TPTP Problem File: SLH0801^1.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/0005_Compiler/prob_00737_033463__5956380_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1418 ( 629 unt; 146 typ; 0 def)
% Number of atoms : 3363 (1433 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 10519 ( 318 ~; 84 |; 218 &;8598 @)
% ( 0 <=>;1301 =>; 0 <=; 0 <~>)
% Maximal formula depth : 30 ( 6 avg)
% Number of types : 26 ( 25 usr)
% Number of type conns : 643 ( 643 >; 0 *; 0 +; 0 <<)
% Number of symbols : 124 ( 121 usr; 10 con; 0-4 aty)
% Number of variables : 3645 ( 178 ^;3299 !; 168 ?;3645 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 16:43:13.060
%------------------------------------------------------------------------------
% Could-be-implicit typings (25)
thf(ty_n_t__Product____Type__Oprod_It__Compiler__Oinstr_Mt__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,
produc5995290525303592096st_int: $tType ).
thf(ty_n_t__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,
produc4189061985984568957nt_int: $tType ).
thf(ty_n_t__Set__Oset_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,
set_Pr2116125220558876360st_int: $tType ).
thf(ty_n_t__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,
produc6425607678544837394st_int: $tType ).
thf(ty_n_t__Set__Oset_It__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,
set_Pr5580431329626101545st_int: $tType ).
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,
produc369741286924889929st_int: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__Int__Oint_J_J,type,
set_Pr5290250543098163609nt_int: $tType ).
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,
produc4435102495419491129nt_int: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__BExp__Obexp_Mt__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J_J,type,
produc1897111610453708512_o_int: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
set_Pr958786334691620121nt_int: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J_J,type,
set_Pr8834758594704517033_o_int: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
product_prod_nat_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
product_prod_int_int: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__String__Ochar_J_J,type,
list_list_char: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J,type,
product_prod_o_int: $tType ).
thf(ty_n_t__List__Olist_It__Compiler__Oinstr_J,type,
list_instr: $tType ).
thf(ty_n_t__List__Olist_It__String__Ochar_J,type,
list_char: $tType ).
thf(ty_n_t__List__Olist_It__Int__Oint_J,type,
list_int: $tType ).
thf(ty_n_t__Compiler__Oinstr,type,
instr: $tType ).
thf(ty_n_t__String__Ochar,type,
char: $tType ).
thf(ty_n_t__BExp__Obexp,type,
bexp: $tType ).
thf(ty_n_t__AExp__Oaexp,type,
aexp: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
thf(ty_n_t__Com__Ocom,type,
com: $tType ).
% Explicit typings (121)
thf(sy_c_AExp_Oaval,type,
aval: aexp > ( list_char > int ) > int ).
thf(sy_c_BExp_Obexp_OAnd,type,
and: bexp > bexp > bexp ).
thf(sy_c_BExp_Obexp_Osize__bexp,type,
size_bexp: bexp > nat ).
thf(sy_c_BExp_Obval,type,
bval: bexp > ( list_char > int ) > $o ).
thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001t__Int__Oint_001t__Int__Oint,type,
bNF_Ca1965613569405424510nt_int: set_Pr958786334691620121nt_int > ( int > int ) > $o ).
thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001t__Int__Oint_001t__Nat__Onat,type,
bNF_Ca1968104039914474786nt_nat: set_Pr958786334691620121nt_int > ( int > nat ) > $o ).
thf(sy_c_Com_Ocom_OAssign,type,
assign: list_char > aexp > com ).
thf(sy_c_Com_Ocom_OIf,type,
if: bexp > com > com > com ).
thf(sy_c_Com_Ocom_OSKIP,type,
skip: com ).
thf(sy_c_Com_Ocom_Osize__com,type,
size_com: com > nat ).
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,
inth_list_char: list_list_char > int > list_char ).
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_Fun_Ofun__upd_001t__List__Olist_It__String__Ochar_J_001t__Int__Oint,type,
fun_up8169409267500384524ar_int: ( list_char > int ) > list_char > int > list_char > int ).
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_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_Otimes__class_Otimes_001t__Int__Oint,type,
times_times_int: int > int > int ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_If_001t__Int__Oint,type,
if_int: $o > int > int > int ).
thf(sy_c_If_001t__List__Olist_It__Int__Oint_J,type,
if_list_int: $o > list_int > list_int > list_int ).
thf(sy_c_If_001t__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,
append_list_char: list_list_char > list_list_char > list_list_char ).
thf(sy_c_List_Obind_001t__Compiler__Oinstr_001t__Compiler__Oinstr,type,
bind_instr_instr: list_instr > ( instr > list_instr ) > list_instr ).
thf(sy_c_List_Obind_001t__Int__Oint_001t__Compiler__Oinstr,type,
bind_int_instr: list_int > ( int > list_instr ) > list_instr ).
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__String__Ochar_J,type,
cons_list_char: list_char > list_list_char > list_list_char ).
thf(sy_c_List_Olist_Omap_001t__Compiler__Oinstr_001t__Compiler__Oinstr,type,
map_instr_instr: ( instr > instr ) > list_instr > list_instr ).
thf(sy_c_List_Olist_Omap_001t__Compiler__Oinstr_001t__Int__Oint,type,
map_instr_int: ( instr > int ) > list_instr > list_int ).
thf(sy_c_List_Olist_Omap_001t__Compiler__Oinstr_001t__List__Olist_It__String__Ochar_J,type,
map_instr_list_char: ( instr > list_char ) > list_instr > list_list_char ).
thf(sy_c_List_Olist_Omap_001t__Int__Oint_001t__Compiler__Oinstr,type,
map_int_instr: ( int > instr ) > list_int > list_instr ).
thf(sy_c_List_Olist_Omap_001t__Int__Oint_001t__Int__Oint,type,
map_int_int: ( int > int ) > list_int > list_int ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__String__Ochar_J_001t__Compiler__Oinstr,type,
map_list_char_instr: ( list_char > instr ) > list_list_char > list_instr ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__String__Ochar_J_001t__List__Olist_It__String__Ochar_J,type,
map_li116305933131242120t_char: ( list_char > list_char ) > list_list_char > list_list_char ).
thf(sy_c_List_Oupto__aux,type,
upto_aux: int > int > list_int > list_int ).
thf(sy_c_List_Oupto__rel,type,
upto_rel: product_prod_int_int > product_prod_int_int > $o ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_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_Osemiring__1__class_Oof__nat__aux_001t__Int__Oint,type,
semiri8420488043553186161ux_int: ( int > int ) > nat > int > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Nat__Onat,type,
semiri8422978514062236437ux_nat: ( nat > nat ) > nat > nat > 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__String__Ochar,type,
size_size_char: char > 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__eq_001_062_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_M_062_It__Int__Oint_M_Eo_J_J,type,
ord_le8207994782381187918_int_o: ( ( list_char > int ) > int > $o ) > ( ( list_char > int ) > int > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_M_062_It__List__Olist_It__Int__Oint_J_M_Eo_J_J,type,
ord_le4781006829409452478_int_o: ( ( list_char > int ) > list_int > $o ) > ( ( list_char > int ) > list_int > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_M_062_It__Int__Oint_M_Eo_J_J,type,
ord_less_eq_o_int_o: ( $o > int > $o ) > ( $o > int > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Int__Oint_M_062_It__Int__Oint_M_Eo_J_J,type,
ord_le6741204236512500942_int_o: ( int > int > $o ) > ( int > int > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Int__Oint_M_062_It__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J_M_Eo_J_J,type,
ord_le8490654098287043871_int_o: ( int > produc369741286924889929st_int > $o ) > ( int > produc369741286924889929st_int > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J_J,type,
ord_le1598226405681992910_int_o: ( product_prod_int_int > product_prod_int_int > $o ) > ( product_prod_int_int > product_prod_int_int > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J,type,
ord_le8369615600986905444_int_o: ( product_prod_int_int > $o ) > ( product_prod_int_int > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__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_M_062_It__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_M_Eo_J_J,type,
ord_le1819137518486948652_int_o: ( produc4189061985984568957nt_int > produc4189061985984568957nt_int > $o ) > ( produc4189061985984568957nt_int > produc4189061985984568957nt_int > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__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_M_Eo_J,type,
ord_le3703957343819488690_int_o: ( produc4189061985984568957nt_int > $o ) > ( produc4189061985984568957nt_int > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__Int__Oint_J_J,type,
ord_le5108646760900237625nt_int: set_Pr5290250543098163609nt_int > set_Pr5290250543098163609nt_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__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,
ord_le5738032636065246921st_int: set_Pr5580431329626101545st_int > set_Pr5580431329626101545st_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J_J,type,
ord_le3982295392052344841_o_int: set_Pr8834758594704517033_o_int > set_Pr8834758594704517033_o_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
ord_le2843351958646193337nt_int: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_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,
ord_le8444298136753458984st_int: set_Pr2116125220558876360st_int > set_Pr2116125220558876360st_int > $o ).
thf(sy_c_Parity_Ounique__euclidean__semiring__with__nat__division__class_Odivides__aux_001t__Int__Oint,type,
unique5329631941980267465ux_int: product_prod_int_int > $o ).
thf(sy_c_Parity_Ounique__euclidean__semiring__with__nat__division__class_Odivides__aux_001t__Nat__Onat,type,
unique5332122412489317741ux_nat: product_prod_nat_nat > $o ).
thf(sy_c_Product__Type_OPair_001_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_001t__Int__Oint,type,
produc5790713362662368625nt_int: ( list_char > int ) > int > produc4435102495419491129nt_int ).
thf(sy_c_Product__Type_OPair_001_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_001t__List__Olist_It__Int__Oint_J,type,
produc8650753666468850689st_int: ( list_char > int ) > list_int > produc369741286924889929st_int ).
thf(sy_c_Product__Type_OPair_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__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__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__Nat__Onat_001t__Nat__Onat,type,
product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).
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__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_Star_Ostar_001t__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__Int__Oint_J,type,
star_P2206551578999486376nt_int: ( produc4435102495419491129nt_int > produc4435102495419491129nt_int > $o ) > produc4435102495419491129nt_int > produc4435102495419491129nt_int > $o ).
thf(sy_c_Star_Ostar_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,
star_P8562542470885916728st_int: ( produc369741286924889929st_int > produc369741286924889929st_int > $o ) > produc369741286924889929st_int > produc369741286924889929st_int > $o ).
thf(sy_c_Star_Ostar_001t__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J,type,
star_P7760428584271161284_o_int: ( product_prod_o_int > product_prod_o_int > $o ) > product_prod_o_int > product_prod_o_int > $o ).
thf(sy_c_Star_Ostar_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
star_P3839882803018024nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > product_prod_int_int > product_prod_int_int > $o ).
thf(sy_c_Star_Ostar_001t__Product____Type__Oprod_It__Int__Oint_Mt__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J_J,type,
star_P707599355569300323st_int: ( produc6425607678544837394st_int > produc6425607678544837394st_int > $o ) > produc6425607678544837394st_int > produc6425607678544837394st_int > $o ).
thf(sy_c_String_Ochar_Osize__char,type,
size_char: char > nat ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
accp_P1096762738010456898nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > product_prod_int_int > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__List__Olist_It__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_Wellfounded_Omeasure_001t__Int__Oint,type,
measure_int: ( int > nat ) > set_Pr958786334691620121nt_int ).
thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__Int__Oint_J,type,
member3599195040465536738nt_int: produc4435102495419491129nt_int > set_Pr5290250543098163609nt_int > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J,type,
member6299553290643319410st_int: produc369741286924889929st_int > set_Pr5580431329626101545st_int > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J,type,
member7847949116333733898_o_int: product_prod_o_int > set_Pr8834758594704517033_o_int > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
member5262025264175285858nt_int: product_prod_int_int > set_Pr958786334691620121nt_int > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Int__Oint_Mt__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J_J,type,
member9100552558438320297st_int: produc6425607678544837394st_int > set_Pr2116125220558876360st_int > $o ).
thf(sy_v_a____,type,
a: aexp ).
thf(sy_v_sa____,type,
sa: list_char > int ).
thf(sy_v_stka____,type,
stka: list_int ).
thf(sy_v_x____,type,
x: list_char ).
% Relevant facts (1264)
thf(fact_0_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 ) ) )
=> ( ! [A: int,Aa: list_char > int,B: list_int] : ( P @ A @ Aa @ B @ A @ Aa @ B )
=> ( ! [A: int,Aa: list_char > int,B: list_int,Ab: int,Ac: list_char > int,Ba: list_int,Ad: int,Ae: list_char > int,Bb: list_int] :
( ( exec1 @ Pa @ ( produc5086643055186798020st_int @ A @ ( produc8650753666468850689st_int @ Aa @ B ) ) @ ( 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 @ A @ Aa @ B @ Ad @ Ae @ Bb ) ) ) )
=> ( P @ X1a @ X1b @ X1c @ X2a @ X2b @ X2c ) ) ) ) ).
% exec_induct
thf(fact_1_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_2_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_3_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_4_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_5_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_6_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_7_com_Oinject_I1_J,axiom,
! [X21: list_char,X22: aexp,Y21: list_char,Y22: aexp] :
( ( ( assign @ X21 @ X22 )
= ( assign @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% com.inject(1)
thf(fact_8_star__step1,axiom,
! [R: produc6425607678544837394st_int > produc6425607678544837394st_int > $o,X: produc6425607678544837394st_int,Y: produc6425607678544837394st_int] :
( ( R @ X @ Y )
=> ( star_P707599355569300323st_int @ R @ X @ Y ) ) ).
% star_step1
thf(fact_9_fun__upd__upd,axiom,
! [F: list_char > int,X: list_char,Y: int,Z: int] :
( ( fun_up8169409267500384524ar_int @ ( fun_up8169409267500384524ar_int @ F @ X @ Y ) @ X @ Z )
= ( fun_up8169409267500384524ar_int @ F @ X @ Z ) ) ).
% fun_upd_upd
thf(fact_10_fun__upd__triv,axiom,
! [F: list_char > int,X: list_char] :
( ( fun_up8169409267500384524ar_int @ F @ X @ ( F @ X ) )
= F ) ).
% fun_upd_triv
thf(fact_11_fun__upd__apply,axiom,
( fun_up8169409267500384524ar_int
= ( ^ [F2: list_char > int,X2: list_char,Y2: int,Z2: list_char] : ( if_int @ ( Z2 = X2 ) @ Y2 @ ( F2 @ Z2 ) ) ) ) ).
% fun_upd_apply
thf(fact_12_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1316708129612266289at_nat @ M )
= ( semiri1316708129612266289at_nat @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_13_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_14_prod_Oinject,axiom,
! [X1: $o,X23: int,Y1: $o,Y23: int] :
( ( ( product_Pair_o_int @ X1 @ X23 )
= ( product_Pair_o_int @ Y1 @ Y23 ) )
= ( ( X1 = Y1 )
& ( X23 = Y23 ) ) ) ).
% prod.inject
thf(fact_15_prod_Oinject,axiom,
! [X1: int,X23: produc369741286924889929st_int,Y1: int,Y23: produc369741286924889929st_int] :
( ( ( produc5086643055186798020st_int @ X1 @ X23 )
= ( produc5086643055186798020st_int @ Y1 @ Y23 ) )
= ( ( X1 = Y1 )
& ( X23 = Y23 ) ) ) ).
% prod.inject
thf(fact_16_prod_Oinject,axiom,
! [X1: int,X23: int,Y1: int,Y23: int] :
( ( ( product_Pair_int_int @ X1 @ X23 )
= ( product_Pair_int_int @ Y1 @ Y23 ) )
= ( ( X1 = Y1 )
& ( X23 = Y23 ) ) ) ).
% prod.inject
thf(fact_17_prod_Oinject,axiom,
! [X1: list_char > int,X23: list_int,Y1: list_char > int,Y23: list_int] :
( ( ( produc8650753666468850689st_int @ X1 @ X23 )
= ( produc8650753666468850689st_int @ Y1 @ Y23 ) )
= ( ( X1 = Y1 )
& ( X23 = Y23 ) ) ) ).
% prod.inject
thf(fact_18_prod_Oinject,axiom,
! [X1: list_char > int,X23: int,Y1: list_char > int,Y23: int] :
( ( ( produc5790713362662368625nt_int @ X1 @ X23 )
= ( produc5790713362662368625nt_int @ Y1 @ Y23 ) )
= ( ( X1 = Y1 )
& ( X23 = Y23 ) ) ) ).
% prod.inject
thf(fact_19_old_Oprod_Oinject,axiom,
! [A2: $o,B2: int,A3: $o,B3: int] :
( ( ( product_Pair_o_int @ A2 @ B2 )
= ( product_Pair_o_int @ A3 @ B3 ) )
= ( ( A2 = A3 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_20_old_Oprod_Oinject,axiom,
! [A2: int,B2: produc369741286924889929st_int,A3: int,B3: produc369741286924889929st_int] :
( ( ( produc5086643055186798020st_int @ A2 @ B2 )
= ( produc5086643055186798020st_int @ A3 @ B3 ) )
= ( ( A2 = A3 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_21_old_Oprod_Oinject,axiom,
! [A2: int,B2: int,A3: int,B3: int] :
( ( ( product_Pair_int_int @ A2 @ B2 )
= ( product_Pair_int_int @ A3 @ B3 ) )
= ( ( A2 = A3 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_22_old_Oprod_Oinject,axiom,
! [A2: list_char > int,B2: list_int,A3: list_char > int,B3: list_int] :
( ( ( produc8650753666468850689st_int @ A2 @ B2 )
= ( produc8650753666468850689st_int @ A3 @ B3 ) )
= ( ( A2 = A3 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_23_old_Oprod_Oinject,axiom,
! [A2: list_char > int,B2: int,A3: list_char > int,B3: int] :
( ( ( produc5790713362662368625nt_int @ A2 @ B2 )
= ( produc5790713362662368625nt_int @ A3 @ B3 ) )
= ( ( A2 = A3 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_24_iexec_Ocases,axiom,
! [X: produc5995290525303592096st_int] :
~ ! [Ins: instr,I: int,S: list_char > int,Stk: list_int] :
( X
!= ( produc1484136438817787418st_int @ Ins @ ( produc5086643055186798020st_int @ I @ ( produc8650753666468850689st_int @ S @ Stk ) ) ) ) ).
% iexec.cases
thf(fact_25_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_26_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_27_Pair__inject,axiom,
! [A2: $o,B2: int,A3: $o,B3: int] :
( ( ( product_Pair_o_int @ A2 @ B2 )
= ( product_Pair_o_int @ A3 @ B3 ) )
=> ~ ( ( A2 = A3 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_28_Pair__inject,axiom,
! [A2: int,B2: produc369741286924889929st_int,A3: int,B3: produc369741286924889929st_int] :
( ( ( produc5086643055186798020st_int @ A2 @ B2 )
= ( produc5086643055186798020st_int @ A3 @ B3 ) )
=> ~ ( ( A2 = A3 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_29_Pair__inject,axiom,
! [A2: int,B2: int,A3: int,B3: int] :
( ( ( product_Pair_int_int @ A2 @ B2 )
= ( product_Pair_int_int @ A3 @ B3 ) )
=> ~ ( ( A2 = A3 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_30_Pair__inject,axiom,
! [A2: list_char > int,B2: list_int,A3: list_char > int,B3: list_int] :
( ( ( produc8650753666468850689st_int @ A2 @ B2 )
= ( produc8650753666468850689st_int @ A3 @ B3 ) )
=> ~ ( ( A2 = A3 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_31_Pair__inject,axiom,
! [A2: list_char > int,B2: int,A3: list_char > int,B3: int] :
( ( ( produc5790713362662368625nt_int @ A2 @ B2 )
= ( produc5790713362662368625nt_int @ A3 @ B3 ) )
=> ~ ( ( A2 = A3 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_32_prod__cases,axiom,
! [P: product_prod_o_int > $o,P2: product_prod_o_int] :
( ! [A: $o,B: int] : ( P @ ( product_Pair_o_int @ A @ B ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_33_prod__cases,axiom,
! [P: produc6425607678544837394st_int > $o,P2: produc6425607678544837394st_int] :
( ! [A: int,B: produc369741286924889929st_int] : ( P @ ( produc5086643055186798020st_int @ A @ B ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_34_prod__cases,axiom,
! [P: product_prod_int_int > $o,P2: product_prod_int_int] :
( ! [A: int,B: int] : ( P @ ( product_Pair_int_int @ A @ B ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_35_prod__cases,axiom,
! [P: produc369741286924889929st_int > $o,P2: produc369741286924889929st_int] :
( ! [A: list_char > int,B: list_int] : ( P @ ( produc8650753666468850689st_int @ A @ B ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_36_prod__cases,axiom,
! [P: produc4435102495419491129nt_int > $o,P2: produc4435102495419491129nt_int] :
( ! [A: list_char > int,B: int] : ( P @ ( produc5790713362662368625nt_int @ A @ B ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_37_surj__pair,axiom,
! [P2: product_prod_o_int] :
? [X3: $o,Y3: int] :
( P2
= ( product_Pair_o_int @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_38_surj__pair,axiom,
! [P2: produc6425607678544837394st_int] :
? [X3: int,Y3: produc369741286924889929st_int] :
( P2
= ( produc5086643055186798020st_int @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_39_surj__pair,axiom,
! [P2: product_prod_int_int] :
? [X3: int,Y3: int] :
( P2
= ( product_Pair_int_int @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_40_surj__pair,axiom,
! [P2: produc369741286924889929st_int] :
? [X3: list_char > int,Y3: list_int] :
( P2
= ( produc8650753666468850689st_int @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_41_surj__pair,axiom,
! [P2: produc4435102495419491129nt_int] :
? [X3: list_char > int,Y3: int] :
( P2
= ( produc5790713362662368625nt_int @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_42_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_o_int] :
~ ! [A: $o,B: int] :
( Y
!= ( product_Pair_o_int @ A @ B ) ) ).
% old.prod.exhaust
thf(fact_43_old_Oprod_Oexhaust,axiom,
! [Y: produc6425607678544837394st_int] :
~ ! [A: int,B: produc369741286924889929st_int] :
( Y
!= ( produc5086643055186798020st_int @ A @ B ) ) ).
% old.prod.exhaust
thf(fact_44_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_int_int] :
~ ! [A: int,B: int] :
( Y
!= ( product_Pair_int_int @ A @ B ) ) ).
% old.prod.exhaust
thf(fact_45_old_Oprod_Oexhaust,axiom,
! [Y: produc369741286924889929st_int] :
~ ! [A: list_char > int,B: list_int] :
( Y
!= ( produc8650753666468850689st_int @ A @ B ) ) ).
% old.prod.exhaust
thf(fact_46_old_Oprod_Oexhaust,axiom,
! [Y: produc4435102495419491129nt_int] :
~ ! [A: list_char > int,B: int] :
( Y
!= ( produc5790713362662368625nt_int @ A @ B ) ) ).
% old.prod.exhaust
thf(fact_47_size__neq__size__imp__neq,axiom,
! [X: char,Y: char] :
( ( ( size_size_char @ X )
!= ( size_size_char @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_48_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_49_size__neq__size__imp__neq,axiom,
! [X: list_list_char,Y: list_list_char] :
( ( ( size_s356637359517785349t_char @ X )
!= ( size_s356637359517785349t_char @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_50_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_51_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_52_fun__upd__idem__iff,axiom,
! [F: list_char > int,X: list_char,Y: int] :
( ( ( fun_up8169409267500384524ar_int @ F @ X @ Y )
= F )
= ( ( F @ X )
= Y ) ) ).
% fun_upd_idem_iff
thf(fact_53_fun__upd__twist,axiom,
! [A2: list_char,C: list_char,M: list_char > int,B2: int,D: int] :
( ( A2 != C )
=> ( ( fun_up8169409267500384524ar_int @ ( fun_up8169409267500384524ar_int @ M @ A2 @ B2 ) @ C @ D )
= ( fun_up8169409267500384524ar_int @ ( fun_up8169409267500384524ar_int @ M @ C @ D ) @ A2 @ B2 ) ) ) ).
% fun_upd_twist
thf(fact_54_fun__upd__other,axiom,
! [Z: list_char,X: list_char,F: list_char > int,Y: int] :
( ( Z != X )
=> ( ( fun_up8169409267500384524ar_int @ F @ X @ Y @ Z )
= ( F @ Z ) ) ) ).
% fun_upd_other
thf(fact_55_fun__upd__same,axiom,
! [F: list_char > int,X: list_char,Y: int] :
( ( fun_up8169409267500384524ar_int @ F @ X @ Y @ X )
= Y ) ).
% fun_upd_same
thf(fact_56_fun__upd__idem,axiom,
! [F: list_char > int,X: list_char,Y: int] :
( ( ( F @ X )
= Y )
=> ( ( fun_up8169409267500384524ar_int @ F @ X @ Y )
= F ) ) ).
% fun_upd_idem
thf(fact_57_fun__upd__eqD,axiom,
! [F: list_char > int,X: list_char,Y: int,G: list_char > int,Z: int] :
( ( ( fun_up8169409267500384524ar_int @ F @ X @ Y )
= ( fun_up8169409267500384524ar_int @ G @ X @ Z ) )
=> ( Y = Z ) ) ).
% fun_upd_eqD
thf(fact_58_fun__upd__def,axiom,
( fun_up8169409267500384524ar_int
= ( ^ [F2: list_char > int,A4: list_char,B4: int,X2: list_char] : ( if_int @ ( X2 = A4 ) @ B4 @ ( F2 @ X2 ) ) ) ) ).
% fun_upd_def
thf(fact_59_star__trans,axiom,
! [R: produc6425607678544837394st_int > produc6425607678544837394st_int > $o,X: produc6425607678544837394st_int,Y: produc6425607678544837394st_int,Z: produc6425607678544837394st_int] :
( ( star_P707599355569300323st_int @ R @ X @ Y )
=> ( ( star_P707599355569300323st_int @ R @ Y @ Z )
=> ( star_P707599355569300323st_int @ R @ X @ Z ) ) ) ).
% star_trans
thf(fact_60_star_Osimps,axiom,
( star_P707599355569300323st_int
= ( ^ [R2: produc6425607678544837394st_int > produc6425607678544837394st_int > $o,A1: produc6425607678544837394st_int,A22: produc6425607678544837394st_int] :
( ? [X2: produc6425607678544837394st_int] :
( ( A1 = X2 )
& ( A22 = X2 ) )
| ? [X2: produc6425607678544837394st_int,Y2: produc6425607678544837394st_int,Z2: produc6425607678544837394st_int] :
( ( A1 = X2 )
& ( A22 = Z2 )
& ( R2 @ X2 @ Y2 )
& ( star_P707599355569300323st_int @ R2 @ Y2 @ Z2 ) ) ) ) ) ).
% star.simps
thf(fact_61_star_Ocases,axiom,
! [R: produc6425607678544837394st_int > produc6425607678544837394st_int > $o,A12: produc6425607678544837394st_int,A23: produc6425607678544837394st_int] :
( ( star_P707599355569300323st_int @ R @ A12 @ A23 )
=> ( ( A23 != A12 )
=> ~ ! [Y3: produc6425607678544837394st_int] :
( ( R @ A12 @ Y3 )
=> ~ ( star_P707599355569300323st_int @ R @ Y3 @ A23 ) ) ) ) ).
% star.cases
thf(fact_62_star_Ostep,axiom,
! [R: produc6425607678544837394st_int > produc6425607678544837394st_int > $o,X: produc6425607678544837394st_int,Y: produc6425607678544837394st_int,Z: produc6425607678544837394st_int] :
( ( R @ X @ Y )
=> ( ( star_P707599355569300323st_int @ R @ Y @ Z )
=> ( star_P707599355569300323st_int @ R @ X @ Z ) ) ) ).
% star.step
thf(fact_63_star_Orefl,axiom,
! [R: produc6425607678544837394st_int > produc6425607678544837394st_int > $o,X: produc6425607678544837394st_int] : ( star_P707599355569300323st_int @ R @ X @ X ) ).
% star.refl
thf(fact_64_prod__induct3,axiom,
! [P: produc6425607678544837394st_int > $o,X: produc6425607678544837394st_int] :
( ! [A: int,B: list_char > int,C2: list_int] : ( P @ ( produc5086643055186798020st_int @ A @ ( produc8650753666468850689st_int @ B @ C2 ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_65_prod__cases3,axiom,
! [Y: produc6425607678544837394st_int] :
~ ! [A: int,B: list_char > int,C2: list_int] :
( Y
!= ( produc5086643055186798020st_int @ A @ ( produc8650753666468850689st_int @ B @ C2 ) ) ) ).
% prod_cases3
thf(fact_66_star__induct,axiom,
! [R: product_prod_o_int > product_prod_o_int > $o,X1a: $o,X1b: int,X2a: $o,X2b: int,P: $o > int > $o > int > $o] :
( ( star_P7760428584271161284_o_int @ R @ ( product_Pair_o_int @ X1a @ X1b ) @ ( product_Pair_o_int @ X2a @ X2b ) )
=> ( ! [A: $o,B: int] : ( P @ A @ B @ A @ B )
=> ( ! [A: $o,B: int,Aa: $o,Ba: int,Ab: $o,Bb: int] :
( ( R @ ( product_Pair_o_int @ A @ B ) @ ( product_Pair_o_int @ Aa @ Ba ) )
=> ( ( star_P7760428584271161284_o_int @ R @ ( product_Pair_o_int @ Aa @ Ba ) @ ( product_Pair_o_int @ Ab @ Bb ) )
=> ( ( P @ Aa @ Ba @ Ab @ Bb )
=> ( P @ A @ B @ Ab @ Bb ) ) ) )
=> ( P @ X1a @ X1b @ X2a @ X2b ) ) ) ) ).
% star_induct
thf(fact_67_star__induct,axiom,
! [R: produc6425607678544837394st_int > produc6425607678544837394st_int > $o,X1a: int,X1b: produc369741286924889929st_int,X2a: int,X2b: produc369741286924889929st_int,P: int > produc369741286924889929st_int > int > produc369741286924889929st_int > $o] :
( ( star_P707599355569300323st_int @ R @ ( produc5086643055186798020st_int @ X1a @ X1b ) @ ( produc5086643055186798020st_int @ X2a @ X2b ) )
=> ( ! [A: int,B: produc369741286924889929st_int] : ( P @ A @ B @ A @ B )
=> ( ! [A: int,B: produc369741286924889929st_int,Aa: int,Ba: produc369741286924889929st_int,Ab: int,Bb: produc369741286924889929st_int] :
( ( R @ ( produc5086643055186798020st_int @ A @ B ) @ ( produc5086643055186798020st_int @ Aa @ Ba ) )
=> ( ( star_P707599355569300323st_int @ R @ ( produc5086643055186798020st_int @ Aa @ Ba ) @ ( produc5086643055186798020st_int @ Ab @ Bb ) )
=> ( ( P @ Aa @ Ba @ Ab @ Bb )
=> ( P @ A @ B @ Ab @ Bb ) ) ) )
=> ( P @ X1a @ X1b @ X2a @ X2b ) ) ) ) ).
% star_induct
thf(fact_68_star__induct,axiom,
! [R: product_prod_int_int > product_prod_int_int > $o,X1a: int,X1b: int,X2a: int,X2b: int,P: int > int > int > int > $o] :
( ( star_P3839882803018024nt_int @ R @ ( product_Pair_int_int @ X1a @ X1b ) @ ( product_Pair_int_int @ X2a @ X2b ) )
=> ( ! [A: int,B: int] : ( P @ A @ B @ A @ B )
=> ( ! [A: int,B: int,Aa: int,Ba: int,Ab: int,Bb: int] :
( ( R @ ( product_Pair_int_int @ A @ B ) @ ( product_Pair_int_int @ Aa @ Ba ) )
=> ( ( star_P3839882803018024nt_int @ R @ ( product_Pair_int_int @ Aa @ Ba ) @ ( product_Pair_int_int @ Ab @ Bb ) )
=> ( ( P @ Aa @ Ba @ Ab @ Bb )
=> ( P @ A @ B @ Ab @ Bb ) ) ) )
=> ( P @ X1a @ X1b @ X2a @ X2b ) ) ) ) ).
% star_induct
thf(fact_69_star__induct,axiom,
! [R: produc369741286924889929st_int > produc369741286924889929st_int > $o,X1a: list_char > int,X1b: list_int,X2a: list_char > int,X2b: list_int,P: ( list_char > int ) > list_int > ( list_char > int ) > list_int > $o] :
( ( star_P8562542470885916728st_int @ R @ ( produc8650753666468850689st_int @ X1a @ X1b ) @ ( produc8650753666468850689st_int @ X2a @ X2b ) )
=> ( ! [A: list_char > int,B: list_int] : ( P @ A @ B @ A @ B )
=> ( ! [A: list_char > int,B: list_int,Aa: list_char > int,Ba: list_int,Ab: list_char > int,Bb: list_int] :
( ( R @ ( produc8650753666468850689st_int @ A @ B ) @ ( produc8650753666468850689st_int @ Aa @ Ba ) )
=> ( ( star_P8562542470885916728st_int @ R @ ( produc8650753666468850689st_int @ Aa @ Ba ) @ ( produc8650753666468850689st_int @ Ab @ Bb ) )
=> ( ( P @ Aa @ Ba @ Ab @ Bb )
=> ( P @ A @ B @ Ab @ Bb ) ) ) )
=> ( P @ X1a @ X1b @ X2a @ X2b ) ) ) ) ).
% star_induct
thf(fact_70_star__induct,axiom,
! [R: produc4435102495419491129nt_int > produc4435102495419491129nt_int > $o,X1a: list_char > int,X1b: int,X2a: list_char > int,X2b: int,P: ( list_char > int ) > int > ( list_char > int ) > int > $o] :
( ( star_P2206551578999486376nt_int @ R @ ( produc5790713362662368625nt_int @ X1a @ X1b ) @ ( produc5790713362662368625nt_int @ X2a @ X2b ) )
=> ( ! [A: list_char > int,B: int] : ( P @ A @ B @ A @ B )
=> ( ! [A: list_char > int,B: int,Aa: list_char > int,Ba: int,Ab: list_char > int,Bb: int] :
( ( R @ ( produc5790713362662368625nt_int @ A @ B ) @ ( produc5790713362662368625nt_int @ Aa @ Ba ) )
=> ( ( star_P2206551578999486376nt_int @ R @ ( produc5790713362662368625nt_int @ Aa @ Ba ) @ ( produc5790713362662368625nt_int @ Ab @ Bb ) )
=> ( ( P @ Aa @ Ba @ Ab @ Bb )
=> ( P @ A @ B @ Ab @ Bb ) ) ) )
=> ( P @ X1a @ X1b @ X2a @ X2b ) ) ) ) ).
% star_induct
thf(fact_71_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_72_acomp__correct,axiom,
! [A2: aexp,S2: list_char > int,Stk2: list_int] : ( star_P707599355569300323st_int @ ( exec1 @ ( acomp @ A2 ) ) @ ( produc5086643055186798020st_int @ zero_zero_int @ ( produc8650753666468850689st_int @ S2 @ Stk2 ) ) @ ( produc5086643055186798020st_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ ( acomp @ A2 ) ) ) @ ( produc8650753666468850689st_int @ S2 @ ( cons_int @ ( aval @ A2 @ S2 ) @ Stk2 ) ) ) ) ).
% acomp_correct
thf(fact_73_pred__equals__eq2,axiom,
! [R3: set_Pr8834758594704517033_o_int,S3: set_Pr8834758594704517033_o_int] :
( ( ( ^ [X2: $o,Y2: int] : ( member7847949116333733898_o_int @ ( product_Pair_o_int @ X2 @ Y2 ) @ R3 ) )
= ( ^ [X2: $o,Y2: int] : ( member7847949116333733898_o_int @ ( product_Pair_o_int @ X2 @ Y2 ) @ S3 ) ) )
= ( R3 = S3 ) ) ).
% pred_equals_eq2
thf(fact_74_pred__equals__eq2,axiom,
! [R3: set_Pr2116125220558876360st_int,S3: set_Pr2116125220558876360st_int] :
( ( ( ^ [X2: int,Y2: produc369741286924889929st_int] : ( member9100552558438320297st_int @ ( produc5086643055186798020st_int @ X2 @ Y2 ) @ R3 ) )
= ( ^ [X2: int,Y2: produc369741286924889929st_int] : ( member9100552558438320297st_int @ ( produc5086643055186798020st_int @ X2 @ Y2 ) @ S3 ) ) )
= ( R3 = S3 ) ) ).
% pred_equals_eq2
thf(fact_75_pred__equals__eq2,axiom,
! [R3: set_Pr958786334691620121nt_int,S3: set_Pr958786334691620121nt_int] :
( ( ( ^ [X2: int,Y2: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y2 ) @ R3 ) )
= ( ^ [X2: int,Y2: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y2 ) @ S3 ) ) )
= ( R3 = S3 ) ) ).
% pred_equals_eq2
thf(fact_76_pred__equals__eq2,axiom,
! [R3: set_Pr5580431329626101545st_int,S3: set_Pr5580431329626101545st_int] :
( ( ( ^ [X2: list_char > int,Y2: list_int] : ( member6299553290643319410st_int @ ( produc8650753666468850689st_int @ X2 @ Y2 ) @ R3 ) )
= ( ^ [X2: list_char > int,Y2: list_int] : ( member6299553290643319410st_int @ ( produc8650753666468850689st_int @ X2 @ Y2 ) @ S3 ) ) )
= ( R3 = S3 ) ) ).
% pred_equals_eq2
thf(fact_77_pred__equals__eq2,axiom,
! [R3: set_Pr5290250543098163609nt_int,S3: set_Pr5290250543098163609nt_int] :
( ( ( ^ [X2: list_char > int,Y2: int] : ( member3599195040465536738nt_int @ ( produc5790713362662368625nt_int @ X2 @ Y2 ) @ R3 ) )
= ( ^ [X2: list_char > int,Y2: int] : ( member3599195040465536738nt_int @ ( produc5790713362662368625nt_int @ X2 @ Y2 ) @ S3 ) ) )
= ( R3 = S3 ) ) ).
% pred_equals_eq2
thf(fact_78_com_Osize__gen_I2_J,axiom,
! [X21: list_char,X22: aexp] :
( ( size_com @ ( assign @ X21 @ X22 ) )
= zero_zero_nat ) ).
% com.size_gen(2)
thf(fact_79_divides__aux__eq,axiom,
! [Q: int,R: int] :
( ( unique5329631941980267465ux_int @ ( product_Pair_int_int @ Q @ R ) )
= ( R = zero_zero_int ) ) ).
% divides_aux_eq
thf(fact_80_divides__aux__eq,axiom,
! [Q: nat,R: nat] :
( ( unique5332122412489317741ux_nat @ ( product_Pair_nat_nat @ Q @ R ) )
= ( R = zero_zero_nat ) ) ).
% divides_aux_eq
thf(fact_81_com_Osize_I7_J,axiom,
! [X21: list_char,X22: aexp] :
( ( size_size_com @ ( assign @ X21 @ X22 ) )
= zero_zero_nat ) ).
% com.size(7)
thf(fact_82_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_83_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_84_int__int__eq,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% int_int_eq
thf(fact_85_int__if,axiom,
! [P: $o,A2: nat,B2: nat] :
( ( P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A2 @ B2 ) )
= ( semiri1314217659103216013at_int @ A2 ) ) )
& ( ~ P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A2 @ B2 ) )
= ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% int_if
thf(fact_86_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_87_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% bot_nat_0.extremum
thf(fact_88_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_89_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_90_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_91_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_92_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_93_verit__la__disequality,axiom,
! [A2: nat,B2: nat] :
( ( A2 = B2 )
| ~ ( ord_less_eq_nat @ A2 @ B2 )
| ~ ( ord_less_eq_nat @ B2 @ A2 ) ) ).
% verit_la_disequality
thf(fact_94_verit__la__disequality,axiom,
! [A2: int,B2: int] :
( ( A2 = B2 )
| ~ ( ord_less_eq_int @ A2 @ B2 )
| ~ ( ord_less_eq_int @ B2 @ A2 ) ) ).
% verit_la_disequality
thf(fact_95_verit__comp__simplify1_I2_J,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_96_verit__comp__simplify1_I2_J,axiom,
! [A2: int] : ( ord_less_eq_int @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_97_pred__subset__eq2,axiom,
! [R3: set_Pr8834758594704517033_o_int,S3: set_Pr8834758594704517033_o_int] :
( ( ord_less_eq_o_int_o
@ ^ [X2: $o,Y2: int] : ( member7847949116333733898_o_int @ ( product_Pair_o_int @ X2 @ Y2 ) @ R3 )
@ ^ [X2: $o,Y2: int] : ( member7847949116333733898_o_int @ ( product_Pair_o_int @ X2 @ Y2 ) @ S3 ) )
= ( ord_le3982295392052344841_o_int @ R3 @ S3 ) ) ).
% pred_subset_eq2
thf(fact_98_pred__subset__eq2,axiom,
! [R3: set_Pr2116125220558876360st_int,S3: set_Pr2116125220558876360st_int] :
( ( ord_le8490654098287043871_int_o
@ ^ [X2: int,Y2: produc369741286924889929st_int] : ( member9100552558438320297st_int @ ( produc5086643055186798020st_int @ X2 @ Y2 ) @ R3 )
@ ^ [X2: int,Y2: produc369741286924889929st_int] : ( member9100552558438320297st_int @ ( produc5086643055186798020st_int @ X2 @ Y2 ) @ S3 ) )
= ( ord_le8444298136753458984st_int @ R3 @ S3 ) ) ).
% pred_subset_eq2
thf(fact_99_pred__subset__eq2,axiom,
! [R3: set_Pr958786334691620121nt_int,S3: set_Pr958786334691620121nt_int] :
( ( ord_le6741204236512500942_int_o
@ ^ [X2: int,Y2: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y2 ) @ R3 )
@ ^ [X2: int,Y2: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y2 ) @ S3 ) )
= ( ord_le2843351958646193337nt_int @ R3 @ S3 ) ) ).
% pred_subset_eq2
thf(fact_100_pred__subset__eq2,axiom,
! [R3: set_Pr5580431329626101545st_int,S3: set_Pr5580431329626101545st_int] :
( ( ord_le4781006829409452478_int_o
@ ^ [X2: list_char > int,Y2: list_int] : ( member6299553290643319410st_int @ ( produc8650753666468850689st_int @ X2 @ Y2 ) @ R3 )
@ ^ [X2: list_char > int,Y2: list_int] : ( member6299553290643319410st_int @ ( produc8650753666468850689st_int @ X2 @ Y2 ) @ S3 ) )
= ( ord_le5738032636065246921st_int @ R3 @ S3 ) ) ).
% pred_subset_eq2
thf(fact_101_pred__subset__eq2,axiom,
! [R3: set_Pr5290250543098163609nt_int,S3: set_Pr5290250543098163609nt_int] :
( ( ord_le8207994782381187918_int_o
@ ^ [X2: list_char > int,Y2: int] : ( member3599195040465536738nt_int @ ( produc5790713362662368625nt_int @ X2 @ Y2 ) @ R3 )
@ ^ [X2: list_char > int,Y2: int] : ( member3599195040465536738nt_int @ ( produc5790713362662368625nt_int @ X2 @ Y2 ) @ S3 ) )
= ( ord_le5108646760900237625nt_int @ R3 @ S3 ) ) ).
% pred_subset_eq2
thf(fact_102_nat__leq__as__int,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_leq_as_int
thf(fact_103_of__nat__mono,axiom,
! [I2: nat,J: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I2 ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).
% of_nat_mono
thf(fact_104_of__nat__mono,axiom,
! [I2: nat,J: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I2 ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).
% of_nat_mono
thf(fact_105_subrelI,axiom,
! [R: set_Pr8834758594704517033_o_int,S2: set_Pr8834758594704517033_o_int] :
( ! [X3: $o,Y3: int] :
( ( member7847949116333733898_o_int @ ( product_Pair_o_int @ X3 @ Y3 ) @ R )
=> ( member7847949116333733898_o_int @ ( product_Pair_o_int @ X3 @ Y3 ) @ S2 ) )
=> ( ord_le3982295392052344841_o_int @ R @ S2 ) ) ).
% subrelI
thf(fact_106_subrelI,axiom,
! [R: set_Pr2116125220558876360st_int,S2: set_Pr2116125220558876360st_int] :
( ! [X3: int,Y3: produc369741286924889929st_int] :
( ( member9100552558438320297st_int @ ( produc5086643055186798020st_int @ X3 @ Y3 ) @ R )
=> ( member9100552558438320297st_int @ ( produc5086643055186798020st_int @ X3 @ Y3 ) @ S2 ) )
=> ( ord_le8444298136753458984st_int @ R @ S2 ) ) ).
% subrelI
thf(fact_107_subrelI,axiom,
! [R: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int] :
( ! [X3: int,Y3: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y3 ) @ R )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y3 ) @ S2 ) )
=> ( ord_le2843351958646193337nt_int @ R @ S2 ) ) ).
% subrelI
thf(fact_108_subrelI,axiom,
! [R: set_Pr5580431329626101545st_int,S2: set_Pr5580431329626101545st_int] :
( ! [X3: list_char > int,Y3: list_int] :
( ( member6299553290643319410st_int @ ( produc8650753666468850689st_int @ X3 @ Y3 ) @ R )
=> ( member6299553290643319410st_int @ ( produc8650753666468850689st_int @ X3 @ Y3 ) @ S2 ) )
=> ( ord_le5738032636065246921st_int @ R @ S2 ) ) ).
% subrelI
thf(fact_109_subrelI,axiom,
! [R: set_Pr5290250543098163609nt_int,S2: set_Pr5290250543098163609nt_int] :
( ! [X3: list_char > int,Y3: int] :
( ( member3599195040465536738nt_int @ ( produc5790713362662368625nt_int @ X3 @ Y3 ) @ R )
=> ( member3599195040465536738nt_int @ ( produc5790713362662368625nt_int @ X3 @ Y3 ) @ S2 ) )
=> ( ord_le5108646760900237625nt_int @ R @ S2 ) ) ).
% subrelI
thf(fact_110_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_111_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_112_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_113_bot__nat__0_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_114_bot__nat__0_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
= ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_115_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_116_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_117_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_118_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_119_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_120_nat__int__comparison_I1_J,axiom,
( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
= ( ^ [A4: nat,B4: nat] :
( ( semiri1314217659103216013at_int @ A4 )
= ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_121_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_122_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_123_size__char__eq__0,axiom,
( size_size_char
= ( ^ [C3: char] : zero_zero_nat ) ) ).
% size_char_eq_0
thf(fact_124_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_125_dual__order_Orefl,axiom,
! [A2: int] : ( ord_less_eq_int @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_126_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_127_order__refl,axiom,
! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% order_refl
thf(fact_128_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_129_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_130_impossible__Cons,axiom,
! [Xs: list_list_char,Ys: list_list_char,X: list_char] :
( ( ord_less_eq_nat @ ( size_s356637359517785349t_char @ Xs ) @ ( size_s356637359517785349t_char @ Ys ) )
=> ( Xs
!= ( cons_list_char @ X @ Ys ) ) ) ).
% impossible_Cons
thf(fact_131_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_132_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_133_com_Osize_I6_J,axiom,
( ( size_size_com @ skip )
= zero_zero_nat ) ).
% com.size(6)
thf(fact_134_com_Osize__gen_I1_J,axiom,
( ( size_com @ skip )
= zero_zero_nat ) ).
% com.size_gen(1)
thf(fact_135_relChain__def,axiom,
( bNF_Ca1968104039914474786nt_nat
= ( ^ [R2: set_Pr958786334691620121nt_int,As: int > nat] :
! [I3: int,J2: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ I3 @ J2 ) @ R2 )
=> ( ord_less_eq_nat @ ( As @ I3 ) @ ( As @ J2 ) ) ) ) ) ).
% relChain_def
thf(fact_136_relChain__def,axiom,
( bNF_Ca1965613569405424510nt_int
= ( ^ [R2: set_Pr958786334691620121nt_int,As: int > int] :
! [I3: int,J2: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ I3 @ J2 ) @ R2 )
=> ( ord_less_eq_int @ ( As @ I3 ) @ ( As @ J2 ) ) ) ) ) ).
% relChain_def
thf(fact_137_exec__appendL,axiom,
! [P: list_instr,I2: int,S2: list_char > int,Stk2: list_int,I4: int,S4: list_char > int,Stk3: list_int,P3: list_instr] :
( ( star_P707599355569300323st_int @ ( exec1 @ P ) @ ( produc5086643055186798020st_int @ I2 @ ( produc8650753666468850689st_int @ S2 @ Stk2 ) ) @ ( produc5086643055186798020st_int @ I4 @ ( produc8650753666468850689st_int @ S4 @ Stk3 ) ) )
=> ( star_P707599355569300323st_int @ ( exec1 @ ( append_instr @ P3 @ P ) ) @ ( produc5086643055186798020st_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P3 ) ) @ I2 ) @ ( produc8650753666468850689st_int @ S2 @ Stk2 ) ) @ ( produc5086643055186798020st_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P3 ) ) @ I4 ) @ ( produc8650753666468850689st_int @ S4 @ Stk3 ) ) ) ) ).
% exec_appendL
thf(fact_138_imp__le__cong,axiom,
! [X: int,X4: int,P: $o,P3: $o] :
( ( X = X4 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X4 )
=> ( P = P3 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> P )
= ( ( ord_less_eq_int @ zero_zero_int @ X4 )
=> P3 ) ) ) ) ).
% imp_le_cong
thf(fact_139_add__right__cancel,axiom,
! [B2: int,A2: int,C: int] :
( ( ( plus_plus_int @ B2 @ A2 )
= ( plus_plus_int @ C @ A2 ) )
= ( B2 = C ) ) ).
% add_right_cancel
thf(fact_140_add__right__cancel,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ( plus_plus_nat @ B2 @ A2 )
= ( plus_plus_nat @ C @ A2 ) )
= ( B2 = C ) ) ).
% add_right_cancel
thf(fact_141_add__left__cancel,axiom,
! [A2: int,B2: int,C: int] :
( ( ( plus_plus_int @ A2 @ B2 )
= ( plus_plus_int @ A2 @ C ) )
= ( B2 = C ) ) ).
% add_left_cancel
thf(fact_142_add__left__cancel,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ( plus_plus_nat @ A2 @ B2 )
= ( plus_plus_nat @ A2 @ C ) )
= ( B2 = C ) ) ).
% add_left_cancel
thf(fact_143_append_Oassoc,axiom,
! [A2: list_instr,B2: list_instr,C: list_instr] :
( ( append_instr @ ( append_instr @ A2 @ B2 ) @ C )
= ( append_instr @ A2 @ ( append_instr @ B2 @ C ) ) ) ).
% append.assoc
thf(fact_144_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_145_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_146_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_147_add__le__cancel__right,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
= ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% add_le_cancel_right
thf(fact_148_add__le__cancel__right,axiom,
! [A2: int,C: int,B2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ C ) )
= ( ord_less_eq_int @ A2 @ B2 ) ) ).
% add_le_cancel_right
thf(fact_149_add__le__cancel__left,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) )
= ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% add_le_cancel_left
thf(fact_150_add__le__cancel__left,axiom,
! [C: int,A2: int,B2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B2 ) )
= ( ord_less_eq_int @ A2 @ B2 ) ) ).
% add_le_cancel_left
thf(fact_151_add__0,axiom,
! [A2: int] :
( ( plus_plus_int @ zero_zero_int @ A2 )
= A2 ) ).
% add_0
thf(fact_152_add__0,axiom,
! [A2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A2 )
= A2 ) ).
% add_0
thf(fact_153_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_154_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_155_add__cancel__right__right,axiom,
! [A2: int,B2: int] :
( ( A2
= ( plus_plus_int @ A2 @ B2 ) )
= ( B2 = zero_zero_int ) ) ).
% add_cancel_right_right
thf(fact_156_add__cancel__right__right,axiom,
! [A2: nat,B2: nat] :
( ( A2
= ( plus_plus_nat @ A2 @ B2 ) )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_157_add__cancel__right__left,axiom,
! [A2: int,B2: int] :
( ( A2
= ( plus_plus_int @ B2 @ A2 ) )
= ( B2 = zero_zero_int ) ) ).
% add_cancel_right_left
thf(fact_158_add__cancel__right__left,axiom,
! [A2: nat,B2: nat] :
( ( A2
= ( plus_plus_nat @ B2 @ A2 ) )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_159_add__cancel__left__right,axiom,
! [A2: int,B2: int] :
( ( ( plus_plus_int @ A2 @ B2 )
= A2 )
= ( B2 = zero_zero_int ) ) ).
% add_cancel_left_right
thf(fact_160_add__cancel__left__right,axiom,
! [A2: nat,B2: nat] :
( ( ( plus_plus_nat @ A2 @ B2 )
= A2 )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_161_add__cancel__left__left,axiom,
! [B2: int,A2: int] :
( ( ( plus_plus_int @ B2 @ A2 )
= A2 )
= ( B2 = zero_zero_int ) ) ).
% add_cancel_left_left
thf(fact_162_add__cancel__left__left,axiom,
! [B2: nat,A2: nat] :
( ( ( plus_plus_nat @ B2 @ A2 )
= A2 )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_163_double__zero__sym,axiom,
! [A2: int] :
( ( zero_zero_int
= ( plus_plus_int @ A2 @ A2 ) )
= ( A2 = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_164_add_Oright__neutral,axiom,
! [A2: int] :
( ( plus_plus_int @ A2 @ zero_zero_int )
= A2 ) ).
% add.right_neutral
thf(fact_165_add_Oright__neutral,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% add.right_neutral
thf(fact_166_double__eq__0__iff,axiom,
! [A2: int] :
( ( ( plus_plus_int @ A2 @ A2 )
= zero_zero_int )
= ( A2 = zero_zero_int ) ) ).
% double_eq_0_iff
thf(fact_167_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_168_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_169_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_170_append__eq__append__conv,axiom,
! [Xs: list_list_char,Ys: list_list_char,Us: list_list_char,Vs: list_list_char] :
( ( ( ( size_s356637359517785349t_char @ Xs )
= ( size_s356637359517785349t_char @ Ys ) )
| ( ( size_s356637359517785349t_char @ Us )
= ( size_s356637359517785349t_char @ Vs ) ) )
=> ( ( ( append_list_char @ Xs @ Us )
= ( append_list_char @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_171_add__le__same__cancel1,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B2 @ A2 ) @ B2 )
= ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_172_add__le__same__cancel1,axiom,
! [B2: int,A2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ B2 @ A2 ) @ B2 )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% add_le_same_cancel1
thf(fact_173_add__le__same__cancel2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B2 ) @ B2 )
= ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_174_add__le__same__cancel2,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ B2 ) @ B2 )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% add_le_same_cancel2
thf(fact_175_le__add__same__cancel1,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ A2 @ B2 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).
% le_add_same_cancel1
thf(fact_176_le__add__same__cancel1,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ ( plus_plus_int @ A2 @ B2 ) )
= ( ord_less_eq_int @ zero_zero_int @ B2 ) ) ).
% le_add_same_cancel1
thf(fact_177_le__add__same__cancel2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ B2 @ A2 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).
% le_add_same_cancel2
thf(fact_178_le__add__same__cancel2,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ ( plus_plus_int @ B2 @ A2 ) )
= ( ord_less_eq_int @ zero_zero_int @ B2 ) ) ).
% le_add_same_cancel2
thf(fact_179_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ A2 ) @ zero_zero_int )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_180_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A2 @ A2 ) )
= ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_181_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_182_le__trans,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I2 @ K ) ) ) ).
% le_trans
thf(fact_183_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_184_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_185_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_186_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B2 ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_187_verit__la__generic,axiom,
! [A2: int,X: int] :
( ( ord_less_eq_int @ A2 @ X )
| ( A2 = X )
| ( ord_less_eq_int @ X @ A2 ) ) ).
% verit_la_generic
thf(fact_188_is__num__normalize_I1_J,axiom,
! [A2: int,B2: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A2 @ B2 ) @ C )
= ( plus_plus_int @ A2 @ ( plus_plus_int @ B2 @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_189_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_190_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_191_add__right__imp__eq,axiom,
! [B2: int,A2: int,C: int] :
( ( ( plus_plus_int @ B2 @ A2 )
= ( plus_plus_int @ C @ A2 ) )
=> ( B2 = C ) ) ).
% add_right_imp_eq
thf(fact_192_add__right__imp__eq,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ( plus_plus_nat @ B2 @ A2 )
= ( plus_plus_nat @ C @ A2 ) )
=> ( B2 = C ) ) ).
% add_right_imp_eq
thf(fact_193_add__left__imp__eq,axiom,
! [A2: int,B2: int,C: int] :
( ( ( plus_plus_int @ A2 @ B2 )
= ( plus_plus_int @ A2 @ C ) )
=> ( B2 = C ) ) ).
% add_left_imp_eq
thf(fact_194_add__left__imp__eq,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ( plus_plus_nat @ A2 @ B2 )
= ( plus_plus_nat @ A2 @ C ) )
=> ( B2 = C ) ) ).
% add_left_imp_eq
thf(fact_195_add_Oleft__commute,axiom,
! [B2: int,A2: int,C: int] :
( ( plus_plus_int @ B2 @ ( plus_plus_int @ A2 @ C ) )
= ( plus_plus_int @ A2 @ ( plus_plus_int @ B2 @ C ) ) ) ).
% add.left_commute
thf(fact_196_add_Oleft__commute,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( plus_plus_nat @ B2 @ ( plus_plus_nat @ A2 @ C ) )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add.left_commute
thf(fact_197_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A4: int,B4: int] : ( plus_plus_int @ B4 @ A4 ) ) ) ).
% add.commute
thf(fact_198_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A4: nat,B4: nat] : ( plus_plus_nat @ B4 @ A4 ) ) ) ).
% add.commute
thf(fact_199_add_Oright__cancel,axiom,
! [B2: int,A2: int,C: int] :
( ( ( plus_plus_int @ B2 @ A2 )
= ( plus_plus_int @ C @ A2 ) )
= ( B2 = C ) ) ).
% add.right_cancel
thf(fact_200_add_Oleft__cancel,axiom,
! [A2: int,B2: int,C: int] :
( ( ( plus_plus_int @ A2 @ B2 )
= ( plus_plus_int @ A2 @ C ) )
= ( B2 = C ) ) ).
% add.left_cancel
thf(fact_201_add_Oassoc,axiom,
! [A2: int,B2: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A2 @ B2 ) @ C )
= ( plus_plus_int @ A2 @ ( plus_plus_int @ B2 @ C ) ) ) ).
% add.assoc
thf(fact_202_add_Oassoc,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B2 ) @ C )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add.assoc
thf(fact_203_group__cancel_Oadd2,axiom,
! [B5: int,K: int,B2: int,A2: int] :
( ( B5
= ( plus_plus_int @ K @ B2 ) )
=> ( ( plus_plus_int @ A2 @ B5 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A2 @ B2 ) ) ) ) ).
% group_cancel.add2
thf(fact_204_group__cancel_Oadd2,axiom,
! [B5: nat,K: nat,B2: nat,A2: nat] :
( ( B5
= ( plus_plus_nat @ K @ B2 ) )
=> ( ( plus_plus_nat @ A2 @ B5 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).
% group_cancel.add2
thf(fact_205_group__cancel_Oadd1,axiom,
! [A5: int,K: int,A2: int,B2: int] :
( ( A5
= ( plus_plus_int @ K @ A2 ) )
=> ( ( plus_plus_int @ A5 @ B2 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A2 @ B2 ) ) ) ) ).
% group_cancel.add1
thf(fact_206_group__cancel_Oadd1,axiom,
! [A5: nat,K: nat,A2: nat,B2: nat] :
( ( A5
= ( plus_plus_nat @ K @ A2 ) )
=> ( ( plus_plus_nat @ A5 @ B2 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).
% group_cancel.add1
thf(fact_207_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I2: int,J: int,K: int,L: int] :
( ( ( I2 = J )
& ( K = L ) )
=> ( ( plus_plus_int @ I2 @ K )
= ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_208_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( I2 = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I2 @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_209_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A2: int,B2: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A2 @ B2 ) @ C )
= ( plus_plus_int @ A2 @ ( plus_plus_int @ B2 @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_210_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B2 ) @ C )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_211_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_212_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_213_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_214_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_215_add__le__imp__le__right,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
=> ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% add_le_imp_le_right
thf(fact_216_add__le__imp__le__right,axiom,
! [A2: int,C: int,B2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ C ) )
=> ( ord_less_eq_int @ A2 @ B2 ) ) ).
% add_le_imp_le_right
thf(fact_217_add__le__imp__le__left,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) )
=> ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% add_le_imp_le_left
thf(fact_218_add__le__imp__le__left,axiom,
! [C: int,A2: int,B2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B2 ) )
=> ( ord_less_eq_int @ A2 @ B2 ) ) ).
% add_le_imp_le_left
thf(fact_219_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
? [C3: nat] :
( B4
= ( plus_plus_nat @ A4 @ C3 ) ) ) ) ).
% le_iff_add
thf(fact_220_add__right__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add_right_mono
thf(fact_221_add__right__mono,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ C ) ) ) ).
% add_right_mono
thf(fact_222_less__eqE,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ~ ! [C2: nat] :
( B2
!= ( plus_plus_nat @ A2 @ C2 ) ) ) ).
% less_eqE
thf(fact_223_add__left__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) ) ) ).
% add_left_mono
thf(fact_224_add__left__mono,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B2 ) ) ) ).
% add_left_mono
thf(fact_225_add__mono,axiom,
! [A2: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).
% add_mono
thf(fact_226_add__mono,axiom,
! [A2: int,B2: int,C: int,D: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ D ) ) ) ) ).
% add_mono
thf(fact_227_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I2 @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_228_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I2: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I2 @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_229_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( I2 = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_230_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I2: int,J: int,K: int,L: int] :
( ( ( I2 = J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_231_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I2 @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_232_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I2: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I2 @ J )
& ( K = L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_233_add_Ogroup__left__neutral,axiom,
! [A2: int] :
( ( plus_plus_int @ zero_zero_int @ A2 )
= A2 ) ).
% add.group_left_neutral
thf(fact_234_add_Ocomm__neutral,axiom,
! [A2: int] :
( ( plus_plus_int @ A2 @ zero_zero_int )
= A2 ) ).
% add.comm_neutral
thf(fact_235_add_Ocomm__neutral,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% add.comm_neutral
thf(fact_236_comm__monoid__add__class_Oadd__0,axiom,
! [A2: int] :
( ( plus_plus_int @ zero_zero_int @ A2 )
= A2 ) ).
% comm_monoid_add_class.add_0
thf(fact_237_comm__monoid__add__class_Oadd__0,axiom,
! [A2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A2 )
= A2 ) ).
% comm_monoid_add_class.add_0
thf(fact_238_verit__sum__simplify,axiom,
! [A2: int] :
( ( plus_plus_int @ A2 @ zero_zero_int )
= A2 ) ).
% verit_sum_simplify
thf(fact_239_verit__sum__simplify,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% verit_sum_simplify
thf(fact_240_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus_int @ K @ zero_zero_int )
= K ) ).
% plus_int_code(1)
thf(fact_241_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_242_zle__iff__zadd,axiom,
( ord_less_eq_int
= ( ^ [W: int,Z2: int] :
? [N3: nat] :
( Z2
= ( plus_plus_int @ W @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_243_exec1__appendR,axiom,
! [P: list_instr,C: produc6425607678544837394st_int,C4: produc6425607678544837394st_int,P3: list_instr] :
( ( exec1 @ P @ C @ C4 )
=> ( exec1 @ ( append_instr @ P @ P3 ) @ C @ C4 ) ) ).
% exec1_appendR
thf(fact_244_add__decreasing,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ B2 ) ) ) ).
% add_decreasing
thf(fact_245_add__decreasing,axiom,
! [A2: int,C: int,B2: int] :
( ( ord_less_eq_int @ A2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ C @ B2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ B2 ) ) ) ).
% add_decreasing
thf(fact_246_add__increasing,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ B2 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_increasing
thf(fact_247_add__increasing,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ord_less_eq_int @ B2 @ ( plus_plus_int @ A2 @ C ) ) ) ) ).
% add_increasing
thf(fact_248_add__decreasing2,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ B2 ) ) ) ).
% add_decreasing2
thf(fact_249_add__decreasing2,axiom,
! [C: int,A2: int,B2: int] :
( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ A2 @ B2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ B2 ) ) ) ).
% add_decreasing2
thf(fact_250_add__increasing2,axiom,
! [C: nat,B2: nat,A2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ord_less_eq_nat @ B2 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_increasing2
thf(fact_251_add__increasing2,axiom,
! [C: int,B2: int,A2: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ B2 @ A2 )
=> ( ord_less_eq_int @ B2 @ ( plus_plus_int @ A2 @ C ) ) ) ) ).
% add_increasing2
thf(fact_252_add__nonneg__nonneg,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_253_add__nonneg__nonneg,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A2 @ B2 ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_254_add__nonpos__nonpos,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_255_add__nonpos__nonpos,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ B2 @ zero_zero_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).
% add_nonpos_nonpos
thf(fact_256_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_257_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_258_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_259_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_260_com_Odistinct_I1_J,axiom,
! [X21: list_char,X22: aexp] :
( skip
!= ( assign @ X21 @ X22 ) ) ).
% com.distinct(1)
thf(fact_261_exec__appendR,axiom,
! [P: list_instr,C: produc6425607678544837394st_int,C4: produc6425607678544837394st_int,P3: list_instr] :
( ( star_P707599355569300323st_int @ ( exec1 @ P ) @ C @ C4 )
=> ( star_P707599355569300323st_int @ ( exec1 @ ( append_instr @ P @ P3 ) ) @ C @ C4 ) ) ).
% exec_appendR
thf(fact_262_exec1__appendL,axiom,
! [P: list_instr,I2: int,S2: list_char > int,Stk2: list_int,I4: int,S4: list_char > int,Stk3: list_int,P3: list_instr] :
( ( exec1 @ P @ ( produc5086643055186798020st_int @ I2 @ ( produc8650753666468850689st_int @ S2 @ Stk2 ) ) @ ( produc5086643055186798020st_int @ I4 @ ( produc8650753666468850689st_int @ S4 @ Stk3 ) ) )
=> ( exec1 @ ( append_instr @ P3 @ P ) @ ( produc5086643055186798020st_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P3 ) ) @ I2 ) @ ( produc8650753666468850689st_int @ S2 @ Stk2 ) ) @ ( produc5086643055186798020st_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P3 ) ) @ I4 ) @ ( produc8650753666468850689st_int @ S4 @ Stk3 ) ) ) ) ).
% exec1_appendL
thf(fact_263_nle__le,axiom,
! [A2: nat,B2: nat] :
( ( ~ ( ord_less_eq_nat @ A2 @ B2 ) )
= ( ( ord_less_eq_nat @ B2 @ A2 )
& ( B2 != A2 ) ) ) ).
% nle_le
thf(fact_264_nle__le,axiom,
! [A2: int,B2: int] :
( ( ~ ( ord_less_eq_int @ A2 @ B2 ) )
= ( ( ord_less_eq_int @ B2 @ A2 )
& ( B2 != A2 ) ) ) ).
% nle_le
thf(fact_265_le__cases3,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z ) )
=> ( ( ( ord_less_eq_nat @ X @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_266_le__cases3,axiom,
! [X: int,Y: int,Z: int] :
( ( ( ord_less_eq_int @ X @ Y )
=> ~ ( ord_less_eq_int @ Y @ Z ) )
=> ( ( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_eq_int @ X @ Z ) )
=> ( ( ( ord_less_eq_int @ X @ Z )
=> ~ ( ord_less_eq_int @ Z @ Y ) )
=> ( ( ( ord_less_eq_int @ Z @ Y )
=> ~ ( ord_less_eq_int @ Y @ X ) )
=> ( ( ( ord_less_eq_int @ Y @ Z )
=> ~ ( ord_less_eq_int @ Z @ X ) )
=> ~ ( ( ord_less_eq_int @ Z @ X )
=> ~ ( ord_less_eq_int @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_267_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
= ( ^ [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
& ( ord_less_eq_nat @ Y2 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_268_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: int,Z3: int] : ( Y4 = Z3 ) )
= ( ^ [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
& ( ord_less_eq_int @ Y2 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_269_ord__eq__le__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( A2 = B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_270_ord__eq__le__trans,axiom,
! [A2: int,B2: int,C: int] :
( ( A2 = B2 )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ord_less_eq_int @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_271_ord__le__eq__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_272_ord__le__eq__trans,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_int @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_273_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_274_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_275_order_Otrans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_276_order_Otrans,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ord_less_eq_int @ A2 @ C ) ) ) ).
% order.trans
thf(fact_277_order__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_eq_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_278_order__trans,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z )
=> ( ord_less_eq_int @ X @ Z ) ) ) ).
% order_trans
thf(fact_279_linorder__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B2: nat] :
( ! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( P @ A @ B ) )
=> ( ! [A: nat,B: nat] :
( ( P @ B @ A )
=> ( P @ A @ B ) )
=> ( P @ A2 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_280_linorder__wlog,axiom,
! [P: int > int > $o,A2: int,B2: int] :
( ! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( P @ A @ B ) )
=> ( ! [A: int,B: int] :
( ( P @ B @ A )
=> ( P @ A @ B ) )
=> ( P @ A2 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_281_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_282_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: int,Z3: int] : ( Y4 = Z3 ) )
= ( ^ [A4: int,B4: int] :
( ( ord_less_eq_int @ B4 @ A4 )
& ( ord_less_eq_int @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_283_dual__order_Oantisym,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_284_dual__order_Oantisym,axiom,
! [B2: int,A2: int] :
( ( ord_less_eq_int @ B2 @ A2 )
=> ( ( ord_less_eq_int @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_285_dual__order_Otrans,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_286_dual__order_Otrans,axiom,
! [B2: int,A2: int,C: int] :
( ( ord_less_eq_int @ B2 @ A2 )
=> ( ( ord_less_eq_int @ C @ B2 )
=> ( ord_less_eq_int @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_287_antisym,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_288_antisym,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_289_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z3: nat] : ( Y4 = Z3 ) )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_290_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: int,Z3: int] : ( Y4 = Z3 ) )
= ( ^ [A4: int,B4: int] :
( ( ord_less_eq_int @ A4 @ B4 )
& ( ord_less_eq_int @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_291_order__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_292_order__subst1,axiom,
! [A2: nat,F: int > nat,B2: int,C: int] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_293_order__subst1,axiom,
! [A2: int,F: nat > int,B2: nat,C: nat] :
( ( ord_less_eq_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_294_order__subst1,axiom,
! [A2: int,F: int > int,B2: int,C: int] :
( ( ord_less_eq_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_295_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_296_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_int @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_297_order__subst2,axiom,
! [A2: int,B2: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_298_order__subst2,axiom,
! [A2: int,B2: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ ( F @ B2 ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_299_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_300_order__eq__refl,axiom,
! [X: int,Y: int] :
( ( X = Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_eq_refl
thf(fact_301_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_302_linorder__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_linear
thf(fact_303_ord__eq__le__subst,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_304_ord__eq__le__subst,axiom,
! [A2: int,F: nat > int,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_305_ord__eq__le__subst,axiom,
! [A2: nat,F: int > nat,B2: int,C: int] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_306_ord__eq__le__subst,axiom,
! [A2: int,F: int > int,B2: int,C: int] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_307_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_308_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_309_ord__le__eq__subst,axiom,
! [A2: int,B2: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_310_ord__le__eq__subst,axiom,
! [A2: int,B2: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_311_linorder__le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_312_linorder__le__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_313_order__antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_314_order__antisym__conv,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_315_not__Cons__self2,axiom,
! [X: int,Xs: list_int] :
( ( cons_int @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_316_not__Cons__self2,axiom,
! [X: instr,Xs: list_instr] :
( ( cons_instr @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_317_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_318_neq__if__length__neq,axiom,
! [Xs: list_list_char,Ys: list_list_char] :
( ( ( size_s356637359517785349t_char @ Xs )
!= ( size_s356637359517785349t_char @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_319_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_instr] :
( ( size_size_list_instr @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_320_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_list_char] :
( ( size_s356637359517785349t_char @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_321_conj__le__cong,axiom,
! [X: int,X4: int,P: $o,P3: $o] :
( ( X = X4 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X4 )
=> ( P = P3 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
& P )
= ( ( ord_less_eq_int @ zero_zero_int @ X4 )
& P3 ) ) ) ) ).
% conj_le_cong
thf(fact_322_size_H__char__eq__0,axiom,
( size_char
= ( ^ [C3: char] : zero_zero_nat ) ) ).
% size'_char_eq_0
thf(fact_323_exec__append__trans,axiom,
! [P: list_instr,S2: list_char > int,Stk2: list_int,I4: int,S4: list_char > int,Stk3: list_int,P3: list_instr,I5: int,S5: list_char > int,Stk4: list_int,J3: int] :
( ( star_P707599355569300323st_int @ ( exec1 @ P ) @ ( produc5086643055186798020st_int @ zero_zero_int @ ( produc8650753666468850689st_int @ S2 @ Stk2 ) ) @ ( produc5086643055186798020st_int @ I4 @ ( produc8650753666468850689st_int @ S4 @ Stk3 ) ) )
=> ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P ) ) @ I4 )
=> ( ( star_P707599355569300323st_int @ ( exec1 @ P3 ) @ ( produc5086643055186798020st_int @ ( minus_minus_int @ I4 @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P ) ) ) @ ( produc8650753666468850689st_int @ S4 @ Stk3 ) ) @ ( produc5086643055186798020st_int @ I5 @ ( produc8650753666468850689st_int @ S5 @ Stk4 ) ) )
=> ( ( J3
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P ) ) @ I5 ) )
=> ( star_P707599355569300323st_int @ ( exec1 @ ( append_instr @ P @ P3 ) ) @ ( produc5086643055186798020st_int @ zero_zero_int @ ( produc8650753666468850689st_int @ S2 @ Stk2 ) ) @ ( produc5086643055186798020st_int @ J3 @ ( produc8650753666468850689st_int @ S5 @ Stk4 ) ) ) ) ) ) ) ).
% exec_append_trans
thf(fact_324_exec__appendL__if,axiom,
! [P3: list_instr,I2: int,P: list_instr,S2: list_char > int,Stk2: list_int,J: int,S4: list_char > int,Stk3: list_int,I4: int] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P3 ) ) @ I2 )
=> ( ( star_P707599355569300323st_int @ ( exec1 @ P ) @ ( produc5086643055186798020st_int @ ( minus_minus_int @ I2 @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P3 ) ) ) @ ( produc8650753666468850689st_int @ S2 @ Stk2 ) ) @ ( produc5086643055186798020st_int @ J @ ( produc8650753666468850689st_int @ S4 @ Stk3 ) ) )
=> ( ( I4
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P3 ) ) @ J ) )
=> ( star_P707599355569300323st_int @ ( exec1 @ ( append_instr @ P3 @ P ) ) @ ( produc5086643055186798020st_int @ I2 @ ( produc8650753666468850689st_int @ S2 @ Stk2 ) ) @ ( produc5086643055186798020st_int @ I4 @ ( produc8650753666468850689st_int @ S4 @ Stk3 ) ) ) ) ) ) ).
% exec_appendL_if
thf(fact_325_exec__Cons__1,axiom,
! [P: list_instr,S2: 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 @ S2 @ Stk2 ) ) @ ( produc5086643055186798020st_int @ J @ ( produc8650753666468850689st_int @ T @ Stk3 ) ) )
=> ( star_P707599355569300323st_int @ ( exec1 @ ( cons_instr @ Ins2 @ P ) ) @ ( produc5086643055186798020st_int @ one_one_int @ ( produc8650753666468850689st_int @ S2 @ Stk2 ) ) @ ( produc5086643055186798020st_int @ ( plus_plus_int @ one_one_int @ J ) @ ( produc8650753666468850689st_int @ T @ Stk3 ) ) ) ) ).
% exec_Cons_1
thf(fact_326_iexec__shift,axiom,
! [N: int,I4: int,S4: list_char > int,Stk3: list_int,Ins2: instr,I2: int,S2: list_char > int,Stk2: list_int] :
( ( ( produc5086643055186798020st_int @ ( plus_plus_int @ N @ I4 ) @ ( produc8650753666468850689st_int @ S4 @ Stk3 ) )
= ( iexec @ Ins2 @ ( produc5086643055186798020st_int @ ( plus_plus_int @ N @ I2 ) @ ( produc8650753666468850689st_int @ S2 @ Stk2 ) ) ) )
= ( ( produc5086643055186798020st_int @ I4 @ ( produc8650753666468850689st_int @ S4 @ Stk3 ) )
= ( iexec @ Ins2 @ ( produc5086643055186798020st_int @ I2 @ ( produc8650753666468850689st_int @ S2 @ Stk2 ) ) ) ) ) ).
% iexec_shift
thf(fact_327_bcomp__correct,axiom,
! [I2: int,B2: bexp,F: $o,S2: list_char > int,Stk2: list_int] :
( ( ord_less_eq_int @ zero_zero_int @ I2 )
=> ( star_P707599355569300323st_int @ ( exec1 @ ( bcomp @ ( produc4047900504771817624_o_int @ B2 @ ( product_Pair_o_int @ F @ I2 ) ) ) ) @ ( produc5086643055186798020st_int @ zero_zero_int @ ( produc8650753666468850689st_int @ S2 @ Stk2 ) )
@ ( produc5086643055186798020st_int
@ ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ ( bcomp @ ( produc4047900504771817624_o_int @ B2 @ ( product_Pair_o_int @ F @ I2 ) ) ) ) )
@ ( if_int
@ ( F
= ( bval @ B2 @ S2 ) )
@ I2
@ zero_zero_int ) )
@ ( produc8650753666468850689st_int @ S2 @ Stk2 ) ) ) ) ).
% bcomp_correct
thf(fact_328_add__0__iff,axiom,
! [B2: int,A2: int] :
( ( B2
= ( plus_plus_int @ B2 @ A2 ) )
= ( A2 = zero_zero_int ) ) ).
% add_0_iff
thf(fact_329_add__0__iff,axiom,
! [B2: nat,A2: nat] :
( ( B2
= ( plus_plus_nat @ B2 @ A2 ) )
= ( A2 = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_330_bind__simps_I2_J,axiom,
! [X: int,Xs: list_int,F: int > list_instr] :
( ( bind_int_instr @ ( cons_int @ X @ Xs ) @ F )
= ( append_instr @ ( F @ X ) @ ( bind_int_instr @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_331_bind__simps_I2_J,axiom,
! [X: instr,Xs: list_instr,F: instr > list_instr] :
( ( bind_instr_instr @ ( cons_instr @ X @ Xs ) @ F )
= ( append_instr @ ( F @ X ) @ ( bind_instr_instr @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_332_diff__self,axiom,
! [A2: int] :
( ( minus_minus_int @ A2 @ A2 )
= zero_zero_int ) ).
% diff_self
thf(fact_333_diff__0__right,axiom,
! [A2: int] :
( ( minus_minus_int @ A2 @ zero_zero_int )
= A2 ) ).
% diff_0_right
thf(fact_334_zero__diff,axiom,
! [A2: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A2 )
= zero_zero_nat ) ).
% zero_diff
thf(fact_335_diff__zero,axiom,
! [A2: int] :
( ( minus_minus_int @ A2 @ zero_zero_int )
= A2 ) ).
% diff_zero
thf(fact_336_diff__zero,axiom,
! [A2: nat] :
( ( minus_minus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% diff_zero
thf(fact_337_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A2: int] :
( ( minus_minus_int @ A2 @ A2 )
= zero_zero_int ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_338_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A2: nat] :
( ( minus_minus_nat @ A2 @ A2 )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_339_add__diff__cancel,axiom,
! [A2: int,B2: int] :
( ( minus_minus_int @ ( plus_plus_int @ A2 @ B2 ) @ B2 )
= A2 ) ).
% add_diff_cancel
thf(fact_340_diff__add__cancel,axiom,
! [A2: int,B2: int] :
( ( plus_plus_int @ ( minus_minus_int @ A2 @ B2 ) @ B2 )
= A2 ) ).
% diff_add_cancel
thf(fact_341_add__diff__cancel__left,axiom,
! [C: int,A2: int,B2: int] :
( ( minus_minus_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B2 ) )
= ( minus_minus_int @ A2 @ B2 ) ) ).
% add_diff_cancel_left
thf(fact_342_add__diff__cancel__left,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) )
= ( minus_minus_nat @ A2 @ B2 ) ) ).
% add_diff_cancel_left
thf(fact_343_add__diff__cancel__left_H,axiom,
! [A2: int,B2: int] :
( ( minus_minus_int @ ( plus_plus_int @ A2 @ B2 ) @ A2 )
= B2 ) ).
% add_diff_cancel_left'
thf(fact_344_add__diff__cancel__left_H,axiom,
! [A2: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ B2 ) @ A2 )
= B2 ) ).
% add_diff_cancel_left'
thf(fact_345_add__diff__cancel__right,axiom,
! [A2: int,C: int,B2: int] :
( ( minus_minus_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ C ) )
= ( minus_minus_int @ A2 @ B2 ) ) ).
% add_diff_cancel_right
thf(fact_346_add__diff__cancel__right,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
= ( minus_minus_nat @ A2 @ B2 ) ) ).
% add_diff_cancel_right
thf(fact_347_add__diff__cancel__right_H,axiom,
! [A2: int,B2: int] :
( ( minus_minus_int @ ( plus_plus_int @ A2 @ B2 ) @ B2 )
= A2 ) ).
% add_diff_cancel_right'
thf(fact_348_add__diff__cancel__right_H,axiom,
! [A2: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ B2 ) @ B2 )
= A2 ) ).
% add_diff_cancel_right'
thf(fact_349_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_350_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_351_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_352_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_353_of__nat__1,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% of_nat_1
thf(fact_354_of__nat__1,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% of_nat_1
thf(fact_355_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_356_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_357_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_358_diff__ge__0__iff__ge,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A2 @ B2 ) )
= ( ord_less_eq_int @ B2 @ A2 ) ) ).
% diff_ge_0_iff_ge
thf(fact_359_diff__add__zero,axiom,
! [A2: nat,B2: nat] :
( ( minus_minus_nat @ A2 @ ( plus_plus_nat @ A2 @ B2 ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_360_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_361_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_362_length__append,axiom,
! [Xs: list_list_char,Ys: list_list_char] :
( ( size_s356637359517785349t_char @ ( append_list_char @ Xs @ Ys ) )
= ( plus_plus_nat @ ( size_s356637359517785349t_char @ Xs ) @ ( size_s356637359517785349t_char @ Ys ) ) ) ).
% length_append
thf(fact_363_diff__left__imp__eq,axiom,
! [A2: int,B2: int,C: int] :
( ( ( minus_minus_int @ A2 @ B2 )
= ( minus_minus_int @ A2 @ C ) )
=> ( B2 = C ) ) ).
% diff_left_imp_eq
thf(fact_364_diff__eq__diff__eq,axiom,
! [A2: int,B2: int,C: int,D: int] :
( ( ( minus_minus_int @ A2 @ B2 )
= ( minus_minus_int @ C @ D ) )
=> ( ( A2 = B2 )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_365_diff__right__commute,axiom,
! [A2: int,C: int,B2: int] :
( ( minus_minus_int @ ( minus_minus_int @ A2 @ C ) @ B2 )
= ( minus_minus_int @ ( minus_minus_int @ A2 @ B2 ) @ C ) ) ).
% diff_right_commute
thf(fact_366_diff__right__commute,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ C ) @ B2 )
= ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B2 ) @ C ) ) ).
% diff_right_commute
thf(fact_367_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
= ( X = one_one_int ) ) ).
% one_reorient
thf(fact_368_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_369_int__le__induct,axiom,
! [I2: int,K: int,P: int > $o] :
( ( ord_less_eq_int @ I2 @ K )
=> ( ( P @ K )
=> ( ! [I: int] :
( ( ord_less_eq_int @ I @ K )
=> ( ( P @ I )
=> ( P @ ( minus_minus_int @ I @ one_one_int ) ) ) )
=> ( P @ I2 ) ) ) ) ).
% int_le_induct
thf(fact_370_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_371_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_372_int__induct,axiom,
! [P: int > $o,K: int,I2: int] :
( ( P @ K )
=> ( ! [I: int] :
( ( ord_less_eq_int @ K @ I )
=> ( ( P @ I )
=> ( P @ ( plus_plus_int @ I @ one_one_int ) ) ) )
=> ( ! [I: int] :
( ( ord_less_eq_int @ I @ K )
=> ( ( P @ I )
=> ( P @ ( minus_minus_int @ I @ one_one_int ) ) ) )
=> ( P @ I2 ) ) ) ) ).
% int_induct
thf(fact_373_diff__mono,axiom,
! [A2: int,B2: int,D: int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ D @ C )
=> ( ord_less_eq_int @ ( minus_minus_int @ A2 @ C ) @ ( minus_minus_int @ B2 @ D ) ) ) ) ).
% diff_mono
thf(fact_374_diff__left__mono,axiom,
! [B2: int,A2: int,C: int] :
( ( ord_less_eq_int @ B2 @ A2 )
=> ( ord_less_eq_int @ ( minus_minus_int @ C @ A2 ) @ ( minus_minus_int @ C @ B2 ) ) ) ).
% diff_left_mono
thf(fact_375_diff__right__mono,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ord_less_eq_int @ ( minus_minus_int @ A2 @ C ) @ ( minus_minus_int @ B2 @ C ) ) ) ).
% diff_right_mono
thf(fact_376_diff__eq__diff__less__eq,axiom,
! [A2: int,B2: int,C: int,D: int] :
( ( ( minus_minus_int @ A2 @ B2 )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_eq_int @ A2 @ B2 )
= ( ord_less_eq_int @ C @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_377_eq__iff__diff__eq__0,axiom,
( ( ^ [Y4: int,Z3: int] : ( Y4 = Z3 ) )
= ( ^ [A4: int,B4: int] :
( ( minus_minus_int @ A4 @ B4 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_378_group__cancel_Osub1,axiom,
! [A5: int,K: int,A2: int,B2: int] :
( ( A5
= ( plus_plus_int @ K @ A2 ) )
=> ( ( minus_minus_int @ A5 @ B2 )
= ( plus_plus_int @ K @ ( minus_minus_int @ A2 @ B2 ) ) ) ) ).
% group_cancel.sub1
thf(fact_379_diff__eq__eq,axiom,
! [A2: int,B2: int,C: int] :
( ( ( minus_minus_int @ A2 @ B2 )
= C )
= ( A2
= ( plus_plus_int @ C @ B2 ) ) ) ).
% diff_eq_eq
thf(fact_380_eq__diff__eq,axiom,
! [A2: int,C: int,B2: int] :
( ( A2
= ( minus_minus_int @ C @ B2 ) )
= ( ( plus_plus_int @ A2 @ B2 )
= C ) ) ).
% eq_diff_eq
thf(fact_381_add__diff__eq,axiom,
! [A2: int,B2: int,C: int] :
( ( plus_plus_int @ A2 @ ( minus_minus_int @ B2 @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A2 @ B2 ) @ C ) ) ).
% add_diff_eq
thf(fact_382_diff__diff__eq2,axiom,
! [A2: int,B2: int,C: int] :
( ( minus_minus_int @ A2 @ ( minus_minus_int @ B2 @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A2 @ C ) @ B2 ) ) ).
% diff_diff_eq2
thf(fact_383_diff__add__eq,axiom,
! [A2: int,B2: int,C: int] :
( ( plus_plus_int @ ( minus_minus_int @ A2 @ B2 ) @ C )
= ( minus_minus_int @ ( plus_plus_int @ A2 @ C ) @ B2 ) ) ).
% diff_add_eq
thf(fact_384_diff__add__eq__diff__diff__swap,axiom,
! [A2: int,B2: int,C: int] :
( ( minus_minus_int @ A2 @ ( plus_plus_int @ B2 @ C ) )
= ( minus_minus_int @ ( minus_minus_int @ A2 @ C ) @ B2 ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_385_add__implies__diff,axiom,
! [C: int,B2: int,A2: int] :
( ( ( plus_plus_int @ C @ B2 )
= A2 )
=> ( C
= ( minus_minus_int @ A2 @ B2 ) ) ) ).
% add_implies_diff
thf(fact_386_add__implies__diff,axiom,
! [C: nat,B2: nat,A2: nat] :
( ( ( plus_plus_nat @ C @ B2 )
= A2 )
=> ( C
= ( minus_minus_nat @ A2 @ B2 ) ) ) ).
% add_implies_diff
thf(fact_387_diff__diff__eq,axiom,
! [A2: int,B2: int,C: int] :
( ( minus_minus_int @ ( minus_minus_int @ A2 @ B2 ) @ C )
= ( minus_minus_int @ A2 @ ( plus_plus_int @ B2 @ C ) ) ) ).
% diff_diff_eq
thf(fact_388_diff__diff__eq,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B2 ) @ C )
= ( minus_minus_nat @ A2 @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% diff_diff_eq
thf(fact_389_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus_int @ K @ zero_zero_int )
= K ) ).
% minus_int_code(1)
thf(fact_390_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_391_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_392_int__diff__cases,axiom,
! [Z: int] :
~ ! [M2: nat,N2: nat] :
( Z
!= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% int_diff_cases
thf(fact_393_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_394_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_395_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N3: nat] :
? [K2: nat] :
( N3
= ( plus_plus_nat @ M3 @ K2 ) ) ) ) ).
% nat_le_iff_add
thf(fact_396_trans__le__add2,axiom,
! [I2: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_397_trans__le__add1,axiom,
! [I2: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_398_add__le__mono1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_399_add__le__mono,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_400_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_401_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_402_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_403_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_404_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_405_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_406_le__iff__diff__le__0,axiom,
( ord_less_eq_int
= ( ^ [A4: int,B4: int] : ( ord_less_eq_int @ ( minus_minus_int @ A4 @ B4 ) @ zero_zero_int ) ) ) ).
% le_iff_diff_le_0
thf(fact_407_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( minus_minus_nat @ B2 @ A2 )
= C )
= ( B2
= ( plus_plus_nat @ C @ A2 ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_408_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( plus_plus_nat @ A2 @ ( minus_minus_nat @ B2 @ A2 ) )
= B2 ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_409_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B2 @ A2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A2 ) @ B2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_410_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A2 )
= ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A2 ) @ C ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_411_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A2 ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_412_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B2 ) @ A2 )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B2 @ A2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_413_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B2 @ A2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B2 ) @ A2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_414_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B2 @ A2 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ B2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_415_le__add__diff,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B2 @ C ) @ A2 ) ) ) ).
% le_add_diff
thf(fact_416_diff__add,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B2 @ A2 ) @ A2 )
= B2 ) ) ).
% diff_add
thf(fact_417_le__diff__eq,axiom,
! [A2: int,C: int,B2: int] :
( ( ord_less_eq_int @ A2 @ ( minus_minus_int @ C @ B2 ) )
= ( ord_less_eq_int @ ( plus_plus_int @ A2 @ B2 ) @ C ) ) ).
% le_diff_eq
thf(fact_418_diff__le__eq,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_eq_int @ ( minus_minus_int @ A2 @ B2 ) @ C )
= ( ord_less_eq_int @ A2 @ ( plus_plus_int @ C @ B2 ) ) ) ).
% diff_le_eq
thf(fact_419_int__ops_I5_J,axiom,
! [A2: nat,B2: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A2 @ B2 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ).
% int_ops(5)
thf(fact_420_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_421_zadd__int__left,axiom,
! [M: nat,N: nat,Z: int] :
( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) ) @ Z ) ) ).
% zadd_int_left
thf(fact_422_odd__nonzero,axiom,
! [Z: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_423_int__ge__induct,axiom,
! [K: int,I2: int,P: int > $o] :
( ( ord_less_eq_int @ K @ I2 )
=> ( ( P @ K )
=> ( ! [I: int] :
( ( ord_less_eq_int @ K @ I )
=> ( ( P @ I )
=> ( P @ ( plus_plus_int @ I @ one_one_int ) ) ) )
=> ( P @ I2 ) ) ) ) ).
% int_ge_induct
thf(fact_424_le__add__diff__inverse,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( plus_plus_nat @ B2 @ ( minus_minus_nat @ A2 @ B2 ) )
= A2 ) ) ).
% le_add_diff_inverse
thf(fact_425_le__add__diff__inverse,axiom,
! [B2: int,A2: int] :
( ( ord_less_eq_int @ B2 @ A2 )
=> ( ( plus_plus_int @ B2 @ ( minus_minus_int @ A2 @ B2 ) )
= A2 ) ) ).
% le_add_diff_inverse
thf(fact_426_le__add__diff__inverse2,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A2 @ B2 ) @ B2 )
= A2 ) ) ).
% le_add_diff_inverse2
thf(fact_427_le__add__diff__inverse2,axiom,
! [B2: int,A2: int] :
( ( ord_less_eq_int @ B2 @ A2 )
=> ( ( plus_plus_int @ ( minus_minus_int @ A2 @ B2 ) @ B2 )
= A2 ) ) ).
% le_add_diff_inverse2
thf(fact_428_of__nat__code,axiom,
( semiri1314217659103216013at_int
= ( ^ [N3: nat] :
( semiri8420488043553186161ux_int
@ ^ [I3: int] : ( plus_plus_int @ I3 @ one_one_int )
@ N3
@ zero_zero_int ) ) ) ).
% of_nat_code
thf(fact_429_of__nat__code,axiom,
( semiri1316708129612266289at_nat
= ( ^ [N3: nat] :
( semiri8422978514062236437ux_nat
@ ^ [I3: nat] : ( plus_plus_nat @ I3 @ one_one_nat )
@ N3
@ zero_zero_nat ) ) ) ).
% of_nat_code
thf(fact_430_add__le__imp__le__diff,axiom,
! [I2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N )
=> ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_431_add__le__imp__le__diff,axiom,
! [I2: int,K: int,N: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ N )
=> ( ord_less_eq_int @ I2 @ ( minus_minus_int @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_432_add__le__add__imp__diff__le,axiom,
! [I2: nat,K: nat,N: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ 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_433_add__le__add__imp__diff__le,axiom,
! [I2: int,K: int,N: int,J: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ I2 @ 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_434_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_435_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_436_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_437_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_438_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_439_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_440_diff__diff__cancel,axiom,
! [I2: nat,N: nat] :
( ( ord_less_eq_nat @ I2 @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I2 ) )
= I2 ) ) ).
% diff_diff_cancel
thf(fact_441_diff__diff__left,axiom,
! [I2: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K )
= ( minus_minus_nat @ I2 @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_442_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_443_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_444_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_445_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I2 )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I2 ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_446_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_447_diff__commute,axiom,
! [I2: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I2 @ K ) @ J ) ) ).
% diff_commute
thf(fact_448_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_449_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_450_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_451_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_452_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_453_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_454_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_455_le__diff__iff_H,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A2 ) @ ( minus_minus_nat @ C @ B2 ) )
= ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% le_diff_iff'
thf(fact_456_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_457_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_458_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_459_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_460_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_461_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_462_Nat_Ole__imp__diff__is__add,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ( minus_minus_nat @ J @ I2 )
= K )
= ( J
= ( plus_plus_nat @ K @ I2 ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_463_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I2 ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I2 ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_464_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J ) @ K )
= ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_465_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_466_le__diff__conv,axiom,
! [J: nat,K: nat,I2: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I2 )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I2 @ K ) ) ) ).
% le_diff_conv
thf(fact_467_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_468_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_469_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_470_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_471_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_472_bcomp_Osimps_I3_J,axiom,
! [B_1: bexp,B_2: bexp,F: $o,I2: int] :
( ( bcomp @ ( produc4047900504771817624_o_int @ ( and @ B_1 @ B_2 ) @ ( product_Pair_o_int @ F @ I2 ) ) )
= ( append_instr @ ( bcomp @ ( produc4047900504771817624_o_int @ B_1 @ ( product_Pair_o_int @ $false @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ ( bcomp @ ( produc4047900504771817624_o_int @ B_2 @ ( product_Pair_o_int @ F @ I2 ) ) ) ) ) @ ( if_int @ F @ zero_zero_int @ I2 ) ) ) ) ) @ ( bcomp @ ( produc4047900504771817624_o_int @ B_2 @ ( product_Pair_o_int @ F @ I2 ) ) ) ) ) ).
% bcomp.simps(3)
thf(fact_473_ccomp_Osimps_I4_J,axiom,
! [B2: bexp,C_1: com,C_2: com] :
( ( ccomp @ ( if @ B2 @ C_1 @ C_2 ) )
= ( append_instr @ ( bcomp @ ( produc4047900504771817624_o_int @ B2 @ ( product_Pair_o_int @ $false @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ ( ccomp @ C_1 ) ) ) @ one_one_int ) ) ) ) @ ( append_instr @ ( ccomp @ C_1 ) @ ( cons_instr @ ( jmp @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ ( ccomp @ C_2 ) ) ) ) @ ( ccomp @ C_2 ) ) ) ) ) ).
% ccomp.simps(4)
thf(fact_474_inth_Osimps,axiom,
! [I2: int,X: list_char,Xs: list_list_char] :
( ( ( I2 = zero_zero_int )
=> ( ( inth_list_char @ ( cons_list_char @ X @ Xs ) @ I2 )
= X ) )
& ( ( I2 != zero_zero_int )
=> ( ( inth_list_char @ ( cons_list_char @ X @ Xs ) @ I2 )
= ( inth_list_char @ Xs @ ( minus_minus_int @ I2 @ one_one_int ) ) ) ) ) ).
% inth.simps
thf(fact_475_inth_Osimps,axiom,
! [I2: int,X: int,Xs: list_int] :
( ( ( I2 = zero_zero_int )
=> ( ( inth_int @ ( cons_int @ X @ Xs ) @ I2 )
= X ) )
& ( ( I2 != zero_zero_int )
=> ( ( inth_int @ ( cons_int @ X @ Xs ) @ I2 )
= ( inth_int @ Xs @ ( minus_minus_int @ I2 @ one_one_int ) ) ) ) ) ).
% inth.simps
thf(fact_476_inth_Osimps,axiom,
! [I2: int,X: instr,Xs: list_instr] :
( ( ( I2 = zero_zero_int )
=> ( ( inth_instr @ ( cons_instr @ X @ Xs ) @ I2 )
= X ) )
& ( ( I2 != zero_zero_int )
=> ( ( inth_instr @ ( cons_instr @ X @ Xs ) @ I2 )
= ( inth_instr @ Xs @ ( minus_minus_int @ I2 @ one_one_int ) ) ) ) ) ).
% inth.simps
thf(fact_477_Euclid__induct,axiom,
! [P: nat > nat > $o,A2: nat,B2: nat] :
( ! [A: nat,B: nat] :
( ( P @ A @ B )
= ( P @ B @ A ) )
=> ( ! [A: nat] : ( P @ A @ zero_zero_nat )
=> ( ! [A: nat,B: nat] :
( ( P @ A @ B )
=> ( P @ A @ ( plus_plus_nat @ A @ B ) ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% Euclid_induct
thf(fact_478_exec1I,axiom,
! [C4: produc6425607678544837394st_int,P: list_instr,I2: int,S2: list_char > int,Stk2: list_int] :
( ( C4
= ( iexec @ ( inth_instr @ P @ I2 ) @ ( produc5086643055186798020st_int @ I2 @ ( produc8650753666468850689st_int @ S2 @ Stk2 ) ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ I2 )
=> ( ( ord_less_int @ I2 @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P ) ) )
=> ( exec1 @ P @ ( produc5086643055186798020st_int @ I2 @ ( produc8650753666468850689st_int @ S2 @ Stk2 ) ) @ C4 ) ) ) ) ).
% exec1I
thf(fact_479_com_Oinject_I3_J,axiom,
! [X41: bexp,X42: com,X43: com,Y41: bexp,Y42: com,Y43: com] :
( ( ( if @ X41 @ X42 @ X43 )
= ( if @ Y41 @ Y42 @ Y43 ) )
= ( ( X41 = Y41 )
& ( X42 = Y42 )
& ( X43 = Y43 ) ) ) ).
% com.inject(3)
thf(fact_480_instr_Oinject_I4_J,axiom,
! [X5: int,Y52: int] :
( ( ( jmp @ X5 )
= ( jmp @ Y52 ) )
= ( X5 = Y52 ) ) ).
% instr.inject(4)
thf(fact_481_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_482_add__less__cancel__left,axiom,
! [C: int,A2: int,B2: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B2 ) )
= ( ord_less_int @ A2 @ B2 ) ) ).
% add_less_cancel_left
thf(fact_483_add__less__cancel__left,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) )
= ( ord_less_nat @ A2 @ B2 ) ) ).
% add_less_cancel_left
thf(fact_484_add__less__cancel__right,axiom,
! [A2: int,C: int,B2: int] :
( ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ C ) )
= ( ord_less_int @ A2 @ B2 ) ) ).
% add_less_cancel_right
thf(fact_485_add__less__cancel__right,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
= ( ord_less_nat @ A2 @ B2 ) ) ).
% add_less_cancel_right
thf(fact_486_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_487_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_488_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A2: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A2 @ A2 ) )
= ( ord_less_int @ zero_zero_int @ A2 ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_489_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A2: int] :
( ( ord_less_int @ ( plus_plus_int @ A2 @ A2 ) @ zero_zero_int )
= ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_490_less__add__same__cancel2,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ ( plus_plus_int @ B2 @ A2 ) )
= ( ord_less_int @ zero_zero_int @ B2 ) ) ).
% less_add_same_cancel2
thf(fact_491_less__add__same__cancel2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ ( plus_plus_nat @ B2 @ A2 ) )
= ( ord_less_nat @ zero_zero_nat @ B2 ) ) ).
% less_add_same_cancel2
thf(fact_492_less__add__same__cancel1,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ ( plus_plus_int @ A2 @ B2 ) )
= ( ord_less_int @ zero_zero_int @ B2 ) ) ).
% less_add_same_cancel1
thf(fact_493_less__add__same__cancel1,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ ( plus_plus_nat @ A2 @ B2 ) )
= ( ord_less_nat @ zero_zero_nat @ B2 ) ) ).
% less_add_same_cancel1
thf(fact_494_add__less__same__cancel2,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ ( plus_plus_int @ A2 @ B2 ) @ B2 )
= ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% add_less_same_cancel2
thf(fact_495_add__less__same__cancel2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ B2 ) @ B2 )
= ( ord_less_nat @ A2 @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_496_add__less__same__cancel1,axiom,
! [B2: int,A2: int] :
( ( ord_less_int @ ( plus_plus_int @ B2 @ A2 ) @ B2 )
= ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% add_less_same_cancel1
thf(fact_497_add__less__same__cancel1,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B2 @ A2 ) @ B2 )
= ( ord_less_nat @ A2 @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_498_diff__gt__0__iff__gt,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A2 @ B2 ) )
= ( ord_less_int @ B2 @ A2 ) ) ).
% diff_gt_0_iff_gt
thf(fact_499_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_500_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_501_zle__add1__eq__le,axiom,
! [W2: int,Z: int] :
( ( ord_less_int @ W2 @ ( plus_plus_int @ Z @ one_one_int ) )
= ( ord_less_eq_int @ W2 @ Z ) ) ).
% zle_add1_eq_le
thf(fact_502_zle__diff1__eq,axiom,
! [W2: int,Z: int] :
( ( ord_less_eq_int @ W2 @ ( minus_minus_int @ Z @ one_one_int ) )
= ( ord_less_int @ W2 @ Z ) ) ).
% zle_diff1_eq
thf(fact_503_inth__append,axiom,
! [I2: int,Xs: list_instr,Ys: list_instr] :
( ( ord_less_eq_int @ zero_zero_int @ I2 )
=> ( ( ( ord_less_int @ I2 @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ Xs ) ) )
=> ( ( inth_instr @ ( append_instr @ Xs @ Ys ) @ I2 )
= ( inth_instr @ Xs @ I2 ) ) )
& ( ~ ( ord_less_int @ I2 @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ Xs ) ) )
=> ( ( inth_instr @ ( append_instr @ Xs @ Ys ) @ I2 )
= ( inth_instr @ Ys @ ( minus_minus_int @ I2 @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ Xs ) ) ) ) ) ) ) ) ).
% inth_append
thf(fact_504_inth__append,axiom,
! [I2: int,Xs: list_list_char,Ys: list_list_char] :
( ( ord_less_eq_int @ zero_zero_int @ I2 )
=> ( ( ( ord_less_int @ I2 @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ Xs ) ) )
=> ( ( inth_list_char @ ( append_list_char @ Xs @ Ys ) @ I2 )
= ( inth_list_char @ Xs @ I2 ) ) )
& ( ~ ( ord_less_int @ I2 @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ Xs ) ) )
=> ( ( inth_list_char @ ( append_list_char @ Xs @ Ys ) @ I2 )
= ( inth_list_char @ Ys @ ( minus_minus_int @ I2 @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ Xs ) ) ) ) ) ) ) ) ).
% inth_append
thf(fact_505_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_506_minf_I7_J,axiom,
! [T: int] :
? [Z4: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z4 )
=> ~ ( ord_less_int @ T @ X6 ) ) ).
% minf(7)
thf(fact_507_minf_I7_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z4 )
=> ~ ( ord_less_nat @ T @ X6 ) ) ).
% minf(7)
thf(fact_508_minf_I5_J,axiom,
! [T: int] :
? [Z4: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z4 )
=> ( ord_less_int @ X6 @ T ) ) ).
% minf(5)
thf(fact_509_minf_I5_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z4 )
=> ( ord_less_nat @ X6 @ T ) ) ).
% minf(5)
thf(fact_510_minf_I4_J,axiom,
! [T: int] :
? [Z4: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z4 )
=> ( X6 != T ) ) ).
% minf(4)
thf(fact_511_minf_I4_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z4 )
=> ( X6 != T ) ) ).
% minf(4)
thf(fact_512_minf_I3_J,axiom,
! [T: int] :
? [Z4: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z4 )
=> ( X6 != T ) ) ).
% minf(3)
thf(fact_513_minf_I3_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z4 )
=> ( X6 != T ) ) ).
% minf(3)
thf(fact_514_minf_I2_J,axiom,
! [P: int > $o,P3: int > $o,Q2: int > $o,Q3: int > $o] :
( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P3 @ X3 ) ) )
=> ( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z5 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z4: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z4 )
=> ( ( ( P @ X6 )
| ( Q2 @ X6 ) )
= ( ( P3 @ X6 )
| ( Q3 @ X6 ) ) ) ) ) ) ).
% minf(2)
thf(fact_515_minf_I2_J,axiom,
! [P: nat > $o,P3: nat > $o,Q2: nat > $o,Q3: nat > $o] :
( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P3 @ X3 ) ) )
=> ( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z4 )
=> ( ( ( P @ X6 )
| ( Q2 @ X6 ) )
= ( ( P3 @ X6 )
| ( Q3 @ X6 ) ) ) ) ) ) ).
% minf(2)
thf(fact_516_minf_I1_J,axiom,
! [P: int > $o,P3: int > $o,Q2: int > $o,Q3: int > $o] :
( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P3 @ X3 ) ) )
=> ( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z5 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z4: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z4 )
=> ( ( ( P @ X6 )
& ( Q2 @ X6 ) )
= ( ( P3 @ X6 )
& ( Q3 @ X6 ) ) ) ) ) ) ).
% minf(1)
thf(fact_517_minf_I1_J,axiom,
! [P: nat > $o,P3: nat > $o,Q2: nat > $o,Q3: nat > $o] :
( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P3 @ X3 ) ) )
=> ( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z4 )
=> ( ( ( P @ X6 )
& ( Q2 @ X6 ) )
= ( ( P3 @ X6 )
& ( Q3 @ X6 ) ) ) ) ) ) ).
% minf(1)
thf(fact_518_pinf_I7_J,axiom,
! [T: int] :
? [Z4: int] :
! [X6: int] :
( ( ord_less_int @ Z4 @ X6 )
=> ( ord_less_int @ T @ X6 ) ) ).
% pinf(7)
thf(fact_519_pinf_I7_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z4 @ X6 )
=> ( ord_less_nat @ T @ X6 ) ) ).
% pinf(7)
thf(fact_520_pinf_I5_J,axiom,
! [T: int] :
? [Z4: int] :
! [X6: int] :
( ( ord_less_int @ Z4 @ X6 )
=> ~ ( ord_less_int @ X6 @ T ) ) ).
% pinf(5)
thf(fact_521_pinf_I5_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z4 @ X6 )
=> ~ ( ord_less_nat @ X6 @ T ) ) ).
% pinf(5)
thf(fact_522_pinf_I4_J,axiom,
! [T: int] :
? [Z4: int] :
! [X6: int] :
( ( ord_less_int @ Z4 @ X6 )
=> ( X6 != T ) ) ).
% pinf(4)
thf(fact_523_pinf_I4_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z4 @ X6 )
=> ( X6 != T ) ) ).
% pinf(4)
thf(fact_524_pinf_I3_J,axiom,
! [T: int] :
? [Z4: int] :
! [X6: int] :
( ( ord_less_int @ Z4 @ X6 )
=> ( X6 != T ) ) ).
% pinf(3)
thf(fact_525_pinf_I3_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z4 @ X6 )
=> ( X6 != T ) ) ).
% pinf(3)
thf(fact_526_pinf_I2_J,axiom,
! [P: int > $o,P3: int > $o,Q2: int > $o,Q3: int > $o] :
( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P3 @ X3 ) ) )
=> ( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ Z5 @ X3 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z4: int] :
! [X6: int] :
( ( ord_less_int @ Z4 @ X6 )
=> ( ( ( P @ X6 )
| ( Q2 @ X6 ) )
= ( ( P3 @ X6 )
| ( Q3 @ X6 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_527_pinf_I2_J,axiom,
! [P: nat > $o,P3: nat > $o,Q2: nat > $o,Q3: nat > $o] :
( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P3 @ X3 ) ) )
=> ( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z4 @ X6 )
=> ( ( ( P @ X6 )
| ( Q2 @ X6 ) )
= ( ( P3 @ X6 )
| ( Q3 @ X6 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_528_pinf_I1_J,axiom,
! [P: int > $o,P3: int > $o,Q2: int > $o,Q3: int > $o] :
( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P3 @ X3 ) ) )
=> ( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ Z5 @ X3 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z4: int] :
! [X6: int] :
( ( ord_less_int @ Z4 @ X6 )
=> ( ( ( P @ X6 )
& ( Q2 @ X6 ) )
= ( ( P3 @ X6 )
& ( Q3 @ X6 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_529_pinf_I1_J,axiom,
! [P: nat > $o,P3: nat > $o,Q2: nat > $o,Q3: nat > $o] :
( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P3 @ X3 ) ) )
=> ( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z4 @ X6 )
=> ( ( ( P @ X6 )
& ( Q2 @ X6 ) )
= ( ( P3 @ X6 )
& ( Q3 @ X6 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_530_lt__ex,axiom,
! [X: int] :
? [Y3: int] : ( ord_less_int @ Y3 @ X ) ).
% lt_ex
thf(fact_531_gt__ex,axiom,
! [X: int] :
? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).
% gt_ex
thf(fact_532_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_533_less__imp__neq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_534_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_535_order_Oasym,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ~ ( ord_less_int @ B2 @ A2 ) ) ).
% order.asym
thf(fact_536_order_Oasym,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ~ ( ord_less_nat @ B2 @ A2 ) ) ).
% order.asym
thf(fact_537_ord__eq__less__trans,axiom,
! [A2: int,B2: int,C: int] :
( ( A2 = B2 )
=> ( ( ord_less_int @ B2 @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_538_ord__eq__less__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( A2 = B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_539_ord__less__eq__trans,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_540_ord__less__eq__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_541_less__induct,axiom,
! [P: nat > $o,A2: nat] :
( ! [X3: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X3 )
=> ( P @ Y5 ) )
=> ( P @ X3 ) )
=> ( P @ A2 ) ) ).
% less_induct
thf(fact_542_antisym__conv3,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_int @ Y @ X )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_543_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_544_linorder__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_545_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_546_dual__order_Oasym,axiom,
! [B2: int,A2: int] :
( ( ord_less_int @ B2 @ A2 )
=> ~ ( ord_less_int @ A2 @ B2 ) ) ).
% dual_order.asym
thf(fact_547_dual__order_Oasym,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ~ ( ord_less_nat @ A2 @ B2 ) ) ).
% dual_order.asym
thf(fact_548_dual__order_Oirrefl,axiom,
! [A2: int] :
~ ( ord_less_int @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_549_dual__order_Oirrefl,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_550_exists__least__iff,axiom,
( ( ^ [P4: nat > $o] :
? [X7: nat] : ( P4 @ X7 ) )
= ( ^ [P5: nat > $o] :
? [N3: nat] :
( ( P5 @ N3 )
& ! [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ~ ( P5 @ M3 ) ) ) ) ) ).
% exists_least_iff
thf(fact_551_linorder__less__wlog,axiom,
! [P: int > int > $o,A2: int,B2: int] :
( ! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( P @ A @ B ) )
=> ( ! [A: int] : ( P @ A @ A )
=> ( ! [A: int,B: int] :
( ( P @ B @ A )
=> ( P @ A @ B ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_552_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B2: nat] :
( ! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( P @ A @ B ) )
=> ( ! [A: nat] : ( P @ A @ A )
=> ( ! [A: nat,B: nat] :
( ( P @ B @ A )
=> ( P @ A @ B ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_553_order_Ostrict__trans,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_int @ B2 @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_554_order_Ostrict__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_555_not__less__iff__gr__or__eq,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ( ord_less_int @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_556_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ( ord_less_nat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_557_dual__order_Ostrict__trans,axiom,
! [B2: int,A2: int,C: int] :
( ( ord_less_int @ B2 @ A2 )
=> ( ( ord_less_int @ C @ B2 )
=> ( ord_less_int @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_558_dual__order_Ostrict__trans,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ( ord_less_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_559_order_Ostrict__implies__not__eq,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( A2 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_560_order_Ostrict__implies__not__eq,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( A2 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_561_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: int,A2: int] :
( ( ord_less_int @ B2 @ A2 )
=> ( A2 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_562_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( A2 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_563_linorder__neqE,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_564_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_565_order__less__asym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_asym
thf(fact_566_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_567_linorder__neq__iff,axiom,
! [X: int,Y: int] :
( ( X != Y )
= ( ( ord_less_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_568_linorder__neq__iff,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
= ( ( ord_less_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_569_order__less__asym_H,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ~ ( ord_less_int @ B2 @ A2 ) ) ).
% order_less_asym'
thf(fact_570_order__less__asym_H,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ~ ( ord_less_nat @ B2 @ A2 ) ) ).
% order_less_asym'
thf(fact_571_order__less__trans,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z )
=> ( ord_less_int @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_572_order__less__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_573_ord__eq__less__subst,axiom,
! [A2: int,F: int > int,B2: int,C: int] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_574_ord__eq__less__subst,axiom,
! [A2: nat,F: int > nat,B2: int,C: int] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_575_ord__eq__less__subst,axiom,
! [A2: int,F: nat > int,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_576_ord__eq__less__subst,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_577_ord__less__eq__subst,axiom,
! [A2: int,B2: int,F: int > int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_578_ord__less__eq__subst,axiom,
! [A2: int,B2: int,F: int > nat,C: nat] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_579_ord__less__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_580_ord__less__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_581_order__less__irrefl,axiom,
! [X: int] :
~ ( ord_less_int @ X @ X ) ).
% order_less_irrefl
thf(fact_582_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_583_order__less__subst1,axiom,
! [A2: int,F: int > int,B2: int,C: int] :
( ( ord_less_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_584_order__less__subst1,axiom,
! [A2: int,F: nat > int,B2: nat,C: nat] :
( ( ord_less_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_585_order__less__subst1,axiom,
! [A2: nat,F: int > nat,B2: int,C: int] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_586_order__less__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_587_order__less__subst2,axiom,
! [A2: int,B2: int,F: int > int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_588_order__less__subst2,axiom,
! [A2: int,B2: int,F: int > nat,C: nat] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_589_order__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_590_order__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_591_order__less__not__sym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_592_order__less__not__sym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_593_order__less__imp__triv,axiom,
! [X: int,Y: int,P: $o] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_594_order__less__imp__triv,axiom,
! [X: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_595_linorder__less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
| ( X = Y )
| ( ord_less_int @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_596_linorder__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_597_order__less__imp__not__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_598_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_599_order__less__imp__not__eq2,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_600_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_601_order__less__imp__not__less,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_602_order__less__imp__not__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_603_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_604_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_605_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_606_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_607_verit__comp__simplify1_I1_J,axiom,
! [A2: int] :
~ ( ord_less_int @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_608_verit__comp__simplify1_I1_J,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_609_verit__comp__simplify1_I3_J,axiom,
! [B3: nat,A3: nat] :
( ( ~ ( ord_less_eq_nat @ B3 @ A3 ) )
= ( ord_less_nat @ A3 @ B3 ) ) ).
% verit_comp_simplify1(3)
thf(fact_610_verit__comp__simplify1_I3_J,axiom,
! [B3: int,A3: int] :
( ( ~ ( ord_less_eq_int @ B3 @ A3 ) )
= ( ord_less_int @ A3 @ B3 ) ) ).
% verit_comp_simplify1(3)
thf(fact_611_order__le__imp__less__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_612_order__le__imp__less__or__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_613_linorder__le__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_614_linorder__le__less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_615_order__less__le__subst2,axiom,
! [A2: int,B2: int,F: int > nat,C: nat] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_616_order__less__le__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_617_order__less__le__subst2,axiom,
! [A2: int,B2: int,F: int > int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ ( F @ B2 ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_618_order__less__le__subst2,axiom,
! [A2: nat,B2: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_int @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_619_order__less__le__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_620_order__less__le__subst1,axiom,
! [A2: int,F: nat > int,B2: nat,C: nat] :
( ( ord_less_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_621_order__less__le__subst1,axiom,
! [A2: nat,F: int > nat,B2: int,C: int] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_622_order__less__le__subst1,axiom,
! [A2: int,F: int > int,B2: int,C: int] :
( ( ord_less_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_623_order__le__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_624_order__le__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_625_order__le__less__subst2,axiom,
! [A2: int,B2: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_626_order__le__less__subst2,axiom,
! [A2: int,B2: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_627_order__le__less__subst1,axiom,
! [A2: nat,F: int > nat,B2: int,C: int] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_628_order__le__less__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_629_order__le__less__subst1,axiom,
! [A2: int,F: int > int,B2: int,C: int] :
( ( ord_less_eq_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_630_order__le__less__subst1,axiom,
! [A2: int,F: nat > int,B2: nat,C: nat] :
( ( ord_less_eq_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_631_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_632_order__less__le__trans,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z )
=> ( ord_less_int @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_633_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_634_order__le__less__trans,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z )
=> ( ord_less_int @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_635_order__neq__le__trans,axiom,
! [A2: nat,B2: nat] :
( ( A2 != B2 )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ord_less_nat @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_636_order__neq__le__trans,axiom,
! [A2: int,B2: int] :
( ( A2 != B2 )
=> ( ( ord_less_eq_int @ A2 @ B2 )
=> ( ord_less_int @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_637_order__le__neq__trans,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_nat @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_638_order__le__neq__trans,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_int @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_639_order__less__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_640_order__less__imp__le,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_641_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_642_linorder__not__less,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_not_less
thf(fact_643_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_644_linorder__not__le,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_eq_int @ X @ Y ) )
= ( ord_less_int @ Y @ X ) ) ).
% linorder_not_le
thf(fact_645_order__less__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
& ( X2 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_646_order__less__le,axiom,
( ord_less_int
= ( ^ [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
& ( X2 != Y2 ) ) ) ) ).
% order_less_le
thf(fact_647_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_648_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ) ).
% order_le_less
thf(fact_649_dual__order_Ostrict__implies__order,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ord_less_eq_nat @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_650_dual__order_Ostrict__implies__order,axiom,
! [B2: int,A2: int] :
( ( ord_less_int @ B2 @ A2 )
=> ( ord_less_eq_int @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_651_order_Ostrict__implies__order,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_652_order_Ostrict__implies__order,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ord_less_eq_int @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_653_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B4: nat,A4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ~ ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_654_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B4: int,A4: int] :
( ( ord_less_eq_int @ B4 @ A4 )
& ~ ( ord_less_eq_int @ A4 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_655_dual__order_Ostrict__trans2,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_656_dual__order_Ostrict__trans2,axiom,
! [B2: int,A2: int,C: int] :
( ( ord_less_int @ B2 @ A2 )
=> ( ( ord_less_eq_int @ C @ B2 )
=> ( ord_less_int @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_657_dual__order_Ostrict__trans1,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_658_dual__order_Ostrict__trans1,axiom,
! [B2: int,A2: int,C: int] :
( ( ord_less_eq_int @ B2 @ A2 )
=> ( ( ord_less_int @ C @ B2 )
=> ( ord_less_int @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_659_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B4: nat,A4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_660_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B4: int,A4: int] :
( ( ord_less_eq_int @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_661_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A4: nat] :
( ( ord_less_nat @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_662_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B4: int,A4: int] :
( ( ord_less_int @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_663_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ~ ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_664_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A4: int,B4: int] :
( ( ord_less_eq_int @ A4 @ B4 )
& ~ ( ord_less_eq_int @ B4 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_665_order_Ostrict__trans2,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_666_order_Ostrict__trans2,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_667_order_Ostrict__trans1,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_668_order_Ostrict__trans1,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_int @ B2 @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_669_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_670_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A4: int,B4: int] :
( ( ord_less_eq_int @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_671_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_nat @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_672_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A4: int,B4: int] :
( ( ord_less_int @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_673_not__le__imp__less,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y @ X )
=> ( ord_less_nat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_674_not__le__imp__less,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_eq_int @ Y @ X )
=> ( ord_less_int @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_675_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
& ~ ( ord_less_eq_nat @ Y2 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_676_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
& ~ ( ord_less_eq_int @ Y2 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_677_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_678_antisym__conv2,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_679_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_680_antisym__conv1,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_681_nless__le,axiom,
! [A2: nat,B2: nat] :
( ( ~ ( ord_less_nat @ A2 @ B2 ) )
= ( ~ ( ord_less_eq_nat @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_682_nless__le,axiom,
! [A2: int,B2: int] :
( ( ~ ( ord_less_int @ A2 @ B2 ) )
= ( ~ ( ord_less_eq_int @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_683_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_684_leI,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% leI
thf(fact_685_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_686_leD,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_int @ X @ Y ) ) ).
% leD
thf(fact_687_pinf_I6_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z4 @ X6 )
=> ~ ( ord_less_eq_nat @ X6 @ T ) ) ).
% pinf(6)
thf(fact_688_pinf_I6_J,axiom,
! [T: int] :
? [Z4: int] :
! [X6: int] :
( ( ord_less_int @ Z4 @ X6 )
=> ~ ( ord_less_eq_int @ X6 @ T ) ) ).
% pinf(6)
thf(fact_689_pinf_I8_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z4 @ X6 )
=> ( ord_less_eq_nat @ T @ X6 ) ) ).
% pinf(8)
thf(fact_690_pinf_I8_J,axiom,
! [T: int] :
? [Z4: int] :
! [X6: int] :
( ( ord_less_int @ Z4 @ X6 )
=> ( ord_less_eq_int @ T @ X6 ) ) ).
% pinf(8)
thf(fact_691_minf_I6_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z4 )
=> ( ord_less_eq_nat @ X6 @ T ) ) ).
% minf(6)
thf(fact_692_minf_I6_J,axiom,
! [T: int] :
? [Z4: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z4 )
=> ( ord_less_eq_int @ X6 @ T ) ) ).
% minf(6)
thf(fact_693_minf_I8_J,axiom,
! [T: nat] :
? [Z4: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z4 )
=> ~ ( ord_less_eq_nat @ T @ X6 ) ) ).
% minf(8)
thf(fact_694_minf_I8_J,axiom,
! [T: int] :
? [Z4: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z4 )
=> ~ ( ord_less_eq_int @ T @ X6 ) ) ).
% minf(8)
thf(fact_695_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_696_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_697_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_698_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_699_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_700_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_701_add__mono__thms__linordered__field_I5_J,axiom,
! [I2: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I2 @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_702_add__mono__thms__linordered__field_I5_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I2 @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_703_add__mono__thms__linordered__field_I2_J,axiom,
! [I2: int,J: int,K: int,L: int] :
( ( ( I2 = J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_704_add__mono__thms__linordered__field_I2_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( I2 = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_705_add__mono__thms__linordered__field_I1_J,axiom,
! [I2: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I2 @ J )
& ( K = L ) )
=> ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_706_add__mono__thms__linordered__field_I1_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I2 @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_707_add__strict__mono,axiom,
! [A2: int,B2: int,C: int,D: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ D ) ) ) ) ).
% add_strict_mono
thf(fact_708_add__strict__mono,axiom,
! [A2: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).
% add_strict_mono
thf(fact_709_add__strict__left__mono,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ord_less_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B2 ) ) ) ).
% add_strict_left_mono
thf(fact_710_add__strict__left__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) ) ) ).
% add_strict_left_mono
thf(fact_711_add__strict__right__mono,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ C ) ) ) ).
% add_strict_right_mono
thf(fact_712_add__strict__right__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add_strict_right_mono
thf(fact_713_add__less__imp__less__left,axiom,
! [C: int,A2: int,B2: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B2 ) )
=> ( ord_less_int @ A2 @ B2 ) ) ).
% add_less_imp_less_left
thf(fact_714_add__less__imp__less__left,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) )
=> ( ord_less_nat @ A2 @ B2 ) ) ).
% add_less_imp_less_left
thf(fact_715_add__less__imp__less__right,axiom,
! [A2: int,C: int,B2: int] :
( ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ C ) )
=> ( ord_less_int @ A2 @ B2 ) ) ).
% add_less_imp_less_right
thf(fact_716_add__less__imp__less__right,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
=> ( ord_less_nat @ A2 @ B2 ) ) ).
% add_less_imp_less_right
thf(fact_717_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_718_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_719_diff__strict__mono,axiom,
! [A2: int,B2: int,D: int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_int @ D @ C )
=> ( ord_less_int @ ( minus_minus_int @ A2 @ C ) @ ( minus_minus_int @ B2 @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_720_diff__eq__diff__less,axiom,
! [A2: int,B2: int,C: int,D: int] :
( ( ( minus_minus_int @ A2 @ B2 )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_int @ A2 @ B2 )
= ( ord_less_int @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_721_diff__strict__left__mono,axiom,
! [B2: int,A2: int,C: int] :
( ( ord_less_int @ B2 @ A2 )
=> ( ord_less_int @ ( minus_minus_int @ C @ A2 ) @ ( minus_minus_int @ C @ B2 ) ) ) ).
% diff_strict_left_mono
thf(fact_722_diff__strict__right__mono,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ord_less_int @ ( minus_minus_int @ A2 @ C ) @ ( minus_minus_int @ B2 @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_723_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_724_instr_Osize_I12_J,axiom,
! [X5: int] :
( ( size_size_instr @ ( jmp @ X5 ) )
= zero_zero_nat ) ).
% instr.size(12)
thf(fact_725_com_Odistinct_I11_J,axiom,
! [X21: list_char,X22: aexp,X41: bexp,X42: com,X43: com] :
( ( assign @ X21 @ X22 )
!= ( if @ X41 @ X42 @ X43 ) ) ).
% com.distinct(11)
thf(fact_726_com_Odistinct_I5_J,axiom,
! [X41: bexp,X42: com,X43: com] :
( skip
!= ( if @ X41 @ X42 @ X43 ) ) ).
% com.distinct(5)
thf(fact_727_add__mono__thms__linordered__field_I4_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I2 @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_728_add__mono__thms__linordered__field_I4_J,axiom,
! [I2: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I2 @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_729_add__mono__thms__linordered__field_I3_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I2 @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_730_add__mono__thms__linordered__field_I3_J,axiom,
! [I2: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I2 @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_731_add__le__less__mono,axiom,
! [A2: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_732_add__le__less__mono,axiom,
! [A2: int,B2: int,C: int,D: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_733_add__less__le__mono,axiom,
! [A2: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_734_add__less__le__mono,axiom,
! [A2: int,B2: int,C: int,D: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_735_pos__add__strict,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ B2 @ C )
=> ( ord_less_int @ B2 @ ( plus_plus_int @ A2 @ C ) ) ) ) ).
% pos_add_strict
thf(fact_736_pos__add__strict,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ B2 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% pos_add_strict
thf(fact_737_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ~ ! [C2: nat] :
( ( B2
= ( plus_plus_nat @ A2 @ C2 ) )
=> ( C2 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_738_add__pos__pos,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A2 @ B2 ) ) ) ) ).
% add_pos_pos
thf(fact_739_add__pos__pos,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).
% add_pos_pos
thf(fact_740_add__neg__neg,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ zero_zero_int )
=> ( ( ord_less_int @ B2 @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).
% add_neg_neg
thf(fact_741_add__neg__neg,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_nat @ B2 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_742_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_743_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_744_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_745_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_746_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_747_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_748_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_749_less__iff__diff__less__0,axiom,
( ord_less_int
= ( ^ [A4: int,B4: int] : ( ord_less_int @ ( minus_minus_int @ A4 @ B4 ) @ zero_zero_int ) ) ) ).
% less_iff_diff_less_0
thf(fact_750_less__add__one,axiom,
! [A2: int] : ( ord_less_int @ A2 @ ( plus_plus_int @ A2 @ one_one_int ) ) ).
% less_add_one
thf(fact_751_less__add__one,axiom,
! [A2: nat] : ( ord_less_nat @ A2 @ ( plus_plus_nat @ A2 @ one_one_nat ) ) ).
% less_add_one
thf(fact_752_add__mono1,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ one_one_int ) @ ( plus_plus_int @ B2 @ one_one_int ) ) ) ).
% add_mono1
thf(fact_753_add__mono1,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ one_one_nat ) @ ( plus_plus_nat @ B2 @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_754_less__diff__eq,axiom,
! [A2: int,C: int,B2: int] :
( ( ord_less_int @ A2 @ ( minus_minus_int @ C @ B2 ) )
= ( ord_less_int @ ( plus_plus_int @ A2 @ B2 ) @ C ) ) ).
% less_diff_eq
thf(fact_755_diff__less__eq,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_int @ ( minus_minus_int @ A2 @ B2 ) @ C )
= ( ord_less_int @ A2 @ ( plus_plus_int @ C @ B2 ) ) ) ).
% diff_less_eq
thf(fact_756_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A2: int,B2: int] :
( ~ ( ord_less_int @ A2 @ B2 )
=> ( ( plus_plus_int @ B2 @ ( minus_minus_int @ A2 @ B2 ) )
= A2 ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_757_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A2: nat,B2: nat] :
( ~ ( ord_less_nat @ A2 @ B2 )
=> ( ( plus_plus_nat @ B2 @ ( minus_minus_nat @ A2 @ B2 ) )
= A2 ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_758_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_759_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_760_int__gr__induct,axiom,
! [K: int,I2: int,P: int > $o] :
( ( ord_less_int @ K @ I2 )
=> ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
=> ( ! [I: int] :
( ( ord_less_int @ K @ I )
=> ( ( P @ I )
=> ( P @ ( plus_plus_int @ I @ one_one_int ) ) ) )
=> ( P @ I2 ) ) ) ) ).
% int_gr_induct
thf(fact_761_zless__add1__eq,axiom,
! [W2: int,Z: int] :
( ( ord_less_int @ W2 @ ( plus_plus_int @ Z @ one_one_int ) )
= ( ( ord_less_int @ W2 @ Z )
| ( W2 = Z ) ) ) ).
% zless_add1_eq
thf(fact_762_int__less__induct,axiom,
! [I2: int,K: int,P: int > $o] :
( ( ord_less_int @ I2 @ K )
=> ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
=> ( ! [I: int] :
( ( ord_less_int @ I @ K )
=> ( ( P @ I )
=> ( P @ ( minus_minus_int @ I @ one_one_int ) ) ) )
=> ( P @ I2 ) ) ) ) ).
% int_less_induct
thf(fact_763_add__neg__nonpos,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_764_add__neg__nonpos,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ B2 @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).
% add_neg_nonpos
thf(fact_765_add__nonneg__pos,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).
% add_nonneg_pos
thf(fact_766_add__nonneg__pos,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A2 @ B2 ) ) ) ) ).
% add_nonneg_pos
thf(fact_767_add__nonpos__neg,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_nat @ B2 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_768_add__nonpos__neg,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ zero_zero_int )
=> ( ( ord_less_int @ B2 @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).
% add_nonpos_neg
thf(fact_769_add__pos__nonneg,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).
% add_pos_nonneg
thf(fact_770_add__pos__nonneg,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A2 @ B2 ) ) ) ) ).
% add_pos_nonneg
thf(fact_771_add__strict__increasing,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_nat @ B2 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_772_add__strict__increasing,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ord_less_int @ B2 @ ( plus_plus_int @ A2 @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_773_add__strict__increasing2,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ B2 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_774_add__strict__increasing2,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ B2 @ C )
=> ( ord_less_int @ B2 @ ( plus_plus_int @ A2 @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_775_zero__less__two,axiom,
ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% zero_less_two
thf(fact_776_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_777_int__one__le__iff__zero__less,axiom,
! [Z: int] :
( ( ord_less_eq_int @ one_one_int @ Z )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% int_one_le_iff_zero_less
thf(fact_778_odd__less__0__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z ) @ zero_zero_int )
= ( ord_less_int @ Z @ zero_zero_int ) ) ).
% odd_less_0_iff
thf(fact_779_add1__zle__eq,axiom,
! [W2: int,Z: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W2 @ one_one_int ) @ Z )
= ( ord_less_int @ W2 @ Z ) ) ).
% add1_zle_eq
thf(fact_780_zless__imp__add1__zle,axiom,
! [W2: int,Z: int] :
( ( ord_less_int @ W2 @ Z )
=> ( ord_less_eq_int @ ( plus_plus_int @ W2 @ one_one_int ) @ Z ) ) ).
% zless_imp_add1_zle
thf(fact_781_le__imp__0__less,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z ) ) ) ).
% le_imp_0_less
thf(fact_782_int__ops_I6_J,axiom,
! [A2: nat,B2: nat] :
( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A2 @ B2 ) )
= zero_zero_int ) )
& ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A2 @ B2 ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).
% int_ops(6)
thf(fact_783_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_784_to__m__state_Osimps,axiom,
( to_m_state
= ( ^ [Xs3: list_list_char,S6: list_char > int,A4: int] :
( if_int
@ ( ( ord_less_int @ zero_zero_int @ A4 )
& ( ord_less_eq_int @ A4 @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ Xs3 ) ) ) )
@ ( S6 @ ( inth_list_char @ Xs3 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ Xs3 ) ) @ A4 ) ) )
@ zero_zero_int ) ) ) ).
% to_m_state.simps
thf(fact_785_upto__aux__rec,axiom,
( upto_aux
= ( ^ [I3: int,J2: int,Js: list_int] : ( if_list_int @ ( ord_less_int @ J2 @ I3 ) @ Js @ ( upto_aux @ I3 @ ( minus_minus_int @ J2 @ one_one_int ) @ ( cons_int @ J2 @ Js ) ) ) ) ) ).
% upto_aux_rec
thf(fact_786_bot__nat__0_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A2 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_787_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_788_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_789_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_790_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_791_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_792_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_793_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_794_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_795_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_796_less__not__refl3,axiom,
! [S2: nat,T: nat] :
( ( ord_less_nat @ S2 @ T )
=> ( S2 != T ) ) ).
% less_not_refl3
thf(fact_797_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_798_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M4: nat] :
( ( ord_less_nat @ M4 @ N2 )
=> ( P @ M4 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_799_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N2 )
& ~ ( P @ M4 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_800_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_801_bot__nat__0_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_802_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_803_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_804_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_805_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_806_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_807_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N2 )
& ~ ( P @ M4 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_808_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_809_length__induct,axiom,
! [P: list_list_char > $o,Xs: list_list_char] :
( ! [Xs2: list_list_char] :
( ! [Ys2: list_list_char] :
( ( ord_less_nat @ ( size_s356637359517785349t_char @ Ys2 ) @ ( size_s356637359517785349t_char @ Xs2 ) )
=> ( P @ Ys2 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_810_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
& ( M3 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_811_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_812_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
| ( M3 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_813_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_814_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_815_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I2: nat,J: nat] :
( ! [I: nat,J4: nat] :
( ( ord_less_nat @ I @ J4 )
=> ( ord_less_nat @ ( F @ I ) @ ( F @ J4 ) ) )
=> ( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_816_add__lessD1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I2 @ J ) @ K )
=> ( ord_less_nat @ I2 @ K ) ) ).
% add_lessD1
thf(fact_817_add__less__mono,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_818_not__add__less1,axiom,
! [I2: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I2 @ J ) @ I2 ) ).
% not_add_less1
thf(fact_819_not__add__less2,axiom,
! [J: nat,I2: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I2 ) @ I2 ) ).
% not_add_less2
thf(fact_820_add__less__mono1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_821_trans__less__add1,axiom,
! [I2: nat,J: nat,M: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ I2 @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_822_trans__less__add2,axiom,
! [I2: nat,J: nat,M: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ I2 @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_823_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_824_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_825_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_826_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
& ! [I6: nat] :
( ( ord_less_nat @ I6 @ K3 )
=> ~ ( P @ I6 ) )
& ( P @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_827_less__imp__add__positive,axiom,
! [I2: nat,J: nat] :
( ( ord_less_nat @ I2 @ J )
=> ? [K3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ( plus_plus_nat @ I2 @ K3 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_828_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_829_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_nat @ ( F @ M2 ) @ ( F @ N2 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_830_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_831_diff__less__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_nat @ ( minus_minus_nat @ A2 @ C ) @ ( minus_minus_nat @ B2 @ C ) ) ) ) ).
% diff_less_mono
thf(fact_832_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_833_less__diff__conv,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_834_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A4: nat,B4: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_835_nat__less__as__int,axiom,
( ord_less_nat
= ( ^ [A4: nat,B4: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_less_as_int
thf(fact_836_nat__diff__split,axiom,
! [P: nat > $o,A2: nat,B2: nat] :
( ( P @ ( minus_minus_nat @ A2 @ B2 ) )
= ( ( ( ord_less_nat @ A2 @ B2 )
=> ( P @ zero_zero_nat ) )
& ! [D2: nat] :
( ( A2
= ( plus_plus_nat @ B2 @ D2 ) )
=> ( P @ D2 ) ) ) ) ).
% nat_diff_split
thf(fact_837_nat__diff__split__asm,axiom,
! [P: nat > $o,A2: nat,B2: nat] :
( ( P @ ( minus_minus_nat @ A2 @ B2 ) )
= ( ~ ( ( ( ord_less_nat @ A2 @ B2 )
& ~ ( P @ zero_zero_nat ) )
| ? [D2: nat] :
( ( A2
= ( plus_plus_nat @ B2 @ D2 ) )
& ~ ( P @ D2 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_838_less__diff__conv2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I2 )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I2 @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_839_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_840_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_841_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_842_JmpE,axiom,
! [I2: int,Pc: int,S2: list_char > int,Stk2: list_int,Cf: produc6425607678544837394st_int] :
( ( iexec_pred @ ( produc1484136438817787418st_int @ ( jmp @ I2 ) @ ( produc5086643055186798020st_int @ Pc @ ( produc8650753666468850689st_int @ S2 @ Stk2 ) ) ) @ Cf )
=> ( Cf
= ( produc5086643055186798020st_int @ ( plus_plus_int @ ( plus_plus_int @ Pc @ one_one_int ) @ I2 ) @ ( produc8650753666468850689st_int @ S2 @ Stk2 ) ) ) ) ).
% JmpE
thf(fact_843_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M5: nat] :
( ( P @ X )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M5 ) )
=> ~ ! [M2: nat] :
( ( P @ M2 )
=> ~ ! [X6: nat] :
( ( P @ X6 )
=> ( ord_less_eq_nat @ X6 @ M2 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_844_iexec__pred_Ocases,axiom,
! [A12: produc5995290525303592096st_int,A23: produc6425607678544837394st_int] :
( ( iexec_pred @ A12 @ A23 )
=> ~ ! [Ins: instr,Cf2: produc6425607678544837394st_int] :
( ( A12
= ( produc1484136438817787418st_int @ Ins @ Cf2 ) )
=> ( A23
!= ( iexec @ Ins @ Cf2 ) ) ) ) ).
% iexec_pred.cases
thf(fact_845_iexec__pred_Osimps,axiom,
( iexec_pred
= ( ^ [A1: produc5995290525303592096st_int,A22: produc6425607678544837394st_int] :
? [Ins3: instr,Cf3: produc6425607678544837394st_int] :
( ( A1
= ( produc1484136438817787418st_int @ Ins3 @ Cf3 ) )
& ( A22
= ( iexec @ Ins3 @ Cf3 ) ) ) ) ) ).
% iexec_pred.simps
thf(fact_846_iexec__pred_Ointros,axiom,
! [Ins2: instr,Cf: produc6425607678544837394st_int] : ( iexec_pred @ ( produc1484136438817787418st_int @ Ins2 @ Cf ) @ ( iexec @ Ins2 @ Cf ) ) ).
% iexec_pred.intros
thf(fact_847_iexec__simp,axiom,
! [Ins2: instr,Cf: produc6425607678544837394st_int,Cf4: produc6425607678544837394st_int] :
( ( iexec_pred @ ( produc1484136438817787418st_int @ Ins2 @ Cf ) @ Cf4 )
= ( Cf4
= ( iexec @ Ins2 @ Cf ) ) ) ).
% iexec_simp
thf(fact_848_LoadIE,axiom,
! [I2: int,Pc: int,S2: list_char > int,Stk2: list_int,Cf: produc6425607678544837394st_int] :
( ( iexec_pred @ ( produc1484136438817787418st_int @ ( loadi @ I2 ) @ ( produc5086643055186798020st_int @ Pc @ ( produc8650753666468850689st_int @ S2 @ Stk2 ) ) ) @ Cf )
=> ( Cf
= ( produc5086643055186798020st_int @ ( plus_plus_int @ Pc @ one_one_int ) @ ( produc8650753666468850689st_int @ S2 @ ( cons_int @ I2 @ Stk2 ) ) ) ) ) ).
% LoadIE
thf(fact_849_LoadE,axiom,
! [X: list_char,Pc: int,S2: list_char > int,Stk2: list_int,Cf: produc6425607678544837394st_int] :
( ( iexec_pred @ ( produc1484136438817787418st_int @ ( load @ X ) @ ( produc5086643055186798020st_int @ Pc @ ( produc8650753666468850689st_int @ S2 @ Stk2 ) ) ) @ Cf )
=> ( Cf
= ( produc5086643055186798020st_int @ ( plus_plus_int @ Pc @ one_one_int ) @ ( produc8650753666468850689st_int @ S2 @ ( cons_int @ ( S2 @ X ) @ Stk2 ) ) ) ) ) ).
% LoadE
thf(fact_850_instr_Oinject_I2_J,axiom,
! [X23: list_char,Y23: list_char] :
( ( ( load @ X23 )
= ( load @ Y23 ) )
= ( X23 = Y23 ) ) ).
% instr.inject(2)
thf(fact_851_instr_Oinject_I1_J,axiom,
! [X1: int,Y1: int] :
( ( ( loadi @ X1 )
= ( loadi @ Y1 ) )
= ( X1 = Y1 ) ) ).
% instr.inject(1)
thf(fact_852_instr_Odistinct_I1_J,axiom,
! [X1: int,X23: list_char] :
( ( loadi @ X1 )
!= ( load @ X23 ) ) ).
% instr.distinct(1)
thf(fact_853_instr_Odistinct_I17_J,axiom,
! [X23: list_char,X5: int] :
( ( load @ X23 )
!= ( jmp @ X5 ) ) ).
% instr.distinct(17)
thf(fact_854_instr_Odistinct_I7_J,axiom,
! [X1: int,X5: int] :
( ( loadi @ X1 )
!= ( jmp @ X5 ) ) ).
% instr.distinct(7)
thf(fact_855_instr_Osize_I9_J,axiom,
! [X23: list_char] :
( ( size_size_instr @ ( load @ X23 ) )
= zero_zero_nat ) ).
% instr.size(9)
thf(fact_856_instr_Osize_I8_J,axiom,
! [X1: int] :
( ( size_size_instr @ ( loadi @ X1 ) )
= zero_zero_nat ) ).
% instr.size(8)
thf(fact_857_exec1__def,axiom,
( exec1
= ( ^ [P5: list_instr,Cf3: produc6425607678544837394st_int,Cf5: produc6425607678544837394st_int] :
( ( iexec_pred @ ( produc1484136438817787418st_int @ ( inth_instr @ P5 @ ( produc8941309049508465880st_int @ Cf3 ) ) @ Cf3 ) @ Cf5 )
& ( ord_less_eq_int @ zero_zero_int @ ( produc8941309049508465880st_int @ Cf3 ) )
& ( ord_less_int @ ( produc8941309049508465880st_int @ Cf3 ) @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P5 ) ) ) ) ) ) ).
% exec1_def
thf(fact_858_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_859_bval_Osimps_I3_J,axiom,
! [B_1: bexp,B_2: bexp,S2: list_char > int] :
( ( bval @ ( and @ B_1 @ B_2 ) @ S2 )
= ( ( bval @ B_1 @ S2 )
& ( bval @ B_2 @ S2 ) ) ) ).
% bval.simps(3)
thf(fact_860_fst__conv,axiom,
! [X1: $o,X23: int] :
( ( product_fst_o_int @ ( product_Pair_o_int @ X1 @ X23 ) )
= X1 ) ).
% fst_conv
thf(fact_861_fst__conv,axiom,
! [X1: int,X23: int] :
( ( product_fst_int_int @ ( product_Pair_int_int @ X1 @ X23 ) )
= X1 ) ).
% fst_conv
thf(fact_862_fst__conv,axiom,
! [X1: list_char > int,X23: list_int] :
( ( produc2155683374303800813st_int @ ( produc8650753666468850689st_int @ X1 @ X23 ) )
= X1 ) ).
% fst_conv
thf(fact_863_fst__conv,axiom,
! [X1: list_char > int,X23: int] :
( ( produc9025023058967219549nt_int @ ( produc5790713362662368625nt_int @ X1 @ X23 ) )
= X1 ) ).
% fst_conv
thf(fact_864_fst__conv,axiom,
! [X1: int,X23: produc369741286924889929st_int] :
( ( produc8941309049508465880st_int @ ( produc5086643055186798020st_int @ X1 @ X23 ) )
= X1 ) ).
% fst_conv
thf(fact_865_fst__eqD,axiom,
! [X: $o,Y: int,A2: $o] :
( ( ( product_fst_o_int @ ( product_Pair_o_int @ X @ Y ) )
= A2 )
=> ( X = A2 ) ) ).
% fst_eqD
thf(fact_866_fst__eqD,axiom,
! [X: int,Y: int,A2: int] :
( ( ( product_fst_int_int @ ( product_Pair_int_int @ X @ Y ) )
= A2 )
=> ( X = A2 ) ) ).
% fst_eqD
thf(fact_867_fst__eqD,axiom,
! [X: list_char > int,Y: list_int,A2: list_char > int] :
( ( ( produc2155683374303800813st_int @ ( produc8650753666468850689st_int @ X @ Y ) )
= A2 )
=> ( X = A2 ) ) ).
% fst_eqD
thf(fact_868_fst__eqD,axiom,
! [X: list_char > int,Y: int,A2: list_char > int] :
( ( ( produc9025023058967219549nt_int @ ( produc5790713362662368625nt_int @ X @ Y ) )
= A2 )
=> ( X = A2 ) ) ).
% fst_eqD
thf(fact_869_fst__eqD,axiom,
! [X: int,Y: produc369741286924889929st_int,A2: int] :
( ( ( produc8941309049508465880st_int @ ( produc5086643055186798020st_int @ X @ Y ) )
= A2 )
=> ( X = A2 ) ) ).
% fst_eqD
thf(fact_870_accp__subset,axiom,
! [R1: produc4189061985984568957nt_int > produc4189061985984568957nt_int > $o,R22: produc4189061985984568957nt_int > produc4189061985984568957nt_int > $o] :
( ( ord_le1819137518486948652_int_o @ R1 @ R22 )
=> ( ord_le3703957343819488690_int_o @ ( accp_P4562858270263085236nt_int @ R22 ) @ ( accp_P4562858270263085236nt_int @ R1 ) ) ) ).
% accp_subset
thf(fact_871_accp__subset,axiom,
! [R1: product_prod_int_int > product_prod_int_int > $o,R22: product_prod_int_int > product_prod_int_int > $o] :
( ( ord_le1598226405681992910_int_o @ R1 @ R22 )
=> ( ord_le8369615600986905444_int_o @ ( accp_P1096762738010456898nt_int @ R22 ) @ ( accp_P1096762738010456898nt_int @ R1 ) ) ) ).
% accp_subset
thf(fact_872_accp__subset__induct,axiom,
! [D3: produc4189061985984568957nt_int > $o,R3: produc4189061985984568957nt_int > produc4189061985984568957nt_int > $o,X: produc4189061985984568957nt_int,P: produc4189061985984568957nt_int > $o] :
( ( ord_le3703957343819488690_int_o @ D3 @ ( accp_P4562858270263085236nt_int @ R3 ) )
=> ( ! [X3: produc4189061985984568957nt_int,Z4: produc4189061985984568957nt_int] :
( ( D3 @ X3 )
=> ( ( R3 @ Z4 @ X3 )
=> ( D3 @ Z4 ) ) )
=> ( ( D3 @ X )
=> ( ! [X3: produc4189061985984568957nt_int] :
( ( D3 @ X3 )
=> ( ! [Z5: produc4189061985984568957nt_int] :
( ( R3 @ Z5 @ X3 )
=> ( P @ Z5 ) )
=> ( P @ X3 ) ) )
=> ( P @ X ) ) ) ) ) ).
% accp_subset_induct
thf(fact_873_accp__subset__induct,axiom,
! [D3: product_prod_int_int > $o,R3: product_prod_int_int > product_prod_int_int > $o,X: product_prod_int_int,P: product_prod_int_int > $o] :
( ( ord_le8369615600986905444_int_o @ D3 @ ( accp_P1096762738010456898nt_int @ R3 ) )
=> ( ! [X3: product_prod_int_int,Z4: product_prod_int_int] :
( ( D3 @ X3 )
=> ( ( R3 @ Z4 @ X3 )
=> ( D3 @ Z4 ) ) )
=> ( ( D3 @ X )
=> ( ! [X3: product_prod_int_int] :
( ( D3 @ X3 )
=> ( ! [Z5: product_prod_int_int] :
( ( R3 @ Z5 @ X3 )
=> ( P @ Z5 ) )
=> ( P @ X3 ) ) )
=> ( P @ X ) ) ) ) ) ).
% accp_subset_induct
thf(fact_874_nat0__intermed__int__val,axiom,
! [N: nat,F: nat > int,K: int] :
( ! [I: nat] :
( ( ord_less_nat @ I @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I @ one_one_nat ) ) @ ( F @ I ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I: nat] :
( ( ord_less_eq_nat @ I @ N )
& ( ( F @ I )
= K ) ) ) ) ) ).
% nat0_intermed_int_val
thf(fact_875_abs__idempotent,axiom,
! [A2: int] :
( ( abs_abs_int @ ( abs_abs_int @ A2 ) )
= ( abs_abs_int @ A2 ) ) ).
% abs_idempotent
thf(fact_876_abs__abs,axiom,
! [A2: int] :
( ( abs_abs_int @ ( abs_abs_int @ A2 ) )
= ( abs_abs_int @ A2 ) ) ).
% abs_abs
thf(fact_877_abs__0__eq,axiom,
! [A2: int] :
( ( zero_zero_int
= ( abs_abs_int @ A2 ) )
= ( A2 = zero_zero_int ) ) ).
% abs_0_eq
thf(fact_878_abs__eq__0,axiom,
! [A2: int] :
( ( ( abs_abs_int @ A2 )
= zero_zero_int )
= ( A2 = zero_zero_int ) ) ).
% abs_eq_0
thf(fact_879_abs__zero,axiom,
( ( abs_abs_int @ zero_zero_int )
= zero_zero_int ) ).
% abs_zero
thf(fact_880_abs__0,axiom,
( ( abs_abs_int @ zero_zero_int )
= zero_zero_int ) ).
% abs_0
thf(fact_881_abs__add__abs,axiom,
! [A2: int,B2: int] :
( ( abs_abs_int @ ( plus_plus_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B2 ) ) )
= ( plus_plus_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B2 ) ) ) ).
% abs_add_abs
thf(fact_882_abs__1,axiom,
( ( abs_abs_int @ one_one_int )
= one_one_int ) ).
% abs_1
thf(fact_883_abs__of__nat,axiom,
! [N: nat] :
( ( abs_abs_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ N ) ) ).
% abs_of_nat
thf(fact_884_abs__of__nonneg,axiom,
! [A2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( abs_abs_int @ A2 )
= A2 ) ) ).
% abs_of_nonneg
thf(fact_885_abs__le__self__iff,axiom,
! [A2: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A2 ) @ A2 )
= ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ).
% abs_le_self_iff
thf(fact_886_abs__le__zero__iff,axiom,
! [A2: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A2 ) @ zero_zero_int )
= ( A2 = zero_zero_int ) ) ).
% abs_le_zero_iff
thf(fact_887_zero__less__abs__iff,axiom,
! [A2: int] :
( ( ord_less_int @ zero_zero_int @ ( abs_abs_int @ A2 ) )
= ( A2 != zero_zero_int ) ) ).
% zero_less_abs_iff
thf(fact_888_zabs__less__one__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( abs_abs_int @ Z ) @ one_one_int )
= ( Z = zero_zero_int ) ) ).
% zabs_less_one_iff
thf(fact_889_abs__le__D1,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A2 ) @ B2 )
=> ( ord_less_eq_int @ A2 @ B2 ) ) ).
% abs_le_D1
thf(fact_890_abs__ge__self,axiom,
! [A2: int] : ( ord_less_eq_int @ A2 @ ( abs_abs_int @ A2 ) ) ).
% abs_ge_self
thf(fact_891_abs__eq__0__iff,axiom,
! [A2: int] :
( ( ( abs_abs_int @ A2 )
= zero_zero_int )
= ( A2 = zero_zero_int ) ) ).
% abs_eq_0_iff
thf(fact_892_abs__one,axiom,
( ( abs_abs_int @ one_one_int )
= one_one_int ) ).
% abs_one
thf(fact_893_abs__minus__commute,axiom,
! [A2: int,B2: int] :
( ( abs_abs_int @ ( minus_minus_int @ A2 @ B2 ) )
= ( abs_abs_int @ ( minus_minus_int @ B2 @ A2 ) ) ) ).
% abs_minus_commute
thf(fact_894_abs__ge__zero,axiom,
! [A2: int] : ( ord_less_eq_int @ zero_zero_int @ ( abs_abs_int @ A2 ) ) ).
% abs_ge_zero
thf(fact_895_abs__not__less__zero,axiom,
! [A2: int] :
~ ( ord_less_int @ ( abs_abs_int @ A2 ) @ zero_zero_int ) ).
% abs_not_less_zero
thf(fact_896_abs__of__pos,axiom,
! [A2: int] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( abs_abs_int @ A2 )
= A2 ) ) ).
% abs_of_pos
thf(fact_897_abs__triangle__ineq,axiom,
! [A2: int,B2: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( plus_plus_int @ A2 @ B2 ) ) @ ( plus_plus_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B2 ) ) ) ).
% abs_triangle_ineq
thf(fact_898_abs__triangle__ineq2__sym,axiom,
! [A2: int,B2: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B2 ) ) @ ( abs_abs_int @ ( minus_minus_int @ B2 @ A2 ) ) ) ).
% abs_triangle_ineq2_sym
thf(fact_899_abs__triangle__ineq3,axiom,
! [A2: int,B2: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B2 ) ) ) @ ( abs_abs_int @ ( minus_minus_int @ A2 @ B2 ) ) ) ).
% abs_triangle_ineq3
thf(fact_900_abs__triangle__ineq2,axiom,
! [A2: int,B2: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B2 ) ) @ ( abs_abs_int @ ( minus_minus_int @ A2 @ B2 ) ) ) ).
% abs_triangle_ineq2
thf(fact_901_abs__diff__le__iff,axiom,
! [X: int,A2: int,R: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ X @ A2 ) ) @ R )
= ( ( ord_less_eq_int @ ( minus_minus_int @ A2 @ R ) @ X )
& ( ord_less_eq_int @ X @ ( plus_plus_int @ A2 @ R ) ) ) ) ).
% abs_diff_le_iff
thf(fact_902_abs__triangle__ineq4,axiom,
! [A2: int,B2: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ A2 @ B2 ) ) @ ( plus_plus_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B2 ) ) ) ).
% abs_triangle_ineq4
thf(fact_903_abs__diff__triangle__ineq,axiom,
! [A2: int,B2: int,C: int,D: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( plus_plus_int @ A2 @ B2 ) @ ( plus_plus_int @ C @ D ) ) ) @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ A2 @ C ) ) @ ( abs_abs_int @ ( minus_minus_int @ B2 @ D ) ) ) ) ).
% abs_diff_triangle_ineq
thf(fact_904_abs__diff__less__iff,axiom,
! [X: int,A2: int,R: int] :
( ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X @ A2 ) ) @ R )
= ( ( ord_less_int @ ( minus_minus_int @ A2 @ R ) @ X )
& ( ord_less_int @ X @ ( plus_plus_int @ A2 @ R ) ) ) ) ).
% abs_diff_less_iff
thf(fact_905_abs__add__one__gt__zero,axiom,
! [X: int] : ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ ( abs_abs_int @ X ) ) ) ).
% abs_add_one_gt_zero
thf(fact_906_in__measure,axiom,
! [X: int,Y: int,F: int > nat] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( measure_int @ F ) )
= ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) ) ).
% in_measure
thf(fact_907_upto_Opinduct,axiom,
! [A0: int,A12: int,P: int > int > $o] :
( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ A0 @ A12 ) )
=> ( ! [I: int,J4: int] :
( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I @ J4 ) )
=> ( ( ( ord_less_eq_int @ I @ J4 )
=> ( P @ ( plus_plus_int @ I @ one_one_int ) @ J4 ) )
=> ( P @ I @ J4 ) ) )
=> ( P @ A0 @ A12 ) ) ) ).
% upto.pinduct
thf(fact_908_bezw__0,axiom,
! [X: nat] :
( ( bezw @ X @ zero_zero_nat )
= ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) ).
% bezw_0
thf(fact_909_nat__ivt__aux,axiom,
! [N: nat,F: nat > int,K: int] :
( ! [I: nat] :
( ( ord_less_nat @ I @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I ) ) @ ( F @ I ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I: nat] :
( ( ord_less_eq_nat @ I @ N )
& ( ( F @ I )
= K ) ) ) ) ) ).
% nat_ivt_aux
thf(fact_910_eq__fst__iff,axiom,
! [A2: $o,P2: product_prod_o_int] :
( ( A2
= ( product_fst_o_int @ P2 ) )
= ( ? [B4: int] :
( P2
= ( product_Pair_o_int @ A2 @ B4 ) ) ) ) ).
% eq_fst_iff
thf(fact_911_eq__fst__iff,axiom,
! [A2: int,P2: product_prod_int_int] :
( ( A2
= ( product_fst_int_int @ P2 ) )
= ( ? [B4: int] :
( P2
= ( product_Pair_int_int @ A2 @ B4 ) ) ) ) ).
% eq_fst_iff
thf(fact_912_eq__fst__iff,axiom,
! [A2: list_char > int,P2: produc369741286924889929st_int] :
( ( A2
= ( produc2155683374303800813st_int @ P2 ) )
= ( ? [B4: list_int] :
( P2
= ( produc8650753666468850689st_int @ A2 @ B4 ) ) ) ) ).
% eq_fst_iff
thf(fact_913_eq__fst__iff,axiom,
! [A2: list_char > int,P2: produc4435102495419491129nt_int] :
( ( A2
= ( produc9025023058967219549nt_int @ P2 ) )
= ( ? [B4: int] :
( P2
= ( produc5790713362662368625nt_int @ A2 @ B4 ) ) ) ) ).
% eq_fst_iff
thf(fact_914_eq__fst__iff,axiom,
! [A2: int,P2: produc6425607678544837394st_int] :
( ( A2
= ( produc8941309049508465880st_int @ P2 ) )
= ( ? [B4: produc369741286924889929st_int] :
( P2
= ( produc5086643055186798020st_int @ A2 @ B4 ) ) ) ) ).
% eq_fst_iff
thf(fact_915_fstI,axiom,
! [X: product_prod_o_int,Y: $o,Z: int] :
( ( X
= ( product_Pair_o_int @ Y @ Z ) )
=> ( ( product_fst_o_int @ X )
= Y ) ) ).
% fstI
thf(fact_916_fstI,axiom,
! [X: product_prod_int_int,Y: int,Z: int] :
( ( X
= ( product_Pair_int_int @ Y @ Z ) )
=> ( ( product_fst_int_int @ X )
= Y ) ) ).
% fstI
thf(fact_917_fstI,axiom,
! [X: produc369741286924889929st_int,Y: list_char > int,Z: list_int] :
( ( X
= ( produc8650753666468850689st_int @ Y @ Z ) )
=> ( ( produc2155683374303800813st_int @ X )
= Y ) ) ).
% fstI
thf(fact_918_fstI,axiom,
! [X: produc4435102495419491129nt_int,Y: list_char > int,Z: int] :
( ( X
= ( produc5790713362662368625nt_int @ Y @ Z ) )
=> ( ( produc9025023058967219549nt_int @ X )
= Y ) ) ).
% fstI
thf(fact_919_fstI,axiom,
! [X: produc6425607678544837394st_int,Y: int,Z: produc369741286924889929st_int] :
( ( X
= ( produc5086643055186798020st_int @ Y @ Z ) )
=> ( ( produc8941309049508465880st_int @ X )
= Y ) ) ).
% fstI
thf(fact_920_nat_Oinject,axiom,
! [X23: nat,Y23: nat] :
( ( ( suc @ X23 )
= ( suc @ Y23 ) )
= ( X23 = Y23 ) ) ).
% nat.inject
thf(fact_921_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_922_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_923_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_924_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_925_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_926_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_927_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_928_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_929_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_930_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_931_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_932_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ M ) )
= ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% of_nat_Suc
thf(fact_933_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ M ) )
= ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M ) ) ) ).
% of_nat_Suc
thf(fact_934_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_935_diff__Suc__diff__eq1,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I2 @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_936_diff__Suc__diff__eq2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I2 )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I2 ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_937_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_938_Nat_OlessE,axiom,
! [I2: nat,K: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( ( K
!= ( suc @ I2 ) )
=> ~ ! [J4: nat] :
( ( ord_less_nat @ I2 @ J4 )
=> ( K
!= ( suc @ J4 ) ) ) ) ) ).
% Nat.lessE
thf(fact_939_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_940_Suc__lessE,axiom,
! [I2: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I2 ) @ K )
=> ~ ! [J4: nat] :
( ( ord_less_nat @ I2 @ J4 )
=> ( K
!= ( suc @ J4 ) ) ) ) ).
% Suc_lessE
thf(fact_941_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_942_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_943_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_944_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
& ( P @ I3 ) ) )
= ( ( P @ N )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ( P @ I3 ) ) ) ) ).
% Ex_less_Suc
thf(fact_945_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_946_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_947_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
=> ( P @ I3 ) ) )
= ( ( P @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( P @ I3 ) ) ) ) ).
% All_less_Suc
thf(fact_948_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_949_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_950_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_951_less__trans__Suc,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I2 ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_952_less__Suc__induct,axiom,
! [I2: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I2 @ J )
=> ( ! [I: nat] : ( P @ I @ ( suc @ I ) )
=> ( ! [I: nat,J4: nat,K3: nat] :
( ( ord_less_nat @ I @ J4 )
=> ( ( ord_less_nat @ J4 @ K3 )
=> ( ( P @ I @ J4 )
=> ( ( P @ J4 @ K3 )
=> ( P @ I @ K3 ) ) ) ) )
=> ( P @ I2 @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_953_strict__inc__induct,axiom,
! [I2: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I2 @ J )
=> ( ! [I: nat] :
( ( J
= ( suc @ I ) )
=> ( P @ I ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( P @ ( suc @ I ) )
=> ( P @ I ) ) )
=> ( P @ I2 ) ) ) ) ).
% strict_inc_induct
thf(fact_954_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_955_nat_Odistinct_I1_J,axiom,
! [X23: nat] :
( zero_zero_nat
!= ( suc @ X23 ) ) ).
% nat.distinct(1)
thf(fact_956_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_957_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_958_nat_OdiscI,axiom,
! [Nat: nat,X23: nat] :
( ( Nat
= ( suc @ X23 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_959_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_960_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_961_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
=> ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
=> ( ! [X3: nat,Y3: nat] :
( ( P @ X3 @ Y3 )
=> ( P @ ( suc @ X3 ) @ ( suc @ Y3 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_962_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_963_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_964_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_965_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_966_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M2: nat] :
( N
= ( suc @ M2 ) ) ) ).
% not0_implies_Suc
thf(fact_967_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_968_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_969_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_970_Suc__le__D,axiom,
! [N: nat,M7: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M7 )
=> ? [M2: nat] :
( M7
= ( suc @ M2 ) ) ) ).
% Suc_le_D
thf(fact_971_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_972_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_973_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_974_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M4: nat] :
( ( ord_less_eq_nat @ ( suc @ M4 ) @ N2 )
=> ( P @ M4 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_975_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_976_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R3: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X3: nat] : ( R3 @ X3 @ X3 )
=> ( ! [X3: nat,Y3: nat,Z4: nat] :
( ( R3 @ X3 @ Y3 )
=> ( ( R3 @ Y3 @ Z4 )
=> ( R3 @ X3 @ Z4 ) ) )
=> ( ! [N2: nat] : ( R3 @ N2 @ ( suc @ N2 ) )
=> ( R3 @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_977_nat__arith_Osuc1,axiom,
! [A5: nat,K: nat,A2: nat] :
( ( A5
= ( plus_plus_nat @ K @ A2 ) )
=> ( ( suc @ A5 )
= ( plus_plus_nat @ K @ ( suc @ A2 ) ) ) ) ).
% nat_arith.suc1
thf(fact_978_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_979_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_980_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I2: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ ( minus_minus_nat @ K @ I2 ) ) ) ) ).
% zero_induct_lemma
thf(fact_981_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_982_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_983_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_984_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_985_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_986_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_987_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_988_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_989_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_990_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_991_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
!= zero_zero_int ) ).
% of_nat_neq_0
thf(fact_992_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
!= zero_zero_nat ) ).
% of_nat_neq_0
thf(fact_993_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( M = zero_zero_nat )
| ? [J2: nat] :
( ( M
= ( suc @ J2 ) )
& ( ord_less_nat @ J2 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_994_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M2: nat] :
( N
= ( suc @ M2 ) ) ) ).
% gr0_implies_Suc
thf(fact_995_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
=> ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( P @ ( suc @ I3 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_996_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_997_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
& ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ( P @ ( suc @ I3 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_998_length__Suc__conv,axiom,
! [Xs: list_int,N: nat] :
( ( ( size_size_list_int @ Xs )
= ( suc @ N ) )
= ( ? [Y2: int,Ys3: list_int] :
( ( Xs
= ( cons_int @ Y2 @ Ys3 ) )
& ( ( size_size_list_int @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_999_length__Suc__conv,axiom,
! [Xs: list_instr,N: nat] :
( ( ( size_size_list_instr @ Xs )
= ( suc @ N ) )
= ( ? [Y2: instr,Ys3: list_instr] :
( ( Xs
= ( cons_instr @ Y2 @ Ys3 ) )
& ( ( size_size_list_instr @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_1000_length__Suc__conv,axiom,
! [Xs: list_list_char,N: nat] :
( ( ( size_s356637359517785349t_char @ Xs )
= ( suc @ N ) )
= ( ? [Y2: list_char,Ys3: list_list_char] :
( ( Xs
= ( cons_list_char @ Y2 @ Ys3 ) )
& ( ( size_s356637359517785349t_char @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_1001_Suc__length__conv,axiom,
! [N: nat,Xs: list_int] :
( ( ( suc @ N )
= ( size_size_list_int @ Xs ) )
= ( ? [Y2: int,Ys3: list_int] :
( ( Xs
= ( cons_int @ Y2 @ Ys3 ) )
& ( ( size_size_list_int @ Ys3 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_1002_Suc__length__conv,axiom,
! [N: nat,Xs: list_instr] :
( ( ( suc @ N )
= ( size_size_list_instr @ Xs ) )
= ( ? [Y2: instr,Ys3: list_instr] :
( ( Xs
= ( cons_instr @ Y2 @ Ys3 ) )
& ( ( size_size_list_instr @ Ys3 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_1003_Suc__length__conv,axiom,
! [N: nat,Xs: list_list_char] :
( ( ( suc @ N )
= ( size_s356637359517785349t_char @ Xs ) )
= ( ? [Y2: list_char,Ys3: list_list_char] :
( ( Xs
= ( cons_list_char @ Y2 @ Ys3 ) )
& ( ( size_s356637359517785349t_char @ Ys3 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_1004_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_1005_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_1006_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_1007_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_1008_dec__induct,axiom,
! [I2: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( P @ I2 )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I2 @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_1009_inc__induct,axiom,
! [I2: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( P @ J )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I2 @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) ) ) )
=> ( P @ I2 ) ) ) ) ).
% inc_induct
thf(fact_1010_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_1011_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_1012_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_1013_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N3: nat] : ( ord_less_eq_nat @ ( suc @ N3 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1014_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_1015_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ? [K3: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K3 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_1016_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M3: nat,N3: nat] :
? [K2: nat] :
( N3
= ( suc @ ( plus_plus_nat @ M3 @ K2 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1017_less__add__Suc2,axiom,
! [I2: nat,M: nat] : ( ord_less_nat @ I2 @ ( suc @ ( plus_plus_nat @ M @ I2 ) ) ) ).
% less_add_Suc2
thf(fact_1018_less__add__Suc1,axiom,
! [I2: nat,M: nat] : ( ord_less_nat @ I2 @ ( suc @ ( plus_plus_nat @ I2 @ M ) ) ) ).
% less_add_Suc1
thf(fact_1019_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_1020_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_1021_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_1022_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_1023_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_1024_Suc__eq__plus1,axiom,
( suc
= ( ^ [N3: nat] : ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_1025_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1026_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_1027_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_1028_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_nat @ K3 @ N )
& ! [I6: nat] :
( ( ord_less_eq_nat @ I6 @ K3 )
=> ~ ( P @ I6 ) )
& ( P @ ( suc @ K3 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1029_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_int] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_int @ Xs ) )
= ( ? [X2: int,Ys3: list_int] :
( ( Xs
= ( cons_int @ X2 @ Ys3 ) )
& ( ord_less_eq_nat @ N @ ( size_size_list_int @ Ys3 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_1030_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_instr] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_instr @ Xs ) )
= ( ? [X2: instr,Ys3: list_instr] :
( ( Xs
= ( cons_instr @ X2 @ Ys3 ) )
& ( ord_less_eq_nat @ N @ ( size_size_list_instr @ Ys3 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_1031_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_list_char] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_s356637359517785349t_char @ Xs ) )
= ( ? [X2: list_char,Ys3: list_list_char] :
( ( Xs
= ( cons_list_char @ X2 @ Ys3 ) )
& ( ord_less_eq_nat @ N @ ( size_s356637359517785349t_char @ Ys3 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_1032_diff__Suc__less,axiom,
! [N: nat,I2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I2 ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_1033_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_1034_int__Suc,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ).
% int_Suc
thf(fact_1035_int__ops_I4_J,axiom,
! [A2: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ A2 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A2 ) @ one_one_int ) ) ).
% int_ops(4)
thf(fact_1036_zless__iff__Suc__zadd,axiom,
( ord_less_int
= ( ^ [W: int,Z2: int] :
? [N3: nat] :
( Z2
= ( plus_plus_int @ W @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ) ).
% zless_iff_Suc_zadd
thf(fact_1037_bexp_Osize_I7_J,axiom,
! [X31: bexp,X32: bexp] :
( ( size_size_bexp @ ( and @ X31 @ X32 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( size_size_bexp @ X31 ) @ ( size_size_bexp @ X32 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% bexp.size(7)
thf(fact_1038_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_1039_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_1040_list_Osize_I4_J,axiom,
! [X21: list_char,X22: list_list_char] :
( ( size_s356637359517785349t_char @ ( cons_list_char @ X21 @ X22 ) )
= ( plus_plus_nat @ ( size_s356637359517785349t_char @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% list.size(4)
thf(fact_1041_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_1042_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_1043_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M3: nat,N3: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ N3 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N3 ) ) ) ) ) ).
% add_eq_if
thf(fact_1044_com_Osize_I9_J,axiom,
! [X41: bexp,X42: com,X43: com] :
( ( size_size_com @ ( if @ X41 @ X42 @ X43 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( size_size_com @ X42 ) @ ( size_size_com @ X43 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% com.size(9)
thf(fact_1045_com_Osize__gen_I4_J,axiom,
! [X41: bexp,X42: com,X43: com] :
( ( size_com @ ( if @ X41 @ X42 @ X43 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( size_com @ X42 ) @ ( size_com @ X43 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% com.size_gen(4)
thf(fact_1046_nat__intermed__int__val,axiom,
! [M: nat,N: nat,F: nat > int,K: int] :
( ! [I: nat] :
( ( ( ord_less_eq_nat @ M @ I )
& ( ord_less_nat @ I @ N ) )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I ) ) @ ( F @ I ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_int @ ( F @ M ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I: nat] :
( ( ord_less_eq_nat @ M @ I )
& ( ord_less_eq_nat @ I @ N )
& ( ( F @ I )
= K ) ) ) ) ) ) ).
% nat_intermed_int_val
thf(fact_1047_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_1048_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_1049_length__Cons,axiom,
! [X: list_char,Xs: list_list_char] :
( ( size_s356637359517785349t_char @ ( cons_list_char @ X @ Xs ) )
= ( suc @ ( size_s356637359517785349t_char @ Xs ) ) ) ).
% length_Cons
thf(fact_1050_bexp_Osize__gen_I3_J,axiom,
! [X31: bexp,X32: bexp] :
( ( size_bexp @ ( and @ X31 @ X32 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( size_bexp @ X31 ) @ ( size_bexp @ X32 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% bexp.size_gen(3)
thf(fact_1051_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M: nat] :
( ! [K3: nat] :
( ( ord_less_nat @ N @ K3 )
=> ( P @ K3 ) )
=> ( ! [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
=> ( ! [I6: nat] :
( ( ord_less_nat @ K3 @ I6 )
=> ( P @ I6 ) )
=> ( P @ K3 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_1052_inth__map,axiom,
! [I2: int,Xs: list_instr,F: instr > instr] :
( ( ord_less_eq_int @ zero_zero_int @ I2 )
=> ( ( ord_less_int @ I2 @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ Xs ) ) )
=> ( ( inth_instr @ ( map_instr_instr @ F @ Xs ) @ I2 )
= ( F @ ( inth_instr @ Xs @ I2 ) ) ) ) ) ).
% inth_map
thf(fact_1053_inth__map,axiom,
! [I2: int,Xs: list_instr,F: instr > list_char] :
( ( ord_less_eq_int @ zero_zero_int @ I2 )
=> ( ( ord_less_int @ I2 @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ Xs ) ) )
=> ( ( inth_list_char @ ( map_instr_list_char @ F @ Xs ) @ I2 )
= ( F @ ( inth_instr @ Xs @ I2 ) ) ) ) ) ).
% inth_map
thf(fact_1054_inth__map,axiom,
! [I2: int,Xs: list_list_char,F: list_char > instr] :
( ( ord_less_eq_int @ zero_zero_int @ I2 )
=> ( ( ord_less_int @ I2 @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ Xs ) ) )
=> ( ( inth_instr @ ( map_list_char_instr @ F @ Xs ) @ I2 )
= ( F @ ( inth_list_char @ Xs @ I2 ) ) ) ) ) ).
% inth_map
thf(fact_1055_inth__map,axiom,
! [I2: int,Xs: list_list_char,F: list_char > list_char] :
( ( ord_less_eq_int @ zero_zero_int @ I2 )
=> ( ( ord_less_int @ I2 @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ Xs ) ) )
=> ( ( inth_list_char @ ( map_li116305933131242120t_char @ F @ Xs ) @ I2 )
= ( F @ ( inth_list_char @ Xs @ I2 ) ) ) ) ) ).
% inth_map
thf(fact_1056_length__map,axiom,
! [F: instr > instr,Xs: list_instr] :
( ( size_size_list_instr @ ( map_instr_instr @ F @ Xs ) )
= ( size_size_list_instr @ Xs ) ) ).
% length_map
thf(fact_1057_length__map,axiom,
! [F: list_char > instr,Xs: list_list_char] :
( ( size_size_list_instr @ ( map_list_char_instr @ F @ Xs ) )
= ( size_s356637359517785349t_char @ Xs ) ) ).
% length_map
thf(fact_1058_length__map,axiom,
! [F: instr > list_char,Xs: list_instr] :
( ( size_s356637359517785349t_char @ ( map_instr_list_char @ F @ Xs ) )
= ( size_size_list_instr @ Xs ) ) ).
% length_map
thf(fact_1059_length__map,axiom,
! [F: list_char > list_char,Xs: list_list_char] :
( ( size_s356637359517785349t_char @ ( map_li116305933131242120t_char @ F @ Xs ) )
= ( size_s356637359517785349t_char @ Xs ) ) ).
% length_map
thf(fact_1060_map__append,axiom,
! [F: instr > instr,Xs: list_instr,Ys: list_instr] :
( ( map_instr_instr @ F @ ( append_instr @ Xs @ Ys ) )
= ( append_instr @ ( map_instr_instr @ F @ Xs ) @ ( map_instr_instr @ F @ Ys ) ) ) ).
% map_append
thf(fact_1061_append__eq__map__conv,axiom,
! [Ys: list_instr,Zs: list_instr,F: instr > instr,Xs: list_instr] :
( ( ( append_instr @ Ys @ Zs )
= ( map_instr_instr @ F @ Xs ) )
= ( ? [Us2: list_instr,Vs2: list_instr] :
( ( Xs
= ( append_instr @ Us2 @ Vs2 ) )
& ( Ys
= ( map_instr_instr @ F @ Us2 ) )
& ( Zs
= ( map_instr_instr @ F @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_1062_map__eq__append__conv,axiom,
! [F: instr > instr,Xs: list_instr,Ys: list_instr,Zs: list_instr] :
( ( ( map_instr_instr @ F @ Xs )
= ( append_instr @ Ys @ Zs ) )
= ( ? [Us2: list_instr,Vs2: list_instr] :
( ( Xs
= ( append_instr @ Us2 @ Vs2 ) )
& ( Ys
= ( map_instr_instr @ F @ Us2 ) )
& ( Zs
= ( map_instr_instr @ F @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_1063_list_Osimps_I9_J,axiom,
! [F: int > int,X21: int,X22: list_int] :
( ( map_int_int @ F @ ( cons_int @ X21 @ X22 ) )
= ( cons_int @ ( F @ X21 ) @ ( map_int_int @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_1064_list_Osimps_I9_J,axiom,
! [F: int > instr,X21: int,X22: list_int] :
( ( map_int_instr @ F @ ( cons_int @ X21 @ X22 ) )
= ( cons_instr @ ( F @ X21 ) @ ( map_int_instr @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_1065_list_Osimps_I9_J,axiom,
! [F: instr > int,X21: instr,X22: list_instr] :
( ( map_instr_int @ F @ ( cons_instr @ X21 @ X22 ) )
= ( cons_int @ ( F @ X21 ) @ ( map_instr_int @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_1066_list_Osimps_I9_J,axiom,
! [F: instr > instr,X21: instr,X22: list_instr] :
( ( map_instr_instr @ F @ ( cons_instr @ X21 @ X22 ) )
= ( cons_instr @ ( F @ X21 ) @ ( map_instr_instr @ F @ X22 ) ) ) ).
% list.simps(9)
thf(fact_1067_Cons__eq__map__D,axiom,
! [X: int,Xs: list_int,F: int > int,Ys: list_int] :
( ( ( cons_int @ X @ Xs )
= ( map_int_int @ F @ Ys ) )
=> ? [Z4: int,Zs2: list_int] :
( ( Ys
= ( cons_int @ Z4 @ Zs2 ) )
& ( X
= ( F @ Z4 ) )
& ( Xs
= ( map_int_int @ F @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_1068_Cons__eq__map__D,axiom,
! [X: int,Xs: list_int,F: instr > int,Ys: list_instr] :
( ( ( cons_int @ X @ Xs )
= ( map_instr_int @ F @ Ys ) )
=> ? [Z4: instr,Zs2: list_instr] :
( ( Ys
= ( cons_instr @ Z4 @ Zs2 ) )
& ( X
= ( F @ Z4 ) )
& ( Xs
= ( map_instr_int @ F @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_1069_Cons__eq__map__D,axiom,
! [X: instr,Xs: list_instr,F: int > instr,Ys: list_int] :
( ( ( cons_instr @ X @ Xs )
= ( map_int_instr @ F @ Ys ) )
=> ? [Z4: int,Zs2: list_int] :
( ( Ys
= ( cons_int @ Z4 @ Zs2 ) )
& ( X
= ( F @ Z4 ) )
& ( Xs
= ( map_int_instr @ F @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_1070_Cons__eq__map__D,axiom,
! [X: instr,Xs: list_instr,F: instr > instr,Ys: list_instr] :
( ( ( cons_instr @ X @ Xs )
= ( map_instr_instr @ F @ Ys ) )
=> ? [Z4: instr,Zs2: list_instr] :
( ( Ys
= ( cons_instr @ Z4 @ Zs2 ) )
& ( X
= ( F @ Z4 ) )
& ( Xs
= ( map_instr_instr @ F @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_1071_map__eq__Cons__D,axiom,
! [F: int > int,Xs: list_int,Y: int,Ys: list_int] :
( ( ( map_int_int @ F @ Xs )
= ( cons_int @ Y @ Ys ) )
=> ? [Z4: int,Zs2: list_int] :
( ( Xs
= ( cons_int @ Z4 @ Zs2 ) )
& ( ( F @ Z4 )
= Y )
& ( ( map_int_int @ F @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_1072_map__eq__Cons__D,axiom,
! [F: instr > int,Xs: list_instr,Y: int,Ys: list_int] :
( ( ( map_instr_int @ F @ Xs )
= ( cons_int @ Y @ Ys ) )
=> ? [Z4: instr,Zs2: list_instr] :
( ( Xs
= ( cons_instr @ Z4 @ Zs2 ) )
& ( ( F @ Z4 )
= Y )
& ( ( map_instr_int @ F @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_1073_map__eq__Cons__D,axiom,
! [F: int > instr,Xs: list_int,Y: instr,Ys: list_instr] :
( ( ( map_int_instr @ F @ Xs )
= ( cons_instr @ Y @ Ys ) )
=> ? [Z4: int,Zs2: list_int] :
( ( Xs
= ( cons_int @ Z4 @ Zs2 ) )
& ( ( F @ Z4 )
= Y )
& ( ( map_int_instr @ F @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_1074_map__eq__Cons__D,axiom,
! [F: instr > instr,Xs: list_instr,Y: instr,Ys: list_instr] :
( ( ( map_instr_instr @ F @ Xs )
= ( cons_instr @ Y @ Ys ) )
=> ? [Z4: instr,Zs2: list_instr] :
( ( Xs
= ( cons_instr @ Z4 @ Zs2 ) )
& ( ( F @ Z4 )
= Y )
& ( ( map_instr_instr @ F @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_1075_Cons__eq__map__conv,axiom,
! [X: int,Xs: list_int,F: int > int,Ys: list_int] :
( ( ( cons_int @ X @ Xs )
= ( map_int_int @ F @ Ys ) )
= ( ? [Z2: int,Zs3: list_int] :
( ( Ys
= ( cons_int @ Z2 @ Zs3 ) )
& ( X
= ( F @ Z2 ) )
& ( Xs
= ( map_int_int @ F @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_1076_Cons__eq__map__conv,axiom,
! [X: int,Xs: list_int,F: instr > int,Ys: list_instr] :
( ( ( cons_int @ X @ Xs )
= ( map_instr_int @ F @ Ys ) )
= ( ? [Z2: instr,Zs3: list_instr] :
( ( Ys
= ( cons_instr @ Z2 @ Zs3 ) )
& ( X
= ( F @ Z2 ) )
& ( Xs
= ( map_instr_int @ F @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_1077_Cons__eq__map__conv,axiom,
! [X: instr,Xs: list_instr,F: int > instr,Ys: list_int] :
( ( ( cons_instr @ X @ Xs )
= ( map_int_instr @ F @ Ys ) )
= ( ? [Z2: int,Zs3: list_int] :
( ( Ys
= ( cons_int @ Z2 @ Zs3 ) )
& ( X
= ( F @ Z2 ) )
& ( Xs
= ( map_int_instr @ F @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_1078_Cons__eq__map__conv,axiom,
! [X: instr,Xs: list_instr,F: instr > instr,Ys: list_instr] :
( ( ( cons_instr @ X @ Xs )
= ( map_instr_instr @ F @ Ys ) )
= ( ? [Z2: instr,Zs3: list_instr] :
( ( Ys
= ( cons_instr @ Z2 @ Zs3 ) )
& ( X
= ( F @ Z2 ) )
& ( Xs
= ( map_instr_instr @ F @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_1079_map__eq__Cons__conv,axiom,
! [F: int > int,Xs: list_int,Y: int,Ys: list_int] :
( ( ( map_int_int @ F @ Xs )
= ( cons_int @ Y @ Ys ) )
= ( ? [Z2: int,Zs3: list_int] :
( ( Xs
= ( cons_int @ Z2 @ Zs3 ) )
& ( ( F @ Z2 )
= Y )
& ( ( map_int_int @ F @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_1080_map__eq__Cons__conv,axiom,
! [F: instr > int,Xs: list_instr,Y: int,Ys: list_int] :
( ( ( map_instr_int @ F @ Xs )
= ( cons_int @ Y @ Ys ) )
= ( ? [Z2: instr,Zs3: list_instr] :
( ( Xs
= ( cons_instr @ Z2 @ Zs3 ) )
& ( ( F @ Z2 )
= Y )
& ( ( map_instr_int @ F @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_1081_map__eq__Cons__conv,axiom,
! [F: int > instr,Xs: list_int,Y: instr,Ys: list_instr] :
( ( ( map_int_instr @ F @ Xs )
= ( cons_instr @ Y @ Ys ) )
= ( ? [Z2: int,Zs3: list_int] :
( ( Xs
= ( cons_int @ Z2 @ Zs3 ) )
& ( ( F @ Z2 )
= Y )
& ( ( map_int_instr @ F @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_1082_map__eq__Cons__conv,axiom,
! [F: instr > instr,Xs: list_instr,Y: instr,Ys: list_instr] :
( ( ( map_instr_instr @ F @ Xs )
= ( cons_instr @ Y @ Ys ) )
= ( ? [Z2: instr,Zs3: list_instr] :
( ( Xs
= ( cons_instr @ Z2 @ Zs3 ) )
& ( ( F @ Z2 )
= Y )
& ( ( map_instr_instr @ F @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_1083_complete__interval,axiom,
! [A2: nat,B2: nat,P: nat > $o] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( P @ A2 )
=> ( ~ ( P @ B2 )
=> ? [C2: nat] :
( ( ord_less_eq_nat @ A2 @ C2 )
& ( ord_less_eq_nat @ C2 @ B2 )
& ! [X6: nat] :
( ( ( ord_less_eq_nat @ A2 @ X6 )
& ( ord_less_nat @ X6 @ C2 ) )
=> ( P @ X6 ) )
& ! [D4: nat] :
( ! [X3: nat] :
( ( ( ord_less_eq_nat @ A2 @ X3 )
& ( ord_less_nat @ X3 @ D4 ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_nat @ D4 @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_1084_complete__interval,axiom,
! [A2: int,B2: int,P: int > $o] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( P @ A2 )
=> ( ~ ( P @ B2 )
=> ? [C2: int] :
( ( ord_less_eq_int @ A2 @ C2 )
& ( ord_less_eq_int @ C2 @ B2 )
& ! [X6: int] :
( ( ( ord_less_eq_int @ A2 @ X6 )
& ( ord_less_int @ X6 @ C2 ) )
=> ( P @ X6 ) )
& ! [D4: int] :
( ! [X3: int] :
( ( ( ord_less_eq_int @ A2 @ X3 )
& ( ord_less_int @ X3 @ D4 ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_int @ D4 @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_1085_decr__lemma,axiom,
! [D: int,X: int,Z: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ord_less_int @ ( minus_minus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z ) ) @ one_one_int ) @ D ) ) @ Z ) ) ).
% decr_lemma
thf(fact_1086_incr__lemma,axiom,
! [D: int,Z: int,X: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ord_less_int @ Z @ ( plus_plus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z ) ) @ one_one_int ) @ D ) ) ) ) ).
% incr_lemma
thf(fact_1087_mult__zero__left,axiom,
! [A2: int] :
( ( times_times_int @ zero_zero_int @ A2 )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_1088_mult__zero__left,axiom,
! [A2: nat] :
( ( times_times_nat @ zero_zero_nat @ A2 )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_1089_mult__zero__right,axiom,
! [A2: int] :
( ( times_times_int @ A2 @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_1090_mult__zero__right,axiom,
! [A2: nat] :
( ( times_times_nat @ A2 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_1091_mult__eq__0__iff,axiom,
! [A2: int,B2: int] :
( ( ( times_times_int @ A2 @ B2 )
= zero_zero_int )
= ( ( A2 = zero_zero_int )
| ( B2 = zero_zero_int ) ) ) ).
% mult_eq_0_iff
thf(fact_1092_mult__eq__0__iff,axiom,
! [A2: nat,B2: nat] :
( ( ( times_times_nat @ A2 @ B2 )
= zero_zero_nat )
= ( ( A2 = zero_zero_nat )
| ( B2 = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_1093_mult__cancel__left,axiom,
! [C: int,A2: int,B2: int] :
( ( ( times_times_int @ C @ A2 )
= ( times_times_int @ C @ B2 ) )
= ( ( C = zero_zero_int )
| ( A2 = B2 ) ) ) ).
% mult_cancel_left
thf(fact_1094_mult__cancel__left,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ( times_times_nat @ C @ A2 )
= ( times_times_nat @ C @ B2 ) )
= ( ( C = zero_zero_nat )
| ( A2 = B2 ) ) ) ).
% mult_cancel_left
thf(fact_1095_mult__cancel__right,axiom,
! [A2: int,C: int,B2: int] :
( ( ( times_times_int @ A2 @ C )
= ( times_times_int @ B2 @ C ) )
= ( ( C = zero_zero_int )
| ( A2 = B2 ) ) ) ).
% mult_cancel_right
thf(fact_1096_mult__cancel__right,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ( times_times_nat @ A2 @ C )
= ( times_times_nat @ B2 @ C ) )
= ( ( C = zero_zero_nat )
| ( A2 = B2 ) ) ) ).
% mult_cancel_right
thf(fact_1097_mult__1,axiom,
! [A2: int] :
( ( times_times_int @ one_one_int @ A2 )
= A2 ) ).
% mult_1
thf(fact_1098_mult__1,axiom,
! [A2: nat] :
( ( times_times_nat @ one_one_nat @ A2 )
= A2 ) ).
% mult_1
thf(fact_1099_mult_Oright__neutral,axiom,
! [A2: int] :
( ( times_times_int @ A2 @ one_one_int )
= A2 ) ).
% mult.right_neutral
thf(fact_1100_mult_Oright__neutral,axiom,
! [A2: nat] :
( ( times_times_nat @ A2 @ one_one_nat )
= A2 ) ).
% mult.right_neutral
thf(fact_1101_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_mult
thf(fact_1102_of__nat__mult,axiom,
! [M: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_mult
thf(fact_1103_abs__mult__self__eq,axiom,
! [A2: int] :
( ( times_times_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ A2 ) )
= ( times_times_int @ A2 @ A2 ) ) ).
% abs_mult_self_eq
thf(fact_1104_mult__cancel__right2,axiom,
! [A2: int,C: int] :
( ( ( times_times_int @ A2 @ C )
= C )
= ( ( C = zero_zero_int )
| ( A2 = one_one_int ) ) ) ).
% mult_cancel_right2
thf(fact_1105_mult__cancel__right1,axiom,
! [C: int,B2: int] :
( ( C
= ( times_times_int @ B2 @ C ) )
= ( ( C = zero_zero_int )
| ( B2 = one_one_int ) ) ) ).
% mult_cancel_right1
thf(fact_1106_mult__cancel__left2,axiom,
! [C: int,A2: int] :
( ( ( times_times_int @ C @ A2 )
= C )
= ( ( C = zero_zero_int )
| ( A2 = one_one_int ) ) ) ).
% mult_cancel_left2
thf(fact_1107_mult__cancel__left1,axiom,
! [C: int,B2: int] :
( ( C
= ( times_times_int @ C @ B2 ) )
= ( ( C = zero_zero_int )
| ( B2 = one_one_int ) ) ) ).
% mult_cancel_left1
thf(fact_1108_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B2 ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_1109_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_1110_zero__le__mult__iff,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ B2 ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
& ( ord_less_eq_int @ zero_zero_int @ B2 ) )
| ( ( ord_less_eq_int @ A2 @ zero_zero_int )
& ( ord_less_eq_int @ B2 @ zero_zero_int ) ) ) ) ).
% zero_le_mult_iff
thf(fact_1111_mult__nonneg__nonpos2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B2 @ A2 ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_1112_mult__nonneg__nonpos2,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ B2 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ B2 @ A2 ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_1113_mult__nonpos__nonneg,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_1114_mult__nonpos__nonneg,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( ord_less_eq_int @ ( times_times_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).
% mult_nonpos_nonneg
thf(fact_1115_mult__nonneg__nonpos,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_1116_mult__nonneg__nonpos,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ B2 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos
thf(fact_1117_mult__nonneg__nonneg,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A2 @ B2 ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_1118_mult__nonneg__nonneg,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ B2 ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_1119_split__mult__neg__le,axiom,
! [A2: nat,B2: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
& ( ord_less_eq_nat @ B2 @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_1120_split__mult__neg__le,axiom,
! [A2: int,B2: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
& ( ord_less_eq_int @ B2 @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A2 @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B2 ) ) )
=> ( ord_less_eq_int @ ( times_times_int @ A2 @ B2 ) @ zero_zero_int ) ) ).
% split_mult_neg_le
thf(fact_1121_mult__le__0__iff,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ ( times_times_int @ A2 @ B2 ) @ zero_zero_int )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
& ( ord_less_eq_int @ B2 @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A2 @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B2 ) ) ) ) ).
% mult_le_0_iff
thf(fact_1122_mult__right__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ C ) ) ) ) ).
% mult_right_mono
thf(fact_1123_mult__right__mono,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ C ) ) ) ) ).
% mult_right_mono
thf(fact_1124_mult__right__mono__neg,axiom,
! [B2: int,A2: int,C: int] :
( ( ord_less_eq_int @ B2 @ A2 )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_1125_mult__left__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B2 ) ) ) ) ).
% mult_left_mono
thf(fact_1126_mult__left__mono,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) ) ) ) ).
% mult_left_mono
thf(fact_1127_mult__nonpos__nonpos,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ B2 @ zero_zero_int )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ B2 ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_1128_mult__left__mono__neg,axiom,
! [B2: int,A2: int,C: int] :
( ( ord_less_eq_int @ B2 @ A2 )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) ) ) ) ).
% mult_left_mono_neg
thf(fact_1129_split__mult__pos__le,axiom,
! [A2: int,B2: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
& ( ord_less_eq_int @ zero_zero_int @ B2 ) )
| ( ( ord_less_eq_int @ A2 @ zero_zero_int )
& ( ord_less_eq_int @ B2 @ zero_zero_int ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ B2 ) ) ) ).
% split_mult_pos_le
thf(fact_1130_zero__le__square,axiom,
! [A2: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ A2 ) ) ).
% zero_le_square
thf(fact_1131_mult__mono_H,axiom,
! [A2: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_1132_mult__mono_H,axiom,
! [A2: int,B2: int,C: int,D: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_1133_mult__mono,axiom,
! [A2: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_1134_mult__mono,axiom,
! [A2: int,B2: int,C: int,D: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_1135_mult__neg__neg,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ zero_zero_int )
=> ( ( ord_less_int @ B2 @ zero_zero_int )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B2 ) ) ) ) ).
% mult_neg_neg
thf(fact_1136_not__square__less__zero,axiom,
! [A2: int] :
~ ( ord_less_int @ ( times_times_int @ A2 @ A2 ) @ zero_zero_int ) ).
% not_square_less_zero
thf(fact_1137_mult__less__0__iff,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ ( times_times_int @ A2 @ B2 ) @ zero_zero_int )
= ( ( ( ord_less_int @ zero_zero_int @ A2 )
& ( ord_less_int @ B2 @ zero_zero_int ) )
| ( ( ord_less_int @ A2 @ zero_zero_int )
& ( ord_less_int @ zero_zero_int @ B2 ) ) ) ) ).
% mult_less_0_iff
thf(fact_1138_mult__neg__pos,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ord_less_int @ ( times_times_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).
% mult_neg_pos
thf(fact_1139_mult__neg__pos,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).
% mult_neg_pos
thf(fact_1140_mult__pos__neg,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ B2 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).
% mult_pos_neg
thf(fact_1141_mult__pos__neg,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ B2 @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg
thf(fact_1142_mult__pos__pos,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B2 ) ) ) ) ).
% mult_pos_pos
thf(fact_1143_mult__pos__pos,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A2 @ B2 ) ) ) ) ).
% mult_pos_pos
thf(fact_1144_mult__pos__neg2,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( ord_less_int @ B2 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ B2 @ A2 ) @ zero_zero_int ) ) ) ).
% mult_pos_neg2
thf(fact_1145_mult__pos__neg2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ( ord_less_nat @ B2 @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B2 @ A2 ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg2
thf(fact_1146_zero__less__mult__iff,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B2 ) )
= ( ( ( ord_less_int @ zero_zero_int @ A2 )
& ( ord_less_int @ zero_zero_int @ B2 ) )
| ( ( ord_less_int @ A2 @ zero_zero_int )
& ( ord_less_int @ B2 @ zero_zero_int ) ) ) ) ).
% zero_less_mult_iff
thf(fact_1147_zero__less__mult__pos,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B2 ) )
=> ( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ord_less_int @ zero_zero_int @ B2 ) ) ) ).
% zero_less_mult_pos
thf(fact_1148_zero__less__mult__pos,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A2 @ B2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ord_less_nat @ zero_zero_nat @ B2 ) ) ) ).
% zero_less_mult_pos
thf(fact_1149_zero__less__mult__pos2,axiom,
! [B2: int,A2: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B2 @ A2 ) )
=> ( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ord_less_int @ zero_zero_int @ B2 ) ) ) ).
% zero_less_mult_pos2
thf(fact_1150_zero__less__mult__pos2,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B2 @ A2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A2 )
=> ( ord_less_nat @ zero_zero_nat @ B2 ) ) ) ).
% zero_less_mult_pos2
thf(fact_1151_mult__less__cancel__left__neg,axiom,
! [C: int,A2: int,B2: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) )
= ( ord_less_int @ B2 @ A2 ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_1152_mult__less__cancel__left__pos,axiom,
! [C: int,A2: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) )
= ( ord_less_int @ A2 @ B2 ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_1153_mult__strict__left__mono__neg,axiom,
! [B2: int,A2: int,C: int] :
( ( ord_less_int @ B2 @ A2 )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_1154_mult__strict__left__mono,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) ) ) ) ).
% mult_strict_left_mono
thf(fact_1155_mult__strict__left__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B2 ) ) ) ) ).
% mult_strict_left_mono
thf(fact_1156_mult__less__cancel__left__disj,axiom,
! [C: int,A2: int,B2: int] :
( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A2 @ B2 ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B2 @ A2 ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_1157_mult__strict__right__mono__neg,axiom,
! [B2: int,A2: int,C: int] :
( ( ord_less_int @ B2 @ A2 )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_1158_mult__strict__right__mono,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_1159_mult__strict__right__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_1160_mult__less__cancel__right__disj,axiom,
! [A2: int,C: int,B2: int] :
( ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A2 @ B2 ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B2 @ A2 ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_1161_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B2 ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_1162_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B2 ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_1163_add__scale__eq__noteq,axiom,
! [R: int,A2: int,B2: int,C: int,D: int] :
( ( R != zero_zero_int )
=> ( ( ( A2 = B2 )
& ( C != D ) )
=> ( ( plus_plus_int @ A2 @ ( times_times_int @ R @ C ) )
!= ( plus_plus_int @ B2 @ ( times_times_int @ R @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_1164_add__scale__eq__noteq,axiom,
! [R: nat,A2: nat,B2: nat,C: nat,D: nat] :
( ( R != zero_zero_nat )
=> ( ( ( A2 = B2 )
& ( C != D ) )
=> ( ( plus_plus_nat @ A2 @ ( times_times_nat @ R @ C ) )
!= ( plus_plus_nat @ B2 @ ( times_times_nat @ R @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_1165_less__1__mult,axiom,
! [M: int,N: int] :
( ( ord_less_int @ one_one_int @ M )
=> ( ( ord_less_int @ one_one_int @ N )
=> ( ord_less_int @ one_one_int @ ( times_times_int @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_1166_less__1__mult,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ M )
=> ( ( ord_less_nat @ one_one_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_1167_square__diff__square__factored,axiom,
! [X: int,Y: int] :
( ( minus_minus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
= ( times_times_int @ ( plus_plus_int @ X @ Y ) @ ( minus_minus_int @ X @ Y ) ) ) ).
% square_diff_square_factored
thf(fact_1168_eq__add__iff2,axiom,
! [A2: int,E: int,C: int,B2: int,D: int] :
( ( ( plus_plus_int @ ( times_times_int @ A2 @ E ) @ C )
= ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ D ) )
= ( C
= ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B2 @ A2 ) @ E ) @ D ) ) ) ).
% eq_add_iff2
thf(fact_1169_eq__add__iff1,axiom,
! [A2: int,E: int,C: int,B2: int,D: int] :
( ( ( plus_plus_int @ ( times_times_int @ A2 @ E ) @ C )
= ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ D ) )
= ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A2 @ B2 ) @ E ) @ C )
= D ) ) ).
% eq_add_iff1
thf(fact_1170_abs__mult__less,axiom,
! [A2: int,C: int,B2: int,D: int] :
( ( ord_less_int @ ( abs_abs_int @ A2 ) @ C )
=> ( ( ord_less_int @ ( abs_abs_int @ B2 ) @ D )
=> ( ord_less_int @ ( times_times_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B2 ) ) @ ( times_times_int @ C @ D ) ) ) ) ).
% abs_mult_less
thf(fact_1171_zmult__zless__mono2,axiom,
! [I2: int,J: int,K: int] :
( ( ord_less_int @ I2 @ J )
=> ( ( ord_less_int @ zero_zero_int @ K )
=> ( ord_less_int @ ( times_times_int @ K @ I2 ) @ ( times_times_int @ K @ J ) ) ) ) ).
% zmult_zless_mono2
thf(fact_1172_abs__zmult__eq__1,axiom,
! [M: int,N: int] :
( ( ( abs_abs_int @ ( times_times_int @ M @ N ) )
= one_one_int )
=> ( ( abs_abs_int @ M )
= one_one_int ) ) ).
% abs_zmult_eq_1
thf(fact_1173_mult__not__zero,axiom,
! [A2: int,B2: int] :
( ( ( times_times_int @ A2 @ B2 )
!= zero_zero_int )
=> ( ( A2 != zero_zero_int )
& ( B2 != zero_zero_int ) ) ) ).
% mult_not_zero
thf(fact_1174_mult__not__zero,axiom,
! [A2: nat,B2: nat] :
( ( ( times_times_nat @ A2 @ B2 )
!= zero_zero_nat )
=> ( ( A2 != zero_zero_nat )
& ( B2 != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_1175_divisors__zero,axiom,
! [A2: int,B2: int] :
( ( ( times_times_int @ A2 @ B2 )
= zero_zero_int )
=> ( ( A2 = zero_zero_int )
| ( B2 = zero_zero_int ) ) ) ).
% divisors_zero
thf(fact_1176_divisors__zero,axiom,
! [A2: nat,B2: nat] :
( ( ( times_times_nat @ A2 @ B2 )
= zero_zero_nat )
=> ( ( A2 = zero_zero_nat )
| ( B2 = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_1177_no__zero__divisors,axiom,
! [A2: int,B2: int] :
( ( A2 != zero_zero_int )
=> ( ( B2 != zero_zero_int )
=> ( ( times_times_int @ A2 @ B2 )
!= zero_zero_int ) ) ) ).
% no_zero_divisors
thf(fact_1178_no__zero__divisors,axiom,
! [A2: nat,B2: nat] :
( ( A2 != zero_zero_nat )
=> ( ( B2 != zero_zero_nat )
=> ( ( times_times_nat @ A2 @ B2 )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_1179_mult__left__cancel,axiom,
! [C: int,A2: int,B2: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ C @ A2 )
= ( times_times_int @ C @ B2 ) )
= ( A2 = B2 ) ) ) ).
% mult_left_cancel
thf(fact_1180_mult__left__cancel,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ C @ A2 )
= ( times_times_nat @ C @ B2 ) )
= ( A2 = B2 ) ) ) ).
% mult_left_cancel
thf(fact_1181_mult__right__cancel,axiom,
! [C: int,A2: int,B2: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ A2 @ C )
= ( times_times_int @ B2 @ C ) )
= ( A2 = B2 ) ) ) ).
% mult_right_cancel
thf(fact_1182_mult__right__cancel,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ A2 @ C )
= ( times_times_nat @ B2 @ C ) )
= ( A2 = B2 ) ) ) ).
% mult_right_cancel
thf(fact_1183_combine__common__factor,axiom,
! [A2: int,E: int,B2: int,C: int] :
( ( plus_plus_int @ ( times_times_int @ A2 @ E ) @ ( plus_plus_int @ ( times_times_int @ B2 @ E ) @ C ) )
= ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A2 @ B2 ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_1184_combine__common__factor,axiom,
! [A2: nat,E: nat,B2: nat,C: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A2 @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B2 @ E ) @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A2 @ B2 ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_1185_distrib__right,axiom,
! [A2: int,B2: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A2 @ B2 ) @ C )
= ( plus_plus_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ C ) ) ) ).
% distrib_right
thf(fact_1186_distrib__right,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A2 @ B2 ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ C ) ) ) ).
% distrib_right
thf(fact_1187_distrib__left,axiom,
! [A2: int,B2: int,C: int] :
( ( times_times_int @ A2 @ ( plus_plus_int @ B2 @ C ) )
= ( plus_plus_int @ ( times_times_int @ A2 @ B2 ) @ ( times_times_int @ A2 @ C ) ) ) ).
% distrib_left
thf(fact_1188_distrib__left,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( times_times_nat @ A2 @ ( plus_plus_nat @ B2 @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ A2 @ B2 ) @ ( times_times_nat @ A2 @ C ) ) ) ).
% distrib_left
thf(fact_1189_comm__semiring__class_Odistrib,axiom,
! [A2: int,B2: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A2 @ B2 ) @ C )
= ( plus_plus_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_1190_comm__semiring__class_Odistrib,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A2 @ B2 ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_1191_ring__class_Oring__distribs_I1_J,axiom,
! [A2: int,B2: int,C: int] :
( ( times_times_int @ A2 @ ( plus_plus_int @ B2 @ C ) )
= ( plus_plus_int @ ( times_times_int @ A2 @ B2 ) @ ( times_times_int @ A2 @ C ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_1192_ring__class_Oring__distribs_I2_J,axiom,
! [A2: int,B2: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A2 @ B2 ) @ C )
= ( plus_plus_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ C ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_1193_crossproduct__eq,axiom,
! [W2: int,Y: int,X: int,Z: int] :
( ( ( plus_plus_int @ ( times_times_int @ W2 @ Y ) @ ( times_times_int @ X @ Z ) )
= ( plus_plus_int @ ( times_times_int @ W2 @ Z ) @ ( times_times_int @ X @ Y ) ) )
= ( ( W2 = X )
| ( Y = Z ) ) ) ).
% crossproduct_eq
thf(fact_1194_crossproduct__eq,axiom,
! [W2: nat,Y: nat,X: nat,Z: nat] :
( ( ( plus_plus_nat @ ( times_times_nat @ W2 @ Y ) @ ( times_times_nat @ X @ Z ) )
= ( plus_plus_nat @ ( times_times_nat @ W2 @ Z ) @ ( times_times_nat @ X @ Y ) ) )
= ( ( W2 = X )
| ( Y = Z ) ) ) ).
% crossproduct_eq
thf(fact_1195_crossproduct__noteq,axiom,
! [A2: int,B2: int,C: int,D: int] :
( ( ( A2 != B2 )
& ( C != D ) )
= ( ( plus_plus_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B2 @ D ) )
!= ( plus_plus_int @ ( times_times_int @ A2 @ D ) @ ( times_times_int @ B2 @ C ) ) ) ) ).
% crossproduct_noteq
thf(fact_1196_crossproduct__noteq,axiom,
! [A2: nat,B2: nat,C: nat,D: nat] :
( ( ( A2 != B2 )
& ( C != D ) )
= ( ( plus_plus_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B2 @ D ) )
!= ( plus_plus_nat @ ( times_times_nat @ A2 @ D ) @ ( times_times_nat @ B2 @ C ) ) ) ) ).
% crossproduct_noteq
thf(fact_1197_mult_Ocomm__neutral,axiom,
! [A2: int] :
( ( times_times_int @ A2 @ one_one_int )
= A2 ) ).
% mult.comm_neutral
thf(fact_1198_mult_Ocomm__neutral,axiom,
! [A2: nat] :
( ( times_times_nat @ A2 @ one_one_nat )
= A2 ) ).
% mult.comm_neutral
thf(fact_1199_comm__monoid__mult__class_Omult__1,axiom,
! [A2: nat] :
( ( times_times_nat @ one_one_nat @ A2 )
= A2 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_1200_times__int__code_I1_J,axiom,
! [K: int] :
( ( times_times_int @ K @ zero_zero_int )
= zero_zero_int ) ).
% times_int_code(1)
thf(fact_1201_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times_int @ zero_zero_int @ L )
= zero_zero_int ) ).
% times_int_code(2)
thf(fact_1202_int__distrib_I1_J,axiom,
! [Z1: int,Z22: int,W2: int] :
( ( times_times_int @ ( plus_plus_int @ Z1 @ Z22 ) @ W2 )
= ( plus_plus_int @ ( times_times_int @ Z1 @ W2 ) @ ( times_times_int @ Z22 @ W2 ) ) ) ).
% int_distrib(1)
thf(fact_1203_int__distrib_I2_J,axiom,
! [W2: int,Z1: int,Z22: int] :
( ( times_times_int @ W2 @ ( plus_plus_int @ Z1 @ Z22 ) )
= ( plus_plus_int @ ( times_times_int @ W2 @ Z1 ) @ ( times_times_int @ W2 @ Z22 ) ) ) ).
% int_distrib(2)
thf(fact_1204_int__distrib_I3_J,axiom,
! [Z1: int,Z22: int,W2: int] :
( ( times_times_int @ ( minus_minus_int @ Z1 @ Z22 ) @ W2 )
= ( minus_minus_int @ ( times_times_int @ Z1 @ W2 ) @ ( times_times_int @ Z22 @ W2 ) ) ) ).
% int_distrib(3)
thf(fact_1205_int__distrib_I4_J,axiom,
! [W2: int,Z1: int,Z22: int] :
( ( times_times_int @ W2 @ ( minus_minus_int @ Z1 @ Z22 ) )
= ( minus_minus_int @ ( times_times_int @ W2 @ Z1 ) @ ( times_times_int @ W2 @ Z22 ) ) ) ).
% int_distrib(4)
thf(fact_1206_pos__zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ( times_times_int @ M @ N )
= one_one_int )
= ( ( M = one_one_int )
& ( N = one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_1207_minusinfinity,axiom,
! [D: int,P1: int > $o,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X3: int,K3: int] :
( ( P1 @ X3 )
= ( P1 @ ( minus_minus_int @ X3 @ ( times_times_int @ K3 @ D ) ) ) )
=> ( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P1 @ X3 ) ) )
=> ( ? [X_12: int] : ( P1 @ X_12 )
=> ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% minusinfinity
thf(fact_1208_plusinfinity,axiom,
! [D: int,P3: int > $o,P: int > $o] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X3: int,K3: int] :
( ( P3 @ X3 )
= ( P3 @ ( minus_minus_int @ X3 @ ( times_times_int @ K3 @ D ) ) ) )
=> ( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P3 @ X3 ) ) )
=> ( ? [X_12: int] : ( P3 @ X_12 )
=> ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% plusinfinity
thf(fact_1209_zmult__zless__mono2__lemma,axiom,
! [I2: int,J: int,K: nat] :
( ( ord_less_int @ I2 @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I2 ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J ) ) ) ) ).
% zmult_zless_mono2_lemma
thf(fact_1210_incr__mult__lemma,axiom,
! [D: int,P: int > $o,K: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X3: int] :
( ( P @ X3 )
=> ( P @ ( plus_plus_int @ X3 @ D ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K )
=> ! [X6: int] :
( ( P @ X6 )
=> ( P @ ( plus_plus_int @ X6 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% incr_mult_lemma
thf(fact_1211_decr__mult__lemma,axiom,
! [D: int,P: int > $o,K: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X3: int] :
( ( P @ X3 )
=> ( P @ ( minus_minus_int @ X3 @ D ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K )
=> ! [X6: int] :
( ( P @ X6 )
=> ( P @ ( minus_minus_int @ X6 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% decr_mult_lemma
thf(fact_1212_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_1213_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_1214_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_1215_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_1216_nat__1__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M @ N ) )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_1217_nat__mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= one_one_nat )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_1218_one__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M @ N ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_1219_mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_1220_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_1221_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_1222_mult__Suc__right,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ M @ ( suc @ N ) )
= ( plus_plus_nat @ M @ ( times_times_nat @ M @ N ) ) ) ).
% mult_Suc_right
thf(fact_1223_one__le__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% one_le_mult_iff
thf(fact_1224_mult__le__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% mult_le_cancel2
thf(fact_1225_mult__less__mono1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_1226_mult__less__mono2,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I2 ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_1227_Suc__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_mult_less_cancel1
thf(fact_1228_Suc__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_mult_le_cancel1
thf(fact_1229_mult__Suc,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ N @ ( times_times_nat @ M @ N ) ) ) ).
% mult_Suc
thf(fact_1230_mult__eq__self__implies__10,axiom,
! [M: nat,N: nat] :
( ( M
= ( times_times_nat @ M @ N ) )
=> ( ( N = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_1231_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_1232_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_1233_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_1234_mult__le__mono,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_1235_mult__le__mono1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_1236_mult__le__mono2,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I2 ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_1237_add__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% add_mult_distrib
thf(fact_1238_add__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% add_mult_distrib2
thf(fact_1239_diff__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% diff_mult_distrib
thf(fact_1240_diff__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_1241_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_1242_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_1243_int__ops_I7_J,axiom,
! [A2: nat,B2: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ A2 @ B2 ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ).
% int_ops(7)
thf(fact_1244_Suc__mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ ( suc @ K ) @ M )
= ( times_times_nat @ ( suc @ K ) @ N ) )
= ( M = N ) ) ).
% Suc_mult_cancel1
thf(fact_1245_n__less__n__mult__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ N @ M ) ) ) ) ).
% n_less_n_mult_m
thf(fact_1246_n__less__m__mult__n,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ M @ N ) ) ) ) ).
% n_less_m_mult_n
thf(fact_1247_one__less__mult,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) ) ) ) ).
% one_less_mult
thf(fact_1248_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M3: nat,N3: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N3 @ ( times_times_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N3 ) ) ) ) ) ).
% mult_eq_if
thf(fact_1249_nat__mult__le__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1250_nat__mult__less__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_1251_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_1252_left__add__mult__distrib,axiom,
! [I2: nat,U: nat,J: nat,K: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I2 @ J ) @ U ) @ K ) ) ).
% left_add_mult_distrib
thf(fact_1253_nat__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_1254_nat__mult__eq__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( M = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_1255_nat__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1256_nat__eq__add__iff1,axiom,
! [J: nat,I2: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I2 )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J ) @ U ) @ M )
= N ) ) ) ).
% nat_eq_add_iff1
thf(fact_1257_nat__eq__add__iff2,axiom,
! [I2: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( M
= ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I2 ) @ U ) @ N ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_1258_nat__le__add__iff1,axiom,
! [J: nat,I2: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I2 )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_le_add_iff1
thf(fact_1259_nat__le__add__iff2,axiom,
! [I2: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I2 ) @ U ) @ N ) ) ) ) ).
% nat_le_add_iff2
thf(fact_1260_nat__diff__add__eq1,axiom,
! [J: nat,I2: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_diff_add_eq1
thf(fact_1261_nat__diff__add__eq2,axiom,
! [I2: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I2 ) @ U ) @ N ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_1262_nat__less__add__iff1,axiom,
! [J: nat,I2: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I2 )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_less_add_iff1
thf(fact_1263_nat__less__add__iff2,axiom,
! [I2: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I2 ) @ U ) @ N ) ) ) ) ).
% nat_less_add_iff2
% Helper facts (7)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
! [X: list_int,Y: list_int] :
( ( if_list_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
! [X: list_int,Y: list_int] :
( ( if_list_int @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
star_P707599355569300323st_int @ ( exec1 @ ( ccomp @ ( assign @ x @ a ) ) ) @ ( produc5086643055186798020st_int @ zero_zero_int @ ( produc8650753666468850689st_int @ sa @ stka ) ) @ ( produc5086643055186798020st_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ ( ccomp @ ( assign @ x @ a ) ) ) ) @ ( produc8650753666468850689st_int @ ( fun_up8169409267500384524ar_int @ sa @ x @ ( aval @ a @ sa ) ) @ stka ) ) ).
%------------------------------------------------------------------------------