ITP001 Axioms: ITP052^5.ax
%------------------------------------------------------------------------------
% File : ITP052^5 : 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 : semi_ring^2.ax [Gau20]
% : HL4052^5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 73 ( 0 unt; 13 typ; 0 def)
% Number of atoms : 2230 ( 114 equ; 0 cnn)
% Maximal formula atoms : 171 ( 30 avg)
% Number of connectives : 5204 ( 0 ~; 0 |; 84 &;4773 @)
% ( 7 <=>; 340 =>; 0 <=; 0 <~>)
% Maximal formula depth : 65 ( 16 avg;4773 nst)
% Number of types : 1 ( 0 usr)
% Number of type conns : 14 ( 14 >; 0 *; 0 +; 0 <<)
% Number of symbols : 43 ( 42 usr; 29 con; 0-2 aty)
% Number of variables : 419 ( 12 ^ 388 !; 19 ?; 419 :)
% SPC : TH0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_ty_2Esemi__ring_2Esemi__ring,type,
ty_2Esemi__ring_2Esemi__ring: del > del ).
thf(tp_c_2Esemi__ring_2Eis__semi__ring,type,
c_2Esemi__ring_2Eis__semi__ring: del > $i ).
thf(mem_c_2Esemi__ring_2Eis__semi__ring,axiom,
! [A_27a: del] : ( mem @ ( c_2Esemi__ring_2Eis__semi__ring @ A_27a ) @ ( arr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) @ bool ) ) ).
thf(tp_c_2Esemi__ring_2Erecordtype_2Esemi__ring,type,
c_2Esemi__ring_2Erecordtype_2Esemi__ring: del > $i ).
thf(mem_c_2Esemi__ring_2Erecordtype_2Esemi__ring,axiom,
! [A_27a: del] : ( mem @ ( c_2Esemi__ring_2Erecordtype_2Esemi__ring @ 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 ) ) @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) ) ) ) ) ) ).
thf(tp_c_2Esemi__ring_2Esemi__ring__CASE,type,
c_2Esemi__ring_2Esemi__ring__CASE: del > del > $i ).
thf(mem_c_2Esemi__ring_2Esemi__ring__CASE,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Esemi__ring_2Esemi__ring__CASE @ A_27a @ A_27b ) @ ( arr @ ( ty_2Esemi__ring_2Esemi__ring @ 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 ) ) @ A_27b ) ) ) ) @ A_27b ) ) ) ).
thf(tp_c_2Esemi__ring_2Esemi__ring__SR0,type,
c_2Esemi__ring_2Esemi__ring__SR0: del > $i ).
thf(mem_c_2Esemi__ring_2Esemi__ring__SR0,axiom,
! [A_27a: del] : ( mem @ ( c_2Esemi__ring_2Esemi__ring__SR0 @ A_27a ) @ ( arr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) @ A_27a ) ) ).
thf(tp_c_2Esemi__ring_2Esemi__ring__SR0__fupd,type,
c_2Esemi__ring_2Esemi__ring__SR0__fupd: del > $i ).
thf(mem_c_2Esemi__ring_2Esemi__ring__SR0__fupd,axiom,
! [A_27a: del] : ( mem @ ( c_2Esemi__ring_2Esemi__ring__SR0__fupd @ A_27a ) @ ( arr @ ( arr @ A_27a @ A_27a ) @ ( arr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) ) ) ) ).
thf(tp_c_2Esemi__ring_2Esemi__ring__SR1,type,
c_2Esemi__ring_2Esemi__ring__SR1: del > $i ).
thf(mem_c_2Esemi__ring_2Esemi__ring__SR1,axiom,
! [A_27a: del] : ( mem @ ( c_2Esemi__ring_2Esemi__ring__SR1 @ A_27a ) @ ( arr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) @ A_27a ) ) ).
thf(tp_c_2Esemi__ring_2Esemi__ring__SR1__fupd,type,
c_2Esemi__ring_2Esemi__ring__SR1__fupd: del > $i ).
thf(mem_c_2Esemi__ring_2Esemi__ring__SR1__fupd,axiom,
! [A_27a: del] : ( mem @ ( c_2Esemi__ring_2Esemi__ring__SR1__fupd @ A_27a ) @ ( arr @ ( arr @ A_27a @ A_27a ) @ ( arr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) ) ) ) ).
thf(tp_c_2Esemi__ring_2Esemi__ring__SRM,type,
c_2Esemi__ring_2Esemi__ring__SRM: del > $i ).
thf(mem_c_2Esemi__ring_2Esemi__ring__SRM,axiom,
! [A_27a: del] : ( mem @ ( c_2Esemi__ring_2Esemi__ring__SRM @ A_27a ) @ ( arr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) ) ).
thf(tp_c_2Esemi__ring_2Esemi__ring__SRM__fupd,type,
c_2Esemi__ring_2Esemi__ring__SRM__fupd: del > $i ).
thf(mem_c_2Esemi__ring_2Esemi__ring__SRM__fupd,axiom,
! [A_27a: del] : ( mem @ ( c_2Esemi__ring_2Esemi__ring__SRM__fupd @ A_27a ) @ ( arr @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) @ ( arr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) ) ) ) ).
thf(tp_c_2Esemi__ring_2Esemi__ring__SRP,type,
c_2Esemi__ring_2Esemi__ring__SRP: del > $i ).
thf(mem_c_2Esemi__ring_2Esemi__ring__SRP,axiom,
! [A_27a: del] : ( mem @ ( c_2Esemi__ring_2Esemi__ring__SRP @ A_27a ) @ ( arr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) ) ).
thf(tp_c_2Esemi__ring_2Esemi__ring__SRP__fupd,type,
c_2Esemi__ring_2Esemi__ring__SRP__fupd: del > $i ).
thf(mem_c_2Esemi__ring_2Esemi__ring__SRP__fupd,axiom,
! [A_27a: del] : ( mem @ ( c_2Esemi__ring_2Esemi__ring__SRP__fupd @ A_27a ) @ ( arr @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) @ ( arr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) ) ) ) ).
thf(tp_c_2Esemi__ring_2Esemi__ring__size,type,
c_2Esemi__ring_2Esemi__ring__size: del > $i ).
thf(mem_c_2Esemi__ring_2Esemi__ring__size,axiom,
! [A_27a: del] : ( mem @ ( c_2Esemi__ring_2Esemi__ring__size @ A_27a ) @ ( arr @ ( arr @ A_27a @ ty_2Enum_2Enum ) @ ( arr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) @ ty_2Enum_2Enum ) ) ) ).
thf(ax_thm_2Esemi__ring_2Esemi__ring__TY__DEF,axiom,
! [A_27a: del] :
? [V0rep: $i] :
( ( mem @ V0rep @ ( arr @ ( ty_2Esemi__ring_2Esemi__ring @ 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 ) ) @ ( arr @ 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 ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) ) ) ) @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
@ ( lam @ ( 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 ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) ) ) )
@ ^ [V1a0_27: $i] :
( 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 ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) ) ) ) @ bool ) )
@ ( lam @ ( 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 ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) ) ) ) @ bool )
@ ^ [V2_27semi__ring_27: $i] :
( 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 ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) ) ) ) )
@ ( lam @ ( 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 ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) ) ) )
@ ^ [V3a0_27: $i] :
( ap
@ ( ap @ c_2Emin_2E_3D_3D_3E
@ ( ap @ ( c_2Ebool_2E_3F @ A_27a )
@ ( lam @ A_27a
@ ^ [V4a0: $i] :
( ap @ ( c_2Ebool_2E_3F @ A_27a )
@ ( lam @ A_27a
@ ^ [V5a1: $i] :
( ap @ ( c_2Ebool_2E_3F @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
@ ( lam @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) )
@ ^ [V6a2: $i] :
( ap @ ( c_2Ebool_2E_3F @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
@ ( lam @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) )
@ ^ [V7a3: $i] :
( 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 ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) ) ) ) ) @ V3a0_27 )
@ ( ap
@ ( ap
@ ( ap
@ ( ap
@ ( lam @ A_27a
@ ^ [V8a0: $i] :
( lam @ A_27a
@ ^ [V9a1: $i] :
( lam @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) )
@ ^ [V10a2: $i] :
( lam @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) )
@ ^ [V11a3: $i] :
( 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 ) ) @ ( arr @ 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 ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) ) ) @ V8a0 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) ) @ V9a1 ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) @ V10a2 ) @ V11a3 ) ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V12n: $i] : ( 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 ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) ) ) ) ) ) ) ) ) )
@ V4a0 )
@ V5a1 )
@ V6a2 )
@ V7a3 ) ) ) ) ) ) ) ) ) ) )
@ ( ap @ V2_27semi__ring_27 @ V3a0_27 ) ) ) ) )
@ ( ap @ V2_27semi__ring_27 @ V1a0_27 ) ) ) ) ) )
@ V0rep ) ) ) ).
thf(ax_thm_2Esemi__ring_2Esemi__ring__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 ) ) )
=> ! [V4f: $i] :
( ( mem @ V4f @ ( arr @ A_27a @ ( arr @ A_27a @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ A_27b ) ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__CASE @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Erecordtype_2Esemi__ring @ A_27a ) @ V0a0 ) @ V1a1 ) @ V2a2 ) @ V3a3 ) ) @ V4f )
= ( ap @ ( ap @ ( ap @ ( ap @ V4f @ V0a0 ) @ V1a1 ) @ V2a2 ) @ V3a3 ) ) ) ) ) ) ) ).
thf(ax_thm_2Esemi__ring_2Esemi__ring__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 ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__size @ A_27a ) @ V0f ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Erecordtype_2Esemi__ring @ A_27a ) @ V1a0 ) @ V2a1 ) @ V3a2 ) @ V4a3 ) ) )
= ( 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 ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Esemi__ring_2Esemi__ring__SR0,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 ) ) )
=> ( ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR0 @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Erecordtype_2Esemi__ring @ A_27a ) @ V0a ) @ V1a0 ) @ V2f ) @ V3f0 ) )
= V0a ) ) ) ) ) ).
thf(ax_thm_2Esemi__ring_2Esemi__ring__SR1,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 ) ) )
=> ( ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR1 @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Erecordtype_2Esemi__ring @ A_27a ) @ V0a ) @ V1a0 ) @ V2f ) @ V3f0 ) )
= V1a0 ) ) ) ) ) ).
thf(ax_thm_2Esemi__ring_2Esemi__ring__SRP,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 ) ) )
=> ( ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Erecordtype_2Esemi__ring @ A_27a ) @ V0a ) @ V1a0 ) @ V2f ) @ V3f0 ) )
= V2f ) ) ) ) ) ).
thf(ax_thm_2Esemi__ring_2Esemi__ring__SRM,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 ) ) )
=> ( ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Erecordtype_2Esemi__ring @ A_27a ) @ V0a ) @ V1a0 ) @ V2f ) @ V3f0 ) )
= V3f0 ) ) ) ) ) ).
thf(conj_thm_2Esemi__ring_2Esemi__ring__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 ) ) )
=> ( ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR0 @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Erecordtype_2Esemi__ring @ A_27a ) @ V0a ) @ V1a0 ) @ V2f ) @ V3f0 ) )
= V0a ) ) ) ) )
& ! [V4a: $i] :
( ( mem @ V4a @ A_27a )
=> ! [V5a0: $i] :
( ( mem @ V5a0 @ A_27a )
=> ! [V6f: $i] :
( ( mem @ V6f @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V7f0: $i] :
( ( mem @ V7f0 @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ( ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR1 @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Erecordtype_2Esemi__ring @ A_27a ) @ V4a ) @ V5a0 ) @ V6f ) @ V7f0 ) )
= V5a0 ) ) ) ) )
& ! [V8a: $i] :
( ( mem @ V8a @ A_27a )
=> ! [V9a0: $i] :
( ( mem @ V9a0 @ A_27a )
=> ! [V10f: $i] :
( ( mem @ V10f @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V11f0: $i] :
( ( mem @ V11f0 @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ( ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Erecordtype_2Esemi__ring @ A_27a ) @ V8a ) @ V9a0 ) @ V10f ) @ V11f0 ) )
= V10f ) ) ) ) )
& ! [V12a: $i] :
( ( mem @ V12a @ A_27a )
=> ! [V13a0: $i] :
( ( mem @ V13a0 @ A_27a )
=> ! [V14f: $i] :
( ( mem @ V14f @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V15f0: $i] :
( ( mem @ V15f0 @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ( ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Erecordtype_2Esemi__ring @ A_27a ) @ V12a ) @ V13a0 ) @ V14f ) @ V15f0 ) )
= V15f0 ) ) ) ) ) ) ).
thf(ax_thm_2Esemi__ring_2Esemi__ring__SR0__fupd,axiom,
! [A_27a: del,V0f1: $i] :
( ( mem @ V0f1 @ ( 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 ) ) )
=> ( ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR0__fupd @ A_27a ) @ V0f1 ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Erecordtype_2Esemi__ring @ A_27a ) @ V1a ) @ V2a0 ) @ V3f ) @ V4f0 ) )
= ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Erecordtype_2Esemi__ring @ A_27a ) @ ( ap @ V0f1 @ V1a ) ) @ V2a0 ) @ V3f ) @ V4f0 ) ) ) ) ) ) ) ).
thf(ax_thm_2Esemi__ring_2Esemi__ring__SR1__fupd,axiom,
! [A_27a: del,V0f1: $i] :
( ( mem @ V0f1 @ ( 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 ) ) )
=> ( ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR1__fupd @ A_27a ) @ V0f1 ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Erecordtype_2Esemi__ring @ A_27a ) @ V1a ) @ V2a0 ) @ V3f ) @ V4f0 ) )
= ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Erecordtype_2Esemi__ring @ A_27a ) @ V1a ) @ ( ap @ V0f1 @ V2a0 ) ) @ V3f ) @ V4f0 ) ) ) ) ) ) ) ).
thf(ax_thm_2Esemi__ring_2Esemi__ring__SRP__fupd,axiom,
! [A_27a: del,V0f1: $i] :
( ( mem @ V0f1 @ ( 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 ) ) )
=> ( ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP__fupd @ A_27a ) @ V0f1 ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Erecordtype_2Esemi__ring @ A_27a ) @ V1a ) @ V2a0 ) @ V3f ) @ V4f0 ) )
= ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Erecordtype_2Esemi__ring @ A_27a ) @ V1a ) @ V2a0 ) @ ( ap @ V0f1 @ V3f ) ) @ V4f0 ) ) ) ) ) ) ) ).
thf(ax_thm_2Esemi__ring_2Esemi__ring__SRM__fupd,axiom,
! [A_27a: del,V0f1: $i] :
( ( mem @ V0f1 @ ( 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 ) ) )
=> ( ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM__fupd @ A_27a ) @ V0f1 ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Erecordtype_2Esemi__ring @ A_27a ) @ V1a ) @ V2a0 ) @ V3f ) @ V4f0 ) )
= ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Erecordtype_2Esemi__ring @ A_27a ) @ V1a ) @ V2a0 ) @ V3f ) @ ( ap @ V0f1 @ V4f0 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esemi__ring_2Esemi__ring__fn__updates,axiom,
! [A_27a: del] :
( ! [V0f1: $i] :
( ( mem @ V0f1 @ ( 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 ) ) )
=> ( ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR0__fupd @ A_27a ) @ V0f1 ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Erecordtype_2Esemi__ring @ A_27a ) @ V1a ) @ V2a0 ) @ V3f ) @ V4f0 ) )
= ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Erecordtype_2Esemi__ring @ A_27a ) @ ( ap @ V0f1 @ V1a ) ) @ V2a0 ) @ V3f ) @ V4f0 ) ) ) ) ) ) )
& ! [V5f1: $i] :
( ( mem @ V5f1 @ ( arr @ A_27a @ A_27a ) )
=> ! [V6a: $i] :
( ( mem @ V6a @ A_27a )
=> ! [V7a0: $i] :
( ( mem @ V7a0 @ A_27a )
=> ! [V8f: $i] :
( ( mem @ V8f @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V9f0: $i] :
( ( mem @ V9f0 @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ( ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR1__fupd @ A_27a ) @ V5f1 ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Erecordtype_2Esemi__ring @ A_27a ) @ V6a ) @ V7a0 ) @ V8f ) @ V9f0 ) )
= ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Erecordtype_2Esemi__ring @ A_27a ) @ V6a ) @ ( ap @ V5f1 @ V7a0 ) ) @ V8f ) @ V9f0 ) ) ) ) ) ) )
& ! [V10f1: $i] :
( ( mem @ V10f1 @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) )
=> ! [V11a: $i] :
( ( mem @ V11a @ A_27a )
=> ! [V12a0: $i] :
( ( mem @ V12a0 @ A_27a )
=> ! [V13f: $i] :
( ( mem @ V13f @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V14f0: $i] :
( ( mem @ V14f0 @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ( ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP__fupd @ A_27a ) @ V10f1 ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Erecordtype_2Esemi__ring @ A_27a ) @ V11a ) @ V12a0 ) @ V13f ) @ V14f0 ) )
= ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Erecordtype_2Esemi__ring @ A_27a ) @ V11a ) @ V12a0 ) @ ( ap @ V10f1 @ V13f ) ) @ V14f0 ) ) ) ) ) ) )
& ! [V15f1: $i] :
( ( mem @ V15f1 @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) )
=> ! [V16a: $i] :
( ( mem @ V16a @ A_27a )
=> ! [V17a0: $i] :
( ( mem @ V17a0 @ A_27a )
=> ! [V18f: $i] :
( ( mem @ V18f @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V19f0: $i] :
( ( mem @ V19f0 @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ( ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM__fupd @ A_27a ) @ V15f1 ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Erecordtype_2Esemi__ring @ A_27a ) @ V16a ) @ V17a0 ) @ V18f ) @ V19f0 ) )
= ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Erecordtype_2Esemi__ring @ A_27a ) @ V16a ) @ V17a0 ) @ V18f ) @ ( ap @ V15f1 @ V19f0 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esemi__ring_2Esemi__ring__accfupds,axiom,
! [A_27a: del] :
( ! [V0s: $i] :
( ( mem @ V0s @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27a ) )
=> ( ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR0 @ A_27a ) @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR1__fupd @ A_27a ) @ V1f ) @ V0s ) )
= ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR0 @ A_27a ) @ V0s ) ) ) )
& ! [V2s: $i] :
( ( mem @ V2s @ ( ty_2Esemi__ring_2Esemi__ring @ 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_2Esemi__ring_2Esemi__ring__SR0 @ A_27a ) @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP__fupd @ A_27a ) @ V3f ) @ V2s ) )
= ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR0 @ A_27a ) @ V2s ) ) ) )
& ! [V4s: $i] :
( ( mem @ V4s @ ( ty_2Esemi__ring_2Esemi__ring @ 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_2Esemi__ring_2Esemi__ring__SR0 @ A_27a ) @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM__fupd @ A_27a ) @ V5f ) @ V4s ) )
= ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR0 @ A_27a ) @ V4s ) ) ) )
& ! [V6s: $i] :
( ( mem @ V6s @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V7f: $i] :
( ( mem @ V7f @ ( arr @ A_27a @ A_27a ) )
=> ( ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR1 @ A_27a ) @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR0__fupd @ A_27a ) @ V7f ) @ V6s ) )
= ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR1 @ A_27a ) @ V6s ) ) ) )
& ! [V8s: $i] :
( ( mem @ V8s @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V9f: $i] :
( ( mem @ V9f @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) )
=> ( ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR1 @ A_27a ) @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP__fupd @ A_27a ) @ V9f ) @ V8s ) )
= ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR1 @ A_27a ) @ V8s ) ) ) )
& ! [V10s: $i] :
( ( mem @ V10s @ ( ty_2Esemi__ring_2Esemi__ring @ 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_2Esemi__ring_2Esemi__ring__SR1 @ A_27a ) @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM__fupd @ A_27a ) @ V11f ) @ V10s ) )
= ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR1 @ A_27a ) @ V10s ) ) ) )
& ! [V12s: $i] :
( ( mem @ V12s @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V13f: $i] :
( ( mem @ V13f @ ( arr @ A_27a @ A_27a ) )
=> ( ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP @ A_27a ) @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR0__fupd @ A_27a ) @ V13f ) @ V12s ) )
= ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP @ A_27a ) @ V12s ) ) ) )
& ! [V14s: $i] :
( ( mem @ V14s @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V15f: $i] :
( ( mem @ V15f @ ( arr @ A_27a @ A_27a ) )
=> ( ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP @ A_27a ) @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR1__fupd @ A_27a ) @ V15f ) @ V14s ) )
= ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP @ A_27a ) @ V14s ) ) ) )
& ! [V16s: $i] :
( ( mem @ V16s @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V17f: $i] :
( ( mem @ V17f @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) )
=> ( ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP @ A_27a ) @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM__fupd @ A_27a ) @ V17f ) @ V16s ) )
= ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP @ A_27a ) @ V16s ) ) ) )
& ! [V18s: $i] :
( ( mem @ V18s @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V19f: $i] :
( ( mem @ V19f @ ( arr @ A_27a @ A_27a ) )
=> ( ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM @ A_27a ) @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR0__fupd @ A_27a ) @ V19f ) @ V18s ) )
= ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM @ A_27a ) @ V18s ) ) ) )
& ! [V20s: $i] :
( ( mem @ V20s @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V21f: $i] :
( ( mem @ V21f @ ( arr @ A_27a @ A_27a ) )
=> ( ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM @ A_27a ) @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR1__fupd @ A_27a ) @ V21f ) @ V20s ) )
= ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM @ A_27a ) @ V20s ) ) ) )
& ! [V22s: $i] :
( ( mem @ V22s @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V23f: $i] :
( ( mem @ V23f @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) )
=> ( ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM @ A_27a ) @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP__fupd @ A_27a ) @ V23f ) @ V22s ) )
= ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM @ A_27a ) @ V22s ) ) ) )
& ! [V24s: $i] :
( ( mem @ V24s @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V25f: $i] :
( ( mem @ V25f @ ( arr @ A_27a @ A_27a ) )
=> ( ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR0 @ A_27a ) @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR0__fupd @ A_27a ) @ V25f ) @ V24s ) )
= ( ap @ V25f @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR0 @ A_27a ) @ V24s ) ) ) ) )
& ! [V26s: $i] :
( ( mem @ V26s @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V27f: $i] :
( ( mem @ V27f @ ( arr @ A_27a @ A_27a ) )
=> ( ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR1 @ A_27a ) @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR1__fupd @ A_27a ) @ V27f ) @ V26s ) )
= ( ap @ V27f @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR1 @ A_27a ) @ V26s ) ) ) ) )
& ! [V28s: $i] :
( ( mem @ V28s @ ( ty_2Esemi__ring_2Esemi__ring @ 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_2Esemi__ring_2Esemi__ring__SRP @ A_27a ) @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP__fupd @ A_27a ) @ V29f ) @ V28s ) )
= ( ap @ V29f @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP @ A_27a ) @ V28s ) ) ) ) )
& ! [V30s: $i] :
( ( mem @ V30s @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V31f: $i] :
( ( mem @ V31f @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) )
=> ( ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM @ A_27a ) @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM__fupd @ A_27a ) @ V31f ) @ V30s ) )
= ( ap @ V31f @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM @ A_27a ) @ V30s ) ) ) ) ) ) ).
thf(conj_thm_2Esemi__ring_2Esemi__ring__fupdfupds,axiom,
! [A_27a: del] :
( ! [V0s: $i] :
( ( mem @ V0s @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27a @ A_27a ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR0__fupd @ A_27a ) @ V2f ) @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR0__fupd @ A_27a ) @ V1g ) @ V0s ) )
= ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR0__fupd @ A_27a ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ A_27a @ A_27a ) @ V2f ) @ V1g ) ) @ V0s ) ) ) ) )
& ! [V3s: $i] :
( ( mem @ V3s @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V4g: $i] :
( ( mem @ V4g @ ( arr @ A_27a @ A_27a ) )
=> ! [V5f: $i] :
( ( mem @ V5f @ ( arr @ A_27a @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR1__fupd @ A_27a ) @ V5f ) @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR1__fupd @ A_27a ) @ V4g ) @ V3s ) )
= ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR1__fupd @ A_27a ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ A_27a @ A_27a ) @ V5f ) @ V4g ) ) @ V3s ) ) ) ) )
& ! [V6s: $i] :
( ( mem @ V6s @ ( ty_2Esemi__ring_2Esemi__ring @ 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_2Esemi__ring_2Esemi__ring__SRP__fupd @ A_27a ) @ V8f ) @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP__fupd @ A_27a ) @ V7g ) @ V6s ) )
= ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP__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 ) ) @ V6s ) ) ) ) )
& ! [V9s: $i] :
( ( mem @ V9s @ ( ty_2Esemi__ring_2Esemi__ring @ 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_2Esemi__ring_2Esemi__ring__SRM__fupd @ A_27a ) @ V11f ) @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM__fupd @ A_27a ) @ V10g ) @ V9s ) )
= ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM__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 ) ) @ V9s ) ) ) ) ) ) ).
thf(conj_thm_2Esemi__ring_2Esemi__ring__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_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR0__fupd @ A_27a ) @ V1f ) ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR0__fupd @ A_27a ) @ V0g ) )
= ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR0__fupd @ A_27a ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ A_27a @ A_27a ) @ V1f ) @ V0g ) ) ) ) )
& ! [V2h: $i] :
( ( mem @ V2h @ ( arr @ A_27b @ ( ty_2Esemi__ring_2Esemi__ring @ 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_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR0__fupd @ A_27a ) @ V4f ) ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR0__fupd @ A_27a ) @ V3g ) ) @ V2h ) )
= ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR0__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_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR1__fupd @ A_27a ) @ V6f ) ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR1__fupd @ A_27a ) @ V5g ) )
= ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR1__fupd @ A_27a ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ A_27a @ A_27a ) @ V6f ) @ V5g ) ) ) ) )
& ! [V7h: $i] :
( ( mem @ V7h @ ( arr @ A_27b @ ( ty_2Esemi__ring_2Esemi__ring @ 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_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR1__fupd @ A_27a ) @ V9f ) ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR1__fupd @ A_27a ) @ V8g ) ) @ V7h ) )
= ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR1__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_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP__fupd @ A_27a ) @ V11f ) ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP__fupd @ A_27a ) @ V10g ) )
= ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP__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_2Esemi__ring_2Esemi__ring @ 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_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP__fupd @ A_27a ) @ V14f ) ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP__fupd @ A_27a ) @ V13g ) ) @ V12h ) )
= ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP__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_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM__fupd @ A_27a ) @ V16f ) ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM__fupd @ A_27a ) @ V15g ) )
= ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM__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_2Esemi__ring_2Esemi__ring @ 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_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM__fupd @ A_27a ) @ V19f ) ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM__fupd @ A_27a ) @ V18g ) ) @ V17h ) )
= ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM__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 ) ) ) ) ) ) ).
thf(conj_thm_2Esemi__ring_2Esemi__ring__fupdcanon,axiom,
! [A_27a: del] :
( ! [V0s: $i] :
( ( mem @ V0s @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27a @ A_27a ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR1__fupd @ A_27a ) @ V2f ) @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR0__fupd @ A_27a ) @ V1g ) @ V0s ) )
= ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR0__fupd @ A_27a ) @ V1g ) @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR1__fupd @ A_27a ) @ V2f ) @ V0s ) ) ) ) ) )
& ! [V3s: $i] :
( ( mem @ V3s @ ( ty_2Esemi__ring_2Esemi__ring @ 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_2Esemi__ring_2Esemi__ring__SRP__fupd @ A_27a ) @ V5f ) @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR0__fupd @ A_27a ) @ V4g ) @ V3s ) )
= ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR0__fupd @ A_27a ) @ V4g ) @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP__fupd @ A_27a ) @ V5f ) @ V3s ) ) ) ) ) )
& ! [V6s: $i] :
( ( mem @ V6s @ ( ty_2Esemi__ring_2Esemi__ring @ 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_2Esemi__ring_2Esemi__ring__SRP__fupd @ A_27a ) @ V8f ) @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR1__fupd @ A_27a ) @ V7g ) @ V6s ) )
= ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR1__fupd @ A_27a ) @ V7g ) @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP__fupd @ A_27a ) @ V8f ) @ V6s ) ) ) ) ) )
& ! [V9s: $i] :
( ( mem @ V9s @ ( ty_2Esemi__ring_2Esemi__ring @ 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_2Esemi__ring_2Esemi__ring__SRM__fupd @ A_27a ) @ V11f ) @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR0__fupd @ A_27a ) @ V10g ) @ V9s ) )
= ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR0__fupd @ A_27a ) @ V10g ) @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM__fupd @ A_27a ) @ V11f ) @ V9s ) ) ) ) ) )
& ! [V12s: $i] :
( ( mem @ V12s @ ( ty_2Esemi__ring_2Esemi__ring @ 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_2Esemi__ring_2Esemi__ring__SRM__fupd @ A_27a ) @ V14f ) @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR1__fupd @ A_27a ) @ V13g ) @ V12s ) )
= ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR1__fupd @ A_27a ) @ V13g ) @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM__fupd @ A_27a ) @ V14f ) @ V12s ) ) ) ) ) )
& ! [V15s: $i] :
( ( mem @ V15s @ ( ty_2Esemi__ring_2Esemi__ring @ 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_2Esemi__ring_2Esemi__ring__SRM__fupd @ A_27a ) @ V17f ) @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP__fupd @ A_27a ) @ V16g ) @ V15s ) )
= ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP__fupd @ A_27a ) @ V16g ) @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM__fupd @ A_27a ) @ V17f ) @ V15s ) ) ) ) ) ) ) ).
thf(conj_thm_2Esemi__ring_2Esemi__ring__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_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR1__fupd @ A_27a ) @ V1f ) ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR0__fupd @ A_27a ) @ V0g ) )
= ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR0__fupd @ A_27a ) @ V0g ) ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR1__fupd @ A_27a ) @ V1f ) ) ) ) )
& ! [V2h: $i] :
( ( mem @ V2h @ ( arr @ A_27b @ ( ty_2Esemi__ring_2Esemi__ring @ 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_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR1__fupd @ A_27a ) @ V4f ) ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR0__fupd @ A_27a ) @ V3g ) ) @ V2h ) )
= ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR0__fupd @ A_27a ) @ V3g ) ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR1__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_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP__fupd @ A_27a ) @ V6f ) ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR0__fupd @ A_27a ) @ V5g ) )
= ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR0__fupd @ A_27a ) @ V5g ) ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP__fupd @ A_27a ) @ V6f ) ) ) ) )
& ! [V7h: $i] :
( ( mem @ V7h @ ( arr @ A_27b @ ( ty_2Esemi__ring_2Esemi__ring @ 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_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP__fupd @ A_27a ) @ V9f ) ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR0__fupd @ A_27a ) @ V8g ) ) @ V7h ) )
= ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR0__fupd @ A_27a ) @ V8g ) ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP__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_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP__fupd @ A_27a ) @ V11f ) ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR1__fupd @ A_27a ) @ V10g ) )
= ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR1__fupd @ A_27a ) @ V10g ) ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP__fupd @ A_27a ) @ V11f ) ) ) ) )
& ! [V12h: $i] :
( ( mem @ V12h @ ( arr @ A_27b @ ( ty_2Esemi__ring_2Esemi__ring @ 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_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP__fupd @ A_27a ) @ V14f ) ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR1__fupd @ A_27a ) @ V13g ) ) @ V12h ) )
= ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR1__fupd @ A_27a ) @ V13g ) ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP__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_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM__fupd @ A_27a ) @ V16f ) ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR0__fupd @ A_27a ) @ V15g ) )
= ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR0__fupd @ A_27a ) @ V15g ) ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM__fupd @ A_27a ) @ V16f ) ) ) ) )
& ! [V17h: $i] :
( ( mem @ V17h @ ( arr @ A_27b @ ( ty_2Esemi__ring_2Esemi__ring @ 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_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM__fupd @ A_27a ) @ V19f ) ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR0__fupd @ A_27a ) @ V18g ) ) @ V17h ) )
= ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR0__fupd @ A_27a ) @ V18g ) ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM__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_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM__fupd @ A_27a ) @ V21f ) ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR1__fupd @ A_27a ) @ V20g ) )
= ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR1__fupd @ A_27a ) @ V20g ) ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM__fupd @ A_27a ) @ V21f ) ) ) ) )
& ! [V22h: $i] :
( ( mem @ V22h @ ( arr @ A_27b @ ( ty_2Esemi__ring_2Esemi__ring @ 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_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM__fupd @ A_27a ) @ V24f ) ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR1__fupd @ A_27a ) @ V23g ) ) @ V22h ) )
= ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR1__fupd @ A_27a ) @ V23g ) ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM__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_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM__fupd @ A_27a ) @ V26f ) ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP__fupd @ A_27a ) @ V25g ) )
= ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP__fupd @ A_27a ) @ V25g ) ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM__fupd @ A_27a ) @ V26f ) ) ) ) )
& ! [V27h: $i] :
( ( mem @ V27h @ ( arr @ A_27b @ ( ty_2Esemi__ring_2Esemi__ring @ 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_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM__fupd @ A_27a ) @ V29f ) ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP__fupd @ A_27a ) @ V28g ) ) @ V27h ) )
= ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP__fupd @ A_27a ) @ V28g ) ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27b @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM__fupd @ A_27a ) @ V29f ) ) @ V27h ) ) ) ) ) ) ) ).
thf(conj_thm_2Esemi__ring_2Esemi__ring__component__equality,axiom,
! [A_27a: del,V0s1: $i] :
( ( mem @ V0s1 @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V1s2: $i] :
( ( mem @ V1s2 @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ( ( V0s1 = V1s2 )
<=> ( ( ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR0 @ A_27a ) @ V0s1 )
= ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR0 @ A_27a ) @ V1s2 ) )
& ( ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR1 @ A_27a ) @ V0s1 )
= ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR1 @ A_27a ) @ V1s2 ) )
& ( ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP @ A_27a ) @ V0s1 )
= ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP @ A_27a ) @ V1s2 ) )
& ( ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM @ A_27a ) @ V0s1 )
= ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM @ A_27a ) @ V1s2 ) ) ) ) ) ) ).
thf(conj_thm_2Esemi__ring_2Esemi__ring__updates__eq__literal,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V1a0: $i] :
( ( mem @ V1a0 @ A_27a )
=> ! [V2a: $i] :
( ( mem @ V2a @ A_27a )
=> ! [V3f0: $i] :
( ( mem @ V3f0 @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V4f: $i] :
( ( mem @ V4f @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ( ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR0__fupd @ A_27a ) @ ( ap @ ( c_2Ecombin_2EK @ A_27a @ A_27a ) @ V1a0 ) ) @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR1__fupd @ A_27a ) @ ( ap @ ( c_2Ecombin_2EK @ A_27a @ A_27a ) @ V2a ) ) @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP__fupd @ A_27a ) @ ( ap @ ( c_2Ecombin_2EK @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) @ V3f0 ) ) @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM__fupd @ A_27a ) @ ( ap @ ( c_2Ecombin_2EK @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) @ V4f ) ) @ V0s ) ) ) )
= ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR0__fupd @ A_27a ) @ ( ap @ ( c_2Ecombin_2EK @ A_27a @ A_27a ) @ V1a0 ) ) @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR1__fupd @ A_27a ) @ ( ap @ ( c_2Ecombin_2EK @ A_27a @ A_27a ) @ V2a ) ) @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP__fupd @ A_27a ) @ ( ap @ ( c_2Ecombin_2EK @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) @ V3f0 ) ) @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM__fupd @ A_27a ) @ ( ap @ ( c_2Ecombin_2EK @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) @ V4f ) ) @ ( c_2Ebool_2EARB @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esemi__ring_2Esemi__ring__literal__nchotomy,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ? [V1a0: $i] :
( ( mem @ V1a0 @ A_27a )
& ? [V2a: $i] :
( ( mem @ V2a @ A_27a )
& ? [V3f0: $i] :
( ( mem @ V3f0 @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
& ? [V4f: $i] :
( ( mem @ V4f @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
& ( V0s
= ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR0__fupd @ A_27a ) @ ( ap @ ( c_2Ecombin_2EK @ A_27a @ A_27a ) @ V1a0 ) ) @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR1__fupd @ A_27a ) @ ( ap @ ( c_2Ecombin_2EK @ A_27a @ A_27a ) @ V2a ) ) @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP__fupd @ A_27a ) @ ( ap @ ( c_2Ecombin_2EK @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) @ V3f0 ) ) @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM__fupd @ A_27a ) @ ( ap @ ( c_2Ecombin_2EK @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) @ V4f ) ) @ ( c_2Ebool_2EARB @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esemi__ring_2EFORALL__semi__ring,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) @ bool ) )
=> ( ! [V1s: $i] :
( ( mem @ V1s @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ( p @ ( ap @ V0P @ V1s ) ) )
<=> ! [V2a0: $i] :
( ( mem @ V2a0 @ A_27a )
=> ! [V3a: $i] :
( ( mem @ V3a @ A_27a )
=> ! [V4f0: $i] :
( ( mem @ V4f0 @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V5f: $i] :
( ( mem @ V5f @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR0__fupd @ A_27a ) @ ( ap @ ( c_2Ecombin_2EK @ A_27a @ A_27a ) @ V2a0 ) ) @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR1__fupd @ A_27a ) @ ( ap @ ( c_2Ecombin_2EK @ A_27a @ A_27a ) @ V3a ) ) @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP__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_2Esemi__ring_2Esemi__ring__SRM__fupd @ A_27a ) @ ( ap @ ( c_2Ecombin_2EK @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) @ V5f ) ) @ ( c_2Ebool_2EARB @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esemi__ring_2EEXISTS__semi__ring,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) @ bool ) )
=> ( ? [V1s: $i] :
( ( mem @ V1s @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
& ( p @ ( ap @ V0P @ V1s ) ) )
<=> ? [V2a0: $i] :
( ( mem @ V2a0 @ A_27a )
& ? [V3a: $i] :
( ( mem @ V3a @ A_27a )
& ? [V4f0: $i] :
( ( mem @ V4f0 @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
& ? [V5f: $i] :
( ( mem @ V5f @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
& ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR0__fupd @ A_27a ) @ ( ap @ ( c_2Ecombin_2EK @ A_27a @ A_27a ) @ V2a0 ) ) @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR1__fupd @ A_27a ) @ ( ap @ ( c_2Ecombin_2EK @ A_27a @ A_27a ) @ V3a ) ) @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP__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_2Esemi__ring_2Esemi__ring__SRM__fupd @ A_27a ) @ ( ap @ ( c_2Ecombin_2EK @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) @ V5f ) ) @ ( c_2Ebool_2EARB @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esemi__ring_2Esemi__ring__literal__11,axiom,
! [A_27a: del,V0a01: $i] :
( ( mem @ V0a01 @ A_27a )
=> ! [V1a1: $i] :
( ( mem @ V1a1 @ A_27a )
=> ! [V2f01: $i] :
( ( mem @ V2f01 @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V3f1: $i] :
( ( mem @ V3f1 @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V4a02: $i] :
( ( mem @ V4a02 @ A_27a )
=> ! [V5a2: $i] :
( ( mem @ V5a2 @ A_27a )
=> ! [V6f02: $i] :
( ( mem @ V6f02 @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V7f2: $i] :
( ( mem @ V7f2 @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ( ( ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR0__fupd @ A_27a ) @ ( ap @ ( c_2Ecombin_2EK @ A_27a @ A_27a ) @ V0a01 ) ) @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR1__fupd @ A_27a ) @ ( ap @ ( c_2Ecombin_2EK @ A_27a @ A_27a ) @ V1a1 ) ) @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP__fupd @ A_27a ) @ ( ap @ ( c_2Ecombin_2EK @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) @ V2f01 ) ) @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM__fupd @ A_27a ) @ ( ap @ ( c_2Ecombin_2EK @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) @ V3f1 ) ) @ ( c_2Ebool_2EARB @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) ) ) ) ) )
= ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR0__fupd @ A_27a ) @ ( ap @ ( c_2Ecombin_2EK @ A_27a @ A_27a ) @ V4a02 ) ) @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR1__fupd @ A_27a ) @ ( ap @ ( c_2Ecombin_2EK @ A_27a @ A_27a ) @ V5a2 ) ) @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP__fupd @ A_27a ) @ ( ap @ ( c_2Ecombin_2EK @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) @ V6f02 ) ) @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM__fupd @ A_27a ) @ ( ap @ ( c_2Ecombin_2EK @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) ) @ V7f2 ) ) @ ( c_2Ebool_2EARB @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) ) ) ) ) ) )
<=> ( ( V0a01 = V4a02 )
& ( V1a1 = V5a2 )
& ( V2f01 = V6f02 )
& ( V3f1 = V7f2 ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esemi__ring_2Edatatype__semi__ring,axiom,
! [A_27a: del,V0record: $i] :
( ( mem @ V0record @ ( arr @ ( ty_2Esemi__ring_2Esemi__ring @ 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 ) ) @ bool ) ) ) ) ) )
=> ! [V1semi__ring: $i] :
( ( mem @ V1semi__ring @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V2SR0: $i] :
( ( mem @ V2SR0 @ A_27a )
=> ! [V3SR1: $i] :
( ( mem @ V3SR1 @ A_27a )
=> ! [V4SRP: $i] :
( ( mem @ V4SRP @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V5SRM: $i] :
( ( mem @ V5SRM @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ( p @ ( ap @ ( c_2Ebool_2EDATATYPE @ bool ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ V0record @ V1semi__ring ) @ V2SR0 ) @ V3SR1 ) @ V4SRP ) @ V5SRM ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esemi__ring_2Esemi__ring__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 ) ) )
=> ! [V4a0_27: $i] :
( ( mem @ V4a0_27 @ A_27a )
=> ! [V5a1_27: $i] :
( ( mem @ V5a1_27 @ A_27a )
=> ! [V6a2_27: $i] :
( ( mem @ V6a2_27 @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ! [V7a3_27: $i] :
( ( mem @ V7a3_27 @ ( arr @ A_27a @ ( arr @ A_27a @ A_27a ) ) )
=> ( ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Erecordtype_2Esemi__ring @ A_27a ) @ V0a0 ) @ V1a1 ) @ V2a2 ) @ V3a3 )
= ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Erecordtype_2Esemi__ring @ A_27a ) @ V4a0_27 ) @ V5a1_27 ) @ V6a2_27 ) @ V7a3_27 ) )
<=> ( ( V0a0 = V4a0_27 )
& ( V1a1 = V5a1_27 )
& ( V2a2 = V6a2_27 )
& ( V3a3 = V7a3_27 ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esemi__ring_2Esemi__ring__nchotomy,axiom,
! [A_27a: del,V0ss: $i] :
( ( mem @ V0ss @ ( ty_2Esemi__ring_2Esemi__ring @ 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 ) ) )
& ( V0ss
= ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Erecordtype_2Esemi__ring @ A_27a ) @ V1a ) @ V2a0 ) @ V3f ) @ V4f0 ) ) ) ) ) ) ) ).
thf(conj_thm_2Esemi__ring_2Esemi__ring__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 ) ) @ A_27b ) ) ) ) )
=> ? [V1fn: $i] :
( ( mem @ V1fn @ ( arr @ ( ty_2Esemi__ring_2Esemi__ring @ 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 ) ) )
=> ( ( ap @ V1fn @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Erecordtype_2Esemi__ring @ A_27a ) @ V2a0 ) @ V3a1 ) @ V4a2 ) @ V5a3 ) )
= ( ap @ ( ap @ ( ap @ ( ap @ V0f @ V2a0 ) @ V3a1 ) @ V4a2 ) @ V5a3 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esemi__ring_2Esemi__ring__induction,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Esemi__ring_2Esemi__ring @ 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 ) ) )
=> ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Erecordtype_2Esemi__ring @ A_27a ) @ V1a ) @ V2a0 ) @ V3f ) @ V4f0 ) ) ) ) ) ) )
=> ! [V5s: $i] :
( ( mem @ V5s @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ( p @ ( ap @ V0P @ V5s ) ) ) ) ) ).
thf(conj_thm_2Esemi__ring_2Esemi__ring__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 ) ) @ A_27b ) ) ) ) )
=> ! [V1M: $i] :
( ( mem @ V1M @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V2M_27: $i] :
( ( mem @ V2M_27 @ ( ty_2Esemi__ring_2Esemi__ring @ 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 ) ) @ 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 ) ) )
=> ( ( V2M_27
= ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Erecordtype_2Esemi__ring @ A_27a ) @ V4a0 ) @ V5a1 ) @ V6a2 ) @ V7a3 ) )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ V3f @ V4a0 ) @ V5a1 ) @ V6a2 ) @ V7a3 )
= ( ap @ ( ap @ ( ap @ ( ap @ V0f_27 @ V4a0 ) @ V5a1 ) @ V6a2 ) @ V7a3 ) ) ) ) ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__CASE @ A_27a @ A_27b ) @ V1M ) @ V3f )
= ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__CASE @ A_27a @ A_27b ) @ V2M_27 ) @ V0f_27 ) ) ) ) ) ) ) ).
thf(conj_thm_2Esemi__ring_2Esemi__ring__case__eq,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ ( ty_2Esemi__ring_2Esemi__ring @ 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 ) ) @ A_27b ) ) ) ) )
=> ! [V2v: $i] :
( ( mem @ V2v @ A_27b )
=> ( ( ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__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 ) ) )
& ( V0x
= ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Erecordtype_2Esemi__ring @ A_27a ) @ V3a ) @ V4a0 ) @ V5f_27 ) @ V6f0 ) )
& ( ( ap @ ( ap @ ( ap @ ( ap @ V1f @ V3a ) @ V4a0 ) @ V5f_27 ) @ V6f0 )
= V2v ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Esemi__ring_2Eis__semi__ring__def,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Esemi__ring_2Eis__semi__ring @ A_27a ) @ V0r ) )
<=> ( ! [V1n: $i] :
( ( mem @ V1n @ A_27a )
=> ! [V2m: $i] :
( ( mem @ V2m @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP @ A_27a ) @ V0r ) @ V1n ) @ V2m )
= ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP @ 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_2Esemi__ring_2Esemi__ring__SRP @ A_27a ) @ V0r ) @ V3n ) @ ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP @ A_27a ) @ V0r ) @ V4m ) @ V5p ) )
= ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP @ A_27a ) @ V0r ) @ ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP @ A_27a ) @ V0r ) @ V3n ) @ V4m ) ) @ V5p ) ) ) ) )
& ! [V6n: $i] :
( ( mem @ V6n @ A_27a )
=> ! [V7m: $i] :
( ( mem @ V7m @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM @ A_27a ) @ V0r ) @ V6n ) @ V7m )
= ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM @ 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_2Esemi__ring_2Esemi__ring__SRM @ A_27a ) @ V0r ) @ V8n ) @ ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM @ A_27a ) @ V0r ) @ V9m ) @ V10p ) )
= ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM @ A_27a ) @ V0r ) @ ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM @ A_27a ) @ V0r ) @ V8n ) @ V9m ) ) @ V10p ) ) ) ) )
& ! [V11n: $i] :
( ( mem @ V11n @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP @ A_27a ) @ V0r ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR0 @ A_27a ) @ V0r ) ) @ V11n )
= V11n ) )
& ! [V12n: $i] :
( ( mem @ V12n @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM @ A_27a ) @ V0r ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR1 @ A_27a ) @ V0r ) ) @ V12n )
= V12n ) )
& ! [V13n: $i] :
( ( mem @ V13n @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM @ A_27a ) @ V0r ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR0 @ A_27a ) @ V0r ) ) @ V13n )
= ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR0 @ 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_2Esemi__ring_2Esemi__ring__SRM @ A_27a ) @ V0r ) @ ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP @ A_27a ) @ V0r ) @ V14n ) @ V15m ) ) @ V16p )
= ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP @ A_27a ) @ V0r ) @ ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM @ A_27a ) @ V0r ) @ V14n ) @ V16p ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM @ A_27a ) @ V0r ) @ V15m ) @ V16p ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esemi__ring_2Eplus__sym,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Esemi__ring_2Eis__semi__ring @ A_27a ) @ V0r ) )
=> ! [V1n: $i] :
( ( mem @ V1n @ A_27a )
=> ! [V2m: $i] :
( ( mem @ V2m @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP @ A_27a ) @ V0r ) @ V1n ) @ V2m )
= ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP @ A_27a ) @ V0r ) @ V2m ) @ V1n ) ) ) ) ) ) ).
thf(conj_thm_2Esemi__ring_2Eplus__assoc,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Esemi__ring_2Eis__semi__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_2Esemi__ring_2Esemi__ring__SRP @ A_27a ) @ V0r ) @ V1n ) @ ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP @ A_27a ) @ V0r ) @ V2m ) @ V3p ) )
= ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP @ A_27a ) @ V0r ) @ ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP @ A_27a ) @ V0r ) @ V1n ) @ V2m ) ) @ V3p ) ) ) ) ) ) ) ).
thf(conj_thm_2Esemi__ring_2Emult__sym,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Esemi__ring_2Eis__semi__ring @ A_27a ) @ V0r ) )
=> ! [V1n: $i] :
( ( mem @ V1n @ A_27a )
=> ! [V2m: $i] :
( ( mem @ V2m @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM @ A_27a ) @ V0r ) @ V1n ) @ V2m )
= ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM @ A_27a ) @ V0r ) @ V2m ) @ V1n ) ) ) ) ) ) ).
thf(conj_thm_2Esemi__ring_2Emult__assoc,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Esemi__ring_2Eis__semi__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_2Esemi__ring_2Esemi__ring__SRM @ A_27a ) @ V0r ) @ V1n ) @ ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM @ A_27a ) @ V0r ) @ V2m ) @ V3p ) )
= ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM @ A_27a ) @ V0r ) @ ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM @ A_27a ) @ V0r ) @ V1n ) @ V2m ) ) @ V3p ) ) ) ) ) ) ) ).
thf(conj_thm_2Esemi__ring_2Eplus__zero__left,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Esemi__ring_2Eis__semi__ring @ A_27a ) @ V0r ) )
=> ! [V1n: $i] :
( ( mem @ V1n @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP @ A_27a ) @ V0r ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR0 @ A_27a ) @ V0r ) ) @ V1n )
= V1n ) ) ) ) ).
thf(conj_thm_2Esemi__ring_2Emult__one__left,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Esemi__ring_2Eis__semi__ring @ A_27a ) @ V0r ) )
=> ! [V1n: $i] :
( ( mem @ V1n @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM @ A_27a ) @ V0r ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR1 @ A_27a ) @ V0r ) ) @ V1n )
= V1n ) ) ) ) ).
thf(conj_thm_2Esemi__ring_2Emult__zero__left,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Esemi__ring_2Eis__semi__ring @ A_27a ) @ V0r ) )
=> ! [V1n: $i] :
( ( mem @ V1n @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM @ A_27a ) @ V0r ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR0 @ A_27a ) @ V0r ) ) @ V1n )
= ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR0 @ A_27a ) @ V0r ) ) ) ) ) ).
thf(conj_thm_2Esemi__ring_2Edistr__left,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Esemi__ring_2Eis__semi__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_2Esemi__ring_2Esemi__ring__SRM @ A_27a ) @ V0r ) @ ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP @ A_27a ) @ V0r ) @ V1n ) @ V2m ) ) @ V3p )
= ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP @ A_27a ) @ V0r ) @ ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM @ A_27a ) @ V0r ) @ V1n ) @ V3p ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM @ A_27a ) @ V0r ) @ V2m ) @ V3p ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esemi__ring_2Eplus__zero__right,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Esemi__ring_2Eis__semi__ring @ A_27a ) @ V0r ) )
=> ! [V1n: $i] :
( ( mem @ V1n @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP @ A_27a ) @ V0r ) @ V1n ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR0 @ A_27a ) @ V0r ) )
= V1n ) ) ) ) ).
thf(conj_thm_2Esemi__ring_2Emult__one__right,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Esemi__ring_2Eis__semi__ring @ A_27a ) @ V0r ) )
=> ! [V1n: $i] :
( ( mem @ V1n @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM @ A_27a ) @ V0r ) @ V1n ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR1 @ A_27a ) @ V0r ) )
= V1n ) ) ) ) ).
thf(conj_thm_2Esemi__ring_2Emult__zero__right,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Esemi__ring_2Eis__semi__ring @ A_27a ) @ V0r ) )
=> ! [V1n: $i] :
( ( mem @ V1n @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM @ A_27a ) @ V0r ) @ V1n ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR0 @ A_27a ) @ V0r ) )
= ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR0 @ A_27a ) @ V0r ) ) ) ) ) ).
thf(conj_thm_2Esemi__ring_2Edistr__right,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Esemi__ring_2Eis__semi__ring @ A_27a ) @ V0r ) )
=> ! [V1m: $i] :
( ( mem @ V1m @ A_27a )
=> ! [V2n: $i] :
( ( mem @ V2n @ A_27a )
=> ! [V3p: $i] :
( ( mem @ V3p @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM @ A_27a ) @ V0r ) @ V1m ) @ ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP @ A_27a ) @ V0r ) @ V2n ) @ V3p ) )
= ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP @ A_27a ) @ V0r ) @ ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM @ A_27a ) @ V0r ) @ V1m ) @ V2n ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM @ A_27a ) @ V0r ) @ V1m ) @ V3p ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Esemi__ring_2Eplus__rotate,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Esemi__ring_2Eis__semi__ring @ A_27a ) @ V0r ) )
=> ! [V1m: $i] :
( ( mem @ V1m @ A_27a )
=> ! [V2n: $i] :
( ( mem @ V2n @ A_27a )
=> ! [V3p: $i] :
( ( mem @ V3p @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP @ A_27a ) @ V0r ) @ ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP @ A_27a ) @ V0r ) @ V1m ) @ V2n ) ) @ V3p )
= ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP @ A_27a ) @ V0r ) @ ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP @ A_27a ) @ V0r ) @ V2n ) @ V3p ) ) @ V1m ) ) ) ) ) ) ) ).
thf(conj_thm_2Esemi__ring_2Eplus__permute,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Esemi__ring_2Eis__semi__ring @ A_27a ) @ V0r ) )
=> ! [V1m: $i] :
( ( mem @ V1m @ A_27a )
=> ! [V2n: $i] :
( ( mem @ V2n @ A_27a )
=> ! [V3p: $i] :
( ( mem @ V3p @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP @ A_27a ) @ V0r ) @ ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP @ A_27a ) @ V0r ) @ V1m ) @ V2n ) ) @ V3p )
= ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP @ A_27a ) @ V0r ) @ ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP @ A_27a ) @ V0r ) @ V1m ) @ V3p ) ) @ V2n ) ) ) ) ) ) ) ).
thf(conj_thm_2Esemi__ring_2Emult__rotate,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Esemi__ring_2Eis__semi__ring @ A_27a ) @ V0r ) )
=> ! [V1m: $i] :
( ( mem @ V1m @ A_27a )
=> ! [V2n: $i] :
( ( mem @ V2n @ A_27a )
=> ! [V3p: $i] :
( ( mem @ V3p @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM @ A_27a ) @ V0r ) @ ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM @ A_27a ) @ V0r ) @ V1m ) @ V2n ) ) @ V3p )
= ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM @ A_27a ) @ V0r ) @ ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM @ A_27a ) @ V0r ) @ V2n ) @ V3p ) ) @ V1m ) ) ) ) ) ) ) ).
thf(conj_thm_2Esemi__ring_2Emult__permute,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Esemi__ring_2Eis__semi__ring @ A_27a ) @ V0r ) )
=> ! [V1m: $i] :
( ( mem @ V1m @ A_27a )
=> ! [V2n: $i] :
( ( mem @ V2n @ A_27a )
=> ! [V3p: $i] :
( ( mem @ V3p @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM @ A_27a ) @ V0r ) @ ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM @ A_27a ) @ V0r ) @ V1m ) @ V2n ) ) @ V3p )
= ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM @ A_27a ) @ V0r ) @ ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM @ A_27a ) @ V0r ) @ V1m ) @ V3p ) ) @ V2n ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------