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 : 42 ( 2 unt; 0 def)
% Number of atoms : 265 ( 4 equ)
% Maximal formula atoms : 14 ( 6 avg)
% Number of connectives : 237 ( 14 ~; 9 |; 23 &)
% ( 27 <=>; 164 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 7 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 6 ( 3 usr; 2 prp; 0-2 aty)
% Number of functors : 9 ( 9 usr; 4 con; 0-2 aty)
% Number of variables : 91 ( 89 !; 2 ?)
% SPC : FOF_SAT_RFO_SEQ
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
fof(mem_c_2EConseqConv_2EASM__MARKER,axiom,
mem(c_2EConseqConv_2EASM__MARKER,arr(bool,arr(bool,bool))) ).
fof(conj_thm_2EConseqConv_2Eforall__eq__thm,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1Q] :
( mem(V1Q,arr(A_27a,bool))
=> ( ! [V2s] :
( mem(V2s,A_27a)
=> ( p(ap(V0P,V2s))
<=> p(ap(V1Q,V2s)) ) )
=> ( ! [V3s] :
( mem(V3s,A_27a)
=> p(ap(V0P,V3s)) )
<=> ! [V4s] :
( mem(V4s,A_27a)
=> p(ap(V1Q,V4s)) ) ) ) ) ) ) ).
fof(conj_thm_2EConseqConv_2Eexists__eq__thm,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(A_27a,bool))
=> ! [V1Q] :
( mem(V1Q,arr(A_27a,bool))
=> ( ! [V2s] :
( mem(V2s,A_27a)
=> ( p(ap(V0P,V2s))
<=> p(ap(V1Q,V2s)) ) )
=> ( ? [V3s] :
( mem(V3s,A_27a)
& p(ap(V0P,V3s)) )
<=> ? [V4s] :
( mem(V4s,A_27a)
& p(ap(V1Q,V4s)) ) ) ) ) ) ) ).
fof(conj_thm_2EConseqConv_2Etrue__imp,axiom,
! [V0t] :
( mem(V0t,bool)
=> ( p(V0t)
=> $true ) ) ).
fof(conj_thm_2EConseqConv_2Efalse__imp,axiom,
! [V0t] :
( mem(V0t,bool)
=> ( $false
=> p(V0t) ) ) ).
fof(conj_thm_2EConseqConv_2ENOT__CLAUSES__X,axiom,
! [V0t] :
( mem(V0t,bool)
=> ( ~ ~ p(V0t)
<=> p(V0t) ) ) ).
fof(conj_thm_2EConseqConv_2ENOT__CLAUSES__T,axiom,
( ~ $true
<=> $false ) ).
fof(conj_thm_2EConseqConv_2ENOT__CLAUSES__F,axiom,
( ~ $false
<=> $true ) ).
fof(conj_thm_2EConseqConv_2EIMP__CONG__conj__strengthen,axiom,
! [V0x] :
( mem(V0x,bool)
=> ! [V1x_27] :
( mem(V1x_27,bool)
=> ! [V2y] :
( mem(V2y,bool)
=> ! [V3y_27] :
( mem(V3y_27,bool)
=> ( ( ( p(V2y)
=> ( p(V1x_27)
=> p(V0x) ) )
& ( p(V1x_27)
=> ( p(V3y_27)
=> p(V2y) ) ) )
=> ( ( p(V1x_27)
& p(V3y_27) )
=> ( p(V0x)
& p(V2y) ) ) ) ) ) ) ) ).
fof(conj_thm_2EConseqConv_2EIMP__CONG__conj__weaken,axiom,
! [V0x] :
( mem(V0x,bool)
=> ! [V1x_27] :
( mem(V1x_27,bool)
=> ! [V2y] :
( mem(V2y,bool)
=> ! [V3y_27] :
( mem(V3y_27,bool)
=> ( ( ( p(V2y)
=> ( p(V0x)
=> p(V1x_27) ) )
& ( p(V1x_27)
=> ( p(V2y)
=> p(V3y_27) ) ) )
=> ( ( p(V0x)
& p(V2y) )
=> ( p(V1x_27)
& p(V3y_27) ) ) ) ) ) ) ) ).
fof(conj_thm_2EConseqConv_2EAND__CLAUSES__TX,axiom,
! [V0t] :
( mem(V0t,bool)
=> ( ( $true
& p(V0t) )
<=> p(V0t) ) ) ).
fof(conj_thm_2EConseqConv_2EAND__CLAUSES__XT,axiom,
! [V0t] :
( mem(V0t,bool)
=> ( ( p(V0t)
& $true )
<=> p(V0t) ) ) ).
fof(conj_thm_2EConseqConv_2EAND__CLAUSES__FX,axiom,
! [V0t] :
( mem(V0t,bool)
=> ( ( $false
& p(V0t) )
<=> $false ) ) ).
fof(conj_thm_2EConseqConv_2EAND__CLAUSES__XF,axiom,
! [V0t] :
( mem(V0t,bool)
=> ( ( p(V0t)
& $false )
<=> $false ) ) ).
fof(conj_thm_2EConseqConv_2EAND__CLAUSES__XX,axiom,
! [V0t] :
( mem(V0t,bool)
=> ( ( p(V0t)
& p(V0t) )
<=> p(V0t) ) ) ).
fof(conj_thm_2EConseqConv_2EIMP__CONG__disj__strengthen,axiom,
! [V0x] :
( mem(V0x,bool)
=> ! [V1x_27] :
( mem(V1x_27,bool)
=> ! [V2y] :
( mem(V2y,bool)
=> ! [V3y_27] :
( mem(V3y_27,bool)
=> ( ( ( ~ p(V2y)
=> ( p(V1x_27)
=> p(V0x) ) )
& ( ~ p(V1x_27)
=> ( p(V3y_27)
=> p(V2y) ) ) )
=> ( ( p(V1x_27)
| p(V3y_27) )
=> ( p(V0x)
| p(V2y) ) ) ) ) ) ) ) ).
fof(conj_thm_2EConseqConv_2EIMP__CONG__disj__weaken,axiom,
! [V0x] :
( mem(V0x,bool)
=> ! [V1x_27] :
( mem(V1x_27,bool)
=> ! [V2y] :
( mem(V2y,bool)
=> ! [V3y_27] :
( mem(V3y_27,bool)
=> ( ( ( ~ p(V2y)
=> ( p(V0x)
=> p(V1x_27) ) )
& ( ~ p(V1x_27)
=> ( p(V2y)
=> p(V3y_27) ) ) )
=> ( ( p(V0x)
| p(V2y) )
=> ( p(V1x_27)
| p(V3y_27) ) ) ) ) ) ) ) ).
fof(conj_thm_2EConseqConv_2EOR__CLAUSES__TX,axiom,
! [V0t] :
( mem(V0t,bool)
=> ( ( $true
| p(V0t) )
<=> $true ) ) ).
fof(conj_thm_2EConseqConv_2EOR__CLAUSES__XT,axiom,
! [V0t] :
( mem(V0t,bool)
=> ( ( p(V0t)
| $true )
<=> $true ) ) ).
fof(conj_thm_2EConseqConv_2EOR__CLAUSES__FX,axiom,
! [V0t] :
( mem(V0t,bool)
=> ( ( $false
| p(V0t) )
<=> p(V0t) ) ) ).
fof(conj_thm_2EConseqConv_2EOR__CLAUSES__XF,axiom,
! [V0t] :
( mem(V0t,bool)
=> ( ( p(V0t)
| $false )
<=> p(V0t) ) ) ).
fof(conj_thm_2EConseqConv_2EOR__CLAUSES__XX,axiom,
! [V0t] :
( mem(V0t,bool)
=> ( ( p(V0t)
| p(V0t) )
<=> p(V0t) ) ) ).
fof(conj_thm_2EConseqConv_2EIMP__CONG__imp__strengthen,axiom,
! [V0x] :
( mem(V0x,bool)
=> ! [V1x_27] :
( mem(V1x_27,bool)
=> ! [V2y] :
( mem(V2y,bool)
=> ! [V3y_27] :
( mem(V3y_27,bool)
=> ( ( ( p(V0x)
=> ( p(V3y_27)
=> p(V2y) ) )
& ( ~ p(V3y_27)
=> ( p(V0x)
=> p(V1x_27) ) ) )
=> ( ( p(V1x_27)
=> p(V3y_27) )
=> ( p(V0x)
=> p(V2y) ) ) ) ) ) ) ) ).
fof(conj_thm_2EConseqConv_2EIMP__CONG__imp__weaken,axiom,
! [V0x] :
( mem(V0x,bool)
=> ! [V1x_27] :
( mem(V1x_27,bool)
=> ! [V2y] :
( mem(V2y,bool)
=> ! [V3y_27] :
( mem(V3y_27,bool)
=> ( ( ( p(V0x)
=> ( p(V2y)
=> p(V3y_27) ) )
& ( ~ p(V3y_27)
=> ( p(V1x_27)
=> p(V0x) ) ) )
=> ( ( p(V0x)
=> p(V2y) )
=> ( p(V1x_27)
=> p(V3y_27) ) ) ) ) ) ) ) ).
fof(conj_thm_2EConseqConv_2EIMP__CONG__simple__imp__strengthen,axiom,
! [V0x] :
( mem(V0x,bool)
=> ! [V1x_27] :
( mem(V1x_27,bool)
=> ! [V2y] :
( mem(V2y,bool)
=> ! [V3y_27] :
( mem(V3y_27,bool)
=> ( ( ( p(V0x)
=> p(V1x_27) )
& ( p(V1x_27)
=> ( p(V3y_27)
=> p(V2y) ) ) )
=> ( ( p(V1x_27)
=> p(V3y_27) )
=> ( p(V0x)
=> p(V2y) ) ) ) ) ) ) ) ).
fof(conj_thm_2EConseqConv_2EIMP__CONG__simple__imp__weaken,axiom,
! [V0x] :
( mem(V0x,bool)
=> ! [V1x_27] :
( mem(V1x_27,bool)
=> ! [V2y] :
( mem(V2y,bool)
=> ! [V3y_27] :
( mem(V3y_27,bool)
=> ( ( ( p(V1x_27)
=> p(V0x) )
& ( p(V1x_27)
=> ( p(V2y)
=> p(V3y_27) ) ) )
=> ( ( p(V0x)
=> p(V2y) )
=> ( p(V1x_27)
=> p(V3y_27) ) ) ) ) ) ) ) ).
fof(conj_thm_2EConseqConv_2EIMP__CLAUSES__TX,axiom,
! [V0t] :
( mem(V0t,bool)
=> ( ( $true
=> p(V0t) )
<=> p(V0t) ) ) ).
fof(conj_thm_2EConseqConv_2EIMP__CLAUSES__XT,axiom,
! [V0t] :
( mem(V0t,bool)
=> ( ( p(V0t)
=> $true )
<=> $true ) ) ).
fof(conj_thm_2EConseqConv_2EIMP__CLAUSES__FX,axiom,
! [V0t] :
( mem(V0t,bool)
=> ( ( $false
=> p(V0t) )
<=> $true ) ) ).
fof(conj_thm_2EConseqConv_2EIMP__CLAUSES__XX,axiom,
! [V0t] :
( mem(V0t,bool)
=> ( ( p(V0t)
=> p(V0t) )
<=> $true ) ) ).
fof(conj_thm_2EConseqConv_2EIMP__CLAUSES__XF,axiom,
! [V0t] :
( mem(V0t,bool)
=> ( ( p(V0t)
=> $false )
<=> ~ p(V0t) ) ) ).
fof(conj_thm_2EConseqConv_2EIMP__CONG__cond__simple,axiom,
! [V0c] :
( mem(V0c,bool)
=> ! [V1x] :
( mem(V1x,bool)
=> ! [V2x_27] :
( mem(V2x_27,bool)
=> ! [V3y] :
( mem(V3y,bool)
=> ! [V4y_27] :
( mem(V4y_27,bool)
=> ( ( ( p(V2x_27)
=> p(V1x) )
& ( p(V4y_27)
=> p(V3y) ) )
=> ( p(ap(ap(ap(c_2Ebool_2ECOND(bool),V0c),V2x_27),V4y_27))
=> p(ap(ap(ap(c_2Ebool_2ECOND(bool),V0c),V1x),V3y)) ) ) ) ) ) ) ) ).
fof(conj_thm_2EConseqConv_2EIMP__CONG__cond,axiom,
! [V0c] :
( mem(V0c,bool)
=> ! [V1x] :
( mem(V1x,bool)
=> ! [V2x_27] :
( mem(V2x_27,bool)
=> ! [V3y] :
( mem(V3y,bool)
=> ! [V4y_27] :
( mem(V4y_27,bool)
=> ( ( ( p(V0c)
=> ( p(V2x_27)
=> p(V1x) ) )
& ( ~ p(V0c)
=> ( p(V4y_27)
=> p(V3y) ) ) )
=> ( p(ap(ap(ap(c_2Ebool_2ECOND(bool),V0c),V2x_27),V4y_27))
=> p(ap(ap(ap(c_2Ebool_2ECOND(bool),V0c),V1x),V3y)) ) ) ) ) ) ) ) ).
fof(conj_thm_2EConseqConv_2ECOND__CLAUSES__CT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0t1] :
( mem(V0t1,A_27a)
=> ! [V1t2] :
( mem(V1t2,A_27a)
=> ap(ap(ap(c_2Ebool_2ECOND(A_27a),c_2Ebool_2ET),V0t1),V1t2) = V0t1 ) ) ) ).
fof(conj_thm_2EConseqConv_2ECOND__CLAUSES__CF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0t1] :
( mem(V0t1,A_27a)
=> ! [V1t2] :
( mem(V1t2,A_27a)
=> ap(ap(ap(c_2Ebool_2ECOND(A_27a),c_2Ebool_2EF),V0t1),V1t2) = V1t2 ) ) ) ).
fof(conj_thm_2EConseqConv_2ECOND__CLAUSES__ID,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0b] :
( mem(V0b,bool)
=> ! [V1t] :
( mem(V1t,A_27a)
=> ap(ap(ap(c_2Ebool_2ECOND(A_27a),V0b),V1t),V1t) = V1t ) ) ) ).
fof(conj_thm_2EConseqConv_2ECOND__CLAUSES__TT,axiom,
! [V0c] :
( mem(V0c,bool)
=> ! [V1x] :
( mem(V1x,bool)
=> ( p(ap(ap(ap(c_2Ebool_2ECOND(bool),V0c),c_2Ebool_2ET),V1x))
<=> ( ~ p(V0c)
=> p(V1x) ) ) ) ) ).
fof(conj_thm_2EConseqConv_2ECOND__CLAUSES__FT,axiom,
! [V0c] :
( mem(V0c,bool)
=> ! [V1x] :
( mem(V1x,bool)
=> ( p(ap(ap(ap(c_2Ebool_2ECOND(bool),V0c),V1x),c_2Ebool_2ET))
<=> ( p(V0c)
=> p(V1x) ) ) ) ) ).
fof(conj_thm_2EConseqConv_2ECOND__CLAUSES__TF,axiom,
! [V0c] :
( mem(V0c,bool)
=> ! [V1x] :
( mem(V1x,bool)
=> ( p(ap(ap(ap(c_2Ebool_2ECOND(bool),V0c),c_2Ebool_2EF),V1x))
<=> ( ~ p(V0c)
& p(V1x) ) ) ) ) ).
fof(conj_thm_2EConseqConv_2ECOND__CLAUSES__FF,axiom,
! [V0c] :
( mem(V0c,bool)
=> ! [V1x] :
( mem(V1x,bool)
=> ( p(ap(ap(ap(c_2Ebool_2ECOND(bool),V0c),V1x),c_2Ebool_2EF))
<=> ( p(V0c)
& p(V1x) ) ) ) ) ).
fof(ax_thm_2EConseqConv_2EASM__MARKER__DEF,axiom,
c_2EConseqConv_2EASM__MARKER = k(bool,i(bool)) ).
fof(conj_thm_2EConseqConv_2EASM__MARKER__THM,axiom,
! [V0y] :
( mem(V0y,bool)
=> ! [V1x] :
( mem(V1x,bool)
=> ( p(ap(ap(c_2EConseqConv_2EASM__MARKER,V0y),V1x))
<=> p(V1x) ) ) ) ).
%------------------------------------------------------------------------------