ITP001 Axioms: ITP008_5.ax
%------------------------------------------------------------------------------
% File : ITP008_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 : combin_2.ax [Gau20]
% : HL4008_5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 121 ( 29 unt; 34 typ; 0 def)
% Number of atoms : 798 ( 80 equ)
% Maximal formula atoms : 11 ( 6 avg)
% Number of connectives : 188 ( 5 ~; 1 |; 9 &)
% ( 13 <=>; 160 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 7 avg)
% Maximal term depth : 2 ( 1 avg)
% Number of FOOLs : 528 ( 528 fml; 0 var)
% Number of types : 1 ( 0 usr)
% Number of type conns : 106 ( 34 >; 72 *; 0 +; 0 <<)
% Number of predicates : 20 ( 19 usr; 8 prp; 0-2 aty)
% Number of functors : 34 ( 34 usr; 0 con; 1-5 aty)
% Number of variables : 371 ( 371 !; 0 ?; 371 :)
% SPC : TF0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
tff(tp_c_2Ecombin_2E_3A_3E,type,
c_2Ecombin_2E_3A_3E: ( del * del ) > $i ).
tff(mem_c_2Ecombin_2E_3A_3E,axiom,
! [A_27a: del,A_27b: del] : mem(c_2Ecombin_2E_3A_3E(A_27a,A_27b),arr(A_27b,arr(arr(A_27b,A_27a),A_27a))) ).
tff(tp_c_2Ecombin_2EASSOC,type,
c_2Ecombin_2EASSOC: del > $i ).
tff(mem_c_2Ecombin_2EASSOC,axiom,
! [A_27a: del] : mem(c_2Ecombin_2EASSOC(A_27a),arr(arr(A_27a,arr(A_27a,A_27a)),bool)) ).
tff(tp_c_2Ecombin_2EC,type,
c_2Ecombin_2EC: ( del * del * del ) > $i ).
tff(mem_c_2Ecombin_2EC,axiom,
! [A_27a: del,A_27b: del,A_27c: del] : mem(c_2Ecombin_2EC(A_27a,A_27b,A_27c),arr(arr(A_27a,arr(A_27b,A_27c)),arr(A_27b,arr(A_27a,A_27c)))) ).
tff(tp_c_2Ecombin_2ECOMM,type,
c_2Ecombin_2ECOMM: ( del * del ) > $i ).
tff(mem_c_2Ecombin_2ECOMM,axiom,
! [A_27a: del,A_27b: del] : mem(c_2Ecombin_2ECOMM(A_27a,A_27b),arr(arr(A_27a,arr(A_27a,A_27b)),bool)) ).
tff(tp_c_2Ecombin_2EFAIL,type,
c_2Ecombin_2EFAIL: ( del * del ) > $i ).
tff(mem_c_2Ecombin_2EFAIL,axiom,
! [A_27a: del,A_27b: del] : mem(c_2Ecombin_2EFAIL(A_27a,A_27b),arr(A_27a,arr(A_27b,A_27a))) ).
tff(tp_c_2Ecombin_2EFCOMM,type,
c_2Ecombin_2EFCOMM: ( del * del * del ) > $i ).
tff(mem_c_2Ecombin_2EFCOMM,axiom,
! [A_27a: del,A_27b: del,A_27c: del] : mem(c_2Ecombin_2EFCOMM(A_27a,A_27b,A_27c),arr(arr(A_27a,arr(A_27b,A_27a)),arr(arr(A_27c,arr(A_27a,A_27a)),bool))) ).
tff(tp_c_2Ecombin_2EI,type,
c_2Ecombin_2EI: del > $i ).
tff(mem_c_2Ecombin_2EI,axiom,
! [A_27a: del] : mem(c_2Ecombin_2EI(A_27a),arr(A_27a,A_27a)) ).
tff(tp_c_2Ecombin_2EK,type,
c_2Ecombin_2EK: ( del * del ) > $i ).
tff(mem_c_2Ecombin_2EK,axiom,
! [A_27a: del,A_27b: del] : mem(c_2Ecombin_2EK(A_27a,A_27b),arr(A_27a,arr(A_27b,A_27a))) ).
tff(tp_c_2Ecombin_2ELEFT__ID,type,
c_2Ecombin_2ELEFT__ID: ( del * del ) > $i ).
tff(mem_c_2Ecombin_2ELEFT__ID,axiom,
! [A_27a: del,A_27b: del] : mem(c_2Ecombin_2ELEFT__ID(A_27a,A_27b),arr(arr(A_27a,arr(A_27b,A_27b)),arr(A_27a,bool))) ).
tff(tp_c_2Ecombin_2EMONOID,type,
c_2Ecombin_2EMONOID: del > $i ).
tff(mem_c_2Ecombin_2EMONOID,axiom,
! [A_27a: del] : mem(c_2Ecombin_2EMONOID(A_27a),arr(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,bool))) ).
tff(tp_c_2Ecombin_2ERIGHT__ID,type,
c_2Ecombin_2ERIGHT__ID: ( del * del ) > $i ).
tff(mem_c_2Ecombin_2ERIGHT__ID,axiom,
! [A_27a: del,A_27b: del] : mem(c_2Ecombin_2ERIGHT__ID(A_27a,A_27b),arr(arr(A_27a,arr(A_27b,A_27a)),arr(A_27b,bool))) ).
tff(tp_c_2Ecombin_2ES,type,
c_2Ecombin_2ES: ( del * del * del ) > $i ).
tff(mem_c_2Ecombin_2ES,axiom,
! [A_27a: del,A_27b: del,A_27c: del] : mem(c_2Ecombin_2ES(A_27a,A_27b,A_27c),arr(arr(A_27a,arr(A_27b,A_27c)),arr(arr(A_27a,A_27b),arr(A_27a,A_27c)))) ).
tff(tp_c_2Ecombin_2EUPDATE,type,
c_2Ecombin_2EUPDATE: ( del * del ) > $i ).
tff(mem_c_2Ecombin_2EUPDATE,axiom,
! [A_27a: del,A_27b: del] : mem(c_2Ecombin_2EUPDATE(A_27a,A_27b),arr(A_27a,arr(A_27b,arr(arr(A_27a,A_27b),arr(A_27a,A_27b))))) ).
tff(tp_c_2Ecombin_2EW,type,
c_2Ecombin_2EW: ( del * del ) > $i ).
tff(mem_c_2Ecombin_2EW,axiom,
! [A_27a: del,A_27b: del] : mem(c_2Ecombin_2EW(A_27a,A_27b),arr(arr(A_27a,arr(A_27a,A_27b)),arr(A_27a,A_27b))) ).
tff(tp_c_2Ecombin_2Eo,type,
c_2Ecombin_2Eo: ( del * del * del ) > $i ).
tff(mem_c_2Ecombin_2Eo,axiom,
! [A_27a: del,A_27b: del,A_27c: del] : mem(c_2Ecombin_2Eo(A_27a,A_27b,A_27c),arr(arr(A_27c,A_27b),arr(arr(A_27a,A_27c),arr(A_27a,A_27b)))) ).
tff(lamtp_f68,type,
f68: ( del * del ) > $i ).
tff(lameq_f68,axiom,
! [A_27b: del,A_27a: del,V0x: $i] : ( ap(f68(A_27b,A_27a),V0x) = k(A_27b,V0x) ) ).
tff(ax_thm_2Ecombin_2EK__DEF,axiom,
! [A_27a: del,A_27b: del] : ( c_2Ecombin_2EK(A_27a,A_27b) = f68(A_27b,A_27a) ) ).
tff(lamtp_f69,type,
f69: ( del * del * del * $i * $i ) > $i ).
tff(lameq_f69,axiom,
! [A_27b: del,A_27c: del,A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,arr(A_27b,A_27c)))
=> ! [V1g: $i] :
( mem(V1g,arr(A_27a,A_27b))
=> ! [V2x: $i] : ( ap(f69(A_27b,A_27c,A_27a,V0f,V1g),V2x) = ap(ap(V0f,V2x),ap(V1g,V2x)) ) ) ) ).
tff(lamtp_f70,type,
f70: ( del * del * del * $i ) > $i ).
tff(lameq_f70,axiom,
! [A_27c: del,A_27a: del,A_27b: del,V0f: $i] :
( mem(V0f,arr(A_27a,arr(A_27b,A_27c)))
=> ! [V1g: $i] : ( ap(f70(A_27c,A_27a,A_27b,V0f),V1g) = f69(A_27b,A_27c,A_27a,V0f,V1g) ) ) ).
tff(lamtp_f71,type,
f71: ( del * del * del ) > $i ).
tff(lameq_f71,axiom,
! [A_27c: del,A_27a: del,A_27b: del,V0f: $i] : ( ap(f71(A_27c,A_27a,A_27b),V0f) = f70(A_27c,A_27a,A_27b,V0f) ) ).
tff(ax_thm_2Ecombin_2ES__DEF,axiom,
! [A_27a: del,A_27b: del,A_27c: del] : ( c_2Ecombin_2ES(A_27a,A_27b,A_27c) = f71(A_27c,A_27a,A_27b) ) ).
tff(ax_thm_2Ecombin_2EI__DEF,axiom,
! [A_27a: del] : ( c_2Ecombin_2EI(A_27a) = ap(ap(c_2Ecombin_2ES(A_27a,arr(A_27a,A_27a),A_27a),c_2Ecombin_2EK(A_27a,arr(A_27a,A_27a))),c_2Ecombin_2EK(A_27a,A_27a)) ) ).
tff(lamtp_f72,type,
f72: ( del * del * del * $i * $i ) > $i ).
tff(lameq_f72,axiom,
! [A_27b: del,A_27c: del,A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,arr(A_27b,A_27c)))
=> ! [V1x: $i] :
( mem(V1x,A_27b)
=> ! [V2y: $i] : ( ap(f72(A_27b,A_27c,A_27a,V0f,V1x),V2y) = ap(ap(V0f,V2y),V1x) ) ) ) ).
tff(lamtp_f73,type,
f73: ( del * del * del * $i ) > $i ).
tff(lameq_f73,axiom,
! [A_27a: del,A_27c: del,A_27b: del,V0f: $i] :
( mem(V0f,arr(A_27a,arr(A_27b,A_27c)))
=> ! [V1x: $i] : ( ap(f73(A_27a,A_27c,A_27b,V0f),V1x) = f72(A_27b,A_27c,A_27a,V0f,V1x) ) ) ).
tff(lamtp_f74,type,
f74: ( del * del * del ) > $i ).
tff(lameq_f74,axiom,
! [A_27a: del,A_27c: del,A_27b: del,V0f: $i] : ( ap(f74(A_27a,A_27c,A_27b),V0f) = f73(A_27a,A_27c,A_27b,V0f) ) ).
tff(ax_thm_2Ecombin_2EC__DEF,axiom,
! [A_27a: del,A_27b: del,A_27c: del] : ( c_2Ecombin_2EC(A_27a,A_27b,A_27c) = f74(A_27a,A_27c,A_27b) ) ).
tff(lamtp_f75,type,
f75: ( del * del * $i ) > $i ).
tff(lameq_f75,axiom,
! [A_27b: del,A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,arr(A_27a,A_27b)))
=> ! [V1x: $i] : ( ap(f75(A_27b,A_27a,V0f),V1x) = ap(ap(V0f,V1x),V1x) ) ) ).
tff(lamtp_f76,type,
f76: ( del * del ) > $i ).
tff(lameq_f76,axiom,
! [A_27b: del,A_27a: del,V0f: $i] : ( ap(f76(A_27b,A_27a),V0f) = f75(A_27b,A_27a,V0f) ) ).
tff(ax_thm_2Ecombin_2EW__DEF,axiom,
! [A_27a: del,A_27b: del] : ( c_2Ecombin_2EW(A_27a,A_27b) = f76(A_27b,A_27a) ) ).
tff(lamtp_f77,type,
f77: ( del * del * del * $i * $i ) > $i ).
tff(lameq_f77,axiom,
! [A_27c: del,A_27b: del,A_27a: del,V1g: $i] :
( mem(V1g,arr(A_27a,A_27c))
=> ! [V0f: $i] :
( mem(V0f,arr(A_27c,A_27b))
=> ! [V2x: $i] : ( ap(f77(A_27c,A_27b,A_27a,V1g,V0f),V2x) = ap(V0f,ap(V1g,V2x)) ) ) ) ).
tff(ax_thm_2Ecombin_2Eo__DEF,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( mem(V0f,arr(A_27c,A_27b))
=> ! [V1g: $i] :
( mem(V1g,arr(A_27a,A_27c))
=> ( ap(ap(c_2Ecombin_2Eo(A_27a,A_27b,A_27c),V0f),V1g) = f77(A_27c,A_27b,A_27a,V1g,V0f) ) ) ) ).
tff(ax_thm_2Ecombin_2EAPP__DEF,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( mem(V0x,A_27b)
=> ! [V1f: $i] :
( mem(V1f,arr(A_27b,A_27a))
=> ( ap(ap(c_2Ecombin_2E_3A_3E(A_27a,A_27b),V0x),V1f) = ap(V1f,V0x) ) ) ) ).
tff(lamtp_f78,type,
f78: ( del * del * $i * $i * $i ) > $i ).
tff(lameq_f78,axiom,
! [A_27b: del,A_27a: del,V1b: $i] :
( mem(V1b,A_27b)
=> ! [V0a: $i] :
( mem(V0a,A_27a)
=> ! [V2f: $i] :
( mem(V2f,arr(A_27a,A_27b))
=> ! [V3c: $i] : ( ap(f78(A_27b,A_27a,V1b,V0a,V2f),V3c) = ap(ap(ap(c_2Ebool_2ECOND(A_27b),ap(ap(c_2Emin_2E_3D(A_27a),V0a),V3c)),V1b),ap(V2f,V3c)) ) ) ) ) ).
tff(lamtp_f79,type,
f79: ( del * del * $i * $i ) > $i ).
tff(lameq_f79,axiom,
! [A_27b: del,A_27a: del,V1b: $i] :
( mem(V1b,A_27b)
=> ! [V0a: $i] :
( mem(V0a,A_27a)
=> ! [V2f: $i] : ( ap(f79(A_27b,A_27a,V1b,V0a),V2f) = f78(A_27b,A_27a,V1b,V0a,V2f) ) ) ) ).
tff(ax_thm_2Ecombin_2EUPDATE__def,axiom,
! [A_27a: del,A_27b: del,V0a: $i] :
( mem(V0a,A_27a)
=> ! [V1b: $i] :
( mem(V1b,A_27b)
=> ( ap(ap(c_2Ecombin_2EUPDATE(A_27a,A_27b),V0a),V1b) = f79(A_27b,A_27a,V1b,V0a) ) ) ) ).
tff(conj_thm_2Ecombin_2Eo__THM,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1g: $i] :
( mem(V1g,arr(A_27c,A_27a))
=> ! [V2x: $i] :
( mem(V2x,A_27c)
=> ( ap(ap(ap(c_2Ecombin_2Eo(A_27c,A_27b,A_27a),V0f),V1g),V2x) = ap(V0f,ap(V1g,V2x)) ) ) ) ) ).
tff(conj_thm_2Ecombin_2Eo__ASSOC,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0f: $i] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1g: $i] :
( mem(V1g,arr(A_27c,A_27a))
=> ! [V2h: $i] :
( mem(V2h,arr(A_27d,A_27c))
=> ( ap(ap(c_2Ecombin_2Eo(A_27d,A_27b,A_27a),V0f),ap(ap(c_2Ecombin_2Eo(A_27d,A_27a,A_27c),V1g),V2h)) = ap(ap(c_2Ecombin_2Eo(A_27d,A_27b,A_27c),ap(ap(c_2Ecombin_2Eo(A_27c,A_27b,A_27a),V0f),V1g)),V2h) ) ) ) ) ).
tff(lamtp_f80,type,
f80: ( del * del * $i ) > $i ).
tff(lameq_f80,axiom,
! [A_27c: del,A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,A_27c))
=> ! [V2x: $i] : ( ap(f80(A_27c,A_27a,V0f),V2x) = ap(V0f,V2x) ) ) ).
tff(lamtp_f81,type,
f81: ( del * del * del * $i * $i ) > $i ).
tff(lameq_f81,axiom,
! [A_27a: del,A_27c: del,A_27b: del,V1g: $i] :
( mem(V1g,arr(A_27b,A_27a))
=> ! [V0f: $i] :
( mem(V0f,arr(A_27a,A_27c))
=> ! [V3x: $i] : ( ap(f81(A_27a,A_27c,A_27b,V1g,V0f),V3x) = ap(V0f,ap(V1g,V3x)) ) ) ) ).
tff(conj_thm_2Ecombin_2Eo__ABS__L,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( mem(V0f,arr(A_27a,A_27c))
=> ! [V1g: $i] :
( mem(V1g,arr(A_27b,A_27a))
=> ( ap(ap(c_2Ecombin_2Eo(A_27b,A_27c,A_27a),f80(A_27c,A_27a,V0f)),V1g) = f81(A_27a,A_27c,A_27b,V1g,V0f) ) ) ) ).
tff(lamtp_f82,type,
f82: ( del * del * $i ) > $i ).
tff(lameq_f82,axiom,
! [A_27c: del,A_27a: del,V1g: $i] :
( mem(V1g,arr(A_27a,A_27c))
=> ! [V2x: $i] : ( ap(f82(A_27c,A_27a,V1g),V2x) = ap(V1g,V2x) ) ) ).
tff(lamtp_f83,type,
f83: ( del * del * del * $i * $i ) > $i ).
tff(lameq_f83,axiom,
! [A_27c: del,A_27b: del,A_27a: del,V1g: $i] :
( mem(V1g,arr(A_27a,A_27c))
=> ! [V0f: $i] :
( mem(V0f,arr(A_27c,A_27b))
=> ! [V3x: $i] : ( ap(f83(A_27c,A_27b,A_27a,V1g,V0f),V3x) = ap(V0f,ap(V1g,V3x)) ) ) ) ).
tff(conj_thm_2Ecombin_2Eo__ABS__R,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( mem(V0f,arr(A_27c,A_27b))
=> ! [V1g: $i] :
( mem(V1g,arr(A_27a,A_27c))
=> ( ap(ap(c_2Ecombin_2Eo(A_27a,A_27b,A_27c),V0f),f82(A_27c,A_27a,V1g)) = f83(A_27c,A_27b,A_27a,V1g,V0f) ) ) ) ).
tff(conj_thm_2Ecombin_2EK__THM,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( mem(V0x,A_27a)
=> ! [V1y: $i] :
( mem(V1y,A_27b)
=> ( ap(ap(c_2Ecombin_2EK(A_27a,A_27b),V0x),V1y) = V0x ) ) ) ).
tff(conj_thm_2Ecombin_2ES__THM,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( mem(V0f,arr(A_27a,arr(A_27b,A_27c)))
=> ! [V1g: $i] :
( mem(V1g,arr(A_27a,A_27b))
=> ! [V2x: $i] :
( mem(V2x,A_27a)
=> ( ap(ap(ap(c_2Ecombin_2ES(A_27a,A_27b,A_27c),V0f),V1g),V2x) = ap(ap(V0f,V2x),ap(V1g,V2x)) ) ) ) ) ).
tff(lamtp_f84,type,
f84: ( del * del * del * $i ) > $i ).
tff(lameq_f84,axiom,
! [A_27b: del,A_27c: del,A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,arr(A_27c,A_27b)))
=> ! [V2x: $i] : ( ap(f84(A_27b,A_27c,A_27a,V0f),V2x) = ap(V0f,V2x) ) ) ).
tff(lamtp_f85,type,
f85: ( del * del * del * $i * $i ) > $i ).
tff(lameq_f85,axiom,
! [A_27c: del,A_27b: del,A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,arr(A_27c,A_27b)))
=> ! [V1g: $i] :
( mem(V1g,arr(A_27a,A_27c))
=> ! [V3x: $i] : ( ap(f85(A_27c,A_27b,A_27a,V0f,V1g),V3x) = ap(ap(V0f,V3x),ap(V1g,V3x)) ) ) ) ).
tff(conj_thm_2Ecombin_2ES__ABS__L,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( mem(V0f,arr(A_27a,arr(A_27c,A_27b)))
=> ! [V1g: $i] :
( mem(V1g,arr(A_27a,A_27c))
=> ( ap(ap(c_2Ecombin_2ES(A_27a,A_27c,A_27b),f84(A_27b,A_27c,A_27a,V0f)),V1g) = f85(A_27c,A_27b,A_27a,V0f,V1g) ) ) ) ).
tff(conj_thm_2Ecombin_2ES__ABS__R,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( mem(V0f,arr(A_27a,arr(A_27c,A_27b)))
=> ! [V1g: $i] :
( mem(V1g,arr(A_27a,A_27c))
=> ( ap(ap(c_2Ecombin_2ES(A_27a,A_27c,A_27b),V0f),f82(A_27c,A_27a,V1g)) = f85(A_27c,A_27b,A_27a,V0f,V1g) ) ) ) ).
tff(conj_thm_2Ecombin_2EC__THM,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( mem(V0f,arr(A_27a,arr(A_27b,A_27c)))
=> ! [V1x: $i] :
( mem(V1x,A_27b)
=> ! [V2y: $i] :
( mem(V2y,A_27a)
=> ( ap(ap(ap(c_2Ecombin_2EC(A_27a,A_27b,A_27c),V0f),V1x),V2y) = ap(ap(V0f,V2y),V1x) ) ) ) ) ).
tff(lamtp_f86,type,
f86: ( del * del * del * $i * $i ) > $i ).
tff(lameq_f86,axiom,
! [A_27c: del,A_27b: del,A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,arr(A_27c,A_27b)))
=> ! [V1y: $i] :
( mem(V1y,A_27c)
=> ! [V3x: $i] : ( ap(f86(A_27c,A_27b,A_27a,V0f,V1y),V3x) = ap(ap(V0f,V3x),V1y) ) ) ) ).
tff(conj_thm_2Ecombin_2EC__ABS__L,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( mem(V0f,arr(A_27a,arr(A_27c,A_27b)))
=> ! [V1y: $i] :
( mem(V1y,A_27c)
=> ( ap(ap(c_2Ecombin_2EC(A_27a,A_27c,A_27b),f84(A_27b,A_27c,A_27a,V0f)),V1y) = f86(A_27c,A_27b,A_27a,V0f,V1y) ) ) ) ).
tff(conj_thm_2Ecombin_2EW__THM,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( mem(V0f,arr(A_27a,arr(A_27a,A_27b)))
=> ! [V1x: $i] :
( mem(V1x,A_27a)
=> ( ap(ap(c_2Ecombin_2EW(A_27a,A_27b),V0f),V1x) = ap(ap(V0f,V1x),V1x) ) ) ) ).
tff(conj_thm_2Ecombin_2EI__THM,axiom,
! [A_27a: del,V0x: $i] :
( mem(V0x,A_27a)
=> ( ap(c_2Ecombin_2EI(A_27a),V0x) = V0x ) ) ).
tff(conj_thm_2Ecombin_2EI__o__ID,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( mem(V0f,arr(A_27a,A_27b))
=> ( ( ap(ap(c_2Ecombin_2Eo(A_27a,A_27b,A_27b),c_2Ecombin_2EI(A_27b)),V0f) = V0f )
& ( ap(ap(c_2Ecombin_2Eo(A_27a,A_27b,A_27a),V0f),c_2Ecombin_2EI(A_27a)) = V0f ) ) ) ).
tff(conj_thm_2Ecombin_2EK__o__THM,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,A_27e: del,A_27f: del] :
( ! [V0f: $i] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1v: $i] :
( mem(V1v,A_27c)
=> ( ap(ap(c_2Ecombin_2Eo(A_27a,A_27c,A_27b),ap(c_2Ecombin_2EK(A_27c,A_27b),V1v)),V0f) = ap(c_2Ecombin_2EK(A_27c,A_27a),V1v) ) ) )
& ! [V2f: $i] :
( mem(V2f,arr(A_27d,A_27e))
=> ! [V3v: $i] :
( mem(V3v,A_27d)
=> ( ap(ap(c_2Ecombin_2Eo(A_27f,A_27e,A_27d),V2f),ap(c_2Ecombin_2EK(A_27d,A_27f),V3v)) = ap(c_2Ecombin_2EK(A_27e,A_27f),ap(V2f,V3v)) ) ) ) ) ).
tff(conj_thm_2Ecombin_2EUPDATE__APPLY,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del] :
( ! [V0a: $i] :
( mem(V0a,A_27a)
=> ! [V1x: $i] :
( mem(V1x,A_27b)
=> ! [V2f: $i] :
( mem(V2f,arr(A_27a,A_27b))
=> ( ap(ap(ap(ap(c_2Ecombin_2EUPDATE(A_27a,A_27b),V0a),V1x),V2f),V0a) = V1x ) ) ) )
& ! [V3a: $i] :
( mem(V3a,A_27c)
=> ! [V4b: $i] :
( mem(V4b,A_27c)
=> ! [V5x: $i] :
( mem(V5x,A_27d)
=> ! [V6f: $i] :
( mem(V6f,arr(A_27c,A_27d))
=> ( ( V3a != V4b )
=> ( ap(ap(ap(ap(c_2Ecombin_2EUPDATE(A_27c,A_27d),V3a),V5x),V6f),V4b) = ap(V6f,V4b) ) ) ) ) ) ) ) ).
tff(conj_thm_2Ecombin_2EAPPLY__UPDATE__THM,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1a: $i] :
( mem(V1a,A_27a)
=> ! [V2b: $i] :
( mem(V2b,A_27b)
=> ! [V3c: $i] :
( mem(V3c,A_27a)
=> ( ap(ap(ap(ap(c_2Ecombin_2EUPDATE(A_27a,A_27b),V1a),V2b),V0f),V3c) = ap(ap(ap(c_2Ebool_2ECOND(A_27b),ap(ap(c_2Emin_2E_3D(A_27a),V1a),V3c)),V2b),ap(V0f,V3c)) ) ) ) ) ) ).
tff(conj_thm_2Ecombin_2EUPDATE__COMMUTES,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1a: $i] :
( mem(V1a,A_27a)
=> ! [V2b: $i] :
( mem(V2b,A_27a)
=> ! [V3c: $i] :
( mem(V3c,A_27b)
=> ! [V4d: $i] :
( mem(V4d,A_27b)
=> ( ( V1a != V2b )
=> ( ap(ap(ap(c_2Ecombin_2EUPDATE(A_27a,A_27b),V1a),V3c),ap(ap(ap(c_2Ecombin_2EUPDATE(A_27a,A_27b),V2b),V4d),V0f)) = ap(ap(ap(c_2Ecombin_2EUPDATE(A_27a,A_27b),V2b),V4d),ap(ap(ap(c_2Ecombin_2EUPDATE(A_27a,A_27b),V1a),V3c),V0f)) ) ) ) ) ) ) ) ).
tff(conj_thm_2Ecombin_2EUPDATE__EQ,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1a: $i] :
( mem(V1a,A_27a)
=> ! [V2b: $i] :
( mem(V2b,A_27b)
=> ! [V3c: $i] :
( mem(V3c,A_27b)
=> ( ap(ap(ap(c_2Ecombin_2EUPDATE(A_27a,A_27b),V1a),V3c),ap(ap(ap(c_2Ecombin_2EUPDATE(A_27a,A_27b),V1a),V2b),V0f)) = ap(ap(ap(c_2Ecombin_2EUPDATE(A_27a,A_27b),V1a),V3c),V0f) ) ) ) ) ) ).
tff(conj_thm_2Ecombin_2EUPDATE__APPLY__ID,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1a: $i] :
( mem(V1a,A_27a)
=> ! [V2b: $i] :
( mem(V2b,A_27b)
=> ( ( ap(V0f,V1a) = V2b )
<=> ( ap(ap(ap(c_2Ecombin_2EUPDATE(A_27a,A_27b),V1a),V2b),V0f) = V0f ) ) ) ) ) ).
tff(conj_thm_2Ecombin_2EUPDATE__APPLY__IMP__ID,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1b: $i] :
( mem(V1b,A_27b)
=> ! [V2a: $i] :
( mem(V2a,A_27a)
=> ( ( ap(V0f,V2a) = V1b )
=> ( ap(ap(ap(c_2Ecombin_2EUPDATE(A_27a,A_27b),V2a),V1b),V0f) = V0f ) ) ) ) ) ).
tff(conj_thm_2Ecombin_2EAPPLY__UPDATE__ID,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1a: $i] :
( mem(V1a,A_27a)
=> ( ap(ap(ap(c_2Ecombin_2EUPDATE(A_27a,A_27b),V1a),ap(V0f,V1a)),V0f) = V0f ) ) ) ).
tff(conj_thm_2Ecombin_2EUPD11__SAME__BASE,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1a: $i] :
( mem(V1a,A_27a)
=> ! [V2b: $i] :
( mem(V2b,A_27a)
=> ! [V3c: $i] :
( mem(V3c,A_27b)
=> ! [V4d: $i] :
( mem(V4d,A_27b)
=> ( ( ap(ap(ap(c_2Ecombin_2EUPDATE(A_27a,A_27b),V1a),V3c),V0f) = ap(ap(ap(c_2Ecombin_2EUPDATE(A_27a,A_27b),V2b),V4d),V0f) )
<=> ( ( ( V1a = V2b )
& ( V3c = V4d ) )
| ( ( V1a != V2b )
& ( ap(ap(ap(c_2Ecombin_2EUPDATE(A_27a,A_27b),V1a),V3c),V0f) = V0f )
& ( ap(ap(ap(c_2Ecombin_2EUPDATE(A_27a,A_27b),V2b),V4d),V0f) = V0f ) ) ) ) ) ) ) ) ) ).
tff(conj_thm_2Ecombin_2ESAME__KEY__UPDATE__DIFFER,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0f: $i] :
( mem(V0f,arr(A_27c,A_27d))
=> ! [V1f1: $i] :
( mem(V1f1,A_27a)
=> ! [V2f2: $i] :
( mem(V2f2,A_27b)
=> ! [V3a: $i] :
( mem(V3a,A_27c)
=> ! [V4b: $i] :
( mem(V4b,A_27d)
=> ! [V5c: $i] :
( mem(V5c,A_27d)
=> ( ( V4b != V5c )
=> ( ap(ap(ap(c_2Ecombin_2EUPDATE(A_27c,A_27d),V3a),V4b),V0f) != ap(ap(ap(c_2Ecombin_2EUPDATE(A_27c,A_27d),V3a),V5c),V0f) ) ) ) ) ) ) ) ) ).
tff(conj_thm_2Ecombin_2EUPD11__SAME__KEY__AND__BASE,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( mem(V0f,arr(A_27a,A_27b))
=> ! [V1a: $i] :
( mem(V1a,A_27a)
=> ! [V2b: $i] :
( mem(V2b,A_27b)
=> ! [V3c: $i] :
( mem(V3c,A_27b)
=> ( ( ap(ap(ap(c_2Ecombin_2EUPDATE(A_27a,A_27b),V1a),V2b),V0f) = ap(ap(ap(c_2Ecombin_2EUPDATE(A_27a,A_27b),V1a),V3c),V0f) )
<=> ( V2b = V3c ) ) ) ) ) ) ).
tff(conj_thm_2Ecombin_2EUPD__SAME__KEY__UNWIND,axiom,
! [A_27a: del,A_27b: del,V0f1: $i] :
( mem(V0f1,arr(A_27a,A_27b))
=> ! [V1f2: $i] :
( mem(V1f2,arr(A_27a,A_27b))
=> ! [V2a: $i] :
( mem(V2a,A_27a)
=> ! [V3b: $i] :
( mem(V3b,A_27b)
=> ! [V4c: $i] :
( mem(V4c,A_27b)
=> ( ( ap(ap(ap(c_2Ecombin_2EUPDATE(A_27a,A_27b),V2a),V3b),V0f1) = ap(ap(ap(c_2Ecombin_2EUPDATE(A_27a,A_27b),V2a),V4c),V1f2) )
=> ( ( V3b = V4c )
& ! [V5v: $i] :
( mem(V5v,A_27b)
=> ( ap(ap(ap(c_2Ecombin_2EUPDATE(A_27a,A_27b),V2a),V5v),V0f1) = ap(ap(ap(c_2Ecombin_2EUPDATE(A_27a,A_27b),V2a),V5v),V1f2) ) ) ) ) ) ) ) ) ) ).
tff(conj_thm_2Ecombin_2EGEN__LET__RAND,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0P: $i] :
( mem(V0P,arr(A_27b,A_27a))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27c,A_27b))
=> ! [V2v: $i] :
( mem(V2v,A_27c)
=> ( ap(V0P,ap(ap(c_2Ebool_2ELET(A_27c,A_27b),V1f),V2v)) = ap(ap(c_2Ebool_2ELET(A_27c,A_27a),ap(ap(c_2Ecombin_2Eo(A_27c,A_27a,A_27b),V0P),V1f)),V2v) ) ) ) ) ).
tff(conj_thm_2Ecombin_2EGEN__LET__RATOR,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( mem(V0f,arr(A_27b,arr(A_27c,A_27a)))
=> ! [V1v: $i] :
( mem(V1v,A_27b)
=> ! [V2x: $i] :
( mem(V2x,A_27c)
=> ( ap(ap(ap(c_2Ebool_2ELET(A_27b,arr(A_27c,A_27a)),V0f),V1v),V2x) = ap(ap(c_2Ebool_2ELET(A_27b,A_27a),ap(ap(c_2Ecombin_2EC(A_27b,A_27c,A_27a),V0f),V2x)),V1v) ) ) ) ) ).
tff(conj_thm_2Ecombin_2ELET__FORALL__ELIM,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,bool))
=> ! [V1v: $i] :
( mem(V1v,A_27a)
=> ( p(ap(ap(c_2Ebool_2ELET(A_27a,bool),V0f),V1v))
<=> p(ap(c_2Ebool_2E_21(A_27a),ap(ap(c_2Ecombin_2ES(A_27a,bool,bool),ap(ap(c_2Ecombin_2Eo(A_27a,arr(bool,bool),bool),c_2Emin_2E_3D_3D_3E),ap(ap(c_2Ecombin_2Eo(A_27a,bool,bool),c_2Emarker_2EAbbrev),ap(ap(c_2Ecombin_2EC(A_27a,A_27a,bool),c_2Emin_2E_3D(A_27a)),V1v)))),V0f))) ) ) ) ).
tff(conj_thm_2Ecombin_2EGEN__literal__case__RAND,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0P: $i] :
( mem(V0P,arr(A_27b,A_27a))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27c,A_27b))
=> ! [V2v: $i] :
( mem(V2v,A_27c)
=> ( ap(V0P,ap(ap(c_2Ebool_2Eliteral__case(A_27c,A_27b),V1f),V2v)) = ap(ap(c_2Ebool_2Eliteral__case(A_27c,A_27a),ap(ap(c_2Ecombin_2Eo(A_27c,A_27a,A_27b),V0P),V1f)),V2v) ) ) ) ) ).
tff(conj_thm_2Ecombin_2EGEN__literal__case__RATOR,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( mem(V0f,arr(A_27b,arr(A_27c,A_27a)))
=> ! [V1v: $i] :
( mem(V1v,A_27b)
=> ! [V2x: $i] :
( mem(V2x,A_27c)
=> ( ap(ap(ap(c_2Ebool_2Eliteral__case(A_27b,arr(A_27c,A_27a)),V0f),V1v),V2x) = ap(ap(c_2Ebool_2Eliteral__case(A_27b,A_27a),ap(ap(c_2Ecombin_2EC(A_27b,A_27c,A_27a),V0f),V2x)),V1v) ) ) ) ) ).
tff(conj_thm_2Ecombin_2Eliteral__case__FORALL__ELIM,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,bool))
=> ! [V1v: $i] :
( mem(V1v,A_27a)
=> ( p(ap(ap(c_2Ebool_2Eliteral__case(A_27a,bool),V0f),V1v))
<=> p(ap(c_2Ebool_2E_21(A_27a),ap(ap(c_2Ecombin_2ES(A_27a,bool,bool),ap(ap(c_2Ecombin_2Eo(A_27a,arr(bool,bool),bool),c_2Emin_2E_3D_3D_3E),ap(ap(c_2Ecombin_2Eo(A_27a,bool,bool),c_2Emarker_2EAbbrev),ap(ap(c_2Ecombin_2EC(A_27a,A_27a,bool),c_2Emin_2E_3D(A_27a)),V1v)))),V0f))) ) ) ) ).
tff(ax_thm_2Ecombin_2EASSOC__DEF,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,arr(A_27a,A_27a)))
=> ( p(ap(c_2Ecombin_2EASSOC(A_27a),V0f))
<=> ! [V1x: $i] :
( mem(V1x,A_27a)
=> ! [V2y: $i] :
( mem(V2y,A_27a)
=> ! [V3z: $i] :
( mem(V3z,A_27a)
=> ( ap(ap(V0f,V1x),ap(ap(V0f,V2y),V3z)) = ap(ap(V0f,ap(ap(V0f,V1x),V2y)),V3z) ) ) ) ) ) ) ).
tff(ax_thm_2Ecombin_2ECOMM__DEF,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( mem(V0f,arr(A_27a,arr(A_27a,A_27b)))
=> ( p(ap(c_2Ecombin_2ECOMM(A_27a,A_27b),V0f))
<=> ! [V1x: $i] :
( mem(V1x,A_27a)
=> ! [V2y: $i] :
( mem(V2y,A_27a)
=> ( ap(ap(V0f,V1x),V2y) = ap(ap(V0f,V2y),V1x) ) ) ) ) ) ).
tff(ax_thm_2Ecombin_2EFCOMM__DEF,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( mem(V0f,arr(A_27a,arr(A_27b,A_27a)))
=> ! [V1g: $i] :
( mem(V1g,arr(A_27c,arr(A_27a,A_27a)))
=> ( p(ap(ap(c_2Ecombin_2EFCOMM(A_27a,A_27b,A_27c),V0f),V1g))
<=> ! [V2x: $i] :
( mem(V2x,A_27c)
=> ! [V3y: $i] :
( mem(V3y,A_27a)
=> ! [V4z: $i] :
( mem(V4z,A_27b)
=> ( ap(ap(V1g,V2x),ap(ap(V0f,V3y),V4z)) = ap(ap(V0f,ap(ap(V1g,V2x),V3y)),V4z) ) ) ) ) ) ) ) ).
tff(ax_thm_2Ecombin_2ERIGHT__ID__DEF,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( mem(V0f,arr(A_27a,arr(A_27b,A_27a)))
=> ! [V1e: $i] :
( mem(V1e,A_27b)
=> ( p(ap(ap(c_2Ecombin_2ERIGHT__ID(A_27a,A_27b),V0f),V1e))
<=> ! [V2x: $i] :
( mem(V2x,A_27a)
=> ( ap(ap(V0f,V2x),V1e) = V2x ) ) ) ) ) ).
tff(ax_thm_2Ecombin_2ELEFT__ID__DEF,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( mem(V0f,arr(A_27a,arr(A_27b,A_27b)))
=> ! [V1e: $i] :
( mem(V1e,A_27a)
=> ( p(ap(ap(c_2Ecombin_2ELEFT__ID(A_27a,A_27b),V0f),V1e))
<=> ! [V2x: $i] :
( mem(V2x,A_27b)
=> ( ap(ap(V0f,V1e),V2x) = V2x ) ) ) ) ) ).
tff(ax_thm_2Ecombin_2EMONOID__DEF,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V1e: $i] :
( mem(V1e,A_27a)
=> ( p(ap(ap(c_2Ecombin_2EMONOID(A_27a),V0f),V1e))
<=> ( p(ap(c_2Ecombin_2EASSOC(A_27a),V0f))
& p(ap(ap(c_2Ecombin_2ERIGHT__ID(A_27a,A_27a),V0f),V1e))
& p(ap(ap(c_2Ecombin_2ELEFT__ID(A_27a,A_27a),V0f),V1e)) ) ) ) ) ).
tff(conj_thm_2Ecombin_2EASSOC__CONJ,axiom,
p(ap(c_2Ecombin_2EASSOC(bool),c_2Ebool_2E_2F_5C)) ).
tff(conj_thm_2Ecombin_2EASSOC__SYM,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,arr(A_27a,A_27a)))
=> ( p(ap(c_2Ecombin_2EASSOC(A_27a),V0f))
<=> ! [V1x: $i] :
( mem(V1x,A_27a)
=> ! [V2y: $i] :
( mem(V2y,A_27a)
=> ! [V3z: $i] :
( mem(V3z,A_27a)
=> ( ap(ap(V0f,ap(ap(V0f,V1x),V2y)),V3z) = ap(ap(V0f,V1x),ap(ap(V0f,V2y),V3z)) ) ) ) ) ) ) ).
tff(conj_thm_2Ecombin_2EASSOC__DISJ,axiom,
p(ap(c_2Ecombin_2EASSOC(bool),c_2Ebool_2E_5C_2F)) ).
tff(conj_thm_2Ecombin_2EFCOMM__ASSOC,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,arr(A_27a,A_27a)))
=> ( p(ap(ap(c_2Ecombin_2EFCOMM(A_27a,A_27a,A_27a),V0f),V0f))
<=> p(ap(c_2Ecombin_2EASSOC(A_27a),V0f)) ) ) ).
tff(conj_thm_2Ecombin_2EMONOID__CONJ__T,axiom,
p(ap(ap(c_2Ecombin_2EMONOID(bool),c_2Ebool_2E_2F_5C),inj__o(fo__c_2Ebool_2ET))) ).
tff(conj_thm_2Ecombin_2EMONOID__DISJ__F,axiom,
p(ap(ap(c_2Ecombin_2EMONOID(bool),c_2Ebool_2E_5C_2F),inj__o(fo__c_2Ebool_2EF))) ).
tff(ax_thm_2Ecombin_2EFAIL__DEF,axiom,
! [A_27a: del,A_27b: del] : ( c_2Ecombin_2EFAIL(A_27a,A_27b) = f68(A_27b,A_27a) ) ).
tff(conj_thm_2Ecombin_2EFAIL__THM,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( mem(V0x,A_27a)
=> ! [V1y: $i] :
( mem(V1y,A_27b)
=> ( ap(ap(c_2Ecombin_2EFAIL(A_27a,A_27b),V0x),V1y) = V0x ) ) ) ).
%------------------------------------------------------------------------------