ITP001 Axioms: ITP028+5.ax
%------------------------------------------------------------------------------
% File : ITP028+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 : fixedPoint+2.ax [Gau20]
% : HL4028+5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 33 ( 0 unt; 0 def)
% Number of atoms : 160 ( 14 equ)
% Maximal formula atoms : 8 ( 4 avg)
% Number of connectives : 127 ( 0 ~; 0 |; 12 &)
% ( 3 <=>; 112 =>; 0 <=; 0 <~>)
% Maximal formula depth : 13 ( 8 avg)
% Maximal term depth : 7 ( 2 avg)
% Number of predicates : 4 ( 3 usr; 0 prp; 1-2 aty)
% Number of functors : 22 ( 22 usr; 1 con; 0-2 aty)
% Number of variables : 95 ( 95 !; 0 ?)
% SPC : FOF_SAT_RFO_SEQ
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
fof(mem_c_2EfixedPoint_2Eclosed,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2EfixedPoint_2Eclosed(A_27a),arr(arr(arr(A_27a,bool),arr(A_27a,bool)),arr(arr(A_27a,bool),bool))) ) ).
fof(mem_c_2EfixedPoint_2Edense,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2EfixedPoint_2Edense(A_27a),arr(arr(arr(A_27a,bool),arr(A_27a,bool)),arr(arr(A_27a,bool),bool))) ) ).
fof(mem_c_2EfixedPoint_2Eempty,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2EfixedPoint_2Eempty(A_27a,A_27b),arr(A_27a,arr(A_27b,bool))) ) ) ).
fof(mem_c_2EfixedPoint_2Efnsum,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2EfixedPoint_2Efnsum(A_27a,A_27b),arr(arr(A_27b,arr(A_27a,bool)),arr(arr(A_27b,arr(A_27a,bool)),arr(A_27b,arr(A_27a,bool))))) ) ) ).
fof(mem_c_2EfixedPoint_2Egfp,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2EfixedPoint_2Egfp(A_27a),arr(arr(arr(A_27a,bool),arr(A_27a,bool)),arr(A_27a,bool))) ) ).
fof(mem_c_2EfixedPoint_2Elfp,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2EfixedPoint_2Elfp(A_27a),arr(arr(arr(A_27a,bool),arr(A_27a,bool)),arr(A_27a,bool))) ) ).
fof(mem_c_2EfixedPoint_2Emonotone,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2EfixedPoint_2Emonotone(A_27a,A_27b),arr(arr(arr(A_27a,bool),arr(A_27b,bool)),bool)) ) ) ).
fof(ax_thm_2EfixedPoint_2Emonotone__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(arr(A_27a,bool),arr(A_27b,bool)))
=> ( p(ap(c_2EfixedPoint_2Emonotone(A_27a,A_27b),V0f))
<=> ! [V1X] :
( mem(V1X,arr(A_27a,bool))
=> ! [V2Y] :
( mem(V2Y,arr(A_27a,bool))
=> ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1X),V2Y))
=> p(ap(ap(c_2Epred__set_2ESUBSET(A_27b),ap(V0f,V1X)),ap(V0f,V2Y))) ) ) ) ) ) ) ) ).
fof(lameq_f310,axiom,
! [A_27a,V0f] :
( mem(V0f,arr(arr(A_27a,bool),arr(A_27a,bool)))
=> ! [V1X] : ap(f310(A_27a,V0f),V1X) = ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),bool),V1X),ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(V0f,V1X)),V1X)) ) ).
fof(ax_thm_2EfixedPoint_2Elfp__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(arr(A_27a,bool),arr(A_27a,bool)))
=> ap(c_2EfixedPoint_2Elfp(A_27a),V0f) = ap(c_2Epred__set_2EBIGINTER(A_27a),ap(c_2Epred__set_2EGSPEC(arr(A_27a,bool),arr(A_27a,bool)),f310(A_27a,V0f))) ) ) ).
fof(lameq_f311,axiom,
! [A_27a,V0f] :
( mem(V0f,arr(arr(A_27a,bool),arr(A_27a,bool)))
=> ! [V1X] : ap(f311(A_27a,V0f),V1X) = ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),bool),V1X),ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1X),ap(V0f,V1X))) ) ).
fof(ax_thm_2EfixedPoint_2Egfp__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(arr(A_27a,bool),arr(A_27a,bool)))
=> ap(c_2EfixedPoint_2Egfp(A_27a),V0f) = ap(c_2Epred__set_2EBIGUNION(A_27a),ap(c_2Epred__set_2EGSPEC(arr(A_27a,bool),arr(A_27a,bool)),f311(A_27a,V0f))) ) ) ).
fof(ax_thm_2EfixedPoint_2Eclosed__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(arr(A_27a,bool),arr(A_27a,bool)))
=> ! [V1X] :
( mem(V1X,arr(A_27a,bool))
=> ( p(ap(ap(c_2EfixedPoint_2Eclosed(A_27a),V0f),V1X))
<=> p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(V0f,V1X)),V1X)) ) ) ) ) ).
fof(ax_thm_2EfixedPoint_2Edense__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(arr(A_27a,bool),arr(A_27a,bool)))
=> ! [V1X] :
( mem(V1X,arr(A_27a,bool))
=> ( p(ap(ap(c_2EfixedPoint_2Edense(A_27a),V0f),V1X))
<=> p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1X),ap(V0f,V1X))) ) ) ) ) ).
fof(conj_thm_2EfixedPoint_2Elfp__least__closed,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(arr(A_27a,bool),arr(A_27a,bool)))
=> ( p(ap(c_2EfixedPoint_2Emonotone(A_27a,A_27a),V0f))
=> ( p(ap(ap(c_2EfixedPoint_2Eclosed(A_27a),V0f),ap(c_2EfixedPoint_2Elfp(A_27a),V0f)))
& ! [V1X] :
( mem(V1X,arr(A_27a,bool))
=> ( p(ap(ap(c_2EfixedPoint_2Eclosed(A_27a),V0f),V1X))
=> p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(c_2EfixedPoint_2Elfp(A_27a),V0f)),V1X)) ) ) ) ) ) ) ).
fof(conj_thm_2EfixedPoint_2Egfp__greatest__dense,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(arr(A_27a,bool),arr(A_27a,bool)))
=> ( p(ap(c_2EfixedPoint_2Emonotone(A_27a,A_27a),V0f))
=> ( p(ap(ap(c_2EfixedPoint_2Edense(A_27a),V0f),ap(c_2EfixedPoint_2Egfp(A_27a),V0f)))
& ! [V1X] :
( mem(V1X,arr(A_27a,bool))
=> ( p(ap(ap(c_2EfixedPoint_2Edense(A_27a),V0f),V1X))
=> p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1X),ap(c_2EfixedPoint_2Egfp(A_27a),V0f))) ) ) ) ) ) ) ).
fof(conj_thm_2EfixedPoint_2Elfp__fixedpoint,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(arr(A_27a,bool),arr(A_27a,bool)))
=> ( p(ap(c_2EfixedPoint_2Emonotone(A_27a,A_27a),V0f))
=> ( ap(c_2EfixedPoint_2Elfp(A_27a),V0f) = ap(V0f,ap(c_2EfixedPoint_2Elfp(A_27a),V0f))
& ! [V1X] :
( mem(V1X,arr(A_27a,bool))
=> ( V1X = ap(V0f,V1X)
=> p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(c_2EfixedPoint_2Elfp(A_27a),V0f)),V1X)) ) ) ) ) ) ) ).
fof(conj_thm_2EfixedPoint_2Egfp__greatest__fixedpoint,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(arr(A_27a,bool),arr(A_27a,bool)))
=> ( p(ap(c_2EfixedPoint_2Emonotone(A_27a,A_27a),V0f))
=> ( ap(c_2EfixedPoint_2Egfp(A_27a),V0f) = ap(V0f,ap(c_2EfixedPoint_2Egfp(A_27a),V0f))
& ! [V1X] :
( mem(V1X,arr(A_27a,bool))
=> ( V1X = ap(V0f,V1X)
=> p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1X),ap(c_2EfixedPoint_2Egfp(A_27a),V0f))) ) ) ) ) ) ) ).
fof(conj_thm_2EfixedPoint_2Elfp__induction,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(arr(A_27a,bool),arr(A_27a,bool)))
=> ( p(ap(c_2EfixedPoint_2Emonotone(A_27a,A_27a),V0f))
=> ! [V1X] :
( mem(V1X,arr(A_27a,bool))
=> ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(V0f,V1X)),V1X))
=> p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(c_2EfixedPoint_2Elfp(A_27a),V0f)),V1X)) ) ) ) ) ) ).
fof(conj_thm_2EfixedPoint_2Egfp__coinduction,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(arr(A_27a,bool),arr(A_27a,bool)))
=> ( p(ap(c_2EfixedPoint_2Emonotone(A_27a,A_27a),V0f))
=> ! [V1X] :
( mem(V1X,arr(A_27a,bool))
=> ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1X),ap(V0f,V1X)))
=> p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1X),ap(c_2EfixedPoint_2Egfp(A_27a),V0f))) ) ) ) ) ) ).
fof(conj_thm_2EfixedPoint_2Elfp__strong__induction,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(arr(A_27a,bool),arr(A_27a,bool)))
=> ( p(ap(c_2EfixedPoint_2Emonotone(A_27a,A_27a),V0f))
=> ! [V1X] :
( mem(V1X,arr(A_27a,bool))
=> ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(V0f,ap(ap(c_2Epred__set_2EINTER(A_27a),V1X),ap(c_2EfixedPoint_2Elfp(A_27a),V0f)))),V1X))
=> p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(c_2EfixedPoint_2Elfp(A_27a),V0f)),V1X)) ) ) ) ) ) ).
fof(conj_thm_2EfixedPoint_2Egfp__strong__coinduction,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(arr(A_27a,bool),arr(A_27a,bool)))
=> ( p(ap(c_2EfixedPoint_2Emonotone(A_27a,A_27a),V0f))
=> ! [V1X] :
( mem(V1X,arr(A_27a,bool))
=> ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1X),ap(V0f,ap(ap(c_2Epred__set_2EUNION(A_27a),V1X),ap(c_2EfixedPoint_2Egfp(A_27a),V0f)))))
=> p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1X),ap(c_2EfixedPoint_2Egfp(A_27a),V0f))) ) ) ) ) ) ).
fof(ax_thm_2EfixedPoint_2Efnsum__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f1] :
( mem(V0f1,arr(A_27b,arr(A_27a,bool)))
=> ! [V1f2] :
( mem(V1f2,arr(A_27b,arr(A_27a,bool)))
=> ! [V2X] :
( mem(V2X,A_27b)
=> ap(ap(ap(c_2EfixedPoint_2Efnsum(A_27a,A_27b),V0f1),V1f2),V2X) = ap(ap(c_2Epred__set_2EUNION(A_27a),ap(V0f1,V2X)),ap(V1f2,V2X)) ) ) ) ) ) ).
fof(conj_thm_2EfixedPoint_2Efnsum__monotone,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f1] :
( mem(V0f1,arr(arr(A_27a,bool),arr(A_27b,bool)))
=> ! [V1f2] :
( mem(V1f2,arr(arr(A_27a,bool),arr(A_27b,bool)))
=> ( ( p(ap(c_2EfixedPoint_2Emonotone(A_27a,A_27b),V0f1))
& p(ap(c_2EfixedPoint_2Emonotone(A_27a,A_27b),V1f2)) )
=> p(ap(c_2EfixedPoint_2Emonotone(A_27a,A_27b),ap(ap(c_2EfixedPoint_2Efnsum(A_27b,arr(A_27a,bool)),V0f1),V1f2))) ) ) ) ) ) ).
fof(ax_thm_2EfixedPoint_2Eempty__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> c_2EfixedPoint_2Eempty(A_27a,A_27b) = k(A_27a,c_2Epred__set_2EEMPTY(A_27b)) ) ) ).
fof(conj_thm_2EfixedPoint_2Eempty__monotone,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> p(ap(c_2EfixedPoint_2Emonotone(A_27a,A_27b),c_2EfixedPoint_2Eempty(arr(A_27a,bool),A_27b))) ) ) ).
fof(conj_thm_2EfixedPoint_2Efnsum__empty,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,arr(A_27b,bool)))
=> ( ap(ap(c_2EfixedPoint_2Efnsum(A_27b,A_27a),V0f),c_2EfixedPoint_2Eempty(A_27a,A_27b)) = V0f
& ap(ap(c_2EfixedPoint_2Efnsum(A_27b,A_27a),c_2EfixedPoint_2Eempty(A_27a,A_27b)),V0f) = V0f ) ) ) ) ).
fof(conj_thm_2EfixedPoint_2Efnsum__ASSOC,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,arr(A_27b,bool)))
=> ! [V1g] :
( mem(V1g,arr(A_27a,arr(A_27b,bool)))
=> ! [V2h] :
( mem(V2h,arr(A_27a,arr(A_27b,bool)))
=> ap(ap(c_2EfixedPoint_2Efnsum(A_27b,A_27a),V0f),ap(ap(c_2EfixedPoint_2Efnsum(A_27b,A_27a),V1g),V2h)) = ap(ap(c_2EfixedPoint_2Efnsum(A_27b,A_27a),ap(ap(c_2EfixedPoint_2Efnsum(A_27b,A_27a),V0f),V1g)),V2h) ) ) ) ) ) ).
fof(conj_thm_2EfixedPoint_2Efnsum__COMM,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,arr(A_27b,bool)))
=> ! [V1g] :
( mem(V1g,arr(A_27a,arr(A_27b,bool)))
=> ap(ap(c_2EfixedPoint_2Efnsum(A_27b,A_27a),V0f),V1g) = ap(ap(c_2EfixedPoint_2Efnsum(A_27b,A_27a),V1g),V0f) ) ) ) ) ).
fof(conj_thm_2EfixedPoint_2Efnsum__SUBSET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(A_27a,arr(A_27b,bool)))
=> ! [V1g] :
( mem(V1g,arr(A_27a,arr(A_27b,bool)))
=> ! [V2X] :
( mem(V2X,A_27a)
=> ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27b),ap(V0f,V2X)),ap(ap(ap(c_2EfixedPoint_2Efnsum(A_27b,A_27a),V0f),V1g),V2X)))
& p(ap(ap(c_2Epred__set_2ESUBSET(A_27b),ap(V1g,V2X)),ap(ap(ap(c_2EfixedPoint_2Efnsum(A_27b,A_27a),V0f),V1g),V2X))) ) ) ) ) ) ) ).
fof(conj_thm_2EfixedPoint_2Elfp__fnsum,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f1] :
( mem(V0f1,arr(arr(A_27a,bool),arr(A_27a,bool)))
=> ! [V1f2] :
( mem(V1f2,arr(arr(A_27a,bool),arr(A_27a,bool)))
=> ( ( p(ap(c_2EfixedPoint_2Emonotone(A_27a,A_27a),V0f1))
& p(ap(c_2EfixedPoint_2Emonotone(A_27a,A_27a),V1f2)) )
=> ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(c_2EfixedPoint_2Elfp(A_27a),V0f1)),ap(c_2EfixedPoint_2Elfp(A_27a),ap(ap(c_2EfixedPoint_2Efnsum(A_27a,arr(A_27a,bool)),V0f1),V1f2))))
& p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(c_2EfixedPoint_2Elfp(A_27a),V1f2)),ap(c_2EfixedPoint_2Elfp(A_27a),ap(ap(c_2EfixedPoint_2Efnsum(A_27a,arr(A_27a,bool)),V0f1),V1f2)))) ) ) ) ) ) ).
fof(conj_thm_2EfixedPoint_2Elfp__rule__applied,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(arr(A_27a,bool),arr(A_27a,bool)))
=> ! [V1X] :
( mem(V1X,arr(A_27a,bool))
=> ! [V2y] :
( mem(V2y,A_27a)
=> ( ( p(ap(c_2EfixedPoint_2Emonotone(A_27a,A_27a),V0f))
& p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1X),ap(c_2EfixedPoint_2Elfp(A_27a),V0f)))
& p(ap(ap(c_2Ebool_2EIN(A_27a),V2y),ap(V0f,V1X))) )
=> p(ap(ap(c_2Ebool_2EIN(A_27a),V2y),ap(c_2EfixedPoint_2Elfp(A_27a),V0f))) ) ) ) ) ) ).
fof(conj_thm_2EfixedPoint_2Elfp__empty,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(arr(A_27a,bool),arr(A_27a,bool)))
=> ! [V1x] :
( mem(V1x,A_27a)
=> ( ( p(ap(c_2EfixedPoint_2Emonotone(A_27a,A_27a),V0f))
& p(ap(ap(c_2Ebool_2EIN(A_27a),V1x),ap(V0f,c_2Epred__set_2EEMPTY(A_27a)))) )
=> p(ap(ap(c_2Ebool_2EIN(A_27a),V1x),ap(c_2EfixedPoint_2Elfp(A_27a),V0f))) ) ) ) ) ).
%------------------------------------------------------------------------------