ITP001 Axioms: ITP056_5.ax
%------------------------------------------------------------------------------
% File : ITP056_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 : transfer_2.ax [Gau20]
% : HL4056_5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 31 ( 9 unt; 10 typ; 0 def)
% Number of atoms : 265 ( 5 equ)
% Maximal formula atoms : 9 ( 8 avg)
% Number of connectives : 62 ( 0 ~; 0 |; 8 &)
% ( 8 <=>; 46 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 9 avg)
% Maximal term depth : 2 ( 1 avg)
% Number of FOOLs : 182 ( 182 fml; 0 var)
% Number of types : 1 ( 0 usr)
% Number of type conns : 26 ( 10 >; 16 *; 0 +; 0 <<)
% Number of predicates : 10 ( 9 usr; 2 prp; 0-2 aty)
% Number of functors : 10 ( 10 usr; 0 con; 2-4 aty)
% Number of variables : 98 ( 96 !; 2 ?; 98 :)
% SPC : TF0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
tff(tp_c_2Etransfer_2EFUN__REL,type,
c_2Etransfer_2EFUN__REL: ( del * del * del * del ) > $i ).
tff(mem_c_2Etransfer_2EFUN__REL,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del] : mem(c_2Etransfer_2EFUN__REL(A_27a,A_27b,A_27c,A_27d),arr(arr(A_27a,arr(A_27b,bool)),arr(arr(A_27c,arr(A_27d,bool)),arr(arr(A_27a,A_27c),arr(arr(A_27b,A_27d),bool))))) ).
tff(tp_c_2Etransfer_2EPAIR__REL,type,
c_2Etransfer_2EPAIR__REL: ( del * del * del * del ) > $i ).
tff(mem_c_2Etransfer_2EPAIR__REL,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del] : mem(c_2Etransfer_2EPAIR__REL(A_27a,A_27b,A_27c,A_27d),arr(arr(A_27a,arr(A_27b,bool)),arr(arr(A_27c,arr(A_27d,bool)),arr(ty_2Epair_2Eprod(A_27a,A_27c),arr(ty_2Epair_2Eprod(A_27b,A_27d),bool))))) ).
tff(tp_c_2Etransfer_2Ebi__unique,type,
c_2Etransfer_2Ebi__unique: ( del * del ) > $i ).
tff(mem_c_2Etransfer_2Ebi__unique,axiom,
! [A_27a: del,A_27b: del] : mem(c_2Etransfer_2Ebi__unique(A_27a,A_27b),arr(arr(A_27a,arr(A_27b,bool)),bool)) ).
tff(tp_c_2Etransfer_2Ebitotal,type,
c_2Etransfer_2Ebitotal: ( del * del ) > $i ).
tff(mem_c_2Etransfer_2Ebitotal,axiom,
! [A_27a: del,A_27b: del] : mem(c_2Etransfer_2Ebitotal(A_27a,A_27b),arr(arr(A_27a,arr(A_27b,bool)),bool)) ).
tff(tp_c_2Etransfer_2Eleft__unique,type,
c_2Etransfer_2Eleft__unique: ( del * del ) > $i ).
tff(mem_c_2Etransfer_2Eleft__unique,axiom,
! [A_27a: del,A_27b: del] : mem(c_2Etransfer_2Eleft__unique(A_27a,A_27b),arr(arr(A_27a,arr(A_27b,bool)),bool)) ).
tff(tp_c_2Etransfer_2Eright__unique,type,
c_2Etransfer_2Eright__unique: ( del * del ) > $i ).
tff(mem_c_2Etransfer_2Eright__unique,axiom,
! [A_27a: del,A_27b: del] : mem(c_2Etransfer_2Eright__unique(A_27a,A_27b),arr(arr(A_27a,arr(A_27b,bool)),bool)) ).
tff(tp_c_2Etransfer_2Esurj,type,
c_2Etransfer_2Esurj: ( del * del ) > $i ).
tff(mem_c_2Etransfer_2Esurj,axiom,
! [A_27a: del,A_27b: del] : mem(c_2Etransfer_2Esurj(A_27a,A_27b),arr(arr(A_27a,arr(A_27b,bool)),bool)) ).
tff(tp_c_2Etransfer_2Etotal,type,
c_2Etransfer_2Etotal: ( del * del ) > $i ).
tff(mem_c_2Etransfer_2Etotal,axiom,
! [A_27a: del,A_27b: del] : mem(c_2Etransfer_2Etotal(A_27a,A_27b),arr(arr(A_27a,arr(A_27b,bool)),bool)) ).
tff(ax_thm_2Etransfer_2Eright__unique__def,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27b,bool)))
=> ( p(ap(c_2Etransfer_2Eright__unique(A_27a,A_27b),V0R))
<=> ! [V1a: $i] :
( mem(V1a,A_27a)
=> ! [V2b1: $i] :
( mem(V2b1,A_27b)
=> ! [V3b2: $i] :
( mem(V3b2,A_27b)
=> ( ( p(ap(ap(V0R,V1a),V2b1))
& p(ap(ap(V0R,V1a),V3b2)) )
=> ( V2b1 = V3b2 ) ) ) ) ) ) ) ).
tff(ax_thm_2Etransfer_2Eleft__unique__def,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27b,bool)))
=> ( p(ap(c_2Etransfer_2Eleft__unique(A_27a,A_27b),V0R))
<=> ! [V1a1: $i] :
( mem(V1a1,A_27a)
=> ! [V2a2: $i] :
( mem(V2a2,A_27a)
=> ! [V3b: $i] :
( mem(V3b,A_27b)
=> ( ( p(ap(ap(V0R,V1a1),V3b))
& p(ap(ap(V0R,V2a2),V3b)) )
=> ( V1a1 = V2a2 ) ) ) ) ) ) ) ).
tff(ax_thm_2Etransfer_2Ebi__unique__def,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27b,bool)))
=> ( p(ap(c_2Etransfer_2Ebi__unique(A_27a,A_27b),V0R))
<=> ( p(ap(c_2Etransfer_2Eleft__unique(A_27a,A_27b),V0R))
& p(ap(c_2Etransfer_2Eright__unique(A_27a,A_27b),V0R)) ) ) ) ).
tff(ax_thm_2Etransfer_2Etotal__def,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27b,bool)))
=> ( p(ap(c_2Etransfer_2Etotal(A_27a,A_27b),V0R))
<=> ! [V1x: $i] :
( mem(V1x,A_27a)
=> ? [V2y: $i] :
( mem(V2y,A_27b)
& p(ap(ap(V0R,V1x),V2y)) ) ) ) ) ).
tff(ax_thm_2Etransfer_2Esurj__def,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27b,bool)))
=> ( p(ap(c_2Etransfer_2Esurj(A_27a,A_27b),V0R))
<=> ! [V1y: $i] :
( mem(V1y,A_27b)
=> ? [V2x: $i] :
( mem(V2x,A_27a)
& p(ap(ap(V0R,V2x),V1y)) ) ) ) ) ).
tff(ax_thm_2Etransfer_2Ebitotal__def,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( mem(V0R,arr(A_27a,arr(A_27b,bool)))
=> ( p(ap(c_2Etransfer_2Ebitotal(A_27a,A_27b),V0R))
<=> ( p(ap(c_2Etransfer_2Etotal(A_27a,A_27b),V0R))
& p(ap(c_2Etransfer_2Esurj(A_27a,A_27b),V0R)) ) ) ) ).
tff(ax_thm_2Etransfer_2EFUN__REL__def,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0AB: $i] :
( mem(V0AB,arr(A_27a,arr(A_27b,bool)))
=> ! [V1CD: $i] :
( mem(V1CD,arr(A_27c,arr(A_27d,bool)))
=> ! [V2f: $i] :
( mem(V2f,arr(A_27a,A_27c))
=> ! [V3g: $i] :
( mem(V3g,arr(A_27b,A_27d))
=> ( p(ap(ap(ap(ap(c_2Etransfer_2EFUN__REL(A_27a,A_27b,A_27c,A_27d),V0AB),V1CD),V2f),V3g))
<=> ! [V4a: $i] :
( mem(V4a,A_27a)
=> ! [V5b: $i] :
( mem(V5b,A_27b)
=> ( p(ap(ap(V0AB,V4a),V5b))
=> p(ap(ap(V1CD,ap(V2f,V4a)),ap(V3g,V5b))) ) ) ) ) ) ) ) ) ).
tff(conj_thm_2Etransfer_2EFUN__REL__COMB,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0AB: $i] :
( mem(V0AB,arr(A_27a,arr(A_27b,bool)))
=> ! [V1CD: $i] :
( mem(V1CD,arr(A_27c,arr(A_27d,bool)))
=> ! [V2f: $i] :
( mem(V2f,arr(A_27a,A_27c))
=> ! [V3g: $i] :
( mem(V3g,arr(A_27b,A_27d))
=> ! [V4a: $i] :
( mem(V4a,A_27a)
=> ! [V5b: $i] :
( mem(V5b,A_27b)
=> ( ( p(ap(ap(ap(ap(c_2Etransfer_2EFUN__REL(A_27a,A_27b,A_27c,A_27d),V0AB),V1CD),V2f),V3g))
& p(ap(ap(V0AB,V4a),V5b)) )
=> p(ap(ap(V1CD,ap(V2f,V4a)),ap(V3g,V5b))) ) ) ) ) ) ) ) ).
tff(lamtp_f978,type,
f978: ( del * del * $i ) > $i ).
tff(lameq_f978,axiom,
! [A_27c: del,A_27a: del,V2f: $i] :
( mem(V2f,arr(A_27a,A_27c))
=> ! [V6a: $i] : ( ap(f978(A_27c,A_27a,V2f),V6a) = ap(V2f,V6a) ) ) ).
tff(lamtp_f979,type,
f979: ( del * del * $i ) > $i ).
tff(lameq_f979,axiom,
! [A_27d: del,A_27b: del,V3g: $i] :
( mem(V3g,arr(A_27b,A_27d))
=> ! [V7b: $i] : ( ap(f979(A_27d,A_27b,V3g),V7b) = ap(V3g,V7b) ) ) ).
tff(conj_thm_2Etransfer_2EFUN__REL__ABS,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0AB: $i] :
( mem(V0AB,arr(A_27a,arr(A_27b,bool)))
=> ! [V1CD: $i] :
( mem(V1CD,arr(A_27c,arr(A_27d,bool)))
=> ! [V2f: $i] :
( mem(V2f,arr(A_27a,A_27c))
=> ! [V3g: $i] :
( mem(V3g,arr(A_27b,A_27d))
=> ( ! [V4a: $i] :
( mem(V4a,A_27a)
=> ! [V5b: $i] :
( mem(V5b,A_27b)
=> ( p(ap(ap(V0AB,V4a),V5b))
=> p(ap(ap(V1CD,ap(V2f,V4a)),ap(V3g,V5b))) ) ) )
=> p(ap(ap(ap(ap(c_2Etransfer_2EFUN__REL(A_27a,A_27b,A_27c,A_27d),V0AB),V1CD),f978(A_27c,A_27a,V2f)),f979(A_27d,A_27b,V3g))) ) ) ) ) ) ).
tff(conj_thm_2Etransfer_2EFUN__REL__EQ2,axiom,
! [A_27a: del,A_27b: del] : ( ap(ap(c_2Etransfer_2EFUN__REL(A_27a,A_27a,A_27b,A_27b),c_2Emin_2E_3D(A_27a)),c_2Emin_2E_3D(A_27b)) = c_2Emin_2E_3D(arr(A_27a,A_27b)) ) ).
tff(ax_thm_2Etransfer_2EPAIR__REL__def,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0AB: $i] :
( mem(V0AB,arr(A_27a,arr(A_27b,bool)))
=> ! [V1CD: $i] :
( mem(V1CD,arr(A_27c,arr(A_27d,bool)))
=> ! [V2a: $i] :
( mem(V2a,A_27a)
=> ! [V3c: $i] :
( mem(V3c,A_27c)
=> ! [V4b: $i] :
( mem(V4b,A_27b)
=> ! [V5d: $i] :
( mem(V5d,A_27d)
=> ( p(ap(ap(ap(ap(c_2Etransfer_2EPAIR__REL(A_27a,A_27b,A_27c,A_27d),V0AB),V1CD),ap(ap(c_2Epair_2E_2C(A_27a,A_27c),V2a),V3c)),ap(ap(c_2Epair_2E_2C(A_27b,A_27d),V4b),V5d)))
<=> ( p(ap(ap(V0AB,V2a),V4b))
& p(ap(ap(V1CD,V3c),V5d)) ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------