length_0_conv
thf(fact_13_length__0__conv,axiom,
! [Xs: list_P419547167332244514st_int] :
( ( ( size_s229347708945616566st_int @ Xs )
= zero_zero_nat )
= ( Xs = nil_Pr749310521696113548st_int ) ) ).
% length_0_conv
thf(fact_14_length__0__conv,axiom,
! [Xs: list_int] :
( ( ( size_size_list_int @ Xs )
= zero_zero_nat )
= ( Xs = nil_int ) ) ).
% length_0_conv
thf(fact_15_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_16_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_17_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_18_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_19_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_20_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_21_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_22_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_23_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_24_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_25_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_26_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_27_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_28_old_Oprod_Oinject,axiom,
! [A: int,B: produc369741286924889929st_int,A2: int,B2: produc369741286924889929st_int] :
( ( ( produc5086643055186798020st_int @ A @ B )
= ( produc5086643055186798020st_int @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_29_old_Oprod_Oinject,axiom,
! [A: int,B: int,A2: int,B2: int] :
( ( ( product_Pair_int_int @ A @ B )
= ( product_Pair_int_int @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_30_old_Oprod_Oinject,axiom,
! [A: com,B: list_char > int,A2: com,B2: list_char > int] :
( ( ( produc5595214716300948949ar_int @ A @ B )
= ( produc5595214716300948949ar_int @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_31_old_Oprod_Oinject,axiom,
! [A: list_char > int,B: list_int,A2: list_char > int,B2: list_int] :
( ( ( produc8650753666468850689st_int @ A @ B )
= ( produc8650753666468850689st_int @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_32_old_Oprod_Oinject,axiom,
! [A: list_char > int,B: int,A2: list_char > int,B2: int] :
( ( ( produc5790713362662368625nt_int @ A @ B )
= ( produc5790713362662368625nt_int @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_33_old_Oprod_Oinject,axiom,
! [A: bexp,B: product_prod_o_int,A2: bexp,B2: product_prod_o_int] :
( ( ( produc4047900504771817624_o_int @ A @ B )
= ( produc4047900504771817624_o_int @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_34_old_Oprod_Oinject,axiom,
! [A: $o,B: int,A2: $o,B2: int] :
( ( ( product_Pair_o_int @ A @ B )
= ( product_Pair_o_int @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_35_prod_Oinject,axiom,
! [X1: int,X2: produc369741286924889929st_int,Y1: int,Y2: produc369741286924889929st_int] :
( ( ( produc5086643055186798020st_int @ X1 @ X2 )
= ( produc5086643055186798020st_int @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_36_prod_Oinject,axiom,
! [X1: int,X2: int,Y1: int,Y2: int] :
( ( ( product_Pair_int_int @ X1 @ X2 )
= ( product_Pair_int_int @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_37_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_38_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_39_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_40_prod_Oinject,axiom,
! [X1: bexp,X2: product_prod_o_int,Y1: bexp,Y2: product_prod_o_int] :
( ( ( produc4047900504771817624_o_int @ X1 @ X2 )
= ( produc4047900504771817624_o_int @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_41_prod_Oinject,axiom,
! [X1: $o,X2: int,Y1: $o,Y2: int] :
( ( ( product_Pair_o_int @ X1 @ X2 )
= ( product_Pair_o_int @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_42_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_43_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_44_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_45_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_46_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_47_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_48_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_49_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_50_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_51_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_52_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1316708129612266289at_nat @ M )
= ( semiri1316708129612266289at_nat @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_53_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_54_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_55_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_56_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_57_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_58_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_59_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_60_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_61_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_62_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_63_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_64_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_65_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_66_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_67_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_68_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_69_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_70_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_71_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_72_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_73_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_74_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_75_diff__zero,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_zero
thf(fact_76_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_77_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_78_diff__0__right,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_0_right
thf(fact_79_diff__self,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% diff_self
thf(fact_80_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_81_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_82_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_83_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_84_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_85_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_86_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_87_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_88_mem__Collect__eq,axiom,
! [A: int,P: int > $o] :
( ( member_int @ A @ ( collect_int @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_89_Collect__mem__eq,axiom,
! [A3: set_int] :
( ( collect_int
@ ^ [X3: int] : ( member_int @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_90_diff__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_91_add__diff__cancel,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_92_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_93_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_94_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_95_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_96_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_97_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_98_of__nat__1,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% of_nat_1
thf(fact_99_of__nat__1,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% of_nat_1
thf(fact_100_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_101_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_102_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_103_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_104_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_105_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_106_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_107_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_108_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_109_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_110_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_111_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_112_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_113_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_114_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_115_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_116_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_117_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_118_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_119_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_120_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_121_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_122_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_123_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_124_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_125_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_126_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_127_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_128_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_129_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_130_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_131_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 ) )
| ? [D: nat] :
( ( A
= ( plus_plus_nat @ B @ D ) )
& ~ ( P @ D ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_132_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 ) )
& ! [D: nat] :
( ( A
= ( plus_plus_nat @ B @ D ) )
=> ( P @ D ) ) ) ) ).
% nat_diff_split
thf(fact_133_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_134_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_135_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_136_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_137_Pair__inject,axiom,
! [A: int,B: produc369741286924889929st_int,A2: int,B2: produc369741286924889929st_int] :
( ( ( produc5086643055186798020st_int @ A @ B )
= ( produc5086643055186798020st_int @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_138_Pair__inject,axiom,
! [A: int,B: int,A2: int,B2: int] :
( ( ( product_Pair_int_int @ A @ B )
= ( product_Pair_int_int @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_139_Pair__inject,axiom,
! [A: com,B: list_char > int,A2: com,B2: list_char > int] :
( ( ( produc5595214716300948949ar_int @ A @ B )
= ( produc5595214716300948949ar_int @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_140_Pair__inject,axiom,
! [A: list_char > int,B: list_int,A2: list_char > int,B2: list_int] :
( ( ( produc8650753666468850689st_int @ A @ B )
= ( produc8650753666468850689st_int @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_141_Pair__inject,axiom,
! [A: list_char > int,B: int,A2: list_char > int,B2: int] :
( ( ( produc5790713362662368625nt_int @ A @ B )
= ( produc5790713362662368625nt_int @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_142_Pair__inject,axiom,
! [A: bexp,B: product_prod_o_int,A2: bexp,B2: product_prod_o_int] :
( ( ( produc4047900504771817624_o_int @ A @ B )
= ( produc4047900504771817624_o_int @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_143_Pair__inject,axiom,
! [A: $o,B: int,A2: $o,B2: int] :
( ( ( product_Pair_o_int @ A @ B )
= ( product_Pair_o_int @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_144_prod__cases,axiom,
! [P: produc6425607678544837394st_int > $o,P2: produc6425607678544837394st_int] :
( ! [A4: int,B3: produc369741286924889929st_int] : ( P @ ( produc5086643055186798020st_int @ A4 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_145_prod__cases,axiom,
! [P: product_prod_int_int > $o,P2: product_prod_int_int] :
( ! [A4: int,B3: int] : ( P @ ( product_Pair_int_int @ A4 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_146_prod__cases,axiom,
! [P: produc2931317944591925149ar_int > $o,P2: produc2931317944591925149ar_int] :
( ! [A4: com,B3: list_char > int] : ( P @ ( produc5595214716300948949ar_int @ A4 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_147_prod__cases,axiom,
! [P: produc369741286924889929st_int > $o,P2: produc369741286924889929st_int] :
( ! [A4: list_char > int,B3: list_int] : ( P @ ( produc8650753666468850689st_int @ A4 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_148_prod__cases,axiom,
! [P: produc4435102495419491129nt_int > $o,P2: produc4435102495419491129nt_int] :
( ! [A4: list_char > int,B3: int] : ( P @ ( produc5790713362662368625nt_int @ A4 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_149_prod__cases,axiom,
! [P: produc1897111610453708512_o_int > $o,P2: produc1897111610453708512_o_int] :
( ! [A4: bexp,B3: product_prod_o_int] : ( P @ ( produc4047900504771817624_o_int @ A4 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_150_prod__cases,axiom,
! [P: product_prod_o_int > $o,P2: product_prod_o_int] :
( ! [A4: $o,B3: int] : ( P @ ( product_Pair_o_int @ A4 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_151_surj__pair,axiom,
! [P2: produc6425607678544837394st_int] :
? [X4: int,Y3: produc369741286924889929st_int] :
( P2
= ( produc5086643055186798020st_int @ X4 @ Y3 ) ) ).
% surj_pair
thf(fact_152_surj__pair,axiom,
! [P2: product_prod_int_int] :
? [X4: int,Y3: int] :
( P2
= ( product_Pair_int_int @ X4 @ Y3 ) ) ).
% surj_pair
thf(fact_153_surj__pair,axiom,
! [P2: produc2931317944591925149ar_int] :
? [X4: com,Y3: list_char > int] :
( P2
= ( produc5595214716300948949ar_int @ X4 @ Y3 ) ) ).
% surj_pair
thf(fact_154_surj__pair,axiom,
! [P2: produc369741286924889929st_int] :
? [X4: list_char > int,Y3: list_int] :
( P2
= ( produc8650753666468850689st_int @ X4 @ Y3 ) ) ).
% surj_pair
thf(fact_155_surj__pair,axiom,
! [P2: produc4435102495419491129nt_int] :
? [X4: list_char > int,Y3: int] :
( P2
= ( produc5790713362662368625nt_int @ X4 @ Y3 ) ) ).
% surj_pair
thf(fact_156_surj__pair,axiom,
! [P2: produc1897111610453708512_o_int] :
? [X4: bexp,Y3: product_prod_o_int] :
( P2
= ( produc4047900504771817624_o_int @ X4 @ Y3 ) ) ).
% surj_pair
thf(fact_157_surj__pair,axiom,
! [P2: product_prod_o_int] :
? [X4: $o,Y3: int] :
( P2
= ( product_Pair_o_int @ X4 @ Y3 ) ) ).
% surj_pair
thf(fact_158_old_Oprod_Oexhaust,axiom,
! [Y: produc6425607678544837394st_int] :
~ ! [A4: int,B3: produc369741286924889929st_int] :
( Y
!= ( produc5086643055186798020st_int @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_159_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_int_int] :
~ ! [A4: int,B3: int] :
( Y
!= ( product_Pair_int_int @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_160_old_Oprod_Oexhaust,axiom,
! [Y: produc2931317944591925149ar_int] :
~ ! [A4: com,B3: list_char > int] :
( Y
!= ( produc5595214716300948949ar_int @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_161_old_Oprod_Oexhaust,axiom,
! [Y: produc369741286924889929st_int] :
~ ! [A4: list_char > int,B3: list_int] :
( Y
!= ( produc8650753666468850689st_int @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_162_old_Oprod_Oexhaust,axiom,
! [Y: produc4435102495419491129nt_int] :
~ ! [A4: list_char > int,B3: int] :
( Y
!= ( produc5790713362662368625nt_int @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_163_old_Oprod_Oexhaust,axiom,
! [Y: produc1897111610453708512_o_int] :
~ ! [A4: bexp,B3: product_prod_o_int] :
( Y
!= ( produc4047900504771817624_o_int @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_164_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_o_int] :
~ ! [A4: $o,B3: int] :
( Y
!= ( product_Pair_o_int @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_165_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_166_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_167_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_168_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_169_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_170_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_171_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A5: int,B4: int] : ( plus_plus_int @ B4 @ A5 ) ) ) ).
% add.commute
thf(fact_172_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A5: nat,B4: nat] : ( plus_plus_nat @ B4 @ A5 ) ) ) ).
% add.commute
thf(fact_173_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_174_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_175_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_176_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_177_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_178_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_179_group__cancel_Oadd1,axiom,
! [A3: int,K: int,A: int,B: int] :
( ( A3
= ( plus_plus_int @ K @ A ) )
=> ( ( plus_plus_int @ A3 @ B )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_180_group__cancel_Oadd1,axiom,
! [A3: nat,K: nat,A: nat,B: nat] :
( ( A3
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A3 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_181_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_182_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_183_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_184_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_185_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_186_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
= ( X = one_one_int ) ) ).
% one_reorient
thf(fact_187_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_188_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_189_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_190_diff__eq__diff__eq,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D2 ) )
=> ( ( A = B )
= ( C = D2 ) ) ) ).
% diff_eq_diff_eq
thf(fact_191_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_192_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_193_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( P @ M2 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_194_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_195_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_196_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_197_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_198_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_199_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_200_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_201_size__neq__size__imp__neq,axiom,
! [X: aexp,Y: aexp] :
( ( ( size_size_aexp @ X )
!= ( size_size_aexp @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_202_size__neq__size__imp__neq,axiom,
! [X: num,Y: num] :
( ( ( size_size_num @ X )
!= ( size_size_num @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_203_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_204_size__neq__size__imp__neq,axiom,
! [X: list_P419547167332244514st_int,Y: list_P419547167332244514st_int] :
( ( ( size_s229347708945616566st_int @ X )
!= ( size_s229347708945616566st_int @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_205_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_206_neq__if__length__neq,axiom,
! [Xs: list_int,Ys: list_int] :
( ( ( size_size_list_int @ Xs )
!= ( size_size_list_int @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_207_neq__if__length__neq,axiom,
! [Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int] :
( ( ( size_s229347708945616566st_int @ Xs )
!= ( size_s229347708945616566st_int @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_208_neq__if__length__neq,axiom,
! [Xs: list_instr,Ys: list_instr] :
( ( ( size_size_list_instr @ Xs )
!= ( size_size_list_instr @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_209_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_int] :
( ( size_size_list_int @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_210_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_P419547167332244514st_int] :
( ( size_s229347708945616566st_int @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_211_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_instr] :
( ( size_size_list_instr @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_212_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_213_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_214_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_215_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_216_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_217_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_218_add_Ogroup__left__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add.group_left_neutral
thf(fact_219_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_220_add_Ocomm__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.comm_neutral
thf(fact_221_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_222_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_223_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_224_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_225_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_226_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_227_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_228_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_229_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_230_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_231_add__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_strict_mono
thf(fact_232_add__strict__mono,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D2 )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).
% add_strict_mono
thf(fact_233_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_234_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_235_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_236_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_237_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_238_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_239_eq__iff__diff__eq__0,axiom,
( ( ^ [Y4: int,Z: int] : ( Y4 = Z ) )
= ( ^ [A5: int,B4: int] :
( ( minus_minus_int @ A5 @ B4 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_240_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_241_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_242_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_243_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_244_diff__eq__diff__less,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D2 ) )
=> ( ( ord_less_int @ A @ B )
= ( ord_less_int @ C @ D2 ) ) ) ).
% diff_eq_diff_less
thf(fact_245_diff__strict__mono,axiom,
! [A: int,B: int,D2: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ D2 @ C )
=> ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D2 ) ) ) ) ).
% diff_strict_mono
thf(fact_246_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_247_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_248_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_249_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_250_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_251_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_252_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_253_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_254_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_255_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_256_group__cancel_Osub1,axiom,
! [A3: int,K: int,A: int,B: int] :
( ( A3
= ( plus_plus_int @ K @ A ) )
=> ( ( minus_minus_int @ A3 @ B )
= ( plus_plus_int @ K @ ( minus_minus_int @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_257_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_258_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_259_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_260_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_261_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_262_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_263_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_264_length__induct,axiom,
! [P: list_int > $o,Xs: list_int] :
( ! [Xs2: list_int] :
( ! [Ys2: list_int] :
( ( ord_less_nat @ ( size_size_list_int @ Ys2 ) @ ( size_size_list_int @ Xs2 ) )
=> ( P @ Ys2 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_265_length__induct,axiom,
! [P: list_P419547167332244514st_int > $o,Xs: list_P419547167332244514st_int] :
( ! [Xs2: list_P419547167332244514st_int] :
( ! [Ys2: list_P419547167332244514st_int] :
( ( ord_less_nat @ ( size_s229347708945616566st_int @ Ys2 ) @ ( size_s229347708945616566st_int @ Xs2 ) )
=> ( P @ Ys2 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_266_length__induct,axiom,
! [P: list_instr > $o,Xs: list_instr] :
( ! [Xs2: list_instr] :
( ! [Ys2: list_instr] :
( ( ord_less_nat @ ( size_size_list_instr @ Ys2 ) @ ( size_size_list_instr @ Xs2 ) )
=> ( P @ Ys2 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_267_prod__induct3,axiom,
! [P: produc6425607678544837394st_int > $o,X: produc6425607678544837394st_int] :
( ! [A4: int,B3: list_char > int,C2: list_int] : ( P @ ( produc5086643055186798020st_int @ A4 @ ( produc8650753666468850689st_int @ B3 @ C2 ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_268_prod__induct3,axiom,
! [P: produc1897111610453708512_o_int > $o,X: produc1897111610453708512_o_int] :
( ! [A4: bexp,B3: $o,C2: int] : ( P @ ( produc4047900504771817624_o_int @ A4 @ ( product_Pair_o_int @ B3 @ C2 ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_269_prod__cases3,axiom,
! [Y: produc6425607678544837394st_int] :
~ ! [A4: int,B3: list_char > int,C2: list_int] :
( Y
!= ( produc5086643055186798020st_int @ A4 @ ( produc8650753666468850689st_int @ B3 @ C2 ) ) ) ).
% prod_cases3
thf(fact_270_prod__cases3,axiom,
! [Y: produc1897111610453708512_o_int] :
~ ! [A4: bexp,B3: $o,C2: int] :
( Y
!= ( produc4047900504771817624_o_int @ A4 @ ( product_Pair_o_int @ B3 @ C2 ) ) ) ).
% prod_cases3
thf(fact_271_fst__conv,axiom,
! [X1: bexp,X2: product_prod_o_int] :
( ( produc2736798884409471108_o_int @ ( produc4047900504771817624_o_int @ X1 @ X2 ) )
= X1 ) ).
% fst_conv
thf(fact_272_fst__conv,axiom,
! [X1: int,X2: produc369741286924889929st_int] :
( ( produc8941309049508465880st_int @ ( produc5086643055186798020st_int @ X1 @ X2 ) )
= X1 ) ).
% fst_conv
thf(fact_273_fst__conv,axiom,
! [X1: $o,X2: int] :
( ( product_fst_o_int @ ( product_Pair_o_int @ X1 @ X2 ) )
= X1 ) ).
% fst_conv
thf(fact_274_fst__conv,axiom,
! [X1: int,X2: int] :
( ( product_fst_int_int @ ( product_Pair_int_int @ X1 @ X2 ) )
= X1 ) ).
% fst_conv
thf(fact_275_fst__conv,axiom,
! [X1: com,X2: list_char > int] :
( ( produc8829524412605799873ar_int @ ( produc5595214716300948949ar_int @ X1 @ X2 ) )
= X1 ) ).
% fst_conv
thf(fact_276_fst__conv,axiom,
! [X1: list_char > int,X2: list_int] :
( ( produc2155683374303800813st_int @ ( produc8650753666468850689st_int @ X1 @ X2 ) )
= X1 ) ).
% fst_conv
thf(fact_277_fst__conv,axiom,
! [X1: list_char > int,X2: int] :
( ( produc9025023058967219549nt_int @ ( produc5790713362662368625nt_int @ X1 @ X2 ) )
= X1 ) ).
% fst_conv
thf(fact_278_fst__eqD,axiom,
! [X: bexp,Y: product_prod_o_int,A: bexp] :
( ( ( produc2736798884409471108_o_int @ ( produc4047900504771817624_o_int @ X @ Y ) )
= A )
=> ( X = A ) ) ).
% fst_eqD
thf(fact_279_fst__eqD,axiom,
! [X: int,Y: produc369741286924889929st_int,A: int] :
( ( ( produc8941309049508465880st_int @ ( produc5086643055186798020st_int @ X @ Y ) )
= A )
=> ( X = A ) ) ).
% fst_eqD
thf(fact_280_fst__eqD,axiom,
! [X: $o,Y: int,A: $o] :
( ( ( product_fst_o_int @ ( product_Pair_o_int @ X @ Y ) )
= A )
=> ( X = A ) ) ).
% fst_eqD
thf(fact_281_fst__eqD,axiom,
! [X: int,Y: int,A: int] :
( ( ( product_fst_int_int @ ( product_Pair_int_int @ X @ Y ) )
= A )
=> ( X = A ) ) ).
% fst_eqD
thf(fact_282_fst__eqD,axiom,
! [X: com,Y: list_char > int,A: com] :
( ( ( produc8829524412605799873ar_int @ ( produc5595214716300948949ar_int @ X @ Y ) )
= A )
=> ( X = A ) ) ).
% fst_eqD
thf(fact_283_fst__eqD,axiom,
! [X: list_char > int,Y: list_int,A: list_char > int] :
( ( ( produc2155683374303800813st_int @ ( produc8650753666468850689st_int @ X @ Y ) )
= A )
=> ( X = A ) ) ).
% fst_eqD
thf(fact_284_fst__eqD,axiom,
! [X: list_char > int,Y: int,A: list_char > int] :
( ( ( produc9025023058967219549nt_int @ ( produc5790713362662368625nt_int @ X @ Y ) )
= A )
=> ( X = A ) ) ).
% fst_eqD
thf(fact_285_snd__conv,axiom,
! [X1: int,X2: produc369741286924889929st_int] :
( ( produc1296517004818028822st_int @ ( produc5086643055186798020st_int @ X1 @ X2 ) )
= X2 ) ).
% snd_conv
thf(fact_286_snd__conv,axiom,
! [X1: list_char > int,X2: list_int] :
( ( produc6244060863719409455st_int @ ( produc8650753666468850689st_int @ X1 @ X2 ) )
= X2 ) ).
% snd_conv
thf(fact_287_snd__conv,axiom,
! [X1: $o,X2: int] :
( ( product_snd_o_int @ ( product_Pair_o_int @ X1 @ X2 ) )
= X2 ) ).
% snd_conv
thf(fact_288_snd__conv,axiom,
! [X1: bexp,X2: product_prod_o_int] :
( ( produc7219675232234870470_o_int @ ( produc4047900504771817624_o_int @ X1 @ X2 ) )
= X2 ) ).
% snd_conv
thf(fact_289_snd__conv,axiom,
! [X1: list_char > int,X2: int] :
( ( produc179445755590504351nt_int @ ( produc5790713362662368625nt_int @ X1 @ X2 ) )
= X2 ) ).
% snd_conv
thf(fact_290_snd__conv,axiom,
! [X1: com,X2: list_char > int] :
( ( produc9207319146083860483ar_int @ ( produc5595214716300948949ar_int @ X1 @ X2 ) )
= X2 ) ).
% snd_conv
thf(fact_291_snd__conv,axiom,
! [X1: int,X2: int] :
( ( product_snd_int_int @ ( product_Pair_int_int @ X1 @ X2 ) )
= X2 ) ).
% snd_conv
thf(fact_292_snd__eqD,axiom,
! [X: int,Y: produc369741286924889929st_int,A: produc369741286924889929st_int] :
( ( ( produc1296517004818028822st_int @ ( produc5086643055186798020st_int @ X @ Y ) )
= A )
=> ( Y = A ) ) ).
% snd_eqD
thf(fact_293_snd__eqD,axiom,
! [X: list_char > int,Y: list_int,A: list_int] :
( ( ( produc6244060863719409455st_int @ ( produc8650753666468850689st_int @ X @ Y ) )
= A )
=> ( Y = A ) ) ).
% snd_eqD
thf(fact_294_snd__eqD,axiom,
! [X: $o,Y: int,A: int] :
( ( ( product_snd_o_int @ ( product_Pair_o_int @ X @ Y ) )
= A )
=> ( Y = A ) ) ).
% snd_eqD
thf(fact_295_snd__eqD,axiom,
! [X: bexp,Y: product_prod_o_int,A: product_prod_o_int] :
( ( ( produc7219675232234870470_o_int @ ( produc4047900504771817624_o_int @ X @ Y ) )
= A )
=> ( Y = A ) ) ).
% snd_eqD
thf(fact_296_snd__eqD,axiom,
! [X: list_char > int,Y: int,A: int] :
( ( ( produc179445755590504351nt_int @ ( produc5790713362662368625nt_int @ X @ Y ) )
= A )
=> ( Y = A ) ) ).
% snd_eqD
thf(fact_297_snd__eqD,axiom,
! [X: com,Y: list_char > int,A: list_char > int] :
( ( ( produc9207319146083860483ar_int @ ( produc5595214716300948949ar_int @ X @ Y ) )
= A )
=> ( Y = A ) ) ).
% snd_eqD
thf(fact_298_snd__eqD,axiom,
! [X: int,Y: int,A: int] :
( ( ( product_snd_int_int @ ( product_Pair_int_int @ X @ Y ) )
= A )
=> ( Y = A ) ) ).
% snd_eqD
thf(fact_299_prod__eq__iff,axiom,
( ( ^ [Y4: produc6425607678544837394st_int,Z: produc6425607678544837394st_int] : ( Y4 = Z ) )
= ( ^ [S2: produc6425607678544837394st_int,T2: produc6425607678544837394st_int] :
( ( ( produc8941309049508465880st_int @ S2 )
= ( produc8941309049508465880st_int @ T2 ) )
& ( ( produc1296517004818028822st_int @ S2 )
= ( produc1296517004818028822st_int @ T2 ) ) ) ) ) ).
% prod_eq_iff
thf(fact_300_prod__eq__iff,axiom,
( ( ^ [Y4: produc369741286924889929st_int,Z: produc369741286924889929st_int] : ( Y4 = Z ) )
= ( ^ [S2: produc369741286924889929st_int,T2: produc369741286924889929st_int] :
( ( ( produc2155683374303800813st_int @ S2 )
= ( produc2155683374303800813st_int @ T2 ) )
& ( ( produc6244060863719409455st_int @ S2 )
= ( produc6244060863719409455st_int @ T2 ) ) ) ) ) ).
% prod_eq_iff
thf(fact_301_prod__eq__iff,axiom,
( ( ^ [Y4: product_prod_o_int,Z: product_prod_o_int] : ( Y4 = Z ) )
= ( ^ [S2: product_prod_o_int,T2: product_prod_o_int] :
( ( ( product_fst_o_int @ S2 )
= ( product_fst_o_int @ T2 ) )
& ( ( product_snd_o_int @ S2 )
= ( product_snd_o_int @ T2 ) ) ) ) ) ).
% prod_eq_iff
thf(fact_302_prod__eq__iff,axiom,
( ( ^ [Y4: produc1897111610453708512_o_int,Z: produc1897111610453708512_o_int] : ( Y4 = Z ) )
= ( ^ [S2: produc1897111610453708512_o_int,T2: produc1897111610453708512_o_int] :
( ( ( produc2736798884409471108_o_int @ S2 )
= ( produc2736798884409471108_o_int @ T2 ) )
& ( ( produc7219675232234870470_o_int @ S2 )
= ( produc7219675232234870470_o_int @ T2 ) ) ) ) ) ).
% prod_eq_iff
thf(fact_303_prod__eq__iff,axiom,
( ( ^ [Y4: produc4435102495419491129nt_int,Z: produc4435102495419491129nt_int] : ( Y4 = Z ) )
= ( ^ [S2: produc4435102495419491129nt_int,T2: produc4435102495419491129nt_int] :
( ( ( produc9025023058967219549nt_int @ S2 )
= ( produc9025023058967219549nt_int @ T2 ) )
& ( ( produc179445755590504351nt_int @ S2 )
= ( produc179445755590504351nt_int @ T2 ) ) ) ) ) ).
% prod_eq_iff
thf(fact_304_prod__eq__iff,axiom,
( ( ^ [Y4: produc2931317944591925149ar_int,Z: produc2931317944591925149ar_int] : ( Y4 = Z ) )
= ( ^ [S2: produc2931317944591925149ar_int,T2: produc2931317944591925149ar_int] :
( ( ( produc8829524412605799873ar_int @ S2 )
= ( produc8829524412605799873ar_int @ T2 ) )
& ( ( produc9207319146083860483ar_int @ S2 )
= ( produc9207319146083860483ar_int @ T2 ) ) ) ) ) ).
% prod_eq_iff
thf(fact_305_prod__eq__iff,axiom,
( ( ^ [Y4: product_prod_int_int,Z: product_prod_int_int] : ( Y4 = Z ) )
= ( ^ [S2: product_prod_int_int,T2: product_prod_int_int] :
( ( ( product_fst_int_int @ S2 )
= ( product_fst_int_int @ T2 ) )
& ( ( product_snd_int_int @ S2 )
= ( product_snd_int_int @ T2 ) ) ) ) ) ).
% prod_eq_iff
thf(fact_306_prod__eqI,axiom,
! [P2: produc6425607678544837394st_int,Q: produc6425607678544837394st_int] :
( ( ( produc8941309049508465880st_int @ P2 )
= ( produc8941309049508465880st_int @ Q ) )
=> ( ( ( produc1296517004818028822st_int @ P2 )
= ( produc1296517004818028822st_int @ Q ) )
=> ( P2 = Q ) ) ) ).
% prod_eqI
thf(fact_307_prod__eqI,axiom,
! [P2: produc369741286924889929st_int,Q: produc369741286924889929st_int] :
( ( ( produc2155683374303800813st_int @ P2 )
= ( produc2155683374303800813st_int @ Q ) )
=> ( ( ( produc6244060863719409455st_int @ P2 )
= ( produc6244060863719409455st_int @ Q ) )
=> ( P2 = Q ) ) ) ).
% prod_eqI
thf(fact_308_prod__eqI,axiom,
! [P2: product_prod_o_int,Q: product_prod_o_int] :
( ( ( product_fst_o_int @ P2 )
= ( product_fst_o_int @ Q ) )
=> ( ( ( product_snd_o_int @ P2 )
= ( product_snd_o_int @ Q ) )
=> ( P2 = Q ) ) ) ).
% prod_eqI
thf(fact_309_prod__eqI,axiom,
! [P2: produc1897111610453708512_o_int,Q: produc1897111610453708512_o_int] :
( ( ( produc2736798884409471108_o_int @ P2 )
= ( produc2736798884409471108_o_int @ Q ) )
=> ( ( ( produc7219675232234870470_o_int @ P2 )
= ( produc7219675232234870470_o_int @ Q ) )
=> ( P2 = Q ) ) ) ).
% prod_eqI
thf(fact_310_prod__eqI,axiom,
! [P2: produc4435102495419491129nt_int,Q: produc4435102495419491129nt_int] :
( ( ( produc9025023058967219549nt_int @ P2 )
= ( produc9025023058967219549nt_int @ Q ) )
=> ( ( ( produc179445755590504351nt_int @ P2 )
= ( produc179445755590504351nt_int @ Q ) )
=> ( P2 = Q ) ) ) ).
% prod_eqI
thf(fact_311_prod__eqI,axiom,
! [P2: produc2931317944591925149ar_int,Q: produc2931317944591925149ar_int] :
( ( ( produc8829524412605799873ar_int @ P2 )
= ( produc8829524412605799873ar_int @ Q ) )
=> ( ( ( produc9207319146083860483ar_int @ P2 )
= ( produc9207319146083860483ar_int @ Q ) )
=> ( P2 = Q ) ) ) ).
% prod_eqI
thf(fact_312_prod__eqI,axiom,
! [P2: product_prod_int_int,Q: product_prod_int_int] :
( ( ( product_fst_int_int @ P2 )
= ( product_fst_int_int @ Q ) )
=> ( ( ( product_snd_int_int @ P2 )
= ( product_snd_int_int @ Q ) )
=> ( P2 = Q ) ) ) ).
% prod_eqI
thf(fact_313_prod_Oexpand,axiom,
! [Prod: produc6425607678544837394st_int,Prod2: produc6425607678544837394st_int] :
( ( ( ( produc8941309049508465880st_int @ Prod )
= ( produc8941309049508465880st_int @ Prod2 ) )
& ( ( produc1296517004818028822st_int @ Prod )
= ( produc1296517004818028822st_int @ Prod2 ) ) )
=> ( Prod = Prod2 ) ) ).
% prod.expand
thf(fact_314_prod_Oexpand,axiom,
! [Prod: produc369741286924889929st_int,Prod2: produc369741286924889929st_int] :
( ( ( ( produc2155683374303800813st_int @ Prod )
= ( produc2155683374303800813st_int @ Prod2 ) )
& ( ( produc6244060863719409455st_int @ Prod )
= ( produc6244060863719409455st_int @ Prod2 ) ) )
=> ( Prod = Prod2 ) ) ).
% prod.expand
thf(fact_315_prod_Oexpand,axiom,
! [Prod: product_prod_o_int,Prod2: product_prod_o_int] :
( ( ( ( product_fst_o_int @ Prod )
= ( product_fst_o_int @ Prod2 ) )
& ( ( product_snd_o_int @ Prod )
= ( product_snd_o_int @ Prod2 ) ) )
=> ( Prod = Prod2 ) ) ).
% prod.expand
thf(fact_316_prod_Oexpand,axiom,
! [Prod: produc1897111610453708512_o_int,Prod2: produc1897111610453708512_o_int] :
( ( ( ( produc2736798884409471108_o_int @ Prod )
= ( produc2736798884409471108_o_int @ Prod2 ) )
& ( ( produc7219675232234870470_o_int @ Prod )
= ( produc7219675232234870470_o_int @ Prod2 ) ) )
=> ( Prod = Prod2 ) ) ).
% prod.expand
thf(fact_317_prod_Oexpand,axiom,
! [Prod: produc4435102495419491129nt_int,Prod2: produc4435102495419491129nt_int] :
( ( ( ( produc9025023058967219549nt_int @ Prod )
= ( produc9025023058967219549nt_int @ Prod2 ) )
& ( ( produc179445755590504351nt_int @ Prod )
= ( produc179445755590504351nt_int @ Prod2 ) ) )
=> ( Prod = Prod2 ) ) ).
% prod.expand
thf(fact_318_prod_Oexpand,axiom,
! [Prod: produc2931317944591925149ar_int,Prod2: produc2931317944591925149ar_int] :
( ( ( ( produc8829524412605799873ar_int @ Prod )
= ( produc8829524412605799873ar_int @ Prod2 ) )
& ( ( produc9207319146083860483ar_int @ Prod )
= ( produc9207319146083860483ar_int @ Prod2 ) ) )
=> ( Prod = Prod2 ) ) ).
% prod.expand
thf(fact_319_prod_Oexpand,axiom,
! [Prod: product_prod_int_int,Prod2: product_prod_int_int] :
( ( ( ( product_fst_int_int @ Prod )
= ( product_fst_int_int @ Prod2 ) )
& ( ( product_snd_int_int @ Prod )
= ( product_snd_int_int @ Prod2 ) ) )
=> ( Prod = Prod2 ) ) ).
% prod.expand
thf(fact_320_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_321_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_322_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C2: nat] :
( ( B
= ( plus_plus_nat @ A @ C2 ) )
=> ( C2 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_323_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_324_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_325_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_326_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_327_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_328_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_329_less__iff__diff__less__0,axiom,
( ord_less_int
= ( ^ [A5: int,B4: int] : ( ord_less_int @ ( minus_minus_int @ A5 @ B4 ) @ zero_zero_int ) ) ) ).
% less_iff_diff_less_0
thf(fact_330_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_331_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_332_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_333_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_334_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_335_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_336_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_337_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_338_list_Osize_I3_J,axiom,
( ( size_size_list_int @ nil_int )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_339_list_Osize_I3_J,axiom,
( ( size_s229347708945616566st_int @ nil_Pr749310521696113548st_int )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_340_list_Osize_I3_J,axiom,
( ( size_size_list_instr @ nil_instr )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_341_list__eq__iff__nth__eq,axiom,
( ( ^ [Y4: list_int,Z: list_int] : ( Y4 = Z ) )
= ( ^ [Xs3: list_int,Ys3: list_int] :
( ( ( size_size_list_int @ Xs3 )
= ( size_size_list_int @ Ys3 ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs3 ) )
=> ( ( nth_int @ Xs3 @ I2 )
= ( nth_int @ Ys3 @ I2 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_342_list__eq__iff__nth__eq,axiom,
( ( ^ [Y4: list_P419547167332244514st_int,Z: list_P419547167332244514st_int] : ( Y4 = Z ) )
= ( ^ [Xs3: list_P419547167332244514st_int,Ys3: list_P419547167332244514st_int] :
( ( ( size_s229347708945616566st_int @ Xs3 )
= ( size_s229347708945616566st_int @ Ys3 ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s229347708945616566st_int @ Xs3 ) )
=> ( ( nth_Pr8327173284404383691st_int @ Xs3 @ I2 )
= ( nth_Pr8327173284404383691st_int @ Ys3 @ I2 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_343_list__eq__iff__nth__eq,axiom,
( ( ^ [Y4: list_instr,Z: list_instr] : ( Y4 = Z ) )
= ( ^ [Xs3: list_instr,Ys3: list_instr] :
( ( ( size_size_list_instr @ Xs3 )
= ( size_size_list_instr @ Ys3 ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_instr @ Xs3 ) )
=> ( ( nth_instr @ Xs3 @ I2 )
= ( nth_instr @ Ys3 @ I2 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_344_Skolem__list__nth,axiom,
! [K: nat,P: nat > int > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ? [X5: int] : ( P @ I2 @ X5 ) ) )
= ( ? [Xs3: list_int] :
( ( ( size_size_list_int @ Xs3 )
= K )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( P @ I2 @ ( nth_int @ Xs3 @ I2 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_345_Skolem__list__nth,axiom,
! [K: nat,P: nat > produc6425607678544837394st_int > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ? [X5: produc6425607678544837394st_int] : ( P @ I2 @ X5 ) ) )
= ( ? [Xs3: list_P419547167332244514st_int] :
( ( ( size_s229347708945616566st_int @ Xs3 )
= K )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( P @ I2 @ ( nth_Pr8327173284404383691st_int @ Xs3 @ I2 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_346_Skolem__list__nth,axiom,
! [K: nat,P: nat > instr > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ? [X5: instr] : ( P @ I2 @ X5 ) ) )
= ( ? [Xs3: list_instr] :
( ( ( size_size_list_instr @ Xs3 )
= K )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( P @ I2 @ ( nth_instr @ Xs3 @ I2 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_347_nth__equalityI,axiom,
! [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 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_348_nth__equalityI,axiom,
! [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 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_349_nth__equalityI,axiom,
! [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 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_350_surjective__pairing,axiom,
! [T: produc6425607678544837394st_int] :
( T
= ( produc5086643055186798020st_int @ ( produc8941309049508465880st_int @ T ) @ ( produc1296517004818028822st_int @ T ) ) ) ).
% surjective_pairing
thf(fact_351_surjective__pairing,axiom,
! [T: produc369741286924889929st_int] :
( T
= ( produc8650753666468850689st_int @ ( produc2155683374303800813st_int @ T ) @ ( produc6244060863719409455st_int @ T ) ) ) ).
% surjective_pairing
thf(fact_352_surjective__pairing,axiom,
! [T: product_prod_o_int] :
( T
= ( product_Pair_o_int @ ( product_fst_o_int @ T ) @ ( product_snd_o_int @ T ) ) ) ).
% surjective_pairing
thf(fact_353_surjective__pairing,axiom,
! [T: produc1897111610453708512_o_int] :
( T
= ( produc4047900504771817624_o_int @ ( produc2736798884409471108_o_int @ T ) @ ( produc7219675232234870470_o_int @ T ) ) ) ).
% surjective_pairing
thf(fact_354_surjective__pairing,axiom,
! [T: produc4435102495419491129nt_int] :
( T
= ( produc5790713362662368625nt_int @ ( produc9025023058967219549nt_int @ T ) @ ( produc179445755590504351nt_int @ T ) ) ) ).
% surjective_pairing
thf(fact_355_surjective__pairing,axiom,
! [T: produc2931317944591925149ar_int] :
( T
= ( produc5595214716300948949ar_int @ ( produc8829524412605799873ar_int @ T ) @ ( produc9207319146083860483ar_int @ T ) ) ) ).
% surjective_pairing
thf(fact_356_surjective__pairing,axiom,
! [T: product_prod_int_int] :
( T
= ( product_Pair_int_int @ ( product_fst_int_int @ T ) @ ( product_snd_int_int @ T ) ) ) ).
% surjective_pairing
thf(fact_357_prod_Oexhaust__sel,axiom,
! [Prod: produc6425607678544837394st_int] :
( Prod
= ( produc5086643055186798020st_int @ ( produc8941309049508465880st_int @ Prod ) @ ( produc1296517004818028822st_int @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_358_prod_Oexhaust__sel,axiom,
! [Prod: produc369741286924889929st_int] :
( Prod
= ( produc8650753666468850689st_int @ ( produc2155683374303800813st_int @ Prod ) @ ( produc6244060863719409455st_int @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_359_prod_Oexhaust__sel,axiom,
! [Prod: product_prod_o_int] :
( Prod
= ( product_Pair_o_int @ ( product_fst_o_int @ Prod ) @ ( product_snd_o_int @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_360_prod_Oexhaust__sel,axiom,
! [Prod: produc1897111610453708512_o_int] :
( Prod
= ( produc4047900504771817624_o_int @ ( produc2736798884409471108_o_int @ Prod ) @ ( produc7219675232234870470_o_int @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_361_prod_Oexhaust__sel,axiom,
! [Prod: produc4435102495419491129nt_int] :
( Prod
= ( produc5790713362662368625nt_int @ ( produc9025023058967219549nt_int @ Prod ) @ ( produc179445755590504351nt_int @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_362_prod_Oexhaust__sel,axiom,
! [Prod: produc2931317944591925149ar_int] :
( Prod
= ( produc5595214716300948949ar_int @ ( produc8829524412605799873ar_int @ Prod ) @ ( produc9207319146083860483ar_int @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_363_prod_Oexhaust__sel,axiom,
! [Prod: product_prod_int_int] :
( Prod
= ( product_Pair_int_int @ ( product_fst_int_int @ Prod ) @ ( product_snd_int_int @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_364_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_365_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_366_zero__less__two,axiom,
ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% zero_less_two
thf(fact_367_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: int > produc369741286924889929st_int > $o,X: int,Y: produc369741286924889929st_int,A: produc6425607678544837394st_int] :
( ( P @ X @ Y )
=> ( ( A
= ( produc5086643055186798020st_int @ X @ Y ) )
=> ( P @ ( produc8941309049508465880st_int @ A ) @ ( produc1296517004818028822st_int @ A ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_368_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: ( list_char > int ) > list_int > $o,X: list_char > int,Y: list_int,A: produc369741286924889929st_int] :
( ( P @ X @ Y )
=> ( ( A
= ( produc8650753666468850689st_int @ X @ Y ) )
=> ( P @ ( produc2155683374303800813st_int @ A ) @ ( produc6244060863719409455st_int @ A ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_369_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: $o > int > $o,X: $o,Y: int,A: product_prod_o_int] :
( ( P @ X @ Y )
=> ( ( A
= ( product_Pair_o_int @ X @ Y ) )
=> ( P @ ( product_fst_o_int @ A ) @ ( product_snd_o_int @ A ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_370_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: bexp > product_prod_o_int > $o,X: bexp,Y: product_prod_o_int,A: produc1897111610453708512_o_int] :
( ( P @ X @ Y )
=> ( ( A
= ( produc4047900504771817624_o_int @ X @ Y ) )
=> ( P @ ( produc2736798884409471108_o_int @ A ) @ ( produc7219675232234870470_o_int @ A ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_371_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: ( list_char > int ) > int > $o,X: list_char > int,Y: int,A: produc4435102495419491129nt_int] :
( ( P @ X @ Y )
=> ( ( A
= ( produc5790713362662368625nt_int @ X @ Y ) )
=> ( P @ ( produc9025023058967219549nt_int @ A ) @ ( produc179445755590504351nt_int @ A ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_372_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: com > ( list_char > int ) > $o,X: com,Y: list_char > int,A: produc2931317944591925149ar_int] :
( ( P @ X @ Y )
=> ( ( A
= ( produc5595214716300948949ar_int @ X @ Y ) )
=> ( P @ ( produc8829524412605799873ar_int @ A ) @ ( produc9207319146083860483ar_int @ A ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_373_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: int > int > $o,X: int,Y: int,A: product_prod_int_int] :
( ( P @ X @ Y )
=> ( ( A
= ( product_Pair_int_int @ X @ Y ) )
=> ( P @ ( product_fst_int_int @ A ) @ ( product_snd_int_int @ A ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_374_conjI__realizer,axiom,
! [P: int > $o,P2: int,Q2: produc369741286924889929st_int > $o,Q: produc369741286924889929st_int] :
( ( P @ P2 )
=> ( ( Q2 @ Q )
=> ( ( P @ ( produc8941309049508465880st_int @ ( produc5086643055186798020st_int @ P2 @ Q ) ) )
& ( Q2 @ ( produc1296517004818028822st_int @ ( produc5086643055186798020st_int @ P2 @ Q ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_375_conjI__realizer,axiom,
! [P: ( list_char > int ) > $o,P2: list_char > int,Q2: list_int > $o,Q: list_int] :
( ( P @ P2 )
=> ( ( Q2 @ Q )
=> ( ( P @ ( produc2155683374303800813st_int @ ( produc8650753666468850689st_int @ P2 @ Q ) ) )
& ( Q2 @ ( produc6244060863719409455st_int @ ( produc8650753666468850689st_int @ P2 @ Q ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_376_conjI__realizer,axiom,
! [P: $o > $o,P2: $o,Q2: int > $o,Q: int] :
( ( P @ P2 )
=> ( ( Q2 @ Q )
=> ( ( P @ ( product_fst_o_int @ ( product_Pair_o_int @ P2 @ Q ) ) )
& ( Q2 @ ( product_snd_o_int @ ( product_Pair_o_int @ P2 @ Q ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_377_conjI__realizer,axiom,
! [P: bexp > $o,P2: bexp,Q2: product_prod_o_int > $o,Q: product_prod_o_int] :
( ( P @ P2 )
=> ( ( Q2 @ Q )
=> ( ( P @ ( produc2736798884409471108_o_int @ ( produc4047900504771817624_o_int @ P2 @ Q ) ) )
& ( Q2 @ ( produc7219675232234870470_o_int @ ( produc4047900504771817624_o_int @ P2 @ Q ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_378_conjI__realizer,axiom,
! [P: ( list_char > int ) > $o,P2: list_char > int,Q2: int > $o,Q: int] :
( ( P @ P2 )
=> ( ( Q2 @ Q )
=> ( ( P @ ( produc9025023058967219549nt_int @ ( produc5790713362662368625nt_int @ P2 @ Q ) ) )
& ( Q2 @ ( produc179445755590504351nt_int @ ( produc5790713362662368625nt_int @ P2 @ Q ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_379_conjI__realizer,axiom,
! [P: com > $o,P2: com,Q2: ( list_char > int ) > $o,Q: list_char > int] :
( ( P @ P2 )
=> ( ( Q2 @ Q )
=> ( ( P @ ( produc8829524412605799873ar_int @ ( produc5595214716300948949ar_int @ P2 @ Q ) ) )
& ( Q2 @ ( produc9207319146083860483ar_int @ ( produc5595214716300948949ar_int @ P2 @ Q ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_380_conjI__realizer,axiom,
! [P: int > $o,P2: int,Q2: int > $o,Q: int] :
( ( P @ P2 )
=> ( ( Q2 @ Q )
=> ( ( P @ ( product_fst_int_int @ ( product_Pair_int_int @ P2 @ Q ) ) )
& ( Q2 @ ( product_snd_int_int @ ( product_Pair_int_int @ P2 @ Q ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_381_exI__realizer,axiom,
! [P: produc369741286924889929st_int > int > $o,Y: produc369741286924889929st_int,X: int] :
( ( P @ Y @ X )
=> ( P @ ( produc1296517004818028822st_int @ ( produc5086643055186798020st_int @ X @ Y ) ) @ ( produc8941309049508465880st_int @ ( produc5086643055186798020st_int @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_382_exI__realizer,axiom,
! [P: list_int > ( list_char > int ) > $o,Y: list_int,X: list_char > int] :
( ( P @ Y @ X )
=> ( P @ ( produc6244060863719409455st_int @ ( produc8650753666468850689st_int @ X @ Y ) ) @ ( produc2155683374303800813st_int @ ( produc8650753666468850689st_int @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_383_exI__realizer,axiom,
! [P: int > $o > $o,Y: int,X: $o] :
( ( P @ Y @ X )
=> ( P @ ( product_snd_o_int @ ( product_Pair_o_int @ X @ Y ) ) @ ( product_fst_o_int @ ( product_Pair_o_int @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_384_exI__realizer,axiom,
! [P: product_prod_o_int > bexp > $o,Y: product_prod_o_int,X: bexp] :
( ( P @ Y @ X )
=> ( P @ ( produc7219675232234870470_o_int @ ( produc4047900504771817624_o_int @ X @ Y ) ) @ ( produc2736798884409471108_o_int @ ( produc4047900504771817624_o_int @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_385_exI__realizer,axiom,
! [P: int > ( list_char > int ) > $o,Y: int,X: list_char > int] :
( ( P @ Y @ X )
=> ( P @ ( produc179445755590504351nt_int @ ( produc5790713362662368625nt_int @ X @ Y ) ) @ ( produc9025023058967219549nt_int @ ( produc5790713362662368625nt_int @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_386_exI__realizer,axiom,
! [P: ( list_char > int ) > com > $o,Y: list_char > int,X: com] :
( ( P @ Y @ X )
=> ( P @ ( produc9207319146083860483ar_int @ ( produc5595214716300948949ar_int @ X @ Y ) ) @ ( produc8829524412605799873ar_int @ ( produc5595214716300948949ar_int @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_387_exI__realizer,axiom,
! [P: int > int > $o,Y: int,X: int] :
( ( P @ Y @ X )
=> ( P @ ( product_snd_int_int @ ( product_Pair_int_int @ X @ Y ) ) @ ( product_fst_int_int @ ( product_Pair_int_int @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_388_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
& ( K
= ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_389_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N2: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% pos_int_cases
thf(fact_390_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_391_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_392_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_393_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_394_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_395_less__add__one,axiom,
! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).
% less_add_one
thf(fact_396_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_397_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_398_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_399_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_400_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_401_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_402_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_403_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_404_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_405_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_406_int__int__eq,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% int_int_eq
thf(fact_407_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_408_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus_int @ K @ zero_zero_int )
= K ) ).
% plus_int_code(1)
thf(fact_409_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus_int @ K @ zero_zero_int )
= K ) ).
% minus_int_code(1)
thf(fact_410_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_411_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_412_odd__nonzero,axiom,
! [Z2: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_413_int__gr__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_int @ K @ I )
=> ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
=> ( ! [I3: int] :
( ( ord_less_int @ K @ I3 )
=> ( ( P @ I3 )
=> ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_gr_induct
thf(fact_414_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_415_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_416_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_417_int__less__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_int @ I @ K )
=> ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
=> ( ! [I3: int] :
( ( ord_less_int @ I3 @ K )
=> ( ( P @ I3 )
=> ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_less_induct
thf(fact_418_int__diff__cases,axiom,
! [Z2: int] :
~ ! [M3: nat,N2: nat] :
( Z2
!= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% int_diff_cases
thf(fact_419_exE__realizer_H,axiom,
! [P: produc369741286924889929st_int > int > $o,P2: produc6425607678544837394st_int] :
( ( P @ ( produc1296517004818028822st_int @ P2 ) @ ( produc8941309049508465880st_int @ P2 ) )
=> ~ ! [X4: int,Y3: produc369741286924889929st_int] :
~ ( P @ Y3 @ X4 ) ) ).
% exE_realizer'
thf(fact_420_exE__realizer_H,axiom,
! [P: list_int > ( list_char > int ) > $o,P2: produc369741286924889929st_int] :
( ( P @ ( produc6244060863719409455st_int @ P2 ) @ ( produc2155683374303800813st_int @ P2 ) )
=> ~ ! [X4: list_char > int,Y3: list_int] :
~ ( P @ Y3 @ X4 ) ) ).
% exE_realizer'
thf(fact_421_exE__realizer_H,axiom,
! [P: int > $o > $o,P2: product_prod_o_int] :
( ( P @ ( product_snd_o_int @ P2 ) @ ( product_fst_o_int @ P2 ) )
=> ~ ! [X4: $o,Y3: int] :
~ ( P @ Y3 @ X4 ) ) ).
% exE_realizer'
thf(fact_422_exE__realizer_H,axiom,
! [P: product_prod_o_int > bexp > $o,P2: produc1897111610453708512_o_int] :
( ( P @ ( produc7219675232234870470_o_int @ P2 ) @ ( produc2736798884409471108_o_int @ P2 ) )
=> ~ ! [X4: bexp,Y3: product_prod_o_int] :
~ ( P @ Y3 @ X4 ) ) ).
% exE_realizer'
thf(fact_423_exE__realizer_H,axiom,
! [P: int > ( list_char > int ) > $o,P2: produc4435102495419491129nt_int] :
( ( P @ ( produc179445755590504351nt_int @ P2 ) @ ( produc9025023058967219549nt_int @ P2 ) )
=> ~ ! [X4: list_char > int,Y3: int] :
~ ( P @ Y3 @ X4 ) ) ).
% exE_realizer'
thf(fact_424_exE__realizer_H,axiom,
! [P: ( list_char > int ) > com > $o,P2: produc2931317944591925149ar_int] :
( ( P @ ( produc9207319146083860483ar_int @ P2 ) @ ( produc8829524412605799873ar_int @ P2 ) )
=> ~ ! [X4: com,Y3: list_char > int] :
~ ( P @ Y3 @ X4 ) ) ).
% exE_realizer'
thf(fact_425_exE__realizer_H,axiom,
! [P: int > int > $o,P2: product_prod_int_int] :
( ( P @ ( product_snd_int_int @ P2 ) @ ( product_fst_int_int @ P2 ) )
=> ~ ! [X4: int,Y3: int] :
~ ( P @ Y3 @ X4 ) ) ).
% exE_realizer'
thf(fact_426_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_427_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_428_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_429_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_430_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_431_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_432_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_433_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A5: nat,B4: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_434_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_435_verit__comp__simplify1_I1_J,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_436_verit__comp__simplify1_I1_J,axiom,
! [A: num] :
~ ( ord_less_num @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_437_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_438_nat__int__comparison_I1_J,axiom,
( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
= ( ^ [A5: nat,B4: nat] :
( ( semiri1314217659103216013at_int @ A5 )
= ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_439_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_440_verit__sum__simplify,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% verit_sum_simplify
thf(fact_441_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_442_size__char__eq__0,axiom,
( size_size_char
= ( ^ [C3: char] : zero_zero_nat ) ) ).
% size_char_eq_0
thf(fact_443_eq__snd__iff,axiom,
! [B: produc369741286924889929st_int,P2: produc6425607678544837394st_int] :
( ( B
= ( produc1296517004818028822st_int @ P2 ) )
= ( ? [A5: int] :
( P2
= ( produc5086643055186798020st_int @ A5 @ B ) ) ) ) ).
% eq_snd_iff
thf(fact_444_eq__snd__iff,axiom,
! [B: list_int,P2: produc369741286924889929st_int] :
( ( B
= ( produc6244060863719409455st_int @ P2 ) )
= ( ? [A5: list_char > int] :
( P2
= ( produc8650753666468850689st_int @ A5 @ B ) ) ) ) ).
% eq_snd_iff
thf(fact_445_eq__snd__iff,axiom,
! [B: int,P2: product_prod_o_int] :
( ( B
= ( product_snd_o_int @ P2 ) )
= ( ? [A5: $o] :
( P2
= ( product_Pair_o_int @ A5 @ B ) ) ) ) ).
% eq_snd_iff
thf(fact_446_eq__snd__iff,axiom,
! [B: product_prod_o_int,P2: produc1897111610453708512_o_int] :
( ( B
= ( produc7219675232234870470_o_int @ P2 ) )
= ( ? [A5: bexp] :
( P2
= ( produc4047900504771817624_o_int @ A5 @ B ) ) ) ) ).
% eq_snd_iff
thf(fact_447_eq__snd__iff,axiom,
! [B: int,P2: produc4435102495419491129nt_int] :
( ( B
= ( produc179445755590504351nt_int @ P2 ) )
= ( ? [A5: list_char > int] :
( P2
= ( produc5790713362662368625nt_int @ A5 @ B ) ) ) ) ).
% eq_snd_iff
thf(fact_448_eq__snd__iff,axiom,
! [B: list_char > int,P2: produc2931317944591925149ar_int] :
( ( B
= ( produc9207319146083860483ar_int @ P2 ) )
= ( ? [A5: com] :
( P2
= ( produc5595214716300948949ar_int @ A5 @ B ) ) ) ) ).
% eq_snd_iff
thf(fact_449_eq__snd__iff,axiom,
! [B: int,P2: product_prod_int_int] :
( ( B
= ( product_snd_int_int @ P2 ) )
= ( ? [A5: int] :
( P2
= ( product_Pair_int_int @ A5 @ B ) ) ) ) ).
% eq_snd_iff
thf(fact_450_sndI,axiom,
! [X: produc6425607678544837394st_int,Y: int,Z2: produc369741286924889929st_int] :
( ( X
= ( produc5086643055186798020st_int @ Y @ Z2 ) )
=> ( ( produc1296517004818028822st_int @ X )
= Z2 ) ) ).
% sndI
thf(fact_451_sndI,axiom,
! [X: produc369741286924889929st_int,Y: list_char > int,Z2: list_int] :
( ( X
= ( produc8650753666468850689st_int @ Y @ Z2 ) )
=> ( ( produc6244060863719409455st_int @ X )
= Z2 ) ) ).
% sndI
thf(fact_452_sndI,axiom,
! [X: product_prod_o_int,Y: $o,Z2: int] :
( ( X
= ( product_Pair_o_int @ Y @ Z2 ) )
=> ( ( product_snd_o_int @ X )
= Z2 ) ) ).
% sndI
thf(fact_453_sndI,axiom,
! [X: produc1897111610453708512_o_int,Y: bexp,Z2: product_prod_o_int] :
( ( X
= ( produc4047900504771817624_o_int @ Y @ Z2 ) )
=> ( ( produc7219675232234870470_o_int @ X )
= Z2 ) ) ).
% sndI
thf(fact_454_sndI,axiom,
! [X: produc4435102495419491129nt_int,Y: list_char > int,Z2: int] :
( ( X
= ( produc5790713362662368625nt_int @ Y @ Z2 ) )
=> ( ( produc179445755590504351nt_int @ X )
= Z2 ) ) ).
% sndI
thf(fact_455_sndI,axiom,
! [X: produc2931317944591925149ar_int,Y: com,Z2: list_char > int] :
( ( X
= ( produc5595214716300948949ar_int @ Y @ Z2 ) )
=> ( ( produc9207319146083860483ar_int @ X )
= Z2 ) ) ).
% sndI
thf(fact_456_sndI,axiom,
! [X: product_prod_int_int,Y: int,Z2: int] :
( ( X
= ( product_Pair_int_int @ Y @ Z2 ) )
=> ( ( product_snd_int_int @ X )
= Z2 ) ) ).
% sndI
thf(fact_457_eq__fst__iff,axiom,
! [A: bexp,P2: produc1897111610453708512_o_int] :
( ( A
= ( produc2736798884409471108_o_int @ P2 ) )
= ( ? [B4: product_prod_o_int] :
( P2
= ( produc4047900504771817624_o_int @ A @ B4 ) ) ) ) ).
% eq_fst_iff
thf(fact_458_eq__fst__iff,axiom,
! [A: int,P2: produc6425607678544837394st_int] :
( ( A
= ( produc8941309049508465880st_int @ P2 ) )
= ( ? [B4: produc369741286924889929st_int] :
( P2
= ( produc5086643055186798020st_int @ A @ B4 ) ) ) ) ).
% eq_fst_iff
thf(fact_459_eq__fst__iff,axiom,
! [A: $o,P2: product_prod_o_int] :
( ( A
= ( product_fst_o_int @ P2 ) )
= ( ? [B4: int] :
( P2
= ( product_Pair_o_int @ A @ B4 ) ) ) ) ).
% eq_fst_iff
thf(fact_460_eq__fst__iff,axiom,
! [A: int,P2: product_prod_int_int] :
( ( A
= ( product_fst_int_int @ P2 ) )
= ( ? [B4: int] :
( P2
= ( product_Pair_int_int @ A @ B4 ) ) ) ) ).
% eq_fst_iff
thf(fact_461_eq__fst__iff,axiom,
! [A: com,P2: produc2931317944591925149ar_int] :
( ( A
= ( produc8829524412605799873ar_int @ P2 ) )
= ( ? [B4: list_char > int] :
( P2
= ( produc5595214716300948949ar_int @ A @ B4 ) ) ) ) ).
% eq_fst_iff
thf(fact_462_eq__fst__iff,axiom,
! [A: list_char > int,P2: produc369741286924889929st_int] :
( ( A
= ( produc2155683374303800813st_int @ P2 ) )
= ( ? [B4: list_int] :
( P2
= ( produc8650753666468850689st_int @ A @ B4 ) ) ) ) ).
% eq_fst_iff
thf(fact_463_eq__fst__iff,axiom,
! [A: list_char > int,P2: produc4435102495419491129nt_int] :
( ( A
= ( produc9025023058967219549nt_int @ P2 ) )
= ( ? [B4: int] :
( P2
= ( produc5790713362662368625nt_int @ A @ B4 ) ) ) ) ).
% eq_fst_iff
thf(fact_464_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_465_append__assoc,axiom,
! [Xs: list_instr,Ys: list_instr,Zs: list_instr] :
( ( append_instr @ ( append_instr @ Xs @ Ys ) @ Zs )
= ( append_instr @ Xs @ ( append_instr @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_466_append__same__eq,axiom,
! [Ys: list_instr,Xs: list_instr,Zs: list_instr] :
( ( ( append_instr @ Ys @ Xs )
= ( append_instr @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_467_same__append__eq,axiom,
! [Xs: list_instr,Ys: list_instr,Zs: list_instr] :
( ( ( append_instr @ Xs @ Ys )
= ( append_instr @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_468_append_Oright__neutral,axiom,
! [A: list_instr] :
( ( append_instr @ A @ nil_instr )
= A ) ).
% append.right_neutral
thf(fact_469_append_Oright__neutral,axiom,
! [A: list_P419547167332244514st_int] :
( ( append5028292981072010935st_int @ A @ nil_Pr749310521696113548st_int )
= A ) ).
% append.right_neutral
thf(fact_470_append_Oright__neutral,axiom,
! [A: list_int] :
( ( append_int @ A @ nil_int )
= A ) ).
% append.right_neutral
thf(fact_471_append__Nil2,axiom,
! [Xs: list_instr] :
( ( append_instr @ Xs @ nil_instr )
= Xs ) ).
% append_Nil2
thf(fact_472_append__Nil2,axiom,
! [Xs: list_P419547167332244514st_int] :
( ( append5028292981072010935st_int @ Xs @ nil_Pr749310521696113548st_int )
= Xs ) ).
% append_Nil2
thf(fact_473_append__Nil2,axiom,
! [Xs: list_int] :
( ( append_int @ Xs @ nil_int )
= Xs ) ).
% append_Nil2
thf(fact_474_append__self__conv,axiom,
! [Xs: list_instr,Ys: list_instr] :
( ( ( append_instr @ Xs @ Ys )
= Xs )
= ( Ys = nil_instr ) ) ).
% append_self_conv
thf(fact_475_append__self__conv,axiom,
! [Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int] :
( ( ( append5028292981072010935st_int @ Xs @ Ys )
= Xs )
= ( Ys = nil_Pr749310521696113548st_int ) ) ).
% append_self_conv
thf(fact_476_append__self__conv,axiom,
! [Xs: list_int,Ys: list_int] :
( ( ( append_int @ Xs @ Ys )
= Xs )
= ( Ys = nil_int ) ) ).
% append_self_conv
thf(fact_477_self__append__conv,axiom,
! [Y: list_instr,Ys: list_instr] :
( ( Y
= ( append_instr @ Y @ Ys ) )
= ( Ys = nil_instr ) ) ).
% self_append_conv
thf(fact_478_self__append__conv,axiom,
! [Y: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int] :
( ( Y
= ( append5028292981072010935st_int @ Y @ Ys ) )
= ( Ys = nil_Pr749310521696113548st_int ) ) ).
% self_append_conv
thf(fact_479_self__append__conv,axiom,
! [Y: list_int,Ys: list_int] :
( ( Y
= ( append_int @ Y @ Ys ) )
= ( Ys = nil_int ) ) ).
% self_append_conv
thf(fact_480_append__self__conv2,axiom,
! [Xs: list_instr,Ys: list_instr] :
( ( ( append_instr @ Xs @ Ys )
= Ys )
= ( Xs = nil_instr ) ) ).
% append_self_conv2
thf(fact_481_append__self__conv2,axiom,
! [Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int] :
( ( ( append5028292981072010935st_int @ Xs @ Ys )
= Ys )
= ( Xs = nil_Pr749310521696113548st_int ) ) ).
% append_self_conv2
thf(fact_482_append__self__conv2,axiom,
! [Xs: list_int,Ys: list_int] :
( ( ( append_int @ Xs @ Ys )
= Ys )
= ( Xs = nil_int ) ) ).
% append_self_conv2
thf(fact_483_self__append__conv2,axiom,
! [Y: list_instr,Xs: list_instr] :
( ( Y
= ( append_instr @ Xs @ Y ) )
= ( Xs = nil_instr ) ) ).
% self_append_conv2
thf(fact_484_self__append__conv2,axiom,
! [Y: list_P419547167332244514st_int,Xs: list_P419547167332244514st_int] :
( ( Y
= ( append5028292981072010935st_int @ Xs @ Y ) )
= ( Xs = nil_Pr749310521696113548st_int ) ) ).
% self_append_conv2
thf(fact_485_self__append__conv2,axiom,
! [Y: list_int,Xs: list_int] :
( ( Y
= ( append_int @ Xs @ Y ) )
= ( Xs = nil_int ) ) ).
% self_append_conv2
thf(fact_486_Nil__is__append__conv,axiom,
! [Xs: list_instr,Ys: list_instr] :
( ( nil_instr
= ( append_instr @ Xs @ Ys ) )
= ( ( Xs = nil_instr )
& ( Ys = nil_instr ) ) ) ).
% Nil_is_append_conv
thf(fact_487_Nil__is__append__conv,axiom,
! [Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int] :
( ( nil_Pr749310521696113548st_int
= ( append5028292981072010935st_int @ Xs @ Ys ) )
= ( ( Xs = nil_Pr749310521696113548st_int )
& ( Ys = nil_Pr749310521696113548st_int ) ) ) ).
% Nil_is_append_conv
thf(fact_488_Nil__is__append__conv,axiom,
! [Xs: list_int,Ys: list_int] :
( ( nil_int
= ( append_int @ Xs @ Ys ) )
= ( ( Xs = nil_int )
& ( Ys = nil_int ) ) ) ).
% Nil_is_append_conv
thf(fact_489_append__is__Nil__conv,axiom,
! [Xs: list_instr,Ys: list_instr] :
( ( ( append_instr @ Xs @ Ys )
= nil_instr )
= ( ( Xs = nil_instr )
& ( Ys = nil_instr ) ) ) ).
% append_is_Nil_conv
thf(fact_490_append__is__Nil__conv,axiom,
! [Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int] :
( ( ( append5028292981072010935st_int @ Xs @ Ys )
= nil_Pr749310521696113548st_int )
= ( ( Xs = nil_Pr749310521696113548st_int )
& ( Ys = nil_Pr749310521696113548st_int ) ) ) ).
% append_is_Nil_conv
thf(fact_491_append__is__Nil__conv,axiom,
! [Xs: list_int,Ys: list_int] :
( ( ( append_int @ Xs @ Ys )
= nil_int )
= ( ( Xs = nil_int )
& ( Ys = nil_int ) ) ) ).
% append_is_Nil_conv
thf(fact_492_append__eq__append__conv,axiom,
! [Xs: list_int,Ys: list_int,Us: list_int,Vs: list_int] :
( ( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys ) )
| ( ( size_size_list_int @ Us )
= ( size_size_list_int @ Vs ) ) )
=> ( ( ( append_int @ Xs @ Us )
= ( append_int @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_493_append__eq__append__conv,axiom,
! [Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int,Us: list_P419547167332244514st_int,Vs: list_P419547167332244514st_int] :
( ( ( ( size_s229347708945616566st_int @ Xs )
= ( size_s229347708945616566st_int @ Ys ) )
| ( ( size_s229347708945616566st_int @ Us )
= ( size_s229347708945616566st_int @ Vs ) ) )
=> ( ( ( append5028292981072010935st_int @ Xs @ Us )
= ( append5028292981072010935st_int @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_494_append__eq__append__conv,axiom,
! [Xs: list_instr,Ys: list_instr,Us: list_instr,Vs: list_instr] :
( ( ( ( size_size_list_instr @ Xs )
= ( size_size_list_instr @ Ys ) )
| ( ( size_size_list_instr @ Us )
= ( size_size_list_instr @ Vs ) ) )
=> ( ( ( append_instr @ Xs @ Us )
= ( append_instr @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_495_length__append,axiom,
! [Xs: list_int,Ys: list_int] :
( ( size_size_list_int @ ( append_int @ Xs @ Ys ) )
= ( plus_plus_nat @ ( size_size_list_int @ Xs ) @ ( size_size_list_int @ Ys ) ) ) ).
% length_append
thf(fact_496_length__append,axiom,
! [Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int] :
( ( size_s229347708945616566st_int @ ( append5028292981072010935st_int @ Xs @ Ys ) )
= ( plus_plus_nat @ ( size_s229347708945616566st_int @ Xs ) @ ( size_s229347708945616566st_int @ Ys ) ) ) ).
% length_append
thf(fact_497_length__append,axiom,
! [Xs: list_instr,Ys: list_instr] :
( ( size_size_list_instr @ ( append_instr @ Xs @ Ys ) )
= ( plus_plus_nat @ ( size_size_list_instr @ Xs ) @ ( size_size_list_instr @ Ys ) ) ) ).
% length_append
thf(fact_498_nth__append__length__plus,axiom,
! [Xs: list_int,Ys: list_int,N: nat] :
( ( nth_int @ ( append_int @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_size_list_int @ Xs ) @ N ) )
= ( nth_int @ Ys @ N ) ) ).
% nth_append_length_plus
thf(fact_499_nth__append__length__plus,axiom,
! [Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int,N: nat] :
( ( nth_Pr8327173284404383691st_int @ ( append5028292981072010935st_int @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_s229347708945616566st_int @ Xs ) @ N ) )
= ( nth_Pr8327173284404383691st_int @ Ys @ N ) ) ).
% nth_append_length_plus
thf(fact_500_nth__append__length__plus,axiom,
! [Xs: list_instr,Ys: list_instr,N: nat] :
( ( nth_instr @ ( append_instr @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_size_list_instr @ Xs ) @ N ) )
= ( nth_instr @ Ys @ N ) ) ).
% nth_append_length_plus
thf(fact_501_append__eq__appendI,axiom,
! [Xs: list_instr,Xs1: list_instr,Zs: list_instr,Ys: list_instr,Us: list_instr] :
( ( ( append_instr @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_instr @ Xs1 @ Us ) )
=> ( ( append_instr @ Xs @ Ys )
= ( append_instr @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_502_append__eq__append__conv2,axiom,
! [Xs: list_instr,Ys: list_instr,Zs: list_instr,Ts: list_instr] :
( ( ( append_instr @ Xs @ Ys )
= ( append_instr @ Zs @ Ts ) )
= ( ? [Us2: list_instr] :
( ( ( Xs
= ( append_instr @ Zs @ Us2 ) )
& ( ( append_instr @ Us2 @ Ys )
= Ts ) )
| ( ( ( append_instr @ Xs @ Us2 )
= Zs )
& ( Ys
= ( append_instr @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_503_append__Nil,axiom,
! [Ys: list_instr] :
( ( append_instr @ nil_instr @ Ys )
= Ys ) ).
% append_Nil
thf(fact_504_append__Nil,axiom,
! [Ys: list_P419547167332244514st_int] :
( ( append5028292981072010935st_int @ nil_Pr749310521696113548st_int @ Ys )
= Ys ) ).
% append_Nil
thf(fact_505_append__Nil,axiom,
! [Ys: list_int] :
( ( append_int @ nil_int @ Ys )
= Ys ) ).
% append_Nil
thf(fact_506_append_Oleft__neutral,axiom,
! [A: list_instr] :
( ( append_instr @ nil_instr @ A )
= A ) ).
% append.left_neutral
thf(fact_507_append_Oleft__neutral,axiom,
! [A: list_P419547167332244514st_int] :
( ( append5028292981072010935st_int @ nil_Pr749310521696113548st_int @ A )
= A ) ).
% append.left_neutral
thf(fact_508_append_Oleft__neutral,axiom,
! [A: list_int] :
( ( append_int @ nil_int @ A )
= A ) ).
% append.left_neutral
thf(fact_509_eq__Nil__appendI,axiom,
! [Xs: list_instr,Ys: list_instr] :
( ( Xs = Ys )
=> ( Xs
= ( append_instr @ nil_instr @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_510_eq__Nil__appendI,axiom,
! [Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int] :
( ( Xs = Ys )
=> ( Xs
= ( append5028292981072010935st_int @ nil_Pr749310521696113548st_int @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_511_eq__Nil__appendI,axiom,
! [Xs: list_int,Ys: list_int] :
( ( Xs = Ys )
=> ( Xs
= ( append_int @ nil_int @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_512_nth__append,axiom,
! [N: nat,Xs: list_int,Ys: list_int] :
( ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
=> ( ( nth_int @ ( append_int @ Xs @ Ys ) @ N )
= ( nth_int @ Xs @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
=> ( ( nth_int @ ( append_int @ Xs @ Ys ) @ N )
= ( nth_int @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_int @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_513_nth__append,axiom,
! [N: nat,Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int] :
( ( ( ord_less_nat @ N @ ( size_s229347708945616566st_int @ Xs ) )
=> ( ( nth_Pr8327173284404383691st_int @ ( append5028292981072010935st_int @ Xs @ Ys ) @ N )
= ( nth_Pr8327173284404383691st_int @ Xs @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_s229347708945616566st_int @ Xs ) )
=> ( ( nth_Pr8327173284404383691st_int @ ( append5028292981072010935st_int @ Xs @ Ys ) @ N )
= ( nth_Pr8327173284404383691st_int @ Ys @ ( minus_minus_nat @ N @ ( size_s229347708945616566st_int @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_514_nth__append,axiom,
! [N: nat,Xs: list_instr,Ys: list_instr] :
( ( ( ord_less_nat @ N @ ( size_size_list_instr @ Xs ) )
=> ( ( nth_instr @ ( append_instr @ Xs @ Ys ) @ N )
= ( nth_instr @ Xs @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_size_list_instr @ Xs ) )
=> ( ( nth_instr @ ( append_instr @ Xs @ Ys ) @ N )
= ( nth_instr @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_instr @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_515_fstI,axiom,
! [X: produc1897111610453708512_o_int,Y: bexp,Z2: product_prod_o_int] :
( ( X
= ( produc4047900504771817624_o_int @ Y @ Z2 ) )
=> ( ( produc2736798884409471108_o_int @ X )
= Y ) ) ).
% fstI
thf(fact_516_fstI,axiom,
! [X: produc6425607678544837394st_int,Y: int,Z2: produc369741286924889929st_int] :
( ( X
= ( produc5086643055186798020st_int @ Y @ Z2 ) )
=> ( ( produc8941309049508465880st_int @ X )
= Y ) ) ).
% fstI
thf(fact_517_fstI,axiom,
! [X: product_prod_o_int,Y: $o,Z2: int] :
( ( X
= ( product_Pair_o_int @ Y @ Z2 ) )
=> ( ( product_fst_o_int @ X )
= Y ) ) ).
% fstI
thf(fact_518_fstI,axiom,
! [X: product_prod_int_int,Y: int,Z2: int] :
( ( X
= ( product_Pair_int_int @ Y @ Z2 ) )
=> ( ( product_fst_int_int @ X )
= Y ) ) ).
% fstI
thf(fact_519_fstI,axiom,
! [X: produc2931317944591925149ar_int,Y: com,Z2: list_char > int] :
( ( X
= ( produc5595214716300948949ar_int @ Y @ Z2 ) )
=> ( ( produc8829524412605799873ar_int @ X )
= Y ) ) ).
% fstI
thf(fact_520_fstI,axiom,
! [X: produc369741286924889929st_int,Y: list_char > int,Z2: list_int] :
( ( X
= ( produc8650753666468850689st_int @ Y @ Z2 ) )
=> ( ( produc2155683374303800813st_int @ X )
= Y ) ) ).
% fstI
thf(fact_521_fstI,axiom,
! [X: produc4435102495419491129nt_int,Y: list_char > int,Z2: int] :
( ( X
= ( produc5790713362662368625nt_int @ Y @ Z2 ) )
=> ( ( produc9025023058967219549nt_int @ X )
= Y ) ) ).
% fstI
thf(fact_522_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_1 ) ) ) @ one_one_int ) ) ) ) @ ( append_instr @ ( ccomp @ c_1 ) @ ( cons_instr @ ( jmp @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ ( ccomp @ c_2 ) ) ) ) @ ( ccomp @ c_2 ) ) ) ) @ cfsa ).
% A
thf(fact_523_size_H__char__eq__0,axiom,
( size_char
= ( ^ [C3: char] : zero_zero_nat ) ) ).
% size'_char_eq_0
thf(fact_524_Euclid__induct,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B3: nat] :
( ( P @ A4 @ B3 )
= ( P @ B3 @ A4 ) )
=> ( ! [A4: nat] : ( P @ A4 @ zero_zero_nat )
=> ( ! [A4: nat,B3: nat] :
( ( P @ A4 @ B3 )
=> ( P @ A4 @ ( plus_plus_nat @ A4 @ B3 ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Euclid_induct
thf(fact_525_add__0__iff,axiom,
! [B: nat,A: nat] :
( ( B
= ( plus_plus_nat @ B @ A ) )
= ( A = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_526_add__0__iff,axiom,
! [B: int,A: int] :
( ( B
= ( plus_plus_int @ B @ A ) )
= ( A = zero_zero_int ) ) ).
% add_0_iff
thf(fact_527_nth__Cons__pos,axiom,
! [N: nat,X: instr,Xs: list_instr] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_instr @ ( cons_instr @ X @ Xs ) @ N )
= ( nth_instr @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_528_nth__Cons__pos,axiom,
! [N: nat,X: int,Xs: list_int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_int @ ( cons_int @ X @ Xs ) @ N )
= ( nth_int @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_529_nth__Cons__pos,axiom,
! [N: nat,X: produc6425607678544837394st_int,Xs: list_P419547167332244514st_int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_Pr8327173284404383691st_int @ ( cons_P4151473878978182108st_int @ X @ Xs ) @ N )
= ( nth_Pr8327173284404383691st_int @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_530_dbl__inc__simps_I2_J,axiom,
( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
= one_one_int ) ).
% dbl_inc_simps(2)
thf(fact_531_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_532_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_533_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_534_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_535_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_536_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_537_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_538_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_539_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_540_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_541_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_542_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_543_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_544_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_545_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_546_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_547_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_548_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_549_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_550_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_551_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_552_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_553_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_554_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_555_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_556_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_557_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_558_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_559_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_560_append1__eq__conv,axiom,
! [Xs: list_instr,X: instr,Ys: list_instr,Y: instr] :
( ( ( append_instr @ Xs @ ( cons_instr @ X @ nil_instr ) )
= ( append_instr @ Ys @ ( cons_instr @ Y @ nil_instr ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_561_append1__eq__conv,axiom,
! [Xs: list_int,X: int,Ys: list_int,Y: int] :
( ( ( append_int @ Xs @ ( cons_int @ X @ nil_int ) )
= ( append_int @ Ys @ ( cons_int @ Y @ nil_int ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_562_append1__eq__conv,axiom,
! [Xs: list_P419547167332244514st_int,X: produc6425607678544837394st_int,Ys: list_P419547167332244514st_int,Y: produc6425607678544837394st_int] :
( ( ( append5028292981072010935st_int @ Xs @ ( cons_P4151473878978182108st_int @ X @ nil_Pr749310521696113548st_int ) )
= ( append5028292981072010935st_int @ Ys @ ( cons_P4151473878978182108st_int @ Y @ nil_Pr749310521696113548st_int ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_563_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_564_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_565_nth__Cons__0,axiom,
! [X: instr,Xs: list_instr] :
( ( nth_instr @ ( cons_instr @ X @ Xs ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_566_nth__Cons__0,axiom,
! [X: int,Xs: list_int] :
( ( nth_int @ ( cons_int @ X @ Xs ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_567_nth__Cons__0,axiom,
! [X: produc6425607678544837394st_int,Xs: list_P419547167332244514st_int] :
( ( nth_Pr8327173284404383691st_int @ ( cons_P4151473878978182108st_int @ X @ Xs ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_568_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_569_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_570_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_571_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_572_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_573_nth__append__length,axiom,
! [Xs: list_int,X: int,Ys: list_int] :
( ( nth_int @ ( append_int @ Xs @ ( cons_int @ X @ Ys ) ) @ ( size_size_list_int @ Xs ) )
= X ) ).
% nth_append_length
thf(fact_574_nth__append__length,axiom,
! [Xs: list_P419547167332244514st_int,X: produc6425607678544837394st_int,Ys: list_P419547167332244514st_int] :
( ( nth_Pr8327173284404383691st_int @ ( append5028292981072010935st_int @ Xs @ ( cons_P4151473878978182108st_int @ X @ Ys ) ) @ ( size_s229347708945616566st_int @ Xs ) )
= X ) ).
% nth_append_length
thf(fact_575_nth__append__length,axiom,
! [Xs: list_instr,X: instr,Ys: list_instr] :
( ( nth_instr @ ( append_instr @ Xs @ ( cons_instr @ X @ Ys ) ) @ ( size_size_list_instr @ Xs ) )
= X ) ).
% nth_append_length
thf(fact_576_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M4: nat] :
( ( P @ X )
=> ( ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M4 ) )
=> ~ ! [M3: nat] :
( ( P @ M3 )
=> ~ ! [X6: nat] :
( ( P @ X6 )
=> ( ord_less_eq_nat @ X6 @ M3 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_577_impossible__Cons,axiom,
! [Xs: list_int,Ys: list_int,X: int] :
( ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ ( size_size_list_int @ Ys ) )
=> ( Xs
!= ( cons_int @ X @ Ys ) ) ) ).
% impossible_Cons
thf(fact_578_impossible__Cons,axiom,
! [Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int,X: produc6425607678544837394st_int] :
( ( ord_less_eq_nat @ ( size_s229347708945616566st_int @ Xs ) @ ( size_s229347708945616566st_int @ Ys ) )
=> ( Xs
!= ( cons_P4151473878978182108st_int @ X @ Ys ) ) ) ).
% impossible_Cons
thf(fact_579_impossible__Cons,axiom,
! [Xs: list_instr,Ys: list_instr,X: instr] :
( ( ord_less_eq_nat @ ( size_size_list_instr @ Xs ) @ ( size_size_list_instr @ Ys ) )
=> ( Xs
!= ( cons_instr @ X @ Ys ) ) ) ).
% impossible_Cons
thf(fact_580_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_581_transpose_Ocases,axiom,
! [X: list_list_instr] :
( ( X != nil_list_instr )
=> ( ! [Xss: list_list_instr] :
( X
!= ( cons_list_instr @ nil_instr @ Xss ) )
=> ~ ! [X4: instr,Xs2: list_instr,Xss: list_list_instr] :
( X
!= ( cons_list_instr @ ( cons_instr @ X4 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_582_transpose_Ocases,axiom,
! [X: list_list_int] :
( ( X != nil_list_int )
=> ( ! [Xss: list_list_int] :
( X
!= ( cons_list_int @ nil_int @ Xss ) )
=> ~ ! [X4: int,Xs2: list_int,Xss: list_list_int] :
( X
!= ( cons_list_int @ ( cons_int @ X4 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_583_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,Xs2: list_P419547167332244514st_int,Xss: list_l6422821129637969586st_int] :
( X
!= ( cons_l8648290059330449388st_int @ ( cons_P4151473878978182108st_int @ X4 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_584_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_585_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_586_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X4: nat] :
( ( P @ X4 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X4 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_587_not__Cons__self2,axiom,
! [X: instr,Xs: list_instr] :
( ( cons_instr @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_588_not__Cons__self2,axiom,
! [X: int,Xs: list_int] :
( ( cons_int @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_589_not__Cons__self2,axiom,
! [X: produc6425607678544837394st_int,Xs: list_P419547167332244514st_int] :
( ( cons_P4151473878978182108st_int @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_590_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_591_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_592_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_593_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_594_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_595_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_596_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_597_verit__la__disequality,axiom,
! [A: num,B: num] :
( ( A = B )
| ~ ( ord_less_eq_num @ A @ B )
| ~ ( ord_less_eq_num @ B @ A ) ) ).
% verit_la_disequality
thf(fact_598_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_599_verit__comp__simplify1_I2_J,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_600_verit__comp__simplify1_I2_J,axiom,
! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_601_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_602_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_603_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_604_splice_Ocases,axiom,
! [X: produc8891168062546637159_instr] :
( ! [Ys4: list_instr] :
( X
!= ( produc7750586281064920791_instr @ nil_instr @ Ys4 ) )
=> ~ ! [X4: instr,Xs2: list_instr,Ys4: list_instr] :
( X
!= ( produc7750586281064920791_instr @ ( cons_instr @ X4 @ Xs2 ) @ Ys4 ) ) ) ).
% splice.cases
thf(fact_605_splice_Ocases,axiom,
! [X: produc1186641810826059865st_int] :
( ! [Ys4: list_int] :
( X
!= ( produc364263696895485585st_int @ nil_int @ Ys4 ) )
=> ~ ! [X4: int,Xs2: list_int,Ys4: list_int] :
( X
!= ( produc364263696895485585st_int @ ( cons_int @ X4 @ Xs2 ) @ Ys4 ) ) ) ).
% splice.cases
thf(fact_606_splice_Ocases,axiom,
! [X: produc5848496107505245989st_int] :
( ! [Ys4: list_P419547167332244514st_int] :
( X
!= ( produc2639222612625641053st_int @ nil_Pr749310521696113548st_int @ Ys4 ) )
=> ~ ! [X4: produc6425607678544837394st_int,Xs2: list_P419547167332244514st_int,Ys4: list_P419547167332244514st_int] :
( X
!= ( produc2639222612625641053st_int @ ( cons_P4151473878978182108st_int @ X4 @ Xs2 ) @ Ys4 ) ) ) ).
% splice.cases
thf(fact_607_shuffles_Ocases,axiom,
! [X: produc8891168062546637159_instr] :
( ! [Ys4: list_instr] :
( X
!= ( produc7750586281064920791_instr @ nil_instr @ Ys4 ) )
=> ( ! [Xs2: list_instr] :
( X
!= ( produc7750586281064920791_instr @ Xs2 @ nil_instr ) )
=> ~ ! [X4: instr,Xs2: list_instr,Y3: instr,Ys4: list_instr] :
( X
!= ( produc7750586281064920791_instr @ ( cons_instr @ X4 @ Xs2 ) @ ( cons_instr @ Y3 @ Ys4 ) ) ) ) ) ).
% shuffles.cases
thf(fact_608_shuffles_Ocases,axiom,
! [X: produc1186641810826059865st_int] :
( ! [Ys4: list_int] :
( X
!= ( produc364263696895485585st_int @ nil_int @ Ys4 ) )
=> ( ! [Xs2: list_int] :
( X
!= ( produc364263696895485585st_int @ Xs2 @ nil_int ) )
=> ~ ! [X4: int,Xs2: list_int,Y3: int,Ys4: list_int] :
( X
!= ( produc364263696895485585st_int @ ( cons_int @ X4 @ Xs2 ) @ ( cons_int @ Y3 @ Ys4 ) ) ) ) ) ).
% shuffles.cases
thf(fact_609_shuffles_Ocases,axiom,
! [X: produc5848496107505245989st_int] :
( ! [Ys4: list_P419547167332244514st_int] :
( X
!= ( produc2639222612625641053st_int @ nil_Pr749310521696113548st_int @ Ys4 ) )
=> ( ! [Xs2: list_P419547167332244514st_int] :
( X
!= ( produc2639222612625641053st_int @ Xs2 @ nil_Pr749310521696113548st_int ) )
=> ~ ! [X4: produc6425607678544837394st_int,Xs2: list_P419547167332244514st_int,Y3: produc6425607678544837394st_int,Ys4: list_P419547167332244514st_int] :
( X
!= ( produc2639222612625641053st_int @ ( cons_P4151473878978182108st_int @ X4 @ Xs2 ) @ ( cons_P4151473878978182108st_int @ Y3 @ Ys4 ) ) ) ) ) ).
% shuffles.cases
thf(fact_610_sorted__wrt_Ocases,axiom,
! [X: produc7133927873007960030_instr] :
( ! [P3: instr > instr > $o] :
( X
!= ( produc7818391707689388878_instr @ P3 @ nil_instr ) )
=> ~ ! [P3: instr > instr > $o,X4: instr,Ys4: list_instr] :
( X
!= ( produc7818391707689388878_instr @ P3 @ ( cons_instr @ X4 @ Ys4 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_611_sorted__wrt_Ocases,axiom,
! [X: produc5834231552977413017st_int] :
( ! [P3: int > int > $o] :
( X
!= ( produc8618682346314911123st_int @ P3 @ nil_int ) )
=> ~ ! [P3: int > int > $o,X4: int,Ys4: list_int] :
( X
!= ( produc8618682346314911123st_int @ P3 @ ( cons_int @ X4 @ Ys4 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_612_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,Ys4: list_P419547167332244514st_int] :
( X
!= ( produc554038597315947693st_int @ P3 @ ( cons_P4151473878978182108st_int @ X4 @ Ys4 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_613_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,Y3: instr,Xs2: list_instr] :
( X
!= ( produc7818391707689388878_instr @ P3 @ ( cons_instr @ X4 @ ( cons_instr @ Y3 @ Xs2 ) ) ) ) ) ) ).
% successively.cases
thf(fact_614_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,Y3: int,Xs2: list_int] :
( X
!= ( produc8618682346314911123st_int @ P3 @ ( cons_int @ X4 @ ( cons_int @ Y3 @ Xs2 ) ) ) ) ) ) ).
% successively.cases
thf(fact_615_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,Y3: produc6425607678544837394st_int,Xs2: list_P419547167332244514st_int] :
( X
!= ( produc554038597315947693st_int @ P3 @ ( cons_P4151473878978182108st_int @ X4 @ ( cons_P4151473878978182108st_int @ Y3 @ Xs2 ) ) ) ) ) ) ).
% successively.cases
thf(fact_616_list__nonempty__induct,axiom,
! [Xs: list_instr,P: list_instr > $o] :
( ( Xs != nil_instr )
=> ( ! [X4: instr] : ( P @ ( cons_instr @ X4 @ nil_instr ) )
=> ( ! [X4: instr,Xs2: list_instr] :
( ( Xs2 != nil_instr )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_instr @ X4 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_617_list__nonempty__induct,axiom,
! [Xs: list_int,P: list_int > $o] :
( ( Xs != nil_int )
=> ( ! [X4: int] : ( P @ ( cons_int @ X4 @ nil_int ) )
=> ( ! [X4: int,Xs2: list_int] :
( ( Xs2 != nil_int )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_int @ X4 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_618_list__nonempty__induct,axiom,
! [Xs: list_P419547167332244514st_int,P: list_P419547167332244514st_int > $o] :
( ( Xs != nil_Pr749310521696113548st_int )
=> ( ! [X4: produc6425607678544837394st_int] : ( P @ ( cons_P4151473878978182108st_int @ X4 @ nil_Pr749310521696113548st_int ) )
=> ( ! [X4: produc6425607678544837394st_int,Xs2: list_P419547167332244514st_int] :
( ( Xs2 != nil_Pr749310521696113548st_int )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_P4151473878978182108st_int @ X4 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_619_list__induct2_H,axiom,
! [P: list_instr > list_instr > $o,Xs: list_instr,Ys: list_instr] :
( ( P @ nil_instr @ nil_instr )
=> ( ! [X4: instr,Xs2: list_instr] : ( P @ ( cons_instr @ X4 @ Xs2 ) @ nil_instr )
=> ( ! [Y3: instr,Ys4: list_instr] : ( P @ nil_instr @ ( cons_instr @ Y3 @ Ys4 ) )
=> ( ! [X4: instr,Xs2: list_instr,Y3: instr,Ys4: list_instr] :
( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_instr @ X4 @ Xs2 ) @ ( cons_instr @ Y3 @ Ys4 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_620_list__induct2_H,axiom,
! [P: list_instr > list_int > $o,Xs: list_instr,Ys: list_int] :
( ( P @ nil_instr @ nil_int )
=> ( ! [X4: instr,Xs2: list_instr] : ( P @ ( cons_instr @ X4 @ Xs2 ) @ nil_int )
=> ( ! [Y3: int,Ys4: list_int] : ( P @ nil_instr @ ( cons_int @ Y3 @ Ys4 ) )
=> ( ! [X4: instr,Xs2: list_instr,Y3: int,Ys4: list_int] :
( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_instr @ X4 @ Xs2 ) @ ( cons_int @ Y3 @ Ys4 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_621_list__induct2_H,axiom,
! [P: list_instr > list_P419547167332244514st_int > $o,Xs: list_instr,Ys: list_P419547167332244514st_int] :
( ( P @ nil_instr @ nil_Pr749310521696113548st_int )
=> ( ! [X4: instr,Xs2: list_instr] : ( P @ ( cons_instr @ X4 @ Xs2 ) @ nil_Pr749310521696113548st_int )
=> ( ! [Y3: produc6425607678544837394st_int,Ys4: list_P419547167332244514st_int] : ( P @ nil_instr @ ( cons_P4151473878978182108st_int @ Y3 @ Ys4 ) )
=> ( ! [X4: instr,Xs2: list_instr,Y3: produc6425607678544837394st_int,Ys4: list_P419547167332244514st_int] :
( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_instr @ X4 @ Xs2 ) @ ( cons_P4151473878978182108st_int @ Y3 @ Ys4 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_622_list__induct2_H,axiom,
! [P: list_int > list_instr > $o,Xs: list_int,Ys: list_instr] :
( ( P @ nil_int @ nil_instr )
=> ( ! [X4: int,Xs2: list_int] : ( P @ ( cons_int @ X4 @ Xs2 ) @ nil_instr )
=> ( ! [Y3: instr,Ys4: list_instr] : ( P @ nil_int @ ( cons_instr @ Y3 @ Ys4 ) )
=> ( ! [X4: int,Xs2: list_int,Y3: instr,Ys4: list_instr] :
( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_int @ X4 @ Xs2 ) @ ( cons_instr @ Y3 @ Ys4 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_623_list__induct2_H,axiom,
! [P: list_int > list_int > $o,Xs: list_int,Ys: list_int] :
( ( P @ nil_int @ nil_int )
=> ( ! [X4: int,Xs2: list_int] : ( P @ ( cons_int @ X4 @ Xs2 ) @ nil_int )
=> ( ! [Y3: int,Ys4: list_int] : ( P @ nil_int @ ( cons_int @ Y3 @ Ys4 ) )
=> ( ! [X4: int,Xs2: list_int,Y3: int,Ys4: list_int] :
( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_int @ X4 @ Xs2 ) @ ( cons_int @ Y3 @ Ys4 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_624_list__induct2_H,axiom,
! [P: list_int > list_P419547167332244514st_int > $o,Xs: list_int,Ys: list_P419547167332244514st_int] :
( ( P @ nil_int @ nil_Pr749310521696113548st_int )
=> ( ! [X4: int,Xs2: list_int] : ( P @ ( cons_int @ X4 @ Xs2 ) @ nil_Pr749310521696113548st_int )
=> ( ! [Y3: produc6425607678544837394st_int,Ys4: list_P419547167332244514st_int] : ( P @ nil_int @ ( cons_P4151473878978182108st_int @ Y3 @ Ys4 ) )
=> ( ! [X4: int,Xs2: list_int,Y3: produc6425607678544837394st_int,Ys4: list_P419547167332244514st_int] :
( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_int @ X4 @ Xs2 ) @ ( cons_P4151473878978182108st_int @ Y3 @ Ys4 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_625_list__induct2_H,axiom,
! [P: list_P419547167332244514st_int > list_instr > $o,Xs: list_P419547167332244514st_int,Ys: list_instr] :
( ( P @ nil_Pr749310521696113548st_int @ nil_instr )
=> ( ! [X4: produc6425607678544837394st_int,Xs2: list_P419547167332244514st_int] : ( P @ ( cons_P4151473878978182108st_int @ X4 @ Xs2 ) @ nil_instr )
=> ( ! [Y3: instr,Ys4: list_instr] : ( P @ nil_Pr749310521696113548st_int @ ( cons_instr @ Y3 @ Ys4 ) )
=> ( ! [X4: produc6425607678544837394st_int,Xs2: list_P419547167332244514st_int,Y3: instr,Ys4: list_instr] :
( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_P4151473878978182108st_int @ X4 @ Xs2 ) @ ( cons_instr @ Y3 @ Ys4 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_626_list__induct2_H,axiom,
! [P: list_P419547167332244514st_int > list_int > $o,Xs: list_P419547167332244514st_int,Ys: list_int] :
( ( P @ nil_Pr749310521696113548st_int @ nil_int )
=> ( ! [X4: produc6425607678544837394st_int,Xs2: list_P419547167332244514st_int] : ( P @ ( cons_P4151473878978182108st_int @ X4 @ Xs2 ) @ nil_int )
=> ( ! [Y3: int,Ys4: list_int] : ( P @ nil_Pr749310521696113548st_int @ ( cons_int @ Y3 @ Ys4 ) )
=> ( ! [X4: produc6425607678544837394st_int,Xs2: list_P419547167332244514st_int,Y3: int,Ys4: list_int] :
( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_P4151473878978182108st_int @ X4 @ Xs2 ) @ ( cons_int @ Y3 @ Ys4 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_627_list__induct2_H,axiom,
! [P: list_P419547167332244514st_int > list_P419547167332244514st_int > $o,Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int] :
( ( P @ nil_Pr749310521696113548st_int @ nil_Pr749310521696113548st_int )
=> ( ! [X4: produc6425607678544837394st_int,Xs2: list_P419547167332244514st_int] : ( P @ ( cons_P4151473878978182108st_int @ X4 @ Xs2 ) @ nil_Pr749310521696113548st_int )
=> ( ! [Y3: produc6425607678544837394st_int,Ys4: list_P419547167332244514st_int] : ( P @ nil_Pr749310521696113548st_int @ ( cons_P4151473878978182108st_int @ Y3 @ Ys4 ) )
=> ( ! [X4: produc6425607678544837394st_int,Xs2: list_P419547167332244514st_int,Y3: produc6425607678544837394st_int,Ys4: list_P419547167332244514st_int] :
( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_P4151473878978182108st_int @ X4 @ Xs2 ) @ ( cons_P4151473878978182108st_int @ Y3 @ Ys4 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_628_neq__Nil__conv,axiom,
! [Xs: list_instr] :
( ( Xs != nil_instr )
= ( ? [Y6: instr,Ys3: list_instr] :
( Xs
= ( cons_instr @ Y6 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_629_neq__Nil__conv,axiom,
! [Xs: list_int] :
( ( Xs != nil_int )
= ( ? [Y6: int,Ys3: list_int] :
( Xs
= ( cons_int @ Y6 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_630_neq__Nil__conv,axiom,
! [Xs: list_P419547167332244514st_int] :
( ( Xs != nil_Pr749310521696113548st_int )
= ( ? [Y6: produc6425607678544837394st_int,Ys3: list_P419547167332244514st_int] :
( Xs
= ( cons_P4151473878978182108st_int @ Y6 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_631_remdups__adj_Ocases,axiom,
! [X: list_instr] :
( ( X != nil_instr )
=> ( ! [X4: instr] :
( X
!= ( cons_instr @ X4 @ nil_instr ) )
=> ~ ! [X4: instr,Y3: instr,Xs2: list_instr] :
( X
!= ( cons_instr @ X4 @ ( cons_instr @ Y3 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_632_remdups__adj_Ocases,axiom,
! [X: list_int] :
( ( X != nil_int )
=> ( ! [X4: int] :
( X
!= ( cons_int @ X4 @ nil_int ) )
=> ~ ! [X4: int,Y3: int,Xs2: list_int] :
( X
!= ( cons_int @ X4 @ ( cons_int @ Y3 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_633_remdups__adj_Ocases,axiom,
! [X: list_P419547167332244514st_int] :
( ( X != nil_Pr749310521696113548st_int )
=> ( ! [X4: produc6425607678544837394st_int] :
( X
!= ( cons_P4151473878978182108st_int @ X4 @ nil_Pr749310521696113548st_int ) )
=> ~ ! [X4: produc6425607678544837394st_int,Y3: produc6425607678544837394st_int,Xs2: list_P419547167332244514st_int] :
( X
!= ( cons_P4151473878978182108st_int @ X4 @ ( cons_P4151473878978182108st_int @ Y3 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_634_min__list_Ocases,axiom,
! [X: list_int] :
( ! [X4: int,Xs2: list_int] :
( X
!= ( cons_int @ X4 @ Xs2 ) )
=> ( X = nil_int ) ) ).
% min_list.cases
thf(fact_635_list_Oexhaust,axiom,
! [Y: list_instr] :
( ( Y != nil_instr )
=> ~ ! [X212: instr,X222: list_instr] :
( Y
!= ( cons_instr @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_636_list_Oexhaust,axiom,
! [Y: list_int] :
( ( Y != nil_int )
=> ~ ! [X212: int,X222: list_int] :
( Y
!= ( cons_int @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_637_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_638_list_OdiscI,axiom,
! [List: list_instr,X21: instr,X22: list_instr] :
( ( List
= ( cons_instr @ X21 @ X22 ) )
=> ( List != nil_instr ) ) ).
% list.discI
thf(fact_639_list_OdiscI,axiom,
! [List: list_int,X21: int,X22: list_int] :
( ( List
= ( cons_int @ X21 @ X22 ) )
=> ( List != nil_int ) ) ).
% list.discI
thf(fact_640_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_641_list_Odistinct_I1_J,axiom,
! [X21: instr,X22: list_instr] :
( nil_instr
!= ( cons_instr @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_642_list_Odistinct_I1_J,axiom,
! [X21: int,X22: list_int] :
( nil_int
!= ( cons_int @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_643_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_644_verit__comp__simplify1_I3_J,axiom,
! [B2: nat,A2: nat] :
( ( ~ ( ord_less_eq_nat @ B2 @ A2 ) )
= ( ord_less_nat @ A2 @ B2 ) ) ).
% verit_comp_simplify1(3)
thf(fact_645_verit__comp__simplify1_I3_J,axiom,
! [B2: int,A2: int] :
( ( ~ ( ord_less_eq_int @ B2 @ A2 ) )
= ( ord_less_int @ A2 @ B2 ) ) ).
% verit_comp_simplify1(3)
thf(fact_646_verit__comp__simplify1_I3_J,axiom,
! [B2: num,A2: num] :
( ( ~ ( ord_less_eq_num @ B2 @ A2 ) )
= ( ord_less_num @ A2 @ B2 ) ) ).
% verit_comp_simplify1(3)
thf(fact_647_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_648_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_649_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_650_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_651_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B4: nat] :
? [C3: nat] :
( B4
= ( plus_plus_nat @ A5 @ C3 ) ) ) ) ).
% le_iff_add
thf(fact_652_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_653_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_654_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C2: nat] :
( B
!= ( plus_plus_nat @ A @ C2 ) ) ) ).
% less_eqE
thf(fact_655_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_656_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_657_add__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_mono
thf(fact_658_add__mono,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).
% add_mono
thf(fact_659_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_660_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_661_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_662_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_663_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_664_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_665_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_666_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_667_diff__eq__diff__less__eq,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D2 ) )
=> ( ( ord_less_eq_int @ A @ B )
= ( ord_less_eq_int @ C @ D2 ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_668_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_669_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_670_diff__mono,axiom,
! [A: int,B: int,D2: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ D2 @ C )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D2 ) ) ) ) ).
% diff_mono
thf(fact_671_Cons__eq__appendI,axiom,
! [X: instr,Xs1: list_instr,Ys: list_instr,Xs: list_instr,Zs: list_instr] :
( ( ( cons_instr @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_instr @ Xs1 @ Zs ) )
=> ( ( cons_instr @ X @ Xs )
= ( append_instr @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_672_Cons__eq__appendI,axiom,
! [X: int,Xs1: list_int,Ys: list_int,Xs: list_int,Zs: list_int] :
( ( ( cons_int @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_int @ Xs1 @ Zs ) )
=> ( ( cons_int @ X @ Xs )
= ( append_int @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_673_Cons__eq__appendI,axiom,
! [X: produc6425607678544837394st_int,Xs1: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int,Xs: list_P419547167332244514st_int,Zs: list_P419547167332244514st_int] :
( ( ( cons_P4151473878978182108st_int @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append5028292981072010935st_int @ Xs1 @ Zs ) )
=> ( ( cons_P4151473878978182108st_int @ X @ Xs )
= ( append5028292981072010935st_int @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_674_append__Cons,axiom,
! [X: instr,Xs: list_instr,Ys: list_instr] :
( ( append_instr @ ( cons_instr @ X @ Xs ) @ Ys )
= ( cons_instr @ X @ ( append_instr @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_675_append__Cons,axiom,
! [X: int,Xs: list_int,Ys: list_int] :
( ( append_int @ ( cons_int @ X @ Xs ) @ Ys )
= ( cons_int @ X @ ( append_int @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_676_append__Cons,axiom,
! [X: produc6425607678544837394st_int,Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int] :
( ( append5028292981072010935st_int @ ( cons_P4151473878978182108st_int @ X @ Xs ) @ Ys )
= ( cons_P4151473878978182108st_int @ X @ ( append5028292981072010935st_int @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_677_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_678_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_679_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_680_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_681_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I3: nat,J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_682_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_683_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_684_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M5: nat,N3: nat] :
( ( ord_less_nat @ M5 @ N3 )
| ( M5 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_685_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_686_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M5: nat,N3: nat] :
( ( ord_less_eq_nat @ M5 @ N3 )
& ( M5 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_687_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M5: nat,N3: nat] :
? [K3: nat] :
( N3
= ( plus_plus_nat @ M5 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_688_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_689_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_690_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_691_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_692_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N2: nat] :
( L
= ( plus_plus_nat @ K @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_693_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_694_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_695_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_696_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_697_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_698_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_699_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_700_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_701_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_702_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_703_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_704_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_705_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_706_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_707_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_708_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_709_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_710_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_711_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_712_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_713_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_714_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_715_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_716_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_717_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_718_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_719_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_720_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_721_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_722_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_723_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_724_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_725_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_726_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_727_list__induct3,axiom,
! [Xs: list_int,Ys: list_int,Zs: list_int,P: list_int > list_int > list_int > $o] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( ( size_size_list_int @ Ys )
= ( size_size_list_int @ Zs ) )
=> ( ( P @ nil_int @ nil_int @ nil_int )
=> ( ! [X4: int,Xs2: list_int,Y3: int,Ys4: list_int,Z3: int,Zs2: list_int] :
( ( ( size_size_list_int @ Xs2 )
= ( size_size_list_int @ Ys4 ) )
=> ( ( ( size_size_list_int @ Ys4 )
= ( size_size_list_int @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs2 )
=> ( P @ ( cons_int @ X4 @ Xs2 ) @ ( cons_int @ Y3 @ Ys4 ) @ ( cons_int @ Z3 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_728_list__induct3,axiom,
! [Xs: list_int,Ys: list_int,Zs: list_instr,P: list_int > list_int > list_instr > $o] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( ( size_size_list_int @ Ys )
= ( size_size_list_instr @ Zs ) )
=> ( ( P @ nil_int @ nil_int @ nil_instr )
=> ( ! [X4: int,Xs2: list_int,Y3: int,Ys4: list_int,Z3: instr,Zs2: list_instr] :
( ( ( size_size_list_int @ Xs2 )
= ( size_size_list_int @ Ys4 ) )
=> ( ( ( size_size_list_int @ Ys4 )
= ( size_size_list_instr @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs2 )
=> ( P @ ( cons_int @ X4 @ Xs2 ) @ ( cons_int @ Y3 @ Ys4 ) @ ( cons_instr @ Z3 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_729_list__induct3,axiom,
! [Xs: list_int,Ys: list_instr,Zs: list_int,P: list_int > list_instr > list_int > $o] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_instr @ Ys ) )
=> ( ( ( size_size_list_instr @ Ys )
= ( size_size_list_int @ Zs ) )
=> ( ( P @ nil_int @ nil_instr @ nil_int )
=> ( ! [X4: int,Xs2: list_int,Y3: instr,Ys4: list_instr,Z3: int,Zs2: list_int] :
( ( ( size_size_list_int @ Xs2 )
= ( size_size_list_instr @ Ys4 ) )
=> ( ( ( size_size_list_instr @ Ys4 )
= ( size_size_list_int @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs2 )
=> ( P @ ( cons_int @ X4 @ Xs2 ) @ ( cons_instr @ Y3 @ Ys4 ) @ ( cons_int @ Z3 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_730_list__induct3,axiom,
! [Xs: list_int,Ys: list_instr,Zs: list_instr,P: list_int > list_instr > list_instr > $o] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_instr @ Ys ) )
=> ( ( ( size_size_list_instr @ Ys )
= ( size_size_list_instr @ Zs ) )
=> ( ( P @ nil_int @ nil_instr @ nil_instr )
=> ( ! [X4: int,Xs2: list_int,Y3: instr,Ys4: list_instr,Z3: instr,Zs2: list_instr] :
( ( ( size_size_list_int @ Xs2 )
= ( size_size_list_instr @ Ys4 ) )
=> ( ( ( size_size_list_instr @ Ys4 )
= ( size_size_list_instr @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs2 )
=> ( P @ ( cons_int @ X4 @ Xs2 ) @ ( cons_instr @ Y3 @ Ys4 ) @ ( cons_instr @ Z3 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_731_list__induct3,axiom,
! [Xs: list_instr,Ys: list_int,Zs: list_int,P: list_instr > list_int > list_int > $o] :
( ( ( size_size_list_instr @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( ( size_size_list_int @ Ys )
= ( size_size_list_int @ Zs ) )
=> ( ( P @ nil_instr @ nil_int @ nil_int )
=> ( ! [X4: instr,Xs2: list_instr,Y3: int,Ys4: list_int,Z3: int,Zs2: list_int] :
( ( ( size_size_list_instr @ Xs2 )
= ( size_size_list_int @ Ys4 ) )
=> ( ( ( size_size_list_int @ Ys4 )
= ( size_size_list_int @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs2 )
=> ( P @ ( cons_instr @ X4 @ Xs2 ) @ ( cons_int @ Y3 @ Ys4 ) @ ( cons_int @ Z3 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_732_list__induct3,axiom,
! [Xs: list_instr,Ys: list_int,Zs: list_instr,P: list_instr > list_int > list_instr > $o] :
( ( ( size_size_list_instr @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( ( size_size_list_int @ Ys )
= ( size_size_list_instr @ Zs ) )
=> ( ( P @ nil_instr @ nil_int @ nil_instr )
=> ( ! [X4: instr,Xs2: list_instr,Y3: int,Ys4: list_int,Z3: instr,Zs2: list_instr] :
( ( ( size_size_list_instr @ Xs2 )
= ( size_size_list_int @ Ys4 ) )
=> ( ( ( size_size_list_int @ Ys4 )
= ( size_size_list_instr @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs2 )
=> ( P @ ( cons_instr @ X4 @ Xs2 ) @ ( cons_int @ Y3 @ Ys4 ) @ ( cons_instr @ Z3 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_733_list__induct3,axiom,
! [Xs: list_instr,Ys: list_instr,Zs: list_int,P: list_instr > list_instr > list_int > $o] :
( ( ( size_size_list_instr @ Xs )
= ( size_size_list_instr @ Ys ) )
=> ( ( ( size_size_list_instr @ Ys )
= ( size_size_list_int @ Zs ) )
=> ( ( P @ nil_instr @ nil_instr @ nil_int )
=> ( ! [X4: instr,Xs2: list_instr,Y3: instr,Ys4: list_instr,Z3: int,Zs2: list_int] :
( ( ( size_size_list_instr @ Xs2 )
= ( size_size_list_instr @ Ys4 ) )
=> ( ( ( size_size_list_instr @ Ys4 )
= ( size_size_list_int @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs2 )
=> ( P @ ( cons_instr @ X4 @ Xs2 ) @ ( cons_instr @ Y3 @ Ys4 ) @ ( cons_int @ Z3 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_734_list__induct3,axiom,
! [Xs: list_instr,Ys: list_instr,Zs: list_instr,P: list_instr > list_instr > list_instr > $o] :
( ( ( size_size_list_instr @ Xs )
= ( size_size_list_instr @ Ys ) )
=> ( ( ( size_size_list_instr @ Ys )
= ( size_size_list_instr @ Zs ) )
=> ( ( P @ nil_instr @ nil_instr @ nil_instr )
=> ( ! [X4: instr,Xs2: list_instr,Y3: instr,Ys4: list_instr,Z3: instr,Zs2: list_instr] :
( ( ( size_size_list_instr @ Xs2 )
= ( size_size_list_instr @ Ys4 ) )
=> ( ( ( size_size_list_instr @ Ys4 )
= ( size_size_list_instr @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs2 )
=> ( P @ ( cons_instr @ X4 @ Xs2 ) @ ( cons_instr @ Y3 @ Ys4 ) @ ( cons_instr @ Z3 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_735_list__induct3,axiom,
! [Xs: list_int,Ys: list_int,Zs: list_P419547167332244514st_int,P: list_int > list_int > list_P419547167332244514st_int > $o] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( ( size_size_list_int @ Ys )
= ( size_s229347708945616566st_int @ Zs ) )
=> ( ( P @ nil_int @ nil_int @ nil_Pr749310521696113548st_int )
=> ( ! [X4: int,Xs2: list_int,Y3: int,Ys4: list_int,Z3: produc6425607678544837394st_int,Zs2: list_P419547167332244514st_int] :
( ( ( size_size_list_int @ Xs2 )
= ( size_size_list_int @ Ys4 ) )
=> ( ( ( size_size_list_int @ Ys4 )
= ( size_s229347708945616566st_int @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs2 )
=> ( P @ ( cons_int @ X4 @ Xs2 ) @ ( cons_int @ Y3 @ Ys4 ) @ ( cons_P4151473878978182108st_int @ Z3 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_736_list__induct3,axiom,
! [Xs: list_int,Ys: list_P419547167332244514st_int,Zs: list_int,P: list_int > list_P419547167332244514st_int > list_int > $o] :
( ( ( size_size_list_int @ Xs )
= ( size_s229347708945616566st_int @ Ys ) )
=> ( ( ( size_s229347708945616566st_int @ Ys )
= ( size_size_list_int @ Zs ) )
=> ( ( P @ nil_int @ nil_Pr749310521696113548st_int @ nil_int )
=> ( ! [X4: int,Xs2: list_int,Y3: produc6425607678544837394st_int,Ys4: list_P419547167332244514st_int,Z3: int,Zs2: list_int] :
( ( ( size_size_list_int @ Xs2 )
= ( size_s229347708945616566st_int @ Ys4 ) )
=> ( ( ( size_s229347708945616566st_int @ Ys4 )
= ( size_size_list_int @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs2 )
=> ( P @ ( cons_int @ X4 @ Xs2 ) @ ( cons_P4151473878978182108st_int @ Y3 @ Ys4 ) @ ( cons_int @ Z3 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_737_list__induct2,axiom,
! [Xs: list_int,Ys: list_int,P: list_int > list_int > $o] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( P @ nil_int @ nil_int )
=> ( ! [X4: int,Xs2: list_int,Y3: int,Ys4: list_int] :
( ( ( size_size_list_int @ Xs2 )
= ( size_size_list_int @ Ys4 ) )
=> ( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_int @ X4 @ Xs2 ) @ ( cons_int @ Y3 @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_738_list__induct2,axiom,
! [Xs: list_int,Ys: list_P419547167332244514st_int,P: list_int > list_P419547167332244514st_int > $o] :
( ( ( size_size_list_int @ Xs )
= ( size_s229347708945616566st_int @ Ys ) )
=> ( ( P @ nil_int @ nil_Pr749310521696113548st_int )
=> ( ! [X4: int,Xs2: list_int,Y3: produc6425607678544837394st_int,Ys4: list_P419547167332244514st_int] :
( ( ( size_size_list_int @ Xs2 )
= ( size_s229347708945616566st_int @ Ys4 ) )
=> ( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_int @ X4 @ Xs2 ) @ ( cons_P4151473878978182108st_int @ Y3 @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_739_list__induct2,axiom,
! [Xs: list_int,Ys: list_instr,P: list_int > list_instr > $o] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_instr @ Ys ) )
=> ( ( P @ nil_int @ nil_instr )
=> ( ! [X4: int,Xs2: list_int,Y3: instr,Ys4: list_instr] :
( ( ( size_size_list_int @ Xs2 )
= ( size_size_list_instr @ Ys4 ) )
=> ( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_int @ X4 @ Xs2 ) @ ( cons_instr @ Y3 @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_740_list__induct2,axiom,
! [Xs: list_P419547167332244514st_int,Ys: list_int,P: list_P419547167332244514st_int > list_int > $o] :
( ( ( size_s229347708945616566st_int @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( P @ nil_Pr749310521696113548st_int @ nil_int )
=> ( ! [X4: produc6425607678544837394st_int,Xs2: list_P419547167332244514st_int,Y3: int,Ys4: list_int] :
( ( ( size_s229347708945616566st_int @ Xs2 )
= ( size_size_list_int @ Ys4 ) )
=> ( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_P4151473878978182108st_int @ X4 @ Xs2 ) @ ( cons_int @ Y3 @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_741_list__induct2,axiom,
! [Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int,P: list_P419547167332244514st_int > list_P419547167332244514st_int > $o] :
( ( ( size_s229347708945616566st_int @ Xs )
= ( size_s229347708945616566st_int @ Ys ) )
=> ( ( P @ nil_Pr749310521696113548st_int @ nil_Pr749310521696113548st_int )
=> ( ! [X4: produc6425607678544837394st_int,Xs2: list_P419547167332244514st_int,Y3: produc6425607678544837394st_int,Ys4: list_P419547167332244514st_int] :
( ( ( size_s229347708945616566st_int @ Xs2 )
= ( size_s229347708945616566st_int @ Ys4 ) )
=> ( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_P4151473878978182108st_int @ X4 @ Xs2 ) @ ( cons_P4151473878978182108st_int @ Y3 @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_742_list__induct2,axiom,
! [Xs: list_P419547167332244514st_int,Ys: list_instr,P: list_P419547167332244514st_int > list_instr > $o] :
( ( ( size_s229347708945616566st_int @ Xs )
= ( size_size_list_instr @ Ys ) )
=> ( ( P @ nil_Pr749310521696113548st_int @ nil_instr )
=> ( ! [X4: produc6425607678544837394st_int,Xs2: list_P419547167332244514st_int,Y3: instr,Ys4: list_instr] :
( ( ( size_s229347708945616566st_int @ Xs2 )
= ( size_size_list_instr @ Ys4 ) )
=> ( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_P4151473878978182108st_int @ X4 @ Xs2 ) @ ( cons_instr @ Y3 @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_743_list__induct2,axiom,
! [Xs: list_instr,Ys: list_int,P: list_instr > list_int > $o] :
( ( ( size_size_list_instr @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( P @ nil_instr @ nil_int )
=> ( ! [X4: instr,Xs2: list_instr,Y3: int,Ys4: list_int] :
( ( ( size_size_list_instr @ Xs2 )
= ( size_size_list_int @ Ys4 ) )
=> ( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_instr @ X4 @ Xs2 ) @ ( cons_int @ Y3 @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_744_list__induct2,axiom,
! [Xs: list_instr,Ys: list_P419547167332244514st_int,P: list_instr > list_P419547167332244514st_int > $o] :
( ( ( size_size_list_instr @ Xs )
= ( size_s229347708945616566st_int @ Ys ) )
=> ( ( P @ nil_instr @ nil_Pr749310521696113548st_int )
=> ( ! [X4: instr,Xs2: list_instr,Y3: produc6425607678544837394st_int,Ys4: list_P419547167332244514st_int] :
( ( ( size_size_list_instr @ Xs2 )
= ( size_s229347708945616566st_int @ Ys4 ) )
=> ( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_instr @ X4 @ Xs2 ) @ ( cons_P4151473878978182108st_int @ Y3 @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_745_list__induct2,axiom,
! [Xs: list_instr,Ys: list_instr,P: list_instr > list_instr > $o] :
( ( ( size_size_list_instr @ Xs )
= ( size_size_list_instr @ Ys ) )
=> ( ( P @ nil_instr @ nil_instr )
=> ( ! [X4: instr,Xs2: list_instr,Y3: instr,Ys4: list_instr] :
( ( ( size_size_list_instr @ Xs2 )
= ( size_size_list_instr @ Ys4 ) )
=> ( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons_instr @ X4 @ Xs2 ) @ ( cons_instr @ Y3 @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_746_add__less__le__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_less_le_mono
thf(fact_747_add__less__le__mono,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D2 )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).
% add_less_le_mono
thf(fact_748_add__le__less__mono,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_le_less_mono
thf(fact_749_add__le__less__mono,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ C @ D2 )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).
% add_le_less_mono
thf(fact_750_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_751_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_752_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_753_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_754_le__iff__diff__le__0,axiom,
( ord_less_eq_int
= ( ^ [A5: int,B4: int] : ( ord_less_eq_int @ ( minus_minus_int @ A5 @ B4 ) @ zero_zero_int ) ) ) ).
% le_iff_diff_le_0
thf(fact_755_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_756_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_757_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_758_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_759_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_760_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_761_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_762_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_763_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_764_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_765_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_766_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_767_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_768_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_769_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_770_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_771_rev__nonempty__induct,axiom,
! [Xs: list_instr,P: list_instr > $o] :
( ( Xs != nil_instr )
=> ( ! [X4: instr] : ( P @ ( cons_instr @ X4 @ nil_instr ) )
=> ( ! [X4: instr,Xs2: list_instr] :
( ( Xs2 != nil_instr )
=> ( ( P @ Xs2 )
=> ( P @ ( append_instr @ Xs2 @ ( cons_instr @ X4 @ nil_instr ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_772_rev__nonempty__induct,axiom,
! [Xs: list_int,P: list_int > $o] :
( ( Xs != nil_int )
=> ( ! [X4: int] : ( P @ ( cons_int @ X4 @ nil_int ) )
=> ( ! [X4: int,Xs2: list_int] :
( ( Xs2 != nil_int )
=> ( ( P @ Xs2 )
=> ( P @ ( append_int @ Xs2 @ ( cons_int @ X4 @ nil_int ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_773_rev__nonempty__induct,axiom,
! [Xs: list_P419547167332244514st_int,P: list_P419547167332244514st_int > $o] :
( ( Xs != nil_Pr749310521696113548st_int )
=> ( ! [X4: produc6425607678544837394st_int] : ( P @ ( cons_P4151473878978182108st_int @ X4 @ nil_Pr749310521696113548st_int ) )
=> ( ! [X4: produc6425607678544837394st_int,Xs2: list_P419547167332244514st_int] :
( ( Xs2 != nil_Pr749310521696113548st_int )
=> ( ( P @ Xs2 )
=> ( P @ ( append5028292981072010935st_int @ Xs2 @ ( cons_P4151473878978182108st_int @ X4 @ nil_Pr749310521696113548st_int ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_774_append__eq__Cons__conv,axiom,
! [Ys: list_instr,Zs: list_instr,X: instr,Xs: list_instr] :
( ( ( append_instr @ Ys @ Zs )
= ( cons_instr @ X @ Xs ) )
= ( ( ( Ys = nil_instr )
& ( Zs
= ( cons_instr @ X @ Xs ) ) )
| ? [Ys5: list_instr] :
( ( Ys
= ( cons_instr @ X @ Ys5 ) )
& ( ( append_instr @ Ys5 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_775_append__eq__Cons__conv,axiom,
! [Ys: list_int,Zs: list_int,X: int,Xs: list_int] :
( ( ( append_int @ Ys @ Zs )
= ( cons_int @ X @ Xs ) )
= ( ( ( Ys = nil_int )
& ( Zs
= ( cons_int @ X @ Xs ) ) )
| ? [Ys5: list_int] :
( ( Ys
= ( cons_int @ X @ Ys5 ) )
& ( ( append_int @ Ys5 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_776_append__eq__Cons__conv,axiom,
! [Ys: list_P419547167332244514st_int,Zs: list_P419547167332244514st_int,X: produc6425607678544837394st_int,Xs: list_P419547167332244514st_int] :
( ( ( append5028292981072010935st_int @ Ys @ Zs )
= ( cons_P4151473878978182108st_int @ X @ Xs ) )
= ( ( ( Ys = nil_Pr749310521696113548st_int )
& ( Zs
= ( cons_P4151473878978182108st_int @ X @ Xs ) ) )
| ? [Ys5: list_P419547167332244514st_int] :
( ( Ys
= ( cons_P4151473878978182108st_int @ X @ Ys5 ) )
& ( ( append5028292981072010935st_int @ Ys5 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_777_Cons__eq__append__conv,axiom,
! [X: instr,Xs: list_instr,Ys: list_instr,Zs: list_instr] :
( ( ( cons_instr @ X @ Xs )
= ( append_instr @ Ys @ Zs ) )
= ( ( ( Ys = nil_instr )
& ( ( cons_instr @ X @ Xs )
= Zs ) )
| ? [Ys5: list_instr] :
( ( ( cons_instr @ X @ Ys5 )
= Ys )
& ( Xs
= ( append_instr @ Ys5 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_778_Cons__eq__append__conv,axiom,
! [X: int,Xs: list_int,Ys: list_int,Zs: list_int] :
( ( ( cons_int @ X @ Xs )
= ( append_int @ Ys @ Zs ) )
= ( ( ( Ys = nil_int )
& ( ( cons_int @ X @ Xs )
= Zs ) )
| ? [Ys5: list_int] :
( ( ( cons_int @ X @ Ys5 )
= Ys )
& ( Xs
= ( append_int @ Ys5 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_779_Cons__eq__append__conv,axiom,
! [X: produc6425607678544837394st_int,Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int,Zs: list_P419547167332244514st_int] :
( ( ( cons_P4151473878978182108st_int @ X @ Xs )
= ( append5028292981072010935st_int @ Ys @ Zs ) )
= ( ( ( Ys = nil_Pr749310521696113548st_int )
& ( ( cons_P4151473878978182108st_int @ X @ Xs )
= Zs ) )
| ? [Ys5: list_P419547167332244514st_int] :
( ( ( cons_P4151473878978182108st_int @ X @ Ys5 )
= Ys )
& ( Xs
= ( append5028292981072010935st_int @ Ys5 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_780_rev__exhaust,axiom,
! [Xs: list_instr] :
( ( Xs != nil_instr )
=> ~ ! [Ys4: list_instr,Y3: instr] :
( Xs
!= ( append_instr @ Ys4 @ ( cons_instr @ Y3 @ nil_instr ) ) ) ) ).
% rev_exhaust
thf(fact_781_rev__exhaust,axiom,
! [Xs: list_int] :
( ( Xs != nil_int )
=> ~ ! [Ys4: list_int,Y3: int] :
( Xs
!= ( append_int @ Ys4 @ ( cons_int @ Y3 @ nil_int ) ) ) ) ).
% rev_exhaust
thf(fact_782_rev__exhaust,axiom,
! [Xs: list_P419547167332244514st_int] :
( ( Xs != nil_Pr749310521696113548st_int )
=> ~ ! [Ys4: list_P419547167332244514st_int,Y3: produc6425607678544837394st_int] :
( Xs
!= ( append5028292981072010935st_int @ Ys4 @ ( cons_P4151473878978182108st_int @ Y3 @ nil_Pr749310521696113548st_int ) ) ) ) ).
% rev_exhaust
thf(fact_783_rev__induct,axiom,
! [P: list_instr > $o,Xs: list_instr] :
( ( P @ nil_instr )
=> ( ! [X4: instr,Xs2: list_instr] :
( ( P @ Xs2 )
=> ( P @ ( append_instr @ Xs2 @ ( cons_instr @ X4 @ nil_instr ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_784_rev__induct,axiom,
! [P: list_int > $o,Xs: list_int] :
( ( P @ nil_int )
=> ( ! [X4: int,Xs2: list_int] :
( ( P @ Xs2 )
=> ( P @ ( append_int @ Xs2 @ ( cons_int @ X4 @ nil_int ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_785_rev__induct,axiom,
! [P: list_P419547167332244514st_int > $o,Xs: list_P419547167332244514st_int] :
( ( P @ nil_Pr749310521696113548st_int )
=> ( ! [X4: produc6425607678544837394st_int,Xs2: list_P419547167332244514st_int] :
( ( P @ Xs2 )
=> ( P @ ( append5028292981072010935st_int @ Xs2 @ ( cons_P4151473878978182108st_int @ X4 @ nil_Pr749310521696113548st_int ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_786_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_787_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_788_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_789_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ( ord_less_nat @ ( F @ M3 ) @ ( F @ N2 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_790_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_791_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_792_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_793_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_794_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_795_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_796_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_797_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_798_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_799_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_800_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_801_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_802_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_803_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_804_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_805_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_806_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_807_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_808_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_809_same__length__different,axiom,
! [Xs: list_int,Ys: list_int] :
( ( Xs != Ys )
=> ( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys ) )
=> ? [Pre: list_int,X4: int,Xs4: list_int,Y3: int,Ys6: list_int] :
( ( X4 != Y3 )
& ( Xs
= ( append_int @ Pre @ ( append_int @ ( cons_int @ X4 @ nil_int ) @ Xs4 ) ) )
& ( Ys
= ( append_int @ Pre @ ( append_int @ ( cons_int @ Y3 @ nil_int ) @ Ys6 ) ) ) ) ) ) ).
% same_length_different
thf(fact_810_same__length__different,axiom,
! [Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int] :
( ( Xs != Ys )
=> ( ( ( size_s229347708945616566st_int @ Xs )
= ( size_s229347708945616566st_int @ Ys ) )
=> ? [Pre: list_P419547167332244514st_int,X4: produc6425607678544837394st_int,Xs4: list_P419547167332244514st_int,Y3: produc6425607678544837394st_int,Ys6: list_P419547167332244514st_int] :
( ( X4 != Y3 )
& ( Xs
= ( append5028292981072010935st_int @ Pre @ ( append5028292981072010935st_int @ ( cons_P4151473878978182108st_int @ X4 @ nil_Pr749310521696113548st_int ) @ Xs4 ) ) )
& ( Ys
= ( append5028292981072010935st_int @ Pre @ ( append5028292981072010935st_int @ ( cons_P4151473878978182108st_int @ Y3 @ nil_Pr749310521696113548st_int ) @ Ys6 ) ) ) ) ) ) ).
% same_length_different
thf(fact_811_same__length__different,axiom,
! [Xs: list_instr,Ys: list_instr] :
( ( Xs != Ys )
=> ( ( ( size_size_list_instr @ Xs )
= ( size_size_list_instr @ Ys ) )
=> ? [Pre: list_instr,X4: instr,Xs4: list_instr,Y3: instr,Ys6: list_instr] :
( ( X4 != Y3 )
& ( Xs
= ( append_instr @ Pre @ ( append_instr @ ( cons_instr @ X4 @ nil_instr ) @ Xs4 ) ) )
& ( Ys
= ( append_instr @ Pre @ ( append_instr @ ( cons_instr @ Y3 @ nil_instr ) @ Ys6 ) ) ) ) ) ) ).
% same_length_different
thf(fact_812_of__nat__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% of_nat_diff
thf(fact_813_of__nat__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).
% of_nat_diff
thf(fact_814_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_815_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_816_nth__Cons_H,axiom,
! [N: nat,X: instr,Xs: list_instr] :
( ( ( N = zero_zero_nat )
=> ( ( nth_instr @ ( cons_instr @ X @ Xs ) @ N )
= X ) )
& ( ( N != zero_zero_nat )
=> ( ( nth_instr @ ( cons_instr @ X @ Xs ) @ N )
= ( nth_instr @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_817_nth__Cons_H,axiom,
! [N: nat,X: int,Xs: list_int] :
( ( ( N = zero_zero_nat )
=> ( ( nth_int @ ( cons_int @ X @ Xs ) @ N )
= X ) )
& ( ( N != zero_zero_nat )
=> ( ( nth_int @ ( cons_int @ X @ Xs ) @ N )
= ( nth_int @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_818_nth__Cons_H,axiom,
! [N: nat,X: produc6425607678544837394st_int,Xs: list_P419547167332244514st_int] :
( ( ( N = zero_zero_nat )
=> ( ( nth_Pr8327173284404383691st_int @ ( cons_P4151473878978182108st_int @ X @ Xs ) @ N )
= X ) )
& ( ( N != zero_zero_nat )
=> ( ( nth_Pr8327173284404383691st_int @ ( cons_P4151473878978182108st_int @ X @ Xs ) @ N )
= ( nth_Pr8327173284404383691st_int @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_819_nth__non__equal__first__eq,axiom,
! [X: instr,Y: instr,Xs: list_instr,N: nat] :
( ( X != Y )
=> ( ( ( nth_instr @ ( cons_instr @ X @ Xs ) @ N )
= Y )
= ( ( ( nth_instr @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
= Y )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_820_nth__non__equal__first__eq,axiom,
! [X: int,Y: int,Xs: list_int,N: nat] :
( ( X != Y )
=> ( ( ( nth_int @ ( cons_int @ X @ Xs ) @ N )
= Y )
= ( ( ( nth_int @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
= Y )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_821_nth__non__equal__first__eq,axiom,
! [X: produc6425607678544837394st_int,Y: produc6425607678544837394st_int,Xs: list_P419547167332244514st_int,N: nat] :
( ( X != Y )
=> ( ( ( nth_Pr8327173284404383691st_int @ ( cons_P4151473878978182108st_int @ X @ Xs ) @ N )
= Y )
= ( ( ( nth_Pr8327173284404383691st_int @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
= Y )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_822_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_823_instr_Oinject_I4_J,axiom,
! [X52: int,Y52: int] :
( ( ( jmp @ X52 )
= ( jmp @ Y52 ) )
= ( X52 = Y52 ) ) ).
% instr.inject(4)
thf(fact_824_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_825_upto__aux__rec,axiom,
( upto_aux
= ( ^ [I2: int,J3: int,Js: list_int] : ( if_list_int @ ( ord_less_int @ J3 @ I2 ) @ Js @ ( upto_aux @ I2 @ ( minus_minus_int @ J3 @ one_one_int ) @ ( cons_int @ J3 @ Js ) ) ) ) ) ).
% upto_aux_rec
thf(fact_826_drop0,axiom,
( ( drop_P7838799151754935884st_int @ zero_zero_nat )
= ( ^ [X3: list_P419547167332244514st_int] : X3 ) ) ).
% drop0
thf(fact_827_drop__drop,axiom,
! [N: nat,M: nat,Xs: list_P419547167332244514st_int] :
( ( drop_P7838799151754935884st_int @ N @ ( drop_P7838799151754935884st_int @ M @ Xs ) )
= ( drop_P7838799151754935884st_int @ ( plus_plus_nat @ N @ M ) @ Xs ) ) ).
% drop_drop
thf(fact_828_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_829_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_830_length__drop,axiom,
! [N: nat,Xs: list_int] :
( ( size_size_list_int @ ( drop_int @ N @ Xs ) )
= ( minus_minus_nat @ ( size_size_list_int @ Xs ) @ N ) ) ).
% length_drop
thf(fact_831_length__drop,axiom,
! [N: nat,Xs: list_P419547167332244514st_int] :
( ( size_s229347708945616566st_int @ ( drop_P7838799151754935884st_int @ N @ Xs ) )
= ( minus_minus_nat @ ( size_s229347708945616566st_int @ Xs ) @ N ) ) ).
% length_drop
thf(fact_832_length__drop,axiom,
! [N: nat,Xs: list_instr] :
( ( size_size_list_instr @ ( drop_instr @ N @ Xs ) )
= ( minus_minus_nat @ ( size_size_list_instr @ Xs ) @ N ) ) ).
% length_drop
thf(fact_833_drop__all,axiom,
! [Xs: list_int,N: nat] :
( ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ N )
=> ( ( drop_int @ N @ Xs )
= nil_int ) ) ).
% drop_all
thf(fact_834_drop__all,axiom,
! [Xs: list_P419547167332244514st_int,N: nat] :
( ( ord_less_eq_nat @ ( size_s229347708945616566st_int @ Xs ) @ N )
=> ( ( drop_P7838799151754935884st_int @ N @ Xs )
= nil_Pr749310521696113548st_int ) ) ).
% drop_all
thf(fact_835_drop__all,axiom,
! [Xs: list_instr,N: nat] :
( ( ord_less_eq_nat @ ( size_size_list_instr @ Xs ) @ N )
=> ( ( drop_instr @ N @ Xs )
= nil_instr ) ) ).
% drop_all
thf(fact_836_drop__eq__Nil,axiom,
! [N: nat,Xs: list_int] :
( ( ( drop_int @ N @ Xs )
= nil_int )
= ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ N ) ) ).
% drop_eq_Nil
thf(fact_837_drop__eq__Nil,axiom,
! [N: nat,Xs: list_P419547167332244514st_int] :
( ( ( drop_P7838799151754935884st_int @ N @ Xs )
= nil_Pr749310521696113548st_int )
= ( ord_less_eq_nat @ ( size_s229347708945616566st_int @ Xs ) @ N ) ) ).
% drop_eq_Nil
thf(fact_838_drop__eq__Nil,axiom,
! [N: nat,Xs: list_instr] :
( ( ( drop_instr @ N @ Xs )
= nil_instr )
= ( ord_less_eq_nat @ ( size_size_list_instr @ Xs ) @ N ) ) ).
% drop_eq_Nil
thf(fact_839_drop__eq__Nil2,axiom,
! [N: nat,Xs: list_int] :
( ( nil_int
= ( drop_int @ N @ Xs ) )
= ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ N ) ) ).
% drop_eq_Nil2
thf(fact_840_drop__eq__Nil2,axiom,
! [N: nat,Xs: list_P419547167332244514st_int] :
( ( nil_Pr749310521696113548st_int
= ( drop_P7838799151754935884st_int @ N @ Xs ) )
= ( ord_less_eq_nat @ ( size_s229347708945616566st_int @ Xs ) @ N ) ) ).
% drop_eq_Nil2
thf(fact_841_drop__eq__Nil2,axiom,
! [N: nat,Xs: list_instr] :
( ( nil_instr
= ( drop_instr @ N @ Xs ) )
= ( ord_less_eq_nat @ ( size_size_list_instr @ Xs ) @ N ) ) ).
% drop_eq_Nil2
thf(fact_842_drop__append,axiom,
! [N: nat,Xs: list_int,Ys: list_int] :
( ( drop_int @ N @ ( append_int @ Xs @ Ys ) )
= ( append_int @ ( drop_int @ N @ Xs ) @ ( drop_int @ ( minus_minus_nat @ N @ ( size_size_list_int @ Xs ) ) @ Ys ) ) ) ).
% drop_append
thf(fact_843_drop__append,axiom,
! [N: nat,Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int] :
( ( drop_P7838799151754935884st_int @ N @ ( append5028292981072010935st_int @ Xs @ Ys ) )
= ( append5028292981072010935st_int @ ( drop_P7838799151754935884st_int @ N @ Xs ) @ ( drop_P7838799151754935884st_int @ ( minus_minus_nat @ N @ ( size_s229347708945616566st_int @ Xs ) ) @ Ys ) ) ) ).
% drop_append
thf(fact_844_drop__append,axiom,
! [N: nat,Xs: list_instr,Ys: list_instr] :
( ( drop_instr @ N @ ( append_instr @ Xs @ Ys ) )
= ( append_instr @ ( drop_instr @ N @ Xs ) @ ( drop_instr @ ( minus_minus_nat @ N @ ( size_size_list_instr @ Xs ) ) @ Ys ) ) ) ).
% drop_append
thf(fact_845_nth__drop,axiom,
! [N: nat,Xs: list_int,I: nat] :
( ( ord_less_eq_nat @ N @ ( size_size_list_int @ Xs ) )
=> ( ( nth_int @ ( drop_int @ N @ Xs ) @ I )
= ( nth_int @ Xs @ ( plus_plus_nat @ N @ I ) ) ) ) ).
% nth_drop
thf(fact_846_nth__drop,axiom,
! [N: nat,Xs: list_P419547167332244514st_int,I: nat] :
( ( ord_less_eq_nat @ N @ ( size_s229347708945616566st_int @ Xs ) )
=> ( ( nth_Pr8327173284404383691st_int @ ( drop_P7838799151754935884st_int @ N @ Xs ) @ I )
= ( nth_Pr8327173284404383691st_int @ Xs @ ( plus_plus_nat @ N @ I ) ) ) ) ).
% nth_drop
thf(fact_847_nth__drop,axiom,
! [N: nat,Xs: list_instr,I: nat] :
( ( ord_less_eq_nat @ N @ ( size_size_list_instr @ Xs ) )
=> ( ( nth_instr @ ( drop_instr @ N @ Xs ) @ I )
= ( nth_instr @ Xs @ ( plus_plus_nat @ N @ I ) ) ) ) ).
% nth_drop
thf(fact_848_imp__le__cong,axiom,
! [X: int,X7: int,P: $o,P4: $o] :
( ( X = X7 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> ( P = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> P )
= ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> P4 ) ) ) ) ).
% imp_le_cong
thf(fact_849_conj__le__cong,axiom,
! [X: int,X7: int,P: $o,P4: $o] :
( ( X = X7 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> ( P = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
& P )
= ( ( ord_less_eq_int @ zero_zero_int @ X7 )
& P4 ) ) ) ) ).
% conj_le_cong
thf(fact_850_drop__0,axiom,
! [Xs: list_P419547167332244514st_int] :
( ( drop_P7838799151754935884st_int @ zero_zero_nat @ Xs )
= Xs ) ).
% drop_0
thf(fact_851_drop__Nil,axiom,
! [N: nat] :
( ( drop_instr @ N @ nil_instr )
= nil_instr ) ).
% drop_Nil
thf(fact_852_drop__Nil,axiom,
! [N: nat] :
( ( drop_P7838799151754935884st_int @ N @ nil_Pr749310521696113548st_int )
= nil_Pr749310521696113548st_int ) ).
% drop_Nil
thf(fact_853_drop__Nil,axiom,
! [N: nat] :
( ( drop_int @ N @ nil_int )
= nil_int ) ).
% drop_Nil
thf(fact_854_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_855_nth__via__drop,axiom,
! [N: nat,Xs: list_instr,Y: instr,Ys: list_instr] :
( ( ( drop_instr @ N @ Xs )
= ( cons_instr @ Y @ Ys ) )
=> ( ( nth_instr @ Xs @ N )
= Y ) ) ).
% nth_via_drop
thf(fact_856_nth__via__drop,axiom,
! [N: nat,Xs: list_int,Y: int,Ys: list_int] :
( ( ( drop_int @ N @ Xs )
= ( cons_int @ Y @ Ys ) )
=> ( ( nth_int @ Xs @ N )
= Y ) ) ).
% nth_via_drop
thf(fact_857_nth__via__drop,axiom,
! [N: nat,Xs: list_P419547167332244514st_int,Y: produc6425607678544837394st_int,Ys: list_P419547167332244514st_int] :
( ( ( drop_P7838799151754935884st_int @ N @ Xs )
= ( cons_P4151473878978182108st_int @ Y @ Ys ) )
=> ( ( nth_Pr8327173284404383691st_int @ Xs @ N )
= Y ) ) ).
% nth_via_drop
thf(fact_858_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B4: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_859_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_860_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ? [N2: nat] :
( K
= ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_861_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ~ ! [N2: nat] :
( K
!= ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% nonneg_int_cases
thf(fact_862_int__ge__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_eq_int @ K @ I )
=> ( ( P @ K )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ K @ I3 )
=> ( ( P @ I3 )
=> ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_ge_induct
thf(fact_863_zle__iff__zadd,axiom,
( ord_less_eq_int
= ( ^ [W2: int,Z4: int] :
? [N3: nat] :
( Z4
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_864_int__le__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_eq_int @ I @ K )
=> ( ( P @ K )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ I3 @ K )
=> ( ( P @ I3 )
=> ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_le_induct
thf(fact_865_minf_I7_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ~ ( ord_less_nat @ T @ X6 ) ) ).
% minf(7)
thf(fact_866_minf_I7_J,axiom,
! [T: int] :
? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z3 )
=> ~ ( ord_less_int @ T @ X6 ) ) ).
% minf(7)
thf(fact_867_minf_I7_J,axiom,
! [T: num] :
? [Z3: num] :
! [X6: num] :
( ( ord_less_num @ X6 @ Z3 )
=> ~ ( ord_less_num @ T @ X6 ) ) ).
% minf(7)
thf(fact_868_minf_I5_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ( ord_less_nat @ X6 @ T ) ) ).
% minf(5)
thf(fact_869_minf_I5_J,axiom,
! [T: int] :
? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z3 )
=> ( ord_less_int @ X6 @ T ) ) ).
% minf(5)
thf(fact_870_minf_I5_J,axiom,
! [T: num] :
? [Z3: num] :
! [X6: num] :
( ( ord_less_num @ X6 @ Z3 )
=> ( ord_less_num @ X6 @ T ) ) ).
% minf(5)
thf(fact_871_minf_I4_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ( X6 != T ) ) ).
% minf(4)
thf(fact_872_minf_I4_J,axiom,
! [T: int] :
? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z3 )
=> ( X6 != T ) ) ).
% minf(4)
thf(fact_873_minf_I4_J,axiom,
! [T: num] :
? [Z3: num] :
! [X6: num] :
( ( ord_less_num @ X6 @ Z3 )
=> ( X6 != T ) ) ).
% minf(4)
thf(fact_874_minf_I3_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ( X6 != T ) ) ).
% minf(3)
thf(fact_875_minf_I3_J,axiom,
! [T: int] :
? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z3 )
=> ( X6 != T ) ) ).
% minf(3)
thf(fact_876_minf_I3_J,axiom,
! [T: num] :
? [Z3: num] :
! [X6: num] :
( ( ord_less_num @ X6 @ Z3 )
=> ( X6 != T ) ) ).
% minf(3)
thf(fact_877_minf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q2: nat > $o,Q3: nat > $o] :
( ? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z5 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z5 )
=> ( ( Q2 @ X4 )
= ( Q3 @ X4 ) ) )
=> ? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ( ( ( P @ X6 )
| ( Q2 @ X6 ) )
= ( ( P4 @ X6 )
| ( Q3 @ X6 ) ) ) ) ) ) ).
% minf(2)
thf(fact_878_minf_I2_J,axiom,
! [P: int > $o,P4: int > $o,Q2: int > $o,Q3: int > $o] :
( ? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z5 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z5 )
=> ( ( Q2 @ X4 )
= ( Q3 @ X4 ) ) )
=> ? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z3 )
=> ( ( ( P @ X6 )
| ( Q2 @ X6 ) )
= ( ( P4 @ X6 )
| ( Q3 @ X6 ) ) ) ) ) ) ).
% minf(2)
thf(fact_879_minf_I2_J,axiom,
! [P: num > $o,P4: num > $o,Q2: num > $o,Q3: num > $o] :
( ? [Z5: num] :
! [X4: num] :
( ( ord_less_num @ X4 @ Z5 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z5: num] :
! [X4: num] :
( ( ord_less_num @ X4 @ Z5 )
=> ( ( Q2 @ X4 )
= ( Q3 @ X4 ) ) )
=> ? [Z3: num] :
! [X6: num] :
( ( ord_less_num @ X6 @ Z3 )
=> ( ( ( P @ X6 )
| ( Q2 @ X6 ) )
= ( ( P4 @ X6 )
| ( Q3 @ X6 ) ) ) ) ) ) ).
% minf(2)
thf(fact_880_minf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q2: nat > $o,Q3: nat > $o] :
( ? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z5 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z5 )
=> ( ( Q2 @ X4 )
= ( Q3 @ X4 ) ) )
=> ? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ( ( ( P @ X6 )
& ( Q2 @ X6 ) )
= ( ( P4 @ X6 )
& ( Q3 @ X6 ) ) ) ) ) ) ).
% minf(1)
thf(fact_881_minf_I1_J,axiom,
! [P: int > $o,P4: int > $o,Q2: int > $o,Q3: int > $o] :
( ? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z5 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z5 )
=> ( ( Q2 @ X4 )
= ( Q3 @ X4 ) ) )
=> ? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z3 )
=> ( ( ( P @ X6 )
& ( Q2 @ X6 ) )
= ( ( P4 @ X6 )
& ( Q3 @ X6 ) ) ) ) ) ) ).
% minf(1)
thf(fact_882_minf_I1_J,axiom,
! [P: num > $o,P4: num > $o,Q2: num > $o,Q3: num > $o] :
( ? [Z5: num] :
! [X4: num] :
( ( ord_less_num @ X4 @ Z5 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z5: num] :
! [X4: num] :
( ( ord_less_num @ X4 @ Z5 )
=> ( ( Q2 @ X4 )
= ( Q3 @ X4 ) ) )
=> ? [Z3: num] :
! [X6: num] :
( ( ord_less_num @ X6 @ Z3 )
=> ( ( ( P @ X6 )
& ( Q2 @ X6 ) )
= ( ( P4 @ X6 )
& ( Q3 @ X6 ) ) ) ) ) ) ).
% minf(1)
thf(fact_883_pinf_I7_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ( ord_less_nat @ T @ X6 ) ) ).
% pinf(7)
thf(fact_884_pinf_I7_J,axiom,
! [T: int] :
? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ Z3 @ X6 )
=> ( ord_less_int @ T @ X6 ) ) ).
% pinf(7)
thf(fact_885_pinf_I7_J,axiom,
! [T: num] :
? [Z3: num] :
! [X6: num] :
( ( ord_less_num @ Z3 @ X6 )
=> ( ord_less_num @ T @ X6 ) ) ).
% pinf(7)
thf(fact_886_pinf_I5_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ~ ( ord_less_nat @ X6 @ T ) ) ).
% pinf(5)
thf(fact_887_pinf_I5_J,axiom,
! [T: int] :
? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ Z3 @ X6 )
=> ~ ( ord_less_int @ X6 @ T ) ) ).
% pinf(5)
thf(fact_888_pinf_I5_J,axiom,
! [T: num] :
? [Z3: num] :
! [X6: num] :
( ( ord_less_num @ Z3 @ X6 )
=> ~ ( ord_less_num @ X6 @ T ) ) ).
% pinf(5)
thf(fact_889_pinf_I4_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ( X6 != T ) ) ).
% pinf(4)
thf(fact_890_pinf_I4_J,axiom,
! [T: int] :
? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ Z3 @ X6 )
=> ( X6 != T ) ) ).
% pinf(4)
thf(fact_891_pinf_I4_J,axiom,
! [T: num] :
? [Z3: num] :
! [X6: num] :
( ( ord_less_num @ Z3 @ X6 )
=> ( X6 != T ) ) ).
% pinf(4)
thf(fact_892_pinf_I3_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ( X6 != T ) ) ).
% pinf(3)
thf(fact_893_pinf_I3_J,axiom,
! [T: int] :
? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ Z3 @ X6 )
=> ( X6 != T ) ) ).
% pinf(3)
thf(fact_894_pinf_I3_J,axiom,
! [T: num] :
? [Z3: num] :
! [X6: num] :
( ( ord_less_num @ Z3 @ X6 )
=> ( X6 != T ) ) ).
% pinf(3)
thf(fact_895_pinf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q2: nat > $o,Q3: nat > $o] :
( ? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z5 @ X4 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z5 @ X4 )
=> ( ( Q2 @ X4 )
= ( Q3 @ X4 ) ) )
=> ? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ( ( ( P @ X6 )
| ( Q2 @ X6 ) )
= ( ( P4 @ X6 )
| ( Q3 @ X6 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_896_pinf_I2_J,axiom,
! [P: int > $o,P4: int > $o,Q2: int > $o,Q3: int > $o] :
( ? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ Z5 @ X4 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ Z5 @ X4 )
=> ( ( Q2 @ X4 )
= ( Q3 @ X4 ) ) )
=> ? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ Z3 @ X6 )
=> ( ( ( P @ X6 )
| ( Q2 @ X6 ) )
= ( ( P4 @ X6 )
| ( Q3 @ X6 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_897_pinf_I2_J,axiom,
! [P: num > $o,P4: num > $o,Q2: num > $o,Q3: num > $o] :
( ? [Z5: num] :
! [X4: num] :
( ( ord_less_num @ Z5 @ X4 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z5: num] :
! [X4: num] :
( ( ord_less_num @ Z5 @ X4 )
=> ( ( Q2 @ X4 )
= ( Q3 @ X4 ) ) )
=> ? [Z3: num] :
! [X6: num] :
( ( ord_less_num @ Z3 @ X6 )
=> ( ( ( P @ X6 )
| ( Q2 @ X6 ) )
= ( ( P4 @ X6 )
| ( Q3 @ X6 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_898_pinf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q2: nat > $o,Q3: nat > $o] :
( ? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z5 @ X4 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z5 @ X4 )
=> ( ( Q2 @ X4 )
= ( Q3 @ X4 ) ) )
=> ? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ( ( ( P @ X6 )
& ( Q2 @ X6 ) )
= ( ( P4 @ X6 )
& ( Q3 @ X6 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_899_pinf_I1_J,axiom,
! [P: int > $o,P4: int > $o,Q2: int > $o,Q3: int > $o] :
( ? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ Z5 @ X4 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ Z5 @ X4 )
=> ( ( Q2 @ X4 )
= ( Q3 @ X4 ) ) )
=> ? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ Z3 @ X6 )
=> ( ( ( P @ X6 )
& ( Q2 @ X6 ) )
= ( ( P4 @ X6 )
& ( Q3 @ X6 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_900_pinf_I1_J,axiom,
! [P: num > $o,P4: num > $o,Q2: num > $o,Q3: num > $o] :
( ? [Z5: num] :
! [X4: num] :
( ( ord_less_num @ Z5 @ X4 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z5: num] :
! [X4: num] :
( ( ord_less_num @ Z5 @ X4 )
=> ( ( Q2 @ X4 )
= ( Q3 @ X4 ) ) )
=> ? [Z3: num] :
! [X6: num] :
( ( ord_less_num @ Z3 @ X6 )
=> ( ( ( P @ X6 )
& ( Q2 @ X6 ) )
= ( ( P4 @ X6 )
& ( Q3 @ X6 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_901_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_902_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_903_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_904_int__induct,axiom,
! [P: int > $o,K: int,I: int] :
( ( P @ K )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ K @ I3 )
=> ( ( P @ I3 )
=> ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ I3 @ K )
=> ( ( P @ I3 )
=> ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_induct
thf(fact_905_iexec_Ocases,axiom,
! [X: produc5995290525303592096st_int] :
~ ! [Ins: instr,I3: int,S3: list_char > int,Stk: list_int] :
( X
!= ( produc1484136438817787418st_int @ Ins @ ( produc5086643055186798020st_int @ I3 @ ( produc8650753666468850689st_int @ S3 @ Stk ) ) ) ) ).
% iexec.cases
thf(fact_906_drop__Cons_H,axiom,
! [N: nat,X: instr,Xs: list_instr] :
( ( ( N = zero_zero_nat )
=> ( ( drop_instr @ N @ ( cons_instr @ X @ Xs ) )
= ( cons_instr @ X @ Xs ) ) )
& ( ( N != zero_zero_nat )
=> ( ( drop_instr @ N @ ( cons_instr @ X @ Xs ) )
= ( drop_instr @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs ) ) ) ) ).
% drop_Cons'
thf(fact_907_drop__Cons_H,axiom,
! [N: nat,X: int,Xs: list_int] :
( ( ( N = zero_zero_nat )
=> ( ( drop_int @ N @ ( cons_int @ X @ Xs ) )
= ( cons_int @ X @ Xs ) ) )
& ( ( N != zero_zero_nat )
=> ( ( drop_int @ N @ ( cons_int @ X @ Xs ) )
= ( drop_int @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs ) ) ) ) ).
% drop_Cons'
thf(fact_908_drop__Cons_H,axiom,
! [N: nat,X: produc6425607678544837394st_int,Xs: list_P419547167332244514st_int] :
( ( ( N = zero_zero_nat )
=> ( ( drop_P7838799151754935884st_int @ N @ ( cons_P4151473878978182108st_int @ X @ Xs ) )
= ( cons_P4151473878978182108st_int @ X @ Xs ) ) )
& ( ( N != zero_zero_nat )
=> ( ( drop_P7838799151754935884st_int @ N @ ( cons_P4151473878978182108st_int @ X @ Xs ) )
= ( drop_P7838799151754935884st_int @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs ) ) ) ) ).
% drop_Cons'
thf(fact_909_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_910_pinf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ~ ( ord_less_eq_nat @ X6 @ T ) ) ).
% pinf(6)
thf(fact_911_pinf_I6_J,axiom,
! [T: int] :
? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ Z3 @ X6 )
=> ~ ( ord_less_eq_int @ X6 @ T ) ) ).
% pinf(6)
thf(fact_912_pinf_I6_J,axiom,
! [T: num] :
? [Z3: num] :
! [X6: num] :
( ( ord_less_num @ Z3 @ X6 )
=> ~ ( ord_less_eq_num @ X6 @ T ) ) ).
% pinf(6)
thf(fact_913_pinf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z3 @ X6 )
=> ( ord_less_eq_nat @ T @ X6 ) ) ).
% pinf(8)
thf(fact_914_pinf_I8_J,axiom,
! [T: int] :
? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ Z3 @ X6 )
=> ( ord_less_eq_int @ T @ X6 ) ) ).
% pinf(8)
thf(fact_915_pinf_I8_J,axiom,
! [T: num] :
? [Z3: num] :
! [X6: num] :
( ( ord_less_num @ Z3 @ X6 )
=> ( ord_less_eq_num @ T @ X6 ) ) ).
% pinf(8)
thf(fact_916_minf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ( ord_less_eq_nat @ X6 @ T ) ) ).
% minf(6)
thf(fact_917_minf_I6_J,axiom,
! [T: int] :
? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z3 )
=> ( ord_less_eq_int @ X6 @ T ) ) ).
% minf(6)
thf(fact_918_minf_I6_J,axiom,
! [T: num] :
? [Z3: num] :
! [X6: num] :
( ( ord_less_num @ X6 @ Z3 )
=> ( ord_less_eq_num @ X6 @ T ) ) ).
% minf(6)
thf(fact_919_minf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z3 )
=> ~ ( ord_less_eq_nat @ T @ X6 ) ) ).
% minf(8)
thf(fact_920_minf_I8_J,axiom,
! [T: int] :
? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z3 )
=> ~ ( ord_less_eq_int @ T @ X6 ) ) ).
% minf(8)
thf(fact_921_minf_I8_J,axiom,
! [T: num] :
? [Z3: num] :
! [X6: num] :
( ( ord_less_num @ X6 @ Z3 )
=> ~ ( ord_less_eq_num @ T @ X6 ) ) ).
% minf(8)
thf(fact_922_instr_Osize_I12_J,axiom,
! [X52: int] :
( ( size_size_instr @ ( jmp @ X52 ) )
= zero_zero_nat ) ).
% instr.size(12)
thf(fact_923_inth__append,axiom,
! [I: int,Xs: list_list_char,Ys: list_list_char] :
( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ Xs ) ) )
=> ( ( inth_list_char @ ( append_list_char @ Xs @ Ys ) @ I )
= ( inth_list_char @ Xs @ I ) ) )
& ( ~ ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ Xs ) ) )
=> ( ( inth_list_char @ ( append_list_char @ Xs @ Ys ) @ I )
= ( inth_list_char @ Ys @ ( minus_minus_int @ I @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ Xs ) ) ) ) ) ) ) ) ).
% inth_append
thf(fact_924_inth__append,axiom,
! [I: int,Xs: list_int,Ys: list_int] :
( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_size_list_int @ Xs ) ) )
=> ( ( inth_int @ ( append_int @ Xs @ Ys ) @ I )
= ( inth_int @ Xs @ I ) ) )
& ( ~ ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_size_list_int @ Xs ) ) )
=> ( ( inth_int @ ( append_int @ Xs @ Ys ) @ I )
= ( inth_int @ Ys @ ( minus_minus_int @ I @ ( semiri1314217659103216013at_int @ ( size_size_list_int @ Xs ) ) ) ) ) ) ) ) ).
% inth_append
thf(fact_925_inth__append,axiom,
! [I: int,Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int] :
( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_s229347708945616566st_int @ Xs ) ) )
=> ( ( inth_P8039235741264645271st_int @ ( append5028292981072010935st_int @ Xs @ Ys ) @ I )
= ( inth_P8039235741264645271st_int @ Xs @ I ) ) )
& ( ~ ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_s229347708945616566st_int @ Xs ) ) )
=> ( ( inth_P8039235741264645271st_int @ ( append5028292981072010935st_int @ Xs @ Ys ) @ I )
= ( inth_P8039235741264645271st_int @ Ys @ ( minus_minus_int @ I @ ( semiri1314217659103216013at_int @ ( size_s229347708945616566st_int @ Xs ) ) ) ) ) ) ) ) ).
% inth_append
thf(fact_926_inth__append,axiom,
! [I: int,Xs: list_instr,Ys: list_instr] :
( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ Xs ) ) )
=> ( ( inth_instr @ ( append_instr @ Xs @ Ys ) @ I )
= ( inth_instr @ Xs @ I ) ) )
& ( ~ ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ Xs ) ) )
=> ( ( inth_instr @ ( append_instr @ Xs @ Ys ) @ I )
= ( inth_instr @ Ys @ ( minus_minus_int @ I @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ Xs ) ) ) ) ) ) ) ) ).
% inth_append
thf(fact_927_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_928_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_929_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_930_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_931_n__lists_Osimps_I1_J,axiom,
! [Xs: list_instr] :
( ( n_lists_instr @ zero_zero_nat @ Xs )
= ( cons_list_instr @ nil_instr @ nil_list_instr ) ) ).
% n_lists.simps(1)
thf(fact_932_n__lists_Osimps_I1_J,axiom,
! [Xs: list_P419547167332244514st_int] :
( ( n_list7311978343830957444st_int @ zero_zero_nat @ Xs )
= ( cons_l8648290059330449388st_int @ nil_Pr749310521696113548st_int @ nil_li4597314861645524892st_int ) ) ).
% n_lists.simps(1)
thf(fact_933_n__lists_Osimps_I1_J,axiom,
! [Xs: list_int] :
( ( n_lists_int @ zero_zero_nat @ Xs )
= ( cons_list_int @ nil_int @ nil_list_int ) ) ).
% n_lists.simps(1)
thf(fact_934_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_935_nat_Oinject,axiom,
! [X2: nat,Y2: nat] :
( ( ( suc @ X2 )
= ( suc @ Y2 ) )
= ( X2 = Y2 ) ) ).
% nat.inject
thf(fact_936_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_937_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_938_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_939_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_940_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_941_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_942_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_943_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_944_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_945_nth__Cons__Suc,axiom,
! [X: instr,Xs: list_instr,N: nat] :
( ( nth_instr @ ( cons_instr @ X @ Xs ) @ ( suc @ N ) )
= ( nth_instr @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_946_nth__Cons__Suc,axiom,
! [X: int,Xs: list_int,N: nat] :
( ( nth_int @ ( cons_int @ X @ Xs ) @ ( suc @ N ) )
= ( nth_int @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_947_nth__Cons__Suc,axiom,
! [X: produc6425607678544837394st_int,Xs: list_P419547167332244514st_int,N: nat] :
( ( nth_Pr8327173284404383691st_int @ ( cons_P4151473878978182108st_int @ X @ Xs ) @ ( suc @ N ) )
= ( nth_Pr8327173284404383691st_int @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_948_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_949_drop__Suc__Cons,axiom,
! [N: nat,X: instr,Xs: list_instr] :
( ( drop_instr @ ( suc @ N ) @ ( cons_instr @ X @ Xs ) )
= ( drop_instr @ N @ Xs ) ) ).
% drop_Suc_Cons
thf(fact_950_drop__Suc__Cons,axiom,
! [N: nat,X: int,Xs: list_int] :
( ( drop_int @ ( suc @ N ) @ ( cons_int @ X @ Xs ) )
= ( drop_int @ N @ Xs ) ) ).
% drop_Suc_Cons
thf(fact_951_drop__Suc__Cons,axiom,
! [N: nat,X: produc6425607678544837394st_int,Xs: list_P419547167332244514st_int] :
( ( drop_P7838799151754935884st_int @ ( suc @ N ) @ ( cons_P4151473878978182108st_int @ X @ Xs ) )
= ( drop_P7838799151754935884st_int @ N @ Xs ) ) ).
% drop_Suc_Cons
thf(fact_952_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ M ) )
= ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% of_nat_Suc
thf(fact_953_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ M ) )
= ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M ) ) ) ).
% of_nat_Suc
thf(fact_954_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_955_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_956_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_957_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_958_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_959_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_960_nat_Odistinct_I1_J,axiom,
! [X2: nat] :
( zero_zero_nat
!= ( suc @ X2 ) ) ).
% nat.distinct(1)
thf(fact_961_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_962_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_963_nat_OdiscI,axiom,
! [Nat: nat,X2: nat] :
( ( Nat
= ( suc @ X2 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_964_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_965_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_966_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X4: nat] : ( P @ X4 @ zero_zero_nat )
=> ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
=> ( ! [X4: nat,Y3: nat] :
( ( P @ X4 @ Y3 )
=> ( P @ ( suc @ X4 ) @ ( suc @ Y3 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_967_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_968_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_969_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_970_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_971_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% not0_implies_Suc
thf(fact_972_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_973_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_974_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_975_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_976_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_977_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_978_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
& ( P @ I2 ) ) )
= ( ( P @ N )
| ? [I2: nat] :
( ( ord_less_nat @ I2 @ N )
& ( P @ I2 ) ) ) ) ).
% Ex_less_Suc
thf(fact_979_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_980_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_981_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
=> ( P @ I2 ) ) )
= ( ( P @ N )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( P @ I2 ) ) ) ) ).
% All_less_Suc
thf(fact_982_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M6: nat] :
( ( M
= ( suc @ M6 ) )
& ( ord_less_nat @ N @ M6 ) ) ) ) ).
% Suc_less_eq2
thf(fact_983_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_984_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_985_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_986_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
=> ( ! [I3: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ J2 @ K2 )
=> ( ( P @ I3 @ J2 )
=> ( ( P @ J2 @ K2 )
=> ( P @ I3 @ K2 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_987_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I3: nat] :
( ( J
= ( suc @ I3 ) )
=> ( P @ I3 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( P @ ( suc @ I3 ) )
=> ( P @ I3 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_988_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_989_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_990_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_991_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_992_Suc__le__D,axiom,
! [N: nat,M7: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M7 )
=> ? [M3: nat] :
( M7
= ( suc @ M3 ) ) ) ).
% Suc_le_D
thf(fact_993_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_994_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_995_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_996_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M2: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
=> ( P @ M2 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_997_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_998_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X4: nat] : ( R @ X4 @ X4 )
=> ( ! [X4: nat,Y3: nat,Z3: nat] :
( ( R @ X4 @ Y3 )
=> ( ( R @ Y3 @ Z3 )
=> ( R @ X4 @ Z3 ) ) )
=> ( ! [N2: nat] : ( R @ N2 @ ( suc @ N2 ) )
=> ( R @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_999_nat__arith_Osuc1,axiom,
! [A3: nat,K: nat,A: nat] :
( ( A3
= ( plus_plus_nat @ K @ A ) )
=> ( ( suc @ A3 )
= ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_1000_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_1001_add__Suc__shift,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_1002_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_1003_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_1004_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_1005_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1006_lift__Suc__antimono__le,axiom,
! [F: nat > int,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_int @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1007_lift__Suc__antimono__le,axiom,
! [F: nat > num,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_num @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_num @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1008_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1009_lift__Suc__mono__le,axiom,
! [F: nat > int,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_int @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1010_lift__Suc__mono__le,axiom,
! [F: nat > num,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_num @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_num @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1011_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_1012_lift__Suc__mono__less__iff,axiom,
! [F: nat > int,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_int @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_1013_lift__Suc__mono__less__iff,axiom,
! [F: nat > num,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less_num @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_num @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_1014_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_1015_lift__Suc__mono__less,axiom,
! [F: nat > int,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_less_int @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_1016_lift__Suc__mono__less,axiom,
! [F: nat > num,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_num @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_less_num @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_1017_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
!= zero_zero_int ) ).
% of_nat_neq_0
thf(fact_1018_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
!= zero_zero_nat ) ).
% of_nat_neq_0
thf(fact_1019_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_1020_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% gr0_implies_Suc
thf(fact_1021_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
=> ( P @ I2 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( P @ ( suc @ I2 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_1022_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1023_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
& ( P @ I2 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I2: nat] :
( ( ord_less_nat @ I2 @ N )
& ( P @ ( suc @ I2 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_1024_Suc__length__conv,axiom,
! [N: nat,Xs: list_int] :
( ( ( suc @ N )
= ( size_size_list_int @ Xs ) )
= ( ? [Y6: int,Ys3: list_int] :
( ( Xs
= ( cons_int @ Y6 @ Ys3 ) )
& ( ( size_size_list_int @ Ys3 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_1025_Suc__length__conv,axiom,
! [N: nat,Xs: list_P419547167332244514st_int] :
( ( ( suc @ N )
= ( size_s229347708945616566st_int @ Xs ) )
= ( ? [Y6: produc6425607678544837394st_int,Ys3: list_P419547167332244514st_int] :
( ( Xs
= ( cons_P4151473878978182108st_int @ Y6 @ Ys3 ) )
& ( ( size_s229347708945616566st_int @ Ys3 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_1026_Suc__length__conv,axiom,
! [N: nat,Xs: list_instr] :
( ( ( suc @ N )
= ( size_size_list_instr @ Xs ) )
= ( ? [Y6: instr,Ys3: list_instr] :
( ( Xs
= ( cons_instr @ Y6 @ Ys3 ) )
& ( ( size_size_list_instr @ Ys3 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_1027_length__Suc__conv,axiom,
! [Xs: list_int,N: nat] :
( ( ( size_size_list_int @ Xs )
= ( suc @ N ) )
= ( ? [Y6: int,Ys3: list_int] :
( ( Xs
= ( cons_int @ Y6 @ Ys3 ) )
& ( ( size_size_list_int @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_1028_length__Suc__conv,axiom,
! [Xs: list_P419547167332244514st_int,N: nat] :
( ( ( size_s229347708945616566st_int @ Xs )
= ( suc @ N ) )
= ( ? [Y6: produc6425607678544837394st_int,Ys3: list_P419547167332244514st_int] :
( ( Xs
= ( cons_P4151473878978182108st_int @ Y6 @ Ys3 ) )
& ( ( size_s229347708945616566st_int @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_1029_length__Suc__conv,axiom,
! [Xs: list_instr,N: nat] :
( ( ( size_size_list_instr @ Xs )
= ( suc @ N ) )
= ( ? [Y6: instr,Ys3: list_instr] :
( ( Xs
= ( cons_instr @ Y6 @ Ys3 ) )
& ( ( size_size_list_instr @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_1030_length__Cons,axiom,
! [X: int,Xs: list_int] :
( ( size_size_list_int @ ( cons_int @ X @ Xs ) )
= ( suc @ ( size_size_list_int @ Xs ) ) ) ).
% length_Cons
thf(fact_1031_length__Cons,axiom,
! [X: produc6425607678544837394st_int,Xs: list_P419547167332244514st_int] :
( ( size_s229347708945616566st_int @ ( cons_P4151473878978182108st_int @ X @ Xs ) )
= ( suc @ ( size_s229347708945616566st_int @ Xs ) ) ) ).
% length_Cons
thf(fact_1032_length__Cons,axiom,
! [X: instr,Xs: list_instr] :
( ( size_size_list_instr @ ( cons_instr @ X @ Xs ) )
= ( suc @ ( size_size_list_instr @ Xs ) ) ) ).
% length_Cons
thf(fact_1033_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_1034_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_1035_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_1036_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N3: nat] : ( ord_less_eq_nat @ ( suc @ N3 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1037_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_1038_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_1039_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_1040_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_1041_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_1042_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_1043_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_1044_less__natE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ! [Q4: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M @ Q4 ) ) ) ) ).
% less_natE
thf(fact_1045_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
% less_add_Suc1
thf(fact_1046_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
% less_add_Suc2
thf(fact_1047_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M5: nat,N3: nat] :
? [K3: nat] :
( N3
= ( suc @ ( plus_plus_nat @ M5 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1048_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_1049_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_1050_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_1051_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_1052_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_1053_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_1054_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1055_Suc__eq__plus1,axiom,
( suc
= ( ^ [N3: nat] : ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_1056_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_1057_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N )
& ! [I4: nat] :
( ( ord_less_eq_nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1058_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_int] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_int @ Xs ) )
= ( ? [X3: int,Ys3: list_int] :
( ( Xs
= ( cons_int @ X3 @ Ys3 ) )
& ( ord_less_eq_nat @ N @ ( size_size_list_int @ Ys3 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_1059_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_P419547167332244514st_int] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_s229347708945616566st_int @ Xs ) )
= ( ? [X3: produc6425607678544837394st_int,Ys3: list_P419547167332244514st_int] :
( ( Xs
= ( cons_P4151473878978182108st_int @ X3 @ Ys3 ) )
& ( ord_less_eq_nat @ N @ ( size_s229347708945616566st_int @ Ys3 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_1060_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_instr] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_instr @ Xs ) )
= ( ? [X3: instr,Ys3: list_instr] :
( ( Xs
= ( cons_instr @ X3 @ Ys3 ) )
& ( ord_less_eq_nat @ N @ ( size_size_list_instr @ Ys3 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_1061_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_1062_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_1063_int__Suc,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ).
% int_Suc
thf(fact_1064_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_1065_zless__iff__Suc__zadd,axiom,
( ord_less_int
= ( ^ [W2: int,Z4: int] :
? [N3: nat] :
( Z4
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ) ).
% zless_iff_Suc_zadd
thf(fact_1066_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_1067_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_1068_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_1069_length__append__singleton,axiom,
! [Xs: list_int,X: int] :
( ( size_size_list_int @ ( append_int @ Xs @ ( cons_int @ X @ nil_int ) ) )
= ( suc @ ( size_size_list_int @ Xs ) ) ) ).
% length_append_singleton
thf(fact_1070_length__append__singleton,axiom,
! [Xs: list_P419547167332244514st_int,X: produc6425607678544837394st_int] :
( ( size_s229347708945616566st_int @ ( append5028292981072010935st_int @ Xs @ ( cons_P4151473878978182108st_int @ X @ nil_Pr749310521696113548st_int ) ) )
= ( suc @ ( size_s229347708945616566st_int @ Xs ) ) ) ).
% length_append_singleton
thf(fact_1071_length__append__singleton,axiom,
! [Xs: list_instr,X: instr] :
( ( size_size_list_instr @ ( append_instr @ Xs @ ( cons_instr @ X @ nil_instr ) ) )
= ( suc @ ( size_size_list_instr @ Xs ) ) ) ).
% length_append_singleton
thf(fact_1072_length__Suc__conv__rev,axiom,
! [Xs: list_int,N: nat] :
( ( ( size_size_list_int @ Xs )
= ( suc @ N ) )
= ( ? [Y6: int,Ys3: list_int] :
( ( Xs
= ( append_int @ Ys3 @ ( cons_int @ Y6 @ nil_int ) ) )
& ( ( size_size_list_int @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_1073_length__Suc__conv__rev,axiom,
! [Xs: list_P419547167332244514st_int,N: nat] :
( ( ( size_s229347708945616566st_int @ Xs )
= ( suc @ N ) )
= ( ? [Y6: produc6425607678544837394st_int,Ys3: list_P419547167332244514st_int] :
( ( Xs
= ( append5028292981072010935st_int @ Ys3 @ ( cons_P4151473878978182108st_int @ Y6 @ nil_Pr749310521696113548st_int ) ) )
& ( ( size_s229347708945616566st_int @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_1074_length__Suc__conv__rev,axiom,
! [Xs: list_instr,N: nat] :
( ( ( size_size_list_instr @ Xs )
= ( suc @ N ) )
= ( ? [Y6: instr,Ys3: list_instr] :
( ( Xs
= ( append_instr @ Ys3 @ ( cons_instr @ Y6 @ nil_instr ) ) )
& ( ( size_size_list_instr @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_1075_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_1076_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_1077_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M5: nat,N3: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ N3 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M5 @ one_one_nat ) @ N3 ) ) ) ) ) ).
% add_eq_if
thf(fact_1078_Cons__nth__drop__Suc,axiom,
! [I: nat,Xs: list_int] :
( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
=> ( ( cons_int @ ( nth_int @ Xs @ I ) @ ( drop_int @ ( suc @ I ) @ Xs ) )
= ( drop_int @ I @ Xs ) ) ) ).
% Cons_nth_drop_Suc
thf(fact_1079_Cons__nth__drop__Suc,axiom,
! [I: nat,Xs: list_P419547167332244514st_int] :
( ( ord_less_nat @ I @ ( size_s229347708945616566st_int @ Xs ) )
=> ( ( cons_P4151473878978182108st_int @ ( nth_Pr8327173284404383691st_int @ Xs @ I ) @ ( drop_P7838799151754935884st_int @ ( suc @ I ) @ Xs ) )
= ( drop_P7838799151754935884st_int @ I @ Xs ) ) ) ).
% Cons_nth_drop_Suc
thf(fact_1080_Cons__nth__drop__Suc,axiom,
! [I: nat,Xs: list_instr] :
( ( ord_less_nat @ I @ ( size_size_list_instr @ Xs ) )
=> ( ( cons_instr @ ( nth_instr @ Xs @ I ) @ ( drop_instr @ ( suc @ I ) @ Xs ) )
= ( drop_instr @ I @ Xs ) ) ) ).
% Cons_nth_drop_Suc
thf(fact_1081_inth_Osimps,axiom,
! [I: int,X: list_char,Xs: list_list_char] :
( ( ( I = zero_zero_int )
=> ( ( inth_list_char @ ( cons_list_char @ X @ Xs ) @ I )
= X ) )
& ( ( I != zero_zero_int )
=> ( ( inth_list_char @ ( cons_list_char @ X @ Xs ) @ I )
= ( inth_list_char @ Xs @ ( minus_minus_int @ I @ one_one_int ) ) ) ) ) ).
% inth.simps
thf(fact_1082_inth_Osimps,axiom,
! [I: int,X: instr,Xs: list_instr] :
( ( ( I = zero_zero_int )
=> ( ( inth_instr @ ( cons_instr @ X @ Xs ) @ I )
= X ) )
& ( ( I != zero_zero_int )
=> ( ( inth_instr @ ( cons_instr @ X @ Xs ) @ I )
= ( inth_instr @ Xs @ ( minus_minus_int @ I @ one_one_int ) ) ) ) ) ).
% inth.simps
thf(fact_1083_inth_Osimps,axiom,
! [I: int,X: int,Xs: list_int] :
( ( ( I = zero_zero_int )
=> ( ( inth_int @ ( cons_int @ X @ Xs ) @ I )
= X ) )
& ( ( I != zero_zero_int )
=> ( ( inth_int @ ( cons_int @ X @ Xs ) @ I )
= ( inth_int @ Xs @ ( minus_minus_int @ I @ one_one_int ) ) ) ) ) ).
% inth.simps
thf(fact_1084_inth_Osimps,axiom,
! [I: int,X: produc6425607678544837394st_int,Xs: list_P419547167332244514st_int] :
( ( ( I = zero_zero_int )
=> ( ( inth_P8039235741264645271st_int @ ( cons_P4151473878978182108st_int @ X @ Xs ) @ I )
= X ) )
& ( ( I != zero_zero_int )
=> ( ( inth_P8039235741264645271st_int @ ( cons_P4151473878978182108st_int @ X @ Xs ) @ I )
= ( inth_P8039235741264645271st_int @ Xs @ ( minus_minus_int @ I @ one_one_int ) ) ) ) ) ).
% inth.simps
thf(fact_1085_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_1086_to__m__state_Osimps,axiom,
( to_m_state
= ( ^ [Xs3: list_list_char,S2: list_char > int,A5: int] :
( if_int
@ ( ( ord_less_int @ zero_zero_int @ A5 )
& ( ord_less_eq_int @ A5 @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ Xs3 ) ) ) )
@ ( S2 @ ( inth_list_char @ Xs3 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ Xs3 ) ) @ A5 ) ) )
@ zero_zero_int ) ) ) ).
% to_m_state.simps
thf(fact_1087_JmpE,axiom,
! [I: int,Pc: int,S: list_char > int,Stk2: list_int,Cf: produc6425607678544837394st_int] :
( ( iexec_pred @ ( produc1484136438817787418st_int @ ( jmp @ I ) @ ( produc5086643055186798020st_int @ Pc @ ( produc8650753666468850689st_int @ S @ Stk2 ) ) ) @ Cf )
=> ( Cf
= ( produc5086643055186798020st_int @ ( plus_plus_int @ ( plus_plus_int @ Pc @ one_one_int ) @ I ) @ ( produc8650753666468850689st_int @ S @ Stk2 ) ) ) ) ).
% JmpE
thf(fact_1088_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_1089_iexec__offset,axiom,
! [Ins2: instr,Pc: int,S: list_char > int,Stk2: list_int,Pc2: int,S4: list_char > int,Stk3: list_int,I: int] :
( ( iexec_pred @ ( produc1484136438817787418st_int @ Ins2 @ ( produc5086643055186798020st_int @ Pc @ ( produc8650753666468850689st_int @ S @ Stk2 ) ) ) @ ( produc5086643055186798020st_int @ Pc2 @ ( produc8650753666468850689st_int @ S4 @ Stk3 ) ) )
=> ( iexec_pred @ ( produc1484136438817787418st_int @ Ins2 @ ( produc5086643055186798020st_int @ ( minus_minus_int @ Pc @ I ) @ ( produc8650753666468850689st_int @ S @ Stk2 ) ) ) @ ( produc5086643055186798020st_int @ ( minus_minus_int @ Pc2 @ I ) @ ( produc8650753666468850689st_int @ S4 @ Stk3 ) ) ) ) ).
% iexec_offset
thf(fact_1090_execl_Osimps_I2_J,axiom,
! [P: list_instr] : ( execl @ P @ nil_Pr749310521696113548st_int ) ).
% execl.simps(2)
thf(fact_1091_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_1092_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_1093_exec1__def,axiom,
( exec1
= ( ^ [P5: list_instr,Cf2: produc6425607678544837394st_int,Cf3: produc6425607678544837394st_int] :
( ( iexec_pred @ ( produc1484136438817787418st_int @ ( inth_instr @ P5 @ ( produc8941309049508465880st_int @ Cf2 ) ) @ Cf2 ) @ Cf3 )
& ( ord_less_eq_int @ zero_zero_int @ ( produc8941309049508465880st_int @ Cf2 ) )
& ( ord_less_int @ ( produc8941309049508465880st_int @ Cf2 ) @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P5 ) ) ) ) ) ) ).
% exec1_def
thf(fact_1094_LoadIE,axiom,
! [I: int,Pc: int,S: list_char > int,Stk2: list_int,Cf: produc6425607678544837394st_int] :
( ( iexec_pred @ ( produc1484136438817787418st_int @ ( loadi @ I ) @ ( produc5086643055186798020st_int @ Pc @ ( produc8650753666468850689st_int @ S @ Stk2 ) ) ) @ Cf )
=> ( Cf
= ( produc5086643055186798020st_int @ ( plus_plus_int @ Pc @ one_one_int ) @ ( produc8650753666468850689st_int @ S @ ( cons_int @ I @ Stk2 ) ) ) ) ) ).
% LoadIE
thf(fact_1095_LoadE,axiom,
! [X: list_char,Pc: int,S: list_char > int,Stk2: list_int,Cf: produc6425607678544837394st_int] :
( ( iexec_pred @ ( produc1484136438817787418st_int @ ( load @ X ) @ ( produc5086643055186798020st_int @ Pc @ ( produc8650753666468850689st_int @ S @ Stk2 ) ) ) @ Cf )
=> ( Cf
= ( produc5086643055186798020st_int @ ( plus_plus_int @ Pc @ one_one_int ) @ ( produc8650753666468850689st_int @ S @ ( cons_int @ ( S @ X ) @ Stk2 ) ) ) ) ) ).
% LoadE
thf(fact_1096_execl_Oelims_I2_J,axiom,
! [X: list_instr,Xa: list_P419547167332244514st_int] :
( ( execl @ X @ Xa )
=> ( ! [Cf4: produc6425607678544837394st_int,Cf5: produc6425607678544837394st_int,Cfs2: list_P419547167332244514st_int] :
( ( Xa
= ( cons_P4151473878978182108st_int @ Cf4 @ ( cons_P4151473878978182108st_int @ Cf5 @ Cfs2 ) ) )
=> ~ ( ( exec1 @ X @ Cf4 @ Cf5 )
& ( execl @ X @ ( cons_P4151473878978182108st_int @ Cf5 @ Cfs2 ) ) ) )
=> ( ( Xa != nil_Pr749310521696113548st_int )
=> ~ ! [V2: produc6425607678544837394st_int] :
( Xa
!= ( cons_P4151473878978182108st_int @ V2 @ nil_Pr749310521696113548st_int ) ) ) ) ) ).
% execl.elims(2)
thf(fact_1097_execl_Oelims_I1_J,axiom,
! [X: list_instr,Xa: list_P419547167332244514st_int,Y: $o] :
( ( ( execl @ X @ Xa )
= Y )
=> ( ! [Cf4: produc6425607678544837394st_int,Cf5: produc6425607678544837394st_int,Cfs2: list_P419547167332244514st_int] :
( ( Xa
= ( cons_P4151473878978182108st_int @ Cf4 @ ( cons_P4151473878978182108st_int @ Cf5 @ Cfs2 ) ) )
=> ( Y
= ( ~ ( ( exec1 @ X @ Cf4 @ Cf5 )
& ( execl @ X @ ( cons_P4151473878978182108st_int @ Cf5 @ Cfs2 ) ) ) ) ) )
=> ( ( ( Xa = nil_Pr749310521696113548st_int )
=> ~ Y )
=> ~ ( ? [V2: produc6425607678544837394st_int] :
( Xa
= ( cons_P4151473878978182108st_int @ V2 @ nil_Pr749310521696113548st_int ) )
=> ~ Y ) ) ) ) ).
% execl.elims(1)
thf(fact_1098_execl_Osimps_I1_J,axiom,
! [P: list_instr,Cf: produc6425607678544837394st_int,Cf6: produc6425607678544837394st_int,Cfs: list_P419547167332244514st_int] :
( ( execl @ P @ ( cons_P4151473878978182108st_int @ Cf @ ( cons_P4151473878978182108st_int @ Cf6 @ Cfs ) ) )
= ( ( exec1 @ P @ Cf @ Cf6 )
& ( execl @ P @ ( cons_P4151473878978182108st_int @ Cf6 @ Cfs ) ) ) ) ).
% execl.simps(1)
thf(fact_1099_execl_Oelims_I3_J,axiom,
! [X: list_instr,Xa: list_P419547167332244514st_int] :
( ~ ( execl @ X @ Xa )
=> ~ ! [Cf4: produc6425607678544837394st_int,Cf5: produc6425607678544837394st_int,Cfs2: list_P419547167332244514st_int] :
( ( Xa
= ( cons_P4151473878978182108st_int @ Cf4 @ ( cons_P4151473878978182108st_int @ Cf5 @ Cfs2 ) ) )
=> ( ( exec1 @ X @ Cf4 @ Cf5 )
& ( execl @ X @ ( cons_P4151473878978182108st_int @ Cf5 @ Cfs2 ) ) ) ) ) ).
% execl.elims(3)
thf(fact_1100_exec1__appendR,axiom,
! [P: list_instr,C: produc6425607678544837394st_int,C4: produc6425607678544837394st_int,P4: list_instr] :
( ( exec1 @ P @ C @ C4 )
=> ( exec1 @ ( append_instr @ P @ P4 ) @ C @ C4 ) ) ).
% exec1_appendR
thf(fact_1101_execl_Ocases,axiom,
! [X: produc4400946599758736694st_int] :
( ! [P3: list_instr,Cf4: produc6425607678544837394st_int,Cf5: produc6425607678544837394st_int,Cfs2: list_P419547167332244514st_int] :
( X
!= ( produc7764413618310988720st_int @ P3 @ ( cons_P4151473878978182108st_int @ Cf4 @ ( cons_P4151473878978182108st_int @ Cf5 @ 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_1102_instr_Odistinct_I17_J,axiom,
! [X2: list_char,X52: int] :
( ( load @ X2 )
!= ( jmp @ X52 ) ) ).
% instr.distinct(17)
thf(fact_1103_instr_Odistinct_I7_J,axiom,
! [X1: int,X52: int] :
( ( loadi @ X1 )
!= ( jmp @ X52 ) ) ).
% instr.distinct(7)
thf(fact_1104_instr_Osize_I9_J,axiom,
! [X2: list_char] :
( ( size_size_instr @ ( load @ X2 ) )
= zero_zero_nat ) ).
% instr.size(9)
thf(fact_1105_instr_Osize_I8_J,axiom,
! [X1: int] :
( ( size_size_instr @ ( loadi @ X1 ) )
= zero_zero_nat ) ).
% instr.size(8)
thf(fact_1106_exec1__appendL,axiom,
! [P: list_instr,I: int,S: list_char > int,Stk2: list_int,I5: int,S4: list_char > int,Stk3: list_int,P4: list_instr] :
( ( exec1 @ P @ ( produc5086643055186798020st_int @ I @ ( produc8650753666468850689st_int @ S @ Stk2 ) ) @ ( produc5086643055186798020st_int @ I5 @ ( produc8650753666468850689st_int @ S4 @ Stk3 ) ) )
=> ( exec1 @ ( append_instr @ P4 @ P ) @ ( produc5086643055186798020st_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P4 ) ) @ I ) @ ( produc8650753666468850689st_int @ S @ Stk2 ) ) @ ( produc5086643055186798020st_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P4 ) ) @ I5 ) @ ( produc8650753666468850689st_int @ S4 @ Stk3 ) ) ) ) ).
% exec1_appendL
thf(fact_1107_exec1I,axiom,
! [C4: produc6425607678544837394st_int,P: list_instr,I: int,S: list_char > int,Stk2: list_int] :
( ( C4
= ( iexec @ ( inth_instr @ P @ I ) @ ( produc5086643055186798020st_int @ I @ ( produc8650753666468850689st_int @ S @ Stk2 ) ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P ) ) )
=> ( exec1 @ P @ ( produc5086643055186798020st_int @ I @ ( produc8650753666468850689st_int @ S @ Stk2 ) ) @ C4 ) ) ) ) ).
% exec1I
thf(fact_1108_exec__append__trans,axiom,
! [P: list_instr,S: list_char > int,Stk2: list_int,I5: int,S4: list_char > int,Stk3: list_int,P4: list_instr,I6: int,S5: list_char > int,Stk4: list_int,J4: int] :
( ( star_P707599355569300323st_int @ ( exec1 @ P ) @ ( produc5086643055186798020st_int @ zero_zero_int @ ( produc8650753666468850689st_int @ S @ Stk2 ) ) @ ( produc5086643055186798020st_int @ I5 @ ( produc8650753666468850689st_int @ S4 @ Stk3 ) ) )
=> ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P ) ) @ I5 )
=> ( ( star_P707599355569300323st_int @ ( exec1 @ P4 ) @ ( produc5086643055186798020st_int @ ( minus_minus_int @ I5 @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P ) ) ) @ ( produc8650753666468850689st_int @ S4 @ Stk3 ) ) @ ( produc5086643055186798020st_int @ I6 @ ( produc8650753666468850689st_int @ S5 @ Stk4 ) ) )
=> ( ( J4
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P ) ) @ I6 ) )
=> ( star_P707599355569300323st_int @ ( exec1 @ ( append_instr @ P @ P4 ) ) @ ( produc5086643055186798020st_int @ zero_zero_int @ ( produc8650753666468850689st_int @ S @ Stk2 ) ) @ ( produc5086643055186798020st_int @ J4 @ ( produc8650753666468850689st_int @ S5 @ Stk4 ) ) ) ) ) ) ) ).
% exec_append_trans
thf(fact_1109_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_1110_ivl__diff,axiom,
! [I: num,N: num,M: num] :
( ( ord_less_eq_num @ I @ N )
=> ( ( minus_minus_set_num @ ( set_or1222409239386451017an_num @ I @ M ) @ ( set_or1222409239386451017an_num @ I @ N ) )
= ( set_or1222409239386451017an_num @ N @ M ) ) ) ).
% ivl_diff
thf(fact_1111_ivl__diff,axiom,
! [I: int,N: int,M: int] :
( ( ord_less_eq_int @ I @ N )
=> ( ( minus_minus_set_int @ ( set_or4662586982721622107an_int @ I @ M ) @ ( set_or4662586982721622107an_int @ I @ N ) )
= ( set_or4662586982721622107an_int @ N @ M ) ) ) ).
% ivl_diff
thf(fact_1112_ivl__diff,axiom,
! [I: nat,N: nat,M: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ I @ M ) @ ( set_or4665077453230672383an_nat @ I @ N ) )
= ( set_or4665077453230672383an_nat @ N @ M ) ) ) ).
% ivl_diff
thf(fact_1113_ivl__subset,axiom,
! [I: num,J: num,M: num,N: num] :
( ( ord_less_eq_set_num @ ( set_or1222409239386451017an_num @ I @ J ) @ ( set_or1222409239386451017an_num @ M @ N ) )
= ( ( ord_less_eq_num @ J @ I )
| ( ( ord_less_eq_num @ M @ I )
& ( ord_less_eq_num @ J @ N ) ) ) ) ).
% ivl_subset
thf(fact_1114_ivl__subset,axiom,
! [I: int,J: int,M: int,N: int] :
( ( ord_less_eq_set_int @ ( set_or4662586982721622107an_int @ I @ J ) @ ( set_or4662586982721622107an_int @ M @ N ) )
= ( ( ord_less_eq_int @ J @ I )
| ( ( ord_less_eq_int @ M @ I )
& ( ord_less_eq_int @ J @ N ) ) ) ) ).
% ivl_subset
thf(fact_1115_ivl__subset,axiom,
! [I: nat,J: nat,M: nat,N: nat] :
( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ I @ J ) @ ( set_or4665077453230672383an_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ J @ I )
| ( ( ord_less_eq_nat @ M @ I )
& ( ord_less_eq_nat @ J @ N ) ) ) ) ).
% ivl_subset
thf(fact_1116_atLeastLessThan__iff,axiom,
! [I: num,L: num,U: num] :
( ( member_num @ I @ ( set_or1222409239386451017an_num @ L @ U ) )
= ( ( ord_less_eq_num @ L @ I )
& ( ord_less_num @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_1117_atLeastLessThan__iff,axiom,
! [I: int,L: int,U: int] :
( ( member_int @ I @ ( set_or4662586982721622107an_int @ L @ U ) )
= ( ( ord_less_eq_int @ L @ I )
& ( ord_less_int @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_1118_atLeastLessThan__iff,axiom,
! [I: nat,L: nat,U: nat] :
( ( member_nat @ I @ ( set_or4665077453230672383an_nat @ L @ U ) )
= ( ( ord_less_eq_nat @ L @ I )
& ( ord_less_nat @ I @ U ) ) ) ).
% atLeastLessThan_iff
thf(fact_1119_iexec__shift,axiom,
! [N: int,I5: int,S4: list_char > int,Stk3: list_int,Ins2: instr,I: int,S: list_char > int,Stk2: list_int] :
( ( ( produc5086643055186798020st_int @ ( plus_plus_int @ N @ I5 ) @ ( produc8650753666468850689st_int @ S4 @ Stk3 ) )
= ( iexec @ Ins2 @ ( produc5086643055186798020st_int @ ( plus_plus_int @ N @ I ) @ ( produc8650753666468850689st_int @ S @ Stk2 ) ) ) )
= ( ( produc5086643055186798020st_int @ I5 @ ( produc8650753666468850689st_int @ S4 @ Stk3 ) )
= ( iexec @ Ins2 @ ( produc5086643055186798020st_int @ I @ ( produc8650753666468850689st_int @ S @ Stk2 ) ) ) ) ) ).
% iexec_shift
thf(fact_1120_exec__Cons__1,axiom,
! [P: list_instr,S: list_char > int,Stk2: list_int,J: int,T: list_char > int,Stk3: list_int,Ins2: instr] :
( ( star_P707599355569300323st_int @ ( exec1 @ P ) @ ( produc5086643055186798020st_int @ zero_zero_int @ ( produc8650753666468850689st_int @ S @ Stk2 ) ) @ ( produc5086643055186798020st_int @ J @ ( produc8650753666468850689st_int @ T @ Stk3 ) ) )
=> ( star_P707599355569300323st_int @ ( exec1 @ ( cons_instr @ Ins2 @ P ) ) @ ( produc5086643055186798020st_int @ one_one_int @ ( produc8650753666468850689st_int @ S @ Stk2 ) ) @ ( produc5086643055186798020st_int @ ( plus_plus_int @ one_one_int @ J ) @ ( produc8650753666468850689st_int @ T @ Stk3 ) ) ) ) ).
% exec_Cons_1
thf(fact_1121_exec__appendL__if,axiom,
! [P4: list_instr,I: int,P: list_instr,S: list_char > int,Stk2: list_int,J: int,S4: list_char > int,Stk3: list_int,I5: int] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P4 ) ) @ I )
=> ( ( star_P707599355569300323st_int @ ( exec1 @ P ) @ ( produc5086643055186798020st_int @ ( minus_minus_int @ I @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P4 ) ) ) @ ( produc8650753666468850689st_int @ S @ Stk2 ) ) @ ( produc5086643055186798020st_int @ J @ ( produc8650753666468850689st_int @ S4 @ Stk3 ) ) )
=> ( ( I5
= ( 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 @ Stk2 ) ) @ ( produc5086643055186798020st_int @ I5 @ ( produc8650753666468850689st_int @ S4 @ Stk3 ) ) ) ) ) ) ).
% exec_appendL_if
thf(fact_1122_atLeastLessThan__subset__iff,axiom,
! [A: num,B: num,C: num,D2: num] :
( ( ord_less_eq_set_num @ ( set_or1222409239386451017an_num @ A @ B ) @ ( set_or1222409239386451017an_num @ C @ D2 ) )
=> ( ( ord_less_eq_num @ B @ A )
| ( ( ord_less_eq_num @ C @ A )
& ( ord_less_eq_num @ B @ D2 ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_1123_atLeastLessThan__subset__iff,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ord_less_eq_set_int @ ( set_or4662586982721622107an_int @ A @ B ) @ ( set_or4662586982721622107an_int @ C @ D2 ) )
=> ( ( ord_less_eq_int @ B @ A )
| ( ( ord_less_eq_int @ C @ A )
& ( ord_less_eq_int @ B @ D2 ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_1124_atLeastLessThan__subset__iff,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_eq_set_nat @ ( set_or4665077453230672383an_nat @ A @ B ) @ ( set_or4665077453230672383an_nat @ C @ D2 ) )
=> ( ( ord_less_eq_nat @ B @ A )
| ( ( ord_less_eq_nat @ C @ A )
& ( ord_less_eq_nat @ B @ D2 ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_1125_atLeastLessThan__inj_I2_J,axiom,
! [A: num,B: num,C: num,D2: num] :
( ( ( set_or1222409239386451017an_num @ A @ B )
= ( set_or1222409239386451017an_num @ C @ D2 ) )
=> ( ( ord_less_num @ A @ B )
=> ( ( ord_less_num @ C @ D2 )
=> ( B = D2 ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_1126_atLeastLessThan__inj_I2_J,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ( set_or4662586982721622107an_int @ A @ B )
= ( set_or4662586982721622107an_int @ C @ D2 ) )
=> ( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D2 )
=> ( B = D2 ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_1127_atLeastLessThan__inj_I2_J,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ( set_or4665077453230672383an_nat @ A @ B )
= ( set_or4665077453230672383an_nat @ C @ D2 ) )
=> ( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( B = D2 ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_1128_atLeastLessThan__inj_I1_J,axiom,
! [A: num,B: num,C: num,D2: num] :
( ( ( set_or1222409239386451017an_num @ A @ B )
= ( set_or1222409239386451017an_num @ C @ D2 ) )
=> ( ( ord_less_num @ A @ B )
=> ( ( ord_less_num @ C @ D2 )
=> ( A = C ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_1129_atLeastLessThan__inj_I1_J,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ( set_or4662586982721622107an_int @ A @ B )
= ( set_or4662586982721622107an_int @ C @ D2 ) )
=> ( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D2 )
=> ( A = C ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_1130_atLeastLessThan__inj_I1_J,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ( set_or4665077453230672383an_nat @ A @ B )
= ( set_or4665077453230672383an_nat @ C @ D2 ) )
=> ( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( A = C ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_1131_Ico__eq__Ico,axiom,
! [L: num,H: num,L2: num,H2: num] :
( ( ( set_or1222409239386451017an_num @ L @ H )
= ( set_or1222409239386451017an_num @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_num @ L @ H )
& ~ ( ord_less_num @ L2 @ H2 ) ) ) ) ).
% Ico_eq_Ico
thf(fact_1132_Ico__eq__Ico,axiom,
! [L: int,H: int,L2: int,H2: int] :
( ( ( set_or4662586982721622107an_int @ L @ H )
= ( set_or4662586982721622107an_int @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_int @ L @ H )
& ~ ( ord_less_int @ L2 @ H2 ) ) ) ) ).
% Ico_eq_Ico
thf(fact_1133_Ico__eq__Ico,axiom,
! [L: nat,H: nat,L2: nat,H2: nat] :
( ( ( set_or4665077453230672383an_nat @ L @ H )
= ( set_or4665077453230672383an_nat @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( ord_less_nat @ L @ H )
& ~ ( ord_less_nat @ L2 @ H2 ) ) ) ) ).
% Ico_eq_Ico
thf(fact_1134_atLeastLessThan__eq__iff,axiom,
! [A: num,B: num,C: num,D2: num] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_num @ C @ D2 )
=> ( ( ( set_or1222409239386451017an_num @ A @ B )
= ( set_or1222409239386451017an_num @ C @ D2 ) )
= ( ( A = C )
& ( B = D2 ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_1135_atLeastLessThan__eq__iff,axiom,
! [A: int,B: int,C: int,D2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D2 )
=> ( ( ( set_or4662586982721622107an_int @ A @ B )
= ( set_or4662586982721622107an_int @ C @ D2 ) )
= ( ( A = C )
& ( B = D2 ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_1136_atLeastLessThan__eq__iff,axiom,
! [A: nat,B: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ( ( set_or4665077453230672383an_nat @ A @ B )
= ( set_or4665077453230672383an_nat @ C @ D2 ) )
= ( ( A = C )
& ( B = D2 ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_1137_exec__appendR,axiom,
! [P: list_instr,C: produc6425607678544837394st_int,C4: produc6425607678544837394st_int,P4: list_instr] :
( ( star_P707599355569300323st_int @ ( exec1 @ P ) @ C @ C4 )
=> ( star_P707599355569300323st_int @ ( exec1 @ ( append_instr @ P @ P4 ) ) @ C @ C4 ) ) ).
% exec_appendR
thf(fact_1138_exec__induct,axiom,
! [Pa: list_instr,X1a: int,X1b: list_char > int,X1c: list_int,X2a: int,X2b: list_char > int,X2c: list_int,P: int > ( list_char > int ) > list_int > int > ( list_char > int ) > list_int > $o] :
( ( star_P707599355569300323st_int @ ( exec1 @ Pa ) @ ( produc5086643055186798020st_int @ X1a @ ( produc8650753666468850689st_int @ X1b @ X1c ) ) @ ( produc5086643055186798020st_int @ X2a @ ( produc8650753666468850689st_int @ X2b @ X2c ) ) )
=> ( ! [A4: int,Aa: list_char > int,B3: list_int] : ( P @ A4 @ Aa @ B3 @ A4 @ Aa @ B3 )
=> ( ! [A4: int,Aa: list_char > int,B3: list_int,Ab: int,Ac: list_char > int,Ba: list_int,Ad: int,Ae: list_char > int,Bb: list_int] :
( ( exec1 @ Pa @ ( produc5086643055186798020st_int @ A4 @ ( produc8650753666468850689st_int @ Aa @ B3 ) ) @ ( produc5086643055186798020st_int @ Ab @ ( produc8650753666468850689st_int @ Ac @ Ba ) ) )
=> ( ( star_P707599355569300323st_int @ ( exec1 @ Pa ) @ ( produc5086643055186798020st_int @ Ab @ ( produc8650753666468850689st_int @ Ac @ Ba ) ) @ ( produc5086643055186798020st_int @ Ad @ ( produc8650753666468850689st_int @ Ae @ Bb ) ) )
=> ( ( P @ Ab @ Ac @ Ba @ Ad @ Ae @ Bb )
=> ( P @ A4 @ Aa @ B3 @ Ad @ Ae @ Bb ) ) ) )
=> ( P @ X1a @ X1b @ X1c @ X2a @ X2b @ X2c ) ) ) ) ).
% exec_induct
thf(fact_1139_exec__appendL,axiom,
! [P: list_instr,I: int,S: list_char > int,Stk2: list_int,I5: int,S4: list_char > int,Stk3: list_int,P4: list_instr] :
( ( star_P707599355569300323st_int @ ( exec1 @ P ) @ ( produc5086643055186798020st_int @ I @ ( produc8650753666468850689st_int @ S @ Stk2 ) ) @ ( produc5086643055186798020st_int @ I5 @ ( produc8650753666468850689st_int @ S4 @ Stk3 ) ) )
=> ( star_P707599355569300323st_int @ ( exec1 @ ( append_instr @ P4 @ P ) ) @ ( produc5086643055186798020st_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P4 ) ) @ I ) @ ( produc8650753666468850689st_int @ S @ Stk2 ) ) @ ( produc5086643055186798020st_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P4 ) ) @ I5 ) @ ( produc8650753666468850689st_int @ S4 @ Stk3 ) ) ) ) ).
% exec_appendL
thf(fact_1140_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_1141_Compiler_Obcomp__correct,axiom,
! [I: int,B: bexp,F: $o,S: list_char > int,Stk2: 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 @ Stk2 ) )
@ ( 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 @ Stk2 ) ) ) ) ).
% Compiler.bcomp_correct
thf(fact_1142_ccomp__bigstep,axiom,
! [C: com,S: list_char > int,T: list_char > int,Stk2: list_int] :
( ( big_big_step @ ( produc5595214716300948949ar_int @ C @ S ) @ T )
=> ( star_P707599355569300323st_int @ ( exec1 @ ( ccomp @ C ) ) @ ( produc5086643055186798020st_int @ zero_zero_int @ ( produc8650753666468850689st_int @ S @ Stk2 ) ) @ ( produc5086643055186798020st_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ ( ccomp @ C ) ) ) @ ( produc8650753666468850689st_int @ T @ Stk2 ) ) ) ) ).
% ccomp_bigstep
thf(fact_1143_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_1144_dual__order_Orefl,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% dual_order.refl
thf(fact_1145_dual__order_Orefl,axiom,
! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% dual_order.refl
thf(fact_1146_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_1147_order__refl,axiom,
! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% order_refl
thf(fact_1148_order__refl,axiom,
! [X: num] : ( ord_less_eq_num @ X @ X ) ).
% order_refl
thf(fact_1149_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_1150_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_1151_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_1152_nle__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_eq_int @ A @ B ) )
= ( ( ord_less_eq_int @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_1153_nle__le,axiom,
! [A: num,B: num] :
( ( ~ ( ord_less_eq_num @ A @ B ) )
= ( ( ord_less_eq_num @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_1154_le__cases3,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_1155_le__cases3,axiom,
! [X: int,Y: int,Z2: int] :
( ( ( ord_less_eq_int @ X @ Y )
=> ~ ( ord_less_eq_int @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_eq_int @ X @ Z2 ) )
=> ( ( ( ord_less_eq_int @ X @ Z2 )
=> ~ ( ord_less_eq_int @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_int @ Z2 @ Y )
=> ~ ( ord_less_eq_int @ Y @ X ) )
=> ( ( ( ord_less_eq_int @ Y @ Z2 )
=> ~ ( ord_less_eq_int @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_int @ Z2 @ X )
=> ~ ( ord_less_eq_int @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_1156_le__cases3,axiom,
! [X: num,Y: num,Z2: num] :
( ( ( ord_less_eq_num @ X @ Y )
=> ~ ( ord_less_eq_num @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_num @ Y @ X )
=> ~ ( ord_less_eq_num @ X @ Z2 ) )
=> ( ( ( ord_less_eq_num @ X @ Z2 )
=> ~ ( ord_less_eq_num @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_num @ Z2 @ Y )
=> ~ ( ord_less_eq_num @ Y @ X ) )
=> ( ( ( ord_less_eq_num @ Y @ Z2 )
=> ~ ( ord_less_eq_num @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_num @ Z2 @ X )
=> ~ ( ord_less_eq_num @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_1157_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
= ( ^ [X3: nat,Y6: nat] :
( ( ord_less_eq_nat @ X3 @ Y6 )
& ( ord_less_eq_nat @ Y6 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_1158_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: int,Z: int] : ( Y4 = Z ) )
= ( ^ [X3: int,Y6: int] :
( ( ord_less_eq_int @ X3 @ Y6 )
& ( ord_less_eq_int @ Y6 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_1159_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: num,Z: num] : ( Y4 = Z ) )
= ( ^ [X3: num,Y6: num] :
( ( ord_less_eq_num @ X3 @ Y6 )
& ( ord_less_eq_num @ Y6 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_1160_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_1161_ord__eq__le__trans,axiom,
! [A: int,B: int,C: int] :
( ( A = B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_1162_ord__eq__le__trans,axiom,
! [A: num,B: num,C: num] :
( ( A = B )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ord_less_eq_num @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_1163_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_1164_ord__le__eq__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_1165_ord__le__eq__trans,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_num @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_1166_order__antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_1167_order__antisym,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_1168_order__antisym,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ord_less_eq_num @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_1169_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_1170_order_Otrans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% order.trans
thf(fact_1171_order_Otrans,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ord_less_eq_num @ A @ C ) ) ) ).
% order.trans
thf(fact_1172_order__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_eq_nat @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_1173_order__trans,axiom,
! [X: int,Y: int,Z2: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z2 )
=> ( ord_less_eq_int @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_1174_order__trans,axiom,
! [X: num,Y: num,Z2: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ord_less_eq_num @ Y @ Z2 )
=> ( ord_less_eq_num @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_1175_linorder__wlog,axiom,
! [P: num > num > $o,A: num,B: num] :
( ! [A4: num,B3: num] :
( ( ord_less_eq_num @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: num,B3: num] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_1176_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_1177_nat0__intermed__int__val,axiom,
! [N: nat,F: nat > int,K: int] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I3 @ one_one_nat ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I3: nat] :
( ( ord_less_eq_nat @ I3 @ N )
& ( ( F @ I3 )
= K ) ) ) ) ) ).
% nat0_intermed_int_val
thf(fact_1178_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_1179_nat__intermed__int__val,axiom,
! [M: nat,N: nat,F: nat > int,K: int] :
( ! [I3: nat] :
( ( ( ord_less_eq_nat @ M @ I3 )
& ( ord_less_nat @ I3 @ N ) )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_int @ ( F @ M ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I3: nat] :
( ( ord_less_eq_nat @ M @ I3 )
& ( ord_less_eq_nat @ I3 @ N )
& ( ( F @ I3 )
= K ) ) ) ) ) ) ).
% nat_intermed_int_val
thf(fact_1180_nat__ivt__aux,axiom,
! [N: nat,F: nat > int,K: int] :
( ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I3: nat] :
( ( ord_less_eq_nat @ I3 @ N )
& ( ( F @ I3 )
= K ) ) ) ) ) ).
% nat_ivt_aux
thf(fact_1181_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ! [I4: nat] :
( ( ord_less_nat @ K2 @ I4 )
=> ( P @ I4 ) )
=> ( P @ K2 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_1182_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_1183_ccomp_Osimps_I1_J,axiom,
( ( ccomp @ skip )
= nil_instr ) ).
% ccomp.simps(1)
thf(fact_1184_com_Osize_I6_J,axiom,
( ( size_size_com @ skip )
= zero_zero_nat ) ).
% com.size(6)
thf(fact_1185_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_1186_bexp_Osize_I5_J,axiom,
! [X1: $o] :
( ( size_size_bexp @ ( bc @ X1 ) )
= zero_zero_nat ) ).
% bexp.size(5)
thf(fact_1187_execl__all__Bc,axiom,
! [V: $o,F: $o,I: int,Cfs: list_P419547167332244514st_int] :
( ( execl_all @ ( if_list_instr @ ( V = F ) @ ( cons_instr @ ( jmp @ I ) @ nil_instr ) @ nil_instr ) @ Cfs )
=> ( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( bpred @ ( produc4047900504771817624_o_int @ ( bc @ V ) @ ( product_Pair_o_int @ F @ I ) ) @ ( nth_Pr8327173284404383691st_int @ Cfs @ zero_zero_nat ) @ ( nth_Pr8327173284404383691st_int @ Cfs @ ( minus_minus_nat @ ( size_s229347708945616566st_int @ Cfs ) @ ( suc @ zero_zero_nat ) ) ) )
& ( ( V = F )
=> ( ( size_s229347708945616566st_int @ Cfs )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
& ( ( V = ~ F )
=> ( ( size_s229347708945616566st_int @ Cfs )
= one_one_nat ) )
& ( ( produc8941309049508465880st_int @ ( nth_Pr8327173284404383691st_int @ Cfs @ zero_zero_nat ) )
= zero_zero_int ) ) ) ) ).
% execl_all_Bc
thf(fact_1188_verit__eq__simplify_I8_J,axiom,
! [X2: num,Y2: num] :
( ( ( bit0 @ X2 )
= ( bit0 @ Y2 ) )
= ( X2 = Y2 ) ) ).
% verit_eq_simplify(8)
thf(fact_1189_Suc__numeral,axiom,
! [N: num] :
( ( suc @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).
% Suc_numeral
thf(fact_1190_add__2__eq__Suc_H,axiom,
! [N: nat] :
( ( plus_plus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( suc @ ( suc @ N ) ) ) ).
% add_2_eq_Suc'
thf(fact_1191_add__2__eq__Suc,axiom,
! [N: nat] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
= ( suc @ ( suc @ N ) ) ) ).
% add_2_eq_Suc
thf(fact_1192_Suc__1,axiom,
( ( suc @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% Suc_1
thf(fact_1193_le__num__One__iff,axiom,
! [X: num] :
( ( ord_less_eq_num @ X @ one )
= ( X = one ) ) ).
% le_num_One_iff
thf(fact_1194_numerals_I1_J,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numerals(1)
thf(fact_1195_nat__1__add__1,axiom,
( ( plus_plus_nat @ one_one_nat @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% nat_1_add_1
thf(fact_1196_add__One__commute,axiom,
! [N: num] :
( ( plus_plus_num @ one @ N )
= ( plus_plus_num @ N @ one ) ) ).
% add_One_commute
thf(fact_1197_verit__eq__simplify_I10_J,axiom,
! [X2: num] :
( one
!= ( bit0 @ X2 ) ) ).
% verit_eq_simplify(10)
thf(fact_1198_Suc__nat__number__of__add,axiom,
! [V: num,N: nat] :
( ( suc @ ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ N ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ one ) ) @ N ) ) ).
% Suc_nat_number_of_add
thf(fact_1199_numeral__1__eq__Suc__0,axiom,
( ( numeral_numeral_nat @ one )
= ( suc @ zero_zero_nat ) ) ).
% numeral_1_eq_Suc_0
thf(fact_1200_numeral__2__eq__2,axiom,
( ( numeral_numeral_nat @ ( bit0 @ one ) )
= ( suc @ ( suc @ zero_zero_nat ) ) ) ).
% numeral_2_eq_2
thf(fact_1201_less__2__cases,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
=> ( ( N = zero_zero_nat )
| ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% less_2_cases
thf(fact_1202_less__2__cases__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( ( N = zero_zero_nat )
| ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% less_2_cases_iff
thf(fact_1203_num_Osize_I4_J,axiom,
( ( size_size_num @ one )
= zero_zero_nat ) ).
% num.size(4)
thf(fact_1204_num_Osize_I5_J,axiom,
! [X2: num] :
( ( size_size_num @ ( bit0 @ X2 ) )
= ( plus_plus_nat @ ( size_size_num @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).
% num.size(5)
thf(fact_1205_nat__induct2,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ( P @ one_one_nat )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( plus_plus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct2
thf(fact_1206_semiring__norm_I69_J,axiom,
! [M: num] :
~ ( ord_less_eq_num @ ( bit0 @ M ) @ one ) ).
% semiring_norm(69)
thf(fact_1207_int__eq__iff__numeral,axiom,
! [M: nat,V: num] :
( ( ( semiri1314217659103216013at_int @ M )
= ( numeral_numeral_int @ V ) )
= ( M
= ( numeral_numeral_nat @ V ) ) ) ).
% int_eq_iff_numeral
thf(fact_1208_semiring__norm_I71_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% semiring_norm(71)
thf(fact_1209_semiring__norm_I68_J,axiom,
! [N: num] : ( ord_less_eq_num @ one @ N ) ).
% semiring_norm(68)
thf(fact_1210_int__ops_I3_J,axiom,
! [N: num] :
( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_int @ N ) ) ).
% int_ops(3)
thf(fact_1211_execl__all__N,axiom,
! [I: int,Cfs: list_P419547167332244514st_int] :
( ( execl_all @ ( cons_instr @ ( loadi @ I ) @ nil_instr ) @ Cfs )
=> ( ( apred @ ( n @ I ) @ ( 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 )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) )
& ( ( produc8941309049508465880st_int @ ( nth_Pr8327173284404383691st_int @ Cfs @ zero_zero_nat ) )
= zero_zero_int ) ) ) ).
% execl_all_N
thf(fact_1212_execl__all__V,axiom,
! [X: list_char,Cfs: list_P419547167332244514st_int] :
( ( execl_all @ ( cons_instr @ ( load @ X ) @ nil_instr ) @ Cfs )
=> ( ( apred @ ( v @ X ) @ ( 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 )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) )
& ( ( produc8941309049508465880st_int @ ( nth_Pr8327173284404383691st_int @ Cfs @ zero_zero_nat ) )
= zero_zero_int ) ) ) ).
% execl_all_V
thf(fact_1213_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_1214_num_Osize__gen_I2_J,axiom,
! [X2: num] :
( ( size_num @ ( bit0 @ X2 ) )
= ( plus_plus_nat @ ( size_num @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).
% num.size_gen(2)
thf(fact_1215_num_Osize__gen_I1_J,axiom,
( ( size_num @ one )
= zero_zero_nat ) ).
% num.size_gen(1)
thf(fact_1216_acomp_Osimps_I2_J,axiom,
! [X: list_char] :
( ( acomp @ ( v @ X ) )
= ( cons_instr @ ( load @ X ) @ nil_instr ) ) ).
% acomp.simps(2)
thf(fact_1217_acomp_Osimps_I1_J,axiom,
! [I: int] :
( ( acomp @ ( n @ I ) )
= ( cons_instr @ ( loadi @ I ) @ nil_instr ) ) ).
% acomp.simps(1)
thf(fact_1218_Compiler_Oacomp__correct,axiom,
! [A: aexp,S: list_char > int,Stk2: list_int] : ( star_P707599355569300323st_int @ ( exec1 @ ( acomp @ A ) ) @ ( produc5086643055186798020st_int @ zero_zero_int @ ( produc8650753666468850689st_int @ S @ Stk2 ) ) @ ( produc5086643055186798020st_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ ( acomp @ A ) ) ) @ ( produc8650753666468850689st_int @ S @ ( cons_int @ ( aval @ A @ S ) @ Stk2 ) ) ) ) ).
% Compiler.acomp_correct
thf(fact_1219_Parity_Oadjust__mod__def,axiom,
( adjust_mod
= ( ^ [L3: num,R2: int] : ( if_int @ ( R2 = zero_zero_int ) @ zero_zero_int @ ( minus_minus_int @ ( numeral_numeral_int @ L3 ) @ R2 ) ) ) ) ).
% Parity.adjust_mod_def
thf(fact_1220_verit__eq__simplify_I9_J,axiom,
! [X32: num,Y32: num] :
( ( ( bit1 @ X32 )
= ( bit1 @ Y32 ) )
= ( X32 = Y32 ) ) ).
% verit_eq_simplify(9)
thf(fact_1221_semiring__norm_I73_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% semiring_norm(73)
thf(fact_1222_semiring__norm_I72_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% semiring_norm(72)
thf(fact_1223_semiring__norm_I70_J,axiom,
! [M: num] :
~ ( ord_less_eq_num @ ( bit1 @ M ) @ one ) ).
% semiring_norm(70)
thf(fact_1224_semiring__norm_I79_J,axiom,
! [M: num,N: num] :
( ( ord_less_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% semiring_norm(79)
thf(fact_1225_semiring__norm_I74_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% semiring_norm(74)
thf(fact_1226_eval__nat__numeral_I3_J,axiom,
! [N: num] :
( ( numeral_numeral_nat @ ( bit1 @ N ) )
= ( suc @ ( numeral_numeral_nat @ ( bit0 @ N ) ) ) ) ).
% eval_nat_numeral(3)
thf(fact_1227_num_Oexhaust,axiom,
! [Y: num] :
( ( Y != one )
=> ( ! [X23: num] :
( Y
!= ( bit0 @ X23 ) )
=> ~ ! [X33: num] :
( Y
!= ( bit1 @ X33 ) ) ) ) ).
% num.exhaust
thf(fact_1228_verit__eq__simplify_I12_J,axiom,
! [X32: num] :
( one
!= ( bit1 @ X32 ) ) ).
% verit_eq_simplify(12)
thf(fact_1229_verit__eq__simplify_I14_J,axiom,
! [X2: num,X32: num] :
( ( bit0 @ X2 )
!= ( bit1 @ X32 ) ) ).
% verit_eq_simplify(14)
thf(fact_1230_numeral__3__eq__3,axiom,
( ( numeral_numeral_nat @ ( bit1 @ one ) )
= ( suc @ ( suc @ ( suc @ zero_zero_nat ) ) ) ) ).
% numeral_3_eq_3
thf(fact_1231_Suc3__eq__add__3,axiom,
! [N: nat] :
( ( suc @ ( suc @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ).
% Suc3_eq_add_3
thf(fact_1232_num_Osize_I6_J,axiom,
! [X32: num] :
( ( size_size_num @ ( bit1 @ X32 ) )
= ( plus_plus_nat @ ( size_size_num @ X32 ) @ ( suc @ zero_zero_nat ) ) ) ).
% num.size(6)
thf(fact_1233_num_Osize__gen_I3_J,axiom,
! [X32: num] :
( ( size_num @ ( bit1 @ X32 ) )
= ( plus_plus_nat @ ( size_num @ X32 ) @ ( suc @ zero_zero_nat ) ) ) ).
% num.size_gen(3)
thf(fact_1234_divmod__BitM__2__eq,axiom,
! [M: num] :
( ( unique5403075989570733136od_int @ ( bitM @ M ) @ ( bit0 @ one ) )
= ( product_Pair_int_int @ ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ one_one_int ) ) ).
% divmod_BitM_2_eq
thf(fact_1235_aexp_Osize_I4_J,axiom,
! [X1: int] :
( ( size_size_aexp @ ( n @ X1 ) )
= zero_zero_nat ) ).
% aexp.size(4)
thf(fact_1236_eval__nat__numeral_I2_J,axiom,
! [N: num] :
( ( numeral_numeral_nat @ ( bit0 @ N ) )
= ( suc @ ( numeral_numeral_nat @ ( bitM @ N ) ) ) ) ).
% eval_nat_numeral(2)
thf(fact_1237_one__plus__BitM,axiom,
! [N: num] :
( ( plus_plus_num @ one @ ( bitM @ N ) )
= ( bit0 @ N ) ) ).
% one_plus_BitM
thf(fact_1238_BitM__plus__one,axiom,
! [N: num] :
( ( plus_plus_num @ ( bitM @ N ) @ one )
= ( bit0 @ N ) ) ).
% BitM_plus_one
thf(fact_1239_aexp_Osize_I5_J,axiom,
! [X2: list_char] :
( ( size_size_aexp @ ( v @ X2 ) )
= zero_zero_nat ) ).
% aexp.size(5)
thf(fact_1240_bezw__0,axiom,
! [X: nat] :
( ( bezw @ X @ zero_zero_nat )
= ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) ).
% bezw_0
thf(fact_1241_even__diff__nat,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) )
= ( ( ord_less_nat @ M @ N )
| ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).
% even_diff_nat
thf(fact_1242_odd__Suc__minus__one,axiom,
! [N: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
% odd_Suc_minus_one
thf(fact_1243_nat__dvd__1__iff__1,axiom,
! [M: nat] :
( ( dvd_dvd_nat @ M @ one_one_nat )
= ( M = one_one_nat ) ) ).
% nat_dvd_1_iff_1
thf(fact_1244_dvd__1__left,axiom,
! [K: nat] : ( dvd_dvd_nat @ ( suc @ zero_zero_nat ) @ K ) ).
% dvd_1_left
thf(fact_1245_dvd__1__iff__1,axiom,
! [M: nat] :
( ( dvd_dvd_nat @ M @ ( suc @ zero_zero_nat ) )
= ( M
= ( suc @ zero_zero_nat ) ) ) ).
% dvd_1_iff_1
thf(fact_1246_nat__dvd__not__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_nat @ M @ N )
=> ~ ( dvd_dvd_nat @ N @ M ) ) ) ).
% nat_dvd_not_less
thf(fact_1247_dvd__pos__nat,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( dvd_dvd_nat @ M @ N )
=> ( ord_less_nat @ zero_zero_nat @ M ) ) ) ).
% dvd_pos_nat
thf(fact_1248_dvd__minus__self,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) )
= ( ( ord_less_nat @ N @ M )
| ( dvd_dvd_nat @ M @ N ) ) ) ).
% dvd_minus_self
thf(fact_1249_less__eq__dvd__minus,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( dvd_dvd_nat @ M @ N )
= ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) ) ) ) ).
% less_eq_dvd_minus
thf(fact_1250_dvd__diffD1,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) )
=> ( ( dvd_dvd_nat @ K @ M )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( dvd_dvd_nat @ K @ N ) ) ) ) ).
% dvd_diffD1
thf(fact_1251_dvd__diffD,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) )
=> ( ( dvd_dvd_nat @ K @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( dvd_dvd_nat @ K @ M ) ) ) ) ).
% dvd_diffD
thf(fact_1252_dvd__imp__le,axiom,
! [K: nat,N: nat] :
( ( dvd_dvd_nat @ K @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ) ).
% dvd_imp_le
thf(fact_1253_gcd__nat_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
=> ( A = zero_zero_nat ) ) ).
% gcd_nat.extremum_uniqueI
thf(fact_1254_gcd__nat_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ( dvd_dvd_nat @ A @ zero_zero_nat )
& ( A != zero_zero_nat ) ) ) ).
% gcd_nat.not_eq_extremum
thf(fact_1255_gcd__nat_Oextremum__unique,axiom,
! [A: nat] :
( ( dvd_dvd_nat @ zero_zero_nat @ A )
= ( A = zero_zero_nat ) ) ).
% gcd_nat.extremum_unique
thf(fact_1256_gcd__nat_Oextremum__strict,axiom,
! [A: nat] :
~ ( ( dvd_dvd_nat @ zero_zero_nat @ A )
& ( zero_zero_nat != A ) ) ).
% gcd_nat.extremum_strict
thf(fact_1257_gcd__nat_Oextremum,axiom,
! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).
% gcd_nat.extremum
thf(fact_1258_dvd__antisym,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ M @ N )
=> ( ( dvd_dvd_nat @ N @ M )
=> ( M = N ) ) ) ).
% dvd_antisym
% 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__Compiler__Oinstr_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Compiler__Oinstr_J_T,axiom,
! [X: list_instr,Y: list_instr] :
( ( if_list_instr @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Compiler__Oinstr_J_T,axiom,
! [X: list_instr,Y: list_instr] :
( ( if_list_instr @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
? [K4: nat] :
( ( ord_less_nat @ K4 @ ( size_s229347708945616566st_int @ cfsa ) )
& ( bpred @ ( produc4047900504771817624_o_int @ b @ ( product_Pair_o_int @ $false @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ ( ccomp @ c_1 ) ) ) @ one_one_int ) ) ) @ ( produc5086643055186798020st_int @ ( minus_minus_int @ ( produc8941309049508465880st_int @ ( nth_Pr8327173284404383691st_int @ cfsa @ zero_zero_nat ) ) @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ nil_instr ) ) ) @ ( produc1296517004818028822st_int @ ( nth_Pr8327173284404383691st_int @ cfsa @ zero_zero_nat ) ) ) @ ( produc5086643055186798020st_int @ ( minus_minus_int @ ( produc8941309049508465880st_int @ ( nth_Pr8327173284404383691st_int @ cfsa @ K4 ) ) @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ nil_instr ) ) ) @ ( produc1296517004818028822st_int @ ( nth_Pr8327173284404383691st_int @ cfsa @ K4 ) ) ) ) ) ).
%------------------------------------------------------------------------------