ITP001 Axioms: ITP072_5.ax
%------------------------------------------------------------------------------
% File : ITP072_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 : quotient_pair_2.ax [Gau20]
% : HL4072_5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 29 ( 3 unt; 5 typ; 0 def)
% Number of atoms : 911 ( 15 equ)
% Maximal formula atoms : 23 ( 31 avg)
% Number of connectives : 210 ( 0 ~; 0 |; 4 &)
% ( 3 <=>; 203 =>; 0 <=; 0 <~>)
% Maximal formula depth : 47 ( 21 avg)
% Maximal term depth : 2 ( 1 avg)
% Number of FOOLs : 677 ( 677 fml; 0 var)
% Number of types : 1 ( 0 usr)
% Number of type conns : 34 ( 5 >; 29 *; 0 +; 0 <<)
% Number of predicates : 21 ( 20 usr; 3 prp; 0-4 aty)
% Number of functors : 5 ( 5 usr; 0 con; 4-9 aty)
% Number of variables : 268 ( 268 !; 0 ?; 268 :)
% SPC : TF0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
tff(tp_c_2Equotient__pair_2E_23_23_23,type,
c_2Equotient__pair_2E_23_23_23: ( del * del * del * del ) > $i ).
tff(mem_c_2Equotient__pair_2E_23_23_23,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del] : mem(c_2Equotient__pair_2E_23_23_23(A_27a,A_27b,A_27c,A_27d),arr(arr(A_27a,arr(A_27c,bool)),arr(arr(A_27b,arr(A_27d,bool)),arr(ty_2Epair_2Eprod(A_27a,A_27b),arr(ty_2Epair_2Eprod(A_27c,A_27d),bool))))) ).
tff(conj_thm_2Equotient__pair_2EPAIR__MAP__I,axiom,
! [A_27a: del,A_27b: del] : ( ap(ap(c_2Epair_2E_23_23(A_27a,A_27b,A_27a,A_27b),c_2Ecombin_2EI(A_27a)),c_2Ecombin_2EI(A_27b)) = c_2Ecombin_2EI(ty_2Epair_2Eprod(A_27a,A_27b)) ) ).
tff(lamtp_f2019,type,
f2019: ( del * del * del * del * $i * $i * $i * $i * $i ) > $i ).
tff(lameq_f2019,axiom,
! [A_27b: del,A_27c: del,A_27a: del,A_27d: del,V2a: $i] :
( mem(V2a,A_27a)
=> ! [V0R1: $i] :
( mem(V0R1,arr(A_27a,arr(A_27c,bool)))
=> ! [V4c: $i] :
( mem(V4c,A_27c)
=> ! [V3b: $i] :
( mem(V3b,A_27b)
=> ! [V1R2: $i] :
( mem(V1R2,arr(A_27b,arr(A_27d,bool)))
=> ! [V5d: $i] : ( ap(f2019(A_27b,A_27c,A_27a,A_27d,V2a,V0R1,V4c,V3b,V1R2),V5d) = ap(ap(c_2Ebool_2E_2F_5C,ap(ap(V0R1,V2a),V4c)),ap(ap(V1R2,V3b),V5d)) ) ) ) ) ) ) ).
tff(lamtp_f2020,type,
f2020: ( del * del * del * del * $i * $i * $i * $i ) > $i ).
tff(lameq_f2020,axiom,
! [A_27d: del,A_27a: del,A_27b: del,A_27c: del,V2a: $i] :
( mem(V2a,A_27a)
=> ! [V0R1: $i] :
( mem(V0R1,arr(A_27a,arr(A_27c,bool)))
=> ! [V3b: $i] :
( mem(V3b,A_27b)
=> ! [V1R2: $i] :
( mem(V1R2,arr(A_27b,arr(A_27d,bool)))
=> ! [V4c: $i] : ( ap(f2020(A_27d,A_27a,A_27b,A_27c,V2a,V0R1,V3b,V1R2),V4c) = f2019(A_27b,A_27c,A_27a,A_27d,V2a,V0R1,V4c,V3b,V1R2) ) ) ) ) ) ).
tff(lamtp_f2021,type,
f2021: ( del * del * del * del * $i * $i * $i ) > $i ).
tff(lameq_f2021,axiom,
! [A_27c: del,A_27d: del,A_27a: del,A_27b: del,V1R2: $i] :
( mem(V1R2,arr(A_27b,arr(A_27d,bool)))
=> ! [V0R1: $i] :
( mem(V0R1,arr(A_27a,arr(A_27c,bool)))
=> ! [V2a: $i] :
( mem(V2a,A_27a)
=> ! [V3b: $i] : ( ap(f2021(A_27c,A_27d,A_27a,A_27b,V1R2,V0R1,V2a),V3b) = ap(c_2Epair_2EUNCURRY(A_27c,A_27d,bool),f2020(A_27d,A_27a,A_27b,A_27c,V2a,V0R1,V3b,V1R2)) ) ) ) ) ).
tff(lamtp_f2022,type,
f2022: ( del * del * del * del * $i * $i ) > $i ).
tff(lameq_f2022,axiom,
! [A_27b: del,A_27d: del,A_27c: del,A_27a: del,V1R2: $i] :
( mem(V1R2,arr(A_27b,arr(A_27d,bool)))
=> ! [V0R1: $i] :
( mem(V0R1,arr(A_27a,arr(A_27c,bool)))
=> ! [V2a: $i] : ( ap(f2022(A_27b,A_27d,A_27c,A_27a,V1R2,V0R1),V2a) = f2021(A_27c,A_27d,A_27a,A_27b,V1R2,V0R1,V2a) ) ) ) ).
tff(ax_thm_2Equotient__pair_2EPAIR__REL,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0R1: $i] :
( mem(V0R1,arr(A_27a,arr(A_27c,bool)))
=> ! [V1R2: $i] :
( mem(V1R2,arr(A_27b,arr(A_27d,bool)))
=> ( ap(ap(c_2Equotient__pair_2E_23_23_23(A_27a,A_27b,A_27c,A_27d),V0R1),V1R2) = ap(c_2Epair_2EUNCURRY(A_27a,A_27b,arr(ty_2Epair_2Eprod(A_27c,A_27d),bool)),f2022(A_27b,A_27d,A_27c,A_27a,V1R2,V0R1)) ) ) ) ).
tff(conj_thm_2Equotient__pair_2EPAIR__REL__THM,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0R1: $i] :
( mem(V0R1,arr(A_27a,arr(A_27c,bool)))
=> ! [V1R2: $i] :
( mem(V1R2,arr(A_27b,arr(A_27d,bool)))
=> ! [V2a: $i] :
( mem(V2a,A_27a)
=> ! [V3b: $i] :
( mem(V3b,A_27b)
=> ! [V4c: $i] :
( mem(V4c,A_27c)
=> ! [V5d: $i] :
( mem(V5d,A_27d)
=> ( p(ap(ap(ap(ap(c_2Equotient__pair_2E_23_23_23(A_27a,A_27b,A_27c,A_27d),V0R1),V1R2),ap(ap(c_2Epair_2E_2C(A_27a,A_27b),V2a),V3b)),ap(ap(c_2Epair_2E_2C(A_27c,A_27d),V4c),V5d)))
<=> ( p(ap(ap(V0R1,V2a),V4c))
& p(ap(ap(V1R2,V3b),V5d)) ) ) ) ) ) ) ) ) ).
tff(conj_thm_2Equotient__pair_2EPAIR__REL__EQ,axiom,
! [A_27a: del,A_27b: del] : ( ap(ap(c_2Equotient__pair_2E_23_23_23(A_27a,A_27b,A_27a,A_27b),c_2Emin_2E_3D(A_27a)),c_2Emin_2E_3D(A_27b)) = c_2Emin_2E_3D(ty_2Epair_2Eprod(A_27a,A_27b)) ) ).
tff(conj_thm_2Equotient__pair_2EPAIR__REL__REFL,axiom,
! [A_27a: del,A_27b: del,V0R1: $i] :
( mem(V0R1,arr(A_27a,arr(A_27a,bool)))
=> ! [V1R2: $i] :
( mem(V1R2,arr(A_27b,arr(A_27b,bool)))
=> ( ( ! [V2x: $i] :
( mem(V2x,A_27a)
=> ! [V3y: $i] :
( mem(V3y,A_27a)
=> ( p(ap(ap(V0R1,V2x),V3y))
<=> ( ap(V0R1,V2x) = ap(V0R1,V3y) ) ) ) )
& ! [V4x: $i] :
( mem(V4x,A_27b)
=> ! [V5y: $i] :
( mem(V5y,A_27b)
=> ( p(ap(ap(V1R2,V4x),V5y))
<=> ( ap(V1R2,V4x) = ap(V1R2,V5y) ) ) ) ) )
=> ! [V6x: $i] :
( mem(V6x,ty_2Epair_2Eprod(A_27a,A_27b))
=> p(ap(ap(ap(ap(c_2Equotient__pair_2E_23_23_23(A_27a,A_27b,A_27a,A_27b),V0R1),V1R2),V6x),V6x)) ) ) ) ) ).
tff(conj_thm_2Equotient__pair_2EPAIR__EQUIV,axiom,
! [A_27a: del,A_27b: del,V0R1: $i] :
( mem(V0R1,arr(A_27a,arr(A_27a,bool)))
=> ! [V1R2: $i] :
( mem(V1R2,arr(A_27b,arr(A_27b,bool)))
=> ( p(ap(c_2Equotient_2EEQUIV(A_27a),V0R1))
=> ( p(ap(c_2Equotient_2EEQUIV(A_27b),V1R2))
=> p(ap(c_2Equotient_2EEQUIV(ty_2Epair_2Eprod(A_27a,A_27b)),ap(ap(c_2Equotient__pair_2E_23_23_23(A_27a,A_27b,A_27a,A_27b),V0R1),V1R2))) ) ) ) ) ).
tff(conj_thm_2Equotient__pair_2EPAIR__QUOTIENT,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0R1: $i] :
( mem(V0R1,arr(A_27a,arr(A_27a,bool)))
=> ! [V1abs1: $i] :
( mem(V1abs1,arr(A_27a,A_27c))
=> ! [V2rep1: $i] :
( mem(V2rep1,arr(A_27c,A_27a))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27a,A_27c),V0R1),V1abs1),V2rep1))
=> ! [V3R2: $i] :
( mem(V3R2,arr(A_27b,arr(A_27b,bool)))
=> ! [V4abs2: $i] :
( mem(V4abs2,arr(A_27b,A_27d))
=> ! [V5rep2: $i] :
( mem(V5rep2,arr(A_27d,A_27b))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27b,A_27d),V3R2),V4abs2),V5rep2))
=> p(ap(ap(ap(c_2Equotient_2EQUOTIENT(ty_2Epair_2Eprod(A_27a,A_27b),ty_2Epair_2Eprod(A_27c,A_27d)),ap(ap(c_2Equotient__pair_2E_23_23_23(A_27a,A_27b,A_27a,A_27b),V0R1),V3R2)),ap(ap(c_2Epair_2E_23_23(A_27a,A_27b,A_27c,A_27d),V1abs1),V4abs2)),ap(ap(c_2Epair_2E_23_23(A_27c,A_27d,A_27a,A_27b),V2rep1),V5rep2))) ) ) ) ) ) ) ) ) ).
tff(conj_thm_2Equotient__pair_2EFST__PRS,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0R1: $i] :
( mem(V0R1,arr(A_27a,arr(A_27a,bool)))
=> ! [V1abs1: $i] :
( mem(V1abs1,arr(A_27a,A_27c))
=> ! [V2rep1: $i] :
( mem(V2rep1,arr(A_27c,A_27a))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27a,A_27c),V0R1),V1abs1),V2rep1))
=> ! [V3R2: $i] :
( mem(V3R2,arr(A_27b,arr(A_27b,bool)))
=> ! [V4abs2: $i] :
( mem(V4abs2,arr(A_27b,A_27d))
=> ! [V5rep2: $i] :
( mem(V5rep2,arr(A_27d,A_27b))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27b,A_27d),V3R2),V4abs2),V5rep2))
=> ! [V6p: $i] :
( mem(V6p,ty_2Epair_2Eprod(A_27c,A_27d))
=> ( ap(c_2Epair_2EFST(A_27c,A_27d),V6p) = ap(V1abs1,ap(c_2Epair_2EFST(A_27a,A_27b),ap(ap(ap(c_2Epair_2E_23_23(A_27c,A_27d,A_27a,A_27b),V2rep1),V5rep2),V6p))) ) ) ) ) ) ) ) ) ) ) ).
tff(conj_thm_2Equotient__pair_2EFST__RSP,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0R1: $i] :
( mem(V0R1,arr(A_27a,arr(A_27a,bool)))
=> ! [V1abs1: $i] :
( mem(V1abs1,arr(A_27a,A_27c))
=> ! [V2rep1: $i] :
( mem(V2rep1,arr(A_27c,A_27a))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27a,A_27c),V0R1),V1abs1),V2rep1))
=> ! [V3R2: $i] :
( mem(V3R2,arr(A_27b,arr(A_27b,bool)))
=> ! [V4abs2: $i] :
( mem(V4abs2,arr(A_27b,A_27d))
=> ! [V5rep2: $i] :
( mem(V5rep2,arr(A_27d,A_27b))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27b,A_27d),V3R2),V4abs2),V5rep2))
=> ! [V6p1: $i] :
( mem(V6p1,ty_2Epair_2Eprod(A_27a,A_27b))
=> ! [V7p2: $i] :
( mem(V7p2,ty_2Epair_2Eprod(A_27a,A_27b))
=> ( p(ap(ap(ap(ap(c_2Equotient__pair_2E_23_23_23(A_27a,A_27b,A_27a,A_27b),V0R1),V3R2),V6p1),V7p2))
=> p(ap(ap(V0R1,ap(c_2Epair_2EFST(A_27a,A_27b),V6p1)),ap(c_2Epair_2EFST(A_27a,A_27b),V7p2))) ) ) ) ) ) ) ) ) ) ) ) ).
tff(conj_thm_2Equotient__pair_2ESND__PRS,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0R1: $i] :
( mem(V0R1,arr(A_27a,arr(A_27a,bool)))
=> ! [V1abs1: $i] :
( mem(V1abs1,arr(A_27a,A_27c))
=> ! [V2rep1: $i] :
( mem(V2rep1,arr(A_27c,A_27a))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27a,A_27c),V0R1),V1abs1),V2rep1))
=> ! [V3R2: $i] :
( mem(V3R2,arr(A_27b,arr(A_27b,bool)))
=> ! [V4abs2: $i] :
( mem(V4abs2,arr(A_27b,A_27d))
=> ! [V5rep2: $i] :
( mem(V5rep2,arr(A_27d,A_27b))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27b,A_27d),V3R2),V4abs2),V5rep2))
=> ! [V6p: $i] :
( mem(V6p,ty_2Epair_2Eprod(A_27c,A_27d))
=> ( ap(c_2Epair_2ESND(A_27c,A_27d),V6p) = ap(V4abs2,ap(c_2Epair_2ESND(A_27a,A_27b),ap(ap(ap(c_2Epair_2E_23_23(A_27c,A_27d,A_27a,A_27b),V2rep1),V5rep2),V6p))) ) ) ) ) ) ) ) ) ) ) ).
tff(conj_thm_2Equotient__pair_2ESND__RSP,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0R1: $i] :
( mem(V0R1,arr(A_27a,arr(A_27a,bool)))
=> ! [V1abs1: $i] :
( mem(V1abs1,arr(A_27a,A_27c))
=> ! [V2rep1: $i] :
( mem(V2rep1,arr(A_27c,A_27a))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27a,A_27c),V0R1),V1abs1),V2rep1))
=> ! [V3R2: $i] :
( mem(V3R2,arr(A_27b,arr(A_27b,bool)))
=> ! [V4abs2: $i] :
( mem(V4abs2,arr(A_27b,A_27d))
=> ! [V5rep2: $i] :
( mem(V5rep2,arr(A_27d,A_27b))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27b,A_27d),V3R2),V4abs2),V5rep2))
=> ! [V6p1: $i] :
( mem(V6p1,ty_2Epair_2Eprod(A_27a,A_27b))
=> ! [V7p2: $i] :
( mem(V7p2,ty_2Epair_2Eprod(A_27a,A_27b))
=> ( p(ap(ap(ap(ap(c_2Equotient__pair_2E_23_23_23(A_27a,A_27b,A_27a,A_27b),V0R1),V3R2),V6p1),V7p2))
=> p(ap(ap(V3R2,ap(c_2Epair_2ESND(A_27a,A_27b),V6p1)),ap(c_2Epair_2ESND(A_27a,A_27b),V7p2))) ) ) ) ) ) ) ) ) ) ) ) ).
tff(conj_thm_2Equotient__pair_2ECOMMA__PRS,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0R1: $i] :
( mem(V0R1,arr(A_27a,arr(A_27a,bool)))
=> ! [V1abs1: $i] :
( mem(V1abs1,arr(A_27a,A_27c))
=> ! [V2rep1: $i] :
( mem(V2rep1,arr(A_27c,A_27a))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27a,A_27c),V0R1),V1abs1),V2rep1))
=> ! [V3R2: $i] :
( mem(V3R2,arr(A_27b,arr(A_27b,bool)))
=> ! [V4abs2: $i] :
( mem(V4abs2,arr(A_27b,A_27d))
=> ! [V5rep2: $i] :
( mem(V5rep2,arr(A_27d,A_27b))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27b,A_27d),V3R2),V4abs2),V5rep2))
=> ! [V6a: $i] :
( mem(V6a,A_27c)
=> ! [V7b: $i] :
( mem(V7b,A_27d)
=> ( ap(ap(c_2Epair_2E_2C(A_27c,A_27d),V6a),V7b) = ap(ap(ap(c_2Epair_2E_23_23(A_27a,A_27b,A_27c,A_27d),V1abs1),V4abs2),ap(ap(c_2Epair_2E_2C(A_27a,A_27b),ap(V2rep1,V6a)),ap(V5rep2,V7b))) ) ) ) ) ) ) ) ) ) ) ) ).
tff(conj_thm_2Equotient__pair_2ECOMMA__RSP,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0R1: $i] :
( mem(V0R1,arr(A_27a,arr(A_27a,bool)))
=> ! [V1abs1: $i] :
( mem(V1abs1,arr(A_27a,A_27c))
=> ! [V2rep1: $i] :
( mem(V2rep1,arr(A_27c,A_27a))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27a,A_27c),V0R1),V1abs1),V2rep1))
=> ! [V3R2: $i] :
( mem(V3R2,arr(A_27b,arr(A_27b,bool)))
=> ! [V4abs2: $i] :
( mem(V4abs2,arr(A_27b,A_27d))
=> ! [V5rep2: $i] :
( mem(V5rep2,arr(A_27d,A_27b))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27b,A_27d),V3R2),V4abs2),V5rep2))
=> ! [V6a1: $i] :
( mem(V6a1,A_27a)
=> ! [V7a2: $i] :
( mem(V7a2,A_27b)
=> ! [V8b1: $i] :
( mem(V8b1,A_27a)
=> ! [V9b2: $i] :
( mem(V9b2,A_27b)
=> ( ( p(ap(ap(V0R1,V6a1),V8b1))
& p(ap(ap(V3R2,V7a2),V9b2)) )
=> p(ap(ap(ap(ap(c_2Equotient__pair_2E_23_23_23(A_27a,A_27b,A_27a,A_27b),V0R1),V3R2),ap(ap(c_2Epair_2E_2C(A_27a,A_27b),V6a1),V7a2)),ap(ap(c_2Epair_2E_2C(A_27a,A_27b),V8b1),V9b2))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
tff(conj_thm_2Equotient__pair_2ECURRY__PRS,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,A_27e: del,A_27f: del,V0R1: $i] :
( mem(V0R1,arr(A_27a,arr(A_27a,bool)))
=> ! [V1abs1: $i] :
( mem(V1abs1,arr(A_27a,A_27d))
=> ! [V2rep1: $i] :
( mem(V2rep1,arr(A_27d,A_27a))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27a,A_27d),V0R1),V1abs1),V2rep1))
=> ! [V3R2: $i] :
( mem(V3R2,arr(A_27b,arr(A_27b,bool)))
=> ! [V4abs2: $i] :
( mem(V4abs2,arr(A_27b,A_27e))
=> ! [V5rep2: $i] :
( mem(V5rep2,arr(A_27e,A_27b))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27b,A_27e),V3R2),V4abs2),V5rep2))
=> ! [V6R3: $i] :
( mem(V6R3,arr(A_27c,arr(A_27c,bool)))
=> ! [V7abs3: $i] :
( mem(V7abs3,arr(A_27c,A_27f))
=> ! [V8rep3: $i] :
( mem(V8rep3,arr(A_27f,A_27c))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27c,A_27f),V6R3),V7abs3),V8rep3))
=> ! [V9f: $i] :
( mem(V9f,arr(ty_2Epair_2Eprod(A_27d,A_27e),A_27f))
=> ! [V10a: $i] :
( mem(V10a,A_27d)
=> ! [V11b: $i] :
( mem(V11b,A_27e)
=> ( ap(ap(ap(c_2Epair_2ECURRY(A_27d,A_27e,A_27f),V9f),V10a),V11b) = ap(V7abs3,ap(ap(ap(c_2Epair_2ECURRY(A_27a,A_27b,A_27c),ap(ap(ap(c_2Equotient_2E_2D_2D_3E(ty_2Epair_2Eprod(A_27a,A_27b),A_27f,ty_2Epair_2Eprod(A_27d,A_27e),A_27c),ap(ap(c_2Epair_2E_23_23(A_27a,A_27b,A_27d,A_27e),V1abs1),V4abs2)),V8rep3),V9f)),ap(V2rep1,V10a)),ap(V5rep2,V11b))) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
tff(conj_thm_2Equotient__pair_2ECURRY__RSP,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,A_27e: del,A_27f: del,V0R1: $i] :
( mem(V0R1,arr(A_27a,arr(A_27a,bool)))
=> ! [V1abs1: $i] :
( mem(V1abs1,arr(A_27a,A_27d))
=> ! [V2rep1: $i] :
( mem(V2rep1,arr(A_27d,A_27a))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27a,A_27d),V0R1),V1abs1),V2rep1))
=> ! [V3R2: $i] :
( mem(V3R2,arr(A_27b,arr(A_27b,bool)))
=> ! [V4abs2: $i] :
( mem(V4abs2,arr(A_27b,A_27e))
=> ! [V5rep2: $i] :
( mem(V5rep2,arr(A_27e,A_27b))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27b,A_27e),V3R2),V4abs2),V5rep2))
=> ! [V6R3: $i] :
( mem(V6R3,arr(A_27c,arr(A_27c,bool)))
=> ! [V7abs3: $i] :
( mem(V7abs3,arr(A_27c,A_27f))
=> ! [V8rep3: $i] :
( mem(V8rep3,arr(A_27f,A_27c))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27c,A_27f),V6R3),V7abs3),V8rep3))
=> ! [V9f1: $i] :
( mem(V9f1,arr(ty_2Epair_2Eprod(A_27a,A_27b),A_27c))
=> ! [V10f2: $i] :
( mem(V10f2,arr(ty_2Epair_2Eprod(A_27a,A_27b),A_27c))
=> ( p(ap(ap(ap(ap(c_2Equotient_2E_3D_3D_3D_3E(ty_2Epair_2Eprod(A_27a,A_27b),A_27c),ap(ap(c_2Equotient__pair_2E_23_23_23(A_27a,A_27b,A_27a,A_27b),V0R1),V3R2)),V6R3),V9f1),V10f2))
=> p(ap(ap(ap(ap(c_2Equotient_2E_3D_3D_3D_3E(A_27a,arr(A_27b,A_27c)),V0R1),ap(ap(c_2Equotient_2E_3D_3D_3D_3E(A_27b,A_27c),V3R2),V6R3)),ap(c_2Epair_2ECURRY(A_27a,A_27b,A_27c),V9f1)),ap(c_2Epair_2ECURRY(A_27a,A_27b,A_27c),V10f2))) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
tff(conj_thm_2Equotient__pair_2EUNCURRY__PRS,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,A_27e: del,A_27f: del,V0R1: $i] :
( mem(V0R1,arr(A_27a,arr(A_27a,bool)))
=> ! [V1abs1: $i] :
( mem(V1abs1,arr(A_27a,A_27d))
=> ! [V2rep1: $i] :
( mem(V2rep1,arr(A_27d,A_27a))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27a,A_27d),V0R1),V1abs1),V2rep1))
=> ! [V3R2: $i] :
( mem(V3R2,arr(A_27b,arr(A_27b,bool)))
=> ! [V4abs2: $i] :
( mem(V4abs2,arr(A_27b,A_27e))
=> ! [V5rep2: $i] :
( mem(V5rep2,arr(A_27e,A_27b))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27b,A_27e),V3R2),V4abs2),V5rep2))
=> ! [V6R3: $i] :
( mem(V6R3,arr(A_27c,arr(A_27c,bool)))
=> ! [V7abs3: $i] :
( mem(V7abs3,arr(A_27c,A_27f))
=> ! [V8rep3: $i] :
( mem(V8rep3,arr(A_27f,A_27c))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27c,A_27f),V6R3),V7abs3),V8rep3))
=> ! [V9f: $i] :
( mem(V9f,arr(A_27d,arr(A_27e,A_27f)))
=> ! [V10p: $i] :
( mem(V10p,ty_2Epair_2Eprod(A_27d,A_27e))
=> ( ap(ap(c_2Epair_2EUNCURRY(A_27d,A_27e,A_27f),V9f),V10p) = ap(V7abs3,ap(ap(c_2Epair_2EUNCURRY(A_27a,A_27b,A_27c),ap(ap(ap(c_2Equotient_2E_2D_2D_3E(A_27a,arr(A_27e,A_27f),A_27d,arr(A_27b,A_27c)),V1abs1),ap(ap(c_2Equotient_2E_2D_2D_3E(A_27b,A_27f,A_27e,A_27c),V4abs2),V8rep3)),V9f)),ap(ap(ap(c_2Epair_2E_23_23(A_27d,A_27e,A_27a,A_27b),V2rep1),V5rep2),V10p))) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
tff(conj_thm_2Equotient__pair_2EUNCURRY__RSP,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,A_27e: del,A_27f: del,V0R1: $i] :
( mem(V0R1,arr(A_27a,arr(A_27a,bool)))
=> ! [V1abs1: $i] :
( mem(V1abs1,arr(A_27a,A_27d))
=> ! [V2rep1: $i] :
( mem(V2rep1,arr(A_27d,A_27a))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27a,A_27d),V0R1),V1abs1),V2rep1))
=> ! [V3R2: $i] :
( mem(V3R2,arr(A_27b,arr(A_27b,bool)))
=> ! [V4abs2: $i] :
( mem(V4abs2,arr(A_27b,A_27e))
=> ! [V5rep2: $i] :
( mem(V5rep2,arr(A_27e,A_27b))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27b,A_27e),V3R2),V4abs2),V5rep2))
=> ! [V6R3: $i] :
( mem(V6R3,arr(A_27c,arr(A_27c,bool)))
=> ! [V7abs3: $i] :
( mem(V7abs3,arr(A_27c,A_27f))
=> ! [V8rep3: $i] :
( mem(V8rep3,arr(A_27f,A_27c))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27c,A_27f),V6R3),V7abs3),V8rep3))
=> ! [V9f1: $i] :
( mem(V9f1,arr(A_27a,arr(A_27b,A_27c)))
=> ! [V10f2: $i] :
( mem(V10f2,arr(A_27a,arr(A_27b,A_27c)))
=> ( p(ap(ap(ap(ap(c_2Equotient_2E_3D_3D_3D_3E(A_27a,arr(A_27b,A_27c)),V0R1),ap(ap(c_2Equotient_2E_3D_3D_3D_3E(A_27b,A_27c),V3R2),V6R3)),V9f1),V10f2))
=> p(ap(ap(ap(ap(c_2Equotient_2E_3D_3D_3D_3E(ty_2Epair_2Eprod(A_27a,A_27b),A_27c),ap(ap(c_2Equotient__pair_2E_23_23_23(A_27a,A_27b,A_27a,A_27b),V0R1),V3R2)),V6R3),ap(c_2Epair_2EUNCURRY(A_27a,A_27b,A_27c),V9f1)),ap(c_2Epair_2EUNCURRY(A_27a,A_27b,A_27c),V10f2))) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
tff(conj_thm_2Equotient__pair_2EPAIR__MAP__PRS,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,A_27e: del,A_27f: del,A_27g: del,A_27h: del,V0R1: $i] :
( mem(V0R1,arr(A_27a,arr(A_27a,bool)))
=> ! [V1abs1: $i] :
( mem(V1abs1,arr(A_27a,A_27e))
=> ! [V2rep1: $i] :
( mem(V2rep1,arr(A_27e,A_27a))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27a,A_27e),V0R1),V1abs1),V2rep1))
=> ! [V3R2: $i] :
( mem(V3R2,arr(A_27b,arr(A_27b,bool)))
=> ! [V4abs2: $i] :
( mem(V4abs2,arr(A_27b,A_27f))
=> ! [V5rep2: $i] :
( mem(V5rep2,arr(A_27f,A_27b))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27b,A_27f),V3R2),V4abs2),V5rep2))
=> ! [V6R3: $i] :
( mem(V6R3,arr(A_27c,arr(A_27c,bool)))
=> ! [V7abs3: $i] :
( mem(V7abs3,arr(A_27c,A_27g))
=> ! [V8rep3: $i] :
( mem(V8rep3,arr(A_27g,A_27c))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27c,A_27g),V6R3),V7abs3),V8rep3))
=> ! [V9R4: $i] :
( mem(V9R4,arr(A_27d,arr(A_27d,bool)))
=> ! [V10abs4: $i] :
( mem(V10abs4,arr(A_27d,A_27h))
=> ! [V11rep4: $i] :
( mem(V11rep4,arr(A_27h,A_27d))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27d,A_27h),V9R4),V10abs4),V11rep4))
=> ! [V12f: $i] :
( mem(V12f,arr(A_27e,A_27f))
=> ! [V13g: $i] :
( mem(V13g,arr(A_27g,A_27h))
=> ( ap(ap(c_2Epair_2E_23_23(A_27e,A_27g,A_27f,A_27h),V12f),V13g) = ap(ap(ap(c_2Equotient_2E_2D_2D_3E(ty_2Epair_2Eprod(A_27e,A_27g),ty_2Epair_2Eprod(A_27b,A_27d),ty_2Epair_2Eprod(A_27a,A_27c),ty_2Epair_2Eprod(A_27f,A_27h)),ap(ap(c_2Epair_2E_23_23(A_27e,A_27g,A_27a,A_27c),V2rep1),V8rep3)),ap(ap(c_2Epair_2E_23_23(A_27b,A_27d,A_27f,A_27h),V4abs2),V10abs4)),ap(ap(c_2Epair_2E_23_23(A_27a,A_27c,A_27b,A_27d),ap(ap(ap(c_2Equotient_2E_2D_2D_3E(A_27a,A_27f,A_27e,A_27b),V1abs1),V5rep2),V12f)),ap(ap(ap(c_2Equotient_2E_2D_2D_3E(A_27c,A_27h,A_27g,A_27d),V7abs3),V11rep4),V13g))) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
tff(conj_thm_2Equotient__pair_2EPAIR__MAP__RSP,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,A_27e: del,A_27f: del,A_27g: del,A_27h: del,V0R1: $i] :
( mem(V0R1,arr(A_27a,arr(A_27a,bool)))
=> ! [V1abs1: $i] :
( mem(V1abs1,arr(A_27a,A_27e))
=> ! [V2rep1: $i] :
( mem(V2rep1,arr(A_27e,A_27a))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27a,A_27e),V0R1),V1abs1),V2rep1))
=> ! [V3R2: $i] :
( mem(V3R2,arr(A_27b,arr(A_27b,bool)))
=> ! [V4abs2: $i] :
( mem(V4abs2,arr(A_27b,A_27f))
=> ! [V5rep2: $i] :
( mem(V5rep2,arr(A_27f,A_27b))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27b,A_27f),V3R2),V4abs2),V5rep2))
=> ! [V6R3: $i] :
( mem(V6R3,arr(A_27c,arr(A_27c,bool)))
=> ! [V7abs3: $i] :
( mem(V7abs3,arr(A_27c,A_27g))
=> ! [V8rep3: $i] :
( mem(V8rep3,arr(A_27g,A_27c))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27c,A_27g),V6R3),V7abs3),V8rep3))
=> ! [V9R4: $i] :
( mem(V9R4,arr(A_27d,arr(A_27d,bool)))
=> ! [V10abs4: $i] :
( mem(V10abs4,arr(A_27d,A_27h))
=> ! [V11rep4: $i] :
( mem(V11rep4,arr(A_27h,A_27d))
=> ( p(ap(ap(ap(c_2Equotient_2EQUOTIENT(A_27d,A_27h),V9R4),V10abs4),V11rep4))
=> ! [V12f1: $i] :
( mem(V12f1,arr(A_27a,A_27b))
=> ! [V13f2: $i] :
( mem(V13f2,arr(A_27a,A_27b))
=> ! [V14g1: $i] :
( mem(V14g1,arr(A_27c,A_27d))
=> ! [V15g2: $i] :
( mem(V15g2,arr(A_27c,A_27d))
=> ( ( p(ap(ap(ap(ap(c_2Equotient_2E_3D_3D_3D_3E(A_27a,A_27b),V0R1),V3R2),V12f1),V13f2))
& p(ap(ap(ap(ap(c_2Equotient_2E_3D_3D_3D_3E(A_27c,A_27d),V6R3),V9R4),V14g1),V15g2)) )
=> p(ap(ap(ap(ap(c_2Equotient_2E_3D_3D_3D_3E(ty_2Epair_2Eprod(A_27a,A_27c),ty_2Epair_2Eprod(A_27b,A_27d)),ap(ap(c_2Equotient__pair_2E_23_23_23(A_27a,A_27c,A_27a,A_27c),V0R1),V6R3)),ap(ap(c_2Equotient__pair_2E_23_23_23(A_27b,A_27d,A_27b,A_27d),V3R2),V9R4)),ap(ap(c_2Epair_2E_23_23(A_27a,A_27c,A_27b,A_27d),V12f1),V14g1)),ap(ap(c_2Epair_2E_23_23(A_27a,A_27c,A_27b,A_27d),V13f2),V15g2))) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------