TPTP Problem File: ITP006_2.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP006_2 : TPTP v9.0.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : HOL4 set theory export of thm_2EquantHeuristics_2EGUESS__RULES__WEAKEN__FORALL__POINT.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_2EquantHeuristics_2EGUESS__RULES__WEAKEN__FORALL__POINT.p [Gau19]
% : HL402501_2.p [TPAP]
% Status : Theorem
% Rating : 0.20 v9.0.0, 0.11 v8.2.0, 0.30 v8.1.0, 0.27 v7.5.0
% Syntax : Number of formulae : 91 ( 26 unt; 33 typ; 0 def)
% Number of atoms : 403 ( 18 equ)
% Maximal formula atoms : 36 ( 4 avg)
% Number of connectives : 223 ( 40 ~; 22 |; 30 &)
% ( 47 <=>; 84 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 5 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of FOOLs : 162 ( 162 fml; 0 var)
% Number of types : 4 ( 2 usr)
% Number of type conns : 34 ( 21 >; 13 *; 0 +; 0 <<)
% Number of predicates : 8 ( 5 usr; 3 prp; 0-2 aty)
% Number of functors : 29 ( 29 usr; 10 con; 0-2 aty)
% Number of variables : 120 ( 115 !; 5 ?; 120 :)
% SPC : TF0_THM_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Bugfixes in axioms and export.
%------------------------------------------------------------------------------
include('Axioms/ITP001/ITP001_2.ax').
%------------------------------------------------------------------------------
tff(stp_o,type,
tp__o: $tType ).
tff(stp_inj_o,type,
inj__o: tp__o > $i ).
tff(stp_surj_o,type,
surj__o: $i > tp__o ).
tff(stp_inj_surj_o,axiom,
! [X: tp__o] : ( surj__o(inj__o(X)) = X ) ).
tff(stp_inj_mem_o,axiom,
! [X: tp__o] : mem(inj__o(X),bool) ).
tff(stp_iso_mem_o,axiom,
! [X: $i] :
( mem(X,bool)
=> ( X = inj__o(surj__o(X)) ) ) ).
tff(tp_c_2Ebool_2ET,type,
c_2Ebool_2ET: $i ).
tff(mem_c_2Ebool_2ET,axiom,
mem(c_2Ebool_2ET,bool) ).
tff(stp_fo_c_2Ebool_2ET,type,
fo__c_2Ebool_2ET: tp__o ).
tff(stp_eq_fo_c_2Ebool_2ET,axiom,
inj__o(fo__c_2Ebool_2ET) = c_2Ebool_2ET ).
tff(ax_true_p,axiom,
p(c_2Ebool_2ET) ).
tff(tp_c_2EquantHeuristics_2EGUESS__FORALL__GAP,type,
c_2EquantHeuristics_2EGUESS__FORALL__GAP: ( del * del ) > $i ).
tff(mem_c_2EquantHeuristics_2EGUESS__FORALL__GAP,axiom,
! [A_27a: del,A_27b: del] : mem(c_2EquantHeuristics_2EGUESS__FORALL__GAP(A_27a,A_27b),arr(arr(A_27a,A_27b),arr(arr(A_27b,bool),bool))) ).
tff(tp_c_2EquantHeuristics_2EGUESS__EXISTS__GAP,type,
c_2EquantHeuristics_2EGUESS__EXISTS__GAP: ( del * del ) > $i ).
tff(mem_c_2EquantHeuristics_2EGUESS__EXISTS__GAP,axiom,
! [A_27a: del,A_27b: del] : mem(c_2EquantHeuristics_2EGUESS__EXISTS__GAP(A_27a,A_27b),arr(arr(A_27a,A_27b),arr(arr(A_27b,bool),bool))) ).
tff(tp_c_2EquantHeuristics_2EGUESS__FORALL__POINT,type,
c_2EquantHeuristics_2EGUESS__FORALL__POINT: ( del * del ) > $i ).
tff(mem_c_2EquantHeuristics_2EGUESS__FORALL__POINT,axiom,
! [A_27a: del,A_27b: del] : mem(c_2EquantHeuristics_2EGUESS__FORALL__POINT(A_27a,A_27b),arr(arr(A_27a,A_27b),arr(arr(A_27b,bool),bool))) ).
tff(tp_c_2EquantHeuristics_2EGUESS__EXISTS__POINT,type,
c_2EquantHeuristics_2EGUESS__EXISTS__POINT: ( del * del ) > $i ).
tff(mem_c_2EquantHeuristics_2EGUESS__EXISTS__POINT,axiom,
! [A_27a: del,A_27b: del] : mem(c_2EquantHeuristics_2EGUESS__EXISTS__POINT(A_27a,A_27b),arr(arr(A_27a,A_27b),arr(arr(A_27b,bool),bool))) ).
tff(tp_c_2EquantHeuristics_2EGUESS__FORALL,type,
c_2EquantHeuristics_2EGUESS__FORALL: ( del * del ) > $i ).
tff(mem_c_2EquantHeuristics_2EGUESS__FORALL,axiom,
! [A_27a: del,A_27b: del] : mem(c_2EquantHeuristics_2EGUESS__FORALL(A_27a,A_27b),arr(arr(A_27a,A_27b),arr(arr(A_27b,bool),bool))) ).
tff(tp_c_2Ebool_2E_3F,type,
c_2Ebool_2E_3F: del > $i ).
tff(mem_c_2Ebool_2E_3F,axiom,
! [A_27a: del] : mem(c_2Ebool_2E_3F(A_27a),arr(arr(A_27a,bool),bool)) ).
tff(ax_ex_p,axiom,
! [A: del,Q: $i] :
( mem(Q,arr(A,bool))
=> ( p(ap(c_2Ebool_2E_3F(A),Q))
<=> ? [X: $i] :
( mem(X,A)
& p(ap(Q,X)) ) ) ) ).
tff(tp_c_2EquantHeuristics_2EGUESS__EXISTS,type,
c_2EquantHeuristics_2EGUESS__EXISTS: ( del * del ) > $i ).
tff(mem_c_2EquantHeuristics_2EGUESS__EXISTS,axiom,
! [A_27a: del,A_27b: del] : mem(c_2EquantHeuristics_2EGUESS__EXISTS(A_27a,A_27b),arr(arr(A_27a,A_27b),arr(arr(A_27b,bool),bool))) ).
tff(tp_c_2Ebool_2EF,type,
c_2Ebool_2EF: $i ).
tff(mem_c_2Ebool_2EF,axiom,
mem(c_2Ebool_2EF,bool) ).
tff(stp_fo_c_2Ebool_2EF,type,
fo__c_2Ebool_2EF: tp__o ).
tff(stp_eq_fo_c_2Ebool_2EF,axiom,
inj__o(fo__c_2Ebool_2EF) = c_2Ebool_2EF ).
tff(ax_false_p,axiom,
~ p(c_2Ebool_2EF) ).
tff(tp_c_2Ebool_2E_5C_2F,type,
c_2Ebool_2E_5C_2F: $i ).
tff(mem_c_2Ebool_2E_5C_2F,axiom,
mem(c_2Ebool_2E_5C_2F,arr(bool,arr(bool,bool))) ).
tff(stp_fo_c_2Ebool_2E_5C_2F,type,
fo__c_2Ebool_2E_5C_2F: ( tp__o * tp__o ) > tp__o ).
tff(stp_eq_fo_c_2Ebool_2E_5C_2F,axiom,
! [X0: tp__o,X1: tp__o] : ( inj__o(fo__c_2Ebool_2E_5C_2F(X0,X1)) = ap(ap(c_2Ebool_2E_5C_2F,inj__o(X0)),inj__o(X1)) ) ).
tff(ax_or_p,axiom,
! [Q: $i] :
( mem(Q,bool)
=> ! [R: $i] :
( mem(R,bool)
=> ( p(ap(ap(c_2Ebool_2E_5C_2F,Q),R))
<=> ( p(Q)
| p(R) ) ) ) ) ).
tff(tp_c_2Ebool_2E_2F_5C,type,
c_2Ebool_2E_2F_5C: $i ).
tff(mem_c_2Ebool_2E_2F_5C,axiom,
mem(c_2Ebool_2E_2F_5C,arr(bool,arr(bool,bool))) ).
tff(stp_fo_c_2Ebool_2E_2F_5C,type,
fo__c_2Ebool_2E_2F_5C: ( tp__o * tp__o ) > tp__o ).
tff(stp_eq_fo_c_2Ebool_2E_2F_5C,axiom,
! [X0: tp__o,X1: tp__o] : ( inj__o(fo__c_2Ebool_2E_2F_5C(X0,X1)) = ap(ap(c_2Ebool_2E_2F_5C,inj__o(X0)),inj__o(X1)) ) ).
tff(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) ) ) ) ) ).
tff(tp_c_2Emin_2E_3D,type,
c_2Emin_2E_3D: del > $i ).
tff(mem_c_2Emin_2E_3D,axiom,
! [A_27a: del] : mem(c_2Emin_2E_3D(A_27a),arr(A_27a,arr(A_27a,bool))) ).
tff(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 ) ) ) ) ).
tff(tp_c_2Ebool_2E_7E,type,
c_2Ebool_2E_7E: $i ).
tff(mem_c_2Ebool_2E_7E,axiom,
mem(c_2Ebool_2E_7E,arr(bool,bool)) ).
tff(stp_fo_c_2Ebool_2E_7E,type,
fo__c_2Ebool_2E_7E: tp__o > tp__o ).
tff(stp_eq_fo_c_2Ebool_2E_7E,axiom,
! [X0: tp__o] : ( inj__o(fo__c_2Ebool_2E_7E(X0)) = ap(c_2Ebool_2E_7E,inj__o(X0)) ) ).
tff(ax_neg_p,axiom,
! [Q: $i] :
( mem(Q,bool)
=> ( p(ap(c_2Ebool_2E_7E,Q))
<=> ~ p(Q) ) ) ).
tff(tp_c_2Emin_2E_3D_3D_3E,type,
c_2Emin_2E_3D_3D_3E: $i ).
tff(mem_c_2Emin_2E_3D_3D_3E,axiom,
mem(c_2Emin_2E_3D_3D_3E,arr(bool,arr(bool,bool))) ).
tff(stp_fo_c_2Emin_2E_3D_3D_3E,type,
fo__c_2Emin_2E_3D_3D_3E: ( tp__o * tp__o ) > tp__o ).
tff(stp_eq_fo_c_2Emin_2E_3D_3D_3E,axiom,
! [X0: tp__o,X1: tp__o] : ( inj__o(fo__c_2Emin_2E_3D_3D_3E(X0,X1)) = ap(ap(c_2Emin_2E_3D_3D_3E,inj__o(X0)),inj__o(X1)) ) ).
tff(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) ) ) ) ) ).
tff(tp_c_2Ebool_2E_21,type,
c_2Ebool_2E_21: del > $i ).
tff(mem_c_2Ebool_2E_21,axiom,
! [A_27a: del] : mem(c_2Ebool_2E_21(A_27a),arr(arr(A_27a,bool),bool)) ).
tff(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)) ) ) ) ).
tff(conj_thm_2Ebool_2ETRUTH,axiom,
$true ).
tff(conj_thm_2Ebool_2EIMP__CLAUSES,axiom,
! [V0t: tp__o] :
( ( ( $true
=> p(inj__o(V0t)) )
<=> p(inj__o(V0t)) )
& ( ( p(inj__o(V0t))
=> $true )
<=> $true )
& ( ( $false
=> p(inj__o(V0t)) )
<=> $true )
& ( ( p(inj__o(V0t))
=> p(inj__o(V0t)) )
<=> $true )
& ( ( p(inj__o(V0t))
=> $false )
<=> ~ p(inj__o(V0t)) ) ) ).
tff(conj_thm_2Ebool_2ENOT__CLAUSES,axiom,
( ! [V0t: tp__o] :
( ~ ~ p(inj__o(V0t))
<=> p(inj__o(V0t)) )
& ( ~ $true
<=> $false )
& ( ~ $false
<=> $true ) ) ).
tff(conj_thm_2Ebool_2EEQ__SYM__EQ,axiom,
! [A_27a: del,V0x: $i] :
( mem(V0x,A_27a)
=> ! [V1y: $i] :
( mem(V1y,A_27a)
=> ( ( V0x = V1y )
<=> ( V1y = V0x ) ) ) ) ).
tff(conj_thm_2Ebool_2EEQ__CLAUSES,axiom,
! [V0t: tp__o] :
( ( ( $true
<=> p(inj__o(V0t)) )
<=> p(inj__o(V0t)) )
& ( ( p(inj__o(V0t))
<=> $true )
<=> p(inj__o(V0t)) )
& ( ( $false
<=> p(inj__o(V0t)) )
<=> ~ p(inj__o(V0t)) )
& ( ( p(inj__o(V0t))
<=> $false )
<=> ~ p(inj__o(V0t)) ) ) ).
tff(conj_thm_2Ebool_2EAND__IMP__INTRO,axiom,
! [V0t1: tp__o,V1t2: tp__o,V2t3: tp__o] :
( ( p(inj__o(V0t1))
=> ( p(inj__o(V1t2))
=> p(inj__o(V2t3)) ) )
<=> ( ( p(inj__o(V0t1))
& p(inj__o(V1t2)) )
=> p(inj__o(V2t3)) ) ) ).
tff(conj_thm_2Ebool_2EIMP__CONG,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(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_2EquantHeuristics_2EGUESS__REWRITES,axiom,
! [A_27a: del,A_27b: del,V0i: $i] :
( mem(V0i,arr(A_27a,A_27b))
=> ! [V1P: $i] :
( mem(V1P,arr(A_27b,bool))
=> ( ( p(ap(ap(c_2EquantHeuristics_2EGUESS__EXISTS(A_27a,A_27b),V0i),V1P))
<=> ! [V2v: $i] :
( mem(V2v,A_27b)
=> ( p(ap(V1P,V2v))
=> ? [V3fv: $i] :
( mem(V3fv,A_27a)
& p(ap(V1P,ap(V0i,V3fv))) ) ) ) )
& ( p(ap(ap(c_2EquantHeuristics_2EGUESS__FORALL(A_27a,A_27b),V0i),V1P))
<=> ! [V4v: $i] :
( mem(V4v,A_27b)
=> ( ~ p(ap(V1P,V4v))
=> ? [V5fv: $i] :
( mem(V5fv,A_27a)
& ~ p(ap(V1P,ap(V0i,V5fv))) ) ) ) )
& ! [V6i: $i] :
( mem(V6i,arr(A_27a,A_27b))
=> ! [V7P: $i] :
( mem(V7P,arr(A_27b,bool))
=> ( p(ap(ap(c_2EquantHeuristics_2EGUESS__EXISTS__POINT(A_27a,A_27b),V6i),V7P))
<=> ! [V8fv: $i] :
( mem(V8fv,A_27a)
=> p(ap(V7P,ap(V6i,V8fv))) ) ) ) )
& ! [V9i: $i] :
( mem(V9i,arr(A_27a,A_27b))
=> ! [V10P: $i] :
( mem(V10P,arr(A_27b,bool))
=> ( p(ap(ap(c_2EquantHeuristics_2EGUESS__FORALL__POINT(A_27a,A_27b),V9i),V10P))
<=> ! [V11fv: $i] :
( mem(V11fv,A_27a)
=> ~ p(ap(V10P,ap(V9i,V11fv))) ) ) ) )
& ! [V12i: $i] :
( mem(V12i,arr(A_27a,A_27b))
=> ! [V13P: $i] :
( mem(V13P,arr(A_27b,bool))
=> ( p(ap(ap(c_2EquantHeuristics_2EGUESS__EXISTS__GAP(A_27a,A_27b),V12i),V13P))
<=> ! [V14v: $i] :
( mem(V14v,A_27b)
=> ( p(ap(V13P,V14v))
=> ? [V15fv: $i] :
( mem(V15fv,A_27a)
& ( V14v = ap(V12i,V15fv) ) ) ) ) ) ) )
& ! [V16i: $i] :
( mem(V16i,arr(A_27a,A_27b))
=> ! [V17P: $i] :
( mem(V17P,arr(A_27b,bool))
=> ( p(ap(ap(c_2EquantHeuristics_2EGUESS__FORALL__GAP(A_27a,A_27b),V16i),V17P))
<=> ! [V18v: $i] :
( mem(V18v,A_27b)
=> ( ~ p(ap(V17P,V18v))
=> ? [V19fv: $i] :
( mem(V19fv,A_27a)
& ( V18v = ap(V16i,V19fv) ) ) ) ) ) ) ) ) ) ) ).
tff(conj_thm_2Esat_2ENOT__NOT,axiom,
! [V0t: tp__o] :
( ~ ~ p(inj__o(V0t))
<=> p(inj__o(V0t)) ) ).
tff(conj_thm_2Esat_2EAND__INV__IMP,axiom,
! [V0A: tp__o] :
( p(inj__o(V0A))
=> ( ~ p(inj__o(V0A))
=> $false ) ) ).
tff(conj_thm_2Esat_2EOR__DUAL2,axiom,
! [V0A: tp__o,V1B: tp__o] :
( ( ~ ( p(inj__o(V0A))
| p(inj__o(V1B)) )
=> $false )
<=> ( ( p(inj__o(V0A))
=> $false )
=> ( ~ p(inj__o(V1B))
=> $false ) ) ) ).
tff(conj_thm_2Esat_2EOR__DUAL3,axiom,
! [V0A: tp__o,V1B: tp__o] :
( ( ~ ( ~ p(inj__o(V0A))
| p(inj__o(V1B)) )
=> $false )
<=> ( p(inj__o(V0A))
=> ( ~ p(inj__o(V1B))
=> $false ) ) ) ).
tff(conj_thm_2Esat_2EAND__INV2,axiom,
! [V0A: tp__o] :
( ( ~ p(inj__o(V0A))
=> $false )
=> ( ( p(inj__o(V0A))
=> $false )
=> $false ) ) ).
tff(conj_thm_2Esat_2Edc__eq,axiom,
! [V0p: tp__o,V1q: tp__o,V2r: tp__o] :
( ( p(inj__o(V0p))
<=> ( p(inj__o(V1q))
<=> p(inj__o(V2r)) ) )
<=> ( ( p(inj__o(V0p))
| p(inj__o(V1q))
| p(inj__o(V2r)) )
& ( p(inj__o(V0p))
| ~ p(inj__o(V2r))
| ~ p(inj__o(V1q)) )
& ( p(inj__o(V1q))
| ~ p(inj__o(V2r))
| ~ p(inj__o(V0p)) )
& ( p(inj__o(V2r))
| ~ p(inj__o(V1q))
| ~ p(inj__o(V0p)) ) ) ) ).
tff(conj_thm_2Esat_2Edc__disj,axiom,
! [V0p: tp__o,V1q: tp__o,V2r: tp__o] :
( ( p(inj__o(V0p))
<=> ( p(inj__o(V1q))
| p(inj__o(V2r)) ) )
<=> ( ( p(inj__o(V0p))
| ~ p(inj__o(V1q)) )
& ( p(inj__o(V0p))
| ~ p(inj__o(V2r)) )
& ( p(inj__o(V1q))
| p(inj__o(V2r))
| ~ p(inj__o(V0p)) ) ) ) ).
tff(conj_thm_2Esat_2Edc__imp,axiom,
! [V0p: tp__o,V1q: tp__o,V2r: tp__o] :
( ( p(inj__o(V0p))
<=> ( p(inj__o(V1q))
=> p(inj__o(V2r)) ) )
<=> ( ( p(inj__o(V0p))
| p(inj__o(V1q)) )
& ( p(inj__o(V0p))
| ~ p(inj__o(V2r)) )
& ( ~ p(inj__o(V1q))
| p(inj__o(V2r))
| ~ p(inj__o(V0p)) ) ) ) ).
tff(conj_thm_2Esat_2Edc__neg,axiom,
! [V0p: tp__o,V1q: tp__o] :
( ( p(inj__o(V0p))
<=> ~ p(inj__o(V1q)) )
<=> ( ( p(inj__o(V0p))
| p(inj__o(V1q)) )
& ( ~ p(inj__o(V1q))
| ~ p(inj__o(V0p)) ) ) ) ).
tff(conj_thm_2Esat_2Epth__ni1,axiom,
! [V0p: tp__o,V1q: tp__o] :
( ~ ( p(inj__o(V0p))
=> p(inj__o(V1q)) )
=> p(inj__o(V0p)) ) ).
tff(conj_thm_2Esat_2Epth__ni2,axiom,
! [V0p: tp__o,V1q: tp__o] :
( ~ ( p(inj__o(V0p))
=> p(inj__o(V1q)) )
=> ~ p(inj__o(V1q)) ) ).
tff(conj_thm_2EquantHeuristics_2EGUESS__RULES__WEAKEN__FORALL__POINT,conjecture,
! [A_27a: del,A_27b: del,V0i: $i] :
( mem(V0i,arr(A_27b,A_27a))
=> ! [V1P: $i] :
( mem(V1P,arr(A_27a,bool))
=> ! [V2Q: $i] :
( mem(V2Q,arr(A_27a,bool))
=> ( ! [V3x: $i] :
( mem(V3x,A_27a)
=> ( p(ap(V2Q,V3x))
=> p(ap(V1P,V3x)) ) )
=> ( p(ap(ap(c_2EquantHeuristics_2EGUESS__FORALL__POINT(A_27b,A_27a),V0i),V1P))
=> p(ap(ap(c_2EquantHeuristics_2EGUESS__FORALL__POINT(A_27b,A_27a),V0i),V2Q)) ) ) ) ) ) ).
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