TPTP Problem File: SLH0796^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : IMP_Compiler_Reuse/0006_Compiler2/prob_00561_028369__6247070_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1521 ( 631 unt; 247 typ; 0 def)
% Number of atoms : 3416 (1429 equ; 0 cnn)
% Maximal formula atoms : 14 ( 2 avg)
% Number of connectives : 10883 ( 361 ~; 57 |; 258 &;8874 @)
% ( 0 <=>;1333 =>; 0 <=; 0 <~>)
% Maximal formula depth : 30 ( 6 avg)
% Number of types : 45 ( 44 usr)
% Number of type conns : 1324 (1324 >; 0 *; 0 +; 0 <<)
% Number of symbols : 206 ( 203 usr; 15 con; 0-4 aty)
% Number of variables : 3860 ( 287 ^;3384 !; 189 ?;3860 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 16:53:06.668
%------------------------------------------------------------------------------
% Could-be-implicit typings (44)
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_J,type,
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thf(ty_n_t__List__Olist_It__Int__Oint_J,type,
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thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
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thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
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thf(ty_n_t__Compiler__Oinstr,type,
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thf(ty_n_t__String__Ochar,type,
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thf(ty_n_t__BExp__Obexp,type,
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thf(ty_n_t__AExp__Oaexp,type,
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thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
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thf(ty_n_t__Com__Ocom,type,
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% Explicit typings (203)
thf(sy_c_AExp_Oaval,type,
aval: aexp > ( list_char > int ) > int ).
thf(sy_c_BExp_Obexp_OAnd,type,
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thf(sy_c_BExp_Obexp_Osize__bexp,type,
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thf(sy_c_BExp_Obval,type,
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thf(sy_c_Big__Step_Obig__step,type,
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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_Compiler2_Oapred,type,
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thf(sy_c_Compiler2_Obpred,type,
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thf(sy_c_Compiler2_Ocpred,type,
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thf(sy_c_Compiler2_Oexecl,type,
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thf(sy_c_Compiler2_Oexecl__all,type,
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thf(sy_c_Compiler2_Oexecl__rel,type,
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thf(sy_c_Compiler2_Ompred,type,
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thf(sy_c_Compiler_Oacomp,type,
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thf(sy_c_Compiler_Obcomp,type,
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thf(sy_c_Compiler_Occomp,type,
ccomp: com > list_instr ).
thf(sy_c_Compiler_Oexec1,type,
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thf(sy_c_Compiler_Oiexec,type,
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thf(sy_c_Compiler_Oiexec__pred,type,
iexec_pred: produc5995290525303592096st_int > produc6425607678544837394st_int > $o ).
thf(sy_c_Compiler_Oinstr_OJMP,type,
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thf(sy_c_Compiler_Oinstr_OLOAD,type,
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thf(sy_c_Compiler_Oinstr_OLOADI,type,
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thf(sy_c_Compiler_Ointh_001t__Compiler__Oinstr,type,
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thf(sy_c_Compiler_Ointh_001t__Int__Oint,type,
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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,
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thf(sy_c_Compiler_Oto__m__state__rel,type,
to_m_state_rel: produc4189061985984568957nt_int > produc4189061985984568957nt_int > $o ).
thf(sy_c_Euclidean__Division_Odivmod__nat,type,
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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_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
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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_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Int_Oint__ge__less__than,type,
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thf(sy_c_Int_Oint__ge__less__than2,type,
int_ge_less_than2: int > set_Pr958786334691620121nt_int ).
thf(sy_c_List_Oappend_001t__Compiler__Oinstr,type,
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thf(sy_c_List_Oappend_001t__Int__Oint,type,
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thf(sy_c_List_Oappend_001t__List__Olist_It__String__Ochar_J,type,
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thf(sy_c_List_Odrop_001t__Product____Type__Oprod_It__Int__Oint_Mt__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J_J,type,
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thf(sy_c_List_Oenumerate_001t__Compiler__Oinstr,type,
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thf(sy_c_List_Oenumerate_001t__Int__Oint,type,
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thf(sy_c_List_Oenumerate_001t__List__Olist_It__String__Ochar_J,type,
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thf(sy_c_List_Olist_OCons_001t__Compiler__Oinstr,type,
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thf(sy_c_List_Olist_OCons_001t__Int__Oint,type,
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thf(sy_c_List_Olist_OCons_001t__List__Olist_It__String__Ochar_J,type,
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thf(sy_c_List_Olist_ONil_001t__Compiler__Oinstr,type,
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thf(sy_c_List_Olist_Omap_001t__Compiler__Oinstr_001t__Compiler__Oinstr,type,
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thf(sy_c_List_Olist_Omap_001t__Compiler__Oinstr_001t__Int__Oint,type,
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thf(sy_c_List_Olist_Omap_001t__Compiler__Oinstr_001t__List__Olist_It__String__Ochar_J,type,
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thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Compiler__Oinstr,type,
produc3129998205563682742_instr: nat > instr > produc2078578717123253124_instr ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Int__Oint,type,
product_Pair_nat_int: nat > int > product_prod_nat_int ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__List__Olist_It__String__Ochar_J,type,
produc120197956887798346t_char: nat > list_char > produc1457211279475724562t_char ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).
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produc2664891610604065915st_int: nat > produc6425607678544837394st_int > produc896087705980016963st_int ).
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Set__Oset_It__Int__Oint_J,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_001t__Int__Oint_001_Eo,type,
produc2452651797061511826_int_o: ( ( list_char > int ) > int > $o ) > produc4435102495419491129nt_int > $o ).
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_001t__Int__Oint_001t__Set__Oset_It__Int__Oint_J,type,
produc3808931903539241832et_int: ( ( list_char > int ) > int > set_int ) > produc4435102495419491129nt_int > set_int ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_001t__List__Olist_It__Int__Oint_J_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J_J_M_Eo_J,type,
produc7737560463583487271_int_o: ( ( list_char > int ) > list_int > produc6425607678544837394st_int > $o ) > produc369741286924889929st_int > produc6425607678544837394st_int > $o ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_001t__List__Olist_It__Int__Oint_J_001_Eo,type,
produc4252051693687794178_int_o: ( ( list_char > int ) > list_int > $o ) > produc369741286924889929st_int > $o ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_001t__List__Olist_It__Int__Oint_J_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,
produc1929222690536411315st_int: ( ( list_char > int ) > list_int > produc369741286924889929st_int ) > produc369741286924889929st_int > produc369741286924889929st_int ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_001t__List__Olist_It__Int__Oint_J_001t__Set__Oset_It__Int__Oint_J,type,
produc384056407671676920et_int: ( ( list_char > int ) > list_int > set_int ) > produc369741286924889929st_int > set_int ).
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produc1573362020775583542_int_o: ( ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o ) > produc2285326912895808259nt_int > $o ).
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Set__Oset_It__Int__Oint_J,type,
produc8665509832508157892et_int: ( ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > set_int ) > produc2285326912895808259nt_int > set_int ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001_Eo_001t__Int__Oint_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J_J_M_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J_J_M_Eo_J_J,type,
produc3125181923677910856_int_o: ( $o > int > produc6425607678544837394st_int > produc6425607678544837394st_int > $o ) > product_prod_o_int > produc6425607678544837394st_int > produc6425607678544837394st_int > $o ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__BExp__Obexp_001t__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J_J_M_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J_J_M_Eo_J_J,type,
produc8845725704633262453_int_o: ( bexp > product_prod_o_int > produc6425607678544837394st_int > produc6425607678544837394st_int > $o ) > produc1897111610453708512_o_int > produc6425607678544837394st_int > produc6425607678544837394st_int > $o ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Com__Ocom_001_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_001_Eo,type,
produc6166317358938442030_int_o: ( com > ( list_char > int ) > $o ) > produc2931317944591925149ar_int > $o ).
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produc2420365882498604339ar_int: ( com > ( list_char > int ) > produc2931317944591925149ar_int ) > produc2931317944591925149ar_int > produc2931317944591925149ar_int ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Com__Ocom_001_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_001t__Set__Oset_It__Int__Oint_J,type,
produc6818752395963765708et_int: ( com > ( list_char > int ) > set_int ) > produc2931317944591925149ar_int > set_int ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001_Eo,type,
produc4947309494688390418_int_o: ( int > int > $o ) > product_prod_int_int > $o ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_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_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J_J_M_Eo_J,type,
produc3978606535191404044_int_o: ( int > produc369741286924889929st_int > produc6425607678544837394st_int > $o ) > produc6425607678544837394st_int > produc6425607678544837394st_int > $o ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_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_001_Eo,type,
produc5824858813793146727_int_o: ( int > produc369741286924889929st_int > $o ) > produc6425607678544837394st_int > $o ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Product__Type_Oprod_Osnd_001_062_It__List__Olist_It__String__Ochar_J_Mt__Int__Oint_J_001t__List__Olist_It__Int__Oint_J,type,
produc6244060863719409455st_int: produc369741286924889929st_int > list_int ).
thf(sy_c_Product__Type_Oprod_Osnd_001_Eo_001t__Int__Oint,type,
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thf(sy_c_Product__Type_Oprod_Osnd_001t__BExp__Obexp_001t__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J,type,
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thf(sy_c_Set_OCollect_001t__Int__Oint,type,
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thf(sy_c_Set_OCollect_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,
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thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(sy_c_Set_OCollect_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,
collec3466707605950672359st_int: ( produc6425607678544837394st_int > $o ) > set_Pr2116125220558876360st_int ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Int__Oint,type,
set_or4662586982721622107an_int: int > int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
set_or4665077453230672383an_nat: nat > nat > set_nat ).
thf(sy_c_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,
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thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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accp_P1482788143903795053st_int: ( produc4400946599758736694st_int > produc4400946599758736694st_int > $o ) > produc4400946599758736694st_int > $o ).
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thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
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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,
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thf(sy_c_member_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(sy_v_P,type,
p: list_instr ).
thf(sy_v_b,type,
b: bexp ).
thf(sy_v_c,type,
c: com ).
thf(sy_v_cfs,type,
cfs: list_P419547167332244514st_int ).
thf(sy_v_f,type,
f: $o ).
thf(sy_v_i,type,
i: int ).
% Relevant facts (1261)
thf(fact_0_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_1_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_2_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_3_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_4_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_5_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_6_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_7_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_8_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_9_of__nat__1,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% of_nat_1
thf(fact_10_of__nat__1,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% of_nat_1
thf(fact_11_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_12_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_13_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_14_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_15_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_16_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_17_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_18_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_19_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_20_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_21_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_22_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_23_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_24_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1316708129612266289at_nat @ M )
= ( semiri1316708129612266289at_nat @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_25_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_26_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_27_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_28_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_29_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_30_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_31_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_32_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_33_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_34_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_35_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_36_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_37_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_38_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_39_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_40_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_41_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_42_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
| ( M2 = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_43_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_44_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
& ( M2 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_45_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_46_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_47_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_48_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_49_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_50_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_51_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_52_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K2 )
=> ~ ( P @ I3 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_53_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_54_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_55_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_56_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_57_mem__Collect__eq,axiom,
! [A: int,P: int > $o] :
( ( member_int @ A @ ( collect_int @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_58_mem__Collect__eq,axiom,
! [A: product_prod_int_int,P: product_prod_int_int > $o] :
( ( member5262025264175285858nt_int @ A @ ( collec213857154873943460nt_int @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_59_Collect__mem__eq,axiom,
! [A2: set_int] :
( ( collect_int
@ ^ [X: int] : ( member_int @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_60_Collect__mem__eq,axiom,
! [A2: set_Pr958786334691620121nt_int] :
( ( collec213857154873943460nt_int
@ ^ [X: product_prod_int_int] : ( member5262025264175285858nt_int @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_61_Collect__cong,axiom,
! [P: product_prod_int_int > $o,Q: product_prod_int_int > $o] :
( ! [X2: product_prod_int_int] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collec213857154873943460nt_int @ P )
= ( collec213857154873943460nt_int @ Q ) ) ) ).
% Collect_cong
thf(fact_62_linorder__neqE__nat,axiom,
! [X3: nat,Y: nat] :
( ( X3 != Y )
=> ( ~ ( ord_less_nat @ X3 @ Y )
=> ( ord_less_nat @ Y @ X3 ) ) ) ).
% linorder_neqE_nat
thf(fact_63_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_64_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( P @ M3 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_65_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_66_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_67_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_68_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_69_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_70_size__neq__size__imp__neq,axiom,
! [X3: list_instr,Y: list_instr] :
( ( ( size_size_list_instr @ X3 )
!= ( size_size_list_instr @ Y ) )
=> ( X3 != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_71_size__neq__size__imp__neq,axiom,
! [X3: list_list_char,Y: list_list_char] :
( ( ( size_s356637359517785349t_char @ X3 )
!= ( size_s356637359517785349t_char @ Y ) )
=> ( X3 != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_72_size__neq__size__imp__neq,axiom,
! [X3: list_int,Y: list_int] :
( ( ( size_size_list_int @ X3 )
!= ( size_size_list_int @ Y ) )
=> ( X3 != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_73_size__neq__size__imp__neq,axiom,
! [X3: bexp,Y: bexp] :
( ( ( size_size_bexp @ X3 )
!= ( size_size_bexp @ Y ) )
=> ( X3 != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_74_size__neq__size__imp__neq,axiom,
! [X3: com,Y: com] :
( ( ( size_size_com @ X3 )
!= ( size_size_com @ Y ) )
=> ( X3 != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_75_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_76_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_77_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_78_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_79_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_80_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_81_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_82_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_83_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_84_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).
% of_nat_mono
thf(fact_85_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).
% of_nat_mono
thf(fact_86_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_87_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_88_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_89_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_90_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_91_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_92_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_93_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_94_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_95_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_96_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_97_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_98_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_99_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_100_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_101_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_102_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_103_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_104_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_105_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_106_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_107_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_108_diff__gt__0__iff__gt,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
= ( ord_less_int @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_109_diff__ge__0__iff__ge,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
= ( ord_less_eq_int @ B @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_110_case__prodI2,axiom,
! [P2: produc2931317944591925149ar_int,C: com > ( list_char > int ) > $o] :
( ! [A3: com,B2: list_char > int] :
( ( P2
= ( produc5595214716300948949ar_int @ A3 @ B2 ) )
=> ( C @ A3 @ B2 ) )
=> ( produc6166317358938442030_int_o @ C @ P2 ) ) ).
% case_prodI2
thf(fact_111_case__prodI2,axiom,
! [P2: produc2285326912895808259nt_int,C: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o] :
( ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B2: product_prod_int_int] :
( ( P2
= ( produc5700946648718959541nt_int @ A3 @ B2 ) )
=> ( C @ A3 @ B2 ) )
=> ( produc1573362020775583542_int_o @ C @ P2 ) ) ).
% case_prodI2
thf(fact_112_case__prodI2,axiom,
! [P2: produc4435102495419491129nt_int,C: ( list_char > int ) > int > $o] :
( ! [A3: list_char > int,B2: int] :
( ( P2
= ( produc5790713362662368625nt_int @ A3 @ B2 ) )
=> ( C @ A3 @ B2 ) )
=> ( produc2452651797061511826_int_o @ C @ P2 ) ) ).
% case_prodI2
thf(fact_113_case__prodI2,axiom,
! [P2: produc7773217078559923341nt_int,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o] :
( ! [A3: int > option6357759511663192854e_term,B2: product_prod_int_int] :
( ( P2
= ( produc4305682042979456191nt_int @ A3 @ B2 ) )
=> ( C @ A3 @ B2 ) )
=> ( produc2558449545302689196_int_o @ C @ P2 ) ) ).
% case_prodI2
thf(fact_114_case__prodI2,axiom,
! [P2: produc6425607678544837394st_int,C: int > produc369741286924889929st_int > $o] :
( ! [A3: int,B2: produc369741286924889929st_int] :
( ( P2
= ( produc5086643055186798020st_int @ A3 @ B2 ) )
=> ( C @ A3 @ B2 ) )
=> ( produc5824858813793146727_int_o @ C @ P2 ) ) ).
% case_prodI2
thf(fact_115_case__prodI2,axiom,
! [P2: product_prod_int_int,C: int > int > $o] :
( ! [A3: int,B2: int] :
( ( P2
= ( product_Pair_int_int @ A3 @ B2 ) )
=> ( C @ A3 @ B2 ) )
=> ( produc4947309494688390418_int_o @ C @ P2 ) ) ).
% case_prodI2
thf(fact_116_case__prodI2,axiom,
! [P2: produc369741286924889929st_int,C: ( list_char > int ) > list_int > $o] :
( ! [A3: list_char > int,B2: list_int] :
( ( P2
= ( produc8650753666468850689st_int @ A3 @ B2 ) )
=> ( C @ A3 @ B2 ) )
=> ( produc4252051693687794178_int_o @ C @ P2 ) ) ).
% case_prodI2
thf(fact_117_case__prodI,axiom,
! [F: com > ( list_char > int ) > $o,A: com,B: list_char > int] :
( ( F @ A @ B )
=> ( produc6166317358938442030_int_o @ F @ ( produc5595214716300948949ar_int @ A @ B ) ) ) ).
% case_prodI
thf(fact_118_case__prodI,axiom,
! [F: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o,A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int] :
( ( F @ A @ B )
=> ( produc1573362020775583542_int_o @ F @ ( produc5700946648718959541nt_int @ A @ B ) ) ) ).
% case_prodI
thf(fact_119_case__prodI,axiom,
! [F: ( list_char > int ) > int > $o,A: list_char > int,B: int] :
( ( F @ A @ B )
=> ( produc2452651797061511826_int_o @ F @ ( produc5790713362662368625nt_int @ A @ B ) ) ) ).
% case_prodI
thf(fact_120_case__prodI,axiom,
! [F: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
( ( F @ A @ B )
=> ( produc2558449545302689196_int_o @ F @ ( produc4305682042979456191nt_int @ A @ B ) ) ) ).
% case_prodI
thf(fact_121_case__prodI,axiom,
! [F: int > produc369741286924889929st_int > $o,A: int,B: produc369741286924889929st_int] :
( ( F @ A @ B )
=> ( produc5824858813793146727_int_o @ F @ ( produc5086643055186798020st_int @ A @ B ) ) ) ).
% case_prodI
thf(fact_122_case__prodI,axiom,
! [F: int > int > $o,A: int,B: int] :
( ( F @ A @ B )
=> ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ A @ B ) ) ) ).
% case_prodI
thf(fact_123_case__prodI,axiom,
! [F: ( list_char > int ) > list_int > $o,A: list_char > int,B: list_int] :
( ( F @ A @ B )
=> ( produc4252051693687794178_int_o @ F @ ( produc8650753666468850689st_int @ A @ B ) ) ) ).
% case_prodI
thf(fact_124_case__prod__conv,axiom,
! [F: int > produc369741286924889929st_int > $o,A: int,B: produc369741286924889929st_int] :
( ( produc5824858813793146727_int_o @ F @ ( produc5086643055186798020st_int @ A @ B ) )
= ( F @ A @ B ) ) ).
% case_prod_conv
thf(fact_125_case__prod__conv,axiom,
! [F: int > produc369741286924889929st_int > produc6425607678544837394st_int > $o,A: int,B: produc369741286924889929st_int] :
( ( produc3978606535191404044_int_o @ F @ ( produc5086643055186798020st_int @ A @ B ) )
= ( F @ A @ B ) ) ).
% case_prod_conv
thf(fact_126_case__prod__conv,axiom,
! [F: int > int > $o,A: int,B: int] :
( ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ A @ B ) )
= ( F @ A @ B ) ) ).
% case_prod_conv
thf(fact_127_case__prod__conv,axiom,
! [F: ( list_char > int ) > list_int > $o,A: list_char > int,B: list_int] :
( ( produc4252051693687794178_int_o @ F @ ( produc8650753666468850689st_int @ A @ B ) )
= ( F @ A @ B ) ) ).
% case_prod_conv
thf(fact_128_case__prod__conv,axiom,
! [F: ( list_char > int ) > list_int > produc6425607678544837394st_int > $o,A: list_char > int,B: list_int] :
( ( produc7737560463583487271_int_o @ F @ ( produc8650753666468850689st_int @ A @ B ) )
= ( F @ A @ B ) ) ).
% case_prod_conv
thf(fact_129_nth__append,axiom,
! [N: nat,Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int] :
( ( ( ord_less_nat @ N @ ( size_s229347708945616566st_int @ Xs ) )
=> ( ( nth_Pr8327173284404383691st_int @ ( append5028292981072010935st_int @ Xs @ Ys ) @ N )
= ( nth_Pr8327173284404383691st_int @ Xs @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_s229347708945616566st_int @ Xs ) )
=> ( ( nth_Pr8327173284404383691st_int @ ( append5028292981072010935st_int @ Xs @ Ys ) @ N )
= ( nth_Pr8327173284404383691st_int @ Ys @ ( minus_minus_nat @ N @ ( size_s229347708945616566st_int @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_130_nth__append,axiom,
! [N: nat,Xs: list_instr,Ys: list_instr] :
( ( ( ord_less_nat @ N @ ( size_size_list_instr @ Xs ) )
=> ( ( nth_instr @ ( append_instr @ Xs @ Ys ) @ N )
= ( nth_instr @ Xs @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_size_list_instr @ Xs ) )
=> ( ( nth_instr @ ( append_instr @ Xs @ Ys ) @ N )
= ( nth_instr @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_instr @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_131_nth__append,axiom,
! [N: nat,Xs: list_list_char,Ys: list_list_char] :
( ( ( ord_less_nat @ N @ ( size_s356637359517785349t_char @ Xs ) )
=> ( ( nth_list_char @ ( append_list_char @ Xs @ Ys ) @ N )
= ( nth_list_char @ Xs @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_s356637359517785349t_char @ Xs ) )
=> ( ( nth_list_char @ ( append_list_char @ Xs @ Ys ) @ N )
= ( nth_list_char @ Ys @ ( minus_minus_nat @ N @ ( size_s356637359517785349t_char @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_132_nth__append,axiom,
! [N: nat,Xs: list_int,Ys: list_int] :
( ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
=> ( ( nth_int @ ( append_int @ Xs @ Ys ) @ N )
= ( nth_int @ Xs @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
=> ( ( nth_int @ ( append_int @ Xs @ Ys ) @ N )
= ( nth_int @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_int @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_133_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_134_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_135_append__eq__append__conv,axiom,
! [Xs: list_int,Ys: list_int,Us: list_int,Vs: list_int] :
( ( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys ) )
| ( ( size_size_list_int @ Us )
= ( size_size_list_int @ Vs ) ) )
=> ( ( ( append_int @ Xs @ Us )
= ( append_int @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_136_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_137_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_138_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_139_diff__zero,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_zero
thf(fact_140_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_141_diff__0__right,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_0_right
thf(fact_142_diff__self,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% diff_self
thf(fact_143_prod_Oinject,axiom,
! [X1: com,X22: list_char > int,Y1: com,Y2: list_char > int] :
( ( ( produc5595214716300948949ar_int @ X1 @ X22 )
= ( produc5595214716300948949ar_int @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_144_prod_Oinject,axiom,
! [X1: produc8551481072490612790e_term > option6357759511663192854e_term,X22: product_prod_int_int,Y1: produc8551481072490612790e_term > option6357759511663192854e_term,Y2: product_prod_int_int] :
( ( ( produc5700946648718959541nt_int @ X1 @ X22 )
= ( produc5700946648718959541nt_int @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_145_prod_Oinject,axiom,
! [X1: list_char > int,X22: list_int,Y1: list_char > int,Y2: list_int] :
( ( ( produc8650753666468850689st_int @ X1 @ X22 )
= ( produc8650753666468850689st_int @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_146_prod_Oinject,axiom,
! [X1: list_char > int,X22: int,Y1: list_char > int,Y2: int] :
( ( ( produc5790713362662368625nt_int @ X1 @ X22 )
= ( produc5790713362662368625nt_int @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_147_prod_Oinject,axiom,
! [X1: int > option6357759511663192854e_term,X22: product_prod_int_int,Y1: int > option6357759511663192854e_term,Y2: product_prod_int_int] :
( ( ( produc4305682042979456191nt_int @ X1 @ X22 )
= ( produc4305682042979456191nt_int @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_148_old_Oprod_Oinject,axiom,
! [A: com,B: list_char > int,A4: com,B3: list_char > int] :
( ( ( produc5595214716300948949ar_int @ A @ B )
= ( produc5595214716300948949ar_int @ A4 @ B3 ) )
= ( ( A = A4 )
& ( B = B3 ) ) ) ).
% old.prod.inject
thf(fact_149_old_Oprod_Oinject,axiom,
! [A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int,A4: produc8551481072490612790e_term > option6357759511663192854e_term,B3: product_prod_int_int] :
( ( ( produc5700946648718959541nt_int @ A @ B )
= ( produc5700946648718959541nt_int @ A4 @ B3 ) )
= ( ( A = A4 )
& ( B = B3 ) ) ) ).
% old.prod.inject
thf(fact_150_old_Oprod_Oinject,axiom,
! [A: list_char > int,B: list_int,A4: list_char > int,B3: list_int] :
( ( ( produc8650753666468850689st_int @ A @ B )
= ( produc8650753666468850689st_int @ A4 @ B3 ) )
= ( ( A = A4 )
& ( B = B3 ) ) ) ).
% old.prod.inject
thf(fact_151_old_Oprod_Oinject,axiom,
! [A: list_char > int,B: int,A4: list_char > int,B3: int] :
( ( ( produc5790713362662368625nt_int @ A @ B )
= ( produc5790713362662368625nt_int @ A4 @ B3 ) )
= ( ( A = A4 )
& ( B = B3 ) ) ) ).
% old.prod.inject
thf(fact_152_old_Oprod_Oinject,axiom,
! [A: int > option6357759511663192854e_term,B: product_prod_int_int,A4: int > option6357759511663192854e_term,B3: product_prod_int_int] :
( ( ( produc4305682042979456191nt_int @ A @ B )
= ( produc4305682042979456191nt_int @ A4 @ B3 ) )
= ( ( A = A4 )
& ( B = B3 ) ) ) ).
% old.prod.inject
thf(fact_153_append_Oassoc,axiom,
! [A: list_instr,B: list_instr,C: list_instr] :
( ( append_instr @ ( append_instr @ A @ B ) @ C )
= ( append_instr @ A @ ( append_instr @ B @ C ) ) ) ).
% append.assoc
thf(fact_154_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_155_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_156_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_157_mem__case__prodI,axiom,
! [Z: int,C: com > ( list_char > int ) > set_int,A: com,B: list_char > int] :
( ( member_int @ Z @ ( C @ A @ B ) )
=> ( member_int @ Z @ ( produc6818752395963765708et_int @ C @ ( produc5595214716300948949ar_int @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_158_mem__case__prodI,axiom,
! [Z: int,C: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > set_int,A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int] :
( ( member_int @ Z @ ( C @ A @ B ) )
=> ( member_int @ Z @ ( produc8665509832508157892et_int @ C @ ( produc5700946648718959541nt_int @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_159_mem__case__prodI,axiom,
! [Z: int,C: ( list_char > int ) > list_int > set_int,A: list_char > int,B: list_int] :
( ( member_int @ Z @ ( C @ A @ B ) )
=> ( member_int @ Z @ ( produc384056407671676920et_int @ C @ ( produc8650753666468850689st_int @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_160_mem__case__prodI,axiom,
! [Z: int,C: ( list_char > int ) > int > set_int,A: list_char > int,B: int] :
( ( member_int @ Z @ ( C @ A @ B ) )
=> ( member_int @ Z @ ( produc3808931903539241832et_int @ C @ ( produc5790713362662368625nt_int @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_161_mem__case__prodI,axiom,
! [Z: int,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_int,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
( ( member_int @ Z @ ( C @ A @ B ) )
=> ( member_int @ Z @ ( produc4111701587417901774et_int @ C @ ( produc4305682042979456191nt_int @ A @ B ) ) ) ) ).
% mem_case_prodI
thf(fact_162_mem__case__prodI2,axiom,
! [P2: produc2931317944591925149ar_int,Z: int,C: com > ( list_char > int ) > set_int] :
( ! [A3: com,B2: list_char > int] :
( ( P2
= ( produc5595214716300948949ar_int @ A3 @ B2 ) )
=> ( member_int @ Z @ ( C @ A3 @ B2 ) ) )
=> ( member_int @ Z @ ( produc6818752395963765708et_int @ C @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_163_mem__case__prodI2,axiom,
! [P2: produc2285326912895808259nt_int,Z: int,C: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > set_int] :
( ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B2: product_prod_int_int] :
( ( P2
= ( produc5700946648718959541nt_int @ A3 @ B2 ) )
=> ( member_int @ Z @ ( C @ A3 @ B2 ) ) )
=> ( member_int @ Z @ ( produc8665509832508157892et_int @ C @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_164_mem__case__prodI2,axiom,
! [P2: produc369741286924889929st_int,Z: int,C: ( list_char > int ) > list_int > set_int] :
( ! [A3: list_char > int,B2: list_int] :
( ( P2
= ( produc8650753666468850689st_int @ A3 @ B2 ) )
=> ( member_int @ Z @ ( C @ A3 @ B2 ) ) )
=> ( member_int @ Z @ ( produc384056407671676920et_int @ C @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_165_mem__case__prodI2,axiom,
! [P2: produc4435102495419491129nt_int,Z: int,C: ( list_char > int ) > int > set_int] :
( ! [A3: list_char > int,B2: int] :
( ( P2
= ( produc5790713362662368625nt_int @ A3 @ B2 ) )
=> ( member_int @ Z @ ( C @ A3 @ B2 ) ) )
=> ( member_int @ Z @ ( produc3808931903539241832et_int @ C @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_166_mem__case__prodI2,axiom,
! [P2: produc7773217078559923341nt_int,Z: int,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_int] :
( ! [A3: int > option6357759511663192854e_term,B2: product_prod_int_int] :
( ( P2
= ( produc4305682042979456191nt_int @ A3 @ B2 ) )
=> ( member_int @ Z @ ( C @ A3 @ B2 ) ) )
=> ( member_int @ Z @ ( produc4111701587417901774et_int @ C @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_167_case__prodI2_H,axiom,
! [P2: produc6425607678544837394st_int,C: int > produc369741286924889929st_int > produc6425607678544837394st_int > $o,X3: produc6425607678544837394st_int] :
( ! [A3: int,B2: produc369741286924889929st_int] :
( ( ( produc5086643055186798020st_int @ A3 @ B2 )
= P2 )
=> ( C @ A3 @ B2 @ X3 ) )
=> ( produc3978606535191404044_int_o @ C @ P2 @ X3 ) ) ).
% case_prodI2'
thf(fact_168_case__prodI2_H,axiom,
! [P2: produc369741286924889929st_int,C: ( list_char > int ) > list_int > produc6425607678544837394st_int > $o,X3: produc6425607678544837394st_int] :
( ! [A3: list_char > int,B2: list_int] :
( ( ( produc8650753666468850689st_int @ A3 @ B2 )
= P2 )
=> ( C @ A3 @ B2 @ X3 ) )
=> ( produc7737560463583487271_int_o @ C @ P2 @ X3 ) ) ).
% case_prodI2'
thf(fact_169_split__part,axiom,
! [P: $o,Q: int > produc369741286924889929st_int > $o] :
( ( produc5824858813793146727_int_o
@ ^ [A5: int,B4: produc369741286924889929st_int] :
( P
& ( Q @ A5 @ B4 ) ) )
= ( ^ [Ab: produc6425607678544837394st_int] :
( P
& ( produc5824858813793146727_int_o @ Q @ Ab ) ) ) ) ).
% split_part
thf(fact_170_split__part,axiom,
! [P: $o,Q: int > int > $o] :
( ( produc4947309494688390418_int_o
@ ^ [A5: int,B4: int] :
( P
& ( Q @ A5 @ B4 ) ) )
= ( ^ [Ab: product_prod_int_int] :
( P
& ( produc4947309494688390418_int_o @ Q @ Ab ) ) ) ) ).
% split_part
thf(fact_171_split__part,axiom,
! [P: $o,Q: ( list_char > int ) > list_int > $o] :
( ( produc4252051693687794178_int_o
@ ^ [A5: list_char > int,B4: list_int] :
( P
& ( Q @ A5 @ B4 ) ) )
= ( ^ [Ab: produc369741286924889929st_int] :
( P
& ( produc4252051693687794178_int_o @ Q @ Ab ) ) ) ) ).
% split_part
thf(fact_172_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_173_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_174_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_175_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_176_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_177_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_178_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_179_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X2: nat] :
( ( P @ X2 )
& ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_180_mem__case__prodE,axiom,
! [Z: int,C: com > ( list_char > int ) > set_int,P2: produc2931317944591925149ar_int] :
( ( member_int @ Z @ ( produc6818752395963765708et_int @ C @ P2 ) )
=> ~ ! [X2: com,Y3: list_char > int] :
( ( P2
= ( produc5595214716300948949ar_int @ X2 @ Y3 ) )
=> ~ ( member_int @ Z @ ( C @ X2 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_181_mem__case__prodE,axiom,
! [Z: int,C: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > set_int,P2: produc2285326912895808259nt_int] :
( ( member_int @ Z @ ( produc8665509832508157892et_int @ C @ P2 ) )
=> ~ ! [X2: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: product_prod_int_int] :
( ( P2
= ( produc5700946648718959541nt_int @ X2 @ Y3 ) )
=> ~ ( member_int @ Z @ ( C @ X2 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_182_mem__case__prodE,axiom,
! [Z: int,C: ( list_char > int ) > list_int > set_int,P2: produc369741286924889929st_int] :
( ( member_int @ Z @ ( produc384056407671676920et_int @ C @ P2 ) )
=> ~ ! [X2: list_char > int,Y3: list_int] :
( ( P2
= ( produc8650753666468850689st_int @ X2 @ Y3 ) )
=> ~ ( member_int @ Z @ ( C @ X2 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_183_mem__case__prodE,axiom,
! [Z: int,C: ( list_char > int ) > int > set_int,P2: produc4435102495419491129nt_int] :
( ( member_int @ Z @ ( produc3808931903539241832et_int @ C @ P2 ) )
=> ~ ! [X2: list_char > int,Y3: int] :
( ( P2
= ( produc5790713362662368625nt_int @ X2 @ Y3 ) )
=> ~ ( member_int @ Z @ ( C @ X2 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_184_mem__case__prodE,axiom,
! [Z: int,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_int,P2: produc7773217078559923341nt_int] :
( ( member_int @ Z @ ( produc4111701587417901774et_int @ C @ P2 ) )
=> ~ ! [X2: int > option6357759511663192854e_term,Y3: product_prod_int_int] :
( ( P2
= ( produc4305682042979456191nt_int @ X2 @ Y3 ) )
=> ~ ( member_int @ Z @ ( C @ X2 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_185_case__prodD_H,axiom,
! [R: int > produc369741286924889929st_int > produc6425607678544837394st_int > $o,A: int,B: produc369741286924889929st_int,C: produc6425607678544837394st_int] :
( ( produc3978606535191404044_int_o @ R @ ( produc5086643055186798020st_int @ A @ B ) @ C )
=> ( R @ A @ B @ C ) ) ).
% case_prodD'
thf(fact_186_case__prodD_H,axiom,
! [R: ( list_char > int ) > list_int > produc6425607678544837394st_int > $o,A: list_char > int,B: list_int,C: produc6425607678544837394st_int] :
( ( produc7737560463583487271_int_o @ R @ ( produc8650753666468850689st_int @ A @ B ) @ C )
=> ( R @ A @ B @ C ) ) ).
% case_prodD'
thf(fact_187_case__prodE_H,axiom,
! [C: int > produc369741286924889929st_int > produc6425607678544837394st_int > $o,P2: produc6425607678544837394st_int,Z: produc6425607678544837394st_int] :
( ( produc3978606535191404044_int_o @ C @ P2 @ Z )
=> ~ ! [X2: int,Y3: produc369741286924889929st_int] :
( ( P2
= ( produc5086643055186798020st_int @ X2 @ Y3 ) )
=> ~ ( C @ X2 @ Y3 @ Z ) ) ) ).
% case_prodE'
thf(fact_188_case__prodE_H,axiom,
! [C: ( list_char > int ) > list_int > produc6425607678544837394st_int > $o,P2: produc369741286924889929st_int,Z: produc6425607678544837394st_int] :
( ( produc7737560463583487271_int_o @ C @ P2 @ Z )
=> ~ ! [X2: list_char > int,Y3: list_int] :
( ( P2
= ( produc8650753666468850689st_int @ X2 @ Y3 ) )
=> ~ ( C @ X2 @ Y3 @ Z ) ) ) ).
% case_prodE'
thf(fact_189_zero__reorient,axiom,
! [X3: int] :
( ( zero_zero_int = X3 )
= ( X3 = zero_zero_int ) ) ).
% zero_reorient
thf(fact_190_zero__reorient,axiom,
! [X3: nat] :
( ( zero_zero_nat = X3 )
= ( X3 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_191_old_Oprod_Oexhaust,axiom,
! [Y: produc2931317944591925149ar_int] :
~ ! [A3: com,B2: list_char > int] :
( Y
!= ( produc5595214716300948949ar_int @ A3 @ B2 ) ) ).
% old.prod.exhaust
thf(fact_192_old_Oprod_Oexhaust,axiom,
! [Y: produc2285326912895808259nt_int] :
~ ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B2: product_prod_int_int] :
( Y
!= ( produc5700946648718959541nt_int @ A3 @ B2 ) ) ).
% old.prod.exhaust
thf(fact_193_old_Oprod_Oexhaust,axiom,
! [Y: produc369741286924889929st_int] :
~ ! [A3: list_char > int,B2: list_int] :
( Y
!= ( produc8650753666468850689st_int @ A3 @ B2 ) ) ).
% old.prod.exhaust
thf(fact_194_old_Oprod_Oexhaust,axiom,
! [Y: produc4435102495419491129nt_int] :
~ ! [A3: list_char > int,B2: int] :
( Y
!= ( produc5790713362662368625nt_int @ A3 @ B2 ) ) ).
% old.prod.exhaust
thf(fact_195_old_Oprod_Oexhaust,axiom,
! [Y: produc7773217078559923341nt_int] :
~ ! [A3: int > option6357759511663192854e_term,B2: product_prod_int_int] :
( Y
!= ( produc4305682042979456191nt_int @ A3 @ B2 ) ) ).
% old.prod.exhaust
thf(fact_196_surj__pair,axiom,
! [P2: produc2931317944591925149ar_int] :
? [X2: com,Y3: list_char > int] :
( P2
= ( produc5595214716300948949ar_int @ X2 @ Y3 ) ) ).
% surj_pair
thf(fact_197_surj__pair,axiom,
! [P2: produc2285326912895808259nt_int] :
? [X2: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: product_prod_int_int] :
( P2
= ( produc5700946648718959541nt_int @ X2 @ Y3 ) ) ).
% surj_pair
thf(fact_198_surj__pair,axiom,
! [P2: produc369741286924889929st_int] :
? [X2: list_char > int,Y3: list_int] :
( P2
= ( produc8650753666468850689st_int @ X2 @ Y3 ) ) ).
% surj_pair
thf(fact_199_surj__pair,axiom,
! [P2: produc4435102495419491129nt_int] :
? [X2: list_char > int,Y3: int] :
( P2
= ( produc5790713362662368625nt_int @ X2 @ Y3 ) ) ).
% surj_pair
thf(fact_200_surj__pair,axiom,
! [P2: produc7773217078559923341nt_int] :
? [X2: int > option6357759511663192854e_term,Y3: product_prod_int_int] :
( P2
= ( produc4305682042979456191nt_int @ X2 @ Y3 ) ) ).
% surj_pair
thf(fact_201_prod__cases,axiom,
! [P: produc2931317944591925149ar_int > $o,P2: produc2931317944591925149ar_int] :
( ! [A3: com,B2: list_char > int] : ( P @ ( produc5595214716300948949ar_int @ A3 @ B2 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_202_prod__cases,axiom,
! [P: produc2285326912895808259nt_int > $o,P2: produc2285326912895808259nt_int] :
( ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B2: product_prod_int_int] : ( P @ ( produc5700946648718959541nt_int @ A3 @ B2 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_203_prod__cases,axiom,
! [P: produc369741286924889929st_int > $o,P2: produc369741286924889929st_int] :
( ! [A3: list_char > int,B2: list_int] : ( P @ ( produc8650753666468850689st_int @ A3 @ B2 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_204_prod__cases,axiom,
! [P: produc4435102495419491129nt_int > $o,P2: produc4435102495419491129nt_int] :
( ! [A3: list_char > int,B2: int] : ( P @ ( produc5790713362662368625nt_int @ A3 @ B2 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_205_prod__cases,axiom,
! [P: produc7773217078559923341nt_int > $o,P2: produc7773217078559923341nt_int] :
( ! [A3: int > option6357759511663192854e_term,B2: product_prod_int_int] : ( P @ ( produc4305682042979456191nt_int @ A3 @ B2 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_206_Pair__inject,axiom,
! [A: com,B: list_char > int,A4: com,B3: list_char > int] :
( ( ( produc5595214716300948949ar_int @ A @ B )
= ( produc5595214716300948949ar_int @ A4 @ B3 ) )
=> ~ ( ( A = A4 )
=> ( B != B3 ) ) ) ).
% Pair_inject
thf(fact_207_Pair__inject,axiom,
! [A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int,A4: produc8551481072490612790e_term > option6357759511663192854e_term,B3: product_prod_int_int] :
( ( ( produc5700946648718959541nt_int @ A @ B )
= ( produc5700946648718959541nt_int @ A4 @ B3 ) )
=> ~ ( ( A = A4 )
=> ( B != B3 ) ) ) ).
% Pair_inject
thf(fact_208_Pair__inject,axiom,
! [A: list_char > int,B: list_int,A4: list_char > int,B3: list_int] :
( ( ( produc8650753666468850689st_int @ A @ B )
= ( produc8650753666468850689st_int @ A4 @ B3 ) )
=> ~ ( ( A = A4 )
=> ( B != B3 ) ) ) ).
% Pair_inject
thf(fact_209_Pair__inject,axiom,
! [A: list_char > int,B: int,A4: list_char > int,B3: int] :
( ( ( produc5790713362662368625nt_int @ A @ B )
= ( produc5790713362662368625nt_int @ A4 @ B3 ) )
=> ~ ( ( A = A4 )
=> ( B != B3 ) ) ) ).
% Pair_inject
thf(fact_210_Pair__inject,axiom,
! [A: int > option6357759511663192854e_term,B: product_prod_int_int,A4: int > option6357759511663192854e_term,B3: product_prod_int_int] :
( ( ( produc4305682042979456191nt_int @ A @ B )
= ( produc4305682042979456191nt_int @ A4 @ B3 ) )
=> ~ ( ( A = A4 )
=> ( B != B3 ) ) ) ).
% Pair_inject
thf(fact_211_one__reorient,axiom,
! [X3: nat] :
( ( one_one_nat = X3 )
= ( X3 = one_one_nat ) ) ).
% one_reorient
thf(fact_212_one__reorient,axiom,
! [X3: int] :
( ( one_one_int = X3 )
= ( X3 = one_one_int ) ) ).
% one_reorient
thf(fact_213_diff__eq__diff__eq,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( A = B )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_214_diff__right__commute,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% diff_right_commute
thf(fact_215_diff__right__commute,axiom,
! [A: int,C: int,B: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B )
= ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% diff_right_commute
thf(fact_216_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_instr] :
( ( size_size_list_instr @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_217_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_list_char] :
( ( size_s356637359517785349t_char @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_218_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_int] :
( ( size_size_list_int @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_219_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_220_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_221_neq__if__length__neq,axiom,
! [Xs: list_int,Ys: list_int] :
( ( ( size_size_list_int @ Xs )
!= ( size_size_list_int @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_222_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_223_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_224_Collect__case__prod__mono,axiom,
! [A2: int > produc369741286924889929st_int > $o,B5: int > produc369741286924889929st_int > $o] :
( ( ord_le8490654098287043871_int_o @ A2 @ B5 )
=> ( ord_le8444298136753458984st_int @ ( collec3466707605950672359st_int @ ( produc5824858813793146727_int_o @ A2 ) ) @ ( collec3466707605950672359st_int @ ( produc5824858813793146727_int_o @ B5 ) ) ) ) ).
% Collect_case_prod_mono
thf(fact_225_Collect__case__prod__mono,axiom,
! [A2: int > int > $o,B5: int > int > $o] :
( ( ord_le6741204236512500942_int_o @ A2 @ B5 )
=> ( ord_le2843351958646193337nt_int @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ A2 ) ) @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ B5 ) ) ) ) ).
% Collect_case_prod_mono
thf(fact_226_Collect__case__prod__mono,axiom,
! [A2: ( list_char > int ) > list_int > $o,B5: ( list_char > int ) > list_int > $o] :
( ( ord_le4781006829409452478_int_o @ A2 @ B5 )
=> ( ord_le5738032636065246921st_int @ ( collec3870426226673543860st_int @ ( produc4252051693687794178_int_o @ A2 ) ) @ ( collec3870426226673543860st_int @ ( produc4252051693687794178_int_o @ B5 ) ) ) ) ).
% Collect_case_prod_mono
thf(fact_227_prod_Ocase__distrib,axiom,
! [H: $o > $o,F: int > produc369741286924889929st_int > $o,Prod: produc6425607678544837394st_int] :
( ( H @ ( produc5824858813793146727_int_o @ F @ Prod ) )
= ( produc5824858813793146727_int_o
@ ^ [X12: int,X23: produc369741286924889929st_int] : ( H @ ( F @ X12 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_228_prod_Ocase__distrib,axiom,
! [H: $o > produc6425607678544837394st_int > $o,F: int > produc369741286924889929st_int > $o,Prod: produc6425607678544837394st_int] :
( ( H @ ( produc5824858813793146727_int_o @ F @ Prod ) )
= ( produc3978606535191404044_int_o
@ ^ [X12: int,X23: produc369741286924889929st_int] : ( H @ ( F @ X12 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_229_prod_Ocase__distrib,axiom,
! [H: ( produc6425607678544837394st_int > $o ) > $o,F: int > produc369741286924889929st_int > produc6425607678544837394st_int > $o,Prod: produc6425607678544837394st_int] :
( ( H @ ( produc3978606535191404044_int_o @ F @ Prod ) )
= ( produc5824858813793146727_int_o
@ ^ [X12: int,X23: produc369741286924889929st_int] : ( H @ ( F @ X12 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_230_prod_Ocase__distrib,axiom,
! [H: ( produc6425607678544837394st_int > $o ) > produc6425607678544837394st_int > $o,F: int > produc369741286924889929st_int > produc6425607678544837394st_int > $o,Prod: produc6425607678544837394st_int] :
( ( H @ ( produc3978606535191404044_int_o @ F @ Prod ) )
= ( produc3978606535191404044_int_o
@ ^ [X12: int,X23: produc369741286924889929st_int] : ( H @ ( F @ X12 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_231_prod_Ocase__distrib,axiom,
! [H: $o > $o,F: int > int > $o,Prod: product_prod_int_int] :
( ( H @ ( produc4947309494688390418_int_o @ F @ Prod ) )
= ( produc4947309494688390418_int_o
@ ^ [X12: int,X23: int] : ( H @ ( F @ X12 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_232_prod_Ocase__distrib,axiom,
! [H: $o > $o,F: ( list_char > int ) > list_int > $o,Prod: produc369741286924889929st_int] :
( ( H @ ( produc4252051693687794178_int_o @ F @ Prod ) )
= ( produc4252051693687794178_int_o
@ ^ [X12: list_char > int,X23: list_int] : ( H @ ( F @ X12 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_233_prod_Ocase__distrib,axiom,
! [H: $o > produc6425607678544837394st_int > $o,F: ( list_char > int ) > list_int > $o,Prod: produc369741286924889929st_int] :
( ( H @ ( produc4252051693687794178_int_o @ F @ Prod ) )
= ( produc7737560463583487271_int_o
@ ^ [X12: list_char > int,X23: list_int] : ( H @ ( F @ X12 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_234_prod_Ocase__distrib,axiom,
! [H: ( produc6425607678544837394st_int > $o ) > $o,F: ( list_char > int ) > list_int > produc6425607678544837394st_int > $o,Prod: produc369741286924889929st_int] :
( ( H @ ( produc7737560463583487271_int_o @ F @ Prod ) )
= ( produc4252051693687794178_int_o
@ ^ [X12: list_char > int,X23: list_int] : ( H @ ( F @ X12 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_235_prod_Ocase__distrib,axiom,
! [H: ( produc6425607678544837394st_int > $o ) > produc6425607678544837394st_int > $o,F: ( list_char > int ) > list_int > produc6425607678544837394st_int > $o,Prod: produc369741286924889929st_int] :
( ( H @ ( produc7737560463583487271_int_o @ F @ Prod ) )
= ( produc7737560463583487271_int_o
@ ^ [X12: list_char > int,X23: list_int] : ( H @ ( F @ X12 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_236_prod_Odisc__eq__case,axiom,
! [Prod: produc6425607678544837394st_int] :
( produc5824858813793146727_int_o
@ ^ [Uu: int,Uv: produc369741286924889929st_int] : $true
@ Prod ) ).
% prod.disc_eq_case
thf(fact_237_prod_Odisc__eq__case,axiom,
! [Prod: product_prod_int_int] :
( produc4947309494688390418_int_o
@ ^ [Uu: int,Uv: int] : $true
@ Prod ) ).
% prod.disc_eq_case
thf(fact_238_prod_Odisc__eq__case,axiom,
! [Prod: produc369741286924889929st_int] :
( produc4252051693687794178_int_o
@ ^ [Uu: list_char > int,Uv: list_int] : $true
@ Prod ) ).
% prod.disc_eq_case
thf(fact_239_zero__le,axiom,
! [X3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X3 ) ).
% zero_le
thf(fact_240_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_241_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_242_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_243_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_244_diff__mono,axiom,
! [A: int,B: int,D: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ D @ C )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% diff_mono
thf(fact_245_diff__left__mono,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_246_diff__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_247_diff__eq__diff__less__eq,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_eq_int @ A @ B )
= ( ord_less_eq_int @ C @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_248_eq__iff__diff__eq__0,axiom,
( ( ^ [Y5: int,Z2: int] : ( Y5 = Z2 ) )
= ( ^ [A5: int,B4: int] :
( ( minus_minus_int @ A5 @ B4 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_249_diff__strict__mono,axiom,
! [A: int,B: int,D: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ D @ C )
=> ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_250_diff__eq__diff__less,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_int @ A @ B )
= ( ord_less_int @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_251_diff__strict__left__mono,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_252_diff__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_253_old_Oprod_Ocase,axiom,
! [F: int > produc369741286924889929st_int > $o,X1: int,X22: produc369741286924889929st_int] :
( ( produc5824858813793146727_int_o @ F @ ( produc5086643055186798020st_int @ X1 @ X22 ) )
= ( F @ X1 @ X22 ) ) ).
% old.prod.case
thf(fact_254_old_Oprod_Ocase,axiom,
! [F: int > produc369741286924889929st_int > produc6425607678544837394st_int > $o,X1: int,X22: produc369741286924889929st_int] :
( ( produc3978606535191404044_int_o @ F @ ( produc5086643055186798020st_int @ X1 @ X22 ) )
= ( F @ X1 @ X22 ) ) ).
% old.prod.case
thf(fact_255_old_Oprod_Ocase,axiom,
! [F: int > int > $o,X1: int,X22: int] :
( ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ X1 @ X22 ) )
= ( F @ X1 @ X22 ) ) ).
% old.prod.case
thf(fact_256_old_Oprod_Ocase,axiom,
! [F: ( list_char > int ) > list_int > $o,X1: list_char > int,X22: list_int] :
( ( produc4252051693687794178_int_o @ F @ ( produc8650753666468850689st_int @ X1 @ X22 ) )
= ( F @ X1 @ X22 ) ) ).
% old.prod.case
thf(fact_257_old_Oprod_Ocase,axiom,
! [F: ( list_char > int ) > list_int > produc6425607678544837394st_int > $o,X1: list_char > int,X22: list_int] :
( ( produc7737560463583487271_int_o @ F @ ( produc8650753666468850689st_int @ X1 @ X22 ) )
= ( F @ X1 @ X22 ) ) ).
% old.prod.case
thf(fact_258_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_259_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_260_length__induct,axiom,
! [P: list_int > $o,Xs: list_int] :
( ! [Xs2: list_int] :
( ! [Ys2: list_int] :
( ( ord_less_nat @ ( size_size_list_int @ Ys2 ) @ ( size_size_list_int @ Xs2 ) )
=> ( P @ Ys2 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_261_prod__cases3,axiom,
! [Y: produc2285326912895808259nt_int] :
~ ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B2: int,C2: int] :
( Y
!= ( produc5700946648718959541nt_int @ A3 @ ( product_Pair_int_int @ B2 @ C2 ) ) ) ).
% prod_cases3
thf(fact_262_prod__cases3,axiom,
! [Y: produc7773217078559923341nt_int] :
~ ! [A3: int > option6357759511663192854e_term,B2: int,C2: int] :
( Y
!= ( produc4305682042979456191nt_int @ A3 @ ( product_Pair_int_int @ B2 @ C2 ) ) ) ).
% prod_cases3
thf(fact_263_prod__induct3,axiom,
! [P: produc2285326912895808259nt_int > $o,X3: produc2285326912895808259nt_int] :
( ! [A3: produc8551481072490612790e_term > option6357759511663192854e_term,B2: int,C2: int] : ( P @ ( produc5700946648718959541nt_int @ A3 @ ( product_Pair_int_int @ B2 @ C2 ) ) )
=> ( P @ X3 ) ) ).
% prod_induct3
thf(fact_264_prod__induct3,axiom,
! [P: produc7773217078559923341nt_int > $o,X3: produc7773217078559923341nt_int] :
( ! [A3: int > option6357759511663192854e_term,B2: int,C2: int] : ( P @ ( produc4305682042979456191nt_int @ A3 @ ( product_Pair_int_int @ B2 @ C2 ) ) )
=> ( P @ X3 ) ) ).
% prod_induct3
thf(fact_265_case__prodE2,axiom,
! [Q: $o > $o,P: int > produc369741286924889929st_int > $o,Z: produc6425607678544837394st_int] :
( ( Q @ ( produc5824858813793146727_int_o @ P @ Z ) )
=> ~ ! [X2: int,Y3: produc369741286924889929st_int] :
( ( Z
= ( produc5086643055186798020st_int @ X2 @ Y3 ) )
=> ~ ( Q @ ( P @ X2 @ Y3 ) ) ) ) ).
% case_prodE2
thf(fact_266_case__prodE2,axiom,
! [Q: ( produc6425607678544837394st_int > $o ) > $o,P: int > produc369741286924889929st_int > produc6425607678544837394st_int > $o,Z: produc6425607678544837394st_int] :
( ( Q @ ( produc3978606535191404044_int_o @ P @ Z ) )
=> ~ ! [X2: int,Y3: produc369741286924889929st_int] :
( ( Z
= ( produc5086643055186798020st_int @ X2 @ Y3 ) )
=> ~ ( Q @ ( P @ X2 @ Y3 ) ) ) ) ).
% case_prodE2
thf(fact_267_case__prodE2,axiom,
! [Q: $o > $o,P: int > int > $o,Z: product_prod_int_int] :
( ( Q @ ( produc4947309494688390418_int_o @ P @ Z ) )
=> ~ ! [X2: int,Y3: int] :
( ( Z
= ( product_Pair_int_int @ X2 @ Y3 ) )
=> ~ ( Q @ ( P @ X2 @ Y3 ) ) ) ) ).
% case_prodE2
thf(fact_268_case__prodE2,axiom,
! [Q: $o > $o,P: ( list_char > int ) > list_int > $o,Z: produc369741286924889929st_int] :
( ( Q @ ( produc4252051693687794178_int_o @ P @ Z ) )
=> ~ ! [X2: list_char > int,Y3: list_int] :
( ( Z
= ( produc8650753666468850689st_int @ X2 @ Y3 ) )
=> ~ ( Q @ ( P @ X2 @ Y3 ) ) ) ) ).
% case_prodE2
thf(fact_269_case__prodE2,axiom,
! [Q: ( produc6425607678544837394st_int > $o ) > $o,P: ( list_char > int ) > list_int > produc6425607678544837394st_int > $o,Z: produc369741286924889929st_int] :
( ( Q @ ( produc7737560463583487271_int_o @ P @ Z ) )
=> ~ ! [X2: list_char > int,Y3: list_int] :
( ( Z
= ( produc8650753666468850689st_int @ X2 @ Y3 ) )
=> ~ ( Q @ ( P @ X2 @ Y3 ) ) ) ) ).
% case_prodE2
thf(fact_270_case__prod__eta,axiom,
! [F: produc6425607678544837394st_int > $o] :
( ( produc5824858813793146727_int_o
@ ^ [X: int,Y6: produc369741286924889929st_int] : ( F @ ( produc5086643055186798020st_int @ X @ Y6 ) ) )
= F ) ).
% case_prod_eta
thf(fact_271_case__prod__eta,axiom,
! [F: produc6425607678544837394st_int > produc6425607678544837394st_int > $o] :
( ( produc3978606535191404044_int_o
@ ^ [X: int,Y6: produc369741286924889929st_int] : ( F @ ( produc5086643055186798020st_int @ X @ Y6 ) ) )
= F ) ).
% case_prod_eta
thf(fact_272_case__prod__eta,axiom,
! [F: product_prod_int_int > $o] :
( ( produc4947309494688390418_int_o
@ ^ [X: int,Y6: int] : ( F @ ( product_Pair_int_int @ X @ Y6 ) ) )
= F ) ).
% case_prod_eta
thf(fact_273_case__prod__eta,axiom,
! [F: produc369741286924889929st_int > $o] :
( ( produc4252051693687794178_int_o
@ ^ [X: list_char > int,Y6: list_int] : ( F @ ( produc8650753666468850689st_int @ X @ Y6 ) ) )
= F ) ).
% case_prod_eta
thf(fact_274_case__prod__eta,axiom,
! [F: produc369741286924889929st_int > produc6425607678544837394st_int > $o] :
( ( produc7737560463583487271_int_o
@ ^ [X: list_char > int,Y6: list_int] : ( F @ ( produc8650753666468850689st_int @ X @ Y6 ) ) )
= F ) ).
% case_prod_eta
thf(fact_275_cond__case__prod__eta,axiom,
! [F: int > produc369741286924889929st_int > $o,G: produc6425607678544837394st_int > $o] :
( ! [X2: int,Y3: produc369741286924889929st_int] :
( ( F @ X2 @ Y3 )
= ( G @ ( produc5086643055186798020st_int @ X2 @ Y3 ) ) )
=> ( ( produc5824858813793146727_int_o @ F )
= G ) ) ).
% cond_case_prod_eta
thf(fact_276_cond__case__prod__eta,axiom,
! [F: int > produc369741286924889929st_int > produc6425607678544837394st_int > $o,G: produc6425607678544837394st_int > produc6425607678544837394st_int > $o] :
( ! [X2: int,Y3: produc369741286924889929st_int] :
( ( F @ X2 @ Y3 )
= ( G @ ( produc5086643055186798020st_int @ X2 @ Y3 ) ) )
=> ( ( produc3978606535191404044_int_o @ F )
= G ) ) ).
% cond_case_prod_eta
thf(fact_277_cond__case__prod__eta,axiom,
! [F: int > int > $o,G: product_prod_int_int > $o] :
( ! [X2: int,Y3: int] :
( ( F @ X2 @ Y3 )
= ( G @ ( product_Pair_int_int @ X2 @ Y3 ) ) )
=> ( ( produc4947309494688390418_int_o @ F )
= G ) ) ).
% cond_case_prod_eta
thf(fact_278_cond__case__prod__eta,axiom,
! [F: ( list_char > int ) > list_int > $o,G: produc369741286924889929st_int > $o] :
( ! [X2: list_char > int,Y3: list_int] :
( ( F @ X2 @ Y3 )
= ( G @ ( produc8650753666468850689st_int @ X2 @ Y3 ) ) )
=> ( ( produc4252051693687794178_int_o @ F )
= G ) ) ).
% cond_case_prod_eta
thf(fact_279_cond__case__prod__eta,axiom,
! [F: ( list_char > int ) > list_int > produc6425607678544837394st_int > $o,G: produc369741286924889929st_int > produc6425607678544837394st_int > $o] :
( ! [X2: list_char > int,Y3: list_int] :
( ( F @ X2 @ Y3 )
= ( G @ ( produc8650753666468850689st_int @ X2 @ Y3 ) ) )
=> ( ( produc7737560463583487271_int_o @ F )
= G ) ) ).
% cond_case_prod_eta
thf(fact_280_case__prodD,axiom,
! [F: com > ( list_char > int ) > $o,A: com,B: list_char > int] :
( ( produc6166317358938442030_int_o @ F @ ( produc5595214716300948949ar_int @ A @ B ) )
=> ( F @ A @ B ) ) ).
% case_prodD
thf(fact_281_case__prodD,axiom,
! [F: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o,A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int] :
( ( produc1573362020775583542_int_o @ F @ ( produc5700946648718959541nt_int @ A @ B ) )
=> ( F @ A @ B ) ) ).
% case_prodD
thf(fact_282_case__prodD,axiom,
! [F: ( list_char > int ) > int > $o,A: list_char > int,B: int] :
( ( produc2452651797061511826_int_o @ F @ ( produc5790713362662368625nt_int @ A @ B ) )
=> ( F @ A @ B ) ) ).
% case_prodD
thf(fact_283_case__prodD,axiom,
! [F: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
( ( produc2558449545302689196_int_o @ F @ ( produc4305682042979456191nt_int @ A @ B ) )
=> ( F @ A @ B ) ) ).
% case_prodD
thf(fact_284_case__prodD,axiom,
! [F: int > produc369741286924889929st_int > $o,A: int,B: produc369741286924889929st_int] :
( ( produc5824858813793146727_int_o @ F @ ( produc5086643055186798020st_int @ A @ B ) )
=> ( F @ A @ B ) ) ).
% case_prodD
thf(fact_285_case__prodD,axiom,
! [F: int > int > $o,A: int,B: int] :
( ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ A @ B ) )
=> ( F @ A @ B ) ) ).
% case_prodD
thf(fact_286_case__prodD,axiom,
! [F: ( list_char > int ) > list_int > $o,A: list_char > int,B: list_int] :
( ( produc4252051693687794178_int_o @ F @ ( produc8650753666468850689st_int @ A @ B ) )
=> ( F @ A @ B ) ) ).
% case_prodD
thf(fact_287_case__prodE,axiom,
! [C: com > ( list_char > int ) > $o,P2: produc2931317944591925149ar_int] :
( ( produc6166317358938442030_int_o @ C @ P2 )
=> ~ ! [X2: com,Y3: list_char > int] :
( ( P2
= ( produc5595214716300948949ar_int @ X2 @ Y3 ) )
=> ~ ( C @ X2 @ Y3 ) ) ) ).
% case_prodE
thf(fact_288_case__prodE,axiom,
! [C: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o,P2: produc2285326912895808259nt_int] :
( ( produc1573362020775583542_int_o @ C @ P2 )
=> ~ ! [X2: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: product_prod_int_int] :
( ( P2
= ( produc5700946648718959541nt_int @ X2 @ Y3 ) )
=> ~ ( C @ X2 @ Y3 ) ) ) ).
% case_prodE
thf(fact_289_case__prodE,axiom,
! [C: ( list_char > int ) > int > $o,P2: produc4435102495419491129nt_int] :
( ( produc2452651797061511826_int_o @ C @ P2 )
=> ~ ! [X2: list_char > int,Y3: int] :
( ( P2
= ( produc5790713362662368625nt_int @ X2 @ Y3 ) )
=> ~ ( C @ X2 @ Y3 ) ) ) ).
% case_prodE
thf(fact_290_case__prodE,axiom,
! [C: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o,P2: produc7773217078559923341nt_int] :
( ( produc2558449545302689196_int_o @ C @ P2 )
=> ~ ! [X2: int > option6357759511663192854e_term,Y3: product_prod_int_int] :
( ( P2
= ( produc4305682042979456191nt_int @ X2 @ Y3 ) )
=> ~ ( C @ X2 @ Y3 ) ) ) ).
% case_prodE
thf(fact_291_case__prodE,axiom,
! [C: int > produc369741286924889929st_int > $o,P2: produc6425607678544837394st_int] :
( ( produc5824858813793146727_int_o @ C @ P2 )
=> ~ ! [X2: int,Y3: produc369741286924889929st_int] :
( ( P2
= ( produc5086643055186798020st_int @ X2 @ Y3 ) )
=> ~ ( C @ X2 @ Y3 ) ) ) ).
% case_prodE
thf(fact_292_case__prodE,axiom,
! [C: int > int > $o,P2: product_prod_int_int] :
( ( produc4947309494688390418_int_o @ C @ P2 )
=> ~ ! [X2: int,Y3: int] :
( ( P2
= ( product_Pair_int_int @ X2 @ Y3 ) )
=> ~ ( C @ X2 @ Y3 ) ) ) ).
% case_prodE
thf(fact_293_case__prodE,axiom,
! [C: ( list_char > int ) > list_int > $o,P2: produc369741286924889929st_int] :
( ( produc4252051693687794178_int_o @ C @ P2 )
=> ~ ! [X2: list_char > int,Y3: list_int] :
( ( P2
= ( produc8650753666468850689st_int @ X2 @ Y3 ) )
=> ~ ( C @ X2 @ Y3 ) ) ) ).
% case_prodE
thf(fact_294_le__iff__diff__le__0,axiom,
( ord_less_eq_int
= ( ^ [A5: int,B4: int] : ( ord_less_eq_int @ ( minus_minus_int @ A5 @ B4 ) @ zero_zero_int ) ) ) ).
% le_iff_diff_le_0
thf(fact_295_less__iff__diff__less__0,axiom,
( ord_less_int
= ( ^ [A5: int,B4: int] : ( ord_less_int @ ( minus_minus_int @ A5 @ B4 ) @ zero_zero_int ) ) ) ).
% less_iff_diff_less_0
thf(fact_296_nth__equalityI,axiom,
! [Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int] :
( ( ( size_s229347708945616566st_int @ Xs )
= ( size_s229347708945616566st_int @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s229347708945616566st_int @ Xs ) )
=> ( ( nth_Pr8327173284404383691st_int @ Xs @ I2 )
= ( nth_Pr8327173284404383691st_int @ Ys @ I2 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_297_nth__equalityI,axiom,
! [Xs: list_instr,Ys: list_instr] :
( ( ( size_size_list_instr @ Xs )
= ( size_size_list_instr @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_instr @ Xs ) )
=> ( ( nth_instr @ Xs @ I2 )
= ( nth_instr @ Ys @ I2 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_298_nth__equalityI,axiom,
! [Xs: list_list_char,Ys: list_list_char] :
( ( ( size_s356637359517785349t_char @ Xs )
= ( size_s356637359517785349t_char @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s356637359517785349t_char @ Xs ) )
=> ( ( nth_list_char @ Xs @ I2 )
= ( nth_list_char @ Ys @ I2 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_299_nth__equalityI,axiom,
! [Xs: list_int,Ys: list_int] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
=> ( ( nth_int @ Xs @ I2 )
= ( nth_int @ Ys @ I2 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_300_Skolem__list__nth,axiom,
! [K: nat,P: nat > produc6425607678544837394st_int > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ? [X4: produc6425607678544837394st_int] : ( P @ I4 @ X4 ) ) )
= ( ? [Xs3: list_P419547167332244514st_int] :
( ( ( size_s229347708945616566st_int @ Xs3 )
= K )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ( P @ I4 @ ( nth_Pr8327173284404383691st_int @ Xs3 @ I4 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_301_Skolem__list__nth,axiom,
! [K: nat,P: nat > instr > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ? [X4: instr] : ( P @ I4 @ X4 ) ) )
= ( ? [Xs3: list_instr] :
( ( ( size_size_list_instr @ Xs3 )
= K )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ( P @ I4 @ ( nth_instr @ Xs3 @ I4 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_302_Skolem__list__nth,axiom,
! [K: nat,P: nat > list_char > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ? [X4: list_char] : ( P @ I4 @ X4 ) ) )
= ( ? [Xs3: list_list_char] :
( ( ( size_s356637359517785349t_char @ Xs3 )
= K )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ( P @ I4 @ ( nth_list_char @ Xs3 @ I4 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_303_Skolem__list__nth,axiom,
! [K: nat,P: nat > int > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ? [X4: int] : ( P @ I4 @ X4 ) ) )
= ( ? [Xs3: list_int] :
( ( ( size_size_list_int @ Xs3 )
= K )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K )
=> ( P @ I4 @ ( nth_int @ Xs3 @ I4 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_304_list__eq__iff__nth__eq,axiom,
( ( ^ [Y5: list_P419547167332244514st_int,Z2: list_P419547167332244514st_int] : ( Y5 = Z2 ) )
= ( ^ [Xs3: list_P419547167332244514st_int,Ys3: list_P419547167332244514st_int] :
( ( ( size_s229347708945616566st_int @ Xs3 )
= ( size_s229347708945616566st_int @ Ys3 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s229347708945616566st_int @ Xs3 ) )
=> ( ( nth_Pr8327173284404383691st_int @ Xs3 @ I4 )
= ( nth_Pr8327173284404383691st_int @ Ys3 @ I4 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_305_list__eq__iff__nth__eq,axiom,
( ( ^ [Y5: list_instr,Z2: list_instr] : ( Y5 = Z2 ) )
= ( ^ [Xs3: list_instr,Ys3: list_instr] :
( ( ( size_size_list_instr @ Xs3 )
= ( size_size_list_instr @ Ys3 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_instr @ Xs3 ) )
=> ( ( nth_instr @ Xs3 @ I4 )
= ( nth_instr @ Ys3 @ I4 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_306_list__eq__iff__nth__eq,axiom,
( ( ^ [Y5: list_list_char,Z2: list_list_char] : ( Y5 = Z2 ) )
= ( ^ [Xs3: list_list_char,Ys3: list_list_char] :
( ( ( size_s356637359517785349t_char @ Xs3 )
= ( size_s356637359517785349t_char @ Ys3 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s356637359517785349t_char @ Xs3 ) )
=> ( ( nth_list_char @ Xs3 @ I4 )
= ( nth_list_char @ Ys3 @ I4 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_307_list__eq__iff__nth__eq,axiom,
( ( ^ [Y5: list_int,Z2: list_int] : ( Y5 = Z2 ) )
= ( ^ [Xs3: list_int,Ys3: list_int] :
( ( ( size_size_list_int @ Xs3 )
= ( size_size_list_int @ Ys3 ) )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs3 ) )
=> ( ( nth_int @ Xs3 @ I4 )
= ( nth_int @ Ys3 @ I4 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_308_zdiff__int__split,axiom,
! [P: int > $o,X3: nat,Y: nat] :
( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X3 @ Y ) ) )
= ( ( ( ord_less_eq_nat @ Y @ X3 )
=> ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X3 ) @ ( semiri1314217659103216013at_int @ Y ) ) ) )
& ( ( ord_less_nat @ X3 @ Y )
=> ( P @ zero_zero_int ) ) ) ) ).
% zdiff_int_split
thf(fact_309_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N3: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% pos_int_cases
thf(fact_310_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
& ( K
= ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_311_nat__leq__as__int,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B4: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_leq_as_int
thf(fact_312_cpred__def,axiom,
( cpred
= ( ^ [C3: com] :
( produc3978606535191404044_int_o
@ ^ [Pc: int] :
( produc7737560463583487271_int_o
@ ^ [S2: list_char > int,Stk: list_int] :
( produc5824858813793146727_int_o
@ ^ [Pc2: int] :
( produc4252051693687794178_int_o
@ ^ [S3: list_char > int,Stk2: list_int] :
( ( Pc2
= ( plus_plus_int @ Pc @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ ( ccomp @ C3 ) ) ) ) )
& ( big_big_step @ ( produc5595214716300948949ar_int @ C3 @ S2 ) @ S3 )
& ( Stk2 = Stk ) ) ) ) ) ) ) ) ).
% cpred_def
thf(fact_313_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ~ ! [N3: nat] :
( K
!= ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% nonneg_int_cases
thf(fact_314_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ? [N3: nat] :
( K
= ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_315_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B4: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_316_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_317_add__right__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_318_add__right__cancel,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_319_add__left__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_320_add__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_321_add__le__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_322_add__le__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_323_add__le__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_324_add__le__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_325_double__eq__0__iff,axiom,
! [A: int] :
( ( ( plus_plus_int @ A @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% double_eq_0_iff
thf(fact_326_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_327_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_328_zero__eq__add__iff__both__eq__0,axiom,
! [X3: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X3 @ Y ) )
= ( ( X3 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_329_add__eq__0__iff__both__eq__0,axiom,
! [X3: nat,Y: nat] :
( ( ( plus_plus_nat @ X3 @ Y )
= zero_zero_nat )
= ( ( X3 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_330_add__cancel__right__right,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ A @ B ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_right
thf(fact_331_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_332_add__cancel__right__left,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ B @ A ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_left
thf(fact_333_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_334_add__cancel__left__right,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_right
thf(fact_335_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_336_add__cancel__left__left,axiom,
! [B: int,A: int] :
( ( ( plus_plus_int @ B @ A )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_left
thf(fact_337_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_338_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_339_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_340_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_341_add__less__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_342_add__less__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_343_add__less__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_344_add__less__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_345_add__diff__cancel__right_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_346_add__diff__cancel__right_H,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_347_add__diff__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_348_add__diff__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_349_add__diff__cancel__left_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_350_add__diff__cancel__left_H,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_351_add__diff__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_352_add__diff__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_353_diff__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_354_add__diff__cancel,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_355_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_356_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_357_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_358_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_359_le__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel2
thf(fact_360_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_361_le__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel1
thf(fact_362_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_363_add__le__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel2
thf(fact_364_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_365_add__le__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel1
thf(fact_366_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_367_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_368_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_369_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_370_less__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel2
thf(fact_371_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_372_less__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel1
thf(fact_373_add__less__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_374_add__less__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel2
thf(fact_375_add__less__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_376_add__less__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel1
thf(fact_377_diff__add__zero,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_378_zle__add1__eq__le,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
= ( ord_less_eq_int @ W @ Z ) ) ).
% zle_add1_eq_le
thf(fact_379_zle__diff1__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z @ one_one_int ) )
= ( ord_less_int @ W @ Z ) ) ).
% zle_diff1_eq
thf(fact_380_add__right__imp__eq,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_381_add__right__imp__eq,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_382_add__left__imp__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_383_add__left__imp__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_384_add_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C ) )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.left_commute
thf(fact_385_add_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_386_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A5: int,B4: int] : ( plus_plus_int @ B4 @ A5 ) ) ) ).
% add.commute
thf(fact_387_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A5: nat,B4: nat] : ( plus_plus_nat @ B4 @ A5 ) ) ) ).
% add.commute
thf(fact_388_add_Oright__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_389_add_Oleft__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_390_add_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.assoc
thf(fact_391_add_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_392_group__cancel_Oadd2,axiom,
! [B5: int,K: int,B: int,A: int] :
( ( B5
= ( plus_plus_int @ K @ B ) )
=> ( ( plus_plus_int @ A @ B5 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_393_group__cancel_Oadd2,axiom,
! [B5: nat,K: nat,B: nat,A: nat] :
( ( B5
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B5 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_394_group__cancel_Oadd1,axiom,
! [A2: int,K: int,A: int,B: int] :
( ( A2
= ( plus_plus_int @ K @ A ) )
=> ( ( plus_plus_int @ A2 @ B )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_395_group__cancel_Oadd1,axiom,
! [A2: nat,K: nat,A: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_396_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_int @ I @ K )
= ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_397_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_398_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_399_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_400_int__less__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_int @ I @ K )
=> ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
=> ( ! [I2: int] :
( ( ord_less_int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_less_induct
thf(fact_401_zless__add1__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
= ( ( ord_less_int @ W @ Z )
| ( W = Z ) ) ) ).
% zless_add1_eq
thf(fact_402_int__gr__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_int @ K @ I )
=> ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
=> ( ! [I2: int] :
( ( ord_less_int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_gr_induct
thf(fact_403_is__num__normalize_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_404_verit__sum__simplify,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% verit_sum_simplify
thf(fact_405_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_406_odd__nonzero,axiom,
! [Z: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_407_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_408_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus_int @ K @ zero_zero_int )
= K ) ).
% plus_int_code(1)
thf(fact_409_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_410_int__induct,axiom,
! [P: int > $o,K: int,I: int] :
( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_induct
thf(fact_411_add1__zle__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z )
= ( ord_less_int @ W @ Z ) ) ).
% add1_zle_eq
thf(fact_412_int__ge__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_eq_int @ K @ I )
=> ( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_ge_induct
thf(fact_413_zless__imp__add1__zle,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ Z )
=> ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z ) ) ).
% zless_imp_add1_zle
thf(fact_414_add__le__imp__le__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_415_add__le__imp__le__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_416_add__le__imp__le__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_417_add__le__imp__le__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_418_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B4: nat] :
? [C3: nat] :
( B4
= ( plus_plus_nat @ A5 @ C3 ) ) ) ) ).
% le_iff_add
thf(fact_419_add__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_right_mono
thf(fact_420_add__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_421_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C2: nat] :
( B
!= ( plus_plus_nat @ A @ C2 ) ) ) ).
% less_eqE
thf(fact_422_add__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_left_mono
thf(fact_423_add__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_424_add__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_mono
thf(fact_425_add__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_426_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_427_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_428_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_429_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_430_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_431_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_432_add_Ogroup__left__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add.group_left_neutral
thf(fact_433_add_Ocomm__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.comm_neutral
thf(fact_434_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_435_comm__monoid__add__class_Oadd__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_436_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_437_add__less__imp__less__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_438_add__less__imp__less__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_439_add__less__imp__less__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_440_add__less__imp__less__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_441_add__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_442_add__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_443_add__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_444_add__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_445_add__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_446_add__strict__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_447_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_448_add__mono__thms__linordered__field_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( K = L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_449_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_450_add__mono__thms__linordered__field_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_451_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_452_add__mono__thms__linordered__field_I5_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_453_diff__diff__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_454_diff__diff__eq,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_455_add__implies__diff,axiom,
! [C: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ C @ B )
= A )
=> ( C
= ( minus_minus_nat @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_456_add__implies__diff,axiom,
! [C: int,B: int,A: int] :
( ( ( plus_plus_int @ C @ B )
= A )
=> ( C
= ( minus_minus_int @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_457_diff__add__eq__diff__diff__swap,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) )
= ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_458_diff__add__eq,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% diff_add_eq
thf(fact_459_diff__diff__eq2,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% diff_diff_eq2
thf(fact_460_add__diff__eq,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% add_diff_eq
thf(fact_461_eq__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( A
= ( minus_minus_int @ C @ B ) )
= ( ( plus_plus_int @ A @ B )
= C ) ) ).
% eq_diff_eq
thf(fact_462_diff__eq__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( minus_minus_int @ A @ B )
= C )
= ( A
= ( plus_plus_int @ C @ B ) ) ) ).
% diff_eq_eq
thf(fact_463_group__cancel_Osub1,axiom,
! [A2: int,K: int,A: int,B: int] :
( ( A2
= ( plus_plus_int @ K @ A ) )
=> ( ( minus_minus_int @ A2 @ B )
= ( plus_plus_int @ K @ ( minus_minus_int @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_464_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_465_zle__iff__zadd,axiom,
( ord_less_eq_int
= ( ^ [W2: int,Z3: int] :
? [N2: nat] :
( Z3
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_466_add__nonpos__eq__0__iff,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_int @ X3 @ zero_zero_int )
=> ( ( ord_less_eq_int @ Y @ zero_zero_int )
=> ( ( ( plus_plus_int @ X3 @ Y )
= zero_zero_int )
= ( ( X3 = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_467_add__nonpos__eq__0__iff,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X3 @ Y )
= zero_zero_nat )
= ( ( X3 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_468_add__nonneg__eq__0__iff,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ( plus_plus_int @ X3 @ Y )
= zero_zero_int )
= ( ( X3 = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_469_add__nonneg__eq__0__iff,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X3 @ Y )
= zero_zero_nat )
= ( ( X3 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_470_add__nonpos__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_nonpos
thf(fact_471_add__nonpos__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_472_add__nonneg__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_473_add__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_474_add__increasing2,axiom,
! [C: int,B: int,A: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_475_add__increasing2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_476_add__decreasing2,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_477_add__decreasing2,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_478_add__increasing,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_479_add__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_480_add__decreasing,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_481_add__decreasing,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_482_add__less__le__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_483_add__less__le__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_484_add__le__less__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_485_add__le__less__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_486_add__mono__thms__linordered__field_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_487_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_488_add__mono__thms__linordered__field_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_489_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_490_pos__add__strict,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_491_pos__add__strict,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_492_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C2: nat] :
( ( B
= ( plus_plus_nat @ A @ C2 ) )
=> ( C2 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_493_add__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_494_add__pos__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_495_add__neg__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_496_add__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_neg_neg
thf(fact_497_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ( ( minus_minus_nat @ B @ A )
= C )
= ( B
= ( plus_plus_nat @ C @ A ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_498_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_499_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_500_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
= ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_501_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_502_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_503_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_504_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_505_le__add__diff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% le_add_diff
thf(fact_506_diff__add,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
= B ) ) ).
% diff_add
thf(fact_507_le__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ A @ ( minus_minus_int @ C @ B ) )
= ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% le_diff_eq
thf(fact_508_diff__le__eq,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ ( minus_minus_int @ A @ B ) @ C )
= ( ord_less_eq_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% diff_le_eq
thf(fact_509_less__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ A @ ( minus_minus_int @ C @ B ) )
= ( ord_less_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% less_diff_eq
thf(fact_510_diff__less__eq,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ ( minus_minus_int @ A @ B ) @ C )
= ( ord_less_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% diff_less_eq
thf(fact_511_add__strict__increasing2,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_512_add__strict__increasing2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_513_add__strict__increasing,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_514_add__strict__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_515_add__pos__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_516_add__pos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_517_add__nonpos__neg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_neg
thf(fact_518_add__nonpos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_519_add__nonneg__pos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_520_add__nonneg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_521_add__neg__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_neg_nonpos
thf(fact_522_add__neg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_523_verit__la__disequality,axiom,
! [A: int,B: int] :
( ( A = B )
| ~ ( ord_less_eq_int @ A @ B )
| ~ ( ord_less_eq_int @ B @ A ) ) ).
% verit_la_disequality
thf(fact_524_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_525_verit__comp__simplify1_I2_J,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_526_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_527_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_528_verit__comp__simplify1_I1_J,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_529_pinf_I1_J,axiom,
! [P: nat > $o,P3: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z4 @ X2 )
=> ( ( P @ X2 )
= ( P3 @ X2 ) ) )
=> ( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z4 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z5: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z5 @ X5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P3 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_530_pinf_I1_J,axiom,
! [P: int > $o,P3: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X2: int] :
( ( ord_less_int @ Z4 @ X2 )
=> ( ( P @ X2 )
= ( P3 @ X2 ) ) )
=> ( ? [Z4: int] :
! [X2: int] :
( ( ord_less_int @ Z4 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z5: int] :
! [X5: int] :
( ( ord_less_int @ Z5 @ X5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P3 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_531_pinf_I2_J,axiom,
! [P: nat > $o,P3: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z4 @ X2 )
=> ( ( P @ X2 )
= ( P3 @ X2 ) ) )
=> ( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z4 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z5: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z5 @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P3 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_532_pinf_I2_J,axiom,
! [P: int > $o,P3: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X2: int] :
( ( ord_less_int @ Z4 @ X2 )
=> ( ( P @ X2 )
= ( P3 @ X2 ) ) )
=> ( ? [Z4: int] :
! [X2: int] :
( ( ord_less_int @ Z4 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z5: int] :
! [X5: int] :
( ( ord_less_int @ Z5 @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P3 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_533_pinf_I3_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z5 @ X5 )
=> ( X5 != T ) ) ).
% pinf(3)
thf(fact_534_pinf_I3_J,axiom,
! [T: int] :
? [Z5: int] :
! [X5: int] :
( ( ord_less_int @ Z5 @ X5 )
=> ( X5 != T ) ) ).
% pinf(3)
thf(fact_535_pinf_I4_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z5 @ X5 )
=> ( X5 != T ) ) ).
% pinf(4)
thf(fact_536_pinf_I4_J,axiom,
! [T: int] :
? [Z5: int] :
! [X5: int] :
( ( ord_less_int @ Z5 @ X5 )
=> ( X5 != T ) ) ).
% pinf(4)
thf(fact_537_pinf_I5_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z5 @ X5 )
=> ~ ( ord_less_nat @ X5 @ T ) ) ).
% pinf(5)
thf(fact_538_pinf_I5_J,axiom,
! [T: int] :
? [Z5: int] :
! [X5: int] :
( ( ord_less_int @ Z5 @ X5 )
=> ~ ( ord_less_int @ X5 @ T ) ) ).
% pinf(5)
thf(fact_539_pinf_I7_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z5 @ X5 )
=> ( ord_less_nat @ T @ X5 ) ) ).
% pinf(7)
thf(fact_540_pinf_I7_J,axiom,
! [T: int] :
? [Z5: int] :
! [X5: int] :
( ( ord_less_int @ Z5 @ X5 )
=> ( ord_less_int @ T @ X5 ) ) ).
% pinf(7)
thf(fact_541_minf_I1_J,axiom,
! [P: nat > $o,P3: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z4 )
=> ( ( P @ X2 )
= ( P3 @ X2 ) ) )
=> ( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z4 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z5: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P3 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(1)
thf(fact_542_minf_I1_J,axiom,
! [P: int > $o,P3: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z4 )
=> ( ( P @ X2 )
= ( P3 @ X2 ) ) )
=> ( ? [Z4: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z4 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z5: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P3 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(1)
thf(fact_543_minf_I2_J,axiom,
! [P: nat > $o,P3: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z4 )
=> ( ( P @ X2 )
= ( P3 @ X2 ) ) )
=> ( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z4 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z5: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P3 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(2)
thf(fact_544_minf_I2_J,axiom,
! [P: int > $o,P3: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z4 )
=> ( ( P @ X2 )
= ( P3 @ X2 ) ) )
=> ( ? [Z4: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z4 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z5: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P3 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(2)
thf(fact_545_minf_I3_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z5 )
=> ( X5 != T ) ) ).
% minf(3)
thf(fact_546_minf_I3_J,axiom,
! [T: int] :
? [Z5: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z5 )
=> ( X5 != T ) ) ).
% minf(3)
thf(fact_547_minf_I4_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z5 )
=> ( X5 != T ) ) ).
% minf(4)
thf(fact_548_minf_I4_J,axiom,
! [T: int] :
? [Z5: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z5 )
=> ( X5 != T ) ) ).
% minf(4)
thf(fact_549_minf_I5_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z5 )
=> ( ord_less_nat @ X5 @ T ) ) ).
% minf(5)
thf(fact_550_minf_I5_J,axiom,
! [T: int] :
? [Z5: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z5 )
=> ( ord_less_int @ X5 @ T ) ) ).
% minf(5)
thf(fact_551_minf_I7_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z5 )
=> ~ ( ord_less_nat @ T @ X5 ) ) ).
% minf(7)
thf(fact_552_minf_I7_J,axiom,
! [T: int] :
? [Z5: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z5 )
=> ~ ( ord_less_int @ T @ X5 ) ) ).
% minf(7)
thf(fact_553_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus_int @ K @ zero_zero_int )
= K ) ).
% minus_int_code(1)
thf(fact_554_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_555_int__le__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_eq_int @ I @ K )
=> ( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_le_induct
thf(fact_556_verit__la__generic,axiom,
! [A: int,X3: int] :
( ( ord_less_eq_int @ A @ X3 )
| ( A = X3 )
| ( ord_less_eq_int @ X3 @ A ) ) ).
% verit_la_generic
thf(fact_557_int__diff__cases,axiom,
! [Z: int] :
~ ! [M4: nat,N3: nat] :
( Z
!= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M4 ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% int_diff_cases
thf(fact_558_int__int__eq,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% int_int_eq
thf(fact_559_int__if,axiom,
! [P: $o,A: nat,B: nat] :
( ( P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
= ( semiri1314217659103216013at_int @ A ) ) )
& ( ~ P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
= ( semiri1314217659103216013at_int @ B ) ) ) ) ).
% int_if
thf(fact_560_nat__int__comparison_I1_J,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [A5: nat,B4: nat] :
( ( semiri1314217659103216013at_int @ A5 )
= ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_561_verit__comp__simplify1_I3_J,axiom,
! [B3: int,A4: int] :
( ( ~ ( ord_less_eq_int @ B3 @ A4 ) )
= ( ord_less_int @ A4 @ B3 ) ) ).
% verit_comp_simplify1(3)
thf(fact_562_verit__comp__simplify1_I3_J,axiom,
! [B3: nat,A4: nat] :
( ( ~ ( ord_less_eq_nat @ B3 @ A4 ) )
= ( ord_less_nat @ A4 @ B3 ) ) ).
% verit_comp_simplify1(3)
thf(fact_563_pinf_I6_J,axiom,
! [T: int] :
? [Z5: int] :
! [X5: int] :
( ( ord_less_int @ Z5 @ X5 )
=> ~ ( ord_less_eq_int @ X5 @ T ) ) ).
% pinf(6)
thf(fact_564_pinf_I6_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z5 @ X5 )
=> ~ ( ord_less_eq_nat @ X5 @ T ) ) ).
% pinf(6)
thf(fact_565_pinf_I8_J,axiom,
! [T: int] :
? [Z5: int] :
! [X5: int] :
( ( ord_less_int @ Z5 @ X5 )
=> ( ord_less_eq_int @ T @ X5 ) ) ).
% pinf(8)
thf(fact_566_pinf_I8_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z5 @ X5 )
=> ( ord_less_eq_nat @ T @ X5 ) ) ).
% pinf(8)
thf(fact_567_minf_I6_J,axiom,
! [T: int] :
? [Z5: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z5 )
=> ( ord_less_eq_int @ X5 @ T ) ) ).
% minf(6)
thf(fact_568_minf_I6_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z5 )
=> ( ord_less_eq_nat @ X5 @ T ) ) ).
% minf(6)
thf(fact_569_minf_I8_J,axiom,
! [T: int] :
? [Z5: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z5 )
=> ~ ( ord_less_eq_int @ T @ X5 ) ) ).
% minf(8)
thf(fact_570_minf_I8_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z5 )
=> ~ ( ord_less_eq_nat @ T @ X5 ) ) ).
% minf(8)
thf(fact_571_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A5: nat,B4: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_572_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_573_conj__le__cong,axiom,
! [X3: int,X6: int,P: $o,P3: $o] :
( ( X3 = X6 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> ( P = P3 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X3 )
& P )
= ( ( ord_less_eq_int @ zero_zero_int @ X6 )
& P3 ) ) ) ) ).
% conj_le_cong
thf(fact_574_imp__le__cong,axiom,
! [X3: int,X6: int,P: $o,P3: $o] :
( ( X3 = X6 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> ( P = P3 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X3 )
=> P )
= ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> P3 ) ) ) ) ).
% imp_le_cong
thf(fact_575_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_576_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_577_nat__less__as__int,axiom,
( ord_less_nat
= ( ^ [A5: nat,B4: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_less_as_int
thf(fact_578_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_579_int__ops_I6_J,axiom,
! [A: nat,B: nat] :
( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
= zero_zero_int ) )
& ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ) ).
% int_ops(6)
thf(fact_580_bpred__def,axiom,
( bpred
= ( produc8845725704633262453_int_o
@ ^ [B4: bexp] :
( produc3125181923677910856_int_o
@ ^ [F2: $o,I4: int] :
( produc3978606535191404044_int_o
@ ^ [Pc: int] :
( produc7737560463583487271_int_o
@ ^ [S2: list_char > int,Stk: list_int] :
( produc5824858813793146727_int_o
@ ^ [Pc2: int] :
( produc4252051693687794178_int_o
@ ^ [S3: list_char > int,Stk2: list_int] :
( ( Pc2
= ( plus_plus_int @ ( plus_plus_int @ Pc @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ ( bcomp @ ( produc4047900504771817624_o_int @ B4 @ ( product_Pair_o_int @ F2 @ I4 ) ) ) ) ) )
@ ( if_int
@ ( ( bval @ B4 @ S2 )
= F2 )
@ I4
@ zero_zero_int ) ) )
& ( S3 = S2 )
& ( Stk2 = Stk ) ) ) ) ) ) ) ) ) ).
% bpred_def
thf(fact_581_le__add__diff__inverse2,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_582_le__add__diff__inverse2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_583_le__add__diff__inverse,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_584_le__add__diff__inverse,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_585_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_586_zero__less__two,axiom,
ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% zero_less_two
thf(fact_587_pred__subset__eq2,axiom,
! [R: set_Pr6231067571260677117ar_int,S4: set_Pr6231067571260677117ar_int] :
( ( ord_le4000858847371626730_int_o
@ ^ [X: com,Y6: list_char > int] : ( member2095410489637970758ar_int @ ( produc5595214716300948949ar_int @ X @ Y6 ) @ R )
@ ^ [X: com,Y6: list_char > int] : ( member2095410489637970758ar_int @ ( produc5595214716300948949ar_int @ X @ Y6 ) @ S4 ) )
= ( ord_le6049463789062751133ar_int @ R @ S4 ) ) ).
% pred_subset_eq2
thf(fact_588_pred__subset__eq2,axiom,
! [R: set_Pr9222295170931077689nt_int,S4: set_Pr9222295170931077689nt_int] :
( ( ord_le5643404153117327598_int_o
@ ^ [X: produc8551481072490612790e_term > option6357759511663192854e_term,Y6: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X @ Y6 ) @ R )
@ ^ [X: produc8551481072490612790e_term > option6357759511663192854e_term,Y6: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X @ Y6 ) @ S4 ) )
= ( ord_le8725513860283290265nt_int @ R @ S4 ) ) ).
% pred_subset_eq2
thf(fact_589_pred__subset__eq2,axiom,
! [R: set_Pr5580431329626101545st_int,S4: set_Pr5580431329626101545st_int] :
( ( ord_le4781006829409452478_int_o
@ ^ [X: list_char > int,Y6: list_int] : ( member6299553290643319410st_int @ ( produc8650753666468850689st_int @ X @ Y6 ) @ R )
@ ^ [X: list_char > int,Y6: list_int] : ( member6299553290643319410st_int @ ( produc8650753666468850689st_int @ X @ Y6 ) @ S4 ) )
= ( ord_le5738032636065246921st_int @ R @ S4 ) ) ).
% pred_subset_eq2
thf(fact_590_pred__subset__eq2,axiom,
! [R: set_Pr5290250543098163609nt_int,S4: set_Pr5290250543098163609nt_int] :
( ( ord_le8207994782381187918_int_o
@ ^ [X: list_char > int,Y6: int] : ( member3599195040465536738nt_int @ ( produc5790713362662368625nt_int @ X @ Y6 ) @ R )
@ ^ [X: list_char > int,Y6: int] : ( member3599195040465536738nt_int @ ( produc5790713362662368625nt_int @ X @ Y6 ) @ S4 ) )
= ( ord_le5108646760900237625nt_int @ R @ S4 ) ) ).
% pred_subset_eq2
thf(fact_591_pred__subset__eq2,axiom,
! [R: set_Pr1872883991513573699nt_int,S4: set_Pr1872883991513573699nt_int] :
( ( ord_le2124322318746777828_int_o
@ ^ [X: int > option6357759511663192854e_term,Y6: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X @ Y6 ) @ R )
@ ^ [X: int > option6357759511663192854e_term,Y6: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X @ Y6 ) @ S4 ) )
= ( ord_le135402666524580259nt_int @ R @ S4 ) ) ).
% pred_subset_eq2
thf(fact_592_of__nat__code,axiom,
( semiri1314217659103216013at_int
= ( ^ [N2: nat] :
( semiri8420488043553186161ux_int
@ ^ [I4: int] : ( plus_plus_int @ I4 @ one_one_int )
@ N2
@ zero_zero_int ) ) ) ).
% of_nat_code
thf(fact_593_of__nat__code,axiom,
( semiri1316708129612266289at_nat
= ( ^ [N2: nat] :
( semiri8422978514062236437ux_nat
@ ^ [I4: nat] : ( plus_plus_nat @ I4 @ one_one_nat )
@ N2
@ zero_zero_nat ) ) ) ).
% of_nat_code
thf(fact_594_inth__append,axiom,
! [I: int,Xs: list_instr,Ys: list_instr] :
( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ Xs ) ) )
=> ( ( inth_instr @ ( append_instr @ Xs @ Ys ) @ I )
= ( inth_instr @ Xs @ I ) ) )
& ( ~ ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ Xs ) ) )
=> ( ( inth_instr @ ( append_instr @ Xs @ Ys ) @ I )
= ( inth_instr @ Ys @ ( minus_minus_int @ I @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ Xs ) ) ) ) ) ) ) ) ).
% inth_append
thf(fact_595_inth__append,axiom,
! [I: int,Xs: list_list_char,Ys: list_list_char] :
( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ Xs ) ) )
=> ( ( inth_list_char @ ( append_list_char @ Xs @ Ys ) @ I )
= ( inth_list_char @ Xs @ I ) ) )
& ( ~ ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ Xs ) ) )
=> ( ( inth_list_char @ ( append_list_char @ Xs @ Ys ) @ I )
= ( inth_list_char @ Ys @ ( minus_minus_int @ I @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ Xs ) ) ) ) ) ) ) ) ).
% inth_append
thf(fact_596_inth__append,axiom,
! [I: int,Xs: list_int,Ys: list_int] :
( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_size_list_int @ Xs ) ) )
=> ( ( inth_int @ ( append_int @ Xs @ Ys ) @ I )
= ( inth_int @ Xs @ I ) ) )
& ( ~ ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_size_list_int @ Xs ) ) )
=> ( ( inth_int @ ( append_int @ Xs @ Ys ) @ I )
= ( inth_int @ Ys @ ( minus_minus_int @ I @ ( semiri1314217659103216013at_int @ ( size_size_list_int @ Xs ) ) ) ) ) ) ) ) ).
% inth_append
thf(fact_597_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_598_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_599_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_600_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_601_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_602_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_603_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_604_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_605_length__append,axiom,
! [Xs: list_int,Ys: list_int] :
( ( size_size_list_int @ ( append_int @ Xs @ Ys ) )
= ( plus_plus_nat @ ( size_size_list_int @ Xs ) @ ( size_size_list_int @ Ys ) ) ) ).
% length_append
thf(fact_606_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_607_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_608_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_609_nth__append__length__plus,axiom,
! [Xs: list_P419547167332244514st_int,Ys: list_P419547167332244514st_int,N: nat] :
( ( nth_Pr8327173284404383691st_int @ ( append5028292981072010935st_int @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_s229347708945616566st_int @ Xs ) @ N ) )
= ( nth_Pr8327173284404383691st_int @ Ys @ N ) ) ).
% nth_append_length_plus
thf(fact_610_nth__append__length__plus,axiom,
! [Xs: list_instr,Ys: list_instr,N: nat] :
( ( nth_instr @ ( append_instr @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_size_list_instr @ Xs ) @ N ) )
= ( nth_instr @ Ys @ N ) ) ).
% nth_append_length_plus
thf(fact_611_nth__append__length__plus,axiom,
! [Xs: list_list_char,Ys: list_list_char,N: nat] :
( ( nth_list_char @ ( append_list_char @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_s356637359517785349t_char @ Xs ) @ N ) )
= ( nth_list_char @ Ys @ N ) ) ).
% nth_append_length_plus
thf(fact_612_nth__append__length__plus,axiom,
! [Xs: list_int,Ys: list_int,N: nat] :
( ( nth_int @ ( append_int @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_size_list_int @ Xs ) @ N ) )
= ( nth_int @ Ys @ N ) ) ).
% nth_append_length_plus
thf(fact_613_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_614_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_615_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_616_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_617_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_618_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_619_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_620_trans__less__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_621_trans__less__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_622_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_623_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_624_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_625_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_626_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_627_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_628_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N3: nat] :
( L
= ( plus_plus_nat @ K @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_629_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_630_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_631_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_632_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_633_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N2: nat] :
? [K3: nat] :
( N2
= ( plus_plus_nat @ M2 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_634_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_635_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_636_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_637_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_638_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I @ K2 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_639_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M4: nat,N3: nat] :
( ( ord_less_nat @ M4 @ N3 )
=> ( ord_less_nat @ ( F @ M4 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_640_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_641_less__diff__conv,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_642_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_643_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K )
= ( J
= ( plus_plus_nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_644_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_645_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_646_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_647_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_648_int__ops_I5_J,axiom,
! [A: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A @ B ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(5)
thf(fact_649_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_650_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_651_nat__diff__split,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P @ zero_zero_nat ) )
& ! [D2: nat] :
( ( A
= ( plus_plus_nat @ B @ D2 ) )
=> ( P @ D2 ) ) ) ) ).
% nat_diff_split
thf(fact_652_nat__diff__split__asm,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P @ zero_zero_nat ) )
| ? [D2: nat] :
( ( A
= ( plus_plus_nat @ B @ D2 ) )
& ~ ( P @ D2 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_653_less__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_654_linorder__neqE__linordered__idom,axiom,
! [X3: int,Y: int] :
( ( X3 != Y )
=> ( ~ ( ord_less_int @ X3 @ Y )
=> ( ord_less_int @ Y @ X3 ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_655_pred__equals__eq2,axiom,
! [R: set_Pr6231067571260677117ar_int,S4: set_Pr6231067571260677117ar_int] :
( ( ( ^ [X: com,Y6: list_char > int] : ( member2095410489637970758ar_int @ ( produc5595214716300948949ar_int @ X @ Y6 ) @ R ) )
= ( ^ [X: com,Y6: list_char > int] : ( member2095410489637970758ar_int @ ( produc5595214716300948949ar_int @ X @ Y6 ) @ S4 ) ) )
= ( R = S4 ) ) ).
% pred_equals_eq2
thf(fact_656_pred__equals__eq2,axiom,
! [R: set_Pr9222295170931077689nt_int,S4: set_Pr9222295170931077689nt_int] :
( ( ( ^ [X: produc8551481072490612790e_term > option6357759511663192854e_term,Y6: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X @ Y6 ) @ R ) )
= ( ^ [X: produc8551481072490612790e_term > option6357759511663192854e_term,Y6: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X @ Y6 ) @ S4 ) ) )
= ( R = S4 ) ) ).
% pred_equals_eq2
thf(fact_657_pred__equals__eq2,axiom,
! [R: set_Pr5580431329626101545st_int,S4: set_Pr5580431329626101545st_int] :
( ( ( ^ [X: list_char > int,Y6: list_int] : ( member6299553290643319410st_int @ ( produc8650753666468850689st_int @ X @ Y6 ) @ R ) )
= ( ^ [X: list_char > int,Y6: list_int] : ( member6299553290643319410st_int @ ( produc8650753666468850689st_int @ X @ Y6 ) @ S4 ) ) )
= ( R = S4 ) ) ).
% pred_equals_eq2
thf(fact_658_pred__equals__eq2,axiom,
! [R: set_Pr5290250543098163609nt_int,S4: set_Pr5290250543098163609nt_int] :
( ( ( ^ [X: list_char > int,Y6: int] : ( member3599195040465536738nt_int @ ( produc5790713362662368625nt_int @ X @ Y6 ) @ R ) )
= ( ^ [X: list_char > int,Y6: int] : ( member3599195040465536738nt_int @ ( produc5790713362662368625nt_int @ X @ Y6 ) @ S4 ) ) )
= ( R = S4 ) ) ).
% pred_equals_eq2
thf(fact_659_pred__equals__eq2,axiom,
! [R: set_Pr1872883991513573699nt_int,S4: set_Pr1872883991513573699nt_int] :
( ( ( ^ [X: int > option6357759511663192854e_term,Y6: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X @ Y6 ) @ R ) )
= ( ^ [X: int > option6357759511663192854e_term,Y6: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X @ Y6 ) @ S4 ) ) )
= ( R = S4 ) ) ).
% pred_equals_eq2
thf(fact_660_pred__subset__eq,axiom,
! [R: set_int,S4: set_int] :
( ( ord_less_eq_int_o
@ ^ [X: int] : ( member_int @ X @ R )
@ ^ [X: int] : ( member_int @ X @ S4 ) )
= ( ord_less_eq_set_int @ R @ S4 ) ) ).
% pred_subset_eq
thf(fact_661_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_662_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_663_subrelI,axiom,
! [R2: set_Pr6231067571260677117ar_int,S: set_Pr6231067571260677117ar_int] :
( ! [X2: com,Y3: list_char > int] :
( ( member2095410489637970758ar_int @ ( produc5595214716300948949ar_int @ X2 @ Y3 ) @ R2 )
=> ( member2095410489637970758ar_int @ ( produc5595214716300948949ar_int @ X2 @ Y3 ) @ S ) )
=> ( ord_le6049463789062751133ar_int @ R2 @ S ) ) ).
% subrelI
thf(fact_664_subrelI,axiom,
! [R2: set_Pr9222295170931077689nt_int,S: set_Pr9222295170931077689nt_int] :
( ! [X2: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: product_prod_int_int] :
( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X2 @ Y3 ) @ R2 )
=> ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X2 @ Y3 ) @ S ) )
=> ( ord_le8725513860283290265nt_int @ R2 @ S ) ) ).
% subrelI
thf(fact_665_subrelI,axiom,
! [R2: set_Pr5580431329626101545st_int,S: set_Pr5580431329626101545st_int] :
( ! [X2: list_char > int,Y3: list_int] :
( ( member6299553290643319410st_int @ ( produc8650753666468850689st_int @ X2 @ Y3 ) @ R2 )
=> ( member6299553290643319410st_int @ ( produc8650753666468850689st_int @ X2 @ Y3 ) @ S ) )
=> ( ord_le5738032636065246921st_int @ R2 @ S ) ) ).
% subrelI
thf(fact_666_subrelI,axiom,
! [R2: set_Pr5290250543098163609nt_int,S: set_Pr5290250543098163609nt_int] :
( ! [X2: list_char > int,Y3: int] :
( ( member3599195040465536738nt_int @ ( produc5790713362662368625nt_int @ X2 @ Y3 ) @ R2 )
=> ( member3599195040465536738nt_int @ ( produc5790713362662368625nt_int @ X2 @ Y3 ) @ S ) )
=> ( ord_le5108646760900237625nt_int @ R2 @ S ) ) ).
% subrelI
thf(fact_667_subrelI,axiom,
! [R2: set_Pr1872883991513573699nt_int,S: set_Pr1872883991513573699nt_int] :
( ! [X2: int > option6357759511663192854e_term,Y3: product_prod_int_int] :
( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X2 @ Y3 ) @ R2 )
=> ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X2 @ Y3 ) @ S ) )
=> ( ord_le135402666524580259nt_int @ R2 @ S ) ) ).
% subrelI
thf(fact_668_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_669_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_670_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_671_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_672_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_673_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_674_add__less__zeroD,axiom,
! [X3: int,Y: int] :
( ( ord_less_int @ ( plus_plus_int @ X3 @ Y ) @ zero_zero_int )
=> ( ( ord_less_int @ X3 @ zero_zero_int )
| ( ord_less_int @ Y @ zero_zero_int ) ) ) ).
% add_less_zeroD
thf(fact_675_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_676_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_677_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_678_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_679_add__le__imp__le__diff,axiom,
! [I: int,K: int,N: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
=> ( ord_less_eq_int @ I @ ( minus_minus_int @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_680_add__le__imp__le__diff,axiom,
! [I: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_681_add__le__add__imp__diff__le,axiom,
! [I: int,K: int,N: int,J: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
=> ( ord_less_eq_int @ ( minus_minus_int @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_682_add__le__add__imp__diff__le,axiom,
! [I: nat,K: nat,N: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_683_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_684_less__add__one,axiom,
! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).
% less_add_one
thf(fact_685_add__mono1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_686_add__mono1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ one_one_int ) @ ( plus_plus_int @ B @ one_one_int ) ) ) ).
% add_mono1
thf(fact_687_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ~ ( ord_less_nat @ A @ B )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_688_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: int,B: int] :
( ~ ( ord_less_int @ A @ B )
=> ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_689_bounded__Max__nat,axiom,
! [P: nat > $o,X3: nat,M5: nat] :
( ( P @ X3 )
=> ( ! [X2: nat] :
( ( P @ X2 )
=> ( ord_less_eq_nat @ X2 @ M5 ) )
=> ~ ! [M4: nat] :
( ( P @ M4 )
=> ~ ! [X5: nat] :
( ( P @ X5 )
=> ( ord_less_eq_nat @ X5 @ M4 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_690_int__ge__less__than__def,axiom,
( int_ge_less_than
= ( ^ [D2: int] :
( collec213857154873943460nt_int
@ ( produc4947309494688390418_int_o
@ ^ [Z6: int,Z3: int] :
( ( ord_less_eq_int @ D2 @ Z6 )
& ( ord_less_int @ Z6 @ Z3 ) ) ) ) ) ) ).
% int_ge_less_than_def
thf(fact_691_int__ge__less__than2__def,axiom,
( int_ge_less_than2
= ( ^ [D2: int] :
( collec213857154873943460nt_int
@ ( produc4947309494688390418_int_o
@ ^ [Z6: int,Z3: int] :
( ( ord_less_eq_int @ D2 @ Z3 )
& ( ord_less_int @ Z6 @ Z3 ) ) ) ) ) ) ).
% int_ge_less_than2_def
thf(fact_692_to__m__state_Osimps,axiom,
( to_m_state
= ( ^ [Xs3: list_list_char,S2: list_char > int,A5: int] :
( if_int
@ ( ( ord_less_int @ zero_zero_int @ A5 )
& ( ord_less_eq_int @ A5 @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ Xs3 ) ) ) )
@ ( S2 @ ( inth_list_char @ Xs3 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ Xs3 ) ) @ A5 ) ) )
@ zero_zero_int ) ) ) ).
% to_m_state.simps
thf(fact_693_to__m__state_Oelims,axiom,
! [X3: list_list_char,Xa: list_char > int,Xb: int,Y: int] :
( ( ( to_m_state @ X3 @ Xa @ Xb )
= Y )
=> ( ( ( ( ord_less_int @ zero_zero_int @ Xb )
& ( ord_less_eq_int @ Xb @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ X3 ) ) ) )
=> ( Y
= ( Xa @ ( inth_list_char @ X3 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ X3 ) ) @ Xb ) ) ) ) )
& ( ~ ( ( ord_less_int @ zero_zero_int @ Xb )
& ( ord_less_eq_int @ Xb @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ X3 ) ) ) )
=> ( Y = zero_zero_int ) ) ) ) ).
% to_m_state.elims
thf(fact_694_order__refl,axiom,
! [X3: int] : ( ord_less_eq_int @ X3 @ X3 ) ).
% order_refl
thf(fact_695_order__refl,axiom,
! [X3: nat] : ( ord_less_eq_nat @ X3 @ X3 ) ).
% order_refl
thf(fact_696_dual__order_Orefl,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% dual_order.refl
thf(fact_697_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_698_nle__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_eq_int @ A @ B ) )
= ( ( ord_less_eq_int @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_699_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_700_le__cases3,axiom,
! [X3: int,Y: int,Z: int] :
( ( ( ord_less_eq_int @ X3 @ Y )
=> ~ ( ord_less_eq_int @ Y @ Z ) )
=> ( ( ( ord_less_eq_int @ Y @ X3 )
=> ~ ( ord_less_eq_int @ X3 @ Z ) )
=> ( ( ( ord_less_eq_int @ X3 @ Z )
=> ~ ( ord_less_eq_int @ Z @ Y ) )
=> ( ( ( ord_less_eq_int @ Z @ Y )
=> ~ ( ord_less_eq_int @ Y @ X3 ) )
=> ( ( ( ord_less_eq_int @ Y @ Z )
=> ~ ( ord_less_eq_int @ Z @ X3 ) )
=> ~ ( ( ord_less_eq_int @ Z @ X3 )
=> ~ ( ord_less_eq_int @ X3 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_701_le__cases3,axiom,
! [X3: nat,Y: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X3 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ Z ) )
=> ( ( ( ord_less_eq_nat @ X3 @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X3 ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X3 ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_702_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: int,Z2: int] : ( Y5 = Z2 ) )
= ( ^ [X: int,Y6: int] :
( ( ord_less_eq_int @ X @ Y6 )
& ( ord_less_eq_int @ Y6 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_703_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [X: nat,Y6: nat] :
( ( ord_less_eq_nat @ X @ Y6 )
& ( ord_less_eq_nat @ Y6 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_704_ord__eq__le__trans,axiom,
! [A: int,B: int,C: int] :
( ( A = B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_705_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_706_ord__le__eq__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_707_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_708_order__antisym,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_int @ X3 @ Y )
=> ( ( ord_less_eq_int @ Y @ X3 )
=> ( X3 = Y ) ) ) ).
% order_antisym
thf(fact_709_order__antisym,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( ord_less_eq_nat @ Y @ X3 )
=> ( X3 = Y ) ) ) ).
% order_antisym
thf(fact_710_order_Otrans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% order.trans
thf(fact_711_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_712_order__trans,axiom,
! [X3: int,Y: int,Z: int] :
( ( ord_less_eq_int @ X3 @ Y )
=> ( ( ord_less_eq_int @ Y @ Z )
=> ( ord_less_eq_int @ X3 @ Z ) ) ) ).
% order_trans
thf(fact_713_order__trans,axiom,
! [X3: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_eq_nat @ X3 @ Z ) ) ) ).
% order_trans
thf(fact_714_linorder__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A3: int,B2: int] :
( ( ord_less_eq_int @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: int,B2: int] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_715_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: nat,B2: nat] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_716_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: int,Z2: int] : ( Y5 = Z2 ) )
= ( ^ [A5: int,B4: int] :
( ( ord_less_eq_int @ B4 @ A5 )
& ( ord_less_eq_int @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_717_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ B4 @ A5 )
& ( ord_less_eq_nat @ A5 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_718_dual__order_Oantisym,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_719_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_720_dual__order_Otrans,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_721_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_722_antisym,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_723_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_724_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: int,Z2: int] : ( Y5 = Z2 ) )
= ( ^ [A5: int,B4: int] :
( ( ord_less_eq_int @ A5 @ B4 )
& ( ord_less_eq_int @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_725_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ A5 @ B4 )
& ( ord_less_eq_nat @ B4 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_726_order__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_727_order__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_728_order__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_729_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_730_order__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_731_order__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_732_order__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_733_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_734_order__eq__refl,axiom,
! [X3: int,Y: int] :
( ( X3 = Y )
=> ( ord_less_eq_int @ X3 @ Y ) ) ).
% order_eq_refl
thf(fact_735_order__eq__refl,axiom,
! [X3: nat,Y: nat] :
( ( X3 = Y )
=> ( ord_less_eq_nat @ X3 @ Y ) ) ).
% order_eq_refl
thf(fact_736_linorder__linear,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_int @ X3 @ Y )
| ( ord_less_eq_int @ Y @ X3 ) ) ).
% linorder_linear
thf(fact_737_linorder__linear,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
| ( ord_less_eq_nat @ Y @ X3 ) ) ).
% linorder_linear
thf(fact_738_ord__eq__le__subst,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_739_ord__eq__le__subst,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_740_ord__eq__le__subst,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_741_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_742_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_743_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_744_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_745_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_746_linorder__le__cases,axiom,
! [X3: int,Y: int] :
( ~ ( ord_less_eq_int @ X3 @ Y )
=> ( ord_less_eq_int @ Y @ X3 ) ) ).
% linorder_le_cases
thf(fact_747_linorder__le__cases,axiom,
! [X3: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ Y @ X3 ) ) ).
% linorder_le_cases
thf(fact_748_order__antisym__conv,axiom,
! [Y: int,X3: int] :
( ( ord_less_eq_int @ Y @ X3 )
=> ( ( ord_less_eq_int @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% order_antisym_conv
thf(fact_749_order__antisym__conv,axiom,
! [Y: nat,X3: nat] :
( ( ord_less_eq_nat @ Y @ X3 )
=> ( ( ord_less_eq_nat @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% order_antisym_conv
thf(fact_750_lt__ex,axiom,
! [X3: int] :
? [Y3: int] : ( ord_less_int @ Y3 @ X3 ) ).
% lt_ex
thf(fact_751_gt__ex,axiom,
! [X3: nat] :
? [X_1: nat] : ( ord_less_nat @ X3 @ X_1 ) ).
% gt_ex
thf(fact_752_gt__ex,axiom,
! [X3: int] :
? [X_1: int] : ( ord_less_int @ X3 @ X_1 ) ).
% gt_ex
thf(fact_753_less__imp__neq,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( X3 != Y ) ) ).
% less_imp_neq
thf(fact_754_less__imp__neq,axiom,
! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
=> ( X3 != Y ) ) ).
% less_imp_neq
thf(fact_755_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_756_order_Oasym,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order.asym
thf(fact_757_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_758_ord__eq__less__trans,axiom,
! [A: int,B: int,C: int] :
( ( A = B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_759_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_760_ord__less__eq__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( B = C )
=> ( ord_less_int @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_761_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X2: nat] :
( ! [Y4: nat] :
( ( ord_less_nat @ Y4 @ X2 )
=> ( P @ Y4 ) )
=> ( P @ X2 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_762_antisym__conv3,axiom,
! [Y: nat,X3: nat] :
( ~ ( ord_less_nat @ Y @ X3 )
=> ( ( ~ ( ord_less_nat @ X3 @ Y ) )
= ( X3 = Y ) ) ) ).
% antisym_conv3
thf(fact_763_antisym__conv3,axiom,
! [Y: int,X3: int] :
( ~ ( ord_less_int @ Y @ X3 )
=> ( ( ~ ( ord_less_int @ X3 @ Y ) )
= ( X3 = Y ) ) ) ).
% antisym_conv3
thf(fact_764_linorder__cases,axiom,
! [X3: nat,Y: nat] :
( ~ ( ord_less_nat @ X3 @ Y )
=> ( ( X3 != Y )
=> ( ord_less_nat @ Y @ X3 ) ) ) ).
% linorder_cases
thf(fact_765_linorder__cases,axiom,
! [X3: int,Y: int] :
( ~ ( ord_less_int @ X3 @ Y )
=> ( ( X3 != Y )
=> ( ord_less_int @ Y @ X3 ) ) ) ).
% linorder_cases
thf(fact_766_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_767_dual__order_Oasym,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ~ ( ord_less_int @ A @ B ) ) ).
% dual_order.asym
thf(fact_768_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_769_dual__order_Oirrefl,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% dual_order.irrefl
thf(fact_770_exists__least__iff,axiom,
( ( ^ [P4: nat > $o] :
? [X7: nat] : ( P4 @ X7 ) )
= ( ^ [P5: nat > $o] :
? [N2: nat] :
( ( P5 @ N2 )
& ! [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ~ ( P5 @ M2 ) ) ) ) ) ).
% exists_least_iff
thf(fact_771_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B2: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: nat] : ( P @ A3 @ A3 )
=> ( ! [A3: nat,B2: nat] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_772_linorder__less__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A3: int,B2: int] :
( ( ord_less_int @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: int] : ( P @ A3 @ A3 )
=> ( ! [A3: int,B2: int] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_773_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_774_order_Ostrict__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_775_not__less__iff__gr__or__eq,axiom,
! [X3: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X3 @ Y ) )
= ( ( ord_less_nat @ Y @ X3 )
| ( X3 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_776_not__less__iff__gr__or__eq,axiom,
! [X3: int,Y: int] :
( ( ~ ( ord_less_int @ X3 @ Y ) )
= ( ( ord_less_int @ Y @ X3 )
| ( X3 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_777_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_778_dual__order_Ostrict__trans,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_779_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_780_order_Ostrict__implies__not__eq,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_781_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_782_dual__order_Ostrict__implies__not__eq,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_783_linorder__neqE,axiom,
! [X3: nat,Y: nat] :
( ( X3 != Y )
=> ( ~ ( ord_less_nat @ X3 @ Y )
=> ( ord_less_nat @ Y @ X3 ) ) ) ).
% linorder_neqE
thf(fact_784_linorder__neqE,axiom,
! [X3: int,Y: int] :
( ( X3 != Y )
=> ( ~ ( ord_less_int @ X3 @ Y )
=> ( ord_less_int @ Y @ X3 ) ) ) ).
% linorder_neqE
thf(fact_785_order__less__asym,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ~ ( ord_less_nat @ Y @ X3 ) ) ).
% order_less_asym
thf(fact_786_order__less__asym,axiom,
! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
=> ~ ( ord_less_int @ Y @ X3 ) ) ).
% order_less_asym
thf(fact_787_linorder__neq__iff,axiom,
! [X3: nat,Y: nat] :
( ( X3 != Y )
= ( ( ord_less_nat @ X3 @ Y )
| ( ord_less_nat @ Y @ X3 ) ) ) ).
% linorder_neq_iff
thf(fact_788_linorder__neq__iff,axiom,
! [X3: int,Y: int] :
( ( X3 != Y )
= ( ( ord_less_int @ X3 @ Y )
| ( ord_less_int @ Y @ X3 ) ) ) ).
% linorder_neq_iff
thf(fact_789_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_790_order__less__asym_H,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order_less_asym'
thf(fact_791_order__less__trans,axiom,
! [X3: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X3 @ Z ) ) ) ).
% order_less_trans
thf(fact_792_order__less__trans,axiom,
! [X3: int,Y: int,Z: int] :
( ( ord_less_int @ X3 @ Y )
=> ( ( ord_less_int @ Y @ Z )
=> ( ord_less_int @ X3 @ Z ) ) ) ).
% order_less_trans
thf(fact_793_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_794_ord__eq__less__subst,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_795_ord__eq__less__subst,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_796_ord__eq__less__subst,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_797_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_798_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_799_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_800_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_801_order__less__irrefl,axiom,
! [X3: nat] :
~ ( ord_less_nat @ X3 @ X3 ) ).
% order_less_irrefl
thf(fact_802_order__less__irrefl,axiom,
! [X3: int] :
~ ( ord_less_int @ X3 @ X3 ) ).
% order_less_irrefl
thf(fact_803_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_804_order__less__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_805_order__less__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_806_order__less__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_807_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_808_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_809_order__less__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_810_order__less__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_811_order__less__not__sym,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ~ ( ord_less_nat @ Y @ X3 ) ) ).
% order_less_not_sym
thf(fact_812_order__less__not__sym,axiom,
! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
=> ~ ( ord_less_int @ Y @ X3 ) ) ).
% order_less_not_sym
thf(fact_813_order__less__imp__triv,axiom,
! [X3: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X3 @ Y )
=> ( ( ord_less_nat @ Y @ X3 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_814_order__less__imp__triv,axiom,
! [X3: int,Y: int,P: $o] :
( ( ord_less_int @ X3 @ Y )
=> ( ( ord_less_int @ Y @ X3 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_815_linorder__less__linear,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
| ( X3 = Y )
| ( ord_less_nat @ Y @ X3 ) ) ).
% linorder_less_linear
thf(fact_816_linorder__less__linear,axiom,
! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
| ( X3 = Y )
| ( ord_less_int @ Y @ X3 ) ) ).
% linorder_less_linear
thf(fact_817_order__less__imp__not__eq,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( X3 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_818_order__less__imp__not__eq,axiom,
! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
=> ( X3 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_819_order__less__imp__not__eq2,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( Y != X3 ) ) ).
% order_less_imp_not_eq2
thf(fact_820_order__less__imp__not__eq2,axiom,
! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
=> ( Y != X3 ) ) ).
% order_less_imp_not_eq2
thf(fact_821_order__less__imp__not__less,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ~ ( ord_less_nat @ Y @ X3 ) ) ).
% order_less_imp_not_less
thf(fact_822_order__less__imp__not__less,axiom,
! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
=> ~ ( ord_less_int @ Y @ X3 ) ) ).
% order_less_imp_not_less
thf(fact_823_leD,axiom,
! [Y: int,X3: int] :
( ( ord_less_eq_int @ Y @ X3 )
=> ~ ( ord_less_int @ X3 @ Y ) ) ).
% leD
thf(fact_824_leD,axiom,
! [Y: nat,X3: nat] :
( ( ord_less_eq_nat @ Y @ X3 )
=> ~ ( ord_less_nat @ X3 @ Y ) ) ).
% leD
thf(fact_825_leI,axiom,
! [X3: int,Y: int] :
( ~ ( ord_less_int @ X3 @ Y )
=> ( ord_less_eq_int @ Y @ X3 ) ) ).
% leI
thf(fact_826_leI,axiom,
! [X3: nat,Y: nat] :
( ~ ( ord_less_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ Y @ X3 ) ) ).
% leI
thf(fact_827_nless__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_int @ A @ B ) )
= ( ~ ( ord_less_eq_int @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_828_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_829_antisym__conv1,axiom,
! [X3: int,Y: int] :
( ~ ( ord_less_int @ X3 @ Y )
=> ( ( ord_less_eq_int @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% antisym_conv1
thf(fact_830_antisym__conv1,axiom,
! [X3: nat,Y: nat] :
( ~ ( ord_less_nat @ X3 @ Y )
=> ( ( ord_less_eq_nat @ X3 @ Y )
= ( X3 = Y ) ) ) ).
% antisym_conv1
thf(fact_831_antisym__conv2,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_int @ X3 @ Y )
=> ( ( ~ ( ord_less_int @ X3 @ Y ) )
= ( X3 = Y ) ) ) ).
% antisym_conv2
thf(fact_832_antisym__conv2,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( ~ ( ord_less_nat @ X3 @ Y ) )
= ( X3 = Y ) ) ) ).
% antisym_conv2
thf(fact_833_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X: int,Y6: int] :
( ( ord_less_eq_int @ X @ Y6 )
& ~ ( ord_less_eq_int @ Y6 @ X ) ) ) ) ).
% less_le_not_le
thf(fact_834_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X: nat,Y6: nat] :
( ( ord_less_eq_nat @ X @ Y6 )
& ~ ( ord_less_eq_nat @ Y6 @ X ) ) ) ) ).
% less_le_not_le
thf(fact_835_not__le__imp__less,axiom,
! [Y: int,X3: int] :
( ~ ( ord_less_eq_int @ Y @ X3 )
=> ( ord_less_int @ X3 @ Y ) ) ).
% not_le_imp_less
thf(fact_836_not__le__imp__less,axiom,
! [Y: nat,X3: nat] :
( ~ ( ord_less_eq_nat @ Y @ X3 )
=> ( ord_less_nat @ X3 @ Y ) ) ).
% not_le_imp_less
thf(fact_837_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A5: int,B4: int] :
( ( ord_less_int @ A5 @ B4 )
| ( A5 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_838_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_nat @ A5 @ B4 )
| ( A5 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_839_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A5: int,B4: int] :
( ( ord_less_eq_int @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_840_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_841_order_Ostrict__trans1,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_842_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_843_order_Ostrict__trans2,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_844_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_845_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A5: int,B4: int] :
( ( ord_less_eq_int @ A5 @ B4 )
& ~ ( ord_less_eq_int @ B4 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_846_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A5: nat,B4: nat] :
( ( ord_less_eq_nat @ A5 @ B4 )
& ~ ( ord_less_eq_nat @ B4 @ A5 ) ) ) ) ).
% order.strict_iff_not
thf(fact_847_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B4: int,A5: int] :
( ( ord_less_int @ B4 @ A5 )
| ( A5 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_848_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A5: nat] :
( ( ord_less_nat @ B4 @ A5 )
| ( A5 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_849_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B4: int,A5: int] :
( ( ord_less_eq_int @ B4 @ A5 )
& ( A5 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_850_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B4: nat,A5: nat] :
( ( ord_less_eq_nat @ B4 @ A5 )
& ( A5 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_851_dual__order_Ostrict__trans1,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_852_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_853_dual__order_Ostrict__trans2,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_854_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_855_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B4: int,A5: int] :
( ( ord_less_eq_int @ B4 @ A5 )
& ~ ( ord_less_eq_int @ A5 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_856_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B4: nat,A5: nat] :
( ( ord_less_eq_nat @ B4 @ A5 )
& ~ ( ord_less_eq_nat @ A5 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_857_order_Ostrict__implies__order,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_eq_int @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_858_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_859_dual__order_Ostrict__implies__order,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_eq_int @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_860_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_861_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X: int,Y6: int] :
( ( ord_less_int @ X @ Y6 )
| ( X = Y6 ) ) ) ) ).
% order_le_less
thf(fact_862_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X: nat,Y6: nat] :
( ( ord_less_nat @ X @ Y6 )
| ( X = Y6 ) ) ) ) ).
% order_le_less
thf(fact_863_order__less__le,axiom,
( ord_less_int
= ( ^ [X: int,Y6: int] :
( ( ord_less_eq_int @ X @ Y6 )
& ( X != Y6 ) ) ) ) ).
% order_less_le
thf(fact_864_order__less__le,axiom,
( ord_less_nat
= ( ^ [X: nat,Y6: nat] :
( ( ord_less_eq_nat @ X @ Y6 )
& ( X != Y6 ) ) ) ) ).
% order_less_le
thf(fact_865_linorder__not__le,axiom,
! [X3: int,Y: int] :
( ( ~ ( ord_less_eq_int @ X3 @ Y ) )
= ( ord_less_int @ Y @ X3 ) ) ).
% linorder_not_le
thf(fact_866_linorder__not__le,axiom,
! [X3: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X3 @ Y ) )
= ( ord_less_nat @ Y @ X3 ) ) ).
% linorder_not_le
thf(fact_867_linorder__not__less,axiom,
! [X3: int,Y: int] :
( ( ~ ( ord_less_int @ X3 @ Y ) )
= ( ord_less_eq_int @ Y @ X3 ) ) ).
% linorder_not_less
thf(fact_868_linorder__not__less,axiom,
! [X3: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X3 @ Y ) )
= ( ord_less_eq_nat @ Y @ X3 ) ) ).
% linorder_not_less
thf(fact_869_order__less__imp__le,axiom,
! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
=> ( ord_less_eq_int @ X3 @ Y ) ) ).
% order_less_imp_le
thf(fact_870_order__less__imp__le,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ord_less_eq_nat @ X3 @ Y ) ) ).
% order_less_imp_le
thf(fact_871_order__le__neq__trans,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( A != B )
=> ( ord_less_int @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_872_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_873_order__neq__le__trans,axiom,
! [A: int,B: int] :
( ( A != B )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_int @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_874_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_875_order__le__less__trans,axiom,
! [X3: int,Y: int,Z: int] :
( ( ord_less_eq_int @ X3 @ Y )
=> ( ( ord_less_int @ Y @ Z )
=> ( ord_less_int @ X3 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_876_order__le__less__trans,axiom,
! [X3: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X3 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_877_order__less__le__trans,axiom,
! [X3: int,Y: int,Z: int] :
( ( ord_less_int @ X3 @ Y )
=> ( ( ord_less_eq_int @ Y @ Z )
=> ( ord_less_int @ X3 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_878_order__less__le__trans,axiom,
! [X3: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X3 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_nat @ X3 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_879_order__le__less__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_880_order__le__less__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_881_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_882_order__le__less__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_883_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_884_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_885_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_886_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_887_order__less__le__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_888_order__less__le__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_889_order__less__le__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_890_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_891_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_892_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_893_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_894_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_895_linorder__le__less__linear,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_int @ X3 @ Y )
| ( ord_less_int @ Y @ X3 ) ) ).
% linorder_le_less_linear
thf(fact_896_linorder__le__less__linear,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
| ( ord_less_nat @ Y @ X3 ) ) ).
% linorder_le_less_linear
thf(fact_897_order__le__imp__less__or__eq,axiom,
! [X3: int,Y: int] :
( ( ord_less_eq_int @ X3 @ Y )
=> ( ( ord_less_int @ X3 @ Y )
| ( X3 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_898_order__le__imp__less__or__eq,axiom,
! [X3: nat,Y: nat] :
( ( ord_less_eq_nat @ X3 @ Y )
=> ( ( ord_less_nat @ X3 @ Y )
| ( X3 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_899_bcomp_Osimps_I3_J,axiom,
! [B_1: bexp,B_2: bexp,F: $o,I: int] :
( ( bcomp @ ( produc4047900504771817624_o_int @ ( and @ B_1 @ B_2 ) @ ( product_Pair_o_int @ F @ I ) ) )
= ( append_instr @ ( bcomp @ ( produc4047900504771817624_o_int @ B_1 @ ( product_Pair_o_int @ $false @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ ( bcomp @ ( produc4047900504771817624_o_int @ B_2 @ ( product_Pair_o_int @ F @ I ) ) ) ) ) @ ( if_int @ F @ zero_zero_int @ I ) ) ) ) ) @ ( bcomp @ ( produc4047900504771817624_o_int @ B_2 @ ( product_Pair_o_int @ F @ I ) ) ) ) ) ).
% bcomp.simps(3)
thf(fact_900_less__eq__set__def,axiom,
( ord_less_eq_set_int
= ( ^ [A6: set_int,B6: set_int] :
( ord_less_eq_int_o
@ ^ [X: int] : ( member_int @ X @ A6 )
@ ^ [X: int] : ( member_int @ X @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_901_case__prod__app,axiom,
( produc3978606535191404044_int_o
= ( ^ [F2: int > produc369741286924889929st_int > produc6425607678544837394st_int > $o,X: produc6425607678544837394st_int,Y6: produc6425607678544837394st_int] :
( produc5824858813793146727_int_o
@ ^ [L2: int,R3: produc369741286924889929st_int] : ( F2 @ L2 @ R3 @ Y6 )
@ X ) ) ) ).
% case_prod_app
thf(fact_902_case__prod__app,axiom,
( produc7737560463583487271_int_o
= ( ^ [F2: ( list_char > int ) > list_int > produc6425607678544837394st_int > $o,X: produc369741286924889929st_int,Y6: produc6425607678544837394st_int] :
( produc4252051693687794178_int_o
@ ^ [L2: list_char > int,R3: list_int] : ( F2 @ L2 @ R3 @ Y6 )
@ X ) ) ) ).
% case_prod_app
thf(fact_903_subsetI,axiom,
! [A2: set_int,B5: set_int] :
( ! [X2: int] :
( ( member_int @ X2 @ A2 )
=> ( member_int @ X2 @ B5 ) )
=> ( ord_less_eq_set_int @ A2 @ B5 ) ) ).
% subsetI
thf(fact_904_in__mono,axiom,
! [A2: set_int,B5: set_int,X3: int] :
( ( ord_less_eq_set_int @ A2 @ B5 )
=> ( ( member_int @ X3 @ A2 )
=> ( member_int @ X3 @ B5 ) ) ) ).
% in_mono
thf(fact_905_subsetD,axiom,
! [A2: set_int,B5: set_int,C: int] :
( ( ord_less_eq_set_int @ A2 @ B5 )
=> ( ( member_int @ C @ A2 )
=> ( member_int @ C @ B5 ) ) ) ).
% subsetD
thf(fact_906_subset__eq,axiom,
( ord_less_eq_set_int
= ( ^ [A6: set_int,B6: set_int] :
! [X: int] :
( ( member_int @ X @ A6 )
=> ( member_int @ X @ B6 ) ) ) ) ).
% subset_eq
thf(fact_907_subset__iff,axiom,
( ord_less_eq_set_int
= ( ^ [A6: set_int,B6: set_int] :
! [T2: int] :
( ( member_int @ T2 @ A6 )
=> ( member_int @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_908_Collect__mono,axiom,
! [P: product_prod_int_int > $o,Q: product_prod_int_int > $o] :
( ! [X2: product_prod_int_int] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_le2843351958646193337nt_int @ ( collec213857154873943460nt_int @ P ) @ ( collec213857154873943460nt_int @ Q ) ) ) ).
% Collect_mono
thf(fact_909_Collect__mono__iff,axiom,
! [P: product_prod_int_int > $o,Q: product_prod_int_int > $o] :
( ( ord_le2843351958646193337nt_int @ ( collec213857154873943460nt_int @ P ) @ ( collec213857154873943460nt_int @ Q ) )
= ( ! [X: product_prod_int_int] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_910_case__prod__Pair__iden,axiom,
! [P2: produc2931317944591925149ar_int] :
( ( produc2420365882498604339ar_int @ produc5595214716300948949ar_int @ P2 )
= P2 ) ).
% case_prod_Pair_iden
thf(fact_911_case__prod__Pair__iden,axiom,
! [P2: produc2285326912895808259nt_int] :
( ( produc8492565224438309093nt_int @ produc5700946648718959541nt_int @ P2 )
= P2 ) ).
% case_prod_Pair_iden
thf(fact_912_case__prod__Pair__iden,axiom,
! [P2: produc369741286924889929st_int] :
( ( produc1929222690536411315st_int @ produc8650753666468850689st_int @ P2 )
= P2 ) ).
% case_prod_Pair_iden
thf(fact_913_case__prod__Pair__iden,axiom,
! [P2: produc4435102495419491129nt_int] :
( ( produc2297300071446086195nt_int @ produc5790713362662368625nt_int @ P2 )
= P2 ) ).
% case_prod_Pair_iden
thf(fact_914_case__prod__Pair__iden,axiom,
! [P2: produc7773217078559923341nt_int] :
( ( produc5122537100556696953nt_int @ produc4305682042979456191nt_int @ P2 )
= P2 ) ).
% case_prod_Pair_iden
thf(fact_915_Collect__subset,axiom,
! [A2: set_int,P: int > $o] :
( ord_less_eq_set_int
@ ( collect_int
@ ^ [X: int] :
( ( member_int @ X @ A2 )
& ( P @ X ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_916_Collect__subset,axiom,
! [A2: set_Pr958786334691620121nt_int,P: product_prod_int_int > $o] :
( ord_le2843351958646193337nt_int
@ ( collec213857154873943460nt_int
@ ^ [X: product_prod_int_int] :
( ( member5262025264175285858nt_int @ X @ A2 )
& ( P @ X ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_917_size__char__eq__0,axiom,
( size_size_char
= ( ^ [C3: char] : zero_zero_nat ) ) ).
% size_char_eq_0
thf(fact_918_to__m__state_Opelims,axiom,
! [X3: list_list_char,Xa: list_char > int,Xb: int,Y: int] :
( ( ( to_m_state @ X3 @ Xa @ Xb )
= Y )
=> ( ( accp_P4562858270263085236nt_int @ to_m_state_rel @ ( produc3965054194175396271nt_int @ X3 @ ( produc5790713362662368625nt_int @ Xa @ Xb ) ) )
=> ~ ( ( ( ( ( ord_less_int @ zero_zero_int @ Xb )
& ( ord_less_eq_int @ Xb @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ X3 ) ) ) )
=> ( Y
= ( Xa @ ( inth_list_char @ X3 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ X3 ) ) @ Xb ) ) ) ) )
& ( ~ ( ( ord_less_int @ zero_zero_int @ Xb )
& ( ord_less_eq_int @ Xb @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ X3 ) ) ) )
=> ( Y = zero_zero_int ) ) )
=> ~ ( accp_P4562858270263085236nt_int @ to_m_state_rel @ ( produc3965054194175396271nt_int @ X3 @ ( produc5790713362662368625nt_int @ Xa @ Xb ) ) ) ) ) ) ).
% to_m_state.pelims
thf(fact_919_bval_Osimps_I3_J,axiom,
! [B_1: bexp,B_2: bexp,S: list_char > int] :
( ( bval @ ( and @ B_1 @ B_2 ) @ S )
= ( ( bval @ B_1 @ S )
& ( bval @ B_2 @ S ) ) ) ).
% bval.simps(3)
thf(fact_920_size_H__char__eq__0,axiom,
( size_char
= ( ^ [C3: char] : zero_zero_nat ) ) ).
% size'_char_eq_0
thf(fact_921_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_922_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_923_accp__subset,axiom,
! [R1: produc4400946599758736694st_int > produc4400946599758736694st_int > $o,R22: produc4400946599758736694st_int > produc4400946599758736694st_int > $o] :
( ( ord_le5253059192728285626_int_o @ R1 @ R22 )
=> ( ord_le166954219300680377_int_o @ ( accp_P1482788143903795053st_int @ R22 ) @ ( accp_P1482788143903795053st_int @ R1 ) ) ) ).
% accp_subset
thf(fact_924_accp__subset__induct,axiom,
! [D3: produc4189061985984568957nt_int > $o,R: produc4189061985984568957nt_int > produc4189061985984568957nt_int > $o,X3: produc4189061985984568957nt_int,P: produc4189061985984568957nt_int > $o] :
( ( ord_le3703957343819488690_int_o @ D3 @ ( accp_P4562858270263085236nt_int @ R ) )
=> ( ! [X2: produc4189061985984568957nt_int,Z5: produc4189061985984568957nt_int] :
( ( D3 @ X2 )
=> ( ( R @ Z5 @ X2 )
=> ( D3 @ Z5 ) ) )
=> ( ( D3 @ X3 )
=> ( ! [X2: produc4189061985984568957nt_int] :
( ( D3 @ X2 )
=> ( ! [Z4: produc4189061985984568957nt_int] :
( ( R @ Z4 @ X2 )
=> ( P @ Z4 ) )
=> ( P @ X2 ) ) )
=> ( P @ X3 ) ) ) ) ) ).
% accp_subset_induct
thf(fact_925_accp__subset__induct,axiom,
! [D3: product_prod_int_int > $o,R: product_prod_int_int > product_prod_int_int > $o,X3: product_prod_int_int,P: product_prod_int_int > $o] :
( ( ord_le8369615600986905444_int_o @ D3 @ ( accp_P1096762738010456898nt_int @ R ) )
=> ( ! [X2: product_prod_int_int,Z5: product_prod_int_int] :
( ( D3 @ X2 )
=> ( ( R @ Z5 @ X2 )
=> ( D3 @ Z5 ) ) )
=> ( ( D3 @ X3 )
=> ( ! [X2: product_prod_int_int] :
( ( D3 @ X2 )
=> ( ! [Z4: product_prod_int_int] :
( ( R @ Z4 @ X2 )
=> ( P @ Z4 ) )
=> ( P @ X2 ) ) )
=> ( P @ X3 ) ) ) ) ) ).
% accp_subset_induct
thf(fact_926_accp__subset__induct,axiom,
! [D3: produc4400946599758736694st_int > $o,R: produc4400946599758736694st_int > produc4400946599758736694st_int > $o,X3: produc4400946599758736694st_int,P: produc4400946599758736694st_int > $o] :
( ( ord_le166954219300680377_int_o @ D3 @ ( accp_P1482788143903795053st_int @ R ) )
=> ( ! [X2: produc4400946599758736694st_int,Z5: produc4400946599758736694st_int] :
( ( D3 @ X2 )
=> ( ( R @ Z5 @ X2 )
=> ( D3 @ Z5 ) ) )
=> ( ( D3 @ X3 )
=> ( ! [X2: produc4400946599758736694st_int] :
( ( D3 @ X2 )
=> ( ! [Z4: produc4400946599758736694st_int] :
( ( R @ Z4 @ X2 )
=> ( P @ Z4 ) )
=> ( P @ X2 ) ) )
=> ( P @ X3 ) ) ) ) ) ).
% accp_subset_induct
thf(fact_927_nat0__intermed__int__val,axiom,
! [N: nat,F: nat > int,K: int] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I2 @ one_one_nat ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I2: nat] :
( ( ord_less_eq_nat @ I2 @ N )
& ( ( F @ I2 )
= K ) ) ) ) ) ).
% nat0_intermed_int_val
thf(fact_928_big__step__determ,axiom,
! [C: com,S: list_char > int,T: list_char > int,U: list_char > int] :
( ( big_big_step @ ( produc5595214716300948949ar_int @ C @ S ) @ T )
=> ( ( big_big_step @ ( produc5595214716300948949ar_int @ C @ S ) @ U )
=> ( U = T ) ) ) ).
% big_step_determ
thf(fact_929_abs__idempotent,axiom,
! [A: int] :
( ( abs_abs_int @ ( abs_abs_int @ A ) )
= ( abs_abs_int @ A ) ) ).
% abs_idempotent
thf(fact_930_abs__abs,axiom,
! [A: int] :
( ( abs_abs_int @ ( abs_abs_int @ A ) )
= ( abs_abs_int @ A ) ) ).
% abs_abs
thf(fact_931_abs__zero,axiom,
( ( abs_abs_int @ zero_zero_int )
= zero_zero_int ) ).
% abs_zero
thf(fact_932_abs__eq__0,axiom,
! [A: int] :
( ( ( abs_abs_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% abs_eq_0
thf(fact_933_abs__0__eq,axiom,
! [A: int] :
( ( zero_zero_int
= ( abs_abs_int @ A ) )
= ( A = zero_zero_int ) ) ).
% abs_0_eq
thf(fact_934_abs__0,axiom,
( ( abs_abs_int @ zero_zero_int )
= zero_zero_int ) ).
% abs_0
thf(fact_935_abs__add__abs,axiom,
! [A: int,B: int] :
( ( abs_abs_int @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) )
= ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% abs_add_abs
thf(fact_936_abs__1,axiom,
( ( abs_abs_int @ one_one_int )
= one_one_int ) ).
% abs_1
thf(fact_937_abs__of__nat,axiom,
! [N: nat] :
( ( abs_abs_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ N ) ) ).
% abs_of_nat
thf(fact_938_abs__of__nonneg,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( abs_abs_int @ A )
= A ) ) ).
% abs_of_nonneg
thf(fact_939_abs__le__self__iff,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ A )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% abs_le_self_iff
thf(fact_940_abs__le__zero__iff,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ zero_zero_int )
= ( A = zero_zero_int ) ) ).
% abs_le_zero_iff
thf(fact_941_zero__less__abs__iff,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( abs_abs_int @ A ) )
= ( A != zero_zero_int ) ) ).
% zero_less_abs_iff
thf(fact_942_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_943_abs__ge__self,axiom,
! [A: int] : ( ord_less_eq_int @ A @ ( abs_abs_int @ A ) ) ).
% abs_ge_self
thf(fact_944_abs__le__D1,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
=> ( ord_less_eq_int @ A @ B ) ) ).
% abs_le_D1
thf(fact_945_abs__eq__0__iff,axiom,
! [A: int] :
( ( ( abs_abs_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% abs_eq_0_iff
thf(fact_946_abs__one,axiom,
( ( abs_abs_int @ one_one_int )
= one_one_int ) ).
% abs_one
thf(fact_947_abs__minus__commute,axiom,
! [A: int,B: int] :
( ( abs_abs_int @ ( minus_minus_int @ A @ B ) )
= ( abs_abs_int @ ( minus_minus_int @ B @ A ) ) ) ).
% abs_minus_commute
thf(fact_948_abs__ge__zero,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( abs_abs_int @ A ) ) ).
% abs_ge_zero
thf(fact_949_abs__of__pos,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( abs_abs_int @ A )
= A ) ) ).
% abs_of_pos
thf(fact_950_abs__not__less__zero,axiom,
! [A: int] :
~ ( ord_less_int @ ( abs_abs_int @ A ) @ zero_zero_int ) ).
% abs_not_less_zero
thf(fact_951_abs__triangle__ineq,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( plus_plus_int @ A @ B ) ) @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% abs_triangle_ineq
thf(fact_952_abs__triangle__ineq2,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) ) ).
% abs_triangle_ineq2
thf(fact_953_abs__triangle__ineq3,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) ) ).
% abs_triangle_ineq3
thf(fact_954_abs__triangle__ineq2__sym,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( abs_abs_int @ ( minus_minus_int @ B @ A ) ) ) ).
% abs_triangle_ineq2_sym
thf(fact_955_abs__diff__le__iff,axiom,
! [X3: int,A: int,R2: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ X3 @ A ) ) @ R2 )
= ( ( ord_less_eq_int @ ( minus_minus_int @ A @ R2 ) @ X3 )
& ( ord_less_eq_int @ X3 @ ( plus_plus_int @ A @ R2 ) ) ) ) ).
% abs_diff_le_iff
thf(fact_956_abs__triangle__ineq4,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% abs_triangle_ineq4
thf(fact_957_abs__diff__triangle__ineq,axiom,
! [A: int,B: int,C: int,D: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ ( plus_plus_int @ C @ D ) ) ) @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ A @ C ) ) @ ( abs_abs_int @ ( minus_minus_int @ B @ D ) ) ) ) ).
% abs_diff_triangle_ineq
thf(fact_958_abs__diff__less__iff,axiom,
! [X3: int,A: int,R2: int] :
( ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X3 @ A ) ) @ R2 )
= ( ( ord_less_int @ ( minus_minus_int @ A @ R2 ) @ X3 )
& ( ord_less_int @ X3 @ ( plus_plus_int @ A @ R2 ) ) ) ) ).
% abs_diff_less_iff
thf(fact_959_abs__add__one__gt__zero,axiom,
! [X3: int] : ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ ( abs_abs_int @ X3 ) ) ) ).
% abs_add_one_gt_zero
thf(fact_960_sim__sym,axiom,
! [C: com,C4: com] :
( ( ! [S2: list_char > int,T2: list_char > int] :
( ( big_big_step @ ( produc5595214716300948949ar_int @ C @ S2 ) @ T2 )
= ( big_big_step @ ( produc5595214716300948949ar_int @ C4 @ S2 ) @ T2 ) ) )
= ( ! [S2: list_char > int,T2: list_char > int] :
( ( big_big_step @ ( produc5595214716300948949ar_int @ C4 @ S2 ) @ T2 )
= ( big_big_step @ ( produc5595214716300948949ar_int @ C @ S2 ) @ T2 ) ) ) ) ).
% sim_sym
thf(fact_961_sim__refl,axiom,
! [C: com,S5: list_char > int,T3: list_char > int] :
( ( big_big_step @ ( produc5595214716300948949ar_int @ C @ S5 ) @ T3 )
= ( big_big_step @ ( produc5595214716300948949ar_int @ C @ S5 ) @ T3 ) ) ).
% sim_refl
thf(fact_962_sim__trans,axiom,
! [C: com,C4: com,C5: com] :
( ! [S6: list_char > int,T4: list_char > int] :
( ( big_big_step @ ( produc5595214716300948949ar_int @ C @ S6 ) @ T4 )
= ( big_big_step @ ( produc5595214716300948949ar_int @ C4 @ S6 ) @ T4 ) )
=> ( ! [S6: list_char > int,T4: list_char > int] :
( ( big_big_step @ ( produc5595214716300948949ar_int @ C4 @ S6 ) @ T4 )
= ( big_big_step @ ( produc5595214716300948949ar_int @ C5 @ S6 ) @ T4 ) )
=> ! [S5: list_char > int,T3: list_char > int] :
( ( big_big_step @ ( produc5595214716300948949ar_int @ C @ S5 ) @ T3 )
= ( big_big_step @ ( produc5595214716300948949ar_int @ C5 @ S5 ) @ T3 ) ) ) ) ).
% sim_trans
thf(fact_963_upto_Opinduct,axiom,
! [A0: int,A1: int,P: int > int > $o] :
( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ A0 @ A1 ) )
=> ( ! [I2: int,J2: int] :
( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I2 @ J2 ) )
=> ( ( ( ord_less_eq_int @ I2 @ J2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) @ J2 ) )
=> ( P @ I2 @ J2 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% upto.pinduct
thf(fact_964_nat__ivt__aux,axiom,
! [N: nat,F: nat > int,K: int] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I2: nat] :
( ( ord_less_eq_nat @ I2 @ N )
& ( ( F @ I2 )
= K ) ) ) ) ) ).
% nat_ivt_aux
thf(fact_965_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_966_nat_Oinject,axiom,
! [X22: nat,Y2: nat] :
( ( ( suc @ X22 )
= ( suc @ Y2 ) )
= ( X22 = Y2 ) ) ).
% nat.inject
thf(fact_967_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_968_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_969_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_970_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_971_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_972_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_973_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_974_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_975_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_976_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_977_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ M ) )
= ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% of_nat_Suc
thf(fact_978_of__nat__Suc,axiom,
! [M: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ M ) )
= ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M ) ) ) ).
% of_nat_Suc
thf(fact_979_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_980_diff__Suc__diff__eq1,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_981_diff__Suc__diff__eq2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_982_Suc__lessI2,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( M
!= ( minus_minus_nat @ N @ one_one_nat ) )
=> ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% Suc_lessI2
thf(fact_983_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_984_small__lazy_H_Ocases,axiom,
! [X3: product_prod_int_int] :
~ ! [D4: int,I2: int] :
( X3
!= ( product_Pair_int_int @ D4 @ I2 ) ) ).
% small_lazy'.cases
thf(fact_985_exhaustive__int_H_Ocases,axiom,
! [X3: produc7773217078559923341nt_int] :
~ ! [F3: int > option6357759511663192854e_term,D4: int,I2: int] :
( X3
!= ( produc4305682042979456191nt_int @ F3 @ ( product_Pair_int_int @ D4 @ I2 ) ) ) ).
% exhaustive_int'.cases
thf(fact_986_full__exhaustive__int_H_Ocases,axiom,
! [X3: produc2285326912895808259nt_int] :
~ ! [F3: produc8551481072490612790e_term > option6357759511663192854e_term,D4: int,I2: int] :
( X3
!= ( produc5700946648718959541nt_int @ F3 @ ( product_Pair_int_int @ D4 @ I2 ) ) ) ).
% full_exhaustive_int'.cases
thf(fact_987_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_988_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_989_Suc__inject,axiom,
! [X3: nat,Y: nat] :
( ( ( suc @ X3 )
= ( suc @ Y ) )
=> ( X3 = Y ) ) ).
% Suc_inject
thf(fact_990_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_991_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_992_nat__arith_Osuc1,axiom,
! [A2: nat,K: nat,A: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( suc @ A2 )
= ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_993_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_994_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_995_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_996_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_997_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_998_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_999_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X2: nat] : ( P @ X2 @ zero_zero_nat )
=> ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
=> ( ! [X2: nat,Y3: nat] :
( ( P @ X2 @ Y3 )
=> ( P @ ( suc @ X2 ) @ ( suc @ Y3 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_1000_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_1001_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_1002_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_1003_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_1004_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% not0_implies_Suc
thf(fact_1005_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_1006_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] :
( ( J
= ( suc @ I2 ) )
=> ( P @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( P @ ( suc @ I2 ) )
=> ( P @ I2 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_1007_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ K2 )
=> ( ( P @ I2 @ J2 )
=> ( ( P @ J2 @ K2 )
=> ( P @ I2 @ K2 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_1008_less__trans__Suc,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_1009_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_1010_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_1011_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_1012_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
=> ( P @ I4 ) ) )
= ( ( P @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( P @ I4 ) ) ) ) ).
% All_less_Suc
thf(fact_1013_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_1014_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_1015_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
& ( P @ I4 ) ) )
= ( ( P @ N )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N )
& ( P @ I4 ) ) ) ) ).
% Ex_less_Suc
thf(fact_1016_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_1017_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_1018_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_1019_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_1020_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_1021_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_1022_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X2: nat] : ( R @ X2 @ X2 )
=> ( ! [X2: nat,Y3: nat,Z5: nat] :
( ( R @ X2 @ Y3 )
=> ( ( R @ Y3 @ Z5 )
=> ( R @ X2 @ Z5 ) ) )
=> ( ! [N3: nat] : ( R @ N3 @ ( suc @ N3 ) )
=> ( R @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1023_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_1024_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N3 )
=> ( P @ M3 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_1025_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_1026_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_1027_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_1028_Suc__le__D,axiom,
! [N: nat,M7: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M7 )
=> ? [M4: nat] :
( M7
= ( suc @ M4 ) ) ) ).
% Suc_le_D
thf(fact_1029_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_1030_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_1031_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_1032_lift__Suc__antimono__le,axiom,
! [F: nat > int,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_int @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1033_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1034_lift__Suc__mono__le,axiom,
! [F: nat > int,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_int @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1035_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1036_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_1037_lift__Suc__mono__less__iff,axiom,
! [F: nat > int,N: nat,M: nat] :
( ! [N3: nat] : ( ord_less_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_int @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_1038_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_1039_lift__Suc__mono__less,axiom,
! [F: nat > int,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_less_int @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_1040_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
!= zero_zero_int ) ).
% of_nat_neq_0
thf(fact_1041_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
!= zero_zero_nat ) ).
% of_nat_neq_0
thf(fact_1042_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( M = zero_zero_nat )
| ? [J3: nat] :
( ( M
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_1043_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% gr0_implies_Suc
thf(fact_1044_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
=> ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( P @ ( suc @ I4 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_1045_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M2: nat] :
( N
= ( suc @ M2 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1046_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
& ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N )
& ( P @ ( suc @ I4 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_1047_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_1048_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_1049_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_1050_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_1051_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_1052_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_1053_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_1054_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_1055_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_1056_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N2: nat] : ( ord_less_eq_nat @ ( suc @ N2 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1057_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_1058_less__natE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ! [Q3: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M @ Q3 ) ) ) ) ).
% less_natE
thf(fact_1059_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
% less_add_Suc1
thf(fact_1060_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
% less_add_Suc2
thf(fact_1061_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M2: nat,N2: nat] :
? [K3: nat] :
( N2
= ( suc @ ( plus_plus_nat @ M2 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1062_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ? [K2: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_1063_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_1064_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_1065_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_1066_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_1067_Suc__eq__plus1,axiom,
( suc
= ( ^ [N2: nat] : ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_1068_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1069_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_1070_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_1071_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N )
& ! [I3: nat] :
( ( ord_less_eq_nat @ I3 @ K2 )
=> ~ ( P @ I3 ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1072_diff__Suc__less,axiom,
! [N: nat,I: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_1073_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_1074_int__Suc,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ).
% int_Suc
thf(fact_1075_int__ops_I4_J,axiom,
! [A: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ A ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ one_one_int ) ) ).
% int_ops(4)
thf(fact_1076_zless__iff__Suc__zadd,axiom,
( ord_less_int
= ( ^ [W2: int,Z3: int] :
? [N2: nat] :
( Z3
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ) ).
% zless_iff_Suc_zadd
thf(fact_1077_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_1078_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_1079_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_1080_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M2: nat,N2: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ N2 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N2 ) ) ) ) ) ).
% add_eq_if
thf(fact_1081_nat__intermed__int__val,axiom,
! [M: nat,N: nat,F: nat > int,K: int] :
( ! [I2: nat] :
( ( ( ord_less_eq_nat @ M @ I2 )
& ( ord_less_nat @ I2 @ N ) )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_int @ ( F @ M ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I2: nat] :
( ( ord_less_eq_nat @ M @ I2 )
& ( ord_less_eq_nat @ I2 @ N )
& ( ( F @ I2 )
= K ) ) ) ) ) ) ).
% nat_intermed_int_val
thf(fact_1082_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_1083_nth__enumerate__eq,axiom,
! [M: nat,Xs: list_P419547167332244514st_int,N: nat] :
( ( ord_less_nat @ M @ ( size_s229347708945616566st_int @ Xs ) )
=> ( ( nth_Pr6118945435929909578st_int @ ( enumer1800257932645307117st_int @ N @ Xs ) @ M )
= ( produc2664891610604065915st_int @ ( plus_plus_nat @ N @ M ) @ ( nth_Pr8327173284404383691st_int @ Xs @ M ) ) ) ) ).
% nth_enumerate_eq
thf(fact_1084_nth__enumerate__eq,axiom,
! [M: nat,Xs: list_instr,N: nat] :
( ( ord_less_nat @ M @ ( size_size_list_instr @ Xs ) )
=> ( ( nth_Pr4329546499383151805_instr @ ( enumerate_instr @ N @ Xs ) @ M )
= ( produc3129998205563682742_instr @ ( plus_plus_nat @ N @ M ) @ ( nth_instr @ Xs @ M ) ) ) ) ).
% nth_enumerate_eq
thf(fact_1085_nth__enumerate__eq,axiom,
! [M: nat,Xs: list_list_char,N: nat] :
( ( ord_less_nat @ M @ ( size_s356637359517785349t_char @ Xs ) )
=> ( ( nth_Pr7433773183981329817t_char @ ( enumerate_list_char @ N @ Xs ) @ M )
= ( produc120197956887798346t_char @ ( plus_plus_nat @ N @ M ) @ ( nth_list_char @ Xs @ M ) ) ) ) ).
% nth_enumerate_eq
thf(fact_1086_nth__enumerate__eq,axiom,
! [M: nat,Xs: list_int,N: nat] :
( ( ord_less_nat @ M @ ( size_size_list_int @ Xs ) )
=> ( ( nth_Pr3440142176431000676at_int @ ( enumerate_int @ N @ Xs ) @ M )
= ( product_Pair_nat_int @ ( plus_plus_nat @ N @ M ) @ ( nth_int @ Xs @ M ) ) ) ) ).
% nth_enumerate_eq
thf(fact_1087_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ! [I3: nat] :
( ( ord_less_nat @ K2 @ I3 )
=> ( P @ I3 ) )
=> ( P @ K2 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_1088_length__enumerate,axiom,
! [N: nat,Xs: list_instr] :
( ( size_s142077898034696872_instr @ ( enumerate_instr @ N @ Xs ) )
= ( size_size_list_instr @ Xs ) ) ).
% length_enumerate
thf(fact_1089_length__enumerate,axiom,
! [N: nat,Xs: list_list_char] :
( ( size_s3402568042334145540t_char @ ( enumerate_list_char @ N @ Xs ) )
= ( size_s356637359517785349t_char @ Xs ) ) ).
% length_enumerate
thf(fact_1090_length__enumerate,axiom,
! [N: nat,Xs: list_int] :
( ( size_s2970893825323803983at_int @ ( enumerate_int @ N @ Xs ) )
= ( size_size_list_int @ Xs ) ) ).
% length_enumerate
thf(fact_1091_enumerate__append__eq,axiom,
! [N: nat,Xs: list_instr,Ys: list_instr] :
( ( enumerate_instr @ N @ ( append_instr @ Xs @ Ys ) )
= ( append227978101926261161_instr @ ( enumerate_instr @ N @ Xs ) @ ( enumerate_instr @ ( plus_plus_nat @ N @ ( size_size_list_instr @ Xs ) ) @ Ys ) ) ) ).
% enumerate_append_eq
thf(fact_1092_enumerate__append__eq,axiom,
! [N: nat,Xs: list_list_char,Ys: list_list_char] :
( ( enumerate_list_char @ N @ ( append_list_char @ Xs @ Ys ) )
= ( append2138297074218479021t_char @ ( enumerate_list_char @ N @ Xs ) @ ( enumerate_list_char @ ( plus_plus_nat @ N @ ( size_s356637359517785349t_char @ Xs ) ) @ Ys ) ) ) ).
% enumerate_append_eq
thf(fact_1093_enumerate__append__eq,axiom,
! [N: nat,Xs: list_int,Ys: list_int] :
( ( enumerate_int @ N @ ( append_int @ Xs @ Ys ) )
= ( append6031344391939132024at_int @ ( enumerate_int @ N @ Xs ) @ ( enumerate_int @ ( plus_plus_nat @ N @ ( size_size_list_int @ Xs ) ) @ Ys ) ) ) ).
% enumerate_append_eq
thf(fact_1094_split__cong,axiom,
! [Q4: produc6425607678544837394st_int,F: int > produc369741286924889929st_int > $o,G: int > produc369741286924889929st_int > $o,P2: produc6425607678544837394st_int] :
( ! [X2: int,Y3: produc369741286924889929st_int] :
( ( ( produc5086643055186798020st_int @ X2 @ Y3 )
= Q4 )
=> ( ( F @ X2 @ Y3 )
= ( G @ X2 @ Y3 ) ) )
=> ( ( P2 = Q4 )
=> ( ( produc5824858813793146727_int_o @ F @ P2 )
= ( produc5824858813793146727_int_o @ G @ Q4 ) ) ) ) ).
% split_cong
thf(fact_1095_split__cong,axiom,
! [Q4: produc6425607678544837394st_int,F: int > produc369741286924889929st_int > produc6425607678544837394st_int > $o,G: int > produc369741286924889929st_int > produc6425607678544837394st_int > $o,P2: produc6425607678544837394st_int] :
( ! [X2: int,Y3: produc369741286924889929st_int] :
( ( ( produc5086643055186798020st_int @ X2 @ Y3 )
= Q4 )
=> ( ( F @ X2 @ Y3 )
= ( G @ X2 @ Y3 ) ) )
=> ( ( P2 = Q4 )
=> ( ( produc3978606535191404044_int_o @ F @ P2 )
= ( produc3978606535191404044_int_o @ G @ Q4 ) ) ) ) ).
% split_cong
thf(fact_1096_split__cong,axiom,
! [Q4: product_prod_int_int,F: int > int > $o,G: int > int > $o,P2: product_prod_int_int] :
( ! [X2: int,Y3: int] :
( ( ( product_Pair_int_int @ X2 @ Y3 )
= Q4 )
=> ( ( F @ X2 @ Y3 )
= ( G @ X2 @ Y3 ) ) )
=> ( ( P2 = Q4 )
=> ( ( produc4947309494688390418_int_o @ F @ P2 )
= ( produc4947309494688390418_int_o @ G @ Q4 ) ) ) ) ).
% split_cong
thf(fact_1097_split__cong,axiom,
! [Q4: produc369741286924889929st_int,F: ( list_char > int ) > list_int > $o,G: ( list_char > int ) > list_int > $o,P2: produc369741286924889929st_int] :
( ! [X2: list_char > int,Y3: list_int] :
( ( ( produc8650753666468850689st_int @ X2 @ Y3 )
= Q4 )
=> ( ( F @ X2 @ Y3 )
= ( G @ X2 @ Y3 ) ) )
=> ( ( P2 = Q4 )
=> ( ( produc4252051693687794178_int_o @ F @ P2 )
= ( produc4252051693687794178_int_o @ G @ Q4 ) ) ) ) ).
% split_cong
thf(fact_1098_split__cong,axiom,
! [Q4: produc369741286924889929st_int,F: ( list_char > int ) > list_int > produc6425607678544837394st_int > $o,G: ( list_char > int ) > list_int > produc6425607678544837394st_int > $o,P2: produc369741286924889929st_int] :
( ! [X2: list_char > int,Y3: list_int] :
( ( ( produc8650753666468850689st_int @ X2 @ Y3 )
= Q4 )
=> ( ( F @ X2 @ Y3 )
= ( G @ X2 @ Y3 ) ) )
=> ( ( P2 = Q4 )
=> ( ( produc7737560463583487271_int_o @ F @ P2 )
= ( produc7737560463583487271_int_o @ G @ Q4 ) ) ) ) ).
% split_cong
thf(fact_1099_Euclid__induct,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B2: nat] :
( ( P @ A3 @ B2 )
= ( P @ B2 @ A3 ) )
=> ( ! [A3: nat] : ( P @ A3 @ zero_zero_nat )
=> ( ! [A3: nat,B2: nat] :
( ( P @ A3 @ B2 )
=> ( P @ A3 @ ( plus_plus_nat @ A3 @ B2 ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Euclid_induct
thf(fact_1100_Euclidean__Division_Odivmod__nat__if,axiom,
( euclidean_divmod_nat
= ( ^ [M2: nat,N2: nat] :
( if_Pro6206227464963214023at_nat
@ ( ( N2 = zero_zero_nat )
| ( ord_less_nat @ M2 @ N2 ) )
@ ( product_Pair_nat_nat @ zero_zero_nat @ M2 )
@ ( produc2626176000494625587at_nat
@ ^ [Q5: nat] : ( product_Pair_nat_nat @ ( suc @ Q5 ) )
@ ( euclidean_divmod_nat @ ( minus_minus_nat @ M2 @ N2 ) @ N2 ) ) ) ) ) ).
% Euclidean_Division.divmod_nat_if
thf(fact_1101_bezw__0,axiom,
! [X3: nat] :
( ( bezw @ X3 @ zero_zero_nat )
= ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) ).
% bezw_0
thf(fact_1102_inth__map,axiom,
! [I: int,Xs: list_instr,F: instr > list_char] :
( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ Xs ) ) )
=> ( ( inth_list_char @ ( map_instr_list_char @ F @ Xs ) @ I )
= ( F @ ( inth_instr @ Xs @ I ) ) ) ) ) ).
% inth_map
thf(fact_1103_inth__map,axiom,
! [I: int,Xs: list_instr,F: instr > instr] :
( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ Xs ) ) )
=> ( ( inth_instr @ ( map_instr_instr @ F @ Xs ) @ I )
= ( F @ ( inth_instr @ Xs @ I ) ) ) ) ) ).
% inth_map
thf(fact_1104_inth__map,axiom,
! [I: int,Xs: list_list_char,F: list_char > list_char] :
( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ Xs ) ) )
=> ( ( inth_list_char @ ( map_li116305933131242120t_char @ F @ Xs ) @ I )
= ( F @ ( inth_list_char @ Xs @ I ) ) ) ) ) ).
% inth_map
thf(fact_1105_inth__map,axiom,
! [I: int,Xs: list_list_char,F: list_char > instr] :
( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_s356637359517785349t_char @ Xs ) ) )
=> ( ( inth_instr @ ( map_list_char_instr @ F @ Xs ) @ I )
= ( F @ ( inth_list_char @ Xs @ I ) ) ) ) ) ).
% inth_map
thf(fact_1106_inth__map,axiom,
! [I: int,Xs: list_int,F: int > list_char] :
( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_size_list_int @ Xs ) ) )
=> ( ( inth_list_char @ ( map_int_list_char @ F @ Xs ) @ I )
= ( F @ ( inth_int @ Xs @ I ) ) ) ) ) ).
% inth_map
thf(fact_1107_inth__map,axiom,
! [I: int,Xs: list_int,F: int > instr] :
( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_size_list_int @ Xs ) ) )
=> ( ( inth_instr @ ( map_int_instr @ F @ Xs ) @ I )
= ( F @ ( inth_int @ Xs @ I ) ) ) ) ) ).
% inth_map
thf(fact_1108_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_1109_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_1110_length__map,axiom,
! [F: int > instr,Xs: list_int] :
( ( size_size_list_instr @ ( map_int_instr @ F @ Xs ) )
= ( size_size_list_int @ Xs ) ) ).
% length_map
thf(fact_1111_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_1112_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_1113_length__map,axiom,
! [F: int > list_char,Xs: list_int] :
( ( size_s356637359517785349t_char @ ( map_int_list_char @ F @ Xs ) )
= ( size_size_list_int @ Xs ) ) ).
% length_map
thf(fact_1114_length__map,axiom,
! [F: instr > int,Xs: list_instr] :
( ( size_size_list_int @ ( map_instr_int @ F @ Xs ) )
= ( size_size_list_instr @ Xs ) ) ).
% length_map
thf(fact_1115_length__map,axiom,
! [F: list_char > int,Xs: list_list_char] :
( ( size_size_list_int @ ( map_list_char_int @ F @ Xs ) )
= ( size_s356637359517785349t_char @ Xs ) ) ).
% length_map
thf(fact_1116_length__map,axiom,
! [F: int > int,Xs: list_int] :
( ( size_size_list_int @ ( map_int_int @ F @ Xs ) )
= ( size_size_list_int @ Xs ) ) ).
% length_map
thf(fact_1117_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_1118_nth__map,axiom,
! [N: nat,Xs: list_P419547167332244514st_int,F: produc6425607678544837394st_int > produc6425607678544837394st_int] :
( ( ord_less_nat @ N @ ( size_s229347708945616566st_int @ Xs ) )
=> ( ( nth_Pr8327173284404383691st_int @ ( map_Pr8625093460839654122st_int @ F @ Xs ) @ N )
= ( F @ ( nth_Pr8327173284404383691st_int @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_1119_nth__map,axiom,
! [N: nat,Xs: list_instr,F: instr > produc6425607678544837394st_int] :
( ( ord_less_nat @ N @ ( size_size_list_instr @ Xs ) )
=> ( ( nth_Pr8327173284404383691st_int @ ( map_in7259965606695656109st_int @ F @ Xs ) @ N )
= ( F @ ( nth_instr @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_1120_nth__map,axiom,
! [N: nat,Xs: list_list_char,F: list_char > produc6425607678544837394st_int] :
( ( ord_less_nat @ N @ ( size_s356637359517785349t_char @ Xs ) )
=> ( ( nth_Pr8327173284404383691st_int @ ( map_li6440205260635360185st_int @ F @ Xs ) @ N )
= ( F @ ( nth_list_char @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_1121_nth__map,axiom,
! [N: nat,Xs: list_int,F: int > produc6425607678544837394st_int] :
( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
=> ( ( nth_Pr8327173284404383691st_int @ ( map_in8512336359594627332st_int @ F @ Xs ) @ N )
= ( F @ ( nth_int @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_1122_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_1123_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_1124_add__0__iff,axiom,
! [B: int,A: int] :
( ( B
= ( plus_plus_int @ B @ A ) )
= ( A = zero_zero_int ) ) ).
% add_0_iff
thf(fact_1125_add__0__iff,axiom,
! [B: nat,A: nat] :
( ( B
= ( plus_plus_nat @ B @ A ) )
= ( A = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_1126_complete__interval,axiom,
! [A: int,B: int,P: int > $o] :
( ( ord_less_int @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C2: int] :
( ( ord_less_eq_int @ A @ C2 )
& ( ord_less_eq_int @ C2 @ B )
& ! [X5: int] :
( ( ( ord_less_eq_int @ A @ X5 )
& ( ord_less_int @ X5 @ C2 ) )
=> ( P @ X5 ) )
& ! [D5: int] :
( ! [X2: int] :
( ( ( ord_less_eq_int @ A @ X2 )
& ( ord_less_int @ X2 @ D5 ) )
=> ( P @ X2 ) )
=> ( ord_less_eq_int @ D5 @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_1127_complete__interval,axiom,
! [A: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C2: nat] :
( ( ord_less_eq_nat @ A @ C2 )
& ( ord_less_eq_nat @ C2 @ B )
& ! [X5: nat] :
( ( ( ord_less_eq_nat @ A @ X5 )
& ( ord_less_nat @ X5 @ C2 ) )
=> ( P @ X5 ) )
& ! [D5: nat] :
( ! [X2: nat] :
( ( ( ord_less_eq_nat @ A @ X2 )
& ( ord_less_nat @ X2 @ D5 ) )
=> ( P @ X2 ) )
=> ( ord_less_eq_nat @ D5 @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_1128_Compiler2_Oacomp__correct,axiom,
! [A: aexp,Cfs: list_P419547167332244514st_int] :
( ( execl_all @ ( acomp @ A ) @ Cfs )
=> ( apred @ A @ ( nth_Pr8327173284404383691st_int @ Cfs @ zero_zero_nat ) @ ( nth_Pr8327173284404383691st_int @ Cfs @ ( minus_minus_nat @ ( size_s229347708945616566st_int @ Cfs ) @ ( suc @ zero_zero_nat ) ) ) ) ) ).
% Compiler2.acomp_correct
thf(fact_1129_acomp__acomp,axiom,
! [A_1: aexp,A_2: aexp,P: list_instr,Cfs: list_P419547167332244514st_int] :
( ( execl_all @ ( append_instr @ ( acomp @ A_1 ) @ ( append_instr @ ( acomp @ A_2 ) @ P ) ) @ Cfs )
=> ( ! [Cfs2: list_P419547167332244514st_int] :
( ( execl_all @ ( acomp @ A_1 ) @ Cfs2 )
=> ( apred @ A_1 @ ( nth_Pr8327173284404383691st_int @ Cfs2 @ zero_zero_nat ) @ ( nth_Pr8327173284404383691st_int @ Cfs2 @ ( minus_minus_nat @ ( size_s229347708945616566st_int @ Cfs2 ) @ one_one_nat ) ) ) )
=> ( ! [Cfs2: list_P419547167332244514st_int] :
( ( execl_all @ ( acomp @ A_2 ) @ Cfs2 )
=> ( apred @ A_2 @ ( nth_Pr8327173284404383691st_int @ Cfs2 @ zero_zero_nat ) @ ( nth_Pr8327173284404383691st_int @ Cfs2 @ ( minus_minus_nat @ ( size_s229347708945616566st_int @ Cfs2 ) @ one_one_nat ) ) ) )
=> ( produc5824858813793146727_int_o
@ ^ [Pc: int] :
( produc4252051693687794178_int_o
@ ^ [S2: list_char > int,Stk: list_int] :
( ( Pc = zero_zero_int )
& ? [K3: nat] :
( ( ord_less_nat @ K3 @ ( size_s229347708945616566st_int @ Cfs ) )
& ( ( nth_Pr8327173284404383691st_int @ Cfs @ K3 )
= ( produc5086643055186798020st_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ ( append_instr @ ( acomp @ A_1 ) @ ( acomp @ A_2 ) ) ) ) @ ( produc8650753666468850689st_int @ S2 @ ( cons_int @ ( aval @ A_2 @ S2 ) @ ( cons_int @ ( aval @ A_1 @ S2 ) @ Stk ) ) ) ) ) ) ) )
@ ( nth_Pr8327173284404383691st_int @ Cfs @ zero_zero_nat ) ) ) ) ) ).
% acomp_acomp
thf(fact_1130_apred__def,axiom,
( apred
= ( ^ [A5: aexp] :
( produc3978606535191404044_int_o
@ ^ [Pc: int] :
( produc7737560463583487271_int_o
@ ^ [S2: list_char > int,Stk: list_int] :
( produc5824858813793146727_int_o
@ ^ [Pc2: int] :
( produc4252051693687794178_int_o
@ ^ [S3: list_char > int,Stk2: list_int] :
( ( Pc2
= ( plus_plus_int @ Pc @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ ( acomp @ A5 ) ) ) ) )
& ( S3 = S2 )
& ( Stk2
= ( cons_int @ ( aval @ A5 @ S2 ) @ Stk ) ) ) ) ) ) ) ) ) ).
% apred_def
thf(fact_1131_list_Oinject,axiom,
! [X21: int,X222: list_int,Y21: int,Y22: list_int] :
( ( ( cons_int @ X21 @ X222 )
= ( cons_int @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X222 = Y22 ) ) ) ).
% list.inject
thf(fact_1132_list_Oinject,axiom,
! [X21: instr,X222: list_instr,Y21: instr,Y22: list_instr] :
( ( ( cons_instr @ X21 @ X222 )
= ( cons_instr @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X222 = Y22 ) ) ) ).
% list.inject
thf(fact_1133_list_Oinject,axiom,
! [X21: produc6425607678544837394st_int,X222: list_P419547167332244514st_int,Y21: produc6425607678544837394st_int,Y22: list_P419547167332244514st_int] :
( ( ( cons_P4151473878978182108st_int @ X21 @ X222 )
= ( cons_P4151473878978182108st_int @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X222 = Y22 ) ) ) ).
% list.inject
thf(fact_1134_nth__Cons__0,axiom,
! [X3: int,Xs: list_int] :
( ( nth_int @ ( cons_int @ X3 @ Xs ) @ zero_zero_nat )
= X3 ) ).
% nth_Cons_0
thf(fact_1135_nth__Cons__0,axiom,
! [X3: instr,Xs: list_instr] :
( ( nth_instr @ ( cons_instr @ X3 @ Xs ) @ zero_zero_nat )
= X3 ) ).
% nth_Cons_0
thf(fact_1136_nth__Cons__0,axiom,
! [X3: produc6425607678544837394st_int,Xs: list_P419547167332244514st_int] :
( ( nth_Pr8327173284404383691st_int @ ( cons_P4151473878978182108st_int @ X3 @ Xs ) @ zero_zero_nat )
= X3 ) ).
% nth_Cons_0
thf(fact_1137_nth__Cons__Suc,axiom,
! [X3: int,Xs: list_int,N: nat] :
( ( nth_int @ ( cons_int @ X3 @ Xs ) @ ( suc @ N ) )
= ( nth_int @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_1138_nth__Cons__Suc,axiom,
! [X3: instr,Xs: list_instr,N: nat] :
( ( nth_instr @ ( cons_instr @ X3 @ Xs ) @ ( suc @ N ) )
= ( nth_instr @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_1139_nth__Cons__Suc,axiom,
! [X3: produc6425607678544837394st_int,Xs: list_P419547167332244514st_int,N: nat] :
( ( nth_Pr8327173284404383691st_int @ ( cons_P4151473878978182108st_int @ X3 @ Xs ) @ ( suc @ N ) )
= ( nth_Pr8327173284404383691st_int @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_1140_nth__append__length,axiom,
! [Xs: list_P419547167332244514st_int,X3: produc6425607678544837394st_int,Ys: list_P419547167332244514st_int] :
( ( nth_Pr8327173284404383691st_int @ ( append5028292981072010935st_int @ Xs @ ( cons_P4151473878978182108st_int @ X3 @ Ys ) ) @ ( size_s229347708945616566st_int @ Xs ) )
= X3 ) ).
% nth_append_length
thf(fact_1141_nth__append__length,axiom,
! [Xs: list_instr,X3: instr,Ys: list_instr] :
( ( nth_instr @ ( append_instr @ Xs @ ( cons_instr @ X3 @ Ys ) ) @ ( size_size_list_instr @ Xs ) )
= X3 ) ).
% nth_append_length
thf(fact_1142_nth__append__length,axiom,
! [Xs: list_list_char,X3: list_char,Ys: list_list_char] :
( ( nth_list_char @ ( append_list_char @ Xs @ ( cons_list_char @ X3 @ Ys ) ) @ ( size_s356637359517785349t_char @ Xs ) )
= X3 ) ).
% nth_append_length
thf(fact_1143_nth__append__length,axiom,
! [Xs: list_int,X3: int,Ys: list_int] :
( ( nth_int @ ( append_int @ Xs @ ( cons_int @ X3 @ Ys ) ) @ ( size_size_list_int @ Xs ) )
= X3 ) ).
% nth_append_length
thf(fact_1144_enumerate__simps_I2_J,axiom,
! [N: nat,X3: int,Xs: list_int] :
( ( enumerate_int @ N @ ( cons_int @ X3 @ Xs ) )
= ( cons_P2335045147070616083at_int @ ( product_Pair_nat_int @ N @ X3 ) @ ( enumerate_int @ ( suc @ N ) @ Xs ) ) ) ).
% enumerate_simps(2)
thf(fact_1145_enumerate__simps_I2_J,axiom,
! [N: nat,X3: instr,Xs: list_instr] :
( ( enumerate_instr @ N @ ( cons_instr @ X3 @ Xs ) )
= ( cons_P4643890853483612494_instr @ ( produc3129998205563682742_instr @ N @ X3 ) @ ( enumerate_instr @ ( suc @ N ) @ Xs ) ) ) ).
% enumerate_simps(2)
thf(fact_1146_enumerate__simps_I2_J,axiom,
! [N: nat,X3: produc6425607678544837394st_int,Xs: list_P419547167332244514st_int] :
( ( enumer1800257932645307117st_int @ N @ ( cons_P4151473878978182108st_int @ X3 @ Xs ) )
= ( cons_P6123078848854619129st_int @ ( produc2664891610604065915st_int @ N @ X3 ) @ ( enumer1800257932645307117st_int @ ( suc @ N ) @ Xs ) ) ) ).
% enumerate_simps(2)
thf(fact_1147_nth__Cons__pos,axiom,
! [N: nat,X3: int,Xs: list_int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_int @ ( cons_int @ X3 @ Xs ) @ N )
= ( nth_int @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_1148_nth__Cons__pos,axiom,
! [N: nat,X3: instr,Xs: list_instr] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_instr @ ( cons_instr @ X3 @ Xs ) @ N )
= ( nth_instr @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_1149_nth__Cons__pos,axiom,
! [N: nat,X3: produc6425607678544837394st_int,Xs: list_P419547167332244514st_int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_Pr8327173284404383691st_int @ ( cons_P4151473878978182108st_int @ X3 @ Xs ) @ N )
= ( nth_Pr8327173284404383691st_int @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_1150_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 ) )
= ( ? [Z3: int,Zs2: list_int] :
( ( Xs
= ( cons_int @ Z3 @ Zs2 ) )
& ( ( F @ Z3 )
= Y )
& ( ( map_int_int @ F @ Zs2 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_1151_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 ) )
= ( ? [Z3: instr,Zs2: list_instr] :
( ( Xs
= ( cons_instr @ Z3 @ Zs2 ) )
& ( ( F @ Z3 )
= Y )
& ( ( map_instr_int @ F @ Zs2 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_1152_map__eq__Cons__conv,axiom,
! [F: produc6425607678544837394st_int > int,Xs: list_P419547167332244514st_int,Y: int,Ys: list_int] :
( ( ( map_Pr3224476576807795204nt_int @ F @ Xs )
= ( cons_int @ Y @ Ys ) )
= ( ? [Z3: produc6425607678544837394st_int,Zs2: list_P419547167332244514st_int] :
( ( Xs
= ( cons_P4151473878978182108st_int @ Z3 @ Zs2 ) )
& ( ( F @ Z3 )
= Y )
& ( ( map_Pr3224476576807795204nt_int @ F @ Zs2 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_1153_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 ) )
= ( ? [Z3: int,Zs2: list_int] :
( ( Xs
= ( cons_int @ Z3 @ Zs2 ) )
& ( ( F @ Z3 )
= Y )
& ( ( map_int_instr @ F @ Zs2 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_1154_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 ) )
= ( ? [Z3: instr,Zs2: list_instr] :
( ( Xs
= ( cons_instr @ Z3 @ Zs2 ) )
& ( ( F @ Z3 )
= Y )
& ( ( map_instr_instr @ F @ Zs2 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_1155_map__eq__Cons__conv,axiom,
! [F: produc6425607678544837394st_int > instr,Xs: list_P419547167332244514st_int,Y: instr,Ys: list_instr] :
( ( ( map_Pr7514435936640407687_instr @ F @ Xs )
= ( cons_instr @ Y @ Ys ) )
= ( ? [Z3: produc6425607678544837394st_int,Zs2: list_P419547167332244514st_int] :
( ( Xs
= ( cons_P4151473878978182108st_int @ Z3 @ Zs2 ) )
& ( ( F @ Z3 )
= Y )
& ( ( map_Pr7514435936640407687_instr @ F @ Zs2 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_1156_map__eq__Cons__conv,axiom,
! [F: int > produc6425607678544837394st_int,Xs: list_int,Y: produc6425607678544837394st_int,Ys: list_P419547167332244514st_int] :
( ( ( map_in8512336359594627332st_int @ F @ Xs )
= ( cons_P4151473878978182108st_int @ Y @ Ys ) )
= ( ? [Z3: int,Zs2: list_int] :
( ( Xs
= ( cons_int @ Z3 @ Zs2 ) )
& ( ( F @ Z3 )
= Y )
& ( ( map_in8512336359594627332st_int @ F @ Zs2 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_1157_map__eq__Cons__conv,axiom,
! [F: instr > produc6425607678544837394st_int,Xs: list_instr,Y: produc6425607678544837394st_int,Ys: list_P419547167332244514st_int] :
( ( ( map_in7259965606695656109st_int @ F @ Xs )
= ( cons_P4151473878978182108st_int @ Y @ Ys ) )
= ( ? [Z3: instr,Zs2: list_instr] :
( ( Xs
= ( cons_instr @ Z3 @ Zs2 ) )
& ( ( F @ Z3 )
= Y )
& ( ( map_in7259965606695656109st_int @ F @ Zs2 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_1158_map__eq__Cons__conv,axiom,
! [F: produc6425607678544837394st_int > produc6425607678544837394st_int,Xs: list_P419547167332244514st_int,Y: produc6425607678544837394st_int,Ys: list_P419547167332244514st_int] :
( ( ( map_Pr8625093460839654122st_int @ F @ Xs )
= ( cons_P4151473878978182108st_int @ Y @ Ys ) )
= ( ? [Z3: produc6425607678544837394st_int,Zs2: list_P419547167332244514st_int] :
( ( Xs
= ( cons_P4151473878978182108st_int @ Z3 @ Zs2 ) )
& ( ( F @ Z3 )
= Y )
& ( ( map_Pr8625093460839654122st_int @ F @ Zs2 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_1159_Cons__eq__map__conv,axiom,
! [X3: int,Xs: list_int,F: int > int,Ys: list_int] :
( ( ( cons_int @ X3 @ Xs )
= ( map_int_int @ F @ Ys ) )
= ( ? [Z3: int,Zs2: list_int] :
( ( Ys
= ( cons_int @ Z3 @ Zs2 ) )
& ( X3
= ( F @ Z3 ) )
& ( Xs
= ( map_int_int @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_1160_Cons__eq__map__conv,axiom,
! [X3: int,Xs: list_int,F: instr > int,Ys: list_instr] :
( ( ( cons_int @ X3 @ Xs )
= ( map_instr_int @ F @ Ys ) )
= ( ? [Z3: instr,Zs2: list_instr] :
( ( Ys
= ( cons_instr @ Z3 @ Zs2 ) )
& ( X3
= ( F @ Z3 ) )
& ( Xs
= ( map_instr_int @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_1161_Cons__eq__map__conv,axiom,
! [X3: int,Xs: list_int,F: produc6425607678544837394st_int > int,Ys: list_P419547167332244514st_int] :
( ( ( cons_int @ X3 @ Xs )
= ( map_Pr3224476576807795204nt_int @ F @ Ys ) )
= ( ? [Z3: produc6425607678544837394st_int,Zs2: list_P419547167332244514st_int] :
( ( Ys
= ( cons_P4151473878978182108st_int @ Z3 @ Zs2 ) )
& ( X3
= ( F @ Z3 ) )
& ( Xs
= ( map_Pr3224476576807795204nt_int @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_1162_Cons__eq__map__conv,axiom,
! [X3: instr,Xs: list_instr,F: int > instr,Ys: list_int] :
( ( ( cons_instr @ X3 @ Xs )
= ( map_int_instr @ F @ Ys ) )
= ( ? [Z3: int,Zs2: list_int] :
( ( Ys
= ( cons_int @ Z3 @ Zs2 ) )
& ( X3
= ( F @ Z3 ) )
& ( Xs
= ( map_int_instr @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_1163_Cons__eq__map__conv,axiom,
! [X3: instr,Xs: list_instr,F: instr > instr,Ys: list_instr] :
( ( ( cons_instr @ X3 @ Xs )
= ( map_instr_instr @ F @ Ys ) )
= ( ? [Z3: instr,Zs2: list_instr] :
( ( Ys
= ( cons_instr @ Z3 @ Zs2 ) )
& ( X3
= ( F @ Z3 ) )
& ( Xs
= ( map_instr_instr @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_1164_Cons__eq__map__conv,axiom,
! [X3: instr,Xs: list_instr,F: produc6425607678544837394st_int > instr,Ys: list_P419547167332244514st_int] :
( ( ( cons_instr @ X3 @ Xs )
= ( map_Pr7514435936640407687_instr @ F @ Ys ) )
= ( ? [Z3: produc6425607678544837394st_int,Zs2: list_P419547167332244514st_int] :
( ( Ys
= ( cons_P4151473878978182108st_int @ Z3 @ Zs2 ) )
& ( X3
= ( F @ Z3 ) )
& ( Xs
= ( map_Pr7514435936640407687_instr @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_1165_Cons__eq__map__conv,axiom,
! [X3: produc6425607678544837394st_int,Xs: list_P419547167332244514st_int,F: int > produc6425607678544837394st_int,Ys: list_int] :
( ( ( cons_P4151473878978182108st_int @ X3 @ Xs )
= ( map_in8512336359594627332st_int @ F @ Ys ) )
= ( ? [Z3: int,Zs2: list_int] :
( ( Ys
= ( cons_int @ Z3 @ Zs2 ) )
& ( X3
= ( F @ Z3 ) )
& ( Xs
= ( map_in8512336359594627332st_int @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_1166_Cons__eq__map__conv,axiom,
! [X3: produc6425607678544837394st_int,Xs: list_P419547167332244514st_int,F: instr > produc6425607678544837394st_int,Ys: list_instr] :
( ( ( cons_P4151473878978182108st_int @ X3 @ Xs )
= ( map_in7259965606695656109st_int @ F @ Ys ) )
= ( ? [Z3: instr,Zs2: list_instr] :
( ( Ys
= ( cons_instr @ Z3 @ Zs2 ) )
& ( X3
= ( F @ Z3 ) )
& ( Xs
= ( map_in7259965606695656109st_int @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_1167_Cons__eq__map__conv,axiom,
! [X3: produc6425607678544837394st_int,Xs: list_P419547167332244514st_int,F: produc6425607678544837394st_int > produc6425607678544837394st_int,Ys: list_P419547167332244514st_int] :
( ( ( cons_P4151473878978182108st_int @ X3 @ Xs )
= ( map_Pr8625093460839654122st_int @ F @ Ys ) )
= ( ? [Z3: produc6425607678544837394st_int,Zs2: list_P419547167332244514st_int] :
( ( Ys
= ( cons_P4151473878978182108st_int @ Z3 @ Zs2 ) )
& ( X3
= ( F @ Z3 ) )
& ( Xs
= ( map_Pr8625093460839654122st_int @ F @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_1168_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 ) )
=> ? [Z5: int,Zs3: list_int] :
( ( Xs
= ( cons_int @ Z5 @ Zs3 ) )
& ( ( F @ Z5 )
= Y )
& ( ( map_int_int @ F @ Zs3 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_1169_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 ) )
=> ? [Z5: instr,Zs3: list_instr] :
( ( Xs
= ( cons_instr @ Z5 @ Zs3 ) )
& ( ( F @ Z5 )
= Y )
& ( ( map_instr_int @ F @ Zs3 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_1170_map__eq__Cons__D,axiom,
! [F: produc6425607678544837394st_int > int,Xs: list_P419547167332244514st_int,Y: int,Ys: list_int] :
( ( ( map_Pr3224476576807795204nt_int @ F @ Xs )
= ( cons_int @ Y @ Ys ) )
=> ? [Z5: produc6425607678544837394st_int,Zs3: list_P419547167332244514st_int] :
( ( Xs
= ( cons_P4151473878978182108st_int @ Z5 @ Zs3 ) )
& ( ( F @ Z5 )
= Y )
& ( ( map_Pr3224476576807795204nt_int @ F @ Zs3 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_1171_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 ) )
=> ? [Z5: int,Zs3: list_int] :
( ( Xs
= ( cons_int @ Z5 @ Zs3 ) )
& ( ( F @ Z5 )
= Y )
& ( ( map_int_instr @ F @ Zs3 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_1172_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 ) )
=> ? [Z5: instr,Zs3: list_instr] :
( ( Xs
= ( cons_instr @ Z5 @ Zs3 ) )
& ( ( F @ Z5 )
= Y )
& ( ( map_instr_instr @ F @ Zs3 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_1173_map__eq__Cons__D,axiom,
! [F: produc6425607678544837394st_int > instr,Xs: list_P419547167332244514st_int,Y: instr,Ys: list_instr] :
( ( ( map_Pr7514435936640407687_instr @ F @ Xs )
= ( cons_instr @ Y @ Ys ) )
=> ? [Z5: produc6425607678544837394st_int,Zs3: list_P419547167332244514st_int] :
( ( Xs
= ( cons_P4151473878978182108st_int @ Z5 @ Zs3 ) )
& ( ( F @ Z5 )
= Y )
& ( ( map_Pr7514435936640407687_instr @ F @ Zs3 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_1174_map__eq__Cons__D,axiom,
! [F: int > produc6425607678544837394st_int,Xs: list_int,Y: produc6425607678544837394st_int,Ys: list_P419547167332244514st_int] :
( ( ( map_in8512336359594627332st_int @ F @ Xs )
= ( cons_P4151473878978182108st_int @ Y @ Ys ) )
=> ? [Z5: int,Zs3: list_int] :
( ( Xs
= ( cons_int @ Z5 @ Zs3 ) )
& ( ( F @ Z5 )
= Y )
& ( ( map_in8512336359594627332st_int @ F @ Zs3 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_1175_map__eq__Cons__D,axiom,
! [F: instr > produc6425607678544837394st_int,Xs: list_instr,Y: produc6425607678544837394st_int,Ys: list_P419547167332244514st_int] :
( ( ( map_in7259965606695656109st_int @ F @ Xs )
= ( cons_P4151473878978182108st_int @ Y @ Ys ) )
=> ? [Z5: instr,Zs3: list_instr] :
( ( Xs
= ( cons_instr @ Z5 @ Zs3 ) )
& ( ( F @ Z5 )
= Y )
& ( ( map_in7259965606695656109st_int @ F @ Zs3 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_1176_map__eq__Cons__D,axiom,
! [F: produc6425607678544837394st_int > produc6425607678544837394st_int,Xs: list_P419547167332244514st_int,Y: produc6425607678544837394st_int,Ys: list_P419547167332244514st_int] :
( ( ( map_Pr8625093460839654122st_int @ F @ Xs )
= ( cons_P4151473878978182108st_int @ Y @ Ys ) )
=> ? [Z5: produc6425607678544837394st_int,Zs3: list_P419547167332244514st_int] :
( ( Xs
= ( cons_P4151473878978182108st_int @ Z5 @ Zs3 ) )
& ( ( F @ Z5 )
= Y )
& ( ( map_Pr8625093460839654122st_int @ F @ Zs3 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_1177_Cons__eq__map__D,axiom,
! [X3: int,Xs: list_int,F: int > int,Ys: list_int] :
( ( ( cons_int @ X3 @ Xs )
= ( map_int_int @ F @ Ys ) )
=> ? [Z5: int,Zs3: list_int] :
( ( Ys
= ( cons_int @ Z5 @ Zs3 ) )
& ( X3
= ( F @ Z5 ) )
& ( Xs
= ( map_int_int @ F @ Zs3 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_1178_Cons__eq__map__D,axiom,
! [X3: int,Xs: list_int,F: instr > int,Ys: list_instr] :
( ( ( cons_int @ X3 @ Xs )
= ( map_instr_int @ F @ Ys ) )
=> ? [Z5: instr,Zs3: list_instr] :
( ( Ys
= ( cons_instr @ Z5 @ Zs3 ) )
& ( X3
= ( F @ Z5 ) )
& ( Xs
= ( map_instr_int @ F @ Zs3 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_1179_Cons__eq__map__D,axiom,
! [X3: int,Xs: list_int,F: produc6425607678544837394st_int > int,Ys: list_P419547167332244514st_int] :
( ( ( cons_int @ X3 @ Xs )
= ( map_Pr3224476576807795204nt_int @ F @ Ys ) )
=> ? [Z5: produc6425607678544837394st_int,Zs3: list_P419547167332244514st_int] :
( ( Ys
= ( cons_P4151473878978182108st_int @ Z5 @ Zs3 ) )
& ( X3
= ( F @ Z5 ) )
& ( Xs
= ( map_Pr3224476576807795204nt_int @ F @ Zs3 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_1180_Cons__eq__map__D,axiom,
! [X3: instr,Xs: list_instr,F: int > instr,Ys: list_int] :
( ( ( cons_instr @ X3 @ Xs )
= ( map_int_instr @ F @ Ys ) )
=> ? [Z5: int,Zs3: list_int] :
( ( Ys
= ( cons_int @ Z5 @ Zs3 ) )
& ( X3
= ( F @ Z5 ) )
& ( Xs
= ( map_int_instr @ F @ Zs3 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_1181_Cons__eq__map__D,axiom,
! [X3: instr,Xs: list_instr,F: instr > instr,Ys: list_instr] :
( ( ( cons_instr @ X3 @ Xs )
= ( map_instr_instr @ F @ Ys ) )
=> ? [Z5: instr,Zs3: list_instr] :
( ( Ys
= ( cons_instr @ Z5 @ Zs3 ) )
& ( X3
= ( F @ Z5 ) )
& ( Xs
= ( map_instr_instr @ F @ Zs3 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_1182_Cons__eq__map__D,axiom,
! [X3: instr,Xs: list_instr,F: produc6425607678544837394st_int > instr,Ys: list_P419547167332244514st_int] :
( ( ( cons_instr @ X3 @ Xs )
= ( map_Pr7514435936640407687_instr @ F @ Ys ) )
=> ? [Z5: produc6425607678544837394st_int,Zs3: list_P419547167332244514st_int] :
( ( Ys
= ( cons_P4151473878978182108st_int @ Z5 @ Zs3 ) )
& ( X3
= ( F @ Z5 ) )
& ( Xs
= ( map_Pr7514435936640407687_instr @ F @ Zs3 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_1183_Cons__eq__map__D,axiom,
! [X3: produc6425607678544837394st_int,Xs: list_P419547167332244514st_int,F: int > produc6425607678544837394st_int,Ys: list_int] :
( ( ( cons_P4151473878978182108st_int @ X3 @ Xs )
= ( map_in8512336359594627332st_int @ F @ Ys ) )
=> ? [Z5: int,Zs3: list_int] :
( ( Ys
= ( cons_int @ Z5 @ Zs3 ) )
& ( X3
= ( F @ Z5 ) )
& ( Xs
= ( map_in8512336359594627332st_int @ F @ Zs3 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_1184_Cons__eq__map__D,axiom,
! [X3: produc6425607678544837394st_int,Xs: list_P419547167332244514st_int,F: instr > produc6425607678544837394st_int,Ys: list_instr] :
( ( ( cons_P4151473878978182108st_int @ X3 @ Xs )
= ( map_in7259965606695656109st_int @ F @ Ys ) )
=> ? [Z5: instr,Zs3: list_instr] :
( ( Ys
= ( cons_instr @ Z5 @ Zs3 ) )
& ( X3
= ( F @ Z5 ) )
& ( Xs
= ( map_in7259965606695656109st_int @ F @ Zs3 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_1185_Cons__eq__map__D,axiom,
! [X3: produc6425607678544837394st_int,Xs: list_P419547167332244514st_int,F: produc6425607678544837394st_int > produc6425607678544837394st_int,Ys: list_P419547167332244514st_int] :
( ( ( cons_P4151473878978182108st_int @ X3 @ Xs )
= ( map_Pr8625093460839654122st_int @ F @ Ys ) )
=> ? [Z5: produc6425607678544837394st_int,Zs3: list_P419547167332244514st_int] :
( ( Ys
= ( cons_P4151473878978182108st_int @ Z5 @ Zs3 ) )
& ( X3
= ( F @ Z5 ) )
& ( Xs
= ( map_Pr8625093460839654122st_int @ F @ Zs3 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_1186_list_Osimps_I9_J,axiom,
! [F: int > int,X21: int,X222: list_int] :
( ( map_int_int @ F @ ( cons_int @ X21 @ X222 ) )
= ( cons_int @ ( F @ X21 ) @ ( map_int_int @ F @ X222 ) ) ) ).
% list.simps(9)
thf(fact_1187_list_Osimps_I9_J,axiom,
! [F: int > instr,X21: int,X222: list_int] :
( ( map_int_instr @ F @ ( cons_int @ X21 @ X222 ) )
= ( cons_instr @ ( F @ X21 ) @ ( map_int_instr @ F @ X222 ) ) ) ).
% list.simps(9)
thf(fact_1188_list_Osimps_I9_J,axiom,
! [F: int > produc6425607678544837394st_int,X21: int,X222: list_int] :
( ( map_in8512336359594627332st_int @ F @ ( cons_int @ X21 @ X222 ) )
= ( cons_P4151473878978182108st_int @ ( F @ X21 ) @ ( map_in8512336359594627332st_int @ F @ X222 ) ) ) ).
% list.simps(9)
thf(fact_1189_list_Osimps_I9_J,axiom,
! [F: instr > int,X21: instr,X222: list_instr] :
( ( map_instr_int @ F @ ( cons_instr @ X21 @ X222 ) )
= ( cons_int @ ( F @ X21 ) @ ( map_instr_int @ F @ X222 ) ) ) ).
% list.simps(9)
thf(fact_1190_list_Osimps_I9_J,axiom,
! [F: instr > instr,X21: instr,X222: list_instr] :
( ( map_instr_instr @ F @ ( cons_instr @ X21 @ X222 ) )
= ( cons_instr @ ( F @ X21 ) @ ( map_instr_instr @ F @ X222 ) ) ) ).
% list.simps(9)
thf(fact_1191_list_Osimps_I9_J,axiom,
! [F: instr > produc6425607678544837394st_int,X21: instr,X222: list_instr] :
( ( map_in7259965606695656109st_int @ F @ ( cons_instr @ X21 @ X222 ) )
= ( cons_P4151473878978182108st_int @ ( F @ X21 ) @ ( map_in7259965606695656109st_int @ F @ X222 ) ) ) ).
% list.simps(9)
thf(fact_1192_list_Osimps_I9_J,axiom,
! [F: produc6425607678544837394st_int > int,X21: produc6425607678544837394st_int,X222: list_P419547167332244514st_int] :
( ( map_Pr3224476576807795204nt_int @ F @ ( cons_P4151473878978182108st_int @ X21 @ X222 ) )
= ( cons_int @ ( F @ X21 ) @ ( map_Pr3224476576807795204nt_int @ F @ X222 ) ) ) ).
% list.simps(9)
thf(fact_1193_list_Osimps_I9_J,axiom,
! [F: produc6425607678544837394st_int > instr,X21: produc6425607678544837394st_int,X222: list_P419547167332244514st_int] :
( ( map_Pr7514435936640407687_instr @ F @ ( cons_P4151473878978182108st_int @ X21 @ X222 ) )
= ( cons_instr @ ( F @ X21 ) @ ( map_Pr7514435936640407687_instr @ F @ X222 ) ) ) ).
% list.simps(9)
thf(fact_1194_list_Osimps_I9_J,axiom,
! [F: produc6425607678544837394st_int > produc6425607678544837394st_int,X21: produc6425607678544837394st_int,X222: list_P419547167332244514st_int] :
( ( map_Pr8625093460839654122st_int @ F @ ( cons_P4151473878978182108st_int @ X21 @ X222 ) )
= ( cons_P4151473878978182108st_int @ ( F @ X21 ) @ ( map_Pr8625093460839654122st_int @ F @ X222 ) ) ) ).
% list.simps(9)
thf(fact_1195_not__Cons__self2,axiom,
! [X3: instr,Xs: list_instr] :
( ( cons_instr @ X3 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_1196_not__Cons__self2,axiom,
! [X3: produc6425607678544837394st_int,Xs: list_P419547167332244514st_int] :
( ( cons_P4151473878978182108st_int @ X3 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_1197_upto__aux__rec,axiom,
( upto_aux
= ( ^ [I4: int,J3: int,Js: list_int] : ( if_list_int @ ( ord_less_int @ J3 @ I4 ) @ Js @ ( upto_aux @ I4 @ ( minus_minus_int @ J3 @ one_one_int ) @ ( cons_int @ J3 @ Js ) ) ) ) ) ).
% upto_aux_rec
thf(fact_1198_iexec__shift,axiom,
! [N: int,I5: int,S7: list_char > int,Stk3: list_int,Ins: instr,I: int,S: list_char > int,Stk4: list_int] :
( ( ( produc5086643055186798020st_int @ ( plus_plus_int @ N @ I5 ) @ ( produc8650753666468850689st_int @ S7 @ Stk3 ) )
= ( iexec @ Ins @ ( produc5086643055186798020st_int @ ( plus_plus_int @ N @ I ) @ ( produc8650753666468850689st_int @ S @ Stk4 ) ) ) )
= ( ( produc5086643055186798020st_int @ I5 @ ( produc8650753666468850689st_int @ S7 @ Stk3 ) )
= ( iexec @ Ins @ ( produc5086643055186798020st_int @ I @ ( produc8650753666468850689st_int @ S @ Stk4 ) ) ) ) ) ).
% iexec_shift
thf(fact_1199_exec1I,axiom,
! [C4: produc6425607678544837394st_int,P: list_instr,I: int,S: list_char > int,Stk4: list_int] :
( ( C4
= ( iexec @ ( inth_instr @ P @ I ) @ ( produc5086643055186798020st_int @ I @ ( produc8650753666468850689st_int @ S @ Stk4 ) ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( ( ord_less_int @ I @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P ) ) )
=> ( exec1 @ P @ ( produc5086643055186798020st_int @ I @ ( produc8650753666468850689st_int @ S @ Stk4 ) ) @ C4 ) ) ) ) ).
% exec1I
thf(fact_1200_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_1201_exec1__appendL,axiom,
! [P: list_instr,I: int,S: list_char > int,Stk4: list_int,I5: int,S7: list_char > int,Stk3: list_int,P3: list_instr] :
( ( exec1 @ P @ ( produc5086643055186798020st_int @ I @ ( produc8650753666468850689st_int @ S @ Stk4 ) ) @ ( produc5086643055186798020st_int @ I5 @ ( produc8650753666468850689st_int @ S7 @ Stk3 ) ) )
=> ( exec1 @ ( append_instr @ P3 @ P ) @ ( produc5086643055186798020st_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P3 ) ) @ I ) @ ( produc8650753666468850689st_int @ S @ Stk4 ) ) @ ( produc5086643055186798020st_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P3 ) ) @ I5 ) @ ( produc8650753666468850689st_int @ S7 @ Stk3 ) ) ) ) ).
% exec1_appendL
thf(fact_1202_Compiler_Obcomp__correct,axiom,
! [I: int,B: bexp,F: $o,S: list_char > int,Stk4: list_int] :
( ( ord_less_eq_int @ zero_zero_int @ I )
=> ( star_P707599355569300323st_int @ ( exec1 @ ( bcomp @ ( produc4047900504771817624_o_int @ B @ ( product_Pair_o_int @ F @ I ) ) ) ) @ ( produc5086643055186798020st_int @ zero_zero_int @ ( produc8650753666468850689st_int @ S @ Stk4 ) )
@ ( produc5086643055186798020st_int
@ ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ ( bcomp @ ( produc4047900504771817624_o_int @ B @ ( product_Pair_o_int @ F @ I ) ) ) ) )
@ ( if_int
@ ( F
= ( bval @ B @ S ) )
@ I
@ zero_zero_int ) )
@ ( produc8650753666468850689st_int @ S @ Stk4 ) ) ) ) ).
% Compiler.bcomp_correct
thf(fact_1203_exec__Cons__1,axiom,
! [P: list_instr,S: list_char > int,Stk4: list_int,J: int,T: list_char > int,Stk3: list_int,Ins: instr] :
( ( star_P707599355569300323st_int @ ( exec1 @ P ) @ ( produc5086643055186798020st_int @ zero_zero_int @ ( produc8650753666468850689st_int @ S @ Stk4 ) ) @ ( produc5086643055186798020st_int @ J @ ( produc8650753666468850689st_int @ T @ Stk3 ) ) )
=> ( star_P707599355569300323st_int @ ( exec1 @ ( cons_instr @ Ins @ P ) ) @ ( produc5086643055186798020st_int @ one_one_int @ ( produc8650753666468850689st_int @ S @ Stk4 ) ) @ ( produc5086643055186798020st_int @ ( plus_plus_int @ one_one_int @ J ) @ ( produc8650753666468850689st_int @ T @ Stk3 ) ) ) ) ).
% exec_Cons_1
thf(fact_1204_exec__appendL__if,axiom,
! [P3: list_instr,I: int,P: list_instr,S: list_char > int,Stk4: list_int,J: int,S7: list_char > int,Stk3: list_int,I5: int] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P3 ) ) @ I )
=> ( ( star_P707599355569300323st_int @ ( exec1 @ P ) @ ( produc5086643055186798020st_int @ ( minus_minus_int @ I @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P3 ) ) ) @ ( produc8650753666468850689st_int @ S @ Stk4 ) ) @ ( produc5086643055186798020st_int @ J @ ( produc8650753666468850689st_int @ S7 @ Stk3 ) ) )
=> ( ( I5
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P3 ) ) @ J ) )
=> ( star_P707599355569300323st_int @ ( exec1 @ ( append_instr @ P3 @ P ) ) @ ( produc5086643055186798020st_int @ I @ ( produc8650753666468850689st_int @ S @ Stk4 ) ) @ ( produc5086643055186798020st_int @ I5 @ ( produc8650753666468850689st_int @ S7 @ Stk3 ) ) ) ) ) ) ).
% exec_appendL_if
thf(fact_1205_Compiler_Oacomp__correct,axiom,
! [A: aexp,S: list_char > int,Stk4: list_int] : ( star_P707599355569300323st_int @ ( exec1 @ ( acomp @ A ) ) @ ( produc5086643055186798020st_int @ zero_zero_int @ ( produc8650753666468850689st_int @ S @ Stk4 ) ) @ ( produc5086643055186798020st_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ ( acomp @ A ) ) ) @ ( produc8650753666468850689st_int @ S @ ( cons_int @ ( aval @ A @ S ) @ Stk4 ) ) ) ) ).
% Compiler.acomp_correct
thf(fact_1206_exec__append__trans,axiom,
! [P: list_instr,S: list_char > int,Stk4: list_int,I5: int,S7: list_char > int,Stk3: list_int,P3: list_instr,I6: int,S8: list_char > int,Stk5: list_int,J4: int] :
( ( star_P707599355569300323st_int @ ( exec1 @ P ) @ ( produc5086643055186798020st_int @ zero_zero_int @ ( produc8650753666468850689st_int @ S @ Stk4 ) ) @ ( produc5086643055186798020st_int @ I5 @ ( produc8650753666468850689st_int @ S7 @ Stk3 ) ) )
=> ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P ) ) @ I5 )
=> ( ( star_P707599355569300323st_int @ ( exec1 @ P3 ) @ ( produc5086643055186798020st_int @ ( minus_minus_int @ I5 @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P ) ) ) @ ( produc8650753666468850689st_int @ S7 @ Stk3 ) ) @ ( produc5086643055186798020st_int @ I6 @ ( produc8650753666468850689st_int @ S8 @ Stk5 ) ) )
=> ( ( J4
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P ) ) @ I6 ) )
=> ( star_P707599355569300323st_int @ ( exec1 @ ( append_instr @ P @ P3 ) ) @ ( produc5086643055186798020st_int @ zero_zero_int @ ( produc8650753666468850689st_int @ S @ Stk4 ) ) @ ( produc5086643055186798020st_int @ J4 @ ( produc8650753666468850689st_int @ S8 @ Stk5 ) ) ) ) ) ) ) ).
% exec_append_trans
thf(fact_1207_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_1208_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 ) ) )
=> ( ! [A3: int,Aa: list_char > int,B2: list_int] : ( P @ A3 @ Aa @ B2 @ A3 @ Aa @ B2 )
=> ( ! [A3: int,Aa: list_char > int,B2: list_int,Ab2: int,Ac: list_char > int,Ba: list_int,Ad: int,Ae: list_char > int,Bb: list_int] :
( ( exec1 @ Pa @ ( produc5086643055186798020st_int @ A3 @ ( produc8650753666468850689st_int @ Aa @ B2 ) ) @ ( produc5086643055186798020st_int @ Ab2 @ ( produc8650753666468850689st_int @ Ac @ Ba ) ) )
=> ( ( star_P707599355569300323st_int @ ( exec1 @ Pa ) @ ( produc5086643055186798020st_int @ Ab2 @ ( produc8650753666468850689st_int @ Ac @ Ba ) ) @ ( produc5086643055186798020st_int @ Ad @ ( produc8650753666468850689st_int @ Ae @ Bb ) ) )
=> ( ( P @ Ab2 @ Ac @ Ba @ Ad @ Ae @ Bb )
=> ( P @ A3 @ Aa @ B2 @ Ad @ Ae @ Bb ) ) ) )
=> ( P @ X1a @ X1b @ X1c @ X2a @ X2b @ X2c ) ) ) ) ).
% exec_induct
thf(fact_1209_exec__appendL,axiom,
! [P: list_instr,I: int,S: list_char > int,Stk4: list_int,I5: int,S7: list_char > int,Stk3: list_int,P3: list_instr] :
( ( star_P707599355569300323st_int @ ( exec1 @ P ) @ ( produc5086643055186798020st_int @ I @ ( produc8650753666468850689st_int @ S @ Stk4 ) ) @ ( produc5086643055186798020st_int @ I5 @ ( produc8650753666468850689st_int @ S7 @ Stk3 ) ) )
=> ( star_P707599355569300323st_int @ ( exec1 @ ( append_instr @ P3 @ P ) ) @ ( produc5086643055186798020st_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P3 ) ) @ I ) @ ( produc8650753666468850689st_int @ S @ Stk4 ) ) @ ( produc5086643055186798020st_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P3 ) ) @ I5 ) @ ( produc8650753666468850689st_int @ S7 @ Stk3 ) ) ) ) ).
% exec_appendL
thf(fact_1210_ccomp__bigstep,axiom,
! [C: com,S: list_char > int,T: list_char > int,Stk4: list_int] :
( ( big_big_step @ ( produc5595214716300948949ar_int @ C @ S ) @ T )
=> ( star_P707599355569300323st_int @ ( exec1 @ ( ccomp @ C ) ) @ ( produc5086643055186798020st_int @ zero_zero_int @ ( produc8650753666468850689st_int @ S @ Stk4 ) ) @ ( produc5086643055186798020st_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ ( ccomp @ C ) ) ) @ ( produc8650753666468850689st_int @ T @ Stk4 ) ) ) ) ).
% ccomp_bigstep
thf(fact_1211_ccomp_Osimps_I4_J,axiom,
! [B: bexp,C_1: com,C_2: com] :
( ( ccomp @ ( if @ B @ C_1 @ C_2 ) )
= ( append_instr @ ( bcomp @ ( produc4047900504771817624_o_int @ B @ ( product_Pair_o_int @ $false @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ ( ccomp @ C_1 ) ) ) @ one_one_int ) ) ) ) @ ( append_instr @ ( ccomp @ C_1 ) @ ( cons_instr @ ( jmp @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ ( ccomp @ C_2 ) ) ) ) @ ( ccomp @ C_2 ) ) ) ) ) ).
% ccomp.simps(4)
thf(fact_1212_instr_Oinject_I4_J,axiom,
! [X52: int,Y52: int] :
( ( ( jmp @ X52 )
= ( jmp @ Y52 ) )
= ( X52 = Y52 ) ) ).
% instr.inject(4)
thf(fact_1213_instr_Osize_I12_J,axiom,
! [X52: int] :
( ( size_size_instr @ ( jmp @ X52 ) )
= zero_zero_nat ) ).
% instr.size(12)
thf(fact_1214_commute__if,axiom,
! [B1: bexp,B22: bexp,C11: com,C12: com,C22: com,S5: list_char > int,T3: list_char > int] :
( ( big_big_step @ ( produc5595214716300948949ar_int @ ( if @ B1 @ ( if @ B22 @ C11 @ C12 ) @ C22 ) @ S5 ) @ T3 )
= ( big_big_step @ ( produc5595214716300948949ar_int @ ( if @ B22 @ ( if @ B1 @ C11 @ C22 ) @ ( if @ B1 @ C12 @ C22 ) ) @ S5 ) @ T3 ) ) ).
% commute_if
thf(fact_1215_triv__if,axiom,
! [B: bexp,C: com,S5: list_char > int,T3: list_char > int] :
( ( big_big_step @ ( produc5595214716300948949ar_int @ ( if @ B @ C @ C ) @ S5 ) @ T3 )
= ( big_big_step @ ( produc5595214716300948949ar_int @ C @ S5 ) @ T3 ) ) ).
% triv_if
thf(fact_1216_IfE,axiom,
! [B: bexp,C1: com,C22: com,S: list_char > int,T: list_char > int] :
( ( big_big_step @ ( produc5595214716300948949ar_int @ ( if @ B @ C1 @ C22 ) @ S ) @ T )
=> ( ( ( bval @ B @ S )
=> ~ ( big_big_step @ ( produc5595214716300948949ar_int @ C1 @ S ) @ T ) )
=> ~ ( ~ ( bval @ B @ S )
=> ~ ( big_big_step @ ( produc5595214716300948949ar_int @ C22 @ S ) @ T ) ) ) ) ).
% IfE
thf(fact_1217_IfFalse,axiom,
! [B: bexp,S: list_char > int,C_2: com,T: list_char > int,C_1: com] :
( ~ ( bval @ B @ S )
=> ( ( big_big_step @ ( produc5595214716300948949ar_int @ C_2 @ S ) @ T )
=> ( big_big_step @ ( produc5595214716300948949ar_int @ ( if @ B @ C_1 @ C_2 ) @ S ) @ T ) ) ) ).
% IfFalse
thf(fact_1218_IfTrue,axiom,
! [B: bexp,S: list_char > int,C_1: com,T: list_char > int,C_2: com] :
( ( bval @ B @ S )
=> ( ( big_big_step @ ( produc5595214716300948949ar_int @ C_1 @ S ) @ T )
=> ( big_big_step @ ( produc5595214716300948949ar_int @ ( if @ B @ C_1 @ C_2 ) @ S ) @ T ) ) ) ).
% IfTrue
thf(fact_1219_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_1220_JmpE,axiom,
! [I: int,Pc3: int,S: list_char > int,Stk4: list_int,Cf: produc6425607678544837394st_int] :
( ( iexec_pred @ ( produc1484136438817787418st_int @ ( jmp @ I ) @ ( produc5086643055186798020st_int @ Pc3 @ ( produc8650753666468850689st_int @ S @ Stk4 ) ) ) @ Cf )
=> ( Cf
= ( produc5086643055186798020st_int @ ( plus_plus_int @ ( plus_plus_int @ Pc3 @ one_one_int ) @ I ) @ ( produc8650753666468850689st_int @ S @ Stk4 ) ) ) ) ).
% JmpE
thf(fact_1221_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_1222_iexec__offset,axiom,
! [Ins: instr,Pc3: int,S: list_char > int,Stk4: list_int,Pc4: int,S7: list_char > int,Stk3: list_int,I: int] :
( ( iexec_pred @ ( produc1484136438817787418st_int @ Ins @ ( produc5086643055186798020st_int @ Pc3 @ ( produc8650753666468850689st_int @ S @ Stk4 ) ) ) @ ( produc5086643055186798020st_int @ Pc4 @ ( produc8650753666468850689st_int @ S7 @ Stk3 ) ) )
=> ( iexec_pred @ ( produc1484136438817787418st_int @ Ins @ ( produc5086643055186798020st_int @ ( minus_minus_int @ Pc3 @ I ) @ ( produc8650753666468850689st_int @ S @ Stk4 ) ) ) @ ( produc5086643055186798020st_int @ ( minus_minus_int @ Pc4 @ I ) @ ( produc8650753666468850689st_int @ S7 @ Stk3 ) ) ) ) ).
% iexec_offset
thf(fact_1223_iexec__pred_Ocases,axiom,
! [A1: produc5995290525303592096st_int,A22: produc6425607678544837394st_int] :
( ( iexec_pred @ A1 @ A22 )
=> ~ ! [Ins2: instr,Cf2: produc6425607678544837394st_int] :
( ( A1
= ( produc1484136438817787418st_int @ Ins2 @ Cf2 ) )
=> ( A22
!= ( iexec @ Ins2 @ Cf2 ) ) ) ) ).
% iexec_pred.cases
thf(fact_1224_iexec__pred_Osimps,axiom,
( iexec_pred
= ( ^ [A12: produc5995290525303592096st_int,A23: produc6425607678544837394st_int] :
? [Ins3: instr,Cf3: produc6425607678544837394st_int] :
( ( A12
= ( produc1484136438817787418st_int @ Ins3 @ Cf3 ) )
& ( A23
= ( iexec @ Ins3 @ Cf3 ) ) ) ) ) ).
% iexec_pred.simps
thf(fact_1225_iexec__pred_Ointros,axiom,
! [Ins: instr,Cf: produc6425607678544837394st_int] : ( iexec_pred @ ( produc1484136438817787418st_int @ Ins @ Cf ) @ ( iexec @ Ins @ Cf ) ) ).
% iexec_pred.intros
thf(fact_1226_iexec__simp,axiom,
! [Ins: instr,Cf: produc6425607678544837394st_int,Cf4: produc6425607678544837394st_int] :
( ( iexec_pred @ ( produc1484136438817787418st_int @ Ins @ Cf ) @ Cf4 )
= ( Cf4
= ( iexec @ Ins @ Cf ) ) ) ).
% iexec_simp
thf(fact_1227_LoadIE,axiom,
! [I: int,Pc3: int,S: list_char > int,Stk4: list_int,Cf: produc6425607678544837394st_int] :
( ( iexec_pred @ ( produc1484136438817787418st_int @ ( loadi @ I ) @ ( produc5086643055186798020st_int @ Pc3 @ ( produc8650753666468850689st_int @ S @ Stk4 ) ) ) @ Cf )
=> ( Cf
= ( produc5086643055186798020st_int @ ( plus_plus_int @ Pc3 @ one_one_int ) @ ( produc8650753666468850689st_int @ S @ ( cons_int @ I @ Stk4 ) ) ) ) ) ).
% LoadIE
thf(fact_1228_LoadE,axiom,
! [X3: list_char,Pc3: int,S: list_char > int,Stk4: list_int,Cf: produc6425607678544837394st_int] :
( ( iexec_pred @ ( produc1484136438817787418st_int @ ( load @ X3 ) @ ( produc5086643055186798020st_int @ Pc3 @ ( produc8650753666468850689st_int @ S @ Stk4 ) ) ) @ Cf )
=> ( Cf
= ( produc5086643055186798020st_int @ ( plus_plus_int @ Pc3 @ one_one_int ) @ ( produc8650753666468850689st_int @ S @ ( cons_int @ ( S @ X3 ) @ Stk4 ) ) ) ) ) ).
% LoadE
thf(fact_1229_instr_Oinject_I2_J,axiom,
! [X22: list_char,Y2: list_char] :
( ( ( load @ X22 )
= ( load @ Y2 ) )
= ( X22 = Y2 ) ) ).
% instr.inject(2)
thf(fact_1230_instr_Oinject_I1_J,axiom,
! [X1: int,Y1: int] :
( ( ( loadi @ X1 )
= ( loadi @ Y1 ) )
= ( X1 = Y1 ) ) ).
% instr.inject(1)
thf(fact_1231_instr_Odistinct_I7_J,axiom,
! [X1: int,X52: int] :
( ( loadi @ X1 )
!= ( jmp @ X52 ) ) ).
% instr.distinct(7)
thf(fact_1232_instr_Odistinct_I17_J,axiom,
! [X22: list_char,X52: int] :
( ( load @ X22 )
!= ( jmp @ X52 ) ) ).
% instr.distinct(17)
thf(fact_1233_instr_Odistinct_I1_J,axiom,
! [X1: int,X22: list_char] :
( ( loadi @ X1 )
!= ( load @ X22 ) ) ).
% instr.distinct(1)
thf(fact_1234_instr_Osize_I9_J,axiom,
! [X22: list_char] :
( ( size_size_instr @ ( load @ X22 ) )
= zero_zero_nat ) ).
% instr.size(9)
thf(fact_1235_instr_Osize_I8_J,axiom,
! [X1: int] :
( ( size_size_instr @ ( loadi @ X1 ) )
= zero_zero_nat ) ).
% instr.size(8)
thf(fact_1236_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_1237_execl__next,axiom,
! [P: list_instr,Cfs: list_P419547167332244514st_int,K: nat] :
( ( execl @ P @ Cfs )
=> ( ( ord_less_nat @ K @ ( size_s229347708945616566st_int @ Cfs ) )
=> ( ( K
!= ( minus_minus_nat @ ( size_s229347708945616566st_int @ Cfs ) @ one_one_nat ) )
=> ( ( iexec_pred @ ( produc1484136438817787418st_int @ ( inth_instr @ P @ ( produc8941309049508465880st_int @ ( nth_Pr8327173284404383691st_int @ Cfs @ K ) ) ) @ ( nth_Pr8327173284404383691st_int @ Cfs @ K ) ) @ ( nth_Pr8327173284404383691st_int @ Cfs @ ( suc @ K ) ) )
& ( ord_less_eq_int @ zero_zero_int @ ( produc8941309049508465880st_int @ ( nth_Pr8327173284404383691st_int @ Cfs @ K ) ) )
& ( ord_less_int @ ( produc8941309049508465880st_int @ ( nth_Pr8327173284404383691st_int @ Cfs @ K ) ) @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P ) ) ) ) ) ) ) ).
% execl_next
thf(fact_1238_execl_Oelims_I3_J,axiom,
! [X3: list_instr,Xa: list_P419547167332244514st_int] :
( ~ ( execl @ X3 @ Xa )
=> ~ ! [Cf2: produc6425607678544837394st_int,Cf6: produc6425607678544837394st_int,Cfs2: list_P419547167332244514st_int] :
( ( Xa
= ( cons_P4151473878978182108st_int @ Cf2 @ ( cons_P4151473878978182108st_int @ Cf6 @ Cfs2 ) ) )
=> ( ( exec1 @ X3 @ Cf2 @ Cf6 )
& ( execl @ X3 @ ( cons_P4151473878978182108st_int @ Cf6 @ Cfs2 ) ) ) ) ) ).
% execl.elims(3)
thf(fact_1239_execl_Osimps_I1_J,axiom,
! [P: list_instr,Cf: produc6425607678544837394st_int,Cf4: produc6425607678544837394st_int,Cfs: list_P419547167332244514st_int] :
( ( execl @ P @ ( cons_P4151473878978182108st_int @ Cf @ ( cons_P4151473878978182108st_int @ Cf4 @ Cfs ) ) )
= ( ( exec1 @ P @ Cf @ Cf4 )
& ( execl @ P @ ( cons_P4151473878978182108st_int @ Cf4 @ Cfs ) ) ) ) ).
% execl.simps(1)
thf(fact_1240_execl__last,axiom,
! [P: list_instr,Cfs: list_P419547167332244514st_int,K: nat] :
( ( execl @ P @ Cfs )
=> ( ( ord_less_nat @ K @ ( size_s229347708945616566st_int @ Cfs ) )
=> ( ~ ( member_int @ ( produc8941309049508465880st_int @ ( nth_Pr8327173284404383691st_int @ Cfs @ K ) ) @ ( set_or4662586982721622107an_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P ) ) ) )
=> ( ( minus_minus_nat @ ( size_s229347708945616566st_int @ Cfs ) @ one_one_nat )
= K ) ) ) ) ).
% execl_last
thf(fact_1241_execl__all__def,axiom,
( execl_all
= ( ^ [P5: list_instr,Cfs3: list_P419547167332244514st_int] :
( ( execl @ P5 @ Cfs3 )
& ( Cfs3 != nil_Pr749310521696113548st_int )
& ( ( produc8941309049508465880st_int @ ( nth_Pr8327173284404383691st_int @ Cfs3 @ zero_zero_nat ) )
= zero_zero_int )
& ~ ( member_int @ ( produc8941309049508465880st_int @ ( nth_Pr8327173284404383691st_int @ Cfs3 @ ( minus_minus_nat @ ( size_s229347708945616566st_int @ Cfs3 ) @ one_one_nat ) ) ) @ ( set_or4662586982721622107an_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ ( size_size_list_instr @ P5 ) ) ) ) ) ) ) ).
% execl_all_def
thf(fact_1242_Compiler2_Obcomp__correct,axiom,
! [X3: produc1897111610453708512_o_int,Cfs: list_P419547167332244514st_int] :
( ( execl_all @ ( bcomp @ X3 ) @ Cfs )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( product_snd_o_int @ ( produc7219675232234870470_o_int @ X3 ) ) )
=> ( bpred @ X3 @ ( nth_Pr8327173284404383691st_int @ Cfs @ zero_zero_nat ) @ ( nth_Pr8327173284404383691st_int @ Cfs @ ( minus_minus_nat @ ( size_s229347708945616566st_int @ Cfs ) @ ( suc @ zero_zero_nat ) ) ) ) ) ) ).
% Compiler2.bcomp_correct
thf(fact_1243_execl_Osimps_I3_J,axiom,
! [P: list_instr,V: produc6425607678544837394st_int] : ( execl @ P @ ( cons_P4151473878978182108st_int @ V @ nil_Pr749310521696113548st_int ) ) ).
% execl.simps(3)
thf(fact_1244_execl_Ocases,axiom,
! [X3: produc4400946599758736694st_int] :
( ! [P6: list_instr,Cf2: produc6425607678544837394st_int,Cf6: produc6425607678544837394st_int,Cfs2: list_P419547167332244514st_int] :
( X3
!= ( produc7764413618310988720st_int @ P6 @ ( cons_P4151473878978182108st_int @ Cf2 @ ( cons_P4151473878978182108st_int @ Cf6 @ Cfs2 ) ) ) )
=> ( ! [P6: list_instr] :
( X3
!= ( produc7764413618310988720st_int @ P6 @ nil_Pr749310521696113548st_int ) )
=> ~ ! [P6: list_instr,V2: produc6425607678544837394st_int] :
( X3
!= ( produc7764413618310988720st_int @ P6 @ ( cons_P4151473878978182108st_int @ V2 @ nil_Pr749310521696113548st_int ) ) ) ) ) ).
% execl.cases
thf(fact_1245_all__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ! [M2: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( P @ M2 ) ) )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
=> ( P @ X ) ) ) ) ).
% all_nat_less_eq
thf(fact_1246_ex__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ? [M2: nat] :
( ( ord_less_nat @ M2 @ N )
& ( P @ M2 ) ) )
= ( ? [X: nat] :
( ( member_nat @ X @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
& ( P @ X ) ) ) ) ).
% ex_nat_less_eq
thf(fact_1247_execl_Osimps_I2_J,axiom,
! [P: list_instr] : ( execl @ P @ nil_Pr749310521696113548st_int ) ).
% execl.simps(2)
thf(fact_1248_execl_Oelims_I1_J,axiom,
! [X3: list_instr,Xa: list_P419547167332244514st_int,Y: $o] :
( ( ( execl @ X3 @ Xa )
= Y )
=> ( ! [Cf2: produc6425607678544837394st_int,Cf6: produc6425607678544837394st_int,Cfs2: list_P419547167332244514st_int] :
( ( Xa
= ( cons_P4151473878978182108st_int @ Cf2 @ ( cons_P4151473878978182108st_int @ Cf6 @ Cfs2 ) ) )
=> ( Y
= ( ~ ( ( exec1 @ X3 @ Cf2 @ Cf6 )
& ( execl @ X3 @ ( cons_P4151473878978182108st_int @ Cf6 @ Cfs2 ) ) ) ) ) )
=> ( ( ( Xa = nil_Pr749310521696113548st_int )
=> ~ Y )
=> ~ ( ? [V2: produc6425607678544837394st_int] :
( Xa
= ( cons_P4151473878978182108st_int @ V2 @ nil_Pr749310521696113548st_int ) )
=> ~ Y ) ) ) ) ).
% execl.elims(1)
thf(fact_1249_execl_Oelims_I2_J,axiom,
! [X3: list_instr,Xa: list_P419547167332244514st_int] :
( ( execl @ X3 @ Xa )
=> ( ! [Cf2: produc6425607678544837394st_int,Cf6: produc6425607678544837394st_int,Cfs2: list_P419547167332244514st_int] :
( ( Xa
= ( cons_P4151473878978182108st_int @ Cf2 @ ( cons_P4151473878978182108st_int @ Cf6 @ Cfs2 ) ) )
=> ~ ( ( exec1 @ X3 @ Cf2 @ Cf6 )
& ( execl @ X3 @ ( cons_P4151473878978182108st_int @ Cf6 @ Cfs2 ) ) ) )
=> ( ( Xa != nil_Pr749310521696113548st_int )
=> ~ ! [V2: produc6425607678544837394st_int] :
( Xa
!= ( cons_P4151473878978182108st_int @ V2 @ nil_Pr749310521696113548st_int ) ) ) ) ) ).
% execl.elims(2)
thf(fact_1250_mpred__merge,axiom,
! [P: list_instr,Cfs: list_P419547167332244514st_int,K: nat,M: nat,N: nat] :
( ( mpred @ P @ Cfs @ K @ M )
=> ( ( mpred @ P @ Cfs @ M @ N )
=> ( mpred @ P @ Cfs @ K @ N ) ) ) ).
% mpred_merge
thf(fact_1251_execl__all__SKIP,axiom,
! [Cfs: list_P419547167332244514st_int] :
( ( execl_all @ nil_instr @ Cfs )
=> ( ( cpred @ skip @ ( nth_Pr8327173284404383691st_int @ Cfs @ zero_zero_nat ) @ ( nth_Pr8327173284404383691st_int @ Cfs @ ( minus_minus_nat @ ( size_s229347708945616566st_int @ Cfs ) @ ( suc @ zero_zero_nat ) ) ) )
& ( ( size_s229347708945616566st_int @ Cfs )
= ( suc @ zero_zero_nat ) ) ) ) ).
% execl_all_SKIP
thf(fact_1252_ccomp_Osimps_I1_J,axiom,
( ( ccomp @ skip )
= nil_instr ) ).
% ccomp.simps(1)
thf(fact_1253_SkipE,axiom,
! [S: list_char > int,T: list_char > int] :
( ( big_big_step @ ( produc5595214716300948949ar_int @ skip @ S ) @ T )
=> ( T = S ) ) ).
% SkipE
thf(fact_1254_Skip,axiom,
! [S: list_char > int] : ( big_big_step @ ( produc5595214716300948949ar_int @ skip @ S ) @ S ) ).
% Skip
thf(fact_1255_com_Osize__gen_I1_J,axiom,
( ( size_com @ skip )
= zero_zero_nat ) ).
% com.size_gen(1)
thf(fact_1256_com_Osize_I6_J,axiom,
( ( size_size_com @ skip )
= zero_zero_nat ) ).
% com.size(6)
thf(fact_1257_execl_Opelims_I1_J,axiom,
! [X3: list_instr,Xa: list_P419547167332244514st_int,Y: $o] :
( ( ( execl @ X3 @ Xa )
= Y )
=> ( ( accp_P1482788143903795053st_int @ execl_rel @ ( produc7764413618310988720st_int @ X3 @ Xa ) )
=> ( ! [Cf2: produc6425607678544837394st_int,Cf6: produc6425607678544837394st_int,Cfs2: list_P419547167332244514st_int] :
( ( Xa
= ( cons_P4151473878978182108st_int @ Cf2 @ ( cons_P4151473878978182108st_int @ Cf6 @ Cfs2 ) ) )
=> ( ( Y
= ( ( exec1 @ X3 @ Cf2 @ Cf6 )
& ( execl @ X3 @ ( cons_P4151473878978182108st_int @ Cf6 @ Cfs2 ) ) ) )
=> ~ ( accp_P1482788143903795053st_int @ execl_rel @ ( produc7764413618310988720st_int @ X3 @ ( cons_P4151473878978182108st_int @ Cf2 @ ( cons_P4151473878978182108st_int @ Cf6 @ Cfs2 ) ) ) ) ) )
=> ( ( ( Xa = nil_Pr749310521696113548st_int )
=> ( Y
=> ~ ( accp_P1482788143903795053st_int @ execl_rel @ ( produc7764413618310988720st_int @ X3 @ nil_Pr749310521696113548st_int ) ) ) )
=> ~ ! [V2: produc6425607678544837394st_int] :
( ( Xa
= ( cons_P4151473878978182108st_int @ V2 @ nil_Pr749310521696113548st_int ) )
=> ( Y
=> ~ ( accp_P1482788143903795053st_int @ execl_rel @ ( produc7764413618310988720st_int @ X3 @ ( cons_P4151473878978182108st_int @ V2 @ nil_Pr749310521696113548st_int ) ) ) ) ) ) ) ) ) ).
% execl.pelims(1)
thf(fact_1258_execl_Opelims_I2_J,axiom,
! [X3: list_instr,Xa: list_P419547167332244514st_int] :
( ( execl @ X3 @ Xa )
=> ( ( accp_P1482788143903795053st_int @ execl_rel @ ( produc7764413618310988720st_int @ X3 @ Xa ) )
=> ( ! [Cf2: produc6425607678544837394st_int,Cf6: produc6425607678544837394st_int,Cfs2: list_P419547167332244514st_int] :
( ( Xa
= ( cons_P4151473878978182108st_int @ Cf2 @ ( cons_P4151473878978182108st_int @ Cf6 @ Cfs2 ) ) )
=> ( ( accp_P1482788143903795053st_int @ execl_rel @ ( produc7764413618310988720st_int @ X3 @ ( cons_P4151473878978182108st_int @ Cf2 @ ( cons_P4151473878978182108st_int @ Cf6 @ Cfs2 ) ) ) )
=> ~ ( ( exec1 @ X3 @ Cf2 @ Cf6 )
& ( execl @ X3 @ ( cons_P4151473878978182108st_int @ Cf6 @ Cfs2 ) ) ) ) )
=> ( ( ( Xa = nil_Pr749310521696113548st_int )
=> ~ ( accp_P1482788143903795053st_int @ execl_rel @ ( produc7764413618310988720st_int @ X3 @ nil_Pr749310521696113548st_int ) ) )
=> ~ ! [V2: produc6425607678544837394st_int] :
( ( Xa
= ( cons_P4151473878978182108st_int @ V2 @ nil_Pr749310521696113548st_int ) )
=> ~ ( accp_P1482788143903795053st_int @ execl_rel @ ( produc7764413618310988720st_int @ X3 @ ( cons_P4151473878978182108st_int @ V2 @ nil_Pr749310521696113548st_int ) ) ) ) ) ) ) ) ).
% execl.pelims(2)
thf(fact_1259_execl_Opelims_I3_J,axiom,
! [X3: list_instr,Xa: list_P419547167332244514st_int] :
( ~ ( execl @ X3 @ Xa )
=> ( ( accp_P1482788143903795053st_int @ execl_rel @ ( produc7764413618310988720st_int @ X3 @ Xa ) )
=> ~ ! [Cf2: produc6425607678544837394st_int,Cf6: produc6425607678544837394st_int,Cfs2: list_P419547167332244514st_int] :
( ( Xa
= ( cons_P4151473878978182108st_int @ Cf2 @ ( cons_P4151473878978182108st_int @ Cf6 @ Cfs2 ) ) )
=> ( ( accp_P1482788143903795053st_int @ execl_rel @ ( produc7764413618310988720st_int @ X3 @ ( cons_P4151473878978182108st_int @ Cf2 @ ( cons_P4151473878978182108st_int @ Cf6 @ Cfs2 ) ) ) )
=> ( ( exec1 @ X3 @ Cf2 @ Cf6 )
& ( execl @ X3 @ ( cons_P4151473878978182108st_int @ Cf6 @ Cfs2 ) ) ) ) ) ) ) ).
% execl.pelims(3)
thf(fact_1260_mpred__drop,axiom,
! [K: nat,Cfs: list_P419547167332244514st_int,P: list_instr,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ ( size_s229347708945616566st_int @ Cfs ) )
=> ( ( ord_less_eq_nat @ ( size_size_list_int @ ( produc6244060863719409455st_int @ ( produc1296517004818028822st_int @ ( nth_Pr8327173284404383691st_int @ Cfs @ zero_zero_nat ) ) ) ) @ ( size_size_list_int @ ( produc6244060863719409455st_int @ ( produc1296517004818028822st_int @ ( nth_Pr8327173284404383691st_int @ Cfs @ K ) ) ) ) )
=> ( ( mpred @ P @ ( drop_P7838799151754935884st_int @ K @ Cfs ) @ M @ N )
=> ( mpred @ P @ Cfs @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) ) ) ) ) ).
% mpred_drop
% Helper facts (9)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X3: int,Y: int] :
( ( if_int @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X3: int,Y: int] :
( ( if_int @ $true @ X3 @ Y )
= X3 ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X3: nat,Y: nat] :
( ( if_nat @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X3: nat,Y: nat] :
( ( if_nat @ $true @ X3 @ Y )
= X3 ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
! [X3: list_int,Y: list_int] :
( ( if_list_int @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
! [X3: list_int,Y: list_int] :
( ( if_list_int @ $true @ X3 @ Y )
= X3 ) ).
thf(help_If_3_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
! [X3: product_prod_nat_nat,Y: product_prod_nat_nat] :
( ( if_Pro6206227464963214023at_nat @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
! [X3: product_prod_nat_nat,Y: product_prod_nat_nat] :
( ( if_Pro6206227464963214023at_nat @ $true @ X3 @ Y )
= X3 ) ).
% Conjectures (4)
thf(conj_0,hypothesis,
execl_all @ ( append_instr @ ( bcomp @ ( produc4047900504771817624_o_int @ b @ ( product_Pair_o_int @ f @ i ) ) ) @ ( append_instr @ ( ccomp @ c ) @ p ) ) @ cfs ).
thf(conj_1,hypothesis,
ord_less_eq_int @ zero_zero_int @ i ).
thf(conj_2,hypothesis,
! [Cfs4: list_P419547167332244514st_int] :
( ( execl_all @ ( ccomp @ c ) @ Cfs4 )
=> ( cpred @ c @ ( nth_Pr8327173284404383691st_int @ Cfs4 @ zero_zero_nat ) @ ( nth_Pr8327173284404383691st_int @ Cfs4 @ ( minus_minus_nat @ ( size_s229347708945616566st_int @ Cfs4 ) @ one_one_nat ) ) ) ) ).
thf(conj_3,conjecture,
( produc5824858813793146727_int_o
@ ^ [Pc: int] :
( produc4252051693687794178_int_o
@ ^ [S2: list_char > int,Stk: list_int] :
( ( Pc = zero_zero_int )
& ( ( ( bval @ b @ S2 )
!= f )
=> ? [K3: nat] :
( ( ord_less_nat @ K3 @ ( size_s229347708945616566st_int @ cfs ) )
& ( produc5824858813793146727_int_o
@ ^ [Pc2: int] :
( produc4252051693687794178_int_o
@ ^ [S3: list_char > int,Stk2: list_int] :
( ( Pc2
= ( semiri1314217659103216013at_int @ ( size_size_list_instr @ ( append_instr @ ( bcomp @ ( produc4047900504771817624_o_int @ b @ ( product_Pair_o_int @ f @ i ) ) ) @ ( ccomp @ c ) ) ) ) )
& ( big_big_step @ ( produc5595214716300948949ar_int @ c @ S2 ) @ S3 )
& ( Stk2 = Stk ) ) )
@ ( nth_Pr8327173284404383691st_int @ cfs @ K3 ) ) ) ) ) )
@ ( nth_Pr8327173284404383691st_int @ cfs @ zero_zero_nat ) ) ).
%------------------------------------------------------------------------------