ITP001 Axioms: ITP004_5.ax
%------------------------------------------------------------------------------
% File : ITP004_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 : ConseqConv_2.ax [Gau20]
% : HL4004_5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 45 ( 3 unt; 2 typ; 0 def)
% Number of atoms : 436 ( 5 equ)
% Maximal formula atoms : 10 ( 9 avg)
% Number of connectives : 161 ( 14 ~; 9 |; 23 &)
% ( 27 <=>; 88 =>; 0 <=; 0 <~>)
% Maximal formula depth : 10 ( 5 avg)
% Maximal term depth : 2 ( 1 avg)
% Number of FOOLs : 246 ( 246 fml; 0 var)
% Number of types : 1 ( 0 usr)
% Number of type conns : 2 ( 1 >; 1 *; 0 +; 0 <<)
% Number of predicates : 16 ( 13 usr; 7 prp; 0-2 aty)
% Number of functors : 2 ( 2 usr; 1 con; 0-2 aty)
% Number of variables : 93 ( 91 !; 2 ?; 93 :)
% SPC : TF0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
tff(tp_c_2EConseqConv_2EASM__MARKER,type,
c_2EConseqConv_2EASM__MARKER: $i ).
tff(mem_c_2EConseqConv_2EASM__MARKER,axiom,
mem(c_2EConseqConv_2EASM__MARKER,arr(bool,arr(bool,bool))) ).
tff(stp_fo_c_2EConseqConv_2EASM__MARKER,type,
fo__c_2EConseqConv_2EASM__MARKER: ( tp__o * tp__o ) > tp__o ).
tff(stp_eq_fo_c_2EConseqConv_2EASM__MARKER,axiom,
! [X0: tp__o,X1: tp__o] : ( inj__o(fo__c_2EConseqConv_2EASM__MARKER(X0,X1)) = ap(ap(c_2EConseqConv_2EASM__MARKER,inj__o(X0)),inj__o(X1)) ) ).
tff(conj_thm_2EConseqConv_2Eforall__eq__thm,axiom,
! [A_27a: del,V0P: $i] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1Q: $i] :
( mem(V1Q,arr(A_27a,bool))
=> ( ! [V2s: $i] :
( mem(V2s,A_27a)
=> ( p(ap(V0P,V2s))
<=> p(ap(V1Q,V2s)) ) )
=> ( ! [V3s: $i] :
( mem(V3s,A_27a)
=> p(ap(V0P,V3s)) )
<=> ! [V4s: $i] :
( mem(V4s,A_27a)
=> p(ap(V1Q,V4s)) ) ) ) ) ) ).
tff(conj_thm_2EConseqConv_2Eexists__eq__thm,axiom,
! [A_27a: del,V0P: $i] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1Q: $i] :
( mem(V1Q,arr(A_27a,bool))
=> ( ! [V2s: $i] :
( mem(V2s,A_27a)
=> ( p(ap(V0P,V2s))
<=> p(ap(V1Q,V2s)) ) )
=> ( ? [V3s: $i] :
( mem(V3s,A_27a)
& p(ap(V0P,V3s)) )
<=> ? [V4s: $i] :
( mem(V4s,A_27a)
& p(ap(V1Q,V4s)) ) ) ) ) ) ).
tff(conj_thm_2EConseqConv_2Etrue__imp,axiom,
! [V0t: tp__o] :
( p(inj__o(V0t))
=> $true ) ).
tff(conj_thm_2EConseqConv_2Efalse__imp,axiom,
! [V0t: tp__o] :
( $false
=> p(inj__o(V0t)) ) ).
tff(conj_thm_2EConseqConv_2ENOT__CLAUSES__X,axiom,
! [V0t: tp__o] :
( ~ ~ p(inj__o(V0t))
<=> p(inj__o(V0t)) ) ).
tff(conj_thm_2EConseqConv_2ENOT__CLAUSES__T,axiom,
( ~ $true
<=> $false ) ).
tff(conj_thm_2EConseqConv_2ENOT__CLAUSES__F,axiom,
( ~ $false
<=> $true ) ).
tff(conj_thm_2EConseqConv_2EIMP__CONG__conj__strengthen,axiom,
! [V0x: tp__o,V1x_27: tp__o,V2y: tp__o,V3y_27: tp__o] :
( ( ( p(inj__o(V2y))
=> ( p(inj__o(V1x_27))
=> p(inj__o(V0x)) ) )
& ( p(inj__o(V1x_27))
=> ( p(inj__o(V3y_27))
=> p(inj__o(V2y)) ) ) )
=> ( ( p(inj__o(V1x_27))
& p(inj__o(V3y_27)) )
=> ( p(inj__o(V0x))
& p(inj__o(V2y)) ) ) ) ).
tff(conj_thm_2EConseqConv_2EIMP__CONG__conj__weaken,axiom,
! [V0x: tp__o,V1x_27: tp__o,V2y: tp__o,V3y_27: tp__o] :
( ( ( p(inj__o(V2y))
=> ( p(inj__o(V0x))
=> p(inj__o(V1x_27)) ) )
& ( p(inj__o(V1x_27))
=> ( p(inj__o(V2y))
=> p(inj__o(V3y_27)) ) ) )
=> ( ( p(inj__o(V0x))
& p(inj__o(V2y)) )
=> ( p(inj__o(V1x_27))
& p(inj__o(V3y_27)) ) ) ) ).
tff(conj_thm_2EConseqConv_2EAND__CLAUSES__TX,axiom,
! [V0t: tp__o] :
( ( $true
& p(inj__o(V0t)) )
<=> p(inj__o(V0t)) ) ).
tff(conj_thm_2EConseqConv_2EAND__CLAUSES__XT,axiom,
! [V0t: tp__o] :
( ( p(inj__o(V0t))
& $true )
<=> p(inj__o(V0t)) ) ).
tff(conj_thm_2EConseqConv_2EAND__CLAUSES__FX,axiom,
! [V0t: tp__o] :
( ( $false
& p(inj__o(V0t)) )
<=> $false ) ).
tff(conj_thm_2EConseqConv_2EAND__CLAUSES__XF,axiom,
! [V0t: tp__o] :
( ( p(inj__o(V0t))
& $false )
<=> $false ) ).
tff(conj_thm_2EConseqConv_2EAND__CLAUSES__XX,axiom,
! [V0t: tp__o] :
( ( p(inj__o(V0t))
& p(inj__o(V0t)) )
<=> p(inj__o(V0t)) ) ).
tff(conj_thm_2EConseqConv_2EIMP__CONG__disj__strengthen,axiom,
! [V0x: tp__o,V1x_27: tp__o,V2y: tp__o,V3y_27: tp__o] :
( ( ( ~ p(inj__o(V2y))
=> ( p(inj__o(V1x_27))
=> p(inj__o(V0x)) ) )
& ( ~ p(inj__o(V1x_27))
=> ( p(inj__o(V3y_27))
=> p(inj__o(V2y)) ) ) )
=> ( ( p(inj__o(V1x_27))
| p(inj__o(V3y_27)) )
=> ( p(inj__o(V0x))
| p(inj__o(V2y)) ) ) ) ).
tff(conj_thm_2EConseqConv_2EIMP__CONG__disj__weaken,axiom,
! [V0x: tp__o,V1x_27: tp__o,V2y: tp__o,V3y_27: tp__o] :
( ( ( ~ p(inj__o(V2y))
=> ( p(inj__o(V0x))
=> p(inj__o(V1x_27)) ) )
& ( ~ p(inj__o(V1x_27))
=> ( p(inj__o(V2y))
=> p(inj__o(V3y_27)) ) ) )
=> ( ( p(inj__o(V0x))
| p(inj__o(V2y)) )
=> ( p(inj__o(V1x_27))
| p(inj__o(V3y_27)) ) ) ) ).
tff(conj_thm_2EConseqConv_2EOR__CLAUSES__TX,axiom,
! [V0t: tp__o] :
( ( $true
| p(inj__o(V0t)) )
<=> $true ) ).
tff(conj_thm_2EConseqConv_2EOR__CLAUSES__XT,axiom,
! [V0t: tp__o] :
( ( p(inj__o(V0t))
| $true )
<=> $true ) ).
tff(conj_thm_2EConseqConv_2EOR__CLAUSES__FX,axiom,
! [V0t: tp__o] :
( ( $false
| p(inj__o(V0t)) )
<=> p(inj__o(V0t)) ) ).
tff(conj_thm_2EConseqConv_2EOR__CLAUSES__XF,axiom,
! [V0t: tp__o] :
( ( p(inj__o(V0t))
| $false )
<=> p(inj__o(V0t)) ) ).
tff(conj_thm_2EConseqConv_2EOR__CLAUSES__XX,axiom,
! [V0t: tp__o] :
( ( p(inj__o(V0t))
| p(inj__o(V0t)) )
<=> p(inj__o(V0t)) ) ).
tff(conj_thm_2EConseqConv_2EIMP__CONG__imp__strengthen,axiom,
! [V0x: tp__o,V1x_27: tp__o,V2y: tp__o,V3y_27: tp__o] :
( ( ( p(inj__o(V0x))
=> ( p(inj__o(V3y_27))
=> p(inj__o(V2y)) ) )
& ( ~ p(inj__o(V3y_27))
=> ( p(inj__o(V0x))
=> p(inj__o(V1x_27)) ) ) )
=> ( ( p(inj__o(V1x_27))
=> p(inj__o(V3y_27)) )
=> ( p(inj__o(V0x))
=> p(inj__o(V2y)) ) ) ) ).
tff(conj_thm_2EConseqConv_2EIMP__CONG__imp__weaken,axiom,
! [V0x: tp__o,V1x_27: tp__o,V2y: tp__o,V3y_27: tp__o] :
( ( ( p(inj__o(V0x))
=> ( p(inj__o(V2y))
=> p(inj__o(V3y_27)) ) )
& ( ~ p(inj__o(V3y_27))
=> ( p(inj__o(V1x_27))
=> p(inj__o(V0x)) ) ) )
=> ( ( p(inj__o(V0x))
=> p(inj__o(V2y)) )
=> ( p(inj__o(V1x_27))
=> p(inj__o(V3y_27)) ) ) ) ).
tff(conj_thm_2EConseqConv_2EIMP__CONG__simple__imp__strengthen,axiom,
! [V0x: tp__o,V1x_27: tp__o,V2y: tp__o,V3y_27: tp__o] :
( ( ( p(inj__o(V0x))
=> p(inj__o(V1x_27)) )
& ( p(inj__o(V1x_27))
=> ( p(inj__o(V3y_27))
=> p(inj__o(V2y)) ) ) )
=> ( ( p(inj__o(V1x_27))
=> p(inj__o(V3y_27)) )
=> ( p(inj__o(V0x))
=> p(inj__o(V2y)) ) ) ) ).
tff(conj_thm_2EConseqConv_2EIMP__CONG__simple__imp__weaken,axiom,
! [V0x: tp__o,V1x_27: tp__o,V2y: tp__o,V3y_27: tp__o] :
( ( ( p(inj__o(V1x_27))
=> p(inj__o(V0x)) )
& ( p(inj__o(V1x_27))
=> ( p(inj__o(V2y))
=> p(inj__o(V3y_27)) ) ) )
=> ( ( p(inj__o(V0x))
=> p(inj__o(V2y)) )
=> ( p(inj__o(V1x_27))
=> p(inj__o(V3y_27)) ) ) ) ).
tff(conj_thm_2EConseqConv_2EIMP__CLAUSES__TX,axiom,
! [V0t: tp__o] :
( ( $true
=> p(inj__o(V0t)) )
<=> p(inj__o(V0t)) ) ).
tff(conj_thm_2EConseqConv_2EIMP__CLAUSES__XT,axiom,
! [V0t: tp__o] :
( ( p(inj__o(V0t))
=> $true )
<=> $true ) ).
tff(conj_thm_2EConseqConv_2EIMP__CLAUSES__FX,axiom,
! [V0t: tp__o] :
( ( $false
=> p(inj__o(V0t)) )
<=> $true ) ).
tff(conj_thm_2EConseqConv_2EIMP__CLAUSES__XX,axiom,
! [V0t: tp__o] :
( ( p(inj__o(V0t))
=> p(inj__o(V0t)) )
<=> $true ) ).
tff(conj_thm_2EConseqConv_2EIMP__CLAUSES__XF,axiom,
! [V0t: tp__o] :
( ( p(inj__o(V0t))
=> $false )
<=> ~ p(inj__o(V0t)) ) ).
tff(conj_thm_2EConseqConv_2EIMP__CONG__cond__simple,axiom,
! [V0c: tp__o,V1x: tp__o,V2x_27: tp__o,V3y: tp__o,V4y_27: tp__o] :
( ( ( p(inj__o(V2x_27))
=> p(inj__o(V1x)) )
& ( p(inj__o(V4y_27))
=> p(inj__o(V3y)) ) )
=> ( p(ap(ap(ap(c_2Ebool_2ECOND(bool),inj__o(V0c)),inj__o(V2x_27)),inj__o(V4y_27)))
=> p(ap(ap(ap(c_2Ebool_2ECOND(bool),inj__o(V0c)),inj__o(V1x)),inj__o(V3y))) ) ) ).
tff(conj_thm_2EConseqConv_2EIMP__CONG__cond,axiom,
! [V0c: tp__o,V1x: tp__o,V2x_27: tp__o,V3y: tp__o,V4y_27: tp__o] :
( ( ( p(inj__o(V0c))
=> ( p(inj__o(V2x_27))
=> p(inj__o(V1x)) ) )
& ( ~ p(inj__o(V0c))
=> ( p(inj__o(V4y_27))
=> p(inj__o(V3y)) ) ) )
=> ( p(ap(ap(ap(c_2Ebool_2ECOND(bool),inj__o(V0c)),inj__o(V2x_27)),inj__o(V4y_27)))
=> p(ap(ap(ap(c_2Ebool_2ECOND(bool),inj__o(V0c)),inj__o(V1x)),inj__o(V3y))) ) ) ).
tff(conj_thm_2EConseqConv_2ECOND__CLAUSES__CT,axiom,
! [A_27a: del,V0t1: $i] :
( mem(V0t1,A_27a)
=> ! [V1t2: $i] :
( mem(V1t2,A_27a)
=> ( ap(ap(ap(c_2Ebool_2ECOND(A_27a),inj__o(fo__c_2Ebool_2ET)),V0t1),V1t2) = V0t1 ) ) ) ).
tff(conj_thm_2EConseqConv_2ECOND__CLAUSES__CF,axiom,
! [A_27a: del,V0t1: $i] :
( mem(V0t1,A_27a)
=> ! [V1t2: $i] :
( mem(V1t2,A_27a)
=> ( ap(ap(ap(c_2Ebool_2ECOND(A_27a),inj__o(fo__c_2Ebool_2EF)),V0t1),V1t2) = V1t2 ) ) ) ).
tff(conj_thm_2EConseqConv_2ECOND__CLAUSES__ID,axiom,
! [A_27a: del,V0b: tp__o,V1t: $i] :
( mem(V1t,A_27a)
=> ( ap(ap(ap(c_2Ebool_2ECOND(A_27a),inj__o(V0b)),V1t),V1t) = V1t ) ) ).
tff(conj_thm_2EConseqConv_2ECOND__CLAUSES__TT,axiom,
! [V0c: tp__o,V1x: tp__o] :
( p(ap(ap(ap(c_2Ebool_2ECOND(bool),inj__o(V0c)),inj__o(fo__c_2Ebool_2ET)),inj__o(V1x)))
<=> ( ~ p(inj__o(V0c))
=> p(inj__o(V1x)) ) ) ).
tff(conj_thm_2EConseqConv_2ECOND__CLAUSES__FT,axiom,
! [V0c: tp__o,V1x: tp__o] :
( p(ap(ap(ap(c_2Ebool_2ECOND(bool),inj__o(V0c)),inj__o(V1x)),inj__o(fo__c_2Ebool_2ET)))
<=> ( p(inj__o(V0c))
=> p(inj__o(V1x)) ) ) ).
tff(conj_thm_2EConseqConv_2ECOND__CLAUSES__TF,axiom,
! [V0c: tp__o,V1x: tp__o] :
( p(ap(ap(ap(c_2Ebool_2ECOND(bool),inj__o(V0c)),inj__o(fo__c_2Ebool_2EF)),inj__o(V1x)))
<=> ( ~ p(inj__o(V0c))
& p(inj__o(V1x)) ) ) ).
tff(conj_thm_2EConseqConv_2ECOND__CLAUSES__FF,axiom,
! [V0c: tp__o,V1x: tp__o] :
( p(ap(ap(ap(c_2Ebool_2ECOND(bool),inj__o(V0c)),inj__o(V1x)),inj__o(fo__c_2Ebool_2EF)))
<=> ( p(inj__o(V0c))
& p(inj__o(V1x)) ) ) ).
tff(ax_thm_2EConseqConv_2EASM__MARKER__DEF,axiom,
c_2EConseqConv_2EASM__MARKER = k(bool,i(bool)) ).
tff(conj_thm_2EConseqConv_2EASM__MARKER__THM,axiom,
! [V0y: tp__o,V1x: tp__o] :
( p(ap(ap(c_2EConseqConv_2EASM__MARKER,inj__o(V0y)),inj__o(V1x)))
<=> p(inj__o(V1x)) ) ).
%------------------------------------------------------------------------------