ITP001 Axioms: ITP069+5.ax
%------------------------------------------------------------------------------
% File : ITP069+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 : primeFactor+2.ax [Gau20]
% : HL4069+5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 11 ( 2 unt; 0 def)
% Number of atoms : 63 ( 11 equ)
% Maximal formula atoms : 14 ( 5 avg)
% Number of connectives : 52 ( 0 ~; 0 |; 16 &)
% ( 0 <=>; 36 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 7 avg)
% Maximal term depth : 5 ( 2 avg)
% Number of predicates : 3 ( 2 usr; 0 prp; 1-2 aty)
% Number of functors : 19 ( 19 usr; 12 con; 0-2 aty)
% Number of variables : 22 ( 21 !; 1 ?)
% SPC : FOF_SAT_RFO_SEQ
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
fof(mem_c_2EprimeFactor_2EPRIME__FACTORS,axiom,
mem(c_2EprimeFactor_2EPRIME__FACTORS,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))) ).
fof(conj_thm_2EprimeFactor_2EPRIME__FACTORS__EXIST,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Eprim__rec_2E_3C,c_2Enum_2E0),V0n))
=> ? [V1b] :
( mem(V1b,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))
& p(ap(c_2Ebag_2EFINITE__BAG(ty_2Enum_2Enum),V1b))
& ! [V2m] :
( mem(V2m,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Ebag_2EBAG__IN(ty_2Enum_2Enum),V2m),V1b))
=> p(ap(c_2Edivides_2Eprime,V2m)) ) )
& V0n = ap(ap(c_2Ebag_2EBAG__GEN__PROD,V1b),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))) ) ) ) ).
fof(ax_thm_2EprimeFactor_2EPRIME__FACTORS__def,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Eprim__rec_2E_3C,c_2Enum_2E0),V0n))
=> ( p(ap(c_2Ebag_2EFINITE__BAG(ty_2Enum_2Enum),ap(c_2EprimeFactor_2EPRIME__FACTORS,V0n)))
& ! [V1m] :
( mem(V1m,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Ebag_2EBAG__IN(ty_2Enum_2Enum),V1m),ap(c_2EprimeFactor_2EPRIME__FACTORS,V0n)))
=> p(ap(c_2Edivides_2Eprime,V1m)) ) )
& V0n = ap(ap(c_2Ebag_2EBAG__GEN__PROD,ap(c_2EprimeFactor_2EPRIME__FACTORS,V0n)),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))) ) ) ) ).
fof(conj_thm_2EprimeFactor_2EUNIQUE__PRIME__FACTORS,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ! [V1b1] :
( mem(V1b1,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))
=> ! [V2b2] :
( mem(V2b2,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))
=> ( ( p(ap(c_2Ebag_2EFINITE__BAG(ty_2Enum_2Enum),V1b1))
& ! [V3m] :
( mem(V3m,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Ebag_2EBAG__IN(ty_2Enum_2Enum),V3m),V1b1))
=> p(ap(c_2Edivides_2Eprime,V3m)) ) )
& V0n = ap(ap(c_2Ebag_2EBAG__GEN__PROD,V1b1),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO)))
& p(ap(c_2Ebag_2EFINITE__BAG(ty_2Enum_2Enum),V2b2))
& ! [V4m] :
( mem(V4m,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Ebag_2EBAG__IN(ty_2Enum_2Enum),V4m),V2b2))
=> p(ap(c_2Edivides_2Eprime,V4m)) ) )
& V0n = ap(ap(c_2Ebag_2EBAG__GEN__PROD,V2b2),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))) )
=> V1b1 = V2b2 ) ) ) ) ).
fof(conj_thm_2EprimeFactor_2EPRIME__FACTORIZATION,axiom,
! [V0n] :
( mem(V0n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Eprim__rec_2E_3C,c_2Enum_2E0),V0n))
=> ! [V1b] :
( mem(V1b,arr(ty_2Enum_2Enum,ty_2Enum_2Enum))
=> ( ( p(ap(c_2Ebag_2EFINITE__BAG(ty_2Enum_2Enum),V1b))
& ! [V2x] :
( mem(V2x,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Ebag_2EBAG__IN(ty_2Enum_2Enum),V2x),V1b))
=> p(ap(c_2Edivides_2Eprime,V2x)) ) )
& ap(ap(c_2Ebag_2EBAG__GEN__PROD,V1b),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))) = V0n )
=> V1b = ap(c_2EprimeFactor_2EPRIME__FACTORS,V0n) ) ) ) ) ).
fof(conj_thm_2EprimeFactor_2EPRIME__FACTORS__1,axiom,
ap(c_2EprimeFactor_2EPRIME__FACTORS,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))) = c_2Ebag_2EEMPTY__BAG(ty_2Enum_2Enum) ).
fof(conj_thm_2EprimeFactor_2EPRIME__FACTOR__DIVIDES,axiom,
! [V0x] :
( mem(V0x,ty_2Enum_2Enum)
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ( ( p(ap(ap(c_2Eprim__rec_2E_3C,c_2Enum_2E0),V1n))
& p(ap(ap(c_2Ebag_2EBAG__IN(ty_2Enum_2Enum),V0x),ap(c_2EprimeFactor_2EPRIME__FACTORS,V1n))) )
=> p(ap(ap(c_2Edivides_2Edivides,V0x),V1n)) ) ) ) ).
fof(conj_thm_2EprimeFactor_2EDIVISOR__IN__PRIME__FACTORS,axiom,
! [V0p] :
( mem(V0p,ty_2Enum_2Enum)
=> ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ( ( p(ap(ap(c_2Eprim__rec_2E_3C,c_2Enum_2E0),V1n))
& p(ap(c_2Edivides_2Eprime,V0p))
& p(ap(ap(c_2Edivides_2Edivides,V0p),V1n)) )
=> p(ap(ap(c_2Ebag_2EBAG__IN(ty_2Enum_2Enum),V0p),ap(c_2EprimeFactor_2EPRIME__FACTORS,V1n))) ) ) ) ).
fof(conj_thm_2EprimeFactor_2EPRIME__FACTORS__MULT,axiom,
! [V0a] :
( mem(V0a,ty_2Enum_2Enum)
=> ! [V1b] :
( mem(V1b,ty_2Enum_2Enum)
=> ( ( p(ap(ap(c_2Eprim__rec_2E_3C,c_2Enum_2E0),V0a))
& p(ap(ap(c_2Eprim__rec_2E_3C,c_2Enum_2E0),V1b)) )
=> ap(c_2EprimeFactor_2EPRIME__FACTORS,ap(ap(c_2Earithmetic_2E_2A,V0a),V1b)) = ap(ap(c_2Ebag_2EBAG__UNION(ty_2Enum_2Enum),ap(c_2EprimeFactor_2EPRIME__FACTORS,V0a)),ap(c_2EprimeFactor_2EPRIME__FACTORS,V1b)) ) ) ) ).
fof(conj_thm_2EprimeFactor_2EFACTORS__prime,axiom,
! [V0p] :
( mem(V0p,ty_2Enum_2Enum)
=> ( p(ap(c_2Edivides_2Eprime,V0p))
=> ap(c_2EprimeFactor_2EPRIME__FACTORS,V0p) = ap(ap(c_2Ebag_2EBAG__INSERT(ty_2Enum_2Enum),V0p),c_2Ebag_2EEMPTY__BAG(ty_2Enum_2Enum)) ) ) ).
fof(conj_thm_2EprimeFactor_2EPRIME__FACTORS__EXP,axiom,
! [V0p] :
( mem(V0p,ty_2Enum_2Enum)
=> ! [V1e] :
( mem(V1e,ty_2Enum_2Enum)
=> ( p(ap(c_2Edivides_2Eprime,V0p))
=> ap(ap(c_2EprimeFactor_2EPRIME__FACTORS,ap(ap(c_2Earithmetic_2EEXP,V0p),V1e)),V0p) = V1e ) ) ) ).
%------------------------------------------------------------------------------