ITP001 Axioms: ITP096^5.ax
%------------------------------------------------------------------------------
% File : ITP096^5 : 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 : canonical^2.ax [Gau20]
% : HL4096^5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 111 ( 1 unt; 29 typ; 0 def)
% Number of atoms : 3621 ( 167 equ; 0 cnn)
% Maximal formula atoms : 277 ( 32 avg)
% Number of connectives : 6802 ( 9 ~; 10 |; 123 &;6212 @)
% ( 8 <=>; 440 =>; 0 <=; 0 <~>)
% Maximal formula depth : 60 ( 14 avg;6212 nst)
% Number of types : 1 ( 0 usr)
% Number of type conns : 31 ( 31 >; 0 *; 0 +; 0 <<)
% Number of symbols : 89 ( 88 usr; 59 con; 0-2 aty)
% Number of variables : 646 ( 35 ^ 585 !; 26 ?; 646 :)
% SPC : TH0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_ty_2Ecanonical_2Ecanonical__sum,type,
ty_2Ecanonical_2Ecanonical__sum: del > del ).
thf(tp_ty_2Ecanonical_2Espolynom,type,
ty_2Ecanonical_2Espolynom: del > del ).
thf(tp_c_2Ecanonical_2ECons__monom,type,
c_2Ecanonical_2ECons__monom: del > $i ).
thf(mem_c_2Ecanonical_2ECons__monom,axiom,
! [A_27a: del] : ( mem @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ ( arr @ A_27a @ ( arr @ ( ty_2Elist_2Elist @ ty_2Equote_2Eindex ) @ ( arr @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) ) ) ) ) ).
thf(tp_c_2Ecanonical_2ECons__varlist,type,
c_2Ecanonical_2ECons__varlist: del > $i ).
thf(mem_c_2Ecanonical_2ECons__varlist,axiom,
! [A_27a: del] : ( mem @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( arr @ ( ty_2Elist_2Elist @ ty_2Equote_2Eindex ) @ ( arr @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) ) ) ) ).
thf(tp_c_2Ecanonical_2ENil__monom,type,
c_2Ecanonical_2ENil__monom: del > $i ).
thf(mem_c_2Ecanonical_2ENil__monom,axiom,
! [A_27a: del] : ( mem @ ( c_2Ecanonical_2ENil__monom @ A_27a ) @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) ) ).
thf(tp_c_2Ecanonical_2ESPconst,type,
c_2Ecanonical_2ESPconst: del > $i ).
thf(mem_c_2Ecanonical_2ESPconst,axiom,
! [A_27a: del] : ( mem @ ( c_2Ecanonical_2ESPconst @ A_27a ) @ ( arr @ A_27a @ ( ty_2Ecanonical_2Espolynom @ A_27a ) ) ) ).
thf(tp_c_2Ecanonical_2ESPmult,type,
c_2Ecanonical_2ESPmult: del > $i ).
thf(mem_c_2Ecanonical_2ESPmult,axiom,
! [A_27a: del] : ( mem @ ( c_2Ecanonical_2ESPmult @ A_27a ) @ ( arr @ ( ty_2Ecanonical_2Espolynom @ A_27a ) @ ( arr @ ( ty_2Ecanonical_2Espolynom @ A_27a ) @ ( ty_2Ecanonical_2Espolynom @ A_27a ) ) ) ) ).
thf(tp_c_2Ecanonical_2ESPplus,type,
c_2Ecanonical_2ESPplus: del > $i ).
thf(mem_c_2Ecanonical_2ESPplus,axiom,
! [A_27a: del] : ( mem @ ( c_2Ecanonical_2ESPplus @ A_27a ) @ ( arr @ ( ty_2Ecanonical_2Espolynom @ A_27a ) @ ( arr @ ( ty_2Ecanonical_2Espolynom @ A_27a ) @ ( ty_2Ecanonical_2Espolynom @ A_27a ) ) ) ) ).
thf(tp_c_2Ecanonical_2ESPvar,type,
c_2Ecanonical_2ESPvar: del > $i ).
thf(mem_c_2Ecanonical_2ESPvar,axiom,
! [A_27a: del] : ( mem @ ( c_2Ecanonical_2ESPvar @ A_27a ) @ ( arr @ ty_2Equote_2Eindex @ ( ty_2Ecanonical_2Espolynom @ A_27a ) ) ) ).
thf(tp_c_2Ecanonical_2Ecanonical__sum__CASE,type,
c_2Ecanonical_2Ecanonical__sum__CASE: del > del > $i ).
thf(mem_c_2Ecanonical_2Ecanonical__sum__CASE,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Ecanonical_2Ecanonical__sum__CASE @ A_27a @ A_27b ) @ ( arr @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) @ ( arr @ A_27b @ ( arr @ ( arr @ A_27a @ ( arr @ ( ty_2Elist_2Elist @ ty_2Equote_2Eindex ) @ ( arr @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) @ A_27b ) ) ) @ ( arr @ ( arr @ ( ty_2Elist_2Elist @ ty_2Equote_2Eindex ) @ ( arr @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) @ A_27b ) ) @ A_27b ) ) ) ) ) ).
thf(tp_c_2Ecanonical_2Ecanonical__sum__merge,type,
c_2Ecanonical_2Ecanonical__sum__merge: del > $i ).
thf(mem_c_2Ecanonical_2Ecanonical__sum__merge,axiom,
! [A_27a: del] : ( mem @ ( c_2Ecanonical_2Ecanonical__sum__merge @ A_27a ) @ ( arr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( arr @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) @ ( arr @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) ) ) ) ) ).
thf(tp_c_2Ecanonical_2Ecanonical__sum__prod,type,
c_2Ecanonical_2Ecanonical__sum__prod: del > $i ).
thf(mem_c_2Ecanonical_2Ecanonical__sum__prod,axiom,
! [A_27a: del] : ( mem @ ( c_2Ecanonical_2Ecanonical__sum__prod @ A_27a ) @ ( arr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( arr @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) @ ( arr @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) ) ) ) ) ).
thf(tp_c_2Ecanonical_2Ecanonical__sum__scalar,type,
c_2Ecanonical_2Ecanonical__sum__scalar: del > $i ).
thf(mem_c_2Ecanonical_2Ecanonical__sum__scalar,axiom,
! [A_27a: del] : ( mem @ ( c_2Ecanonical_2Ecanonical__sum__scalar @ A_27a ) @ ( arr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( arr @ A_27a @ ( arr @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) ) ) ) ) ).
thf(tp_c_2Ecanonical_2Ecanonical__sum__scalar2,type,
c_2Ecanonical_2Ecanonical__sum__scalar2: del > $i ).
thf(mem_c_2Ecanonical_2Ecanonical__sum__scalar2,axiom,
! [A_27a: del] : ( mem @ ( c_2Ecanonical_2Ecanonical__sum__scalar2 @ A_27a ) @ ( arr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( arr @ ( ty_2Elist_2Elist @ ty_2Equote_2Eindex ) @ ( arr @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) ) ) ) ) ).
thf(tp_c_2Ecanonical_2Ecanonical__sum__scalar3,type,
c_2Ecanonical_2Ecanonical__sum__scalar3: del > $i ).
thf(mem_c_2Ecanonical_2Ecanonical__sum__scalar3,axiom,
! [A_27a: del] : ( mem @ ( c_2Ecanonical_2Ecanonical__sum__scalar3 @ A_27a ) @ ( arr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( arr @ A_27a @ ( arr @ ( ty_2Elist_2Elist @ ty_2Equote_2Eindex ) @ ( arr @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) ) ) ) ) ) ).
thf(tp_c_2Ecanonical_2Ecanonical__sum__simplify,type,
c_2Ecanonical_2Ecanonical__sum__simplify: del > $i ).
thf(mem_c_2Ecanonical_2Ecanonical__sum__simplify,axiom,
! [A_27a: del] : ( mem @ ( c_2Ecanonical_2Ecanonical__sum__simplify @ A_27a ) @ ( arr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( arr @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) ) ) ) ).
thf(tp_c_2Ecanonical_2Ecanonical__sum__size,type,
c_2Ecanonical_2Ecanonical__sum__size: del > $i ).
thf(mem_c_2Ecanonical_2Ecanonical__sum__size,axiom,
! [A_27a: del] : ( mem @ ( c_2Ecanonical_2Ecanonical__sum__size @ A_27a ) @ ( arr @ ( arr @ A_27a @ ty_2Enum_2Enum ) @ ( arr @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) @ ty_2Enum_2Enum ) ) ) ).
thf(tp_c_2Ecanonical_2Eics__aux,type,
c_2Ecanonical_2Eics__aux: del > $i ).
thf(mem_c_2Ecanonical_2Eics__aux,axiom,
! [A_27a: del] : ( mem @ ( c_2Ecanonical_2Eics__aux @ A_27a ) @ ( arr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( arr @ ( ty_2Equote_2Evarmap @ A_27a ) @ ( arr @ A_27a @ ( arr @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) @ A_27a ) ) ) ) ) ).
thf(tp_c_2Ecanonical_2Einterp__cs,type,
c_2Ecanonical_2Einterp__cs: del > $i ).
thf(mem_c_2Ecanonical_2Einterp__cs,axiom,
! [A_27a: del] : ( mem @ ( c_2Ecanonical_2Einterp__cs @ A_27a ) @ ( arr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( arr @ ( ty_2Equote_2Evarmap @ A_27a ) @ ( arr @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) @ A_27a ) ) ) ) ).
thf(tp_c_2Ecanonical_2Einterp__m,type,
c_2Ecanonical_2Einterp__m: del > $i ).
thf(mem_c_2Ecanonical_2Einterp__m,axiom,
! [A_27a: del] : ( mem @ ( c_2Ecanonical_2Einterp__m @ A_27a ) @ ( arr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( arr @ ( ty_2Equote_2Evarmap @ A_27a ) @ ( arr @ A_27a @ ( arr @ ( ty_2Elist_2Elist @ ty_2Equote_2Eindex ) @ A_27a ) ) ) ) ) ).
thf(tp_c_2Ecanonical_2Einterp__sp,type,
c_2Ecanonical_2Einterp__sp: del > $i ).
thf(mem_c_2Ecanonical_2Einterp__sp,axiom,
! [A_27a: del] : ( mem @ ( c_2Ecanonical_2Einterp__sp @ A_27a ) @ ( arr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( arr @ ( ty_2Equote_2Evarmap @ A_27a ) @ ( arr @ ( ty_2Ecanonical_2Espolynom @ A_27a ) @ A_27a ) ) ) ) ).
thf(tp_c_2Ecanonical_2Einterp__vl,type,
c_2Ecanonical_2Einterp__vl: del > $i ).
thf(mem_c_2Ecanonical_2Einterp__vl,axiom,
! [A_27a: del] : ( mem @ ( c_2Ecanonical_2Einterp__vl @ A_27a ) @ ( arr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( arr @ ( ty_2Equote_2Evarmap @ A_27a ) @ ( arr @ ( ty_2Elist_2Elist @ ty_2Equote_2Eindex ) @ A_27a ) ) ) ) ).
thf(tp_c_2Ecanonical_2Eivl__aux,type,
c_2Ecanonical_2Eivl__aux: del > $i ).
thf(mem_c_2Ecanonical_2Eivl__aux,axiom,
! [A_27a: del] : ( mem @ ( c_2Ecanonical_2Eivl__aux @ A_27a ) @ ( arr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( arr @ ( ty_2Equote_2Evarmap @ A_27a ) @ ( arr @ ty_2Equote_2Eindex @ ( arr @ ( ty_2Elist_2Elist @ ty_2Equote_2Eindex ) @ A_27a ) ) ) ) ) ).
thf(tp_c_2Ecanonical_2Emonom__insert,type,
c_2Ecanonical_2Emonom__insert: del > $i ).
thf(mem_c_2Ecanonical_2Emonom__insert,axiom,
! [A_27a: del] : ( mem @ ( c_2Ecanonical_2Emonom__insert @ A_27a ) @ ( arr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( arr @ A_27a @ ( arr @ ( ty_2Elist_2Elist @ ty_2Equote_2Eindex ) @ ( arr @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) ) ) ) ) ) ).
thf(tp_c_2Ecanonical_2Espolynom__CASE,type,
c_2Ecanonical_2Espolynom__CASE: del > del > $i ).
thf(mem_c_2Ecanonical_2Espolynom__CASE,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Ecanonical_2Espolynom__CASE @ A_27a @ A_27b ) @ ( arr @ ( ty_2Ecanonical_2Espolynom @ A_27a ) @ ( arr @ ( arr @ ty_2Equote_2Eindex @ A_27b ) @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ ( arr @ ( ty_2Ecanonical_2Espolynom @ A_27a ) @ ( arr @ ( ty_2Ecanonical_2Espolynom @ A_27a ) @ A_27b ) ) @ ( arr @ ( arr @ ( ty_2Ecanonical_2Espolynom @ A_27a ) @ ( arr @ ( ty_2Ecanonical_2Espolynom @ A_27a ) @ A_27b ) ) @ A_27b ) ) ) ) ) ) ).
thf(tp_c_2Ecanonical_2Espolynom__normalize,type,
c_2Ecanonical_2Espolynom__normalize: del > $i ).
thf(mem_c_2Ecanonical_2Espolynom__normalize,axiom,
! [A_27a: del] : ( mem @ ( c_2Ecanonical_2Espolynom__normalize @ A_27a ) @ ( arr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( arr @ ( ty_2Ecanonical_2Espolynom @ A_27a ) @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) ) ) ) ).
thf(tp_c_2Ecanonical_2Espolynom__simplify,type,
c_2Ecanonical_2Espolynom__simplify: del > $i ).
thf(mem_c_2Ecanonical_2Espolynom__simplify,axiom,
! [A_27a: del] : ( mem @ ( c_2Ecanonical_2Espolynom__simplify @ A_27a ) @ ( arr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( arr @ ( ty_2Ecanonical_2Espolynom @ A_27a ) @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) ) ) ) ).
thf(tp_c_2Ecanonical_2Espolynom__size,type,
c_2Ecanonical_2Espolynom__size: del > $i ).
thf(mem_c_2Ecanonical_2Espolynom__size,axiom,
! [A_27a: del] : ( mem @ ( c_2Ecanonical_2Espolynom__size @ A_27a ) @ ( arr @ ( arr @ A_27a @ ty_2Enum_2Enum ) @ ( arr @ ( ty_2Ecanonical_2Espolynom @ A_27a ) @ ty_2Enum_2Enum ) ) ) ).
thf(tp_c_2Ecanonical_2Evarlist__insert,type,
c_2Ecanonical_2Evarlist__insert: del > $i ).
thf(mem_c_2Ecanonical_2Evarlist__insert,axiom,
! [A_27a: del] : ( mem @ ( c_2Ecanonical_2Evarlist__insert @ A_27a ) @ ( arr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( arr @ ( ty_2Elist_2Elist @ ty_2Equote_2Eindex ) @ ( arr @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) ) ) ) ) ).
thf(ax_thm_2Ecanonical_2Ecanonical__sum__TY__DEF,axiom,
! [A_27a: del] :
? [V0rep: $i] :
( ( mem @ V0rep @ ( arr @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) @ ( ty_2Eind__type_2Erecspace @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Elist_2Elist @ ty_2Equote_2Eindex ) ) ) ) )
& ( p
@ ( ap
@ ( ap @ ( c_2Ebool_2ETYPE__DEFINITION @ ( ty_2Eind__type_2Erecspace @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Elist_2Elist @ ty_2Equote_2Eindex ) ) ) @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
@ ( lam @ ( ty_2Eind__type_2Erecspace @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Elist_2Elist @ ty_2Equote_2Eindex ) ) )
@ ^ [V1a0_27: $i] :
( ap @ ( c_2Ebool_2E_21 @ ( arr @ ( ty_2Eind__type_2Erecspace @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Elist_2Elist @ ty_2Equote_2Eindex ) ) ) @ bool ) )
@ ( lam @ ( arr @ ( ty_2Eind__type_2Erecspace @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Elist_2Elist @ ty_2Equote_2Eindex ) ) ) @ bool )
@ ^ [V2_27canonical__sum_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_2Elist_2Elist @ ty_2Equote_2Eindex ) ) ) )
@ ( lam @ ( ty_2Eind__type_2Erecspace @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Elist_2Elist @ ty_2Equote_2Eindex ) ) )
@ ^ [V3a0_27: $i] :
( ap
@ ( ap @ c_2Emin_2E_3D_3D_3E
@ ( ap
@ ( ap @ c_2Ebool_2E_5C_2F
@ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Eind__type_2Erecspace @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Elist_2Elist @ ty_2Equote_2Eindex ) ) ) ) @ V3a0_27 )
@ ( ap @ ( ap @ ( ap @ ( c_2Eind__type_2ECONSTR @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Elist_2Elist @ ty_2Equote_2Eindex ) ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ ( ty_2Elist_2Elist @ ty_2Equote_2Eindex ) ) @ ( c_2Ebool_2EARB @ A_27a ) ) @ ( c_2Ebool_2EARB @ ( ty_2Elist_2Elist @ ty_2Equote_2Eindex ) ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V4n: $i] : ( c_2Eind__type_2EBOTTOM @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Elist_2Elist @ ty_2Equote_2Eindex ) ) ) ) ) ) )
@ ( ap
@ ( ap @ c_2Ebool_2E_5C_2F
@ ( ap @ ( c_2Ebool_2E_3F @ A_27a )
@ ( lam @ A_27a
@ ^ [V5a0: $i] :
( ap @ ( c_2Ebool_2E_3F @ ( ty_2Elist_2Elist @ ty_2Equote_2Eindex ) )
@ ( lam @ ( ty_2Elist_2Elist @ ty_2Equote_2Eindex )
@ ^ [V6a1: $i] :
( ap @ ( c_2Ebool_2E_3F @ ( ty_2Eind__type_2Erecspace @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Elist_2Elist @ ty_2Equote_2Eindex ) ) ) )
@ ( lam @ ( ty_2Eind__type_2Erecspace @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Elist_2Elist @ ty_2Equote_2Eindex ) ) )
@ ^ [V7a2: $i] :
( ap
@ ( ap @ c_2Ebool_2E_2F_5C
@ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Eind__type_2Erecspace @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Elist_2Elist @ ty_2Equote_2Eindex ) ) ) ) @ V3a0_27 )
@ ( ap
@ ( ap
@ ( ap
@ ( lam @ A_27a
@ ^ [V8a0: $i] :
( lam @ ( ty_2Elist_2Elist @ ty_2Equote_2Eindex )
@ ^ [V9a1: $i] :
( lam @ ( ty_2Eind__type_2Erecspace @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Elist_2Elist @ ty_2Equote_2Eindex ) ) )
@ ^ [V10a2: $i] :
( ap @ ( ap @ ( ap @ ( c_2Eind__type_2ECONSTR @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Elist_2Elist @ ty_2Equote_2Eindex ) ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ ( ty_2Elist_2Elist @ ty_2Equote_2Eindex ) ) @ V8a0 ) @ V9a1 ) )
@ ( ap @ ( ap @ ( c_2Eind__type_2EFCONS @ ( ty_2Eind__type_2Erecspace @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Elist_2Elist @ ty_2Equote_2Eindex ) ) ) ) @ V10a2 )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V11n: $i] : ( c_2Eind__type_2EBOTTOM @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Elist_2Elist @ ty_2Equote_2Eindex ) ) ) ) ) ) ) ) )
@ V5a0 )
@ V6a1 )
@ V7a2 ) ) )
@ ( ap @ V2_27canonical__sum_27 @ V7a2 ) ) ) ) ) ) ) ) )
@ ( ap @ ( c_2Ebool_2E_3F @ ( ty_2Elist_2Elist @ ty_2Equote_2Eindex ) )
@ ( lam @ ( ty_2Elist_2Elist @ ty_2Equote_2Eindex )
@ ^ [V12a0: $i] :
( ap @ ( c_2Ebool_2E_3F @ ( ty_2Eind__type_2Erecspace @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Elist_2Elist @ ty_2Equote_2Eindex ) ) ) )
@ ( lam @ ( ty_2Eind__type_2Erecspace @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Elist_2Elist @ ty_2Equote_2Eindex ) ) )
@ ^ [V13a1: $i] :
( ap
@ ( ap @ c_2Ebool_2E_2F_5C
@ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Eind__type_2Erecspace @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Elist_2Elist @ ty_2Equote_2Eindex ) ) ) ) @ V3a0_27 )
@ ( ap
@ ( ap
@ ( lam @ ( ty_2Elist_2Elist @ ty_2Equote_2Eindex )
@ ^ [V14a0: $i] :
( lam @ ( ty_2Eind__type_2Erecspace @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Elist_2Elist @ ty_2Equote_2Eindex ) ) )
@ ^ [V15a1: $i] :
( ap @ ( ap @ ( ap @ ( c_2Eind__type_2ECONSTR @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Elist_2Elist @ ty_2Equote_2Eindex ) ) ) @ ( ap @ c_2Enum_2ESUC @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ ( ty_2Elist_2Elist @ ty_2Equote_2Eindex ) ) @ ( c_2Ebool_2EARB @ A_27a ) ) @ V14a0 ) )
@ ( ap @ ( ap @ ( c_2Eind__type_2EFCONS @ ( ty_2Eind__type_2Erecspace @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Elist_2Elist @ ty_2Equote_2Eindex ) ) ) ) @ V15a1 )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V16n: $i] : ( c_2Eind__type_2EBOTTOM @ ( ty_2Epair_2Eprod @ A_27a @ ( ty_2Elist_2Elist @ ty_2Equote_2Eindex ) ) ) ) ) ) ) )
@ V12a0 )
@ V13a1 ) ) )
@ ( ap @ V2_27canonical__sum_27 @ V13a1 ) ) ) ) ) ) ) ) )
@ ( ap @ V2_27canonical__sum_27 @ V3a0_27 ) ) ) ) )
@ ( ap @ V2_27canonical__sum_27 @ V1a0_27 ) ) ) ) ) )
@ V0rep ) ) ) ).
thf(ax_thm_2Ecanonical_2Ecanonical__sum__case__def,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0v: $i] :
( ( mem @ V0v @ A_27b )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ( arr @ ( ty_2Elist_2Elist @ ty_2Equote_2Eindex ) @ ( arr @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) @ A_27b ) ) ) )
=> ! [V2f1: $i] :
( ( mem @ V2f1 @ ( arr @ ( ty_2Elist_2Elist @ ty_2Equote_2Eindex ) @ ( arr @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) @ A_27b ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Ecanonical__sum__CASE @ A_27a @ A_27b ) @ ( c_2Ecanonical_2ENil__monom @ A_27a ) ) @ V0v ) @ V1f ) @ V2f1 )
= V0v ) ) ) )
& ! [V3a0: $i] :
( ( mem @ V3a0 @ A_27a )
=> ! [V4a1: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V5a2: $i] :
( ( mem @ V5a2 @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ! [V6v: $i] :
( ( mem @ V6v @ A_27b )
=> ! [V7f: $i] :
( ( mem @ V7f @ ( arr @ A_27a @ ( arr @ ( ty_2Elist_2Elist @ ty_2Equote_2Eindex ) @ ( arr @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) @ A_27b ) ) ) )
=> ! [V8f1: $i] :
( ( mem @ V8f1 @ ( arr @ ( ty_2Elist_2Elist @ ty_2Equote_2Eindex ) @ ( arr @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) @ A_27b ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Ecanonical__sum__CASE @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ V3a0 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V4a1 ) ) @ V5a2 ) ) @ V6v ) @ V7f ) @ V8f1 )
= ( ap @ ( ap @ ( ap @ V7f @ V3a0 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V4a1 ) ) @ V5a2 ) ) ) ) ) ) )
& ! [V9a0: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V10a1: $i] :
( ( mem @ V10a1 @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ! [V11v: $i] :
( ( mem @ V11v @ A_27b )
=> ! [V12f: $i] :
( ( mem @ V12f @ ( arr @ A_27a @ ( arr @ ( ty_2Elist_2Elist @ ty_2Equote_2Eindex ) @ ( arr @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) @ A_27b ) ) ) )
=> ! [V13f1: $i] :
( ( mem @ V13f1 @ ( arr @ ( ty_2Elist_2Elist @ ty_2Equote_2Eindex ) @ ( arr @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) @ A_27b ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Ecanonical__sum__CASE @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V9a0 ) ) @ V10a1 ) ) @ V11v ) @ V12f ) @ V13f1 )
= ( ap @ ( ap @ V13f1 @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V9a0 ) ) @ V10a1 ) ) ) ) ) ) ) ).
thf(ax_thm_2Ecanonical_2Ecanonical__sum__size__def,axiom,
! [A_27a: del] :
( ! [V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Ecanonical_2Ecanonical__sum__size @ A_27a ) @ V0f ) @ ( c_2Ecanonical_2ENil__monom @ A_27a ) ) )
= fo__c_2Enum_2E0 ) )
& ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V2a0: $i] :
( ( mem @ V2a0 @ A_27a )
=> ! [V3a1: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V4a2: $i] :
( ( mem @ V4a2 @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Ecanonical_2Ecanonical__sum__size @ A_27a ) @ V1f ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ V2a0 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V3a1 ) ) @ V4a2 ) ) )
= ( 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 @ V1f @ V2a0 ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( ap @ ( c_2Elist_2Elist__size @ ty_2Equote_2Eindex ) @ c_2Equote_2Eindex__size ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V3a1 ) ) ) @ ( ap @ ( ap @ ( c_2Ecanonical_2Ecanonical__sum__size @ A_27a ) @ V1f ) @ V4a2 ) ) ) ) ) ) ) ) )
& ! [V5f: $i] :
( ( mem @ V5f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V6a0: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V7a1: $i] :
( ( mem @ V7a1 @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Ecanonical_2Ecanonical__sum__size @ A_27a ) @ V5f ) @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V6a0 ) ) @ V7a1 ) ) )
= ( 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 @ ( ap @ ( c_2Elist_2Elist__size @ ty_2Equote_2Eindex ) @ c_2Equote_2Eindex__size ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V6a0 ) ) ) @ ( ap @ ( ap @ ( c_2Ecanonical_2Ecanonical__sum__size @ A_27a ) @ V5f ) @ V7a1 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecanonical_2Edatatype__canonical__sum,axiom,
! [A_27a: del,V0canonical__sum: $i] :
( ( mem @ V0canonical__sum @ ( arr @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ ( ty_2Elist_2Elist @ ty_2Equote_2Eindex ) @ ( arr @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) ) ) ) @ ( arr @ ( arr @ ( ty_2Elist_2Elist @ ty_2Equote_2Eindex ) @ ( arr @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) ) ) @ bool ) ) ) )
=> ( p @ ( ap @ ( c_2Ebool_2EDATATYPE @ bool ) @ ( ap @ ( ap @ ( ap @ V0canonical__sum @ ( c_2Ecanonical_2ENil__monom @ A_27a ) ) @ ( c_2Ecanonical_2ECons__monom @ A_27a ) ) @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) ) ) ) ) ).
thf(conj_thm_2Ecanonical_2Ecanonical__sum__11,axiom,
! [A_27a: del] :
( ! [V0a0: $i] :
( ( mem @ V0a0 @ A_27a )
=> ! [V1a1: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V2a2: $i] :
( ( mem @ V2a2 @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ! [V3a0_27: $i] :
( ( mem @ V3a0_27 @ A_27a )
=> ! [V4a1_27: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V5a2_27: $i] :
( ( mem @ V5a2_27 @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ( ( ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ V0a0 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V1a1 ) ) @ V2a2 )
= ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ V3a0_27 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V4a1_27 ) ) @ V5a2_27 ) )
<=> ( ( V0a0 = V3a0_27 )
& ( V1a1 = V4a1_27 )
& ( V2a2 = V5a2_27 ) ) ) ) ) ) )
& ! [V6a0: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V7a1: $i] :
( ( mem @ V7a1 @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ! [V8a0_27: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V9a1_27: $i] :
( ( mem @ V9a1_27 @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ( ( ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V6a0 ) ) @ V7a1 )
= ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V8a0_27 ) ) @ V9a1_27 ) )
<=> ( ( V6a0 = V8a0_27 )
& ( V7a1 = V9a1_27 ) ) ) ) ) ) ).
thf(conj_thm_2Ecanonical_2Ecanonical__sum__distinct,axiom,
! [A_27a: del] :
( ! [V0a2: $i] :
( ( mem @ V0a2 @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ! [V1a1: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V2a0: $i] :
( ( mem @ V2a0 @ A_27a )
=> ( ( c_2Ecanonical_2ENil__monom @ A_27a )
!= ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ V2a0 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V1a1 ) ) @ V0a2 ) ) ) )
& ! [V3a1: $i] :
( ( mem @ V3a1 @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ! [V4a0: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex] :
( ( c_2Ecanonical_2ENil__monom @ A_27a )
!= ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V4a0 ) ) @ V3a1 ) ) )
& ! [V5a2: $i] :
( ( mem @ V5a2 @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ! [V6a1_27: $i] :
( ( mem @ V6a1_27 @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ! [V7a1: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V8a0_27: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V9a0: $i] :
( ( mem @ V9a0 @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ V9a0 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V7a1 ) ) @ V5a2 )
!= ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V8a0_27 ) ) @ V6a1_27 ) ) ) ) ) ) ).
thf(conj_thm_2Ecanonical_2Ecanonical__sum__nchotomy,axiom,
! [A_27a: del,V0cc: $i] :
( ( mem @ V0cc @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ( ( V0cc
= ( c_2Ecanonical_2ENil__monom @ A_27a ) )
| ? [V1a: $i] :
( ( mem @ V1a @ A_27a )
& ? [V2l: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V3c: $i] :
( ( mem @ V3c @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
& ( V0cc
= ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ V1a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V2l ) ) @ V3c ) ) ) )
| ? [V4l: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V5c: $i] :
( ( mem @ V5c @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
& ( V0cc
= ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V4l ) ) @ V5c ) ) ) ) ) ).
thf(conj_thm_2Ecanonical_2Ecanonical__sum__Axiom,axiom,
! [A_27a: del,A_27b: del,V0f0: $i] :
( ( mem @ V0f0 @ A_27b )
=> ! [V1f1: $i] :
( ( mem @ V1f1 @ ( arr @ A_27a @ ( arr @ ( ty_2Elist_2Elist @ ty_2Equote_2Eindex ) @ ( arr @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) @ ( arr @ A_27b @ A_27b ) ) ) ) )
=> ! [V2f2: $i] :
( ( mem @ V2f2 @ ( arr @ ( ty_2Elist_2Elist @ ty_2Equote_2Eindex ) @ ( arr @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) @ ( arr @ A_27b @ A_27b ) ) ) )
=> ? [V3fn: $i] :
( ( mem @ V3fn @ ( arr @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) @ A_27b ) )
& ( ( ap @ V3fn @ ( c_2Ecanonical_2ENil__monom @ A_27a ) )
= V0f0 )
& ! [V4a0: $i] :
( ( mem @ V4a0 @ A_27a )
=> ! [V5a1: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V6a2: $i] :
( ( mem @ V6a2 @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ( ( ap @ V3fn @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ V4a0 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V5a1 ) ) @ V6a2 ) )
= ( ap @ ( ap @ ( ap @ ( ap @ V1f1 @ V4a0 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V5a1 ) ) @ V6a2 ) @ ( ap @ V3fn @ V6a2 ) ) ) ) )
& ! [V7a0: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V8a1: $i] :
( ( mem @ V8a1 @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ( ( ap @ V3fn @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V7a0 ) ) @ V8a1 ) )
= ( ap @ ( ap @ ( ap @ V2f2 @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V7a0 ) ) @ V8a1 ) @ ( ap @ V3fn @ V8a1 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecanonical_2Ecanonical__sum__induction,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) @ bool ) )
=> ( ( ( p @ ( ap @ V0P @ ( c_2Ecanonical_2ENil__monom @ A_27a ) ) )
& ! [V1c: $i] :
( ( mem @ V1c @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ( ( p @ ( ap @ V0P @ V1c ) )
=> ! [V2l: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V3a: $i] :
( ( mem @ V3a @ A_27a )
=> ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ V3a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V2l ) ) @ V1c ) ) ) ) ) )
& ! [V4c: $i] :
( ( mem @ V4c @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ( ( p @ ( ap @ V0P @ V4c ) )
=> ! [V5l: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex] : ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V5l ) ) @ V4c ) ) ) ) ) )
=> ! [V6c: $i] :
( ( mem @ V6c @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ( p @ ( ap @ V0P @ V6c ) ) ) ) ) ).
thf(conj_thm_2Ecanonical_2Ecanonical__sum__case__cong,axiom,
! [A_27a: del,A_27b: del,V0v_27: $i] :
( ( mem @ V0v_27 @ A_27b )
=> ! [V1f_27: $i] :
( ( mem @ V1f_27 @ ( arr @ A_27a @ ( arr @ ( ty_2Elist_2Elist @ ty_2Equote_2Eindex ) @ ( arr @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) @ A_27b ) ) ) )
=> ! [V2f1_27: $i] :
( ( mem @ V2f1_27 @ ( arr @ ( ty_2Elist_2Elist @ ty_2Equote_2Eindex ) @ ( arr @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) @ A_27b ) ) )
=> ! [V3M: $i] :
( ( mem @ V3M @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ! [V4M_27: $i] :
( ( mem @ V4M_27 @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ! [V5v: $i] :
( ( mem @ V5v @ A_27b )
=> ! [V6f: $i] :
( ( mem @ V6f @ ( arr @ A_27a @ ( arr @ ( ty_2Elist_2Elist @ ty_2Equote_2Eindex ) @ ( arr @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) @ A_27b ) ) ) )
=> ! [V7f1: $i] :
( ( mem @ V7f1 @ ( arr @ ( ty_2Elist_2Elist @ ty_2Equote_2Eindex ) @ ( arr @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) @ A_27b ) ) )
=> ( ( ( V3M = V4M_27 )
& ( ( V4M_27
= ( c_2Ecanonical_2ENil__monom @ A_27a ) )
=> ( V5v = V0v_27 ) )
& ! [V8a0: $i] :
( ( mem @ V8a0 @ A_27a )
=> ! [V9a1: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V10a2: $i] :
( ( mem @ V10a2 @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ( ( V4M_27
= ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ V8a0 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V9a1 ) ) @ V10a2 ) )
=> ( ( ap @ ( ap @ ( ap @ V6f @ V8a0 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V9a1 ) ) @ V10a2 )
= ( ap @ ( ap @ ( ap @ V1f_27 @ V8a0 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V9a1 ) ) @ V10a2 ) ) ) ) )
& ! [V11a0: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V12a1: $i] :
( ( mem @ V12a1 @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ( ( V4M_27
= ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V11a0 ) ) @ V12a1 ) )
=> ( ( ap @ ( ap @ V7f1 @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V11a0 ) ) @ V12a1 )
= ( ap @ ( ap @ V2f1_27 @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V11a0 ) ) @ V12a1 ) ) ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Ecanonical__sum__CASE @ A_27a @ A_27b ) @ V3M ) @ V5v ) @ V6f ) @ V7f1 )
= ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Ecanonical__sum__CASE @ A_27a @ A_27b ) @ V4M_27 ) @ V0v_27 ) @ V1f_27 ) @ V2f1_27 ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecanonical_2Ecanonical__sum__case__eq,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ! [V1v: $i] :
( ( mem @ V1v @ A_27b )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ ( arr @ ( ty_2Elist_2Elist @ ty_2Equote_2Eindex ) @ ( arr @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) @ A_27b ) ) ) )
=> ! [V3f1: $i] :
( ( mem @ V3f1 @ ( arr @ ( ty_2Elist_2Elist @ ty_2Equote_2Eindex ) @ ( arr @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) @ A_27b ) ) )
=> ! [V4v_27: $i] :
( ( mem @ V4v_27 @ A_27b )
=> ( ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Ecanonical__sum__CASE @ A_27a @ A_27b ) @ V0x ) @ V1v ) @ V2f ) @ V3f1 )
= V4v_27 )
<=> ( ( ( V0x
= ( c_2Ecanonical_2ENil__monom @ A_27a ) )
& ( V1v = V4v_27 ) )
| ? [V5a: $i] :
( ( mem @ V5a @ A_27a )
& ? [V6l: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V7c: $i] :
( ( mem @ V7c @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
& ( V0x
= ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ V5a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V6l ) ) @ V7c ) )
& ( ( ap @ ( ap @ ( ap @ V2f @ V5a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V6l ) ) @ V7c )
= V4v_27 ) ) )
| ? [V8l: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V9c: $i] :
( ( mem @ V9c @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
& ( V0x
= ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V8l ) ) @ V9c ) )
& ( ( ap @ ( ap @ V3f1 @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V8l ) ) @ V9c )
= V4v_27 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecanonical_2Ecanonical__sum__merge__ind,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( arr @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) @ ( arr @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) @ bool ) ) ) )
=> ( ( ! [V1sr: $i] :
( ( mem @ V1sr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V2c1: $i] :
( ( mem @ V2c1 @ A_27a )
=> ! [V3l1: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V4t1: $i] :
( ( mem @ V4t1 @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ! [V5c2: $i] :
( ( mem @ V5c2 @ A_27a )
=> ! [V6l2: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V7t2: $i] :
( ( mem @ V7t2 @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ( ( ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2EternaryComparisons_2Elist__compare @ ty_2Equote_2Eindex @ ty_2Equote_2Eindex ) @ c_2Equote_2Eindex__compare ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V3l1 ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V6l2 ) ) )
= fo__c_2EternaryComparisons_2EEQUAL )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ V1sr ) @ V4t1 ) @ V7t2 ) ) )
& ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2EternaryComparisons_2Elist__compare @ ty_2Equote_2Eindex @ ty_2Equote_2Eindex ) @ c_2Equote_2Eindex__compare ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V3l1 ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V6l2 ) ) )
= fo__c_2EternaryComparisons_2EGREATER )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ V1sr ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ V2c1 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V3l1 ) ) @ V4t1 ) ) @ V7t2 ) ) )
& ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2EternaryComparisons_2Elist__compare @ ty_2Equote_2Eindex @ ty_2Equote_2Eindex ) @ c_2Equote_2Eindex__compare ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V3l1 ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V6l2 ) ) )
= fo__c_2EternaryComparisons_2ELESS )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ V1sr ) @ V4t1 ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ V5c2 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V6l2 ) ) @ V7t2 ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ V1sr ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ V2c1 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V3l1 ) ) @ V4t1 ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ V5c2 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V6l2 ) ) @ V7t2 ) ) ) ) ) ) ) ) )
& ! [V8sr: $i] :
( ( mem @ V8sr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V9c1: $i] :
( ( mem @ V9c1 @ A_27a )
=> ! [V10l1: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V11t1: $i] :
( ( mem @ V11t1 @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ! [V12l2: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V13t2: $i] :
( ( mem @ V13t2 @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ( ( ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2EternaryComparisons_2Elist__compare @ ty_2Equote_2Eindex @ ty_2Equote_2Eindex ) @ c_2Equote_2Eindex__compare ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V10l1 ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V12l2 ) ) )
= fo__c_2EternaryComparisons_2EEQUAL )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ V8sr ) @ V11t1 ) @ V13t2 ) ) )
& ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2EternaryComparisons_2Elist__compare @ ty_2Equote_2Eindex @ ty_2Equote_2Eindex ) @ c_2Equote_2Eindex__compare ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V10l1 ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V12l2 ) ) )
= fo__c_2EternaryComparisons_2EGREATER )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ V8sr ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ V9c1 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V10l1 ) ) @ V11t1 ) ) @ V13t2 ) ) )
& ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2EternaryComparisons_2Elist__compare @ ty_2Equote_2Eindex @ ty_2Equote_2Eindex ) @ c_2Equote_2Eindex__compare ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V10l1 ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V12l2 ) ) )
= fo__c_2EternaryComparisons_2ELESS )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ V8sr ) @ V11t1 ) @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V12l2 ) ) @ V13t2 ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ V8sr ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ V9c1 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V10l1 ) ) @ V11t1 ) ) @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V12l2 ) ) @ V13t2 ) ) ) ) ) ) ) )
& ! [V14sr: $i] :
( ( mem @ V14sr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V15l1: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V16t1: $i] :
( ( mem @ V16t1 @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ! [V17c2: $i] :
( ( mem @ V17c2 @ A_27a )
=> ! [V18l2: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V19t2: $i] :
( ( mem @ V19t2 @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ( ( ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2EternaryComparisons_2Elist__compare @ ty_2Equote_2Eindex @ ty_2Equote_2Eindex ) @ c_2Equote_2Eindex__compare ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V15l1 ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V18l2 ) ) )
= fo__c_2EternaryComparisons_2EEQUAL )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ V14sr ) @ V16t1 ) @ V19t2 ) ) )
& ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2EternaryComparisons_2Elist__compare @ ty_2Equote_2Eindex @ ty_2Equote_2Eindex ) @ c_2Equote_2Eindex__compare ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V15l1 ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V18l2 ) ) )
= fo__c_2EternaryComparisons_2EGREATER )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ V14sr ) @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V15l1 ) ) @ V16t1 ) ) @ V19t2 ) ) )
& ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2EternaryComparisons_2Elist__compare @ ty_2Equote_2Eindex @ ty_2Equote_2Eindex ) @ c_2Equote_2Eindex__compare ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V15l1 ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V18l2 ) ) )
= fo__c_2EternaryComparisons_2ELESS )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ V14sr ) @ V16t1 ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ V17c2 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V18l2 ) ) @ V19t2 ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ V14sr ) @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V15l1 ) ) @ V16t1 ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ V17c2 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V18l2 ) ) @ V19t2 ) ) ) ) ) ) ) )
& ! [V20sr: $i] :
( ( mem @ V20sr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V21l1: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V22t1: $i] :
( ( mem @ V22t1 @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ! [V23l2: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V24t2: $i] :
( ( mem @ V24t2 @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ( ( ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2EternaryComparisons_2Elist__compare @ ty_2Equote_2Eindex @ ty_2Equote_2Eindex ) @ c_2Equote_2Eindex__compare ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V21l1 ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V23l2 ) ) )
= fo__c_2EternaryComparisons_2EEQUAL )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ V20sr ) @ V22t1 ) @ V24t2 ) ) )
& ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2EternaryComparisons_2Elist__compare @ ty_2Equote_2Eindex @ ty_2Equote_2Eindex ) @ c_2Equote_2Eindex__compare ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V21l1 ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V23l2 ) ) )
= fo__c_2EternaryComparisons_2EGREATER )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ V20sr ) @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V21l1 ) ) @ V22t1 ) ) @ V24t2 ) ) )
& ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2EternaryComparisons_2Elist__compare @ ty_2Equote_2Eindex @ ty_2Equote_2Eindex ) @ c_2Equote_2Eindex__compare ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V21l1 ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V23l2 ) ) )
= fo__c_2EternaryComparisons_2ELESS )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ V20sr ) @ V22t1 ) @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V23l2 ) ) @ V24t2 ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ V20sr ) @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V21l1 ) ) @ V22t1 ) ) @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V23l2 ) ) @ V24t2 ) ) ) ) ) ) )
& ! [V25sr: $i] :
( ( mem @ V25sr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V26s1: $i] :
( ( mem @ V26s1 @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ V25sr ) @ V26s1 ) @ ( c_2Ecanonical_2ENil__monom @ A_27a ) ) ) ) )
& ! [V27sr: $i] :
( ( mem @ V27sr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V28v4: $i] :
( ( mem @ V28v4 @ A_27a )
=> ! [V29v5: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V30v6: $i] :
( ( mem @ V30v6 @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ V27sr ) @ ( c_2Ecanonical_2ENil__monom @ A_27a ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ V28v4 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V29v5 ) ) @ V30v6 ) ) ) ) ) )
& ! [V31sr: $i] :
( ( mem @ V31sr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V32v7: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V33v8: $i] :
( ( mem @ V33v8 @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ V31sr ) @ ( c_2Ecanonical_2ENil__monom @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V32v7 ) ) @ V33v8 ) ) ) ) ) )
=> ! [V34v: $i] :
( ( mem @ V34v @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V35v1: $i] :
( ( mem @ V35v1 @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ! [V36v2: $i] :
( ( mem @ V36v2 @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ V34v ) @ V35v1 ) @ V36v2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecanonical_2Ecanonical__sum__merge__def,axiom,
! [A_27a: del] :
( ! [V0t2: $i] :
( ( mem @ V0t2 @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ! [V1t1: $i] :
( ( mem @ V1t1 @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ! [V2sr: $i] :
( ( mem @ V2sr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V3l2: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V4l1: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V5c2: $i] :
( ( mem @ V5c2 @ A_27a )
=> ! [V6c1: $i] :
( ( mem @ V6c1 @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Ecanonical__sum__merge @ A_27a ) @ V2sr ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ V6c1 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V4l1 ) ) @ V1t1 ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ V5c2 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V3l2 ) ) @ V0t2 ) )
= ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EternaryComparisons_2Eordering__CASE @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) ) @ ( ap @ ( ap @ ( ap @ ( c_2EternaryComparisons_2Elist__compare @ ty_2Equote_2Eindex @ ty_2Equote_2Eindex ) @ c_2Equote_2Eindex__compare ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V4l1 ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V3l2 ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ V6c1 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V4l1 ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Ecanonical__sum__merge @ A_27a ) @ V2sr ) @ V1t1 ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ V5c2 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V3l2 ) ) @ V0t2 ) ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP @ A_27a ) @ V2sr ) @ V6c1 ) @ V5c2 ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V4l1 ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Ecanonical__sum__merge @ A_27a ) @ V2sr ) @ V1t1 ) @ V0t2 ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ V5c2 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V3l2 ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Ecanonical__sum__merge @ A_27a ) @ V2sr ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ V6c1 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V4l1 ) ) @ V1t1 ) ) @ V0t2 ) ) ) ) ) ) ) ) )
& ! [V7t2: $i] :
( ( mem @ V7t2 @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ! [V8t1: $i] :
( ( mem @ V8t1 @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ! [V9sr: $i] :
( ( mem @ V9sr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V10l2: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V11l1: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V12c1: $i] :
( ( mem @ V12c1 @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Ecanonical__sum__merge @ A_27a ) @ V9sr ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ V12c1 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V11l1 ) ) @ V8t1 ) ) @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V10l2 ) ) @ V7t2 ) )
= ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EternaryComparisons_2Eordering__CASE @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) ) @ ( ap @ ( ap @ ( ap @ ( c_2EternaryComparisons_2Elist__compare @ ty_2Equote_2Eindex @ ty_2Equote_2Eindex ) @ c_2Equote_2Eindex__compare ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V11l1 ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V10l2 ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ V12c1 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V11l1 ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Ecanonical__sum__merge @ A_27a ) @ V9sr ) @ V8t1 ) @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V10l2 ) ) @ V7t2 ) ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP @ A_27a ) @ V9sr ) @ V12c1 ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR1 @ A_27a ) @ V9sr ) ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V11l1 ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Ecanonical__sum__merge @ A_27a ) @ V9sr ) @ V8t1 ) @ V7t2 ) ) ) @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V10l2 ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Ecanonical__sum__merge @ A_27a ) @ V9sr ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ V12c1 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V11l1 ) ) @ V8t1 ) ) @ V7t2 ) ) ) ) ) ) ) )
& ! [V13t2: $i] :
( ( mem @ V13t2 @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ! [V14t1: $i] :
( ( mem @ V14t1 @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ! [V15sr: $i] :
( ( mem @ V15sr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V16l2: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V17l1: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V18c2: $i] :
( ( mem @ V18c2 @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Ecanonical__sum__merge @ A_27a ) @ V15sr ) @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V17l1 ) ) @ V14t1 ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ V18c2 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V16l2 ) ) @ V13t2 ) )
= ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EternaryComparisons_2Eordering__CASE @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) ) @ ( ap @ ( ap @ ( ap @ ( c_2EternaryComparisons_2Elist__compare @ ty_2Equote_2Eindex @ ty_2Equote_2Eindex ) @ c_2Equote_2Eindex__compare ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V17l1 ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V16l2 ) ) ) @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V17l1 ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Ecanonical__sum__merge @ A_27a ) @ V15sr ) @ V14t1 ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ V18c2 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V16l2 ) ) @ V13t2 ) ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP @ A_27a ) @ V15sr ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR1 @ A_27a ) @ V15sr ) ) @ V18c2 ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V17l1 ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Ecanonical__sum__merge @ A_27a ) @ V15sr ) @ V14t1 ) @ V13t2 ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ V18c2 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V16l2 ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Ecanonical__sum__merge @ A_27a ) @ V15sr ) @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V17l1 ) ) @ V14t1 ) ) @ V13t2 ) ) ) ) ) ) ) )
& ! [V19t2: $i] :
( ( mem @ V19t2 @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ! [V20t1: $i] :
( ( mem @ V20t1 @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ! [V21sr: $i] :
( ( mem @ V21sr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V22l2: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V23l1: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex] :
( ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Ecanonical__sum__merge @ A_27a ) @ V21sr ) @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V23l1 ) ) @ V20t1 ) ) @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V22l2 ) ) @ V19t2 ) )
= ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EternaryComparisons_2Eordering__CASE @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) ) @ ( ap @ ( ap @ ( ap @ ( c_2EternaryComparisons_2Elist__compare @ ty_2Equote_2Eindex @ ty_2Equote_2Eindex ) @ c_2Equote_2Eindex__compare ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V23l1 ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V22l2 ) ) ) @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V23l1 ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Ecanonical__sum__merge @ A_27a ) @ V21sr ) @ V20t1 ) @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V22l2 ) ) @ V19t2 ) ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP @ A_27a ) @ V21sr ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR1 @ A_27a ) @ V21sr ) ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR1 @ A_27a ) @ V21sr ) ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V23l1 ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Ecanonical__sum__merge @ A_27a ) @ V21sr ) @ V20t1 ) @ V19t2 ) ) ) @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V22l2 ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Ecanonical__sum__merge @ A_27a ) @ V21sr ) @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V23l1 ) ) @ V20t1 ) ) @ V19t2 ) ) ) ) ) ) )
& ! [V24sr: $i] :
( ( mem @ V24sr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V25s1: $i] :
( ( mem @ V25s1 @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Ecanonical__sum__merge @ A_27a ) @ V24sr ) @ V25s1 ) @ ( c_2Ecanonical_2ENil__monom @ A_27a ) )
= V25s1 ) ) )
& ! [V26v6: $i] :
( ( mem @ V26v6 @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ! [V27v5: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V28v4: $i] :
( ( mem @ V28v4 @ A_27a )
=> ! [V29sr: $i] :
( ( mem @ V29sr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Ecanonical__sum__merge @ A_27a ) @ V29sr ) @ ( c_2Ecanonical_2ENil__monom @ A_27a ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ V28v4 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V27v5 ) ) @ V26v6 ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ V28v4 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V27v5 ) ) @ V26v6 ) ) ) ) )
& ! [V30v8: $i] :
( ( mem @ V30v8 @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ! [V31v7: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V32sr: $i] :
( ( mem @ V32sr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Ecanonical__sum__merge @ A_27a ) @ V32sr ) @ ( c_2Ecanonical_2ENil__monom @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V31v7 ) ) @ V30v8 ) )
= ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V31v7 ) ) @ V30v8 ) ) ) ) ) ).
thf(conj_thm_2Ecanonical_2Emonom__insert__ind,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( arr @ A_27a @ ( arr @ ( ty_2Elist_2Elist @ ty_2Equote_2Eindex ) @ ( arr @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) @ bool ) ) ) ) )
=> ( ( ! [V1sr: $i] :
( ( mem @ V1sr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V2c1: $i] :
( ( mem @ V2c1 @ A_27a )
=> ! [V3l1: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V4c2: $i] :
( ( mem @ V4c2 @ A_27a )
=> ! [V5l2: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V6t2: $i] :
( ( mem @ V6t2 @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ( ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2EternaryComparisons_2Elist__compare @ ty_2Equote_2Eindex @ ty_2Equote_2Eindex ) @ c_2Equote_2Eindex__compare ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V3l1 ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V5l2 ) ) )
= fo__c_2EternaryComparisons_2EGREATER )
=> ( p @ ( ap @ ( ap @ ( ap @ ( ap @ V0P @ V1sr ) @ V2c1 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V3l1 ) ) @ V6t2 ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( ap @ V0P @ V1sr ) @ V2c1 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V3l1 ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ V4c2 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V5l2 ) ) @ V6t2 ) ) ) ) ) ) ) )
& ! [V7sr: $i] :
( ( mem @ V7sr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V8c1: $i] :
( ( mem @ V8c1 @ A_27a )
=> ! [V9l1: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V10l2: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V11t2: $i] :
( ( mem @ V11t2 @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ( ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2EternaryComparisons_2Elist__compare @ ty_2Equote_2Eindex @ ty_2Equote_2Eindex ) @ c_2Equote_2Eindex__compare ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V9l1 ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V10l2 ) ) )
= fo__c_2EternaryComparisons_2EGREATER )
=> ( p @ ( ap @ ( ap @ ( ap @ ( ap @ V0P @ V7sr ) @ V8c1 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V9l1 ) ) @ V11t2 ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( ap @ V0P @ V7sr ) @ V8c1 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V9l1 ) ) @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V10l2 ) ) @ V11t2 ) ) ) ) ) ) )
& ! [V12sr: $i] :
( ( mem @ V12sr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V13c1: $i] :
( ( mem @ V13c1 @ A_27a )
=> ! [V14l1: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex] : ( p @ ( ap @ ( ap @ ( ap @ ( ap @ V0P @ V12sr ) @ V13c1 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V14l1 ) ) @ ( c_2Ecanonical_2ENil__monom @ A_27a ) ) ) ) ) )
=> ! [V15v: $i] :
( ( mem @ V15v @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V16v1: $i] :
( ( mem @ V16v1 @ A_27a )
=> ! [V17v2: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V18v3: $i] :
( ( mem @ V18v3 @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( ap @ V0P @ V15v ) @ V16v1 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V17v2 ) ) @ V18v3 ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecanonical_2Emonom__insert__def,axiom,
! [A_27a: del] :
( ! [V0t2: $i] :
( ( mem @ V0t2 @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ! [V1sr: $i] :
( ( mem @ V1sr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V2l2: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V3l1: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V4c2: $i] :
( ( mem @ V4c2 @ A_27a )
=> ! [V5c1: $i] :
( ( mem @ V5c1 @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Emonom__insert @ A_27a ) @ V1sr ) @ V5c1 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V3l1 ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ V4c2 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V2l2 ) ) @ V0t2 ) )
= ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EternaryComparisons_2Eordering__CASE @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) ) @ ( ap @ ( ap @ ( ap @ ( c_2EternaryComparisons_2Elist__compare @ ty_2Equote_2Eindex @ ty_2Equote_2Eindex ) @ c_2Equote_2Eindex__compare ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V3l1 ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V2l2 ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ V5c1 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V3l1 ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ V4c2 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V2l2 ) ) @ V0t2 ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP @ A_27a ) @ V1sr ) @ V5c1 ) @ V4c2 ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V3l1 ) ) @ V0t2 ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ V4c2 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V2l2 ) ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Emonom__insert @ A_27a ) @ V1sr ) @ V5c1 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V3l1 ) ) @ V0t2 ) ) ) ) ) ) ) )
& ! [V6t2: $i] :
( ( mem @ V6t2 @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ! [V7sr: $i] :
( ( mem @ V7sr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V8l2: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V9l1: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V10c1: $i] :
( ( mem @ V10c1 @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Emonom__insert @ A_27a ) @ V7sr ) @ V10c1 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V9l1 ) ) @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V8l2 ) ) @ V6t2 ) )
= ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EternaryComparisons_2Eordering__CASE @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) ) @ ( ap @ ( ap @ ( ap @ ( c_2EternaryComparisons_2Elist__compare @ ty_2Equote_2Eindex @ ty_2Equote_2Eindex ) @ c_2Equote_2Eindex__compare ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V9l1 ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V8l2 ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ V10c1 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V9l1 ) ) @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V8l2 ) ) @ V6t2 ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP @ A_27a ) @ V7sr ) @ V10c1 ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR1 @ A_27a ) @ V7sr ) ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V9l1 ) ) @ V6t2 ) ) @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V8l2 ) ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Emonom__insert @ A_27a ) @ V7sr ) @ V10c1 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V9l1 ) ) @ V6t2 ) ) ) ) ) ) )
& ! [V11sr: $i] :
( ( mem @ V11sr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V12l1: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V13c1: $i] :
( ( mem @ V13c1 @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Emonom__insert @ A_27a ) @ V11sr ) @ V13c1 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V12l1 ) ) @ ( c_2Ecanonical_2ENil__monom @ A_27a ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ V13c1 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V12l1 ) ) @ ( c_2Ecanonical_2ENil__monom @ A_27a ) ) ) ) ) ) ).
thf(conj_thm_2Ecanonical_2Evarlist__insert__ind,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) @ ( arr @ ( ty_2Elist_2Elist @ ty_2Equote_2Eindex ) @ ( arr @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) @ bool ) ) ) )
=> ( ( ! [V1sr: $i] :
( ( mem @ V1sr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V2l1: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V3c2: $i] :
( ( mem @ V3c2 @ A_27a )
=> ! [V4l2: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V5t2: $i] :
( ( mem @ V5t2 @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ( ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2EternaryComparisons_2Elist__compare @ ty_2Equote_2Eindex @ ty_2Equote_2Eindex ) @ c_2Equote_2Eindex__compare ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V2l1 ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V4l2 ) ) )
= fo__c_2EternaryComparisons_2EGREATER )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ V1sr ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V2l1 ) ) @ V5t2 ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ V1sr ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V2l1 ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ V3c2 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V4l2 ) ) @ V5t2 ) ) ) ) ) ) )
& ! [V6sr: $i] :
( ( mem @ V6sr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V7l1: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V8l2: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V9t2: $i] :
( ( mem @ V9t2 @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ( ( ( ( surj__ty_2EternaryComparisons_2Eordering @ ( ap @ ( ap @ ( ap @ ( c_2EternaryComparisons_2Elist__compare @ ty_2Equote_2Eindex @ ty_2Equote_2Eindex ) @ c_2Equote_2Eindex__compare ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V7l1 ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V8l2 ) ) )
= fo__c_2EternaryComparisons_2EGREATER )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ V6sr ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V7l1 ) ) @ V9t2 ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ V6sr ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V7l1 ) ) @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V8l2 ) ) @ V9t2 ) ) ) ) ) )
& ! [V10sr: $i] :
( ( mem @ V10sr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V11l1: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex] : ( p @ ( ap @ ( ap @ ( ap @ V0P @ V10sr ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V11l1 ) ) @ ( c_2Ecanonical_2ENil__monom @ A_27a ) ) ) ) )
=> ! [V12v: $i] :
( ( mem @ V12v @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V13v1: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V14v2: $i] :
( ( mem @ V14v2 @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ V12v ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V13v1 ) ) @ V14v2 ) ) ) ) ) ) ).
thf(conj_thm_2Ecanonical_2Evarlist__insert__def,axiom,
! [A_27a: del] :
( ! [V0t2: $i] :
( ( mem @ V0t2 @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ! [V1sr: $i] :
( ( mem @ V1sr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V2l2: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V3l1: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V4c2: $i] :
( ( mem @ V4c2 @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Evarlist__insert @ A_27a ) @ V1sr ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V3l1 ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ V4c2 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V2l2 ) ) @ V0t2 ) )
= ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EternaryComparisons_2Eordering__CASE @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) ) @ ( ap @ ( ap @ ( ap @ ( c_2EternaryComparisons_2Elist__compare @ ty_2Equote_2Eindex @ ty_2Equote_2Eindex ) @ c_2Equote_2Eindex__compare ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V3l1 ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V2l2 ) ) ) @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V3l1 ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ V4c2 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V2l2 ) ) @ V0t2 ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP @ A_27a ) @ V1sr ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR1 @ A_27a ) @ V1sr ) ) @ V4c2 ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V3l1 ) ) @ V0t2 ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ V4c2 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V2l2 ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Evarlist__insert @ A_27a ) @ V1sr ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V3l1 ) ) @ V0t2 ) ) ) ) ) ) )
& ! [V5t2: $i] :
( ( mem @ V5t2 @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ! [V6sr: $i] :
( ( mem @ V6sr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V7l2: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V8l1: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex] :
( ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Evarlist__insert @ A_27a ) @ V6sr ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V8l1 ) ) @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V7l2 ) ) @ V5t2 ) )
= ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EternaryComparisons_2Eordering__CASE @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) ) @ ( ap @ ( ap @ ( ap @ ( c_2EternaryComparisons_2Elist__compare @ ty_2Equote_2Eindex @ ty_2Equote_2Eindex ) @ c_2Equote_2Eindex__compare ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V8l1 ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V7l2 ) ) ) @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V8l1 ) ) @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V7l2 ) ) @ V5t2 ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP @ A_27a ) @ V6sr ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR1 @ A_27a ) @ V6sr ) ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR1 @ A_27a ) @ V6sr ) ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V8l1 ) ) @ V5t2 ) ) @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V7l2 ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Evarlist__insert @ A_27a ) @ V6sr ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V8l1 ) ) @ V5t2 ) ) ) ) ) )
& ! [V9sr: $i] :
( ( mem @ V9sr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V10l1: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex] :
( ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Evarlist__insert @ A_27a ) @ V9sr ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V10l1 ) ) @ ( c_2Ecanonical_2ENil__monom @ A_27a ) )
= ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V10l1 ) ) @ ( c_2Ecanonical_2ENil__monom @ A_27a ) ) ) ) ) ).
thf(ax_thm_2Ecanonical_2Ecanonical__sum__scalar__def,axiom,
! [A_27a: del] :
( ! [V0sr: $i] :
( ( mem @ V0sr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V1c0: $i] :
( ( mem @ V1c0 @ A_27a )
=> ! [V2c: $i] :
( ( mem @ V2c @ A_27a )
=> ! [V3l: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V4t: $i] :
( ( mem @ V4t @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Ecanonical__sum__scalar @ A_27a ) @ V0sr ) @ V1c0 ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ V2c ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V3l ) ) @ V4t ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM @ A_27a ) @ V0sr ) @ V1c0 ) @ V2c ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V3l ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Ecanonical__sum__scalar @ A_27a ) @ V0sr ) @ V1c0 ) @ V4t ) ) ) ) ) ) )
& ! [V5sr: $i] :
( ( mem @ V5sr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V6c0: $i] :
( ( mem @ V6c0 @ A_27a )
=> ! [V7l: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V8t: $i] :
( ( mem @ V8t @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Ecanonical__sum__scalar @ A_27a ) @ V5sr ) @ V6c0 ) @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V7l ) ) @ V8t ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ V6c0 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V7l ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Ecanonical__sum__scalar @ A_27a ) @ V5sr ) @ V6c0 ) @ V8t ) ) ) ) ) )
& ! [V9sr: $i] :
( ( mem @ V9sr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V10c0: $i] :
( ( mem @ V10c0 @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Ecanonical__sum__scalar @ A_27a ) @ V9sr ) @ V10c0 ) @ ( c_2Ecanonical_2ENil__monom @ A_27a ) )
= ( c_2Ecanonical_2ENil__monom @ A_27a ) ) ) ) ) ).
thf(ax_thm_2Ecanonical_2Ecanonical__sum__scalar2__def,axiom,
! [A_27a: del] :
( ! [V0sr: $i] :
( ( mem @ V0sr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V1l0: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V2c: $i] :
( ( mem @ V2c @ A_27a )
=> ! [V3l: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V4t: $i] :
( ( mem @ V4t @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Ecanonical__sum__scalar2 @ A_27a ) @ V0sr ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V1l0 ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ V2c ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V3l ) ) @ V4t ) )
= ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Emonom__insert @ A_27a ) @ V0sr ) @ V2c ) @ ( ap @ ( ap @ ( ap @ ( c_2EternaryComparisons_2Elist__merge @ ty_2Equote_2Eindex ) @ c_2Equote_2Eindex__lt ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V1l0 ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V3l ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Ecanonical__sum__scalar2 @ A_27a ) @ V0sr ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V1l0 ) ) @ V4t ) ) ) ) ) )
& ! [V5sr: $i] :
( ( mem @ V5sr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V6l0: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V7l: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V8t: $i] :
( ( mem @ V8t @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Ecanonical__sum__scalar2 @ A_27a ) @ V5sr ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V6l0 ) ) @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V7l ) ) @ V8t ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Evarlist__insert @ A_27a ) @ V5sr ) @ ( ap @ ( ap @ ( ap @ ( c_2EternaryComparisons_2Elist__merge @ ty_2Equote_2Eindex ) @ c_2Equote_2Eindex__lt ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V6l0 ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V7l ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Ecanonical__sum__scalar2 @ A_27a ) @ V5sr ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V6l0 ) ) @ V8t ) ) ) ) )
& ! [V9sr: $i] :
( ( mem @ V9sr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V10l0: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex] :
( ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Ecanonical__sum__scalar2 @ A_27a ) @ V9sr ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V10l0 ) ) @ ( c_2Ecanonical_2ENil__monom @ A_27a ) )
= ( c_2Ecanonical_2ENil__monom @ A_27a ) ) ) ) ).
thf(ax_thm_2Ecanonical_2Ecanonical__sum__scalar3__def,axiom,
! [A_27a: del] :
( ! [V0sr: $i] :
( ( mem @ V0sr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V1c0: $i] :
( ( mem @ V1c0 @ A_27a )
=> ! [V2l0: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V3c: $i] :
( ( mem @ V3c @ A_27a )
=> ! [V4l: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V5t: $i] :
( ( mem @ V5t @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Ecanonical__sum__scalar3 @ A_27a ) @ V0sr ) @ V1c0 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V2l0 ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ V3c ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V4l ) ) @ V5t ) )
= ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Emonom__insert @ A_27a ) @ V0sr ) @ ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM @ A_27a ) @ V0sr ) @ V1c0 ) @ V3c ) ) @ ( ap @ ( ap @ ( ap @ ( c_2EternaryComparisons_2Elist__merge @ ty_2Equote_2Eindex ) @ c_2Equote_2Eindex__lt ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V2l0 ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V4l ) ) ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Ecanonical__sum__scalar3 @ A_27a ) @ V0sr ) @ V1c0 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V2l0 ) ) @ V5t ) ) ) ) ) ) )
& ! [V6sr: $i] :
( ( mem @ V6sr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V7c0: $i] :
( ( mem @ V7c0 @ A_27a )
=> ! [V8l0: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V9l: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V10t: $i] :
( ( mem @ V10t @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Ecanonical__sum__scalar3 @ A_27a ) @ V6sr ) @ V7c0 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V8l0 ) ) @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V9l ) ) @ V10t ) )
= ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Emonom__insert @ A_27a ) @ V6sr ) @ V7c0 ) @ ( ap @ ( ap @ ( ap @ ( c_2EternaryComparisons_2Elist__merge @ ty_2Equote_2Eindex ) @ c_2Equote_2Eindex__lt ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V8l0 ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V9l ) ) ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Ecanonical__sum__scalar3 @ A_27a ) @ V6sr ) @ V7c0 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V8l0 ) ) @ V10t ) ) ) ) ) )
& ! [V11sr: $i] :
( ( mem @ V11sr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V12c0: $i] :
( ( mem @ V12c0 @ A_27a )
=> ! [V13l0: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex] :
( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Ecanonical__sum__scalar3 @ A_27a ) @ V11sr ) @ V12c0 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V13l0 ) ) @ ( c_2Ecanonical_2ENil__monom @ A_27a ) )
= ( c_2Ecanonical_2ENil__monom @ A_27a ) ) ) ) ) ).
thf(ax_thm_2Ecanonical_2Ecanonical__sum__prod__def,axiom,
! [A_27a: del] :
( ! [V0sr: $i] :
( ( mem @ V0sr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V1c1: $i] :
( ( mem @ V1c1 @ A_27a )
=> ! [V2l1: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V3t1: $i] :
( ( mem @ V3t1 @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ! [V4s2: $i] :
( ( mem @ V4s2 @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Ecanonical__sum__prod @ A_27a ) @ V0sr ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ V1c1 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V2l1 ) ) @ V3t1 ) ) @ V4s2 )
= ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Ecanonical__sum__merge @ A_27a ) @ V0sr ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Ecanonical__sum__scalar3 @ A_27a ) @ V0sr ) @ V1c1 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V2l1 ) ) @ V4s2 ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Ecanonical__sum__prod @ A_27a ) @ V0sr ) @ V3t1 ) @ V4s2 ) ) ) ) ) ) )
& ! [V5sr: $i] :
( ( mem @ V5sr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V6l1: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V7t1: $i] :
( ( mem @ V7t1 @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ! [V8s2: $i] :
( ( mem @ V8s2 @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Ecanonical__sum__prod @ A_27a ) @ V5sr ) @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V6l1 ) ) @ V7t1 ) ) @ V8s2 )
= ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Ecanonical__sum__merge @ A_27a ) @ V5sr ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Ecanonical__sum__scalar2 @ A_27a ) @ V5sr ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V6l1 ) ) @ V8s2 ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Ecanonical__sum__prod @ A_27a ) @ V5sr ) @ V7t1 ) @ V8s2 ) ) ) ) ) )
& ! [V9sr: $i] :
( ( mem @ V9sr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V10s2: $i] :
( ( mem @ V10s2 @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Ecanonical__sum__prod @ A_27a ) @ V9sr ) @ ( c_2Ecanonical_2ENil__monom @ A_27a ) ) @ V10s2 )
= ( c_2Ecanonical_2ENil__monom @ A_27a ) ) ) ) ) ).
thf(ax_thm_2Ecanonical_2Ecanonical__sum__simplify__def,axiom,
! [A_27a: del] :
( ! [V0sr: $i] :
( ( mem @ V0sr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V1c: $i] :
( ( mem @ V1c @ A_27a )
=> ! [V2l: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V3t: $i] :
( ( mem @ V3t @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Ecanonical_2Ecanonical__sum__simplify @ A_27a ) @ V0sr ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ V1c ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V2l ) ) @ V3t ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V1c ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR0 @ A_27a ) @ V0sr ) ) ) @ ( ap @ ( ap @ ( c_2Ecanonical_2Ecanonical__sum__simplify @ A_27a ) @ V0sr ) @ V3t ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V1c ) @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR1 @ A_27a ) @ V0sr ) ) ) @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V2l ) ) @ ( ap @ ( ap @ ( c_2Ecanonical_2Ecanonical__sum__simplify @ A_27a ) @ V0sr ) @ V3t ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ V1c ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V2l ) ) @ ( ap @ ( ap @ ( c_2Ecanonical_2Ecanonical__sum__simplify @ A_27a ) @ V0sr ) @ V3t ) ) ) ) ) ) ) )
& ! [V4sr: $i] :
( ( mem @ V4sr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V5l: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V6t: $i] :
( ( mem @ V6t @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Ecanonical_2Ecanonical__sum__simplify @ A_27a ) @ V4sr ) @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V5l ) ) @ V6t ) )
= ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V5l ) ) @ ( ap @ ( ap @ ( c_2Ecanonical_2Ecanonical__sum__simplify @ A_27a ) @ V4sr ) @ V6t ) ) ) ) )
& ! [V7sr: $i] :
( ( mem @ V7sr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Ecanonical_2Ecanonical__sum__simplify @ A_27a ) @ V7sr ) @ ( c_2Ecanonical_2ENil__monom @ A_27a ) )
= ( c_2Ecanonical_2ENil__monom @ A_27a ) ) ) ) ).
thf(ax_thm_2Ecanonical_2Eivl__aux__def,axiom,
! [A_27a: del] :
( ! [V0sr: $i] :
( ( mem @ V0sr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V1vm: $i] :
( ( mem @ V1vm @ ( ty_2Equote_2Evarmap @ A_27a ) )
=> ! [V2x: tp__ty_2Equote_2Eindex] :
( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Eivl__aux @ A_27a ) @ V0sr ) @ V1vm ) @ ( inj__ty_2Equote_2Eindex @ V2x ) ) @ ( c_2Elist_2ENIL @ ty_2Equote_2Eindex ) )
= ( ap @ ( ap @ ( c_2Equote_2Evarmap__find @ A_27a ) @ ( inj__ty_2Equote_2Eindex @ V2x ) ) @ V1vm ) ) ) )
& ! [V3sr: $i] :
( ( mem @ V3sr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V4vm: $i] :
( ( mem @ V4vm @ ( ty_2Equote_2Evarmap @ A_27a ) )
=> ! [V5x: tp__ty_2Equote_2Eindex,V6x_27: tp__ty_2Equote_2Eindex,V7t_27: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex] :
( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Eivl__aux @ A_27a ) @ V3sr ) @ V4vm ) @ ( inj__ty_2Equote_2Eindex @ V5x ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ty_2Equote_2Eindex ) @ ( inj__ty_2Equote_2Eindex @ V6x_27 ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V7t_27 ) ) )
= ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM @ A_27a ) @ V3sr ) @ ( ap @ ( ap @ ( c_2Equote_2Evarmap__find @ A_27a ) @ ( inj__ty_2Equote_2Eindex @ V5x ) ) @ V4vm ) ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Eivl__aux @ A_27a ) @ V3sr ) @ V4vm ) @ ( inj__ty_2Equote_2Eindex @ V6x_27 ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V7t_27 ) ) ) ) ) ) ) ).
thf(ax_thm_2Ecanonical_2Einterp__vl__def,axiom,
! [A_27a: del] :
( ! [V0sr: $i] :
( ( mem @ V0sr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V1vm: $i] :
( ( mem @ V1vm @ ( ty_2Equote_2Evarmap @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Einterp__vl @ A_27a ) @ V0sr ) @ V1vm ) @ ( c_2Elist_2ENIL @ ty_2Equote_2Eindex ) )
= ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR1 @ A_27a ) @ V0sr ) ) ) )
& ! [V2sr: $i] :
( ( mem @ V2sr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V3vm: $i] :
( ( mem @ V3vm @ ( ty_2Equote_2Evarmap @ A_27a ) )
=> ! [V4x: tp__ty_2Equote_2Eindex,V5t: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex] :
( ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Einterp__vl @ A_27a ) @ V2sr ) @ V3vm ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ty_2Equote_2Eindex ) @ ( inj__ty_2Equote_2Eindex @ V4x ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V5t ) ) )
= ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Eivl__aux @ A_27a ) @ V2sr ) @ V3vm ) @ ( inj__ty_2Equote_2Eindex @ V4x ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V5t ) ) ) ) ) ) ).
thf(ax_thm_2Ecanonical_2Einterp__m__def,axiom,
! [A_27a: del] :
( ! [V0sr: $i] :
( ( mem @ V0sr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V1vm: $i] :
( ( mem @ V1vm @ ( ty_2Equote_2Evarmap @ A_27a ) )
=> ! [V2c: $i] :
( ( mem @ V2c @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Einterp__m @ A_27a ) @ V0sr ) @ V1vm ) @ V2c ) @ ( c_2Elist_2ENIL @ ty_2Equote_2Eindex ) )
= V2c ) ) ) )
& ! [V3sr: $i] :
( ( mem @ V3sr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V4vm: $i] :
( ( mem @ V4vm @ ( ty_2Equote_2Evarmap @ A_27a ) )
=> ! [V5c: $i] :
( ( mem @ V5c @ A_27a )
=> ! [V6x: tp__ty_2Equote_2Eindex,V7t: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex] :
( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Einterp__m @ A_27a ) @ V3sr ) @ V4vm ) @ V5c ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ty_2Equote_2Eindex ) @ ( inj__ty_2Equote_2Eindex @ V6x ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V7t ) ) )
= ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM @ A_27a ) @ V3sr ) @ V5c ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Eivl__aux @ A_27a ) @ V3sr ) @ V4vm ) @ ( inj__ty_2Equote_2Eindex @ V6x ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V7t ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Ecanonical_2Eics__aux__def,axiom,
! [A_27a: del] :
( ! [V0sr: $i] :
( ( mem @ V0sr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V1vm: $i] :
( ( mem @ V1vm @ ( ty_2Equote_2Evarmap @ A_27a ) )
=> ! [V2a: $i] :
( ( mem @ V2a @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Eics__aux @ A_27a ) @ V0sr ) @ V1vm ) @ V2a ) @ ( c_2Ecanonical_2ENil__monom @ A_27a ) )
= V2a ) ) ) )
& ! [V3sr: $i] :
( ( mem @ V3sr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V4vm: $i] :
( ( mem @ V4vm @ ( ty_2Equote_2Evarmap @ A_27a ) )
=> ! [V5a: $i] :
( ( mem @ V5a @ A_27a )
=> ! [V6l: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V7t: $i] :
( ( mem @ V7t @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Eics__aux @ A_27a ) @ V3sr ) @ V4vm ) @ V5a ) @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V6l ) ) @ V7t ) )
= ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP @ A_27a ) @ V3sr ) @ V5a ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Eics__aux @ A_27a ) @ V3sr ) @ V4vm ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Einterp__vl @ A_27a ) @ V3sr ) @ V4vm ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V6l ) ) ) @ V7t ) ) ) ) ) ) )
& ! [V8sr: $i] :
( ( mem @ V8sr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V9vm: $i] :
( ( mem @ V9vm @ ( ty_2Equote_2Evarmap @ A_27a ) )
=> ! [V10a: $i] :
( ( mem @ V10a @ A_27a )
=> ! [V11c: $i] :
( ( mem @ V11c @ A_27a )
=> ! [V12l: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V13t: $i] :
( ( mem @ V13t @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Eics__aux @ A_27a ) @ V8sr ) @ V9vm ) @ V10a ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ V11c ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V12l ) ) @ V13t ) )
= ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP @ A_27a ) @ V8sr ) @ V10a ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Eics__aux @ A_27a ) @ V8sr ) @ V9vm ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Einterp__m @ A_27a ) @ V8sr ) @ V9vm ) @ V11c ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V12l ) ) ) @ V13t ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Ecanonical_2Einterp__cs__def,axiom,
! [A_27a: del] :
( ! [V0sr: $i] :
( ( mem @ V0sr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V1vm: $i] :
( ( mem @ V1vm @ ( ty_2Equote_2Evarmap @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Einterp__cs @ A_27a ) @ V0sr ) @ V1vm ) @ ( c_2Ecanonical_2ENil__monom @ A_27a ) )
= ( ap @ ( c_2Esemi__ring_2Esemi__ring__SR0 @ A_27a ) @ V0sr ) ) ) )
& ! [V2sr: $i] :
( ( mem @ V2sr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V3vm: $i] :
( ( mem @ V3vm @ ( ty_2Equote_2Evarmap @ A_27a ) )
=> ! [V4l: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V5t: $i] :
( ( mem @ V5t @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Einterp__cs @ A_27a ) @ V2sr ) @ V3vm ) @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V4l ) ) @ V5t ) )
= ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Eics__aux @ A_27a ) @ V2sr ) @ V3vm ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Einterp__vl @ A_27a ) @ V2sr ) @ V3vm ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V4l ) ) ) @ V5t ) ) ) ) )
& ! [V6sr: $i] :
( ( mem @ V6sr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V7vm: $i] :
( ( mem @ V7vm @ ( ty_2Equote_2Evarmap @ A_27a ) )
=> ! [V8c: $i] :
( ( mem @ V8c @ A_27a )
=> ! [V9l: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V10t: $i] :
( ( mem @ V10t @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Einterp__cs @ A_27a ) @ V6sr ) @ V7vm ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ V8c ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V9l ) ) @ V10t ) )
= ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Eics__aux @ A_27a ) @ V6sr ) @ V7vm ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Einterp__m @ A_27a ) @ V6sr ) @ V7vm ) @ V8c ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V9l ) ) ) @ V10t ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecanonical_2Eivl__aux__ok,axiom,
! [A_27a: del,V0sr: $i] :
( ( mem @ V0sr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Esemi__ring_2Eis__semi__ring @ A_27a ) @ V0sr ) )
=> ! [V1vm: $i] :
( ( mem @ V1vm @ ( ty_2Equote_2Evarmap @ A_27a ) )
=> ! [V2v: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V3i: tp__ty_2Equote_2Eindex] :
( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Eivl__aux @ A_27a ) @ V0sr ) @ V1vm ) @ ( inj__ty_2Equote_2Eindex @ V3i ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V2v ) )
= ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM @ A_27a ) @ V0sr ) @ ( ap @ ( ap @ ( c_2Equote_2Evarmap__find @ A_27a ) @ ( inj__ty_2Equote_2Eindex @ V3i ) ) @ V1vm ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Einterp__vl @ A_27a ) @ V0sr ) @ V1vm ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V2v ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecanonical_2Evarlist__merge__ok,axiom,
! [A_27a: del,V0sr: $i] :
( ( mem @ V0sr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Esemi__ring_2Eis__semi__ring @ A_27a ) @ V0sr ) )
=> ! [V1vm: $i] :
( ( mem @ V1vm @ ( ty_2Equote_2Evarmap @ A_27a ) )
=> ! [V2x: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V3y: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex] :
( ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Einterp__vl @ A_27a ) @ V0sr ) @ V1vm ) @ ( ap @ ( ap @ ( ap @ ( c_2EternaryComparisons_2Elist__merge @ ty_2Equote_2Eindex ) @ c_2Equote_2Eindex__lt ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V2x ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V3y ) ) )
= ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM @ A_27a ) @ V0sr ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Einterp__vl @ A_27a ) @ V0sr ) @ V1vm ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V2x ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Einterp__vl @ A_27a ) @ V0sr ) @ V1vm ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V3y ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecanonical_2Eics__aux__ok,axiom,
! [A_27a: del,V0sr: $i] :
( ( mem @ V0sr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Esemi__ring_2Eis__semi__ring @ A_27a ) @ V0sr ) )
=> ! [V1vm: $i] :
( ( mem @ V1vm @ ( ty_2Equote_2Evarmap @ A_27a ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3s: $i] :
( ( mem @ V3s @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Eics__aux @ A_27a ) @ V0sr ) @ V1vm ) @ V2x ) @ V3s )
= ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP @ A_27a ) @ V0sr ) @ V2x ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Einterp__cs @ A_27a ) @ V0sr ) @ V1vm ) @ V3s ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecanonical_2Einterp__m__ok,axiom,
! [A_27a: del,V0sr: $i] :
( ( mem @ V0sr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Esemi__ring_2Eis__semi__ring @ A_27a ) @ V0sr ) )
=> ! [V1vm: $i] :
( ( mem @ V1vm @ ( ty_2Equote_2Evarmap @ A_27a ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3l: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex] :
( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Einterp__m @ A_27a ) @ V0sr ) @ V1vm ) @ V2x ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V3l ) )
= ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM @ A_27a ) @ V0sr ) @ V2x ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Einterp__vl @ A_27a ) @ V0sr ) @ V1vm ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V3l ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecanonical_2Ecanonical__sum__merge__ok,axiom,
! [A_27a: del,V0sr: $i] :
( ( mem @ V0sr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Esemi__ring_2Eis__semi__ring @ A_27a ) @ V0sr ) )
=> ! [V1vm: $i] :
( ( mem @ V1vm @ ( ty_2Equote_2Evarmap @ A_27a ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ! [V3y: $i] :
( ( mem @ V3y @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Einterp__cs @ A_27a ) @ V0sr ) @ V1vm ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Ecanonical__sum__merge @ A_27a ) @ V0sr ) @ V2x ) @ V3y ) )
= ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP @ A_27a ) @ V0sr ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Einterp__cs @ A_27a ) @ V0sr ) @ V1vm ) @ V2x ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Einterp__cs @ A_27a ) @ V0sr ) @ V1vm ) @ V3y ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecanonical_2Emonom__insert__ok,axiom,
! [A_27a: del,V0sr: $i] :
( ( mem @ V0sr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Esemi__ring_2Eis__semi__ring @ A_27a ) @ V0sr ) )
=> ! [V1vm: $i] :
( ( mem @ V1vm @ ( ty_2Equote_2Evarmap @ A_27a ) )
=> ! [V2a: $i] :
( ( mem @ V2a @ A_27a )
=> ! [V3l: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V4s: $i] :
( ( mem @ V4s @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Einterp__cs @ A_27a ) @ V0sr ) @ V1vm ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Emonom__insert @ A_27a ) @ V0sr ) @ V2a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V3l ) ) @ V4s ) )
= ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP @ A_27a ) @ V0sr ) @ ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM @ A_27a ) @ V0sr ) @ V2a ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Einterp__vl @ A_27a ) @ V0sr ) @ V1vm ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V3l ) ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Einterp__cs @ A_27a ) @ V0sr ) @ V1vm ) @ V4s ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecanonical_2Evarlist__insert__ok,axiom,
! [A_27a: del,V0sr: $i] :
( ( mem @ V0sr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Esemi__ring_2Eis__semi__ring @ A_27a ) @ V0sr ) )
=> ! [V1vm: $i] :
( ( mem @ V1vm @ ( ty_2Equote_2Evarmap @ A_27a ) )
=> ! [V2l: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V3s: $i] :
( ( mem @ V3s @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Einterp__cs @ A_27a ) @ V0sr ) @ V1vm ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Evarlist__insert @ A_27a ) @ V0sr ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V2l ) ) @ V3s ) )
= ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP @ A_27a ) @ V0sr ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Einterp__vl @ A_27a ) @ V0sr ) @ V1vm ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V2l ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Einterp__cs @ A_27a ) @ V0sr ) @ V1vm ) @ V3s ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecanonical_2Ecanonical__sum__scalar__ok,axiom,
! [A_27a: del,V0sr: $i] :
( ( mem @ V0sr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Esemi__ring_2Eis__semi__ring @ A_27a ) @ V0sr ) )
=> ! [V1vm: $i] :
( ( mem @ V1vm @ ( ty_2Equote_2Evarmap @ A_27a ) )
=> ! [V2a: $i] :
( ( mem @ V2a @ A_27a )
=> ! [V3s: $i] :
( ( mem @ V3s @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Einterp__cs @ A_27a ) @ V0sr ) @ V1vm ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Ecanonical__sum__scalar @ A_27a ) @ V0sr ) @ V2a ) @ V3s ) )
= ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM @ A_27a ) @ V0sr ) @ V2a ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Einterp__cs @ A_27a ) @ V0sr ) @ V1vm ) @ V3s ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecanonical_2Ecanonical__sum__scalar2__ok,axiom,
! [A_27a: del,V0sr: $i] :
( ( mem @ V0sr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Esemi__ring_2Eis__semi__ring @ A_27a ) @ V0sr ) )
=> ! [V1vm: $i] :
( ( mem @ V1vm @ ( ty_2Equote_2Evarmap @ A_27a ) )
=> ! [V2l: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V3s: $i] :
( ( mem @ V3s @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Einterp__cs @ A_27a ) @ V0sr ) @ V1vm ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Ecanonical__sum__scalar2 @ A_27a ) @ V0sr ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V2l ) ) @ V3s ) )
= ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM @ A_27a ) @ V0sr ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Einterp__vl @ A_27a ) @ V0sr ) @ V1vm ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V2l ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Einterp__cs @ A_27a ) @ V0sr ) @ V1vm ) @ V3s ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecanonical_2Ecanonical__sum__scalar3__ok,axiom,
! [A_27a: del,V0sr: $i] :
( ( mem @ V0sr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Esemi__ring_2Eis__semi__ring @ A_27a ) @ V0sr ) )
=> ! [V1vm: $i] :
( ( mem @ V1vm @ ( ty_2Equote_2Evarmap @ A_27a ) )
=> ! [V2c: $i] :
( ( mem @ V2c @ A_27a )
=> ! [V3l: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V4s: $i] :
( ( mem @ V4s @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Einterp__cs @ A_27a ) @ V0sr ) @ V1vm ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Ecanonical__sum__scalar3 @ A_27a ) @ V0sr ) @ V2c ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V3l ) ) @ V4s ) )
= ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM @ A_27a ) @ V0sr ) @ ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM @ A_27a ) @ V0sr ) @ V2c ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Einterp__vl @ A_27a ) @ V0sr ) @ V1vm ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V3l ) ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Einterp__cs @ A_27a ) @ V0sr ) @ V1vm ) @ V4s ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecanonical_2Ecanonical__sum__prod__ok,axiom,
! [A_27a: del,V0sr: $i] :
( ( mem @ V0sr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Esemi__ring_2Eis__semi__ring @ A_27a ) @ V0sr ) )
=> ! [V1vm: $i] :
( ( mem @ V1vm @ ( ty_2Equote_2Evarmap @ A_27a ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ! [V3y: $i] :
( ( mem @ V3y @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Einterp__cs @ A_27a ) @ V0sr ) @ V1vm ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Ecanonical__sum__prod @ A_27a ) @ V0sr ) @ V2x ) @ V3y ) )
= ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM @ A_27a ) @ V0sr ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Einterp__cs @ A_27a ) @ V0sr ) @ V1vm ) @ V2x ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Einterp__cs @ A_27a ) @ V0sr ) @ V1vm ) @ V3y ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecanonical_2Ecanonical__sum__simplify__ok,axiom,
! [A_27a: del,V0sr: $i] :
( ( mem @ V0sr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Esemi__ring_2Eis__semi__ring @ A_27a ) @ V0sr ) )
=> ! [V1vm: $i] :
( ( mem @ V1vm @ ( ty_2Equote_2Evarmap @ A_27a ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Einterp__cs @ A_27a ) @ V0sr ) @ V1vm ) @ ( ap @ ( ap @ ( c_2Ecanonical_2Ecanonical__sum__simplify @ A_27a ) @ V0sr ) @ V2s ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Einterp__cs @ A_27a ) @ V0sr ) @ V1vm ) @ V2s ) ) ) ) ) ) ).
thf(ax_thm_2Ecanonical_2Espolynom__TY__DEF,axiom,
! [A_27a: del] :
? [V0rep: $i] :
( ( mem @ V0rep @ ( arr @ ( ty_2Ecanonical_2Espolynom @ A_27a ) @ ( ty_2Eind__type_2Erecspace @ ( ty_2Epair_2Eprod @ ty_2Equote_2Eindex @ A_27a ) ) ) )
& ( p
@ ( ap
@ ( ap @ ( c_2Ebool_2ETYPE__DEFINITION @ ( ty_2Eind__type_2Erecspace @ ( ty_2Epair_2Eprod @ ty_2Equote_2Eindex @ A_27a ) ) @ ( ty_2Ecanonical_2Espolynom @ A_27a ) )
@ ( lam @ ( ty_2Eind__type_2Erecspace @ ( ty_2Epair_2Eprod @ ty_2Equote_2Eindex @ A_27a ) )
@ ^ [V1a0_27: $i] :
( ap @ ( c_2Ebool_2E_21 @ ( arr @ ( ty_2Eind__type_2Erecspace @ ( ty_2Epair_2Eprod @ ty_2Equote_2Eindex @ A_27a ) ) @ bool ) )
@ ( lam @ ( arr @ ( ty_2Eind__type_2Erecspace @ ( ty_2Epair_2Eprod @ ty_2Equote_2Eindex @ A_27a ) ) @ bool )
@ ^ [V2_27spolynom_27: $i] :
( ap
@ ( ap @ c_2Emin_2E_3D_3D_3E
@ ( ap @ ( c_2Ebool_2E_21 @ ( ty_2Eind__type_2Erecspace @ ( ty_2Epair_2Eprod @ ty_2Equote_2Eindex @ A_27a ) ) )
@ ( lam @ ( ty_2Eind__type_2Erecspace @ ( ty_2Epair_2Eprod @ ty_2Equote_2Eindex @ A_27a ) )
@ ^ [V3a0_27: $i] :
( ap
@ ( ap @ c_2Emin_2E_3D_3D_3E
@ ( ap
@ ( ap @ c_2Ebool_2E_5C_2F
@ ( ap @ ( c_2Ebool_2E_3F @ ty_2Equote_2Eindex )
@ ( lam @ ty_2Equote_2Eindex
@ ^ [V4a: $i] :
( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Eind__type_2Erecspace @ ( ty_2Epair_2Eprod @ ty_2Equote_2Eindex @ A_27a ) ) ) @ V3a0_27 )
@ ( ap
@ ( lam @ ty_2Equote_2Eindex
@ ^ [V5a: $i] :
( ap @ ( ap @ ( ap @ ( c_2Eind__type_2ECONSTR @ ( ty_2Epair_2Eprod @ ty_2Equote_2Eindex @ A_27a ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Equote_2Eindex @ A_27a ) @ V5a ) @ ( c_2Ebool_2EARB @ A_27a ) ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V6n: $i] : ( c_2Eind__type_2EBOTTOM @ ( ty_2Epair_2Eprod @ ty_2Equote_2Eindex @ A_27a ) ) ) ) )
@ V4a ) ) ) ) )
@ ( ap
@ ( ap @ c_2Ebool_2E_5C_2F
@ ( ap @ ( c_2Ebool_2E_3F @ A_27a )
@ ( lam @ A_27a
@ ^ [V7a: $i] :
( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Eind__type_2Erecspace @ ( ty_2Epair_2Eprod @ ty_2Equote_2Eindex @ A_27a ) ) ) @ V3a0_27 )
@ ( ap
@ ( lam @ A_27a
@ ^ [V8a: $i] :
( ap @ ( ap @ ( ap @ ( c_2Eind__type_2ECONSTR @ ( ty_2Epair_2Eprod @ ty_2Equote_2Eindex @ A_27a ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Equote_2Eindex @ A_27a ) @ ( c_2Ebool_2EARB @ ty_2Equote_2Eindex ) ) @ V8a ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V9n: $i] : ( c_2Eind__type_2EBOTTOM @ ( ty_2Epair_2Eprod @ ty_2Equote_2Eindex @ A_27a ) ) ) ) )
@ V7a ) ) ) ) )
@ ( ap
@ ( ap @ c_2Ebool_2E_5C_2F
@ ( ap @ ( c_2Ebool_2E_3F @ ( ty_2Eind__type_2Erecspace @ ( ty_2Epair_2Eprod @ ty_2Equote_2Eindex @ A_27a ) ) )
@ ( lam @ ( ty_2Eind__type_2Erecspace @ ( ty_2Epair_2Eprod @ ty_2Equote_2Eindex @ A_27a ) )
@ ^ [V10a0: $i] :
( ap @ ( c_2Ebool_2E_3F @ ( ty_2Eind__type_2Erecspace @ ( ty_2Epair_2Eprod @ ty_2Equote_2Eindex @ A_27a ) ) )
@ ( lam @ ( ty_2Eind__type_2Erecspace @ ( ty_2Epair_2Eprod @ ty_2Equote_2Eindex @ A_27a ) )
@ ^ [V11a1: $i] :
( ap
@ ( ap @ c_2Ebool_2E_2F_5C
@ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Eind__type_2Erecspace @ ( ty_2Epair_2Eprod @ ty_2Equote_2Eindex @ A_27a ) ) ) @ V3a0_27 )
@ ( ap
@ ( ap
@ ( lam @ ( ty_2Eind__type_2Erecspace @ ( ty_2Epair_2Eprod @ ty_2Equote_2Eindex @ A_27a ) )
@ ^ [V12a0: $i] :
( lam @ ( ty_2Eind__type_2Erecspace @ ( ty_2Epair_2Eprod @ ty_2Equote_2Eindex @ A_27a ) )
@ ^ [V13a1: $i] :
( ap @ ( ap @ ( ap @ ( c_2Eind__type_2ECONSTR @ ( ty_2Epair_2Eprod @ ty_2Equote_2Eindex @ A_27a ) ) @ ( ap @ c_2Enum_2ESUC @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Equote_2Eindex @ A_27a ) @ ( c_2Ebool_2EARB @ ty_2Equote_2Eindex ) ) @ ( c_2Ebool_2EARB @ A_27a ) ) )
@ ( ap @ ( ap @ ( c_2Eind__type_2EFCONS @ ( ty_2Eind__type_2Erecspace @ ( ty_2Epair_2Eprod @ ty_2Equote_2Eindex @ A_27a ) ) ) @ V12a0 )
@ ( ap @ ( ap @ ( c_2Eind__type_2EFCONS @ ( ty_2Eind__type_2Erecspace @ ( ty_2Epair_2Eprod @ ty_2Equote_2Eindex @ A_27a ) ) ) @ V13a1 )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V14n: $i] : ( c_2Eind__type_2EBOTTOM @ ( ty_2Epair_2Eprod @ ty_2Equote_2Eindex @ A_27a ) ) ) ) ) ) ) )
@ V10a0 )
@ V11a1 ) ) )
@ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ V2_27spolynom_27 @ V10a0 ) ) @ ( ap @ V2_27spolynom_27 @ V11a1 ) ) ) ) ) ) ) )
@ ( ap @ ( c_2Ebool_2E_3F @ ( ty_2Eind__type_2Erecspace @ ( ty_2Epair_2Eprod @ ty_2Equote_2Eindex @ A_27a ) ) )
@ ( lam @ ( ty_2Eind__type_2Erecspace @ ( ty_2Epair_2Eprod @ ty_2Equote_2Eindex @ A_27a ) )
@ ^ [V15a0: $i] :
( ap @ ( c_2Ebool_2E_3F @ ( ty_2Eind__type_2Erecspace @ ( ty_2Epair_2Eprod @ ty_2Equote_2Eindex @ A_27a ) ) )
@ ( lam @ ( ty_2Eind__type_2Erecspace @ ( ty_2Epair_2Eprod @ ty_2Equote_2Eindex @ A_27a ) )
@ ^ [V16a1: $i] :
( ap
@ ( ap @ c_2Ebool_2E_2F_5C
@ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Eind__type_2Erecspace @ ( ty_2Epair_2Eprod @ ty_2Equote_2Eindex @ A_27a ) ) ) @ V3a0_27 )
@ ( ap
@ ( ap
@ ( lam @ ( ty_2Eind__type_2Erecspace @ ( ty_2Epair_2Eprod @ ty_2Equote_2Eindex @ A_27a ) )
@ ^ [V17a0: $i] :
( lam @ ( ty_2Eind__type_2Erecspace @ ( ty_2Epair_2Eprod @ ty_2Equote_2Eindex @ A_27a ) )
@ ^ [V18a1: $i] :
( ap @ ( ap @ ( ap @ ( c_2Eind__type_2ECONSTR @ ( ty_2Epair_2Eprod @ ty_2Equote_2Eindex @ A_27a ) ) @ ( ap @ c_2Enum_2ESUC @ ( ap @ c_2Enum_2ESUC @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Equote_2Eindex @ A_27a ) @ ( c_2Ebool_2EARB @ ty_2Equote_2Eindex ) ) @ ( c_2Ebool_2EARB @ A_27a ) ) )
@ ( ap @ ( ap @ ( c_2Eind__type_2EFCONS @ ( ty_2Eind__type_2Erecspace @ ( ty_2Epair_2Eprod @ ty_2Equote_2Eindex @ A_27a ) ) ) @ V17a0 )
@ ( ap @ ( ap @ ( c_2Eind__type_2EFCONS @ ( ty_2Eind__type_2Erecspace @ ( ty_2Epair_2Eprod @ ty_2Equote_2Eindex @ A_27a ) ) ) @ V18a1 )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V19n: $i] : ( c_2Eind__type_2EBOTTOM @ ( ty_2Epair_2Eprod @ ty_2Equote_2Eindex @ A_27a ) ) ) ) ) ) ) )
@ V15a0 )
@ V16a1 ) ) )
@ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ V2_27spolynom_27 @ V15a0 ) ) @ ( ap @ V2_27spolynom_27 @ V16a1 ) ) ) ) ) ) ) ) ) ) )
@ ( ap @ V2_27spolynom_27 @ V3a0_27 ) ) ) ) )
@ ( ap @ V2_27spolynom_27 @ V1a0_27 ) ) ) ) ) )
@ V0rep ) ) ) ).
thf(ax_thm_2Ecanonical_2Espolynom__case__def,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0a: tp__ty_2Equote_2Eindex,V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Equote_2Eindex @ A_27b ) )
=> ! [V2f1: $i] :
( ( mem @ V2f1 @ ( arr @ A_27a @ A_27b ) )
=> ! [V3f2: $i] :
( ( mem @ V3f2 @ ( arr @ ( ty_2Ecanonical_2Espolynom @ A_27a ) @ ( arr @ ( ty_2Ecanonical_2Espolynom @ A_27a ) @ A_27b ) ) )
=> ! [V4f3: $i] :
( ( mem @ V4f3 @ ( arr @ ( ty_2Ecanonical_2Espolynom @ A_27a ) @ ( arr @ ( ty_2Ecanonical_2Espolynom @ A_27a ) @ A_27b ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Espolynom__CASE @ A_27a @ A_27b ) @ ( ap @ ( c_2Ecanonical_2ESPvar @ A_27a ) @ ( inj__ty_2Equote_2Eindex @ V0a ) ) ) @ V1f ) @ V2f1 ) @ V3f2 ) @ V4f3 )
= ( ap @ V1f @ ( inj__ty_2Equote_2Eindex @ V0a ) ) ) ) ) ) )
& ! [V5a: $i] :
( ( mem @ V5a @ A_27a )
=> ! [V6f: $i] :
( ( mem @ V6f @ ( arr @ ty_2Equote_2Eindex @ A_27b ) )
=> ! [V7f1: $i] :
( ( mem @ V7f1 @ ( arr @ A_27a @ A_27b ) )
=> ! [V8f2: $i] :
( ( mem @ V8f2 @ ( arr @ ( ty_2Ecanonical_2Espolynom @ A_27a ) @ ( arr @ ( ty_2Ecanonical_2Espolynom @ A_27a ) @ A_27b ) ) )
=> ! [V9f3: $i] :
( ( mem @ V9f3 @ ( arr @ ( ty_2Ecanonical_2Espolynom @ A_27a ) @ ( arr @ ( ty_2Ecanonical_2Espolynom @ A_27a ) @ A_27b ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Espolynom__CASE @ A_27a @ A_27b ) @ ( ap @ ( c_2Ecanonical_2ESPconst @ A_27a ) @ V5a ) ) @ V6f ) @ V7f1 ) @ V8f2 ) @ V9f3 )
= ( ap @ V7f1 @ V5a ) ) ) ) ) ) )
& ! [V10a0: $i] :
( ( mem @ V10a0 @ ( ty_2Ecanonical_2Espolynom @ A_27a ) )
=> ! [V11a1: $i] :
( ( mem @ V11a1 @ ( ty_2Ecanonical_2Espolynom @ A_27a ) )
=> ! [V12f: $i] :
( ( mem @ V12f @ ( arr @ ty_2Equote_2Eindex @ A_27b ) )
=> ! [V13f1: $i] :
( ( mem @ V13f1 @ ( arr @ A_27a @ A_27b ) )
=> ! [V14f2: $i] :
( ( mem @ V14f2 @ ( arr @ ( ty_2Ecanonical_2Espolynom @ A_27a ) @ ( arr @ ( ty_2Ecanonical_2Espolynom @ A_27a ) @ A_27b ) ) )
=> ! [V15f3: $i] :
( ( mem @ V15f3 @ ( arr @ ( ty_2Ecanonical_2Espolynom @ A_27a ) @ ( arr @ ( ty_2Ecanonical_2Espolynom @ A_27a ) @ A_27b ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Espolynom__CASE @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Ecanonical_2ESPplus @ A_27a ) @ V10a0 ) @ V11a1 ) ) @ V12f ) @ V13f1 ) @ V14f2 ) @ V15f3 )
= ( ap @ ( ap @ V14f2 @ V10a0 ) @ V11a1 ) ) ) ) ) ) ) )
& ! [V16a0: $i] :
( ( mem @ V16a0 @ ( ty_2Ecanonical_2Espolynom @ A_27a ) )
=> ! [V17a1: $i] :
( ( mem @ V17a1 @ ( ty_2Ecanonical_2Espolynom @ A_27a ) )
=> ! [V18f: $i] :
( ( mem @ V18f @ ( arr @ ty_2Equote_2Eindex @ A_27b ) )
=> ! [V19f1: $i] :
( ( mem @ V19f1 @ ( arr @ A_27a @ A_27b ) )
=> ! [V20f2: $i] :
( ( mem @ V20f2 @ ( arr @ ( ty_2Ecanonical_2Espolynom @ A_27a ) @ ( arr @ ( ty_2Ecanonical_2Espolynom @ A_27a ) @ A_27b ) ) )
=> ! [V21f3: $i] :
( ( mem @ V21f3 @ ( arr @ ( ty_2Ecanonical_2Espolynom @ A_27a ) @ ( arr @ ( ty_2Ecanonical_2Espolynom @ A_27a ) @ A_27b ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Espolynom__CASE @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Ecanonical_2ESPmult @ A_27a ) @ V16a0 ) @ V17a1 ) ) @ V18f ) @ V19f1 ) @ V20f2 ) @ V21f3 )
= ( ap @ ( ap @ V21f3 @ V16a0 ) @ V17a1 ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Ecanonical_2Espolynom__size__def,axiom,
! [A_27a: del] :
( ! [V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1a: tp__ty_2Equote_2Eindex] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Ecanonical_2Espolynom__size @ A_27a ) @ V0f ) @ ( ap @ ( c_2Ecanonical_2ESPvar @ A_27a ) @ ( inj__ty_2Equote_2Eindex @ V1a ) ) ) )
= ( 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 @ c_2Equote_2Eindex__size @ ( inj__ty_2Equote_2Eindex @ V1a ) ) ) ) ) )
& ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V3a: $i] :
( ( mem @ V3a @ A_27a )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Ecanonical_2Espolynom__size @ A_27a ) @ V2f ) @ ( ap @ ( c_2Ecanonical_2ESPconst @ A_27a ) @ V3a ) ) )
= ( 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 @ V2f @ V3a ) ) ) ) ) )
& ! [V4f: $i] :
( ( mem @ V4f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V5a0: $i] :
( ( mem @ V5a0 @ ( ty_2Ecanonical_2Espolynom @ A_27a ) )
=> ! [V6a1: $i] :
( ( mem @ V6a1 @ ( ty_2Ecanonical_2Espolynom @ A_27a ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Ecanonical_2Espolynom__size @ A_27a ) @ V4f ) @ ( ap @ ( ap @ ( c_2Ecanonical_2ESPplus @ A_27a ) @ V5a0 ) @ V6a1 ) ) )
= ( 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 @ ( ap @ ( c_2Ecanonical_2Espolynom__size @ A_27a ) @ V4f ) @ V5a0 ) ) @ ( ap @ ( ap @ ( c_2Ecanonical_2Espolynom__size @ A_27a ) @ V4f ) @ V6a1 ) ) ) ) ) ) ) )
& ! [V7f: $i] :
( ( mem @ V7f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V8a0: $i] :
( ( mem @ V8a0 @ ( ty_2Ecanonical_2Espolynom @ A_27a ) )
=> ! [V9a1: $i] :
( ( mem @ V9a1 @ ( ty_2Ecanonical_2Espolynom @ A_27a ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Ecanonical_2Espolynom__size @ A_27a ) @ V7f ) @ ( ap @ ( ap @ ( c_2Ecanonical_2ESPmult @ A_27a ) @ V8a0 ) @ V9a1 ) ) )
= ( 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 @ ( ap @ ( c_2Ecanonical_2Espolynom__size @ A_27a ) @ V7f ) @ V8a0 ) ) @ ( ap @ ( ap @ ( c_2Ecanonical_2Espolynom__size @ A_27a ) @ V7f ) @ V9a1 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecanonical_2Edatatype__spolynom,axiom,
! [A_27a: del,V0spolynom: $i] :
( ( mem @ V0spolynom @ ( arr @ ( arr @ ty_2Equote_2Eindex @ ( ty_2Ecanonical_2Espolynom @ A_27a ) ) @ ( arr @ ( arr @ A_27a @ ( ty_2Ecanonical_2Espolynom @ A_27a ) ) @ ( arr @ ( arr @ ( ty_2Ecanonical_2Espolynom @ A_27a ) @ ( arr @ ( ty_2Ecanonical_2Espolynom @ A_27a ) @ ( ty_2Ecanonical_2Espolynom @ A_27a ) ) ) @ ( arr @ ( arr @ ( ty_2Ecanonical_2Espolynom @ A_27a ) @ ( arr @ ( ty_2Ecanonical_2Espolynom @ A_27a ) @ ( ty_2Ecanonical_2Espolynom @ A_27a ) ) ) @ bool ) ) ) ) )
=> ( p @ ( ap @ ( c_2Ebool_2EDATATYPE @ bool ) @ ( ap @ ( ap @ ( ap @ ( ap @ V0spolynom @ ( c_2Ecanonical_2ESPvar @ A_27a ) ) @ ( c_2Ecanonical_2ESPconst @ A_27a ) ) @ ( c_2Ecanonical_2ESPplus @ A_27a ) ) @ ( c_2Ecanonical_2ESPmult @ A_27a ) ) ) ) ) ).
thf(conj_thm_2Ecanonical_2Espolynom__11,axiom,
! [A_27a: del] :
( ! [V0a: tp__ty_2Equote_2Eindex,V1a_27: tp__ty_2Equote_2Eindex] :
( ( ( ap @ ( c_2Ecanonical_2ESPvar @ A_27a ) @ ( inj__ty_2Equote_2Eindex @ V0a ) )
= ( ap @ ( c_2Ecanonical_2ESPvar @ A_27a ) @ ( inj__ty_2Equote_2Eindex @ V1a_27 ) ) )
<=> ( V0a = V1a_27 ) )
& ! [V2a: $i] :
( ( mem @ V2a @ A_27a )
=> ! [V3a_27: $i] :
( ( mem @ V3a_27 @ A_27a )
=> ( ( ( ap @ ( c_2Ecanonical_2ESPconst @ A_27a ) @ V2a )
= ( ap @ ( c_2Ecanonical_2ESPconst @ A_27a ) @ V3a_27 ) )
<=> ( V2a = V3a_27 ) ) ) )
& ! [V4a0: $i] :
( ( mem @ V4a0 @ ( ty_2Ecanonical_2Espolynom @ A_27a ) )
=> ! [V5a1: $i] :
( ( mem @ V5a1 @ ( ty_2Ecanonical_2Espolynom @ A_27a ) )
=> ! [V6a0_27: $i] :
( ( mem @ V6a0_27 @ ( ty_2Ecanonical_2Espolynom @ A_27a ) )
=> ! [V7a1_27: $i] :
( ( mem @ V7a1_27 @ ( ty_2Ecanonical_2Espolynom @ A_27a ) )
=> ( ( ( ap @ ( ap @ ( c_2Ecanonical_2ESPplus @ A_27a ) @ V4a0 ) @ V5a1 )
= ( ap @ ( ap @ ( c_2Ecanonical_2ESPplus @ A_27a ) @ V6a0_27 ) @ V7a1_27 ) )
<=> ( ( V4a0 = V6a0_27 )
& ( V5a1 = V7a1_27 ) ) ) ) ) ) )
& ! [V8a0: $i] :
( ( mem @ V8a0 @ ( ty_2Ecanonical_2Espolynom @ A_27a ) )
=> ! [V9a1: $i] :
( ( mem @ V9a1 @ ( ty_2Ecanonical_2Espolynom @ A_27a ) )
=> ! [V10a0_27: $i] :
( ( mem @ V10a0_27 @ ( ty_2Ecanonical_2Espolynom @ A_27a ) )
=> ! [V11a1_27: $i] :
( ( mem @ V11a1_27 @ ( ty_2Ecanonical_2Espolynom @ A_27a ) )
=> ( ( ( ap @ ( ap @ ( c_2Ecanonical_2ESPmult @ A_27a ) @ V8a0 ) @ V9a1 )
= ( ap @ ( ap @ ( c_2Ecanonical_2ESPmult @ A_27a ) @ V10a0_27 ) @ V11a1_27 ) )
<=> ( ( V8a0 = V10a0_27 )
& ( V9a1 = V11a1_27 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecanonical_2Espolynom__distinct,axiom,
! [A_27a: del] :
( ! [V0a_27: $i] :
( ( mem @ V0a_27 @ A_27a )
=> ! [V1a: tp__ty_2Equote_2Eindex] :
( ( ap @ ( c_2Ecanonical_2ESPvar @ A_27a ) @ ( inj__ty_2Equote_2Eindex @ V1a ) )
!= ( ap @ ( c_2Ecanonical_2ESPconst @ A_27a ) @ V0a_27 ) ) )
& ! [V2a1: $i] :
( ( mem @ V2a1 @ ( ty_2Ecanonical_2Espolynom @ A_27a ) )
=> ! [V3a0: $i] :
( ( mem @ V3a0 @ ( ty_2Ecanonical_2Espolynom @ A_27a ) )
=> ! [V4a: tp__ty_2Equote_2Eindex] :
( ( ap @ ( c_2Ecanonical_2ESPvar @ A_27a ) @ ( inj__ty_2Equote_2Eindex @ V4a ) )
!= ( ap @ ( ap @ ( c_2Ecanonical_2ESPplus @ A_27a ) @ V3a0 ) @ V2a1 ) ) ) )
& ! [V5a1: $i] :
( ( mem @ V5a1 @ ( ty_2Ecanonical_2Espolynom @ A_27a ) )
=> ! [V6a0: $i] :
( ( mem @ V6a0 @ ( ty_2Ecanonical_2Espolynom @ A_27a ) )
=> ! [V7a: tp__ty_2Equote_2Eindex] :
( ( ap @ ( c_2Ecanonical_2ESPvar @ A_27a ) @ ( inj__ty_2Equote_2Eindex @ V7a ) )
!= ( ap @ ( ap @ ( c_2Ecanonical_2ESPmult @ A_27a ) @ V6a0 ) @ V5a1 ) ) ) )
& ! [V8a1: $i] :
( ( mem @ V8a1 @ ( ty_2Ecanonical_2Espolynom @ A_27a ) )
=> ! [V9a0: $i] :
( ( mem @ V9a0 @ ( ty_2Ecanonical_2Espolynom @ A_27a ) )
=> ! [V10a: $i] :
( ( mem @ V10a @ A_27a )
=> ( ( ap @ ( c_2Ecanonical_2ESPconst @ A_27a ) @ V10a )
!= ( ap @ ( ap @ ( c_2Ecanonical_2ESPplus @ A_27a ) @ V9a0 ) @ V8a1 ) ) ) ) )
& ! [V11a1: $i] :
( ( mem @ V11a1 @ ( ty_2Ecanonical_2Espolynom @ A_27a ) )
=> ! [V12a0: $i] :
( ( mem @ V12a0 @ ( ty_2Ecanonical_2Espolynom @ A_27a ) )
=> ! [V13a: $i] :
( ( mem @ V13a @ A_27a )
=> ( ( ap @ ( c_2Ecanonical_2ESPconst @ A_27a ) @ V13a )
!= ( ap @ ( ap @ ( c_2Ecanonical_2ESPmult @ A_27a ) @ V12a0 ) @ V11a1 ) ) ) ) )
& ! [V14a1_27: $i] :
( ( mem @ V14a1_27 @ ( ty_2Ecanonical_2Espolynom @ A_27a ) )
=> ! [V15a1: $i] :
( ( mem @ V15a1 @ ( ty_2Ecanonical_2Espolynom @ A_27a ) )
=> ! [V16a0_27: $i] :
( ( mem @ V16a0_27 @ ( ty_2Ecanonical_2Espolynom @ A_27a ) )
=> ! [V17a0: $i] :
( ( mem @ V17a0 @ ( ty_2Ecanonical_2Espolynom @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Ecanonical_2ESPplus @ A_27a ) @ V17a0 ) @ V15a1 )
!= ( ap @ ( ap @ ( c_2Ecanonical_2ESPmult @ A_27a ) @ V16a0_27 ) @ V14a1_27 ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecanonical_2Espolynom__nchotomy,axiom,
! [A_27a: del,V0ss: $i] :
( ( mem @ V0ss @ ( ty_2Ecanonical_2Espolynom @ A_27a ) )
=> ( ? [V1i: tp__ty_2Equote_2Eindex] :
( V0ss
= ( ap @ ( c_2Ecanonical_2ESPvar @ A_27a ) @ ( inj__ty_2Equote_2Eindex @ V1i ) ) )
| ? [V2a: $i] :
( ( mem @ V2a @ A_27a )
& ( V0ss
= ( ap @ ( c_2Ecanonical_2ESPconst @ A_27a ) @ V2a ) ) )
| ? [V3s: $i] :
( ( mem @ V3s @ ( ty_2Ecanonical_2Espolynom @ A_27a ) )
& ? [V4s0: $i] :
( ( mem @ V4s0 @ ( ty_2Ecanonical_2Espolynom @ A_27a ) )
& ( V0ss
= ( ap @ ( ap @ ( c_2Ecanonical_2ESPplus @ A_27a ) @ V3s ) @ V4s0 ) ) ) )
| ? [V5s: $i] :
( ( mem @ V5s @ ( ty_2Ecanonical_2Espolynom @ A_27a ) )
& ? [V6s0: $i] :
( ( mem @ V6s0 @ ( ty_2Ecanonical_2Espolynom @ A_27a ) )
& ( V0ss
= ( ap @ ( ap @ ( c_2Ecanonical_2ESPmult @ A_27a ) @ V5s ) @ V6s0 ) ) ) ) ) ) ).
thf(conj_thm_2Ecanonical_2Espolynom__Axiom,axiom,
! [A_27a: del,A_27b: del,V0f0: $i] :
( ( mem @ V0f0 @ ( arr @ ty_2Equote_2Eindex @ A_27b ) )
=> ! [V1f1: $i] :
( ( mem @ V1f1 @ ( arr @ A_27a @ A_27b ) )
=> ! [V2f2: $i] :
( ( mem @ V2f2 @ ( arr @ ( ty_2Ecanonical_2Espolynom @ A_27a ) @ ( arr @ ( ty_2Ecanonical_2Espolynom @ A_27a ) @ ( arr @ A_27b @ ( arr @ A_27b @ A_27b ) ) ) ) )
=> ! [V3f3: $i] :
( ( mem @ V3f3 @ ( arr @ ( ty_2Ecanonical_2Espolynom @ A_27a ) @ ( arr @ ( ty_2Ecanonical_2Espolynom @ A_27a ) @ ( arr @ A_27b @ ( arr @ A_27b @ A_27b ) ) ) ) )
=> ? [V4fn: $i] :
( ( mem @ V4fn @ ( arr @ ( ty_2Ecanonical_2Espolynom @ A_27a ) @ A_27b ) )
& ! [V5a: tp__ty_2Equote_2Eindex] :
( ( ap @ V4fn @ ( ap @ ( c_2Ecanonical_2ESPvar @ A_27a ) @ ( inj__ty_2Equote_2Eindex @ V5a ) ) )
= ( ap @ V0f0 @ ( inj__ty_2Equote_2Eindex @ V5a ) ) )
& ! [V6a: $i] :
( ( mem @ V6a @ A_27a )
=> ( ( ap @ V4fn @ ( ap @ ( c_2Ecanonical_2ESPconst @ A_27a ) @ V6a ) )
= ( ap @ V1f1 @ V6a ) ) )
& ! [V7a0: $i] :
( ( mem @ V7a0 @ ( ty_2Ecanonical_2Espolynom @ A_27a ) )
=> ! [V8a1: $i] :
( ( mem @ V8a1 @ ( ty_2Ecanonical_2Espolynom @ A_27a ) )
=> ( ( ap @ V4fn @ ( ap @ ( ap @ ( c_2Ecanonical_2ESPplus @ A_27a ) @ V7a0 ) @ V8a1 ) )
= ( ap @ ( ap @ ( ap @ ( ap @ V2f2 @ V7a0 ) @ V8a1 ) @ ( ap @ V4fn @ V7a0 ) ) @ ( ap @ V4fn @ V8a1 ) ) ) ) )
& ! [V9a0: $i] :
( ( mem @ V9a0 @ ( ty_2Ecanonical_2Espolynom @ A_27a ) )
=> ! [V10a1: $i] :
( ( mem @ V10a1 @ ( ty_2Ecanonical_2Espolynom @ A_27a ) )
=> ( ( ap @ V4fn @ ( ap @ ( ap @ ( c_2Ecanonical_2ESPmult @ A_27a ) @ V9a0 ) @ V10a1 ) )
= ( ap @ ( ap @ ( ap @ ( ap @ V3f3 @ V9a0 ) @ V10a1 ) @ ( ap @ V4fn @ V9a0 ) ) @ ( ap @ V4fn @ V10a1 ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecanonical_2Espolynom__induction,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Ecanonical_2Espolynom @ A_27a ) @ bool ) )
=> ( ( ! [V1i: tp__ty_2Equote_2Eindex] : ( p @ ( ap @ V0P @ ( ap @ ( c_2Ecanonical_2ESPvar @ A_27a ) @ ( inj__ty_2Equote_2Eindex @ V1i ) ) ) )
& ! [V2a: $i] :
( ( mem @ V2a @ A_27a )
=> ( p @ ( ap @ V0P @ ( ap @ ( c_2Ecanonical_2ESPconst @ A_27a ) @ V2a ) ) ) )
& ! [V3s: $i] :
( ( mem @ V3s @ ( ty_2Ecanonical_2Espolynom @ A_27a ) )
=> ! [V4s0: $i] :
( ( mem @ V4s0 @ ( ty_2Ecanonical_2Espolynom @ A_27a ) )
=> ( ( ( p @ ( ap @ V0P @ V3s ) )
& ( p @ ( ap @ V0P @ V4s0 ) ) )
=> ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Ecanonical_2ESPplus @ A_27a ) @ V3s ) @ V4s0 ) ) ) ) ) )
& ! [V5s: $i] :
( ( mem @ V5s @ ( ty_2Ecanonical_2Espolynom @ A_27a ) )
=> ! [V6s0: $i] :
( ( mem @ V6s0 @ ( ty_2Ecanonical_2Espolynom @ A_27a ) )
=> ( ( ( p @ ( ap @ V0P @ V5s ) )
& ( p @ ( ap @ V0P @ V6s0 ) ) )
=> ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Ecanonical_2ESPmult @ A_27a ) @ V5s ) @ V6s0 ) ) ) ) ) ) )
=> ! [V7s: $i] :
( ( mem @ V7s @ ( ty_2Ecanonical_2Espolynom @ A_27a ) )
=> ( p @ ( ap @ V0P @ V7s ) ) ) ) ) ).
thf(conj_thm_2Ecanonical_2Espolynom__case__cong,axiom,
! [A_27a: del,A_27b: del,V0f_27: $i] :
( ( mem @ V0f_27 @ ( arr @ ty_2Equote_2Eindex @ A_27b ) )
=> ! [V1f1_27: $i] :
( ( mem @ V1f1_27 @ ( arr @ A_27a @ A_27b ) )
=> ! [V2f2_27: $i] :
( ( mem @ V2f2_27 @ ( arr @ ( ty_2Ecanonical_2Espolynom @ A_27a ) @ ( arr @ ( ty_2Ecanonical_2Espolynom @ A_27a ) @ A_27b ) ) )
=> ! [V3f3_27: $i] :
( ( mem @ V3f3_27 @ ( arr @ ( ty_2Ecanonical_2Espolynom @ A_27a ) @ ( arr @ ( ty_2Ecanonical_2Espolynom @ A_27a ) @ A_27b ) ) )
=> ! [V4M: $i] :
( ( mem @ V4M @ ( ty_2Ecanonical_2Espolynom @ A_27a ) )
=> ! [V5M_27: $i] :
( ( mem @ V5M_27 @ ( ty_2Ecanonical_2Espolynom @ A_27a ) )
=> ! [V6f: $i] :
( ( mem @ V6f @ ( arr @ ty_2Equote_2Eindex @ A_27b ) )
=> ! [V7f1: $i] :
( ( mem @ V7f1 @ ( arr @ A_27a @ A_27b ) )
=> ! [V8f2: $i] :
( ( mem @ V8f2 @ ( arr @ ( ty_2Ecanonical_2Espolynom @ A_27a ) @ ( arr @ ( ty_2Ecanonical_2Espolynom @ A_27a ) @ A_27b ) ) )
=> ! [V9f3: $i] :
( ( mem @ V9f3 @ ( arr @ ( ty_2Ecanonical_2Espolynom @ A_27a ) @ ( arr @ ( ty_2Ecanonical_2Espolynom @ A_27a ) @ A_27b ) ) )
=> ( ( ( V4M = V5M_27 )
& ! [V10a: tp__ty_2Equote_2Eindex] :
( ( V5M_27
= ( ap @ ( c_2Ecanonical_2ESPvar @ A_27a ) @ ( inj__ty_2Equote_2Eindex @ V10a ) ) )
=> ( ( ap @ V6f @ ( inj__ty_2Equote_2Eindex @ V10a ) )
= ( ap @ V0f_27 @ ( inj__ty_2Equote_2Eindex @ V10a ) ) ) )
& ! [V11a: $i] :
( ( mem @ V11a @ A_27a )
=> ( ( V5M_27
= ( ap @ ( c_2Ecanonical_2ESPconst @ A_27a ) @ V11a ) )
=> ( ( ap @ V7f1 @ V11a )
= ( ap @ V1f1_27 @ V11a ) ) ) )
& ! [V12a0: $i] :
( ( mem @ V12a0 @ ( ty_2Ecanonical_2Espolynom @ A_27a ) )
=> ! [V13a1: $i] :
( ( mem @ V13a1 @ ( ty_2Ecanonical_2Espolynom @ A_27a ) )
=> ( ( V5M_27
= ( ap @ ( ap @ ( c_2Ecanonical_2ESPplus @ A_27a ) @ V12a0 ) @ V13a1 ) )
=> ( ( ap @ ( ap @ V8f2 @ V12a0 ) @ V13a1 )
= ( ap @ ( ap @ V2f2_27 @ V12a0 ) @ V13a1 ) ) ) ) )
& ! [V14a0: $i] :
( ( mem @ V14a0 @ ( ty_2Ecanonical_2Espolynom @ A_27a ) )
=> ! [V15a1: $i] :
( ( mem @ V15a1 @ ( ty_2Ecanonical_2Espolynom @ A_27a ) )
=> ( ( V5M_27
= ( ap @ ( ap @ ( c_2Ecanonical_2ESPmult @ A_27a ) @ V14a0 ) @ V15a1 ) )
=> ( ( ap @ ( ap @ V9f3 @ V14a0 ) @ V15a1 )
= ( ap @ ( ap @ V3f3_27 @ V14a0 ) @ V15a1 ) ) ) ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Espolynom__CASE @ A_27a @ A_27b ) @ V4M ) @ V6f ) @ V7f1 ) @ V8f2 ) @ V9f3 )
= ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Espolynom__CASE @ A_27a @ A_27b ) @ V5M_27 ) @ V0f_27 ) @ V1f1_27 ) @ V2f2_27 ) @ V3f3_27 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecanonical_2Espolynom__case__eq,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ ( ty_2Ecanonical_2Espolynom @ A_27a ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Equote_2Eindex @ A_27b ) )
=> ! [V2f1: $i] :
( ( mem @ V2f1 @ ( arr @ A_27a @ A_27b ) )
=> ! [V3f2: $i] :
( ( mem @ V3f2 @ ( arr @ ( ty_2Ecanonical_2Espolynom @ A_27a ) @ ( arr @ ( ty_2Ecanonical_2Espolynom @ A_27a ) @ A_27b ) ) )
=> ! [V4f3: $i] :
( ( mem @ V4f3 @ ( arr @ ( ty_2Ecanonical_2Espolynom @ A_27a ) @ ( arr @ ( ty_2Ecanonical_2Espolynom @ A_27a ) @ A_27b ) ) )
=> ! [V5v: $i] :
( ( mem @ V5v @ A_27b )
=> ( ( ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Espolynom__CASE @ A_27a @ A_27b ) @ V0x ) @ V1f ) @ V2f1 ) @ V3f2 ) @ V4f3 )
= V5v )
<=> ( ? [V6i: tp__ty_2Equote_2Eindex] :
( ( V0x
= ( ap @ ( c_2Ecanonical_2ESPvar @ A_27a ) @ ( inj__ty_2Equote_2Eindex @ V6i ) ) )
& ( ( ap @ V1f @ ( inj__ty_2Equote_2Eindex @ V6i ) )
= V5v ) )
| ? [V7a: $i] :
( ( mem @ V7a @ A_27a )
& ( V0x
= ( ap @ ( c_2Ecanonical_2ESPconst @ A_27a ) @ V7a ) )
& ( ( ap @ V2f1 @ V7a )
= V5v ) )
| ? [V8s: $i] :
( ( mem @ V8s @ ( ty_2Ecanonical_2Espolynom @ A_27a ) )
& ? [V9s0: $i] :
( ( mem @ V9s0 @ ( ty_2Ecanonical_2Espolynom @ A_27a ) )
& ( V0x
= ( ap @ ( ap @ ( c_2Ecanonical_2ESPplus @ A_27a ) @ V8s ) @ V9s0 ) )
& ( ( ap @ ( ap @ V3f2 @ V8s ) @ V9s0 )
= V5v ) ) )
| ? [V10s: $i] :
( ( mem @ V10s @ ( ty_2Ecanonical_2Espolynom @ A_27a ) )
& ? [V11s0: $i] :
( ( mem @ V11s0 @ ( ty_2Ecanonical_2Espolynom @ A_27a ) )
& ( V0x
= ( ap @ ( ap @ ( c_2Ecanonical_2ESPmult @ A_27a ) @ V10s ) @ V11s0 ) )
& ( ( ap @ ( ap @ V4f3 @ V10s ) @ V11s0 )
= V5v ) ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Ecanonical_2Espolynom__normalize__def,axiom,
! [A_27a: del] :
( ! [V0sr: $i] :
( ( mem @ V0sr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V1i: tp__ty_2Equote_2Eindex] :
( ( ap @ ( ap @ ( c_2Ecanonical_2Espolynom__normalize @ A_27a ) @ V0sr ) @ ( ap @ ( c_2Ecanonical_2ESPvar @ A_27a ) @ ( inj__ty_2Equote_2Eindex @ V1i ) ) )
= ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ty_2Equote_2Eindex ) @ ( inj__ty_2Equote_2Eindex @ V1i ) ) @ ( c_2Elist_2ENIL @ ty_2Equote_2Eindex ) ) ) @ ( c_2Ecanonical_2ENil__monom @ A_27a ) ) ) )
& ! [V2sr: $i] :
( ( mem @ V2sr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V3c: $i] :
( ( mem @ V3c @ A_27a )
=> ( ( ap @ ( ap @ ( c_2Ecanonical_2Espolynom__normalize @ A_27a ) @ V2sr ) @ ( ap @ ( c_2Ecanonical_2ESPconst @ A_27a ) @ V3c ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ V3c ) @ ( c_2Elist_2ENIL @ ty_2Equote_2Eindex ) ) @ ( c_2Ecanonical_2ENil__monom @ A_27a ) ) ) ) )
& ! [V4sr: $i] :
( ( mem @ V4sr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V5l: $i] :
( ( mem @ V5l @ ( ty_2Ecanonical_2Espolynom @ A_27a ) )
=> ! [V6r: $i] :
( ( mem @ V6r @ ( ty_2Ecanonical_2Espolynom @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Ecanonical_2Espolynom__normalize @ A_27a ) @ V4sr ) @ ( ap @ ( ap @ ( c_2Ecanonical_2ESPplus @ A_27a ) @ V5l ) @ V6r ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Ecanonical__sum__merge @ A_27a ) @ V4sr ) @ ( ap @ ( ap @ ( c_2Ecanonical_2Espolynom__normalize @ A_27a ) @ V4sr ) @ V5l ) ) @ ( ap @ ( ap @ ( c_2Ecanonical_2Espolynom__normalize @ A_27a ) @ V4sr ) @ V6r ) ) ) ) ) )
& ! [V7sr: $i] :
( ( mem @ V7sr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V8l: $i] :
( ( mem @ V8l @ ( ty_2Ecanonical_2Espolynom @ A_27a ) )
=> ! [V9r: $i] :
( ( mem @ V9r @ ( ty_2Ecanonical_2Espolynom @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Ecanonical_2Espolynom__normalize @ A_27a ) @ V7sr ) @ ( ap @ ( ap @ ( c_2Ecanonical_2ESPmult @ A_27a ) @ V8l ) @ V9r ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Ecanonical__sum__prod @ A_27a ) @ V7sr ) @ ( ap @ ( ap @ ( c_2Ecanonical_2Espolynom__normalize @ A_27a ) @ V7sr ) @ V8l ) ) @ ( ap @ ( ap @ ( c_2Ecanonical_2Espolynom__normalize @ A_27a ) @ V7sr ) @ V9r ) ) ) ) ) ) ) ).
thf(ax_thm_2Ecanonical_2Espolynom__simplify__def,axiom,
! [A_27a: del,V0sr: $i] :
( ( mem @ V0sr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ ( ty_2Ecanonical_2Espolynom @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Ecanonical_2Espolynom__simplify @ A_27a ) @ V0sr ) @ V1x )
= ( ap @ ( ap @ ( c_2Ecanonical_2Ecanonical__sum__simplify @ A_27a ) @ V0sr ) @ ( ap @ ( ap @ ( c_2Ecanonical_2Espolynom__normalize @ A_27a ) @ V0sr ) @ V1x ) ) ) ) ) ).
thf(ax_thm_2Ecanonical_2Einterp__sp__def,axiom,
! [A_27a: del] :
( ! [V0sr: $i] :
( ( mem @ V0sr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V1vm: $i] :
( ( mem @ V1vm @ ( ty_2Equote_2Evarmap @ A_27a ) )
=> ! [V2c: $i] :
( ( mem @ V2c @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Einterp__sp @ A_27a ) @ V0sr ) @ V1vm ) @ ( ap @ ( c_2Ecanonical_2ESPconst @ A_27a ) @ V2c ) )
= V2c ) ) ) )
& ! [V3sr: $i] :
( ( mem @ V3sr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V4vm: $i] :
( ( mem @ V4vm @ ( ty_2Equote_2Evarmap @ A_27a ) )
=> ! [V5i: tp__ty_2Equote_2Eindex] :
( ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Einterp__sp @ A_27a ) @ V3sr ) @ V4vm ) @ ( ap @ ( c_2Ecanonical_2ESPvar @ A_27a ) @ ( inj__ty_2Equote_2Eindex @ V5i ) ) )
= ( ap @ ( ap @ ( c_2Equote_2Evarmap__find @ A_27a ) @ ( inj__ty_2Equote_2Eindex @ V5i ) ) @ V4vm ) ) ) )
& ! [V6sr: $i] :
( ( mem @ V6sr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V7vm: $i] :
( ( mem @ V7vm @ ( ty_2Equote_2Evarmap @ A_27a ) )
=> ! [V8p1: $i] :
( ( mem @ V8p1 @ ( ty_2Ecanonical_2Espolynom @ A_27a ) )
=> ! [V9p2: $i] :
( ( mem @ V9p2 @ ( ty_2Ecanonical_2Espolynom @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Einterp__sp @ A_27a ) @ V6sr ) @ V7vm ) @ ( ap @ ( ap @ ( c_2Ecanonical_2ESPplus @ A_27a ) @ V8p1 ) @ V9p2 ) )
= ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRP @ A_27a ) @ V6sr ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Einterp__sp @ A_27a ) @ V6sr ) @ V7vm ) @ V8p1 ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Einterp__sp @ A_27a ) @ V6sr ) @ V7vm ) @ V9p2 ) ) ) ) ) ) )
& ! [V10sr: $i] :
( ( mem @ V10sr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ! [V11vm: $i] :
( ( mem @ V11vm @ ( ty_2Equote_2Evarmap @ A_27a ) )
=> ! [V12p1: $i] :
( ( mem @ V12p1 @ ( ty_2Ecanonical_2Espolynom @ A_27a ) )
=> ! [V13p2: $i] :
( ( mem @ V13p2 @ ( ty_2Ecanonical_2Espolynom @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Einterp__sp @ A_27a ) @ V10sr ) @ V11vm ) @ ( ap @ ( ap @ ( c_2Ecanonical_2ESPmult @ A_27a ) @ V12p1 ) @ V13p2 ) )
= ( ap @ ( ap @ ( ap @ ( c_2Esemi__ring_2Esemi__ring__SRM @ A_27a ) @ V10sr ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Einterp__sp @ A_27a ) @ V10sr ) @ V11vm ) @ V12p1 ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Einterp__sp @ A_27a ) @ V10sr ) @ V11vm ) @ V13p2 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ecanonical_2Espolynomial__normalize__ok,axiom,
! [A_27a: del,V0sr: $i] :
( ( mem @ V0sr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Esemi__ring_2Eis__semi__ring @ A_27a ) @ V0sr ) )
=> ! [V1vm: $i] :
( ( mem @ V1vm @ ( ty_2Equote_2Evarmap @ A_27a ) )
=> ! [V2p: $i] :
( ( mem @ V2p @ ( ty_2Ecanonical_2Espolynom @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Einterp__cs @ A_27a ) @ V0sr ) @ V1vm ) @ ( ap @ ( ap @ ( c_2Ecanonical_2Espolynom__normalize @ A_27a ) @ V0sr ) @ V2p ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Einterp__sp @ A_27a ) @ V0sr ) @ V1vm ) @ V2p ) ) ) ) ) ) ).
thf(conj_thm_2Ecanonical_2Espolynomial__simplify__ok,axiom,
! [A_27a: del,V0sr: $i] :
( ( mem @ V0sr @ ( ty_2Esemi__ring_2Esemi__ring @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Esemi__ring_2Eis__semi__ring @ A_27a ) @ V0sr ) )
=> ! [V1vm: $i] :
( ( mem @ V1vm @ ( ty_2Equote_2Evarmap @ A_27a ) )
=> ! [V2p: $i] :
( ( mem @ V2p @ ( ty_2Ecanonical_2Espolynom @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Einterp__cs @ A_27a ) @ V0sr ) @ V1vm ) @ ( ap @ ( ap @ ( c_2Ecanonical_2Espolynom__simplify @ A_27a ) @ V0sr ) @ V2p ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2Einterp__sp @ A_27a ) @ V0sr ) @ V1vm ) @ V2p ) ) ) ) ) ) ).
%------------------------------------------------------------------------------