TPTP Problem File: ITP004^2.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : ITP004^2 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : HOL4 set theory export of thm_2Epred__set_2EREST__SUBSET.p, bushy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau19] Gauthier (2019), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : thm_2Epred__set_2EREST__SUBSET.p [Gau19]
% : HL401501^2.p [TPAP]
% Status : Theorem
% Rating : 0.40 v8.2.0, 0.46 v8.1.0, 0.36 v7.5.0
% Syntax : Number of formulae : 45 ( 1 unt; 19 typ; 0 def)
% Number of atoms : 167 ( 8 equ; 0 cnn)
% Maximal formula atoms : 16 ( 6 avg)
% Number of connectives : 300 ( 2 ~; 0 |; 2 &; 260 @)
% ( 7 <=>; 29 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 8 avg)
% Number of types : 3 ( 1 usr)
% Number of type conns : 20 ( 20 >; 0 *; 0 +; 0 <<)
% Number of symbols : 24 ( 23 usr; 10 con; 0-2 aty)
% Number of variables : 49 ( 0 ^; 49 !; 0 ?; 49 :)
% SPC : TH0_THM_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Bugfixes in axioms and export.
%------------------------------------------------------------------------------
include('Axioms/ITP001/ITP001^2.ax').
%------------------------------------------------------------------------------
thf(tp_c_2Emin_2E_3D_3D_3E,type,
c_2Emin_2E_3D_3D_3E: $i ).
thf(mem_c_2Emin_2E_3D_3D_3E,axiom,
mem @ c_2Emin_2E_3D_3D_3E @ ( arr @ bool @ ( arr @ bool @ bool ) ) ).
thf(ax_imp_p,axiom,
! [Q: $i] :
( ( mem @ Q @ bool )
=> ! [R: $i] :
( ( mem @ R @ bool )
=> ( ( p @ ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ Q ) @ R ) )
<=> ( ( p @ Q )
=> ( p @ R ) ) ) ) ) ).
thf(tp_c_2Epred__set_2ESUBSET,type,
c_2Epred__set_2ESUBSET: del > $i ).
thf(mem_c_2Epred__set_2ESUBSET,axiom,
! [A_27a: del] : ( mem @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) ) ).
thf(tp_c_2Ebool_2E_7E,type,
c_2Ebool_2E_7E: $i ).
thf(mem_c_2Ebool_2E_7E,axiom,
mem @ c_2Ebool_2E_7E @ ( arr @ bool @ bool ) ).
thf(ax_neg_p,axiom,
! [Q: $i] :
( ( mem @ Q @ bool )
=> ( ( p @ ( ap @ c_2Ebool_2E_7E @ Q ) )
<=> ~ ( p @ Q ) ) ) ).
thf(tp_c_2Ebool_2E_2F_5C,type,
c_2Ebool_2E_2F_5C: $i ).
thf(mem_c_2Ebool_2E_2F_5C,axiom,
mem @ c_2Ebool_2E_2F_5C @ ( arr @ bool @ ( arr @ bool @ bool ) ) ).
thf(ax_and_p,axiom,
! [Q: $i] :
( ( mem @ Q @ bool )
=> ! [R: $i] :
( ( mem @ R @ bool )
=> ( ( p @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ Q ) @ R ) )
<=> ( ( p @ Q )
& ( p @ R ) ) ) ) ) ).
thf(tp_c_2Ebool_2EIN,type,
c_2Ebool_2EIN: del > $i ).
thf(mem_c_2Ebool_2EIN,axiom,
! [A_27a: del] : ( mem @ ( c_2Ebool_2EIN @ A_27a ) @ ( arr @ A_27a @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) ) ).
thf(tp_c_2Epred__set_2ECHOICE,type,
c_2Epred__set_2ECHOICE: del > $i ).
thf(mem_c_2Epred__set_2ECHOICE,axiom,
! [A_27a: del] : ( mem @ ( c_2Epred__set_2ECHOICE @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ A_27a ) ) ).
thf(tp_c_2Epred__set_2EDELETE,type,
c_2Epred__set_2EDELETE: del > $i ).
thf(mem_c_2Epred__set_2EDELETE,axiom,
! [A_27a: del] : ( mem @ ( c_2Epred__set_2EDELETE @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) ) ).
thf(tp_c_2Epred__set_2EREST,type,
c_2Epred__set_2EREST: del > $i ).
thf(mem_c_2Epred__set_2EREST,axiom,
! [A_27a: del] : ( mem @ ( c_2Epred__set_2EREST @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ bool ) ) ) ).
thf(tp_c_2Emin_2E_3D,type,
c_2Emin_2E_3D: del > $i ).
thf(mem_c_2Emin_2E_3D,axiom,
! [A_27a: del] : ( mem @ ( c_2Emin_2E_3D @ A_27a ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) ).
thf(ax_eq_p,axiom,
! [A: del,X: $i] :
( ( mem @ X @ A )
=> ! [Y: $i] :
( ( mem @ Y @ A )
=> ( ( p @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A ) @ X ) @ Y ) )
<=> ( X = Y ) ) ) ) ).
thf(tp_c_2Ebool_2E_21,type,
c_2Ebool_2E_21: del > $i ).
thf(mem_c_2Ebool_2E_21,axiom,
! [A_27a: del] : ( mem @ ( c_2Ebool_2E_21 @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) ).
thf(ax_all_p,axiom,
! [A: del,Q: $i] :
( ( mem @ Q @ ( arr @ A @ bool ) )
=> ( ( p @ ( ap @ ( c_2Ebool_2E_21 @ A ) @ Q ) )
<=> ! [X: $i] :
( ( mem @ X @ A )
=> ( p @ ( ap @ Q @ X ) ) ) ) ) ).
thf(ax_thm_2Epred__set_2ESUBSET__DEF,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V0s ) @ V1t ) )
<=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V0s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1t ) ) ) ) ) ) ) ).
thf(conj_thm_2Epred__set_2EIN__DELETE,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1x ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ V0s ) @ V2y ) ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1x ) @ V0s ) )
& ( V1x != V2y ) ) ) ) ) ) ).
thf(ax_thm_2Epred__set_2EREST__DEF,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( c_2Epred__set_2EREST @ A_27a ) @ V0s )
= ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ V0s ) @ ( ap @ ( c_2Epred__set_2ECHOICE @ A_27a ) @ V0s ) ) ) ) ).
thf(conj_thm_2Epred__set_2EREST__SUBSET,conjecture,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( c_2Epred__set_2EREST @ A_27a ) @ V0s ) ) @ V0s ) ) ) ).
%------------------------------------------------------------------------------