ITP001 Axioms: ITP111^5.ax
%------------------------------------------------------------------------------
% File : ITP111^5 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 set theory export, chainy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau20] Gauthier (2020), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : intto^2.ax [Gau20]
% : HL4111^5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 29 ( 19 unt; 6 typ; 0 def)
% Number of atoms : 234 ( 20 equ; 0 cnn)
% Maximal formula atoms : 6 ( 8 avg)
% Number of connectives : 237 ( 2 ~; 0 |; 0 &; 234 @)
% ( 0 <=>; 1 =>; 0 <=; 0 <~>)
% Maximal formula depth : 5 ( 3 avg; 234 nst)
% Number of types : 2 ( 1 usr)
% Number of type conns : 4 ( 4 >; 0 *; 0 +; 0 <<)
% Number of symbols : 34 ( 33 usr; 30 con; 0-2 aty)
% Number of variables : 27 ( 0 ^ 27 !; 0 ?; 27 :)
% SPC : TH0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_c_2Eintto_2EintOrd,type,
c_2Eintto_2EintOrd: $i ).
thf(mem_c_2Eintto_2EintOrd,axiom,
mem @ c_2Eintto_2EintOrd @ ( arr @ ty_2Einteger_2Eint @ ( arr @ ty_2Einteger_2Eint @ ty_2EternaryComparisons_2Eordering ) ) ).
thf(stp_fo_c_2Eintto_2EintOrd,type,
fo__c_2Eintto_2EintOrd: tp__ty_2Einteger_2Eint > tp__ty_2Einteger_2Eint > tp__ty_2EternaryComparisons_2Eordering ).
thf(stp_eq_fo_c_2Eintto_2EintOrd,axiom,
! [X0: tp__ty_2Einteger_2Eint,X1: tp__ty_2Einteger_2Eint] :
( ( inj__ty_2EternaryComparisons_2Eordering @ ( fo__c_2Eintto_2EintOrd @ X0 @ X1 ) )
= ( ap @ ( ap @ c_2Eintto_2EintOrd @ ( inj__ty_2Einteger_2Eint @ X0 ) ) @ ( inj__ty_2Einteger_2Eint @ X1 ) ) ) ).
thf(stp_c_ty_2Etoto_2Etoto_ty_2Einteger_2Eint,type,
tp__c_ty_2Etoto_2Etoto_ty_2Einteger_2Eint: $tType ).
thf(stp_inj_c_ty_2Etoto_2Etoto_ty_2Einteger_2Eint,type,
inj__c_ty_2Etoto_2Etoto_ty_2Einteger_2Eint: tp__c_ty_2Etoto_2Etoto_ty_2Einteger_2Eint > $i ).
thf(stp_surj_c_ty_2Etoto_2Etoto_ty_2Einteger_2Eint,type,
surj__c_ty_2Etoto_2Etoto_ty_2Einteger_2Eint: $i > tp__c_ty_2Etoto_2Etoto_ty_2Einteger_2Eint ).
thf(stp_inj_surj_c_ty_2Etoto_2Etoto_ty_2Einteger_2Eint,axiom,
! [X: tp__c_ty_2Etoto_2Etoto_ty_2Einteger_2Eint] :
( ( surj__c_ty_2Etoto_2Etoto_ty_2Einteger_2Eint @ ( inj__c_ty_2Etoto_2Etoto_ty_2Einteger_2Eint @ X ) )
= X ) ).
thf(stp_inj_mem_c_ty_2Etoto_2Etoto_ty_2Einteger_2Eint,axiom,
! [X: tp__c_ty_2Etoto_2Etoto_ty_2Einteger_2Eint] : ( mem @ ( inj__c_ty_2Etoto_2Etoto_ty_2Einteger_2Eint @ X ) @ ( ty_2Etoto_2Etoto @ ty_2Einteger_2Eint ) ) ).
thf(stp_iso_mem_c_ty_2Etoto_2Etoto_ty_2Einteger_2Eint,axiom,
! [X: $i] :
( ( mem @ X @ ( ty_2Etoto_2Etoto @ ty_2Einteger_2Eint ) )
=> ( X
= ( inj__c_ty_2Etoto_2Etoto_ty_2Einteger_2Eint @ ( surj__c_ty_2Etoto_2Etoto_ty_2Einteger_2Eint @ X ) ) ) ) ).
thf(tp_c_2Eintto_2Eintto,type,
c_2Eintto_2Eintto: $i ).
thf(mem_c_2Eintto_2Eintto,axiom,
mem @ c_2Eintto_2Eintto @ ( ty_2Etoto_2Etoto @ ty_2Einteger_2Eint ) ).
thf(ax_thm_2Eintto_2EintOrd,axiom,
( c_2Eintto_2EintOrd
= ( ap @ ( c_2Etoto_2ETO__of__LinearOrder @ ty_2Einteger_2Eint ) @ c_2Einteger_2Eint__lt ) ) ).
thf(ax_thm_2Eintto_2Eintto,axiom,
( ( surj__c_ty_2Etoto_2Etoto_ty_2Einteger_2Eint @ c_2Eintto_2Eintto )
= ( surj__c_ty_2Etoto_2Etoto_ty_2Einteger_2Eint @ ( ap @ ( c_2Etoto_2ETO @ ty_2Einteger_2Eint ) @ c_2Eintto_2EintOrd ) ) ) ).
thf(conj_thm_2Eintto_2Eapintto__thm,axiom,
( ( ap @ ( c_2Etoto_2Eapto @ ty_2Einteger_2Eint ) @ c_2Eintto_2Eintto )
= c_2Eintto_2EintOrd ) ).
thf(conj_thm_2Eintto_2Epos__pos__thm,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ c_2Eintto_2EintOrd @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V0m ) ) ) @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
= ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ c_2Etoto_2EnumOrd @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ).
thf(conj_thm_2Eintto_2Eneg__neg__thm,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ c_2Eintto_2EintOrd @ ( ap @ c_2Einteger_2Eint__neg @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V0m ) ) ) ) @ ( ap @ c_2Einteger_2Eint__neg @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) )
= ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ c_2Etoto_2EnumOrd @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( inj__ty_2Enum_2Enum @ V0m ) ) ) ) ).
thf(conj_thm_2Eintto_2EBIT1__nz,axiom,
! [V0n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
!= fo__c_2Enum_2E0 ) ).
thf(conj_thm_2Eintto_2EBIT2__nz,axiom,
! [V0n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
!= fo__c_2Enum_2E0 ) ).
thf(conj_thm_2Eintto_2Eneg__lt__BIT1__thm,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ c_2Eintto_2EintOrd @ ( ap @ c_2Einteger_2Eint__neg @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V0m ) ) ) ) @ ( ap @ c_2Einteger_2Eint__of__num @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) )
= fo__c_2EternaryComparisons_2ELESS ) ).
thf(conj_thm_2Eintto_2Eneg__lt__BIT2__thm,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ c_2Eintto_2EintOrd @ ( ap @ c_2Einteger_2Eint__neg @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V0m ) ) ) ) @ ( ap @ c_2Einteger_2Eint__of__num @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) )
= fo__c_2EternaryComparisons_2ELESS ) ).
thf(conj_thm_2Eintto_2Eneg__BIT1__lt__thm,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ c_2Eintto_2EintOrd @ ( ap @ c_2Einteger_2Eint__neg @ ( ap @ c_2Einteger_2Eint__of__num @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V0m ) ) ) ) ) @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
= fo__c_2EternaryComparisons_2ELESS ) ).
thf(conj_thm_2Eintto_2Eneg__BIT2__lt__thm,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ c_2Eintto_2EintOrd @ ( ap @ c_2Einteger_2Eint__neg @ ( ap @ c_2Einteger_2Eint__of__num @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V0m ) ) ) ) ) @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
= fo__c_2EternaryComparisons_2ELESS ) ).
thf(conj_thm_2Eintto_2Eneg__ZERO__eq__ZERO__thm,axiom,
( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ c_2Eintto_2EintOrd @ ( ap @ c_2Einteger_2Eint__neg @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) )
= fo__c_2EternaryComparisons_2EEQUAL ) ).
thf(conj_thm_2Eintto_2EBIT1__gt__neg__thm,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ c_2Eintto_2EintOrd @ ( ap @ c_2Einteger_2Eint__of__num @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V0m ) ) ) ) @ ( ap @ c_2Einteger_2Eint__neg @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) )
= fo__c_2EternaryComparisons_2EGREATER ) ).
thf(conj_thm_2Eintto_2EBIT2__gt__neg__thm,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ c_2Eintto_2EintOrd @ ( ap @ c_2Einteger_2Eint__of__num @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V0m ) ) ) ) @ ( ap @ c_2Einteger_2Eint__neg @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) )
= fo__c_2EternaryComparisons_2EGREATER ) ).
thf(conj_thm_2Eintto_2Egt__neg__BIT1__thm,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ c_2Eintto_2EintOrd @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V0m ) ) ) @ ( ap @ c_2Einteger_2Eint__neg @ ( ap @ c_2Einteger_2Eint__of__num @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ) )
= fo__c_2EternaryComparisons_2EGREATER ) ).
thf(conj_thm_2Eintto_2Egt__neg__BIT2__thm,axiom,
! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ c_2Eintto_2EintOrd @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ V0m ) ) ) @ ( ap @ c_2Einteger_2Eint__neg @ ( ap @ c_2Einteger_2Eint__of__num @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ) )
= fo__c_2EternaryComparisons_2EGREATER ) ).
thf(conj_thm_2Eintto_2EZERO__eq__neg__ZERO__thm,axiom,
( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ c_2Eintto_2EintOrd @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) @ ( ap @ c_2Einteger_2Eint__neg @ ( ap @ c_2Einteger_2Eint__of__num @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) )
= fo__c_2EternaryComparisons_2EEQUAL ) ).
%------------------------------------------------------------------------------