ITP001 Axioms: ITP074_5.ax
%------------------------------------------------------------------------------
% File : ITP074_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 : ring_2.ax [Gau20]
% : HL4074_5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 106 ( 17 unt; 29 typ; 0 def)
% Number of atoms : 2988 ( 156 equ)
% Maximal formula atoms : 75 ( 28 avg)
% Number of connectives : 584 ( 0 ~; 0 |; 117 &)
% ( 7 <=>; 460 =>; 0 <=; 0 <~>)
% Maximal formula depth : 30 ( 10 avg)
% Maximal term depth : 2 ( 1 avg)
% Number of FOOLs : 2327 (2327 fml; 0 var)
% Number of types : 1 ( 0 usr)
% Number of type conns : 57 ( 29 >; 28 *; 0 +; 0 <<)
% Number of predicates : 33 ( 32 usr; 9 prp; 0-3 aty)
% Number of functors : 29 ( 29 usr; 0 con; 1-6 aty)
% Number of variables : 563 ( 540 !; 23 ?; 563 :)
% SPC : TF0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
tff(tp_ty_2Ering_2Ering,type,
ty_2Ering_2Ering: del > del ).
tff(tp_c_2Ering_2Eis__ring,type,
c_2Ering_2Eis__ring: del > $i ).
tff(mem_c_2Ering_2Eis__ring,axiom,
! [A_27a: del] : mem(c_2Ering_2Eis__ring(A_27a),arr(ty_2Ering_2Ering(A_27a),bool)) ).
tff(tp_c_2Ering_2Erecordtype_2Ering,type,
c_2Ering_2Erecordtype_2Ering: del > $i ).
tff(mem_c_2Ering_2Erecordtype_2Ering,axiom,
! [A_27a: del] : mem(c_2Ering_2Erecordtype_2Ering(A_27a),arr(A_27a,arr(A_27a,arr(arr(A_27a,arr(A_27a,A_27a)),arr(arr(A_27a,arr(A_27a,A_27a)),arr(arr(A_27a,A_27a),ty_2Ering_2Ering(A_27a))))))) ).
tff(tp_c_2Ering_2Ering__CASE,type,
c_2Ering_2Ering__CASE: ( del * del ) > $i ).
tff(mem_c_2Ering_2Ering__CASE,axiom,
! [A_27a: del,A_27b: del] : mem(c_2Ering_2Ering__CASE(A_27a,A_27b),arr(ty_2Ering_2Ering(A_27a),arr(arr(A_27a,arr(A_27a,arr(arr(A_27a,arr(A_27a,A_27a)),arr(arr(A_27a,arr(A_27a,A_27a)),arr(arr(A_27a,A_27a),A_27b))))),A_27b))) ).
tff(tp_c_2Ering_2Ering__R0,type,
c_2Ering_2Ering__R0: del > $i ).
tff(mem_c_2Ering_2Ering__R0,axiom,
! [A_27a: del] : mem(c_2Ering_2Ering__R0(A_27a),arr(ty_2Ering_2Ering(A_27a),A_27a)) ).
tff(tp_c_2Ering_2Ering__R0__fupd,type,
c_2Ering_2Ering__R0__fupd: del > $i ).
tff(mem_c_2Ering_2Ering__R0__fupd,axiom,
! [A_27a: del] : mem(c_2Ering_2Ering__R0__fupd(A_27a),arr(arr(A_27a,A_27a),arr(ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)))) ).
tff(tp_c_2Ering_2Ering__R1,type,
c_2Ering_2Ering__R1: del > $i ).
tff(mem_c_2Ering_2Ering__R1,axiom,
! [A_27a: del] : mem(c_2Ering_2Ering__R1(A_27a),arr(ty_2Ering_2Ering(A_27a),A_27a)) ).
tff(tp_c_2Ering_2Ering__R1__fupd,type,
c_2Ering_2Ering__R1__fupd: del > $i ).
tff(mem_c_2Ering_2Ering__R1__fupd,axiom,
! [A_27a: del] : mem(c_2Ering_2Ering__R1__fupd(A_27a),arr(arr(A_27a,A_27a),arr(ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)))) ).
tff(tp_c_2Ering_2Ering__RM,type,
c_2Ering_2Ering__RM: del > $i ).
tff(mem_c_2Ering_2Ering__RM,axiom,
! [A_27a: del] : mem(c_2Ering_2Ering__RM(A_27a),arr(ty_2Ering_2Ering(A_27a),arr(A_27a,arr(A_27a,A_27a)))) ).
tff(tp_c_2Ering_2Ering__RM__fupd,type,
c_2Ering_2Ering__RM__fupd: del > $i ).
tff(mem_c_2Ering_2Ering__RM__fupd,axiom,
! [A_27a: del] : mem(c_2Ering_2Ering__RM__fupd(A_27a),arr(arr(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,arr(A_27a,A_27a))),arr(ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)))) ).
tff(tp_c_2Ering_2Ering__RN,type,
c_2Ering_2Ering__RN: del > $i ).
tff(mem_c_2Ering_2Ering__RN,axiom,
! [A_27a: del] : mem(c_2Ering_2Ering__RN(A_27a),arr(ty_2Ering_2Ering(A_27a),arr(A_27a,A_27a))) ).
tff(tp_c_2Ering_2Ering__RN__fupd,type,
c_2Ering_2Ering__RN__fupd: del > $i ).
tff(mem_c_2Ering_2Ering__RN__fupd,axiom,
! [A_27a: del] : mem(c_2Ering_2Ering__RN__fupd(A_27a),arr(arr(arr(A_27a,A_27a),arr(A_27a,A_27a)),arr(ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)))) ).
tff(tp_c_2Ering_2Ering__RP,type,
c_2Ering_2Ering__RP: del > $i ).
tff(mem_c_2Ering_2Ering__RP,axiom,
! [A_27a: del] : mem(c_2Ering_2Ering__RP(A_27a),arr(ty_2Ering_2Ering(A_27a),arr(A_27a,arr(A_27a,A_27a)))) ).
tff(tp_c_2Ering_2Ering__RP__fupd,type,
c_2Ering_2Ering__RP__fupd: del > $i ).
tff(mem_c_2Ering_2Ering__RP__fupd,axiom,
! [A_27a: del] : mem(c_2Ering_2Ering__RP__fupd(A_27a),arr(arr(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,arr(A_27a,A_27a))),arr(ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)))) ).
tff(tp_c_2Ering_2Ering__size,type,
c_2Ering_2Ering__size: del > $i ).
tff(mem_c_2Ering_2Ering__size,axiom,
! [A_27a: del] : mem(c_2Ering_2Ering__size(A_27a),arr(arr(A_27a,ty_2Enum_2Enum),arr(ty_2Ering_2Ering(A_27a),ty_2Enum_2Enum))) ).
tff(tp_c_2Ering_2Esemi__ring__of,type,
c_2Ering_2Esemi__ring__of: del > $i ).
tff(mem_c_2Ering_2Esemi__ring__of,axiom,
! [A_27a: del] : mem(c_2Ering_2Esemi__ring__of(A_27a),arr(ty_2Ering_2Ering(A_27a),ty_2Esemi__ring_2Esemi__ring(A_27a))) ).
tff(lamtp_f1835,type,
f1835: ( del * $i * $i * $i * $i ) > $i ).
tff(lameq_f1835,axiom,
! [A_27a: del,V10a1: $i] :
( mem(V10a1,A_27a)
=> ! [V12a3: $i] :
( mem(V12a3,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V11a2: $i] :
( mem(V11a2,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V9a0: $i] :
( mem(V9a0,A_27a)
=> ! [V13a4: $i] : ( ap(f1835(A_27a,V10a1,V12a3,V11a2,V9a0),V13a4) = ap(ap(ap(c_2Eind__type_2ECONSTR(ty_2Epair_2Eprod(A_27a,ty_2Epair_2Eprod(A_27a,ty_2Epair_2Eprod(arr(A_27a,arr(A_27a,A_27a)),ty_2Epair_2Eprod(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,A_27a)))))),inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)),ap(ap(c_2Epair_2E_2C(A_27a,ty_2Epair_2Eprod(A_27a,ty_2Epair_2Eprod(arr(A_27a,arr(A_27a,A_27a)),ty_2Epair_2Eprod(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,A_27a))))),V9a0),ap(ap(c_2Epair_2E_2C(A_27a,ty_2Epair_2Eprod(arr(A_27a,arr(A_27a,A_27a)),ty_2Epair_2Eprod(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,A_27a)))),V10a1),ap(ap(c_2Epair_2E_2C(arr(A_27a,arr(A_27a,A_27a)),ty_2Epair_2Eprod(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,A_27a))),V11a2),ap(ap(c_2Epair_2E_2C(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,A_27a)),V12a3),V13a4))))),k(ty_2Enum_2Enum,c_2Eind__type_2EBOTTOM(ty_2Epair_2Eprod(A_27a,ty_2Epair_2Eprod(A_27a,ty_2Epair_2Eprod(arr(A_27a,arr(A_27a,A_27a)),ty_2Epair_2Eprod(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,A_27a)))))))) ) ) ) ) ) ).
tff(lamtp_f1836,type,
f1836: ( del * $i * $i * $i ) > $i ).
tff(lameq_f1836,axiom,
! [A_27a: del,V10a1: $i] :
( mem(V10a1,A_27a)
=> ! [V11a2: $i] :
( mem(V11a2,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V9a0: $i] :
( mem(V9a0,A_27a)
=> ! [V12a3: $i] : ( ap(f1836(A_27a,V10a1,V11a2,V9a0),V12a3) = f1835(A_27a,V10a1,V12a3,V11a2,V9a0) ) ) ) ) ).
tff(lamtp_f1837,type,
f1837: ( del * $i * $i ) > $i ).
tff(lameq_f1837,axiom,
! [A_27a: del,V10a1: $i] :
( mem(V10a1,A_27a)
=> ! [V9a0: $i] :
( mem(V9a0,A_27a)
=> ! [V11a2: $i] : ( ap(f1837(A_27a,V10a1,V9a0),V11a2) = f1836(A_27a,V10a1,V11a2,V9a0) ) ) ) ).
tff(lamtp_f1838,type,
f1838: ( del * $i ) > $i ).
tff(lameq_f1838,axiom,
! [A_27a: del,V9a0: $i] :
( mem(V9a0,A_27a)
=> ! [V10a1: $i] : ( ap(f1838(A_27a,V9a0),V10a1) = f1837(A_27a,V10a1,V9a0) ) ) ).
tff(lamtp_f1839,type,
f1839: del > $i ).
tff(lameq_f1839,axiom,
! [A_27a: del,V9a0: $i] : ( ap(f1839(A_27a),V9a0) = f1838(A_27a,V9a0) ) ).
tff(lamtp_f1840,type,
f1840: ( del * $i * $i * $i * $i * $i ) > $i ).
tff(lameq_f1840,axiom,
! [A_27a: del,V6a2: $i] :
( mem(V6a2,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V4a0: $i] :
( mem(V4a0,A_27a)
=> ! [V5a1: $i] :
( mem(V5a1,A_27a)
=> ! [V7a3: $i] :
( mem(V7a3,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V3a0_27: $i] :
( mem(V3a0_27,ty_2Eind__type_2Erecspace(ty_2Epair_2Eprod(A_27a,ty_2Epair_2Eprod(A_27a,ty_2Epair_2Eprod(arr(A_27a,arr(A_27a,A_27a)),ty_2Epair_2Eprod(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,A_27a)))))))
=> ! [V8a4: $i] : ( ap(f1840(A_27a,V6a2,V4a0,V5a1,V7a3,V3a0_27),V8a4) = ap(ap(c_2Emin_2E_3D(ty_2Eind__type_2Erecspace(ty_2Epair_2Eprod(A_27a,ty_2Epair_2Eprod(A_27a,ty_2Epair_2Eprod(arr(A_27a,arr(A_27a,A_27a)),ty_2Epair_2Eprod(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,A_27a))))))),V3a0_27),ap(ap(ap(ap(ap(f1839(A_27a),V4a0),V5a1),V6a2),V7a3),V8a4)) ) ) ) ) ) ) ).
tff(lamtp_f1841,type,
f1841: ( del * $i * $i * $i * $i ) > $i ).
tff(lameq_f1841,axiom,
! [A_27a: del,V3a0_27: $i] :
( mem(V3a0_27,ty_2Eind__type_2Erecspace(ty_2Epair_2Eprod(A_27a,ty_2Epair_2Eprod(A_27a,ty_2Epair_2Eprod(arr(A_27a,arr(A_27a,A_27a)),ty_2Epair_2Eprod(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,A_27a)))))))
=> ! [V5a1: $i] :
( mem(V5a1,A_27a)
=> ! [V4a0: $i] :
( mem(V4a0,A_27a)
=> ! [V6a2: $i] :
( mem(V6a2,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V7a3: $i] : ( ap(f1841(A_27a,V3a0_27,V5a1,V4a0,V6a2),V7a3) = ap(c_2Ebool_2E_3F(arr(A_27a,A_27a)),f1840(A_27a,V6a2,V4a0,V5a1,V7a3,V3a0_27)) ) ) ) ) ) ).
tff(lamtp_f1842,type,
f1842: ( del * $i * $i * $i ) > $i ).
tff(lameq_f1842,axiom,
! [A_27a: del,V4a0: $i] :
( mem(V4a0,A_27a)
=> ! [V5a1: $i] :
( mem(V5a1,A_27a)
=> ! [V3a0_27: $i] :
( mem(V3a0_27,ty_2Eind__type_2Erecspace(ty_2Epair_2Eprod(A_27a,ty_2Epair_2Eprod(A_27a,ty_2Epair_2Eprod(arr(A_27a,arr(A_27a,A_27a)),ty_2Epair_2Eprod(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,A_27a)))))))
=> ! [V6a2: $i] : ( ap(f1842(A_27a,V4a0,V5a1,V3a0_27),V6a2) = ap(c_2Ebool_2E_3F(arr(A_27a,arr(A_27a,A_27a))),f1841(A_27a,V3a0_27,V5a1,V4a0,V6a2)) ) ) ) ) ).
tff(lamtp_f1843,type,
f1843: ( del * $i * $i ) > $i ).
tff(lameq_f1843,axiom,
! [A_27a: del,V3a0_27: $i] :
( mem(V3a0_27,ty_2Eind__type_2Erecspace(ty_2Epair_2Eprod(A_27a,ty_2Epair_2Eprod(A_27a,ty_2Epair_2Eprod(arr(A_27a,arr(A_27a,A_27a)),ty_2Epair_2Eprod(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,A_27a)))))))
=> ! [V4a0: $i] :
( mem(V4a0,A_27a)
=> ! [V5a1: $i] : ( ap(f1843(A_27a,V3a0_27,V4a0),V5a1) = ap(c_2Ebool_2E_3F(arr(A_27a,arr(A_27a,A_27a))),f1842(A_27a,V4a0,V5a1,V3a0_27)) ) ) ) ).
tff(lamtp_f1844,type,
f1844: ( del * $i ) > $i ).
tff(lameq_f1844,axiom,
! [A_27a: del,V3a0_27: $i] :
( mem(V3a0_27,ty_2Eind__type_2Erecspace(ty_2Epair_2Eprod(A_27a,ty_2Epair_2Eprod(A_27a,ty_2Epair_2Eprod(arr(A_27a,arr(A_27a,A_27a)),ty_2Epair_2Eprod(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,A_27a)))))))
=> ! [V4a0: $i] : ( ap(f1844(A_27a,V3a0_27),V4a0) = ap(c_2Ebool_2E_3F(A_27a),f1843(A_27a,V3a0_27,V4a0)) ) ) ).
tff(lamtp_f1845,type,
f1845: ( del * $i ) > $i ).
tff(lameq_f1845,axiom,
! [A_27a: del,V2_27ring_27: $i] :
( mem(V2_27ring_27,arr(ty_2Eind__type_2Erecspace(ty_2Epair_2Eprod(A_27a,ty_2Epair_2Eprod(A_27a,ty_2Epair_2Eprod(arr(A_27a,arr(A_27a,A_27a)),ty_2Epair_2Eprod(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,A_27a)))))),bool))
=> ! [V3a0_27: $i] : ( ap(f1845(A_27a,V2_27ring_27),V3a0_27) = ap(ap(c_2Emin_2E_3D_3D_3E,ap(c_2Ebool_2E_3F(A_27a),f1844(A_27a,V3a0_27))),ap(V2_27ring_27,V3a0_27)) ) ) ).
tff(lamtp_f1846,type,
f1846: ( del * $i ) > $i ).
tff(lameq_f1846,axiom,
! [A_27a: del,V1a0_27: $i] :
( mem(V1a0_27,ty_2Eind__type_2Erecspace(ty_2Epair_2Eprod(A_27a,ty_2Epair_2Eprod(A_27a,ty_2Epair_2Eprod(arr(A_27a,arr(A_27a,A_27a)),ty_2Epair_2Eprod(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,A_27a)))))))
=> ! [V2_27ring_27: $i] : ( ap(f1846(A_27a,V1a0_27),V2_27ring_27) = ap(ap(c_2Emin_2E_3D_3D_3E,ap(c_2Ebool_2E_21(ty_2Eind__type_2Erecspace(ty_2Epair_2Eprod(A_27a,ty_2Epair_2Eprod(A_27a,ty_2Epair_2Eprod(arr(A_27a,arr(A_27a,A_27a)),ty_2Epair_2Eprod(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,A_27a))))))),f1845(A_27a,V2_27ring_27))),ap(V2_27ring_27,V1a0_27)) ) ) ).
tff(lamtp_f1847,type,
f1847: del > $i ).
tff(lameq_f1847,axiom,
! [A_27a: del,V1a0_27: $i] : ( ap(f1847(A_27a),V1a0_27) = ap(c_2Ebool_2E_21(arr(ty_2Eind__type_2Erecspace(ty_2Epair_2Eprod(A_27a,ty_2Epair_2Eprod(A_27a,ty_2Epair_2Eprod(arr(A_27a,arr(A_27a,A_27a)),ty_2Epair_2Eprod(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,A_27a)))))),bool)),f1846(A_27a,V1a0_27)) ) ).
tff(ax_thm_2Ering_2Ering__TY__DEF,axiom,
! [A_27a: del] :
? [V0rep: $i] :
( mem(V0rep,arr(ty_2Ering_2Ering(A_27a),ty_2Eind__type_2Erecspace(ty_2Epair_2Eprod(A_27a,ty_2Epair_2Eprod(A_27a,ty_2Epair_2Eprod(arr(A_27a,arr(A_27a,A_27a)),ty_2Epair_2Eprod(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,A_27a))))))))
& p(ap(ap(c_2Ebool_2ETYPE__DEFINITION(ty_2Eind__type_2Erecspace(ty_2Epair_2Eprod(A_27a,ty_2Epair_2Eprod(A_27a,ty_2Epair_2Eprod(arr(A_27a,arr(A_27a,A_27a)),ty_2Epair_2Eprod(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,A_27a)))))),ty_2Ering_2Ering(A_27a)),f1847(A_27a)),V0rep)) ) ).
tff(ax_thm_2Ering_2Ering__case__def,axiom,
! [A_27a: del,A_27b: del,V0a0: $i] :
( mem(V0a0,A_27a)
=> ! [V1a1: $i] :
( mem(V1a1,A_27a)
=> ! [V2a2: $i] :
( mem(V2a2,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V3a3: $i] :
( mem(V3a3,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V4a4: $i] :
( mem(V4a4,arr(A_27a,A_27a))
=> ! [V5f: $i] :
( mem(V5f,arr(A_27a,arr(A_27a,arr(arr(A_27a,arr(A_27a,A_27a)),arr(arr(A_27a,arr(A_27a,A_27a)),arr(arr(A_27a,A_27a),A_27b))))))
=> ( ap(ap(c_2Ering_2Ering__CASE(A_27a,A_27b),ap(ap(ap(ap(ap(c_2Ering_2Erecordtype_2Ering(A_27a),V0a0),V1a1),V2a2),V3a3),V4a4)),V5f) = ap(ap(ap(ap(ap(V5f,V0a0),V1a1),V2a2),V3a3),V4a4) ) ) ) ) ) ) ) ).
tff(ax_thm_2Ering_2Ering__size__def,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1a0: $i] :
( mem(V1a0,A_27a)
=> ! [V2a1: $i] :
( mem(V2a1,A_27a)
=> ! [V3a2: $i] :
( mem(V3a2,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V4a3: $i] :
( mem(V4a3,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V5a4: $i] :
( mem(V5a4,arr(A_27a,A_27a))
=> ( surj__ty_2Enum_2Enum(ap(ap(c_2Ering_2Ering__size(A_27a),V0f),ap(ap(ap(ap(ap(c_2Ering_2Erecordtype_2Ering(A_27a),V1a0),V2a1),V3a2),V4a3),V5a4))) = surj__ty_2Enum_2Enum(ap(ap(c_2Earithmetic_2E_2B,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))),ap(ap(c_2Earithmetic_2E_2B,ap(V0f,V1a0)),ap(V0f,V2a1)))) ) ) ) ) ) ) ) ).
tff(ax_thm_2Ering_2Ering__R0,axiom,
! [A_27a: del,V0a: $i] :
( mem(V0a,A_27a)
=> ! [V1a0: $i] :
( mem(V1a0,A_27a)
=> ! [V2f: $i] :
( mem(V2f,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V3f0: $i] :
( mem(V3f0,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V4f1: $i] :
( mem(V4f1,arr(A_27a,A_27a))
=> ( ap(c_2Ering_2Ering__R0(A_27a),ap(ap(ap(ap(ap(c_2Ering_2Erecordtype_2Ering(A_27a),V0a),V1a0),V2f),V3f0),V4f1)) = V0a ) ) ) ) ) ) ).
tff(ax_thm_2Ering_2Ering__R1,axiom,
! [A_27a: del,V0a: $i] :
( mem(V0a,A_27a)
=> ! [V1a0: $i] :
( mem(V1a0,A_27a)
=> ! [V2f: $i] :
( mem(V2f,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V3f0: $i] :
( mem(V3f0,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V4f1: $i] :
( mem(V4f1,arr(A_27a,A_27a))
=> ( ap(c_2Ering_2Ering__R1(A_27a),ap(ap(ap(ap(ap(c_2Ering_2Erecordtype_2Ering(A_27a),V0a),V1a0),V2f),V3f0),V4f1)) = V1a0 ) ) ) ) ) ) ).
tff(ax_thm_2Ering_2Ering__RP,axiom,
! [A_27a: del,V0a: $i] :
( mem(V0a,A_27a)
=> ! [V1a0: $i] :
( mem(V1a0,A_27a)
=> ! [V2f: $i] :
( mem(V2f,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V3f0: $i] :
( mem(V3f0,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V4f1: $i] :
( mem(V4f1,arr(A_27a,A_27a))
=> ( ap(c_2Ering_2Ering__RP(A_27a),ap(ap(ap(ap(ap(c_2Ering_2Erecordtype_2Ering(A_27a),V0a),V1a0),V2f),V3f0),V4f1)) = V2f ) ) ) ) ) ) ).
tff(ax_thm_2Ering_2Ering__RM,axiom,
! [A_27a: del,V0a: $i] :
( mem(V0a,A_27a)
=> ! [V1a0: $i] :
( mem(V1a0,A_27a)
=> ! [V2f: $i] :
( mem(V2f,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V3f0: $i] :
( mem(V3f0,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V4f1: $i] :
( mem(V4f1,arr(A_27a,A_27a))
=> ( ap(c_2Ering_2Ering__RM(A_27a),ap(ap(ap(ap(ap(c_2Ering_2Erecordtype_2Ering(A_27a),V0a),V1a0),V2f),V3f0),V4f1)) = V3f0 ) ) ) ) ) ) ).
tff(ax_thm_2Ering_2Ering__RN,axiom,
! [A_27a: del,V0a: $i] :
( mem(V0a,A_27a)
=> ! [V1a0: $i] :
( mem(V1a0,A_27a)
=> ! [V2f: $i] :
( mem(V2f,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V3f0: $i] :
( mem(V3f0,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V4f1: $i] :
( mem(V4f1,arr(A_27a,A_27a))
=> ( ap(c_2Ering_2Ering__RN(A_27a),ap(ap(ap(ap(ap(c_2Ering_2Erecordtype_2Ering(A_27a),V0a),V1a0),V2f),V3f0),V4f1)) = V4f1 ) ) ) ) ) ) ).
tff(conj_thm_2Ering_2Ering__accessors,axiom,
! [A_27a: del] :
( ! [V0a: $i] :
( mem(V0a,A_27a)
=> ! [V1a0: $i] :
( mem(V1a0,A_27a)
=> ! [V2f: $i] :
( mem(V2f,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V3f0: $i] :
( mem(V3f0,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V4f1: $i] :
( mem(V4f1,arr(A_27a,A_27a))
=> ( ap(c_2Ering_2Ering__R0(A_27a),ap(ap(ap(ap(ap(c_2Ering_2Erecordtype_2Ering(A_27a),V0a),V1a0),V2f),V3f0),V4f1)) = V0a ) ) ) ) ) )
& ! [V5a: $i] :
( mem(V5a,A_27a)
=> ! [V6a0: $i] :
( mem(V6a0,A_27a)
=> ! [V7f: $i] :
( mem(V7f,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V8f0: $i] :
( mem(V8f0,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V9f1: $i] :
( mem(V9f1,arr(A_27a,A_27a))
=> ( ap(c_2Ering_2Ering__R1(A_27a),ap(ap(ap(ap(ap(c_2Ering_2Erecordtype_2Ering(A_27a),V5a),V6a0),V7f),V8f0),V9f1)) = V6a0 ) ) ) ) ) )
& ! [V10a: $i] :
( mem(V10a,A_27a)
=> ! [V11a0: $i] :
( mem(V11a0,A_27a)
=> ! [V12f: $i] :
( mem(V12f,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V13f0: $i] :
( mem(V13f0,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V14f1: $i] :
( mem(V14f1,arr(A_27a,A_27a))
=> ( ap(c_2Ering_2Ering__RP(A_27a),ap(ap(ap(ap(ap(c_2Ering_2Erecordtype_2Ering(A_27a),V10a),V11a0),V12f),V13f0),V14f1)) = V12f ) ) ) ) ) )
& ! [V15a: $i] :
( mem(V15a,A_27a)
=> ! [V16a0: $i] :
( mem(V16a0,A_27a)
=> ! [V17f: $i] :
( mem(V17f,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V18f0: $i] :
( mem(V18f0,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V19f1: $i] :
( mem(V19f1,arr(A_27a,A_27a))
=> ( ap(c_2Ering_2Ering__RM(A_27a),ap(ap(ap(ap(ap(c_2Ering_2Erecordtype_2Ering(A_27a),V15a),V16a0),V17f),V18f0),V19f1)) = V18f0 ) ) ) ) ) )
& ! [V20a: $i] :
( mem(V20a,A_27a)
=> ! [V21a0: $i] :
( mem(V21a0,A_27a)
=> ! [V22f: $i] :
( mem(V22f,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V23f0: $i] :
( mem(V23f0,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V24f1: $i] :
( mem(V24f1,arr(A_27a,A_27a))
=> ( ap(c_2Ering_2Ering__RN(A_27a),ap(ap(ap(ap(ap(c_2Ering_2Erecordtype_2Ering(A_27a),V20a),V21a0),V22f),V23f0),V24f1)) = V24f1 ) ) ) ) ) ) ) ).
tff(ax_thm_2Ering_2Ering__R0__fupd,axiom,
! [A_27a: del,V0f2: $i] :
( mem(V0f2,arr(A_27a,A_27a))
=> ! [V1a: $i] :
( mem(V1a,A_27a)
=> ! [V2a0: $i] :
( mem(V2a0,A_27a)
=> ! [V3f: $i] :
( mem(V3f,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V4f0: $i] :
( mem(V4f0,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V5f1: $i] :
( mem(V5f1,arr(A_27a,A_27a))
=> ( ap(ap(c_2Ering_2Ering__R0__fupd(A_27a),V0f2),ap(ap(ap(ap(ap(c_2Ering_2Erecordtype_2Ering(A_27a),V1a),V2a0),V3f),V4f0),V5f1)) = ap(ap(ap(ap(ap(c_2Ering_2Erecordtype_2Ering(A_27a),ap(V0f2,V1a)),V2a0),V3f),V4f0),V5f1) ) ) ) ) ) ) ) ).
tff(ax_thm_2Ering_2Ering__R1__fupd,axiom,
! [A_27a: del,V0f2: $i] :
( mem(V0f2,arr(A_27a,A_27a))
=> ! [V1a: $i] :
( mem(V1a,A_27a)
=> ! [V2a0: $i] :
( mem(V2a0,A_27a)
=> ! [V3f: $i] :
( mem(V3f,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V4f0: $i] :
( mem(V4f0,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V5f1: $i] :
( mem(V5f1,arr(A_27a,A_27a))
=> ( ap(ap(c_2Ering_2Ering__R1__fupd(A_27a),V0f2),ap(ap(ap(ap(ap(c_2Ering_2Erecordtype_2Ering(A_27a),V1a),V2a0),V3f),V4f0),V5f1)) = ap(ap(ap(ap(ap(c_2Ering_2Erecordtype_2Ering(A_27a),V1a),ap(V0f2,V2a0)),V3f),V4f0),V5f1) ) ) ) ) ) ) ) ).
tff(ax_thm_2Ering_2Ering__RP__fupd,axiom,
! [A_27a: del,V0f2: $i] :
( mem(V0f2,arr(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,arr(A_27a,A_27a))))
=> ! [V1a: $i] :
( mem(V1a,A_27a)
=> ! [V2a0: $i] :
( mem(V2a0,A_27a)
=> ! [V3f: $i] :
( mem(V3f,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V4f0: $i] :
( mem(V4f0,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V5f1: $i] :
( mem(V5f1,arr(A_27a,A_27a))
=> ( ap(ap(c_2Ering_2Ering__RP__fupd(A_27a),V0f2),ap(ap(ap(ap(ap(c_2Ering_2Erecordtype_2Ering(A_27a),V1a),V2a0),V3f),V4f0),V5f1)) = ap(ap(ap(ap(ap(c_2Ering_2Erecordtype_2Ering(A_27a),V1a),V2a0),ap(V0f2,V3f)),V4f0),V5f1) ) ) ) ) ) ) ) ).
tff(ax_thm_2Ering_2Ering__RM__fupd,axiom,
! [A_27a: del,V0f2: $i] :
( mem(V0f2,arr(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,arr(A_27a,A_27a))))
=> ! [V1a: $i] :
( mem(V1a,A_27a)
=> ! [V2a0: $i] :
( mem(V2a0,A_27a)
=> ! [V3f: $i] :
( mem(V3f,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V4f0: $i] :
( mem(V4f0,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V5f1: $i] :
( mem(V5f1,arr(A_27a,A_27a))
=> ( ap(ap(c_2Ering_2Ering__RM__fupd(A_27a),V0f2),ap(ap(ap(ap(ap(c_2Ering_2Erecordtype_2Ering(A_27a),V1a),V2a0),V3f),V4f0),V5f1)) = ap(ap(ap(ap(ap(c_2Ering_2Erecordtype_2Ering(A_27a),V1a),V2a0),V3f),ap(V0f2,V4f0)),V5f1) ) ) ) ) ) ) ) ).
tff(ax_thm_2Ering_2Ering__RN__fupd,axiom,
! [A_27a: del,V0f2: $i] :
( mem(V0f2,arr(arr(A_27a,A_27a),arr(A_27a,A_27a)))
=> ! [V1a: $i] :
( mem(V1a,A_27a)
=> ! [V2a0: $i] :
( mem(V2a0,A_27a)
=> ! [V3f: $i] :
( mem(V3f,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V4f0: $i] :
( mem(V4f0,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V5f1: $i] :
( mem(V5f1,arr(A_27a,A_27a))
=> ( ap(ap(c_2Ering_2Ering__RN__fupd(A_27a),V0f2),ap(ap(ap(ap(ap(c_2Ering_2Erecordtype_2Ering(A_27a),V1a),V2a0),V3f),V4f0),V5f1)) = ap(ap(ap(ap(ap(c_2Ering_2Erecordtype_2Ering(A_27a),V1a),V2a0),V3f),V4f0),ap(V0f2,V5f1)) ) ) ) ) ) ) ) ).
tff(conj_thm_2Ering_2Ering__fn__updates,axiom,
! [A_27a: del] :
( ! [V0f2: $i] :
( mem(V0f2,arr(A_27a,A_27a))
=> ! [V1a: $i] :
( mem(V1a,A_27a)
=> ! [V2a0: $i] :
( mem(V2a0,A_27a)
=> ! [V3f: $i] :
( mem(V3f,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V4f0: $i] :
( mem(V4f0,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V5f1: $i] :
( mem(V5f1,arr(A_27a,A_27a))
=> ( ap(ap(c_2Ering_2Ering__R0__fupd(A_27a),V0f2),ap(ap(ap(ap(ap(c_2Ering_2Erecordtype_2Ering(A_27a),V1a),V2a0),V3f),V4f0),V5f1)) = ap(ap(ap(ap(ap(c_2Ering_2Erecordtype_2Ering(A_27a),ap(V0f2,V1a)),V2a0),V3f),V4f0),V5f1) ) ) ) ) ) ) )
& ! [V6f2: $i] :
( mem(V6f2,arr(A_27a,A_27a))
=> ! [V7a: $i] :
( mem(V7a,A_27a)
=> ! [V8a0: $i] :
( mem(V8a0,A_27a)
=> ! [V9f: $i] :
( mem(V9f,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V10f0: $i] :
( mem(V10f0,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V11f1: $i] :
( mem(V11f1,arr(A_27a,A_27a))
=> ( ap(ap(c_2Ering_2Ering__R1__fupd(A_27a),V6f2),ap(ap(ap(ap(ap(c_2Ering_2Erecordtype_2Ering(A_27a),V7a),V8a0),V9f),V10f0),V11f1)) = ap(ap(ap(ap(ap(c_2Ering_2Erecordtype_2Ering(A_27a),V7a),ap(V6f2,V8a0)),V9f),V10f0),V11f1) ) ) ) ) ) ) )
& ! [V12f2: $i] :
( mem(V12f2,arr(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,arr(A_27a,A_27a))))
=> ! [V13a: $i] :
( mem(V13a,A_27a)
=> ! [V14a0: $i] :
( mem(V14a0,A_27a)
=> ! [V15f: $i] :
( mem(V15f,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V16f0: $i] :
( mem(V16f0,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V17f1: $i] :
( mem(V17f1,arr(A_27a,A_27a))
=> ( ap(ap(c_2Ering_2Ering__RP__fupd(A_27a),V12f2),ap(ap(ap(ap(ap(c_2Ering_2Erecordtype_2Ering(A_27a),V13a),V14a0),V15f),V16f0),V17f1)) = ap(ap(ap(ap(ap(c_2Ering_2Erecordtype_2Ering(A_27a),V13a),V14a0),ap(V12f2,V15f)),V16f0),V17f1) ) ) ) ) ) ) )
& ! [V18f2: $i] :
( mem(V18f2,arr(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,arr(A_27a,A_27a))))
=> ! [V19a: $i] :
( mem(V19a,A_27a)
=> ! [V20a0: $i] :
( mem(V20a0,A_27a)
=> ! [V21f: $i] :
( mem(V21f,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V22f0: $i] :
( mem(V22f0,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V23f1: $i] :
( mem(V23f1,arr(A_27a,A_27a))
=> ( ap(ap(c_2Ering_2Ering__RM__fupd(A_27a),V18f2),ap(ap(ap(ap(ap(c_2Ering_2Erecordtype_2Ering(A_27a),V19a),V20a0),V21f),V22f0),V23f1)) = ap(ap(ap(ap(ap(c_2Ering_2Erecordtype_2Ering(A_27a),V19a),V20a0),V21f),ap(V18f2,V22f0)),V23f1) ) ) ) ) ) ) )
& ! [V24f2: $i] :
( mem(V24f2,arr(arr(A_27a,A_27a),arr(A_27a,A_27a)))
=> ! [V25a: $i] :
( mem(V25a,A_27a)
=> ! [V26a0: $i] :
( mem(V26a0,A_27a)
=> ! [V27f: $i] :
( mem(V27f,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V28f0: $i] :
( mem(V28f0,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V29f1: $i] :
( mem(V29f1,arr(A_27a,A_27a))
=> ( ap(ap(c_2Ering_2Ering__RN__fupd(A_27a),V24f2),ap(ap(ap(ap(ap(c_2Ering_2Erecordtype_2Ering(A_27a),V25a),V26a0),V27f),V28f0),V29f1)) = ap(ap(ap(ap(ap(c_2Ering_2Erecordtype_2Ering(A_27a),V25a),V26a0),V27f),V28f0),ap(V24f2,V29f1)) ) ) ) ) ) ) ) ) ).
tff(conj_thm_2Ering_2Ering__accfupds,axiom,
! [A_27a: del] :
( ! [V0r: $i] :
( mem(V0r,ty_2Ering_2Ering(A_27a))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,A_27a))
=> ( ap(c_2Ering_2Ering__R0(A_27a),ap(ap(c_2Ering_2Ering__R1__fupd(A_27a),V1f),V0r)) = ap(c_2Ering_2Ering__R0(A_27a),V0r) ) ) )
& ! [V2r: $i] :
( mem(V2r,ty_2Ering_2Ering(A_27a))
=> ! [V3f: $i] :
( mem(V3f,arr(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,arr(A_27a,A_27a))))
=> ( ap(c_2Ering_2Ering__R0(A_27a),ap(ap(c_2Ering_2Ering__RP__fupd(A_27a),V3f),V2r)) = ap(c_2Ering_2Ering__R0(A_27a),V2r) ) ) )
& ! [V4r: $i] :
( mem(V4r,ty_2Ering_2Ering(A_27a))
=> ! [V5f: $i] :
( mem(V5f,arr(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,arr(A_27a,A_27a))))
=> ( ap(c_2Ering_2Ering__R0(A_27a),ap(ap(c_2Ering_2Ering__RM__fupd(A_27a),V5f),V4r)) = ap(c_2Ering_2Ering__R0(A_27a),V4r) ) ) )
& ! [V6r: $i] :
( mem(V6r,ty_2Ering_2Ering(A_27a))
=> ! [V7f: $i] :
( mem(V7f,arr(arr(A_27a,A_27a),arr(A_27a,A_27a)))
=> ( ap(c_2Ering_2Ering__R0(A_27a),ap(ap(c_2Ering_2Ering__RN__fupd(A_27a),V7f),V6r)) = ap(c_2Ering_2Ering__R0(A_27a),V6r) ) ) )
& ! [V8r: $i] :
( mem(V8r,ty_2Ering_2Ering(A_27a))
=> ! [V9f: $i] :
( mem(V9f,arr(A_27a,A_27a))
=> ( ap(c_2Ering_2Ering__R1(A_27a),ap(ap(c_2Ering_2Ering__R0__fupd(A_27a),V9f),V8r)) = ap(c_2Ering_2Ering__R1(A_27a),V8r) ) ) )
& ! [V10r: $i] :
( mem(V10r,ty_2Ering_2Ering(A_27a))
=> ! [V11f: $i] :
( mem(V11f,arr(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,arr(A_27a,A_27a))))
=> ( ap(c_2Ering_2Ering__R1(A_27a),ap(ap(c_2Ering_2Ering__RP__fupd(A_27a),V11f),V10r)) = ap(c_2Ering_2Ering__R1(A_27a),V10r) ) ) )
& ! [V12r: $i] :
( mem(V12r,ty_2Ering_2Ering(A_27a))
=> ! [V13f: $i] :
( mem(V13f,arr(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,arr(A_27a,A_27a))))
=> ( ap(c_2Ering_2Ering__R1(A_27a),ap(ap(c_2Ering_2Ering__RM__fupd(A_27a),V13f),V12r)) = ap(c_2Ering_2Ering__R1(A_27a),V12r) ) ) )
& ! [V14r: $i] :
( mem(V14r,ty_2Ering_2Ering(A_27a))
=> ! [V15f: $i] :
( mem(V15f,arr(arr(A_27a,A_27a),arr(A_27a,A_27a)))
=> ( ap(c_2Ering_2Ering__R1(A_27a),ap(ap(c_2Ering_2Ering__RN__fupd(A_27a),V15f),V14r)) = ap(c_2Ering_2Ering__R1(A_27a),V14r) ) ) )
& ! [V16r: $i] :
( mem(V16r,ty_2Ering_2Ering(A_27a))
=> ! [V17f: $i] :
( mem(V17f,arr(A_27a,A_27a))
=> ( ap(c_2Ering_2Ering__RP(A_27a),ap(ap(c_2Ering_2Ering__R0__fupd(A_27a),V17f),V16r)) = ap(c_2Ering_2Ering__RP(A_27a),V16r) ) ) )
& ! [V18r: $i] :
( mem(V18r,ty_2Ering_2Ering(A_27a))
=> ! [V19f: $i] :
( mem(V19f,arr(A_27a,A_27a))
=> ( ap(c_2Ering_2Ering__RP(A_27a),ap(ap(c_2Ering_2Ering__R1__fupd(A_27a),V19f),V18r)) = ap(c_2Ering_2Ering__RP(A_27a),V18r) ) ) )
& ! [V20r: $i] :
( mem(V20r,ty_2Ering_2Ering(A_27a))
=> ! [V21f: $i] :
( mem(V21f,arr(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,arr(A_27a,A_27a))))
=> ( ap(c_2Ering_2Ering__RP(A_27a),ap(ap(c_2Ering_2Ering__RM__fupd(A_27a),V21f),V20r)) = ap(c_2Ering_2Ering__RP(A_27a),V20r) ) ) )
& ! [V22r: $i] :
( mem(V22r,ty_2Ering_2Ering(A_27a))
=> ! [V23f: $i] :
( mem(V23f,arr(arr(A_27a,A_27a),arr(A_27a,A_27a)))
=> ( ap(c_2Ering_2Ering__RP(A_27a),ap(ap(c_2Ering_2Ering__RN__fupd(A_27a),V23f),V22r)) = ap(c_2Ering_2Ering__RP(A_27a),V22r) ) ) )
& ! [V24r: $i] :
( mem(V24r,ty_2Ering_2Ering(A_27a))
=> ! [V25f: $i] :
( mem(V25f,arr(A_27a,A_27a))
=> ( ap(c_2Ering_2Ering__RM(A_27a),ap(ap(c_2Ering_2Ering__R0__fupd(A_27a),V25f),V24r)) = ap(c_2Ering_2Ering__RM(A_27a),V24r) ) ) )
& ! [V26r: $i] :
( mem(V26r,ty_2Ering_2Ering(A_27a))
=> ! [V27f: $i] :
( mem(V27f,arr(A_27a,A_27a))
=> ( ap(c_2Ering_2Ering__RM(A_27a),ap(ap(c_2Ering_2Ering__R1__fupd(A_27a),V27f),V26r)) = ap(c_2Ering_2Ering__RM(A_27a),V26r) ) ) )
& ! [V28r: $i] :
( mem(V28r,ty_2Ering_2Ering(A_27a))
=> ! [V29f: $i] :
( mem(V29f,arr(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,arr(A_27a,A_27a))))
=> ( ap(c_2Ering_2Ering__RM(A_27a),ap(ap(c_2Ering_2Ering__RP__fupd(A_27a),V29f),V28r)) = ap(c_2Ering_2Ering__RM(A_27a),V28r) ) ) )
& ! [V30r: $i] :
( mem(V30r,ty_2Ering_2Ering(A_27a))
=> ! [V31f: $i] :
( mem(V31f,arr(arr(A_27a,A_27a),arr(A_27a,A_27a)))
=> ( ap(c_2Ering_2Ering__RM(A_27a),ap(ap(c_2Ering_2Ering__RN__fupd(A_27a),V31f),V30r)) = ap(c_2Ering_2Ering__RM(A_27a),V30r) ) ) )
& ! [V32r: $i] :
( mem(V32r,ty_2Ering_2Ering(A_27a))
=> ! [V33f: $i] :
( mem(V33f,arr(A_27a,A_27a))
=> ( ap(c_2Ering_2Ering__RN(A_27a),ap(ap(c_2Ering_2Ering__R0__fupd(A_27a),V33f),V32r)) = ap(c_2Ering_2Ering__RN(A_27a),V32r) ) ) )
& ! [V34r: $i] :
( mem(V34r,ty_2Ering_2Ering(A_27a))
=> ! [V35f: $i] :
( mem(V35f,arr(A_27a,A_27a))
=> ( ap(c_2Ering_2Ering__RN(A_27a),ap(ap(c_2Ering_2Ering__R1__fupd(A_27a),V35f),V34r)) = ap(c_2Ering_2Ering__RN(A_27a),V34r) ) ) )
& ! [V36r: $i] :
( mem(V36r,ty_2Ering_2Ering(A_27a))
=> ! [V37f: $i] :
( mem(V37f,arr(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,arr(A_27a,A_27a))))
=> ( ap(c_2Ering_2Ering__RN(A_27a),ap(ap(c_2Ering_2Ering__RP__fupd(A_27a),V37f),V36r)) = ap(c_2Ering_2Ering__RN(A_27a),V36r) ) ) )
& ! [V38r: $i] :
( mem(V38r,ty_2Ering_2Ering(A_27a))
=> ! [V39f: $i] :
( mem(V39f,arr(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,arr(A_27a,A_27a))))
=> ( ap(c_2Ering_2Ering__RN(A_27a),ap(ap(c_2Ering_2Ering__RM__fupd(A_27a),V39f),V38r)) = ap(c_2Ering_2Ering__RN(A_27a),V38r) ) ) )
& ! [V40r: $i] :
( mem(V40r,ty_2Ering_2Ering(A_27a))
=> ! [V41f: $i] :
( mem(V41f,arr(A_27a,A_27a))
=> ( ap(c_2Ering_2Ering__R0(A_27a),ap(ap(c_2Ering_2Ering__R0__fupd(A_27a),V41f),V40r)) = ap(V41f,ap(c_2Ering_2Ering__R0(A_27a),V40r)) ) ) )
& ! [V42r: $i] :
( mem(V42r,ty_2Ering_2Ering(A_27a))
=> ! [V43f: $i] :
( mem(V43f,arr(A_27a,A_27a))
=> ( ap(c_2Ering_2Ering__R1(A_27a),ap(ap(c_2Ering_2Ering__R1__fupd(A_27a),V43f),V42r)) = ap(V43f,ap(c_2Ering_2Ering__R1(A_27a),V42r)) ) ) )
& ! [V44r: $i] :
( mem(V44r,ty_2Ering_2Ering(A_27a))
=> ! [V45f: $i] :
( mem(V45f,arr(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,arr(A_27a,A_27a))))
=> ( ap(c_2Ering_2Ering__RP(A_27a),ap(ap(c_2Ering_2Ering__RP__fupd(A_27a),V45f),V44r)) = ap(V45f,ap(c_2Ering_2Ering__RP(A_27a),V44r)) ) ) )
& ! [V46r: $i] :
( mem(V46r,ty_2Ering_2Ering(A_27a))
=> ! [V47f: $i] :
( mem(V47f,arr(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,arr(A_27a,A_27a))))
=> ( ap(c_2Ering_2Ering__RM(A_27a),ap(ap(c_2Ering_2Ering__RM__fupd(A_27a),V47f),V46r)) = ap(V47f,ap(c_2Ering_2Ering__RM(A_27a),V46r)) ) ) )
& ! [V48r: $i] :
( mem(V48r,ty_2Ering_2Ering(A_27a))
=> ! [V49f: $i] :
( mem(V49f,arr(arr(A_27a,A_27a),arr(A_27a,A_27a)))
=> ( ap(c_2Ering_2Ering__RN(A_27a),ap(ap(c_2Ering_2Ering__RN__fupd(A_27a),V49f),V48r)) = ap(V49f,ap(c_2Ering_2Ering__RN(A_27a),V48r)) ) ) ) ) ).
tff(conj_thm_2Ering_2Ering__fupdfupds,axiom,
! [A_27a: del] :
( ! [V0r: $i] :
( mem(V0r,ty_2Ering_2Ering(A_27a))
=> ! [V1g: $i] :
( mem(V1g,arr(A_27a,A_27a))
=> ! [V2f: $i] :
( mem(V2f,arr(A_27a,A_27a))
=> ( ap(ap(c_2Ering_2Ering__R0__fupd(A_27a),V2f),ap(ap(c_2Ering_2Ering__R0__fupd(A_27a),V1g),V0r)) = ap(ap(c_2Ering_2Ering__R0__fupd(A_27a),ap(ap(c_2Ecombin_2Eo(A_27a,A_27a,A_27a),V2f),V1g)),V0r) ) ) ) )
& ! [V3r: $i] :
( mem(V3r,ty_2Ering_2Ering(A_27a))
=> ! [V4g: $i] :
( mem(V4g,arr(A_27a,A_27a))
=> ! [V5f: $i] :
( mem(V5f,arr(A_27a,A_27a))
=> ( ap(ap(c_2Ering_2Ering__R1__fupd(A_27a),V5f),ap(ap(c_2Ering_2Ering__R1__fupd(A_27a),V4g),V3r)) = ap(ap(c_2Ering_2Ering__R1__fupd(A_27a),ap(ap(c_2Ecombin_2Eo(A_27a,A_27a,A_27a),V5f),V4g)),V3r) ) ) ) )
& ! [V6r: $i] :
( mem(V6r,ty_2Ering_2Ering(A_27a))
=> ! [V7g: $i] :
( mem(V7g,arr(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,arr(A_27a,A_27a))))
=> ! [V8f: $i] :
( mem(V8f,arr(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,arr(A_27a,A_27a))))
=> ( ap(ap(c_2Ering_2Ering__RP__fupd(A_27a),V8f),ap(ap(c_2Ering_2Ering__RP__fupd(A_27a),V7g),V6r)) = ap(ap(c_2Ering_2Ering__RP__fupd(A_27a),ap(ap(c_2Ecombin_2Eo(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,arr(A_27a,A_27a))),V8f),V7g)),V6r) ) ) ) )
& ! [V9r: $i] :
( mem(V9r,ty_2Ering_2Ering(A_27a))
=> ! [V10g: $i] :
( mem(V10g,arr(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,arr(A_27a,A_27a))))
=> ! [V11f: $i] :
( mem(V11f,arr(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,arr(A_27a,A_27a))))
=> ( ap(ap(c_2Ering_2Ering__RM__fupd(A_27a),V11f),ap(ap(c_2Ering_2Ering__RM__fupd(A_27a),V10g),V9r)) = ap(ap(c_2Ering_2Ering__RM__fupd(A_27a),ap(ap(c_2Ecombin_2Eo(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,arr(A_27a,A_27a))),V11f),V10g)),V9r) ) ) ) )
& ! [V12r: $i] :
( mem(V12r,ty_2Ering_2Ering(A_27a))
=> ! [V13g: $i] :
( mem(V13g,arr(arr(A_27a,A_27a),arr(A_27a,A_27a)))
=> ! [V14f: $i] :
( mem(V14f,arr(arr(A_27a,A_27a),arr(A_27a,A_27a)))
=> ( ap(ap(c_2Ering_2Ering__RN__fupd(A_27a),V14f),ap(ap(c_2Ering_2Ering__RN__fupd(A_27a),V13g),V12r)) = ap(ap(c_2Ering_2Ering__RN__fupd(A_27a),ap(ap(c_2Ecombin_2Eo(arr(A_27a,A_27a),arr(A_27a,A_27a),arr(A_27a,A_27a)),V14f),V13g)),V12r) ) ) ) ) ) ).
tff(conj_thm_2Ering_2Ering__fupdfupds__comp,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0g: $i] :
( mem(V0g,arr(A_27a,A_27a))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,A_27a))
=> ( ap(ap(c_2Ecombin_2Eo(ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)),ap(c_2Ering_2Ering__R0__fupd(A_27a),V1f)),ap(c_2Ering_2Ering__R0__fupd(A_27a),V0g)) = ap(c_2Ering_2Ering__R0__fupd(A_27a),ap(ap(c_2Ecombin_2Eo(A_27a,A_27a,A_27a),V1f),V0g)) ) ) )
& ! [V2h: $i] :
( mem(V2h,arr(A_27b,ty_2Ering_2Ering(A_27a)))
=> ! [V3g: $i] :
( mem(V3g,arr(A_27a,A_27a))
=> ! [V4f: $i] :
( mem(V4f,arr(A_27a,A_27a))
=> ( ap(ap(c_2Ecombin_2Eo(A_27b,ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)),ap(c_2Ering_2Ering__R0__fupd(A_27a),V4f)),ap(ap(c_2Ecombin_2Eo(A_27b,ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)),ap(c_2Ering_2Ering__R0__fupd(A_27a),V3g)),V2h)) = ap(ap(c_2Ecombin_2Eo(A_27b,ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)),ap(c_2Ering_2Ering__R0__fupd(A_27a),ap(ap(c_2Ecombin_2Eo(A_27a,A_27a,A_27a),V4f),V3g))),V2h) ) ) ) )
& ! [V5g: $i] :
( mem(V5g,arr(A_27a,A_27a))
=> ! [V6f: $i] :
( mem(V6f,arr(A_27a,A_27a))
=> ( ap(ap(c_2Ecombin_2Eo(ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)),ap(c_2Ering_2Ering__R1__fupd(A_27a),V6f)),ap(c_2Ering_2Ering__R1__fupd(A_27a),V5g)) = ap(c_2Ering_2Ering__R1__fupd(A_27a),ap(ap(c_2Ecombin_2Eo(A_27a,A_27a,A_27a),V6f),V5g)) ) ) )
& ! [V7h: $i] :
( mem(V7h,arr(A_27b,ty_2Ering_2Ering(A_27a)))
=> ! [V8g: $i] :
( mem(V8g,arr(A_27a,A_27a))
=> ! [V9f: $i] :
( mem(V9f,arr(A_27a,A_27a))
=> ( ap(ap(c_2Ecombin_2Eo(A_27b,ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)),ap(c_2Ering_2Ering__R1__fupd(A_27a),V9f)),ap(ap(c_2Ecombin_2Eo(A_27b,ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)),ap(c_2Ering_2Ering__R1__fupd(A_27a),V8g)),V7h)) = ap(ap(c_2Ecombin_2Eo(A_27b,ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)),ap(c_2Ering_2Ering__R1__fupd(A_27a),ap(ap(c_2Ecombin_2Eo(A_27a,A_27a,A_27a),V9f),V8g))),V7h) ) ) ) )
& ! [V10g: $i] :
( mem(V10g,arr(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,arr(A_27a,A_27a))))
=> ! [V11f: $i] :
( mem(V11f,arr(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,arr(A_27a,A_27a))))
=> ( ap(ap(c_2Ecombin_2Eo(ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)),ap(c_2Ering_2Ering__RP__fupd(A_27a),V11f)),ap(c_2Ering_2Ering__RP__fupd(A_27a),V10g)) = ap(c_2Ering_2Ering__RP__fupd(A_27a),ap(ap(c_2Ecombin_2Eo(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,arr(A_27a,A_27a))),V11f),V10g)) ) ) )
& ! [V12h: $i] :
( mem(V12h,arr(A_27b,ty_2Ering_2Ering(A_27a)))
=> ! [V13g: $i] :
( mem(V13g,arr(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,arr(A_27a,A_27a))))
=> ! [V14f: $i] :
( mem(V14f,arr(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,arr(A_27a,A_27a))))
=> ( ap(ap(c_2Ecombin_2Eo(A_27b,ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)),ap(c_2Ering_2Ering__RP__fupd(A_27a),V14f)),ap(ap(c_2Ecombin_2Eo(A_27b,ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)),ap(c_2Ering_2Ering__RP__fupd(A_27a),V13g)),V12h)) = ap(ap(c_2Ecombin_2Eo(A_27b,ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)),ap(c_2Ering_2Ering__RP__fupd(A_27a),ap(ap(c_2Ecombin_2Eo(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,arr(A_27a,A_27a))),V14f),V13g))),V12h) ) ) ) )
& ! [V15g: $i] :
( mem(V15g,arr(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,arr(A_27a,A_27a))))
=> ! [V16f: $i] :
( mem(V16f,arr(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,arr(A_27a,A_27a))))
=> ( ap(ap(c_2Ecombin_2Eo(ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)),ap(c_2Ering_2Ering__RM__fupd(A_27a),V16f)),ap(c_2Ering_2Ering__RM__fupd(A_27a),V15g)) = ap(c_2Ering_2Ering__RM__fupd(A_27a),ap(ap(c_2Ecombin_2Eo(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,arr(A_27a,A_27a))),V16f),V15g)) ) ) )
& ! [V17h: $i] :
( mem(V17h,arr(A_27b,ty_2Ering_2Ering(A_27a)))
=> ! [V18g: $i] :
( mem(V18g,arr(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,arr(A_27a,A_27a))))
=> ! [V19f: $i] :
( mem(V19f,arr(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,arr(A_27a,A_27a))))
=> ( ap(ap(c_2Ecombin_2Eo(A_27b,ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)),ap(c_2Ering_2Ering__RM__fupd(A_27a),V19f)),ap(ap(c_2Ecombin_2Eo(A_27b,ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)),ap(c_2Ering_2Ering__RM__fupd(A_27a),V18g)),V17h)) = ap(ap(c_2Ecombin_2Eo(A_27b,ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)),ap(c_2Ering_2Ering__RM__fupd(A_27a),ap(ap(c_2Ecombin_2Eo(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,arr(A_27a,A_27a))),V19f),V18g))),V17h) ) ) ) )
& ! [V20g: $i] :
( mem(V20g,arr(arr(A_27a,A_27a),arr(A_27a,A_27a)))
=> ! [V21f: $i] :
( mem(V21f,arr(arr(A_27a,A_27a),arr(A_27a,A_27a)))
=> ( ap(ap(c_2Ecombin_2Eo(ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)),ap(c_2Ering_2Ering__RN__fupd(A_27a),V21f)),ap(c_2Ering_2Ering__RN__fupd(A_27a),V20g)) = ap(c_2Ering_2Ering__RN__fupd(A_27a),ap(ap(c_2Ecombin_2Eo(arr(A_27a,A_27a),arr(A_27a,A_27a),arr(A_27a,A_27a)),V21f),V20g)) ) ) )
& ! [V22h: $i] :
( mem(V22h,arr(A_27b,ty_2Ering_2Ering(A_27a)))
=> ! [V23g: $i] :
( mem(V23g,arr(arr(A_27a,A_27a),arr(A_27a,A_27a)))
=> ! [V24f: $i] :
( mem(V24f,arr(arr(A_27a,A_27a),arr(A_27a,A_27a)))
=> ( ap(ap(c_2Ecombin_2Eo(A_27b,ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)),ap(c_2Ering_2Ering__RN__fupd(A_27a),V24f)),ap(ap(c_2Ecombin_2Eo(A_27b,ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)),ap(c_2Ering_2Ering__RN__fupd(A_27a),V23g)),V22h)) = ap(ap(c_2Ecombin_2Eo(A_27b,ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)),ap(c_2Ering_2Ering__RN__fupd(A_27a),ap(ap(c_2Ecombin_2Eo(arr(A_27a,A_27a),arr(A_27a,A_27a),arr(A_27a,A_27a)),V24f),V23g))),V22h) ) ) ) ) ) ).
tff(conj_thm_2Ering_2Ering__fupdcanon,axiom,
! [A_27a: del] :
( ! [V0r: $i] :
( mem(V0r,ty_2Ering_2Ering(A_27a))
=> ! [V1g: $i] :
( mem(V1g,arr(A_27a,A_27a))
=> ! [V2f: $i] :
( mem(V2f,arr(A_27a,A_27a))
=> ( ap(ap(c_2Ering_2Ering__R1__fupd(A_27a),V2f),ap(ap(c_2Ering_2Ering__R0__fupd(A_27a),V1g),V0r)) = ap(ap(c_2Ering_2Ering__R0__fupd(A_27a),V1g),ap(ap(c_2Ering_2Ering__R1__fupd(A_27a),V2f),V0r)) ) ) ) )
& ! [V3r: $i] :
( mem(V3r,ty_2Ering_2Ering(A_27a))
=> ! [V4g: $i] :
( mem(V4g,arr(A_27a,A_27a))
=> ! [V5f: $i] :
( mem(V5f,arr(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,arr(A_27a,A_27a))))
=> ( ap(ap(c_2Ering_2Ering__RP__fupd(A_27a),V5f),ap(ap(c_2Ering_2Ering__R0__fupd(A_27a),V4g),V3r)) = ap(ap(c_2Ering_2Ering__R0__fupd(A_27a),V4g),ap(ap(c_2Ering_2Ering__RP__fupd(A_27a),V5f),V3r)) ) ) ) )
& ! [V6r: $i] :
( mem(V6r,ty_2Ering_2Ering(A_27a))
=> ! [V7g: $i] :
( mem(V7g,arr(A_27a,A_27a))
=> ! [V8f: $i] :
( mem(V8f,arr(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,arr(A_27a,A_27a))))
=> ( ap(ap(c_2Ering_2Ering__RP__fupd(A_27a),V8f),ap(ap(c_2Ering_2Ering__R1__fupd(A_27a),V7g),V6r)) = ap(ap(c_2Ering_2Ering__R1__fupd(A_27a),V7g),ap(ap(c_2Ering_2Ering__RP__fupd(A_27a),V8f),V6r)) ) ) ) )
& ! [V9r: $i] :
( mem(V9r,ty_2Ering_2Ering(A_27a))
=> ! [V10g: $i] :
( mem(V10g,arr(A_27a,A_27a))
=> ! [V11f: $i] :
( mem(V11f,arr(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,arr(A_27a,A_27a))))
=> ( ap(ap(c_2Ering_2Ering__RM__fupd(A_27a),V11f),ap(ap(c_2Ering_2Ering__R0__fupd(A_27a),V10g),V9r)) = ap(ap(c_2Ering_2Ering__R0__fupd(A_27a),V10g),ap(ap(c_2Ering_2Ering__RM__fupd(A_27a),V11f),V9r)) ) ) ) )
& ! [V12r: $i] :
( mem(V12r,ty_2Ering_2Ering(A_27a))
=> ! [V13g: $i] :
( mem(V13g,arr(A_27a,A_27a))
=> ! [V14f: $i] :
( mem(V14f,arr(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,arr(A_27a,A_27a))))
=> ( ap(ap(c_2Ering_2Ering__RM__fupd(A_27a),V14f),ap(ap(c_2Ering_2Ering__R1__fupd(A_27a),V13g),V12r)) = ap(ap(c_2Ering_2Ering__R1__fupd(A_27a),V13g),ap(ap(c_2Ering_2Ering__RM__fupd(A_27a),V14f),V12r)) ) ) ) )
& ! [V15r: $i] :
( mem(V15r,ty_2Ering_2Ering(A_27a))
=> ! [V16g: $i] :
( mem(V16g,arr(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,arr(A_27a,A_27a))))
=> ! [V17f: $i] :
( mem(V17f,arr(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,arr(A_27a,A_27a))))
=> ( ap(ap(c_2Ering_2Ering__RM__fupd(A_27a),V17f),ap(ap(c_2Ering_2Ering__RP__fupd(A_27a),V16g),V15r)) = ap(ap(c_2Ering_2Ering__RP__fupd(A_27a),V16g),ap(ap(c_2Ering_2Ering__RM__fupd(A_27a),V17f),V15r)) ) ) ) )
& ! [V18r: $i] :
( mem(V18r,ty_2Ering_2Ering(A_27a))
=> ! [V19g: $i] :
( mem(V19g,arr(A_27a,A_27a))
=> ! [V20f: $i] :
( mem(V20f,arr(arr(A_27a,A_27a),arr(A_27a,A_27a)))
=> ( ap(ap(c_2Ering_2Ering__RN__fupd(A_27a),V20f),ap(ap(c_2Ering_2Ering__R0__fupd(A_27a),V19g),V18r)) = ap(ap(c_2Ering_2Ering__R0__fupd(A_27a),V19g),ap(ap(c_2Ering_2Ering__RN__fupd(A_27a),V20f),V18r)) ) ) ) )
& ! [V21r: $i] :
( mem(V21r,ty_2Ering_2Ering(A_27a))
=> ! [V22g: $i] :
( mem(V22g,arr(A_27a,A_27a))
=> ! [V23f: $i] :
( mem(V23f,arr(arr(A_27a,A_27a),arr(A_27a,A_27a)))
=> ( ap(ap(c_2Ering_2Ering__RN__fupd(A_27a),V23f),ap(ap(c_2Ering_2Ering__R1__fupd(A_27a),V22g),V21r)) = ap(ap(c_2Ering_2Ering__R1__fupd(A_27a),V22g),ap(ap(c_2Ering_2Ering__RN__fupd(A_27a),V23f),V21r)) ) ) ) )
& ! [V24r: $i] :
( mem(V24r,ty_2Ering_2Ering(A_27a))
=> ! [V25g: $i] :
( mem(V25g,arr(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,arr(A_27a,A_27a))))
=> ! [V26f: $i] :
( mem(V26f,arr(arr(A_27a,A_27a),arr(A_27a,A_27a)))
=> ( ap(ap(c_2Ering_2Ering__RN__fupd(A_27a),V26f),ap(ap(c_2Ering_2Ering__RP__fupd(A_27a),V25g),V24r)) = ap(ap(c_2Ering_2Ering__RP__fupd(A_27a),V25g),ap(ap(c_2Ering_2Ering__RN__fupd(A_27a),V26f),V24r)) ) ) ) )
& ! [V27r: $i] :
( mem(V27r,ty_2Ering_2Ering(A_27a))
=> ! [V28g: $i] :
( mem(V28g,arr(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,arr(A_27a,A_27a))))
=> ! [V29f: $i] :
( mem(V29f,arr(arr(A_27a,A_27a),arr(A_27a,A_27a)))
=> ( ap(ap(c_2Ering_2Ering__RN__fupd(A_27a),V29f),ap(ap(c_2Ering_2Ering__RM__fupd(A_27a),V28g),V27r)) = ap(ap(c_2Ering_2Ering__RM__fupd(A_27a),V28g),ap(ap(c_2Ering_2Ering__RN__fupd(A_27a),V29f),V27r)) ) ) ) ) ) ).
tff(conj_thm_2Ering_2Ering__fupdcanon__comp,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0g: $i] :
( mem(V0g,arr(A_27a,A_27a))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,A_27a))
=> ( ap(ap(c_2Ecombin_2Eo(ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)),ap(c_2Ering_2Ering__R1__fupd(A_27a),V1f)),ap(c_2Ering_2Ering__R0__fupd(A_27a),V0g)) = ap(ap(c_2Ecombin_2Eo(ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)),ap(c_2Ering_2Ering__R0__fupd(A_27a),V0g)),ap(c_2Ering_2Ering__R1__fupd(A_27a),V1f)) ) ) )
& ! [V2h: $i] :
( mem(V2h,arr(A_27b,ty_2Ering_2Ering(A_27a)))
=> ! [V3g: $i] :
( mem(V3g,arr(A_27a,A_27a))
=> ! [V4f: $i] :
( mem(V4f,arr(A_27a,A_27a))
=> ( ap(ap(c_2Ecombin_2Eo(A_27b,ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)),ap(c_2Ering_2Ering__R1__fupd(A_27a),V4f)),ap(ap(c_2Ecombin_2Eo(A_27b,ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)),ap(c_2Ering_2Ering__R0__fupd(A_27a),V3g)),V2h)) = ap(ap(c_2Ecombin_2Eo(A_27b,ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)),ap(c_2Ering_2Ering__R0__fupd(A_27a),V3g)),ap(ap(c_2Ecombin_2Eo(A_27b,ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)),ap(c_2Ering_2Ering__R1__fupd(A_27a),V4f)),V2h)) ) ) ) )
& ! [V5g: $i] :
( mem(V5g,arr(A_27a,A_27a))
=> ! [V6f: $i] :
( mem(V6f,arr(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,arr(A_27a,A_27a))))
=> ( ap(ap(c_2Ecombin_2Eo(ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)),ap(c_2Ering_2Ering__RP__fupd(A_27a),V6f)),ap(c_2Ering_2Ering__R0__fupd(A_27a),V5g)) = ap(ap(c_2Ecombin_2Eo(ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)),ap(c_2Ering_2Ering__R0__fupd(A_27a),V5g)),ap(c_2Ering_2Ering__RP__fupd(A_27a),V6f)) ) ) )
& ! [V7h: $i] :
( mem(V7h,arr(A_27b,ty_2Ering_2Ering(A_27a)))
=> ! [V8g: $i] :
( mem(V8g,arr(A_27a,A_27a))
=> ! [V9f: $i] :
( mem(V9f,arr(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,arr(A_27a,A_27a))))
=> ( ap(ap(c_2Ecombin_2Eo(A_27b,ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)),ap(c_2Ering_2Ering__RP__fupd(A_27a),V9f)),ap(ap(c_2Ecombin_2Eo(A_27b,ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)),ap(c_2Ering_2Ering__R0__fupd(A_27a),V8g)),V7h)) = ap(ap(c_2Ecombin_2Eo(A_27b,ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)),ap(c_2Ering_2Ering__R0__fupd(A_27a),V8g)),ap(ap(c_2Ecombin_2Eo(A_27b,ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)),ap(c_2Ering_2Ering__RP__fupd(A_27a),V9f)),V7h)) ) ) ) )
& ! [V10g: $i] :
( mem(V10g,arr(A_27a,A_27a))
=> ! [V11f: $i] :
( mem(V11f,arr(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,arr(A_27a,A_27a))))
=> ( ap(ap(c_2Ecombin_2Eo(ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)),ap(c_2Ering_2Ering__RP__fupd(A_27a),V11f)),ap(c_2Ering_2Ering__R1__fupd(A_27a),V10g)) = ap(ap(c_2Ecombin_2Eo(ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)),ap(c_2Ering_2Ering__R1__fupd(A_27a),V10g)),ap(c_2Ering_2Ering__RP__fupd(A_27a),V11f)) ) ) )
& ! [V12h: $i] :
( mem(V12h,arr(A_27b,ty_2Ering_2Ering(A_27a)))
=> ! [V13g: $i] :
( mem(V13g,arr(A_27a,A_27a))
=> ! [V14f: $i] :
( mem(V14f,arr(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,arr(A_27a,A_27a))))
=> ( ap(ap(c_2Ecombin_2Eo(A_27b,ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)),ap(c_2Ering_2Ering__RP__fupd(A_27a),V14f)),ap(ap(c_2Ecombin_2Eo(A_27b,ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)),ap(c_2Ering_2Ering__R1__fupd(A_27a),V13g)),V12h)) = ap(ap(c_2Ecombin_2Eo(A_27b,ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)),ap(c_2Ering_2Ering__R1__fupd(A_27a),V13g)),ap(ap(c_2Ecombin_2Eo(A_27b,ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)),ap(c_2Ering_2Ering__RP__fupd(A_27a),V14f)),V12h)) ) ) ) )
& ! [V15g: $i] :
( mem(V15g,arr(A_27a,A_27a))
=> ! [V16f: $i] :
( mem(V16f,arr(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,arr(A_27a,A_27a))))
=> ( ap(ap(c_2Ecombin_2Eo(ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)),ap(c_2Ering_2Ering__RM__fupd(A_27a),V16f)),ap(c_2Ering_2Ering__R0__fupd(A_27a),V15g)) = ap(ap(c_2Ecombin_2Eo(ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)),ap(c_2Ering_2Ering__R0__fupd(A_27a),V15g)),ap(c_2Ering_2Ering__RM__fupd(A_27a),V16f)) ) ) )
& ! [V17h: $i] :
( mem(V17h,arr(A_27b,ty_2Ering_2Ering(A_27a)))
=> ! [V18g: $i] :
( mem(V18g,arr(A_27a,A_27a))
=> ! [V19f: $i] :
( mem(V19f,arr(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,arr(A_27a,A_27a))))
=> ( ap(ap(c_2Ecombin_2Eo(A_27b,ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)),ap(c_2Ering_2Ering__RM__fupd(A_27a),V19f)),ap(ap(c_2Ecombin_2Eo(A_27b,ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)),ap(c_2Ering_2Ering__R0__fupd(A_27a),V18g)),V17h)) = ap(ap(c_2Ecombin_2Eo(A_27b,ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)),ap(c_2Ering_2Ering__R0__fupd(A_27a),V18g)),ap(ap(c_2Ecombin_2Eo(A_27b,ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)),ap(c_2Ering_2Ering__RM__fupd(A_27a),V19f)),V17h)) ) ) ) )
& ! [V20g: $i] :
( mem(V20g,arr(A_27a,A_27a))
=> ! [V21f: $i] :
( mem(V21f,arr(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,arr(A_27a,A_27a))))
=> ( ap(ap(c_2Ecombin_2Eo(ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)),ap(c_2Ering_2Ering__RM__fupd(A_27a),V21f)),ap(c_2Ering_2Ering__R1__fupd(A_27a),V20g)) = ap(ap(c_2Ecombin_2Eo(ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)),ap(c_2Ering_2Ering__R1__fupd(A_27a),V20g)),ap(c_2Ering_2Ering__RM__fupd(A_27a),V21f)) ) ) )
& ! [V22h: $i] :
( mem(V22h,arr(A_27b,ty_2Ering_2Ering(A_27a)))
=> ! [V23g: $i] :
( mem(V23g,arr(A_27a,A_27a))
=> ! [V24f: $i] :
( mem(V24f,arr(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,arr(A_27a,A_27a))))
=> ( ap(ap(c_2Ecombin_2Eo(A_27b,ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)),ap(c_2Ering_2Ering__RM__fupd(A_27a),V24f)),ap(ap(c_2Ecombin_2Eo(A_27b,ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)),ap(c_2Ering_2Ering__R1__fupd(A_27a),V23g)),V22h)) = ap(ap(c_2Ecombin_2Eo(A_27b,ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)),ap(c_2Ering_2Ering__R1__fupd(A_27a),V23g)),ap(ap(c_2Ecombin_2Eo(A_27b,ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)),ap(c_2Ering_2Ering__RM__fupd(A_27a),V24f)),V22h)) ) ) ) )
& ! [V25g: $i] :
( mem(V25g,arr(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,arr(A_27a,A_27a))))
=> ! [V26f: $i] :
( mem(V26f,arr(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,arr(A_27a,A_27a))))
=> ( ap(ap(c_2Ecombin_2Eo(ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)),ap(c_2Ering_2Ering__RM__fupd(A_27a),V26f)),ap(c_2Ering_2Ering__RP__fupd(A_27a),V25g)) = ap(ap(c_2Ecombin_2Eo(ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)),ap(c_2Ering_2Ering__RP__fupd(A_27a),V25g)),ap(c_2Ering_2Ering__RM__fupd(A_27a),V26f)) ) ) )
& ! [V27h: $i] :
( mem(V27h,arr(A_27b,ty_2Ering_2Ering(A_27a)))
=> ! [V28g: $i] :
( mem(V28g,arr(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,arr(A_27a,A_27a))))
=> ! [V29f: $i] :
( mem(V29f,arr(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,arr(A_27a,A_27a))))
=> ( ap(ap(c_2Ecombin_2Eo(A_27b,ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)),ap(c_2Ering_2Ering__RM__fupd(A_27a),V29f)),ap(ap(c_2Ecombin_2Eo(A_27b,ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)),ap(c_2Ering_2Ering__RP__fupd(A_27a),V28g)),V27h)) = ap(ap(c_2Ecombin_2Eo(A_27b,ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)),ap(c_2Ering_2Ering__RP__fupd(A_27a),V28g)),ap(ap(c_2Ecombin_2Eo(A_27b,ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)),ap(c_2Ering_2Ering__RM__fupd(A_27a),V29f)),V27h)) ) ) ) )
& ! [V30g: $i] :
( mem(V30g,arr(A_27a,A_27a))
=> ! [V31f: $i] :
( mem(V31f,arr(arr(A_27a,A_27a),arr(A_27a,A_27a)))
=> ( ap(ap(c_2Ecombin_2Eo(ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)),ap(c_2Ering_2Ering__RN__fupd(A_27a),V31f)),ap(c_2Ering_2Ering__R0__fupd(A_27a),V30g)) = ap(ap(c_2Ecombin_2Eo(ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)),ap(c_2Ering_2Ering__R0__fupd(A_27a),V30g)),ap(c_2Ering_2Ering__RN__fupd(A_27a),V31f)) ) ) )
& ! [V32h: $i] :
( mem(V32h,arr(A_27b,ty_2Ering_2Ering(A_27a)))
=> ! [V33g: $i] :
( mem(V33g,arr(A_27a,A_27a))
=> ! [V34f: $i] :
( mem(V34f,arr(arr(A_27a,A_27a),arr(A_27a,A_27a)))
=> ( ap(ap(c_2Ecombin_2Eo(A_27b,ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)),ap(c_2Ering_2Ering__RN__fupd(A_27a),V34f)),ap(ap(c_2Ecombin_2Eo(A_27b,ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)),ap(c_2Ering_2Ering__R0__fupd(A_27a),V33g)),V32h)) = ap(ap(c_2Ecombin_2Eo(A_27b,ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)),ap(c_2Ering_2Ering__R0__fupd(A_27a),V33g)),ap(ap(c_2Ecombin_2Eo(A_27b,ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)),ap(c_2Ering_2Ering__RN__fupd(A_27a),V34f)),V32h)) ) ) ) )
& ! [V35g: $i] :
( mem(V35g,arr(A_27a,A_27a))
=> ! [V36f: $i] :
( mem(V36f,arr(arr(A_27a,A_27a),arr(A_27a,A_27a)))
=> ( ap(ap(c_2Ecombin_2Eo(ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)),ap(c_2Ering_2Ering__RN__fupd(A_27a),V36f)),ap(c_2Ering_2Ering__R1__fupd(A_27a),V35g)) = ap(ap(c_2Ecombin_2Eo(ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)),ap(c_2Ering_2Ering__R1__fupd(A_27a),V35g)),ap(c_2Ering_2Ering__RN__fupd(A_27a),V36f)) ) ) )
& ! [V37h: $i] :
( mem(V37h,arr(A_27b,ty_2Ering_2Ering(A_27a)))
=> ! [V38g: $i] :
( mem(V38g,arr(A_27a,A_27a))
=> ! [V39f: $i] :
( mem(V39f,arr(arr(A_27a,A_27a),arr(A_27a,A_27a)))
=> ( ap(ap(c_2Ecombin_2Eo(A_27b,ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)),ap(c_2Ering_2Ering__RN__fupd(A_27a),V39f)),ap(ap(c_2Ecombin_2Eo(A_27b,ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)),ap(c_2Ering_2Ering__R1__fupd(A_27a),V38g)),V37h)) = ap(ap(c_2Ecombin_2Eo(A_27b,ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)),ap(c_2Ering_2Ering__R1__fupd(A_27a),V38g)),ap(ap(c_2Ecombin_2Eo(A_27b,ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)),ap(c_2Ering_2Ering__RN__fupd(A_27a),V39f)),V37h)) ) ) ) )
& ! [V40g: $i] :
( mem(V40g,arr(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,arr(A_27a,A_27a))))
=> ! [V41f: $i] :
( mem(V41f,arr(arr(A_27a,A_27a),arr(A_27a,A_27a)))
=> ( ap(ap(c_2Ecombin_2Eo(ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)),ap(c_2Ering_2Ering__RN__fupd(A_27a),V41f)),ap(c_2Ering_2Ering__RP__fupd(A_27a),V40g)) = ap(ap(c_2Ecombin_2Eo(ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)),ap(c_2Ering_2Ering__RP__fupd(A_27a),V40g)),ap(c_2Ering_2Ering__RN__fupd(A_27a),V41f)) ) ) )
& ! [V42h: $i] :
( mem(V42h,arr(A_27b,ty_2Ering_2Ering(A_27a)))
=> ! [V43g: $i] :
( mem(V43g,arr(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,arr(A_27a,A_27a))))
=> ! [V44f: $i] :
( mem(V44f,arr(arr(A_27a,A_27a),arr(A_27a,A_27a)))
=> ( ap(ap(c_2Ecombin_2Eo(A_27b,ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)),ap(c_2Ering_2Ering__RN__fupd(A_27a),V44f)),ap(ap(c_2Ecombin_2Eo(A_27b,ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)),ap(c_2Ering_2Ering__RP__fupd(A_27a),V43g)),V42h)) = ap(ap(c_2Ecombin_2Eo(A_27b,ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)),ap(c_2Ering_2Ering__RP__fupd(A_27a),V43g)),ap(ap(c_2Ecombin_2Eo(A_27b,ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)),ap(c_2Ering_2Ering__RN__fupd(A_27a),V44f)),V42h)) ) ) ) )
& ! [V45g: $i] :
( mem(V45g,arr(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,arr(A_27a,A_27a))))
=> ! [V46f: $i] :
( mem(V46f,arr(arr(A_27a,A_27a),arr(A_27a,A_27a)))
=> ( ap(ap(c_2Ecombin_2Eo(ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)),ap(c_2Ering_2Ering__RN__fupd(A_27a),V46f)),ap(c_2Ering_2Ering__RM__fupd(A_27a),V45g)) = ap(ap(c_2Ecombin_2Eo(ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)),ap(c_2Ering_2Ering__RM__fupd(A_27a),V45g)),ap(c_2Ering_2Ering__RN__fupd(A_27a),V46f)) ) ) )
& ! [V47h: $i] :
( mem(V47h,arr(A_27b,ty_2Ering_2Ering(A_27a)))
=> ! [V48g: $i] :
( mem(V48g,arr(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,arr(A_27a,A_27a))))
=> ! [V49f: $i] :
( mem(V49f,arr(arr(A_27a,A_27a),arr(A_27a,A_27a)))
=> ( ap(ap(c_2Ecombin_2Eo(A_27b,ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)),ap(c_2Ering_2Ering__RN__fupd(A_27a),V49f)),ap(ap(c_2Ecombin_2Eo(A_27b,ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)),ap(c_2Ering_2Ering__RM__fupd(A_27a),V48g)),V47h)) = ap(ap(c_2Ecombin_2Eo(A_27b,ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)),ap(c_2Ering_2Ering__RM__fupd(A_27a),V48g)),ap(ap(c_2Ecombin_2Eo(A_27b,ty_2Ering_2Ering(A_27a),ty_2Ering_2Ering(A_27a)),ap(c_2Ering_2Ering__RN__fupd(A_27a),V49f)),V47h)) ) ) ) ) ) ).
tff(conj_thm_2Ering_2Ering__component__equality,axiom,
! [A_27a: del,V0r1: $i] :
( mem(V0r1,ty_2Ering_2Ering(A_27a))
=> ! [V1r2: $i] :
( mem(V1r2,ty_2Ering_2Ering(A_27a))
=> ( ( V0r1 = V1r2 )
<=> ( ( ap(c_2Ering_2Ering__R0(A_27a),V0r1) = ap(c_2Ering_2Ering__R0(A_27a),V1r2) )
& ( ap(c_2Ering_2Ering__R1(A_27a),V0r1) = ap(c_2Ering_2Ering__R1(A_27a),V1r2) )
& ( ap(c_2Ering_2Ering__RP(A_27a),V0r1) = ap(c_2Ering_2Ering__RP(A_27a),V1r2) )
& ( ap(c_2Ering_2Ering__RM(A_27a),V0r1) = ap(c_2Ering_2Ering__RM(A_27a),V1r2) )
& ( ap(c_2Ering_2Ering__RN(A_27a),V0r1) = ap(c_2Ering_2Ering__RN(A_27a),V1r2) ) ) ) ) ) ).
tff(conj_thm_2Ering_2Ering__updates__eq__literal,axiom,
! [A_27a: del,V0r: $i] :
( mem(V0r,ty_2Ering_2Ering(A_27a))
=> ! [V1a0: $i] :
( mem(V1a0,A_27a)
=> ! [V2a: $i] :
( mem(V2a,A_27a)
=> ! [V3f1: $i] :
( mem(V3f1,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V4f0: $i] :
( mem(V4f0,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V5f: $i] :
( mem(V5f,arr(A_27a,A_27a))
=> ( ap(ap(c_2Ering_2Ering__R0__fupd(A_27a),ap(c_2Ecombin_2EK(A_27a,A_27a),V1a0)),ap(ap(c_2Ering_2Ering__R1__fupd(A_27a),ap(c_2Ecombin_2EK(A_27a,A_27a),V2a)),ap(ap(c_2Ering_2Ering__RP__fupd(A_27a),ap(c_2Ecombin_2EK(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,arr(A_27a,A_27a))),V3f1)),ap(ap(c_2Ering_2Ering__RM__fupd(A_27a),ap(c_2Ecombin_2EK(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,arr(A_27a,A_27a))),V4f0)),ap(ap(c_2Ering_2Ering__RN__fupd(A_27a),ap(c_2Ecombin_2EK(arr(A_27a,A_27a),arr(A_27a,A_27a)),V5f)),V0r))))) = ap(ap(c_2Ering_2Ering__R0__fupd(A_27a),ap(c_2Ecombin_2EK(A_27a,A_27a),V1a0)),ap(ap(c_2Ering_2Ering__R1__fupd(A_27a),ap(c_2Ecombin_2EK(A_27a,A_27a),V2a)),ap(ap(c_2Ering_2Ering__RP__fupd(A_27a),ap(c_2Ecombin_2EK(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,arr(A_27a,A_27a))),V3f1)),ap(ap(c_2Ering_2Ering__RM__fupd(A_27a),ap(c_2Ecombin_2EK(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,arr(A_27a,A_27a))),V4f0)),ap(ap(c_2Ering_2Ering__RN__fupd(A_27a),ap(c_2Ecombin_2EK(arr(A_27a,A_27a),arr(A_27a,A_27a)),V5f)),c_2Ebool_2EARB(ty_2Ering_2Ering(A_27a))))))) ) ) ) ) ) ) ) ).
tff(conj_thm_2Ering_2Ering__literal__nchotomy,axiom,
! [A_27a: del,V0r: $i] :
( mem(V0r,ty_2Ering_2Ering(A_27a))
=> ? [V1a0: $i] :
( mem(V1a0,A_27a)
& ? [V2a: $i] :
( mem(V2a,A_27a)
& ? [V3f1: $i] :
( mem(V3f1,arr(A_27a,arr(A_27a,A_27a)))
& ? [V4f0: $i] :
( mem(V4f0,arr(A_27a,arr(A_27a,A_27a)))
& ? [V5f: $i] :
( mem(V5f,arr(A_27a,A_27a))
& ( V0r = ap(ap(c_2Ering_2Ering__R0__fupd(A_27a),ap(c_2Ecombin_2EK(A_27a,A_27a),V1a0)),ap(ap(c_2Ering_2Ering__R1__fupd(A_27a),ap(c_2Ecombin_2EK(A_27a,A_27a),V2a)),ap(ap(c_2Ering_2Ering__RP__fupd(A_27a),ap(c_2Ecombin_2EK(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,arr(A_27a,A_27a))),V3f1)),ap(ap(c_2Ering_2Ering__RM__fupd(A_27a),ap(c_2Ecombin_2EK(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,arr(A_27a,A_27a))),V4f0)),ap(ap(c_2Ering_2Ering__RN__fupd(A_27a),ap(c_2Ecombin_2EK(arr(A_27a,A_27a),arr(A_27a,A_27a)),V5f)),c_2Ebool_2EARB(ty_2Ering_2Ering(A_27a))))))) ) ) ) ) ) ) ) ).
tff(conj_thm_2Ering_2EFORALL__ring,axiom,
! [A_27a: del,V0P: $i] :
( mem(V0P,arr(ty_2Ering_2Ering(A_27a),bool))
=> ( ! [V1r: $i] :
( mem(V1r,ty_2Ering_2Ering(A_27a))
=> p(ap(V0P,V1r)) )
<=> ! [V2a0: $i] :
( mem(V2a0,A_27a)
=> ! [V3a: $i] :
( mem(V3a,A_27a)
=> ! [V4f1: $i] :
( mem(V4f1,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V5f0: $i] :
( mem(V5f0,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V6f: $i] :
( mem(V6f,arr(A_27a,A_27a))
=> p(ap(V0P,ap(ap(c_2Ering_2Ering__R0__fupd(A_27a),ap(c_2Ecombin_2EK(A_27a,A_27a),V2a0)),ap(ap(c_2Ering_2Ering__R1__fupd(A_27a),ap(c_2Ecombin_2EK(A_27a,A_27a),V3a)),ap(ap(c_2Ering_2Ering__RP__fupd(A_27a),ap(c_2Ecombin_2EK(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,arr(A_27a,A_27a))),V4f1)),ap(ap(c_2Ering_2Ering__RM__fupd(A_27a),ap(c_2Ecombin_2EK(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,arr(A_27a,A_27a))),V5f0)),ap(ap(c_2Ering_2Ering__RN__fupd(A_27a),ap(c_2Ecombin_2EK(arr(A_27a,A_27a),arr(A_27a,A_27a)),V6f)),c_2Ebool_2EARB(ty_2Ering_2Ering(A_27a))))))))) ) ) ) ) ) ) ) ).
tff(conj_thm_2Ering_2EEXISTS__ring,axiom,
! [A_27a: del,V0P: $i] :
( mem(V0P,arr(ty_2Ering_2Ering(A_27a),bool))
=> ( ? [V1r: $i] :
( mem(V1r,ty_2Ering_2Ering(A_27a))
& p(ap(V0P,V1r)) )
<=> ? [V2a0: $i] :
( mem(V2a0,A_27a)
& ? [V3a: $i] :
( mem(V3a,A_27a)
& ? [V4f1: $i] :
( mem(V4f1,arr(A_27a,arr(A_27a,A_27a)))
& ? [V5f0: $i] :
( mem(V5f0,arr(A_27a,arr(A_27a,A_27a)))
& ? [V6f: $i] :
( mem(V6f,arr(A_27a,A_27a))
& p(ap(V0P,ap(ap(c_2Ering_2Ering__R0__fupd(A_27a),ap(c_2Ecombin_2EK(A_27a,A_27a),V2a0)),ap(ap(c_2Ering_2Ering__R1__fupd(A_27a),ap(c_2Ecombin_2EK(A_27a,A_27a),V3a)),ap(ap(c_2Ering_2Ering__RP__fupd(A_27a),ap(c_2Ecombin_2EK(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,arr(A_27a,A_27a))),V4f1)),ap(ap(c_2Ering_2Ering__RM__fupd(A_27a),ap(c_2Ecombin_2EK(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,arr(A_27a,A_27a))),V5f0)),ap(ap(c_2Ering_2Ering__RN__fupd(A_27a),ap(c_2Ecombin_2EK(arr(A_27a,A_27a),arr(A_27a,A_27a)),V6f)),c_2Ebool_2EARB(ty_2Ering_2Ering(A_27a))))))))) ) ) ) ) ) ) ) ).
tff(conj_thm_2Ering_2Ering__literal__11,axiom,
! [A_27a: del,V0a01: $i] :
( mem(V0a01,A_27a)
=> ! [V1a1: $i] :
( mem(V1a1,A_27a)
=> ! [V2f11: $i] :
( mem(V2f11,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V3f01: $i] :
( mem(V3f01,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V4f1: $i] :
( mem(V4f1,arr(A_27a,A_27a))
=> ! [V5a02: $i] :
( mem(V5a02,A_27a)
=> ! [V6a2: $i] :
( mem(V6a2,A_27a)
=> ! [V7f12: $i] :
( mem(V7f12,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V8f02: $i] :
( mem(V8f02,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V9f2: $i] :
( mem(V9f2,arr(A_27a,A_27a))
=> ( ( ap(ap(c_2Ering_2Ering__R0__fupd(A_27a),ap(c_2Ecombin_2EK(A_27a,A_27a),V0a01)),ap(ap(c_2Ering_2Ering__R1__fupd(A_27a),ap(c_2Ecombin_2EK(A_27a,A_27a),V1a1)),ap(ap(c_2Ering_2Ering__RP__fupd(A_27a),ap(c_2Ecombin_2EK(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,arr(A_27a,A_27a))),V2f11)),ap(ap(c_2Ering_2Ering__RM__fupd(A_27a),ap(c_2Ecombin_2EK(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,arr(A_27a,A_27a))),V3f01)),ap(ap(c_2Ering_2Ering__RN__fupd(A_27a),ap(c_2Ecombin_2EK(arr(A_27a,A_27a),arr(A_27a,A_27a)),V4f1)),c_2Ebool_2EARB(ty_2Ering_2Ering(A_27a))))))) = ap(ap(c_2Ering_2Ering__R0__fupd(A_27a),ap(c_2Ecombin_2EK(A_27a,A_27a),V5a02)),ap(ap(c_2Ering_2Ering__R1__fupd(A_27a),ap(c_2Ecombin_2EK(A_27a,A_27a),V6a2)),ap(ap(c_2Ering_2Ering__RP__fupd(A_27a),ap(c_2Ecombin_2EK(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,arr(A_27a,A_27a))),V7f12)),ap(ap(c_2Ering_2Ering__RM__fupd(A_27a),ap(c_2Ecombin_2EK(arr(A_27a,arr(A_27a,A_27a)),arr(A_27a,arr(A_27a,A_27a))),V8f02)),ap(ap(c_2Ering_2Ering__RN__fupd(A_27a),ap(c_2Ecombin_2EK(arr(A_27a,A_27a),arr(A_27a,A_27a)),V9f2)),c_2Ebool_2EARB(ty_2Ering_2Ering(A_27a))))))) )
<=> ( ( V0a01 = V5a02 )
& ( V1a1 = V6a2 )
& ( V2f11 = V7f12 )
& ( V3f01 = V8f02 )
& ( V4f1 = V9f2 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
tff(conj_thm_2Ering_2Edatatype__ring,axiom,
! [A_27a: del,V0record: $i] :
( mem(V0record,arr(ty_2Ering_2Ering(A_27a),arr(A_27a,arr(A_27a,arr(arr(A_27a,arr(A_27a,A_27a)),arr(arr(A_27a,arr(A_27a,A_27a)),arr(arr(A_27a,A_27a),bool)))))))
=> ! [V1ring: $i] :
( mem(V1ring,ty_2Ering_2Ering(A_27a))
=> ! [V2R0: $i] :
( mem(V2R0,A_27a)
=> ! [V3R1: $i] :
( mem(V3R1,A_27a)
=> ! [V4RP: $i] :
( mem(V4RP,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V5RM: $i] :
( mem(V5RM,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V6RN: $i] :
( mem(V6RN,arr(A_27a,A_27a))
=> p(ap(c_2Ebool_2EDATATYPE(bool),ap(ap(ap(ap(ap(ap(V0record,V1ring),V2R0),V3R1),V4RP),V5RM),V6RN))) ) ) ) ) ) ) ) ).
tff(conj_thm_2Ering_2Ering__11,axiom,
! [A_27a: del,V0a0: $i] :
( mem(V0a0,A_27a)
=> ! [V1a1: $i] :
( mem(V1a1,A_27a)
=> ! [V2a2: $i] :
( mem(V2a2,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V3a3: $i] :
( mem(V3a3,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V4a4: $i] :
( mem(V4a4,arr(A_27a,A_27a))
=> ! [V5a0_27: $i] :
( mem(V5a0_27,A_27a)
=> ! [V6a1_27: $i] :
( mem(V6a1_27,A_27a)
=> ! [V7a2_27: $i] :
( mem(V7a2_27,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V8a3_27: $i] :
( mem(V8a3_27,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V9a4_27: $i] :
( mem(V9a4_27,arr(A_27a,A_27a))
=> ( ( ap(ap(ap(ap(ap(c_2Ering_2Erecordtype_2Ering(A_27a),V0a0),V1a1),V2a2),V3a3),V4a4) = ap(ap(ap(ap(ap(c_2Ering_2Erecordtype_2Ering(A_27a),V5a0_27),V6a1_27),V7a2_27),V8a3_27),V9a4_27) )
<=> ( ( V0a0 = V5a0_27 )
& ( V1a1 = V6a1_27 )
& ( V2a2 = V7a2_27 )
& ( V3a3 = V8a3_27 )
& ( V4a4 = V9a4_27 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
tff(conj_thm_2Ering_2Ering__nchotomy,axiom,
! [A_27a: del,V0rr: $i] :
( mem(V0rr,ty_2Ering_2Ering(A_27a))
=> ? [V1a: $i] :
( mem(V1a,A_27a)
& ? [V2a0: $i] :
( mem(V2a0,A_27a)
& ? [V3f: $i] :
( mem(V3f,arr(A_27a,arr(A_27a,A_27a)))
& ? [V4f0: $i] :
( mem(V4f0,arr(A_27a,arr(A_27a,A_27a)))
& ? [V5f1: $i] :
( mem(V5f1,arr(A_27a,A_27a))
& ( V0rr = ap(ap(ap(ap(ap(c_2Ering_2Erecordtype_2Ering(A_27a),V1a),V2a0),V3f),V4f0),V5f1) ) ) ) ) ) ) ) ).
tff(conj_thm_2Ering_2Ering__Axiom,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( mem(V0f,arr(A_27a,arr(A_27a,arr(arr(A_27a,arr(A_27a,A_27a)),arr(arr(A_27a,arr(A_27a,A_27a)),arr(arr(A_27a,A_27a),A_27b))))))
=> ? [V1fn: $i] :
( mem(V1fn,arr(ty_2Ering_2Ering(A_27a),A_27b))
& ! [V2a0: $i] :
( mem(V2a0,A_27a)
=> ! [V3a1: $i] :
( mem(V3a1,A_27a)
=> ! [V4a2: $i] :
( mem(V4a2,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V5a3: $i] :
( mem(V5a3,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V6a4: $i] :
( mem(V6a4,arr(A_27a,A_27a))
=> ( ap(V1fn,ap(ap(ap(ap(ap(c_2Ering_2Erecordtype_2Ering(A_27a),V2a0),V3a1),V4a2),V5a3),V6a4)) = ap(ap(ap(ap(ap(V0f,V2a0),V3a1),V4a2),V5a3),V6a4) ) ) ) ) ) ) ) ) ).
tff(conj_thm_2Ering_2Ering__induction,axiom,
! [A_27a: del,V0P: $i] :
( mem(V0P,arr(ty_2Ering_2Ering(A_27a),bool))
=> ( ! [V1a: $i] :
( mem(V1a,A_27a)
=> ! [V2a0: $i] :
( mem(V2a0,A_27a)
=> ! [V3f: $i] :
( mem(V3f,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V4f0: $i] :
( mem(V4f0,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V5f1: $i] :
( mem(V5f1,arr(A_27a,A_27a))
=> p(ap(V0P,ap(ap(ap(ap(ap(c_2Ering_2Erecordtype_2Ering(A_27a),V1a),V2a0),V3f),V4f0),V5f1))) ) ) ) ) )
=> ! [V6r: $i] :
( mem(V6r,ty_2Ering_2Ering(A_27a))
=> p(ap(V0P,V6r)) ) ) ) ).
tff(conj_thm_2Ering_2Ering__case__cong,axiom,
! [A_27a: del,A_27b: del,V0f_27: $i] :
( mem(V0f_27,arr(A_27a,arr(A_27a,arr(arr(A_27a,arr(A_27a,A_27a)),arr(arr(A_27a,arr(A_27a,A_27a)),arr(arr(A_27a,A_27a),A_27b))))))
=> ! [V1M: $i] :
( mem(V1M,ty_2Ering_2Ering(A_27a))
=> ! [V2M_27: $i] :
( mem(V2M_27,ty_2Ering_2Ering(A_27a))
=> ! [V3f: $i] :
( mem(V3f,arr(A_27a,arr(A_27a,arr(arr(A_27a,arr(A_27a,A_27a)),arr(arr(A_27a,arr(A_27a,A_27a)),arr(arr(A_27a,A_27a),A_27b))))))
=> ( ( ( V1M = V2M_27 )
& ! [V4a0: $i] :
( mem(V4a0,A_27a)
=> ! [V5a1: $i] :
( mem(V5a1,A_27a)
=> ! [V6a2: $i] :
( mem(V6a2,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V7a3: $i] :
( mem(V7a3,arr(A_27a,arr(A_27a,A_27a)))
=> ! [V8a4: $i] :
( mem(V8a4,arr(A_27a,A_27a))
=> ( ( V2M_27 = ap(ap(ap(ap(ap(c_2Ering_2Erecordtype_2Ering(A_27a),V4a0),V5a1),V6a2),V7a3),V8a4) )
=> ( ap(ap(ap(ap(ap(V3f,V4a0),V5a1),V6a2),V7a3),V8a4) = ap(ap(ap(ap(ap(V0f_27,V4a0),V5a1),V6a2),V7a3),V8a4) ) ) ) ) ) ) ) )
=> ( ap(ap(c_2Ering_2Ering__CASE(A_27a,A_27b),V1M),V3f) = ap(ap(c_2Ering_2Ering__CASE(A_27a,A_27b),V2M_27),V0f_27) ) ) ) ) ) ) ).
tff(conj_thm_2Ering_2Ering__case__eq,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( mem(V0x,ty_2Ering_2Ering(A_27a))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,arr(A_27a,arr(arr(A_27a,arr(A_27a,A_27a)),arr(arr(A_27a,arr(A_27a,A_27a)),arr(arr(A_27a,A_27a),A_27b))))))
=> ! [V2v: $i] :
( mem(V2v,A_27b)
=> ( ( ap(ap(c_2Ering_2Ering__CASE(A_27a,A_27b),V0x),V1f) = V2v )
<=> ? [V3a: $i] :
( mem(V3a,A_27a)
& ? [V4a0: $i] :
( mem(V4a0,A_27a)
& ? [V5f_27: $i] :
( mem(V5f_27,arr(A_27a,arr(A_27a,A_27a)))
& ? [V6f0: $i] :
( mem(V6f0,arr(A_27a,arr(A_27a,A_27a)))
& ? [V7f1: $i] :
( mem(V7f1,arr(A_27a,A_27a))
& ( V0x = ap(ap(ap(ap(ap(c_2Ering_2Erecordtype_2Ering(A_27a),V3a),V4a0),V5f_27),V6f0),V7f1) )
& ( ap(ap(ap(ap(ap(V1f,V3a),V4a0),V5f_27),V6f0),V7f1) = V2v ) ) ) ) ) ) ) ) ) ) ).
tff(ax_thm_2Ering_2Eis__ring__def,axiom,
! [A_27a: del,V0r: $i] :
( mem(V0r,ty_2Ering_2Ering(A_27a))
=> ( p(ap(c_2Ering_2Eis__ring(A_27a),V0r))
<=> ( ! [V1n: $i] :
( mem(V1n,A_27a)
=> ! [V2m: $i] :
( mem(V2m,A_27a)
=> ( ap(ap(ap(c_2Ering_2Ering__RP(A_27a),V0r),V1n),V2m) = ap(ap(ap(c_2Ering_2Ering__RP(A_27a),V0r),V2m),V1n) ) ) )
& ! [V3n: $i] :
( mem(V3n,A_27a)
=> ! [V4m: $i] :
( mem(V4m,A_27a)
=> ! [V5p: $i] :
( mem(V5p,A_27a)
=> ( ap(ap(ap(c_2Ering_2Ering__RP(A_27a),V0r),V3n),ap(ap(ap(c_2Ering_2Ering__RP(A_27a),V0r),V4m),V5p)) = ap(ap(ap(c_2Ering_2Ering__RP(A_27a),V0r),ap(ap(ap(c_2Ering_2Ering__RP(A_27a),V0r),V3n),V4m)),V5p) ) ) ) )
& ! [V6n: $i] :
( mem(V6n,A_27a)
=> ! [V7m: $i] :
( mem(V7m,A_27a)
=> ( ap(ap(ap(c_2Ering_2Ering__RM(A_27a),V0r),V6n),V7m) = ap(ap(ap(c_2Ering_2Ering__RM(A_27a),V0r),V7m),V6n) ) ) )
& ! [V8n: $i] :
( mem(V8n,A_27a)
=> ! [V9m: $i] :
( mem(V9m,A_27a)
=> ! [V10p: $i] :
( mem(V10p,A_27a)
=> ( ap(ap(ap(c_2Ering_2Ering__RM(A_27a),V0r),V8n),ap(ap(ap(c_2Ering_2Ering__RM(A_27a),V0r),V9m),V10p)) = ap(ap(ap(c_2Ering_2Ering__RM(A_27a),V0r),ap(ap(ap(c_2Ering_2Ering__RM(A_27a),V0r),V8n),V9m)),V10p) ) ) ) )
& ! [V11n: $i] :
( mem(V11n,A_27a)
=> ( ap(ap(ap(c_2Ering_2Ering__RP(A_27a),V0r),ap(c_2Ering_2Ering__R0(A_27a),V0r)),V11n) = V11n ) )
& ! [V12n: $i] :
( mem(V12n,A_27a)
=> ( ap(ap(ap(c_2Ering_2Ering__RM(A_27a),V0r),ap(c_2Ering_2Ering__R1(A_27a),V0r)),V12n) = V12n ) )
& ! [V13n: $i] :
( mem(V13n,A_27a)
=> ( ap(ap(ap(c_2Ering_2Ering__RP(A_27a),V0r),V13n),ap(ap(c_2Ering_2Ering__RN(A_27a),V0r),V13n)) = ap(c_2Ering_2Ering__R0(A_27a),V0r) ) )
& ! [V14n: $i] :
( mem(V14n,A_27a)
=> ! [V15m: $i] :
( mem(V15m,A_27a)
=> ! [V16p: $i] :
( mem(V16p,A_27a)
=> ( ap(ap(ap(c_2Ering_2Ering__RM(A_27a),V0r),ap(ap(ap(c_2Ering_2Ering__RP(A_27a),V0r),V14n),V15m)),V16p) = ap(ap(ap(c_2Ering_2Ering__RP(A_27a),V0r),ap(ap(ap(c_2Ering_2Ering__RM(A_27a),V0r),V14n),V16p)),ap(ap(ap(c_2Ering_2Ering__RM(A_27a),V0r),V15m),V16p)) ) ) ) ) ) ) ) ).
tff(conj_thm_2Ering_2Eplus__sym,axiom,
! [A_27a: del,V0r: $i] :
( mem(V0r,ty_2Ering_2Ering(A_27a))
=> ( p(ap(c_2Ering_2Eis__ring(A_27a),V0r))
=> ! [V1n: $i] :
( mem(V1n,A_27a)
=> ! [V2m: $i] :
( mem(V2m,A_27a)
=> ( ap(ap(ap(c_2Ering_2Ering__RP(A_27a),V0r),V1n),V2m) = ap(ap(ap(c_2Ering_2Ering__RP(A_27a),V0r),V2m),V1n) ) ) ) ) ) ).
tff(conj_thm_2Ering_2Eplus__assoc,axiom,
! [A_27a: del,V0r: $i] :
( mem(V0r,ty_2Ering_2Ering(A_27a))
=> ( p(ap(c_2Ering_2Eis__ring(A_27a),V0r))
=> ! [V1n: $i] :
( mem(V1n,A_27a)
=> ! [V2m: $i] :
( mem(V2m,A_27a)
=> ! [V3p: $i] :
( mem(V3p,A_27a)
=> ( ap(ap(ap(c_2Ering_2Ering__RP(A_27a),V0r),V1n),ap(ap(ap(c_2Ering_2Ering__RP(A_27a),V0r),V2m),V3p)) = ap(ap(ap(c_2Ering_2Ering__RP(A_27a),V0r),ap(ap(ap(c_2Ering_2Ering__RP(A_27a),V0r),V1n),V2m)),V3p) ) ) ) ) ) ) ).
tff(conj_thm_2Ering_2Emult__sym,axiom,
! [A_27a: del,V0r: $i] :
( mem(V0r,ty_2Ering_2Ering(A_27a))
=> ( p(ap(c_2Ering_2Eis__ring(A_27a),V0r))
=> ! [V1n: $i] :
( mem(V1n,A_27a)
=> ! [V2m: $i] :
( mem(V2m,A_27a)
=> ( ap(ap(ap(c_2Ering_2Ering__RM(A_27a),V0r),V1n),V2m) = ap(ap(ap(c_2Ering_2Ering__RM(A_27a),V0r),V2m),V1n) ) ) ) ) ) ).
tff(conj_thm_2Ering_2Emult__assoc,axiom,
! [A_27a: del,V0r: $i] :
( mem(V0r,ty_2Ering_2Ering(A_27a))
=> ( p(ap(c_2Ering_2Eis__ring(A_27a),V0r))
=> ! [V1n: $i] :
( mem(V1n,A_27a)
=> ! [V2m: $i] :
( mem(V2m,A_27a)
=> ! [V3p: $i] :
( mem(V3p,A_27a)
=> ( ap(ap(ap(c_2Ering_2Ering__RM(A_27a),V0r),V1n),ap(ap(ap(c_2Ering_2Ering__RM(A_27a),V0r),V2m),V3p)) = ap(ap(ap(c_2Ering_2Ering__RM(A_27a),V0r),ap(ap(ap(c_2Ering_2Ering__RM(A_27a),V0r),V1n),V2m)),V3p) ) ) ) ) ) ) ).
tff(conj_thm_2Ering_2Eplus__zero__left,axiom,
! [A_27a: del,V0r: $i] :
( mem(V0r,ty_2Ering_2Ering(A_27a))
=> ( p(ap(c_2Ering_2Eis__ring(A_27a),V0r))
=> ! [V1n: $i] :
( mem(V1n,A_27a)
=> ( ap(ap(ap(c_2Ering_2Ering__RP(A_27a),V0r),ap(c_2Ering_2Ering__R0(A_27a),V0r)),V1n) = V1n ) ) ) ) ).
tff(conj_thm_2Ering_2Emult__one__left,axiom,
! [A_27a: del,V0r: $i] :
( mem(V0r,ty_2Ering_2Ering(A_27a))
=> ( p(ap(c_2Ering_2Eis__ring(A_27a),V0r))
=> ! [V1n: $i] :
( mem(V1n,A_27a)
=> ( ap(ap(ap(c_2Ering_2Ering__RM(A_27a),V0r),ap(c_2Ering_2Ering__R1(A_27a),V0r)),V1n) = V1n ) ) ) ) ).
tff(conj_thm_2Ering_2Eopp__def,axiom,
! [A_27a: del,V0r: $i] :
( mem(V0r,ty_2Ering_2Ering(A_27a))
=> ( p(ap(c_2Ering_2Eis__ring(A_27a),V0r))
=> ! [V1n: $i] :
( mem(V1n,A_27a)
=> ( ap(ap(ap(c_2Ering_2Ering__RP(A_27a),V0r),V1n),ap(ap(c_2Ering_2Ering__RN(A_27a),V0r),V1n)) = ap(c_2Ering_2Ering__R0(A_27a),V0r) ) ) ) ) ).
tff(conj_thm_2Ering_2Edistr__left,axiom,
! [A_27a: del,V0r: $i] :
( mem(V0r,ty_2Ering_2Ering(A_27a))
=> ( p(ap(c_2Ering_2Eis__ring(A_27a),V0r))
=> ! [V1n: $i] :
( mem(V1n,A_27a)
=> ! [V2m: $i] :
( mem(V2m,A_27a)
=> ! [V3p: $i] :
( mem(V3p,A_27a)
=> ( ap(ap(ap(c_2Ering_2Ering__RM(A_27a),V0r),ap(ap(ap(c_2Ering_2Ering__RP(A_27a),V0r),V1n),V2m)),V3p) = ap(ap(ap(c_2Ering_2Ering__RP(A_27a),V0r),ap(ap(ap(c_2Ering_2Ering__RM(A_27a),V0r),V1n),V3p)),ap(ap(ap(c_2Ering_2Ering__RM(A_27a),V0r),V2m),V3p)) ) ) ) ) ) ) ).
tff(conj_thm_2Ering_2Eplus__zero__right,axiom,
! [A_27a: del,V0r: $i] :
( mem(V0r,ty_2Ering_2Ering(A_27a))
=> ( p(ap(c_2Ering_2Eis__ring(A_27a),V0r))
=> ! [V1n: $i] :
( mem(V1n,A_27a)
=> ( ap(ap(ap(c_2Ering_2Ering__RP(A_27a),V0r),V1n),ap(c_2Ering_2Ering__R0(A_27a),V0r)) = V1n ) ) ) ) ).
tff(conj_thm_2Ering_2Emult__zero__left,axiom,
! [A_27a: del,V0r: $i] :
( mem(V0r,ty_2Ering_2Ering(A_27a))
=> ( p(ap(c_2Ering_2Eis__ring(A_27a),V0r))
=> ! [V1n: $i] :
( mem(V1n,A_27a)
=> ( ap(ap(ap(c_2Ering_2Ering__RM(A_27a),V0r),ap(c_2Ering_2Ering__R0(A_27a),V0r)),V1n) = ap(c_2Ering_2Ering__R0(A_27a),V0r) ) ) ) ) ).
tff(conj_thm_2Ering_2Emult__zero__right,axiom,
! [A_27a: del,V0r: $i] :
( mem(V0r,ty_2Ering_2Ering(A_27a))
=> ( p(ap(c_2Ering_2Eis__ring(A_27a),V0r))
=> ! [V1n: $i] :
( mem(V1n,A_27a)
=> ( ap(ap(ap(c_2Ering_2Ering__RM(A_27a),V0r),V1n),ap(c_2Ering_2Ering__R0(A_27a),V0r)) = ap(c_2Ering_2Ering__R0(A_27a),V0r) ) ) ) ) ).
tff(ax_thm_2Ering_2Esemi__ring__of__def,axiom,
! [A_27a: del,V0r: $i] :
( mem(V0r,ty_2Ering_2Ering(A_27a))
=> ( ap(c_2Ering_2Esemi__ring__of(A_27a),V0r) = ap(ap(ap(ap(c_2Esemi__ring_2Erecordtype_2Esemi__ring(A_27a),ap(c_2Ering_2Ering__R0(A_27a),V0r)),ap(c_2Ering_2Ering__R1(A_27a),V0r)),ap(c_2Ering_2Ering__RP(A_27a),V0r)),ap(c_2Ering_2Ering__RM(A_27a),V0r)) ) ) ).
tff(conj_thm_2Ering_2Ering__is__semi__ring,axiom,
! [A_27a: del,V0r: $i] :
( mem(V0r,ty_2Ering_2Ering(A_27a))
=> ( p(ap(c_2Ering_2Eis__ring(A_27a),V0r))
=> p(ap(c_2Esemi__ring_2Eis__semi__ring(A_27a),ap(c_2Ering_2Esemi__ring__of(A_27a),V0r))) ) ) ).
tff(conj_thm_2Ering_2Emult__one__right,axiom,
! [A_27a: del,V0r: $i] :
( mem(V0r,ty_2Ering_2Ering(A_27a))
=> ( p(ap(c_2Ering_2Eis__ring(A_27a),V0r))
=> ! [V1n: $i] :
( mem(V1n,A_27a)
=> ( ap(ap(ap(c_2Ering_2Ering__RM(A_27a),V0r),V1n),ap(c_2Ering_2Ering__R1(A_27a),V0r)) = V1n ) ) ) ) ).
tff(conj_thm_2Ering_2Eneg__mult,axiom,
! [A_27a: del,V0r: $i] :
( mem(V0r,ty_2Ering_2Ering(A_27a))
=> ( p(ap(c_2Ering_2Eis__ring(A_27a),V0r))
=> ! [V1a: $i] :
( mem(V1a,A_27a)
=> ! [V2b: $i] :
( mem(V2b,A_27a)
=> ( ap(ap(ap(c_2Ering_2Ering__RM(A_27a),V0r),ap(ap(c_2Ering_2Ering__RN(A_27a),V0r),V1a)),V2b) = ap(ap(c_2Ering_2Ering__RN(A_27a),V0r),ap(ap(ap(c_2Ering_2Ering__RM(A_27a),V0r),V1a),V2b)) ) ) ) ) ) ).
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