ITP001 Axioms: ITP114^5.ax
%------------------------------------------------------------------------------
% File : ITP114^5 : 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 : ringNorm^2.ax [Gau20]
% : HL4114^5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 101 ( 0 unt; 27 typ; 0 def)
% Number of atoms : 2764 ( 143 equ; 0 cnn)
% Maximal formula atoms : 338 ( 27 avg)
% Number of connectives : 5077 ( 10 ~; 8 |; 98 &;4665 @)
% ( 6 <=>; 290 =>; 0 <=; 0 <~>)
% Maximal formula depth : 56 ( 11 avg;4665 nst)
% Number of types : 1 ( 0 usr)
% Number of type conns : 28 ( 28 >; 0 *; 0 +; 0 <<)
% Number of symbols : 109 ( 108 usr; 81 con; 0-2 aty)
% Number of variables : 458 ( 22 ^ 420 !; 16 ?; 458 :)
% SPC : TH0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_ty_2EringNorm_2Epolynom,type,
ty_2EringNorm_2Epolynom: del > del ).
thf(tp_c_2EringNorm_2EPconst,type,
c_2EringNorm_2EPconst: del > $i ).
thf(mem_c_2EringNorm_2EPconst,axiom,
! [A_27a: del] : ( mem @ ( c_2EringNorm_2EPconst @ A_27a ) @ ( arr @ A_27a @ ( ty_2EringNorm_2Epolynom @ A_27a ) ) ) ).
thf(tp_c_2EringNorm_2EPmult,type,
c_2EringNorm_2EPmult: del > $i ).
thf(mem_c_2EringNorm_2EPmult,axiom,
! [A_27a: del] : ( mem @ ( c_2EringNorm_2EPmult @ A_27a ) @ ( arr @ ( ty_2EringNorm_2Epolynom @ A_27a ) @ ( arr @ ( ty_2EringNorm_2Epolynom @ A_27a ) @ ( ty_2EringNorm_2Epolynom @ A_27a ) ) ) ) ).
thf(tp_c_2EringNorm_2EPopp,type,
c_2EringNorm_2EPopp: del > $i ).
thf(mem_c_2EringNorm_2EPopp,axiom,
! [A_27a: del] : ( mem @ ( c_2EringNorm_2EPopp @ A_27a ) @ ( arr @ ( ty_2EringNorm_2Epolynom @ A_27a ) @ ( ty_2EringNorm_2Epolynom @ A_27a ) ) ) ).
thf(tp_c_2EringNorm_2EPplus,type,
c_2EringNorm_2EPplus: del > $i ).
thf(mem_c_2EringNorm_2EPplus,axiom,
! [A_27a: del] : ( mem @ ( c_2EringNorm_2EPplus @ A_27a ) @ ( arr @ ( ty_2EringNorm_2Epolynom @ A_27a ) @ ( arr @ ( ty_2EringNorm_2Epolynom @ A_27a ) @ ( ty_2EringNorm_2Epolynom @ A_27a ) ) ) ) ).
thf(tp_c_2EringNorm_2EPvar,type,
c_2EringNorm_2EPvar: del > $i ).
thf(mem_c_2EringNorm_2EPvar,axiom,
! [A_27a: del] : ( mem @ ( c_2EringNorm_2EPvar @ A_27a ) @ ( arr @ ty_2Equote_2Eindex @ ( ty_2EringNorm_2Epolynom @ A_27a ) ) ) ).
thf(tp_c_2EringNorm_2Einterp__p,type,
c_2EringNorm_2Einterp__p: del > $i ).
thf(mem_c_2EringNorm_2Einterp__p,axiom,
! [A_27a: del] : ( mem @ ( c_2EringNorm_2Einterp__p @ A_27a ) @ ( arr @ ( ty_2Ering_2Ering @ A_27a ) @ ( arr @ ( ty_2Equote_2Evarmap @ A_27a ) @ ( arr @ ( ty_2EringNorm_2Epolynom @ A_27a ) @ A_27a ) ) ) ) ).
thf(tp_c_2EringNorm_2Epolynom__CASE,type,
c_2EringNorm_2Epolynom__CASE: del > del > $i ).
thf(mem_c_2EringNorm_2Epolynom__CASE,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2EringNorm_2Epolynom__CASE @ A_27a @ A_27b ) @ ( arr @ ( ty_2EringNorm_2Epolynom @ A_27a ) @ ( arr @ ( arr @ ty_2Equote_2Eindex @ A_27b ) @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ ( arr @ ( ty_2EringNorm_2Epolynom @ A_27a ) @ ( arr @ ( ty_2EringNorm_2Epolynom @ A_27a ) @ A_27b ) ) @ ( arr @ ( arr @ ( ty_2EringNorm_2Epolynom @ A_27a ) @ ( arr @ ( ty_2EringNorm_2Epolynom @ A_27a ) @ A_27b ) ) @ ( arr @ ( arr @ ( ty_2EringNorm_2Epolynom @ A_27a ) @ A_27b ) @ A_27b ) ) ) ) ) ) ) ).
thf(tp_c_2EringNorm_2Epolynom__normalize,type,
c_2EringNorm_2Epolynom__normalize: del > $i ).
thf(mem_c_2EringNorm_2Epolynom__normalize,axiom,
! [A_27a: del] : ( mem @ ( c_2EringNorm_2Epolynom__normalize @ A_27a ) @ ( arr @ ( ty_2Ering_2Ering @ A_27a ) @ ( arr @ ( ty_2EringNorm_2Epolynom @ A_27a ) @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) ) ) ) ).
thf(tp_c_2EringNorm_2Epolynom__simplify,type,
c_2EringNorm_2Epolynom__simplify: del > $i ).
thf(mem_c_2EringNorm_2Epolynom__simplify,axiom,
! [A_27a: del] : ( mem @ ( c_2EringNorm_2Epolynom__simplify @ A_27a ) @ ( arr @ ( ty_2Ering_2Ering @ A_27a ) @ ( arr @ ( ty_2EringNorm_2Epolynom @ A_27a ) @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) ) ) ) ).
thf(tp_c_2EringNorm_2Epolynom__size,type,
c_2EringNorm_2Epolynom__size: del > $i ).
thf(mem_c_2EringNorm_2Epolynom__size,axiom,
! [A_27a: del] : ( mem @ ( c_2EringNorm_2Epolynom__size @ A_27a ) @ ( arr @ ( arr @ A_27a @ ty_2Enum_2Enum ) @ ( arr @ ( ty_2EringNorm_2Epolynom @ A_27a ) @ ty_2Enum_2Enum ) ) ) ).
thf(tp_c_2EringNorm_2Er__canonical__sum__merge,type,
c_2EringNorm_2Er__canonical__sum__merge: del > $i ).
thf(mem_c_2EringNorm_2Er__canonical__sum__merge,axiom,
! [A_27a: del] : ( mem @ ( c_2EringNorm_2Er__canonical__sum__merge @ A_27a ) @ ( arr @ ( ty_2Ering_2Ering @ 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_2EringNorm_2Er__canonical__sum__prod,type,
c_2EringNorm_2Er__canonical__sum__prod: del > $i ).
thf(mem_c_2EringNorm_2Er__canonical__sum__prod,axiom,
! [A_27a: del] : ( mem @ ( c_2EringNorm_2Er__canonical__sum__prod @ A_27a ) @ ( arr @ ( ty_2Ering_2Ering @ 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_2EringNorm_2Er__canonical__sum__scalar,type,
c_2EringNorm_2Er__canonical__sum__scalar: del > $i ).
thf(mem_c_2EringNorm_2Er__canonical__sum__scalar,axiom,
! [A_27a: del] : ( mem @ ( c_2EringNorm_2Er__canonical__sum__scalar @ A_27a ) @ ( arr @ ( ty_2Ering_2Ering @ A_27a ) @ ( arr @ A_27a @ ( arr @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) ) ) ) ) ).
thf(tp_c_2EringNorm_2Er__canonical__sum__scalar2,type,
c_2EringNorm_2Er__canonical__sum__scalar2: del > $i ).
thf(mem_c_2EringNorm_2Er__canonical__sum__scalar2,axiom,
! [A_27a: del] : ( mem @ ( c_2EringNorm_2Er__canonical__sum__scalar2 @ A_27a ) @ ( arr @ ( ty_2Ering_2Ering @ 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_2EringNorm_2Er__canonical__sum__scalar3,type,
c_2EringNorm_2Er__canonical__sum__scalar3: del > $i ).
thf(mem_c_2EringNorm_2Er__canonical__sum__scalar3,axiom,
! [A_27a: del] : ( mem @ ( c_2EringNorm_2Er__canonical__sum__scalar3 @ A_27a ) @ ( arr @ ( ty_2Ering_2Ering @ 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_2EringNorm_2Er__canonical__sum__simplify,type,
c_2EringNorm_2Er__canonical__sum__simplify: del > $i ).
thf(mem_c_2EringNorm_2Er__canonical__sum__simplify,axiom,
! [A_27a: del] : ( mem @ ( c_2EringNorm_2Er__canonical__sum__simplify @ A_27a ) @ ( arr @ ( ty_2Ering_2Ering @ A_27a ) @ ( arr @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) ) ) ) ).
thf(tp_c_2EringNorm_2Er__ics__aux,type,
c_2EringNorm_2Er__ics__aux: del > $i ).
thf(mem_c_2EringNorm_2Er__ics__aux,axiom,
! [A_27a: del] : ( mem @ ( c_2EringNorm_2Er__ics__aux @ A_27a ) @ ( arr @ ( ty_2Ering_2Ering @ A_27a ) @ ( arr @ ( ty_2Equote_2Evarmap @ A_27a ) @ ( arr @ A_27a @ ( arr @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) @ A_27a ) ) ) ) ) ).
thf(tp_c_2EringNorm_2Er__interp__cs,type,
c_2EringNorm_2Er__interp__cs: del > $i ).
thf(mem_c_2EringNorm_2Er__interp__cs,axiom,
! [A_27a: del] : ( mem @ ( c_2EringNorm_2Er__interp__cs @ A_27a ) @ ( arr @ ( ty_2Ering_2Ering @ A_27a ) @ ( arr @ ( ty_2Equote_2Evarmap @ A_27a ) @ ( arr @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) @ A_27a ) ) ) ) ).
thf(tp_c_2EringNorm_2Er__interp__m,type,
c_2EringNorm_2Er__interp__m: del > $i ).
thf(mem_c_2EringNorm_2Er__interp__m,axiom,
! [A_27a: del] : ( mem @ ( c_2EringNorm_2Er__interp__m @ A_27a ) @ ( arr @ ( ty_2Ering_2Ering @ A_27a ) @ ( arr @ ( ty_2Equote_2Evarmap @ A_27a ) @ ( arr @ A_27a @ ( arr @ ( ty_2Elist_2Elist @ ty_2Equote_2Eindex ) @ A_27a ) ) ) ) ) ).
thf(tp_c_2EringNorm_2Er__interp__sp,type,
c_2EringNorm_2Er__interp__sp: del > $i ).
thf(mem_c_2EringNorm_2Er__interp__sp,axiom,
! [A_27a: del] : ( mem @ ( c_2EringNorm_2Er__interp__sp @ A_27a ) @ ( arr @ ( ty_2Ering_2Ering @ A_27a ) @ ( arr @ ( ty_2Equote_2Evarmap @ A_27a ) @ ( arr @ ( ty_2Ecanonical_2Espolynom @ A_27a ) @ A_27a ) ) ) ) ).
thf(tp_c_2EringNorm_2Er__interp__vl,type,
c_2EringNorm_2Er__interp__vl: del > $i ).
thf(mem_c_2EringNorm_2Er__interp__vl,axiom,
! [A_27a: del] : ( mem @ ( c_2EringNorm_2Er__interp__vl @ A_27a ) @ ( arr @ ( ty_2Ering_2Ering @ A_27a ) @ ( arr @ ( ty_2Equote_2Evarmap @ A_27a ) @ ( arr @ ( ty_2Elist_2Elist @ ty_2Equote_2Eindex ) @ A_27a ) ) ) ) ).
thf(tp_c_2EringNorm_2Er__ivl__aux,type,
c_2EringNorm_2Er__ivl__aux: del > $i ).
thf(mem_c_2EringNorm_2Er__ivl__aux,axiom,
! [A_27a: del] : ( mem @ ( c_2EringNorm_2Er__ivl__aux @ A_27a ) @ ( arr @ ( ty_2Ering_2Ering @ A_27a ) @ ( arr @ ( ty_2Equote_2Evarmap @ A_27a ) @ ( arr @ ty_2Equote_2Eindex @ ( arr @ ( ty_2Elist_2Elist @ ty_2Equote_2Eindex ) @ A_27a ) ) ) ) ) ).
thf(tp_c_2EringNorm_2Er__monom__insert,type,
c_2EringNorm_2Er__monom__insert: del > $i ).
thf(mem_c_2EringNorm_2Er__monom__insert,axiom,
! [A_27a: del] : ( mem @ ( c_2EringNorm_2Er__monom__insert @ A_27a ) @ ( arr @ ( ty_2Ering_2Ering @ 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_2EringNorm_2Er__spolynom__normalize,type,
c_2EringNorm_2Er__spolynom__normalize: del > $i ).
thf(mem_c_2EringNorm_2Er__spolynom__normalize,axiom,
! [A_27a: del] : ( mem @ ( c_2EringNorm_2Er__spolynom__normalize @ A_27a ) @ ( arr @ ( ty_2Ering_2Ering @ A_27a ) @ ( arr @ ( ty_2Ecanonical_2Espolynom @ A_27a ) @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) ) ) ) ).
thf(tp_c_2EringNorm_2Er__spolynom__simplify,type,
c_2EringNorm_2Er__spolynom__simplify: del > $i ).
thf(mem_c_2EringNorm_2Er__spolynom__simplify,axiom,
! [A_27a: del] : ( mem @ ( c_2EringNorm_2Er__spolynom__simplify @ A_27a ) @ ( arr @ ( ty_2Ering_2Ering @ A_27a ) @ ( arr @ ( ty_2Ecanonical_2Espolynom @ A_27a ) @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) ) ) ) ).
thf(tp_c_2EringNorm_2Er__varlist__insert,type,
c_2EringNorm_2Er__varlist__insert: del > $i ).
thf(mem_c_2EringNorm_2Er__varlist__insert,axiom,
! [A_27a: del] : ( mem @ ( c_2EringNorm_2Er__varlist__insert @ A_27a ) @ ( arr @ ( ty_2Ering_2Ering @ 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_2EringNorm_2Er__interp__sp__def,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ( ( ap @ ( c_2EringNorm_2Er__interp__sp @ A_27a ) @ V0r )
= ( ap @ ( c_2Ecanonical_2Einterp__sp @ A_27a ) @ ( ap @ ( c_2Ering_2Esemi__ring__of @ A_27a ) @ V0r ) ) ) ) ).
thf(ax_thm_2EringNorm_2Er__spolynom__simplify__def,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ( ( ap @ ( c_2EringNorm_2Er__spolynom__simplify @ A_27a ) @ V0r )
= ( ap @ ( c_2Ecanonical_2Espolynom__simplify @ A_27a ) @ ( ap @ ( c_2Ering_2Esemi__ring__of @ A_27a ) @ V0r ) ) ) ) ).
thf(ax_thm_2EringNorm_2Er__spolynom__normalize__def,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ( ( ap @ ( c_2EringNorm_2Er__spolynom__normalize @ A_27a ) @ V0r )
= ( ap @ ( c_2Ecanonical_2Espolynom__normalize @ A_27a ) @ ( ap @ ( c_2Ering_2Esemi__ring__of @ A_27a ) @ V0r ) ) ) ) ).
thf(ax_thm_2EringNorm_2Er__interp__cs__def,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ( ( ap @ ( c_2EringNorm_2Er__interp__cs @ A_27a ) @ V0r )
= ( ap @ ( c_2Ecanonical_2Einterp__cs @ A_27a ) @ ( ap @ ( c_2Ering_2Esemi__ring__of @ A_27a ) @ V0r ) ) ) ) ).
thf(ax_thm_2EringNorm_2Er__ics__aux__def,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ( ( ap @ ( c_2EringNorm_2Er__ics__aux @ A_27a ) @ V0r )
= ( ap @ ( c_2Ecanonical_2Eics__aux @ A_27a ) @ ( ap @ ( c_2Ering_2Esemi__ring__of @ A_27a ) @ V0r ) ) ) ) ).
thf(ax_thm_2EringNorm_2Er__interp__m__def,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ( ( ap @ ( c_2EringNorm_2Er__interp__m @ A_27a ) @ V0r )
= ( ap @ ( c_2Ecanonical_2Einterp__m @ A_27a ) @ ( ap @ ( c_2Ering_2Esemi__ring__of @ A_27a ) @ V0r ) ) ) ) ).
thf(ax_thm_2EringNorm_2Er__interp__vl__def,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ( ( ap @ ( c_2EringNorm_2Er__interp__vl @ A_27a ) @ V0r )
= ( ap @ ( c_2Ecanonical_2Einterp__vl @ A_27a ) @ ( ap @ ( c_2Ering_2Esemi__ring__of @ A_27a ) @ V0r ) ) ) ) ).
thf(ax_thm_2EringNorm_2Er__ivl__aux__def,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ( ( ap @ ( c_2EringNorm_2Er__ivl__aux @ A_27a ) @ V0r )
= ( ap @ ( c_2Ecanonical_2Eivl__aux @ A_27a ) @ ( ap @ ( c_2Ering_2Esemi__ring__of @ A_27a ) @ V0r ) ) ) ) ).
thf(ax_thm_2EringNorm_2Er__canonical__sum__simplify__def,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ( ( ap @ ( c_2EringNorm_2Er__canonical__sum__simplify @ A_27a ) @ V0r )
= ( ap @ ( c_2Ecanonical_2Ecanonical__sum__simplify @ A_27a ) @ ( ap @ ( c_2Ering_2Esemi__ring__of @ A_27a ) @ V0r ) ) ) ) ).
thf(ax_thm_2EringNorm_2Er__canonical__sum__prod__def,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ( ( ap @ ( c_2EringNorm_2Er__canonical__sum__prod @ A_27a ) @ V0r )
= ( ap @ ( c_2Ecanonical_2Ecanonical__sum__prod @ A_27a ) @ ( ap @ ( c_2Ering_2Esemi__ring__of @ A_27a ) @ V0r ) ) ) ) ).
thf(ax_thm_2EringNorm_2Er__canonical__sum__scalar3__def,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ( ( ap @ ( c_2EringNorm_2Er__canonical__sum__scalar3 @ A_27a ) @ V0r )
= ( ap @ ( c_2Ecanonical_2Ecanonical__sum__scalar3 @ A_27a ) @ ( ap @ ( c_2Ering_2Esemi__ring__of @ A_27a ) @ V0r ) ) ) ) ).
thf(ax_thm_2EringNorm_2Er__canonical__sum__scalar2__def,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ( ( ap @ ( c_2EringNorm_2Er__canonical__sum__scalar2 @ A_27a ) @ V0r )
= ( ap @ ( c_2Ecanonical_2Ecanonical__sum__scalar2 @ A_27a ) @ ( ap @ ( c_2Ering_2Esemi__ring__of @ A_27a ) @ V0r ) ) ) ) ).
thf(ax_thm_2EringNorm_2Er__canonical__sum__scalar__def,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ( ( ap @ ( c_2EringNorm_2Er__canonical__sum__scalar @ A_27a ) @ V0r )
= ( ap @ ( c_2Ecanonical_2Ecanonical__sum__scalar @ A_27a ) @ ( ap @ ( c_2Ering_2Esemi__ring__of @ A_27a ) @ V0r ) ) ) ) ).
thf(ax_thm_2EringNorm_2Er__varlist__insert__def,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ( ( ap @ ( c_2EringNorm_2Er__varlist__insert @ A_27a ) @ V0r )
= ( ap @ ( c_2Ecanonical_2Evarlist__insert @ A_27a ) @ ( ap @ ( c_2Ering_2Esemi__ring__of @ A_27a ) @ V0r ) ) ) ) ).
thf(ax_thm_2EringNorm_2Er__monom__insert__def,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ( ( ap @ ( c_2EringNorm_2Er__monom__insert @ A_27a ) @ V0r )
= ( ap @ ( c_2Ecanonical_2Emonom__insert @ A_27a ) @ ( ap @ ( c_2Ering_2Esemi__ring__of @ A_27a ) @ V0r ) ) ) ) ).
thf(ax_thm_2EringNorm_2Er__canonical__sum__merge__def,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ( ( ap @ ( c_2EringNorm_2Er__canonical__sum__merge @ A_27a ) @ V0r )
= ( ap @ ( c_2Ecanonical_2Ecanonical__sum__merge @ A_27a ) @ ( ap @ ( c_2Ering_2Esemi__ring__of @ A_27a ) @ V0r ) ) ) ) ).
thf(conj_thm_2EringNorm_2Einterp__sp__def,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ( ! [V1vm: $i] :
( ( mem @ V1vm @ ( ty_2Equote_2Evarmap @ A_27a ) )
=> ! [V2c: $i] :
( ( mem @ V2c @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Er__interp__sp @ A_27a ) @ V0r ) @ V1vm ) @ ( ap @ ( c_2Ecanonical_2ESPconst @ A_27a ) @ V2c ) )
= V2c ) ) )
& ! [V3vm: $i] :
( ( mem @ V3vm @ ( ty_2Equote_2Evarmap @ A_27a ) )
=> ! [V4i: tp__ty_2Equote_2Eindex] :
( ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Er__interp__sp @ A_27a ) @ V0r ) @ V3vm ) @ ( ap @ ( c_2Ecanonical_2ESPvar @ A_27a ) @ ( inj__ty_2Equote_2Eindex @ V4i ) ) )
= ( ap @ ( ap @ ( c_2Equote_2Evarmap__find @ A_27a ) @ ( inj__ty_2Equote_2Eindex @ V4i ) ) @ V3vm ) ) )
& ! [V5vm: $i] :
( ( mem @ V5vm @ ( ty_2Equote_2Evarmap @ A_27a ) )
=> ! [V6p1: $i] :
( ( mem @ V6p1 @ ( ty_2Ecanonical_2Espolynom @ A_27a ) )
=> ! [V7p2: $i] :
( ( mem @ V7p2 @ ( ty_2Ecanonical_2Espolynom @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Er__interp__sp @ A_27a ) @ V0r ) @ V5vm ) @ ( ap @ ( ap @ ( c_2Ecanonical_2ESPplus @ A_27a ) @ V6p1 ) @ V7p2 ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ering_2Ering__RP @ A_27a ) @ V0r ) @ ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Er__interp__sp @ A_27a ) @ V0r ) @ V5vm ) @ V6p1 ) ) @ ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Er__interp__sp @ A_27a ) @ V0r ) @ V5vm ) @ V7p2 ) ) ) ) ) )
& ! [V8vm: $i] :
( ( mem @ V8vm @ ( ty_2Equote_2Evarmap @ A_27a ) )
=> ! [V9p1: $i] :
( ( mem @ V9p1 @ ( ty_2Ecanonical_2Espolynom @ A_27a ) )
=> ! [V10p2: $i] :
( ( mem @ V10p2 @ ( ty_2Ecanonical_2Espolynom @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Er__interp__sp @ A_27a ) @ V0r ) @ V8vm ) @ ( ap @ ( ap @ ( c_2Ecanonical_2ESPmult @ A_27a ) @ V9p1 ) @ V10p2 ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ering_2Ering__RM @ A_27a ) @ V0r ) @ ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Er__interp__sp @ A_27a ) @ V0r ) @ V8vm ) @ V9p1 ) ) @ ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Er__interp__sp @ A_27a ) @ V0r ) @ V8vm ) @ V10p2 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2EringNorm_2Ecanonical__sum__merge__def,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ( ! [V1t2: $i] :
( ( mem @ V1t2 @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ! [V2t1: $i] :
( ( mem @ V2t1 @ ( ty_2Ecanonical_2Ecanonical__sum @ 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_2EringNorm_2Er__canonical__sum__merge @ A_27a ) @ V0r ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ V6c1 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V4l1 ) ) @ V2t1 ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ V5c2 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V3l2 ) ) @ V1t2 ) )
= ( 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_2EringNorm_2Er__canonical__sum__merge @ A_27a ) @ V0r ) @ V2t1 ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ V5c2 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V3l2 ) ) @ V1t2 ) ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Ering_2Ering__RP @ A_27a ) @ V0r ) @ V6c1 ) @ V5c2 ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V4l1 ) ) @ ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Er__canonical__sum__merge @ A_27a ) @ V0r ) @ V2t1 ) @ V1t2 ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ V5c2 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V3l2 ) ) @ ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Er__canonical__sum__merge @ A_27a ) @ V0r ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ V6c1 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V4l1 ) ) @ V2t1 ) ) @ V1t2 ) ) ) ) ) ) ) )
& ! [V7t2: $i] :
( ( mem @ V7t2 @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ! [V8t1: $i] :
( ( mem @ V8t1 @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ! [V9l2: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V10l1: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V11c1: $i] :
( ( mem @ V11c1 @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Er__canonical__sum__merge @ A_27a ) @ V0r ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ V11c1 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V10l1 ) ) @ V8t1 ) ) @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V9l2 ) ) @ 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 @ V10l1 ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V9l2 ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ V11c1 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V10l1 ) ) @ ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Er__canonical__sum__merge @ A_27a ) @ V0r ) @ V8t1 ) @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V9l2 ) ) @ V7t2 ) ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Ering_2Ering__RP @ A_27a ) @ V0r ) @ V11c1 ) @ ( ap @ ( c_2Ering_2Ering__R1 @ A_27a ) @ V0r ) ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V10l1 ) ) @ ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Er__canonical__sum__merge @ A_27a ) @ V0r ) @ V8t1 ) @ V7t2 ) ) ) @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V9l2 ) ) @ ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Er__canonical__sum__merge @ A_27a ) @ V0r ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ V11c1 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V10l1 ) ) @ V8t1 ) ) @ V7t2 ) ) ) ) ) ) )
& ! [V12t2: $i] :
( ( mem @ V12t2 @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ! [V13t1: $i] :
( ( mem @ V13t1 @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ! [V14l2: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V15l1: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V16c2: $i] :
( ( mem @ V16c2 @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Er__canonical__sum__merge @ A_27a ) @ V0r ) @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V15l1 ) ) @ V13t1 ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ V16c2 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V14l2 ) ) @ V12t2 ) )
= ( 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 @ V15l1 ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V14l2 ) ) ) @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V15l1 ) ) @ ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Er__canonical__sum__merge @ A_27a ) @ V0r ) @ V13t1 ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ V16c2 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V14l2 ) ) @ V12t2 ) ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Ering_2Ering__RP @ A_27a ) @ V0r ) @ ( ap @ ( c_2Ering_2Ering__R1 @ A_27a ) @ V0r ) ) @ V16c2 ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V15l1 ) ) @ ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Er__canonical__sum__merge @ A_27a ) @ V0r ) @ V13t1 ) @ V12t2 ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ V16c2 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V14l2 ) ) @ ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Er__canonical__sum__merge @ A_27a ) @ V0r ) @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V15l1 ) ) @ V13t1 ) ) @ V12t2 ) ) ) ) ) ) )
& ! [V17t2: $i] :
( ( mem @ V17t2 @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ! [V18t1: $i] :
( ( mem @ V18t1 @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ! [V19l2: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V20l1: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex] :
( ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Er__canonical__sum__merge @ A_27a ) @ V0r ) @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V20l1 ) ) @ V18t1 ) ) @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V19l2 ) ) @ V17t2 ) )
= ( 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 @ V20l1 ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V19l2 ) ) ) @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V20l1 ) ) @ ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Er__canonical__sum__merge @ A_27a ) @ V0r ) @ V18t1 ) @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V19l2 ) ) @ V17t2 ) ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Ering_2Ering__RP @ A_27a ) @ V0r ) @ ( ap @ ( c_2Ering_2Ering__R1 @ A_27a ) @ V0r ) ) @ ( ap @ ( c_2Ering_2Ering__R1 @ A_27a ) @ V0r ) ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V20l1 ) ) @ ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Er__canonical__sum__merge @ A_27a ) @ V0r ) @ V18t1 ) @ V17t2 ) ) ) @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V19l2 ) ) @ ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Er__canonical__sum__merge @ A_27a ) @ V0r ) @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V20l1 ) ) @ V18t1 ) ) @ V17t2 ) ) ) ) ) )
& ! [V21s1: $i] :
( ( mem @ V21s1 @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Er__canonical__sum__merge @ A_27a ) @ V0r ) @ V21s1 ) @ ( c_2Ecanonical_2ENil__monom @ A_27a ) )
= V21s1 ) )
& ! [V22v6: $i] :
( ( mem @ V22v6 @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ! [V23v5: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V24v4: $i] :
( ( mem @ V24v4 @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Er__canonical__sum__merge @ A_27a ) @ V0r ) @ ( c_2Ecanonical_2ENil__monom @ A_27a ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ V24v4 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V23v5 ) ) @ V22v6 ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ V24v4 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V23v5 ) ) @ V22v6 ) ) ) )
& ! [V25v8: $i] :
( ( mem @ V25v8 @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ! [V26v7: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex] :
( ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Er__canonical__sum__merge @ A_27a ) @ V0r ) @ ( c_2Ecanonical_2ENil__monom @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V26v7 ) ) @ V25v8 ) )
= ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V26v7 ) ) @ V25v8 ) ) ) ) ) ).
thf(conj_thm_2EringNorm_2Emonom__insert__def,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ( ! [V1t2: $i] :
( ( mem @ V1t2 @ ( ty_2Ecanonical_2Ecanonical__sum @ 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_2EringNorm_2Er__monom__insert @ A_27a ) @ V0r ) @ 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 ) ) @ V1t2 ) )
= ( 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 ) ) @ V1t2 ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Ering_2Ering__RP @ A_27a ) @ V0r ) @ V5c1 ) @ V4c2 ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V3l1 ) ) @ V1t2 ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ V4c2 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V2l2 ) ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Er__monom__insert @ A_27a ) @ V0r ) @ V5c1 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V3l1 ) ) @ V1t2 ) ) ) ) ) ) )
& ! [V6t2: $i] :
( ( mem @ V6t2 @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ! [V7l2: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V8l1: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V9c1: $i] :
( ( mem @ V9c1 @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Er__monom__insert @ A_27a ) @ V0r ) @ V9c1 ) @ ( 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 ) ) @ 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 @ V8l1 ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V7l2 ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ V9c1 ) @ ( 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 ) ) @ V6t2 ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Ering_2Ering__RP @ A_27a ) @ V0r ) @ V9c1 ) @ ( ap @ ( c_2Ering_2Ering__R1 @ A_27a ) @ V0r ) ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V8l1 ) ) @ V6t2 ) ) @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V7l2 ) ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Er__monom__insert @ A_27a ) @ V0r ) @ V9c1 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V8l1 ) ) @ V6t2 ) ) ) ) ) )
& ! [V10l1: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V11c1: $i] :
( ( mem @ V11c1 @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Er__monom__insert @ A_27a ) @ V0r ) @ V11c1 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V10l1 ) ) @ ( c_2Ecanonical_2ENil__monom @ A_27a ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ V11c1 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V10l1 ) ) @ ( c_2Ecanonical_2ENil__monom @ A_27a ) ) ) ) ) ) ).
thf(conj_thm_2EringNorm_2Evarlist__insert__def,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ( ! [V1t2: $i] :
( ( mem @ V1t2 @ ( ty_2Ecanonical_2Ecanonical__sum @ 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_2EringNorm_2Er__varlist__insert @ A_27a ) @ V0r ) @ ( 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 ) ) @ V1t2 ) )
= ( 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 ) ) @ V1t2 ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Ering_2Ering__RP @ A_27a ) @ V0r ) @ ( ap @ ( c_2Ering_2Ering__R1 @ A_27a ) @ V0r ) ) @ V4c2 ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V3l1 ) ) @ V1t2 ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ V4c2 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V2l2 ) ) @ ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Er__varlist__insert @ A_27a ) @ V0r ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V3l1 ) ) @ V1t2 ) ) ) ) ) )
& ! [V5t2: $i] :
( ( mem @ V5t2 @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ! [V6l2: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V7l1: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex] :
( ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Er__varlist__insert @ A_27a ) @ V0r ) @ ( 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 @ V6l2 ) ) @ 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 @ V7l1 ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V6l2 ) ) ) @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( 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 @ V6l2 ) ) @ V5t2 ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Ering_2Ering__RP @ A_27a ) @ V0r ) @ ( ap @ ( c_2Ering_2Ering__R1 @ A_27a ) @ V0r ) ) @ ( ap @ ( c_2Ering_2Ering__R1 @ A_27a ) @ V0r ) ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V7l1 ) ) @ V5t2 ) ) @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V6l2 ) ) @ ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Er__varlist__insert @ A_27a ) @ V0r ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V7l1 ) ) @ V5t2 ) ) ) ) )
& ! [V8l1: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex] :
( ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Er__varlist__insert @ A_27a ) @ V0r ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V8l1 ) ) @ ( c_2Ecanonical_2ENil__monom @ A_27a ) )
= ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V8l1 ) ) @ ( c_2Ecanonical_2ENil__monom @ A_27a ) ) ) ) ) ).
thf(conj_thm_2EringNorm_2Ecanonical__sum__scalar__def,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( ty_2Ering_2Ering @ 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_2EringNorm_2Er__canonical__sum__scalar @ A_27a ) @ V0r ) @ 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_2Ering_2Ering__RM @ A_27a ) @ V0r ) @ V1c0 ) @ V2c ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V3l ) ) @ ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Er__canonical__sum__scalar @ A_27a ) @ V0r ) @ V1c0 ) @ V4t ) ) ) ) ) )
& ! [V5c0: $i] :
( ( mem @ V5c0 @ A_27a )
=> ! [V6l: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V7t: $i] :
( ( mem @ V7t @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Er__canonical__sum__scalar @ A_27a ) @ V0r ) @ V5c0 ) @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V6l ) ) @ V7t ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ V5c0 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V6l ) ) @ ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Er__canonical__sum__scalar @ A_27a ) @ V0r ) @ V5c0 ) @ V7t ) ) ) ) )
& ! [V8c0: $i] :
( ( mem @ V8c0 @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Er__canonical__sum__scalar @ A_27a ) @ V0r ) @ V8c0 ) @ ( c_2Ecanonical_2ENil__monom @ A_27a ) )
= ( c_2Ecanonical_2ENil__monom @ A_27a ) ) ) ) ) ).
thf(conj_thm_2EringNorm_2Ecanonical__sum__scalar2__def,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( ty_2Ering_2Ering @ 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_2EringNorm_2Er__canonical__sum__scalar2 @ A_27a ) @ V0r ) @ ( 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_2EringNorm_2Er__monom__insert @ A_27a ) @ V0r ) @ 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_2EringNorm_2Er__canonical__sum__scalar2 @ A_27a ) @ V0r ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V1l0 ) ) @ V4t ) ) ) ) )
& ! [V5l0: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V6l: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V7t: $i] :
( ( mem @ V7t @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Er__canonical__sum__scalar2 @ A_27a ) @ V0r ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V5l0 ) ) @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V6l ) ) @ V7t ) )
= ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Er__varlist__insert @ A_27a ) @ V0r ) @ ( ap @ ( ap @ ( ap @ ( c_2EternaryComparisons_2Elist__merge @ ty_2Equote_2Eindex ) @ c_2Equote_2Eindex__lt ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V5l0 ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V6l ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Er__canonical__sum__scalar2 @ A_27a ) @ V0r ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V5l0 ) ) @ V7t ) ) ) )
& ! [V8l0: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex] :
( ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Er__canonical__sum__scalar2 @ A_27a ) @ V0r ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V8l0 ) ) @ ( c_2Ecanonical_2ENil__monom @ A_27a ) )
= ( c_2Ecanonical_2ENil__monom @ A_27a ) ) ) ) ).
thf(conj_thm_2EringNorm_2Ecanonical__sum__scalar3__def,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( ty_2Ering_2Ering @ 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_2EringNorm_2Er__canonical__sum__scalar3 @ A_27a ) @ V0r ) @ 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_2EringNorm_2Er__monom__insert @ A_27a ) @ V0r ) @ ( ap @ ( ap @ ( ap @ ( c_2Ering_2Ering__RM @ A_27a ) @ V0r ) @ 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_2EringNorm_2Er__canonical__sum__scalar3 @ A_27a ) @ V0r ) @ V1c0 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V2l0 ) ) @ V5t ) ) ) ) ) )
& ! [V6c0: $i] :
( ( mem @ V6c0 @ A_27a )
=> ! [V7l0: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V8l: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V9t: $i] :
( ( mem @ V9t @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Er__canonical__sum__scalar3 @ A_27a ) @ V0r ) @ V6c0 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V7l0 ) ) @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V8l ) ) @ V9t ) )
= ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Er__monom__insert @ A_27a ) @ V0r ) @ V6c0 ) @ ( ap @ ( ap @ ( ap @ ( c_2EternaryComparisons_2Elist__merge @ ty_2Equote_2Eindex ) @ c_2Equote_2Eindex__lt ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V7l0 ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V8l ) ) ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Er__canonical__sum__scalar3 @ A_27a ) @ V0r ) @ V6c0 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V7l0 ) ) @ V9t ) ) ) ) )
& ! [V10c0: $i] :
( ( mem @ V10c0 @ A_27a )
=> ! [V11l0: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex] :
( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Er__canonical__sum__scalar3 @ A_27a ) @ V0r ) @ V10c0 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V11l0 ) ) @ ( c_2Ecanonical_2ENil__monom @ A_27a ) )
= ( c_2Ecanonical_2ENil__monom @ A_27a ) ) ) ) ) ).
thf(conj_thm_2EringNorm_2Ecanonical__sum__prod__def,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( ty_2Ering_2Ering @ 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_2EringNorm_2Er__canonical__sum__prod @ A_27a ) @ V0r ) @ ( 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_2EringNorm_2Er__canonical__sum__merge @ A_27a ) @ V0r ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Er__canonical__sum__scalar3 @ A_27a ) @ V0r ) @ V1c1 ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V2l1 ) ) @ V4s2 ) ) @ ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Er__canonical__sum__prod @ A_27a ) @ V0r ) @ V3t1 ) @ V4s2 ) ) ) ) ) )
& ! [V5l1: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V6t1: $i] :
( ( mem @ V6t1 @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ! [V7s2: $i] :
( ( mem @ V7s2 @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Er__canonical__sum__prod @ A_27a ) @ V0r ) @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V5l1 ) ) @ V6t1 ) ) @ V7s2 )
= ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Er__canonical__sum__merge @ A_27a ) @ V0r ) @ ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Er__canonical__sum__scalar2 @ A_27a ) @ V0r ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V5l1 ) ) @ V7s2 ) ) @ ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Er__canonical__sum__prod @ A_27a ) @ V0r ) @ V6t1 ) @ V7s2 ) ) ) ) )
& ! [V8s2: $i] :
( ( mem @ V8s2 @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Er__canonical__sum__prod @ A_27a ) @ V0r ) @ ( c_2Ecanonical_2ENil__monom @ A_27a ) ) @ V8s2 )
= ( c_2Ecanonical_2ENil__monom @ A_27a ) ) ) ) ) ).
thf(conj_thm_2EringNorm_2Ecanonical__sum__simplify__def,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( ty_2Ering_2Ering @ 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_2EringNorm_2Er__canonical__sum__simplify @ A_27a ) @ V0r ) @ ( 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_2Ering_2Ering__R0 @ A_27a ) @ V0r ) ) ) @ ( ap @ ( ap @ ( c_2EringNorm_2Er__canonical__sum__simplify @ A_27a ) @ V0r ) @ V3t ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V1c ) @ ( ap @ ( c_2Ering_2Ering__R1 @ A_27a ) @ V0r ) ) ) @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V2l ) ) @ ( ap @ ( ap @ ( c_2EringNorm_2Er__canonical__sum__simplify @ A_27a ) @ V0r ) @ V3t ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ V1c ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V2l ) ) @ ( ap @ ( ap @ ( c_2EringNorm_2Er__canonical__sum__simplify @ A_27a ) @ V0r ) @ V3t ) ) ) ) ) ) )
& ! [V4l: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V5t: $i] :
( ( mem @ V5t @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2EringNorm_2Er__canonical__sum__simplify @ A_27a ) @ V0r ) @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V4l ) ) @ V5t ) )
= ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V4l ) ) @ ( ap @ ( ap @ ( c_2EringNorm_2Er__canonical__sum__simplify @ A_27a ) @ V0r ) @ V5t ) ) ) )
& ( ( ap @ ( ap @ ( c_2EringNorm_2Er__canonical__sum__simplify @ A_27a ) @ V0r ) @ ( c_2Ecanonical_2ENil__monom @ A_27a ) )
= ( c_2Ecanonical_2ENil__monom @ A_27a ) ) ) ) ).
thf(conj_thm_2EringNorm_2Eivl__aux__def,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ( ! [V1vm: $i] :
( ( mem @ V1vm @ ( ty_2Equote_2Evarmap @ A_27a ) )
=> ! [V2x: tp__ty_2Equote_2Eindex] :
( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Er__ivl__aux @ A_27a ) @ V0r ) @ 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 ) ) )
& ! [V3vm: $i] :
( ( mem @ V3vm @ ( ty_2Equote_2Evarmap @ A_27a ) )
=> ! [V4x: tp__ty_2Equote_2Eindex,V5x_27: tp__ty_2Equote_2Eindex,V6t_27: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex] :
( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Er__ivl__aux @ A_27a ) @ V0r ) @ V3vm ) @ ( inj__ty_2Equote_2Eindex @ V4x ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ty_2Equote_2Eindex ) @ ( inj__ty_2Equote_2Eindex @ V5x_27 ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V6t_27 ) ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ering_2Ering__RM @ A_27a ) @ V0r ) @ ( ap @ ( ap @ ( c_2Equote_2Evarmap__find @ A_27a ) @ ( inj__ty_2Equote_2Eindex @ V4x ) ) @ V3vm ) ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Er__ivl__aux @ A_27a ) @ V0r ) @ V3vm ) @ ( inj__ty_2Equote_2Eindex @ V5x_27 ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V6t_27 ) ) ) ) ) ) ) ).
thf(conj_thm_2EringNorm_2Einterp__vl__def,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ( ! [V1vm: $i] :
( ( mem @ V1vm @ ( ty_2Equote_2Evarmap @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Er__interp__vl @ A_27a ) @ V0r ) @ V1vm ) @ ( c_2Elist_2ENIL @ ty_2Equote_2Eindex ) )
= ( ap @ ( c_2Ering_2Ering__R1 @ A_27a ) @ V0r ) ) )
& ! [V2vm: $i] :
( ( mem @ V2vm @ ( ty_2Equote_2Evarmap @ A_27a ) )
=> ! [V3x: tp__ty_2Equote_2Eindex,V4t: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex] :
( ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Er__interp__vl @ A_27a ) @ V0r ) @ V2vm ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ty_2Equote_2Eindex ) @ ( inj__ty_2Equote_2Eindex @ V3x ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V4t ) ) )
= ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Er__ivl__aux @ A_27a ) @ V0r ) @ V2vm ) @ ( inj__ty_2Equote_2Eindex @ V3x ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V4t ) ) ) ) ) ) ).
thf(conj_thm_2EringNorm_2Einterp__m__def,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ( ! [V1vm: $i] :
( ( mem @ V1vm @ ( ty_2Equote_2Evarmap @ A_27a ) )
=> ! [V2c: $i] :
( ( mem @ V2c @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Er__interp__m @ A_27a ) @ V0r ) @ V1vm ) @ V2c ) @ ( c_2Elist_2ENIL @ ty_2Equote_2Eindex ) )
= V2c ) ) )
& ! [V3vm: $i] :
( ( mem @ V3vm @ ( ty_2Equote_2Evarmap @ A_27a ) )
=> ! [V4c: $i] :
( ( mem @ V4c @ A_27a )
=> ! [V5x: tp__ty_2Equote_2Eindex,V6t: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex] :
( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Er__interp__m @ A_27a ) @ V0r ) @ V3vm ) @ V4c ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ ty_2Equote_2Eindex ) @ ( inj__ty_2Equote_2Eindex @ V5x ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V6t ) ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ering_2Ering__RM @ A_27a ) @ V0r ) @ V4c ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Er__ivl__aux @ A_27a ) @ V0r ) @ V3vm ) @ ( inj__ty_2Equote_2Eindex @ V5x ) ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V6t ) ) ) ) ) ) ) ) ).
thf(conj_thm_2EringNorm_2Eics__aux__def,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ( ! [V1vm: $i] :
( ( mem @ V1vm @ ( ty_2Equote_2Evarmap @ A_27a ) )
=> ! [V2a: $i] :
( ( mem @ V2a @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Er__ics__aux @ A_27a ) @ V0r ) @ V1vm ) @ V2a ) @ ( c_2Ecanonical_2ENil__monom @ A_27a ) )
= V2a ) ) )
& ! [V3vm: $i] :
( ( mem @ V3vm @ ( ty_2Equote_2Evarmap @ A_27a ) )
=> ! [V4a: $i] :
( ( mem @ V4a @ A_27a )
=> ! [V5l: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V6t: $i] :
( ( mem @ V6t @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Er__ics__aux @ A_27a ) @ V0r ) @ V3vm ) @ V4a ) @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V5l ) ) @ V6t ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ering_2Ering__RP @ A_27a ) @ V0r ) @ V4a ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Er__ics__aux @ A_27a ) @ V0r ) @ V3vm ) @ ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Er__interp__vl @ A_27a ) @ V0r ) @ V3vm ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V5l ) ) ) @ V6t ) ) ) ) ) )
& ! [V7vm: $i] :
( ( mem @ V7vm @ ( ty_2Equote_2Evarmap @ A_27a ) )
=> ! [V8a: $i] :
( ( mem @ V8a @ A_27a )
=> ! [V9c: $i] :
( ( mem @ V9c @ A_27a )
=> ! [V10l: tp__c_ty_2Elist_2Elist_ty_2Equote_2Eindex,V11t: $i] :
( ( mem @ V11t @ ( ty_2Ecanonical_2Ecanonical__sum @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Er__ics__aux @ A_27a ) @ V0r ) @ V7vm ) @ V8a ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ V9c ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V10l ) ) @ V11t ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ering_2Ering__RP @ A_27a ) @ V0r ) @ V8a ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Er__ics__aux @ A_27a ) @ V0r ) @ V7vm ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Er__interp__m @ A_27a ) @ V0r ) @ V7vm ) @ V9c ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V10l ) ) ) @ V11t ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2EringNorm_2Einterp__cs__def,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ( ! [V1vm: $i] :
( ( mem @ V1vm @ ( ty_2Equote_2Evarmap @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Er__interp__cs @ A_27a ) @ V0r ) @ V1vm ) @ ( c_2Ecanonical_2ENil__monom @ A_27a ) )
= ( ap @ ( c_2Ering_2Ering__R0 @ A_27a ) @ V0r ) ) )
& ! [V2vm: $i] :
( ( mem @ V2vm @ ( ty_2Equote_2Evarmap @ 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_2EringNorm_2Er__interp__cs @ A_27a ) @ V0r ) @ V2vm ) @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V3l ) ) @ V4t ) )
= ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Er__ics__aux @ A_27a ) @ V0r ) @ V2vm ) @ ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Er__interp__vl @ A_27a ) @ V0r ) @ V2vm ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V3l ) ) ) @ V4t ) ) ) )
& ! [V5vm: $i] :
( ( mem @ V5vm @ ( ty_2Equote_2Evarmap @ A_27a ) )
=> ! [V6c: $i] :
( ( mem @ V6c @ 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_2EringNorm_2Er__interp__cs @ A_27a ) @ V0r ) @ V5vm ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ V6c ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V7l ) ) @ V8t ) )
= ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Er__ics__aux @ A_27a ) @ V0r ) @ V5vm ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Er__interp__m @ A_27a ) @ V0r ) @ V5vm ) @ V6c ) @ ( inj__c_ty_2Elist_2Elist_ty_2Equote_2Eindex @ V7l ) ) ) @ V8t ) ) ) ) ) ) ) ).
thf(conj_thm_2EringNorm_2Espolynom__normalize__def,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ( ! [V1i: tp__ty_2Equote_2Eindex] :
( ( ap @ ( ap @ ( c_2EringNorm_2Er__spolynom__normalize @ A_27a ) @ V0r ) @ ( 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 ) ) )
& ! [V2c: $i] :
( ( mem @ V2c @ A_27a )
=> ( ( ap @ ( ap @ ( c_2EringNorm_2Er__spolynom__normalize @ A_27a ) @ V0r ) @ ( ap @ ( c_2Ecanonical_2ESPconst @ A_27a ) @ V2c ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ V2c ) @ ( c_2Elist_2ENIL @ ty_2Equote_2Eindex ) ) @ ( c_2Ecanonical_2ENil__monom @ A_27a ) ) ) )
& ! [V3l: $i] :
( ( mem @ V3l @ ( ty_2Ecanonical_2Espolynom @ A_27a ) )
=> ! [V4r_27: $i] :
( ( mem @ V4r_27 @ ( ty_2Ecanonical_2Espolynom @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2EringNorm_2Er__spolynom__normalize @ A_27a ) @ V0r ) @ ( ap @ ( ap @ ( c_2Ecanonical_2ESPplus @ A_27a ) @ V3l ) @ V4r_27 ) )
= ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Er__canonical__sum__merge @ A_27a ) @ V0r ) @ ( ap @ ( ap @ ( c_2EringNorm_2Er__spolynom__normalize @ A_27a ) @ V0r ) @ V3l ) ) @ ( ap @ ( ap @ ( c_2EringNorm_2Er__spolynom__normalize @ A_27a ) @ V0r ) @ V4r_27 ) ) ) ) )
& ! [V5l: $i] :
( ( mem @ V5l @ ( ty_2Ecanonical_2Espolynom @ A_27a ) )
=> ! [V6r_27: $i] :
( ( mem @ V6r_27 @ ( ty_2Ecanonical_2Espolynom @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2EringNorm_2Er__spolynom__normalize @ A_27a ) @ V0r ) @ ( ap @ ( ap @ ( c_2Ecanonical_2ESPmult @ A_27a ) @ V5l ) @ V6r_27 ) )
= ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Er__canonical__sum__prod @ A_27a ) @ V0r ) @ ( ap @ ( ap @ ( c_2EringNorm_2Er__spolynom__normalize @ A_27a ) @ V0r ) @ V5l ) ) @ ( ap @ ( ap @ ( c_2EringNorm_2Er__spolynom__normalize @ A_27a ) @ V0r ) @ V6r_27 ) ) ) ) ) ) ) ).
thf(conj_thm_2EringNorm_2Espolynom__simplify__def,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ ( ty_2Ecanonical_2Espolynom @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2EringNorm_2Er__spolynom__simplify @ A_27a ) @ V0r ) @ V1x )
= ( ap @ ( ap @ ( c_2EringNorm_2Er__canonical__sum__simplify @ A_27a ) @ V0r ) @ ( ap @ ( ap @ ( c_2EringNorm_2Er__spolynom__normalize @ A_27a ) @ V0r ) @ V1x ) ) ) ) ) ).
thf(ax_thm_2EringNorm_2Epolynom__TY__DEF,axiom,
! [A_27a: del] :
? [V0rep: $i] :
( ( mem @ V0rep @ ( arr @ ( ty_2EringNorm_2Epolynom @ 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_2EringNorm_2Epolynom @ 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_27polynom_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_27polynom_27 @ V10a0 ) ) @ ( ap @ V2_27polynom_27 @ V11a1 ) ) ) ) ) ) ) )
@ ( 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 ) )
@ ^ [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_27polynom_27 @ V15a0 ) ) @ ( ap @ V2_27polynom_27 @ V16a1 ) ) ) ) ) ) ) )
@ ( 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 ) )
@ ^ [V20a: $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
@ ( lam @ ( ty_2Eind__type_2Erecspace @ ( ty_2Epair_2Eprod @ ty_2Equote_2Eindex @ A_27a ) )
@ ^ [V21a: $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 @ ( 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 ) ) ) @ V21a )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V22n: $i] : ( c_2Eind__type_2EBOTTOM @ ( ty_2Epair_2Eprod @ ty_2Equote_2Eindex @ A_27a ) ) ) ) ) )
@ V20a ) ) )
@ ( ap @ V2_27polynom_27 @ V20a ) ) ) ) ) ) ) ) )
@ ( ap @ V2_27polynom_27 @ V3a0_27 ) ) ) ) )
@ ( ap @ V2_27polynom_27 @ V1a0_27 ) ) ) ) ) )
@ V0rep ) ) ) ).
thf(ax_thm_2EringNorm_2Epolynom__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_2EringNorm_2Epolynom @ A_27a ) @ ( arr @ ( ty_2EringNorm_2Epolynom @ A_27a ) @ A_27b ) ) )
=> ! [V4f3: $i] :
( ( mem @ V4f3 @ ( arr @ ( ty_2EringNorm_2Epolynom @ A_27a ) @ ( arr @ ( ty_2EringNorm_2Epolynom @ A_27a ) @ A_27b ) ) )
=> ! [V5f4: $i] :
( ( mem @ V5f4 @ ( arr @ ( ty_2EringNorm_2Epolynom @ A_27a ) @ A_27b ) )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Epolynom__CASE @ A_27a @ A_27b ) @ ( ap @ ( c_2EringNorm_2EPvar @ A_27a ) @ ( inj__ty_2Equote_2Eindex @ V0a ) ) ) @ V1f ) @ V2f1 ) @ V3f2 ) @ V4f3 ) @ V5f4 )
= ( ap @ V1f @ ( inj__ty_2Equote_2Eindex @ V0a ) ) ) ) ) ) ) )
& ! [V6a: $i] :
( ( mem @ V6a @ A_27a )
=> ! [V7f: $i] :
( ( mem @ V7f @ ( arr @ ty_2Equote_2Eindex @ A_27b ) )
=> ! [V8f1: $i] :
( ( mem @ V8f1 @ ( arr @ A_27a @ A_27b ) )
=> ! [V9f2: $i] :
( ( mem @ V9f2 @ ( arr @ ( ty_2EringNorm_2Epolynom @ A_27a ) @ ( arr @ ( ty_2EringNorm_2Epolynom @ A_27a ) @ A_27b ) ) )
=> ! [V10f3: $i] :
( ( mem @ V10f3 @ ( arr @ ( ty_2EringNorm_2Epolynom @ A_27a ) @ ( arr @ ( ty_2EringNorm_2Epolynom @ A_27a ) @ A_27b ) ) )
=> ! [V11f4: $i] :
( ( mem @ V11f4 @ ( arr @ ( ty_2EringNorm_2Epolynom @ A_27a ) @ A_27b ) )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Epolynom__CASE @ A_27a @ A_27b ) @ ( ap @ ( c_2EringNorm_2EPconst @ A_27a ) @ V6a ) ) @ V7f ) @ V8f1 ) @ V9f2 ) @ V10f3 ) @ V11f4 )
= ( ap @ V8f1 @ V6a ) ) ) ) ) ) ) )
& ! [V12a0: $i] :
( ( mem @ V12a0 @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
=> ! [V13a1: $i] :
( ( mem @ V13a1 @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
=> ! [V14f: $i] :
( ( mem @ V14f @ ( arr @ ty_2Equote_2Eindex @ A_27b ) )
=> ! [V15f1: $i] :
( ( mem @ V15f1 @ ( arr @ A_27a @ A_27b ) )
=> ! [V16f2: $i] :
( ( mem @ V16f2 @ ( arr @ ( ty_2EringNorm_2Epolynom @ A_27a ) @ ( arr @ ( ty_2EringNorm_2Epolynom @ A_27a ) @ A_27b ) ) )
=> ! [V17f3: $i] :
( ( mem @ V17f3 @ ( arr @ ( ty_2EringNorm_2Epolynom @ A_27a ) @ ( arr @ ( ty_2EringNorm_2Epolynom @ A_27a ) @ A_27b ) ) )
=> ! [V18f4: $i] :
( ( mem @ V18f4 @ ( arr @ ( ty_2EringNorm_2Epolynom @ A_27a ) @ A_27b ) )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Epolynom__CASE @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2EringNorm_2EPplus @ A_27a ) @ V12a0 ) @ V13a1 ) ) @ V14f ) @ V15f1 ) @ V16f2 ) @ V17f3 ) @ V18f4 )
= ( ap @ ( ap @ V16f2 @ V12a0 ) @ V13a1 ) ) ) ) ) ) ) ) )
& ! [V19a0: $i] :
( ( mem @ V19a0 @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
=> ! [V20a1: $i] :
( ( mem @ V20a1 @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
=> ! [V21f: $i] :
( ( mem @ V21f @ ( arr @ ty_2Equote_2Eindex @ A_27b ) )
=> ! [V22f1: $i] :
( ( mem @ V22f1 @ ( arr @ A_27a @ A_27b ) )
=> ! [V23f2: $i] :
( ( mem @ V23f2 @ ( arr @ ( ty_2EringNorm_2Epolynom @ A_27a ) @ ( arr @ ( ty_2EringNorm_2Epolynom @ A_27a ) @ A_27b ) ) )
=> ! [V24f3: $i] :
( ( mem @ V24f3 @ ( arr @ ( ty_2EringNorm_2Epolynom @ A_27a ) @ ( arr @ ( ty_2EringNorm_2Epolynom @ A_27a ) @ A_27b ) ) )
=> ! [V25f4: $i] :
( ( mem @ V25f4 @ ( arr @ ( ty_2EringNorm_2Epolynom @ A_27a ) @ A_27b ) )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Epolynom__CASE @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2EringNorm_2EPmult @ A_27a ) @ V19a0 ) @ V20a1 ) ) @ V21f ) @ V22f1 ) @ V23f2 ) @ V24f3 ) @ V25f4 )
= ( ap @ ( ap @ V24f3 @ V19a0 ) @ V20a1 ) ) ) ) ) ) ) ) )
& ! [V26a: $i] :
( ( mem @ V26a @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
=> ! [V27f: $i] :
( ( mem @ V27f @ ( arr @ ty_2Equote_2Eindex @ A_27b ) )
=> ! [V28f1: $i] :
( ( mem @ V28f1 @ ( arr @ A_27a @ A_27b ) )
=> ! [V29f2: $i] :
( ( mem @ V29f2 @ ( arr @ ( ty_2EringNorm_2Epolynom @ A_27a ) @ ( arr @ ( ty_2EringNorm_2Epolynom @ A_27a ) @ A_27b ) ) )
=> ! [V30f3: $i] :
( ( mem @ V30f3 @ ( arr @ ( ty_2EringNorm_2Epolynom @ A_27a ) @ ( arr @ ( ty_2EringNorm_2Epolynom @ A_27a ) @ A_27b ) ) )
=> ! [V31f4: $i] :
( ( mem @ V31f4 @ ( arr @ ( ty_2EringNorm_2Epolynom @ A_27a ) @ A_27b ) )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Epolynom__CASE @ A_27a @ A_27b ) @ ( ap @ ( c_2EringNorm_2EPopp @ A_27a ) @ V26a ) ) @ V27f ) @ V28f1 ) @ V29f2 ) @ V30f3 ) @ V31f4 )
= ( ap @ V31f4 @ V26a ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2EringNorm_2Epolynom__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_2EringNorm_2Epolynom__size @ A_27a ) @ V0f ) @ ( ap @ ( c_2EringNorm_2EPvar @ 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_2EringNorm_2Epolynom__size @ A_27a ) @ V2f ) @ ( ap @ ( c_2EringNorm_2EPconst @ 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_2EringNorm_2Epolynom @ A_27a ) )
=> ! [V6a1: $i] :
( ( mem @ V6a1 @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2EringNorm_2Epolynom__size @ A_27a ) @ V4f ) @ ( ap @ ( ap @ ( c_2EringNorm_2EPplus @ 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_2EringNorm_2Epolynom__size @ A_27a ) @ V4f ) @ V5a0 ) ) @ ( ap @ ( ap @ ( c_2EringNorm_2Epolynom__size @ A_27a ) @ V4f ) @ V6a1 ) ) ) ) ) ) ) )
& ! [V7f: $i] :
( ( mem @ V7f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V8a0: $i] :
( ( mem @ V8a0 @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
=> ! [V9a1: $i] :
( ( mem @ V9a1 @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2EringNorm_2Epolynom__size @ A_27a ) @ V7f ) @ ( ap @ ( ap @ ( c_2EringNorm_2EPmult @ 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_2EringNorm_2Epolynom__size @ A_27a ) @ V7f ) @ V8a0 ) ) @ ( ap @ ( ap @ ( c_2EringNorm_2Epolynom__size @ A_27a ) @ V7f ) @ V9a1 ) ) ) ) ) ) ) )
& ! [V10f: $i] :
( ( mem @ V10f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V11a: $i] :
( ( mem @ V11a @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2EringNorm_2Epolynom__size @ A_27a ) @ V10f ) @ ( ap @ ( c_2EringNorm_2EPopp @ A_27a ) @ V11a ) ) )
= ( 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_2EringNorm_2Epolynom__size @ A_27a ) @ V10f ) @ V11a ) ) ) ) ) ) ) ).
thf(conj_thm_2EringNorm_2Edatatype__polynom,axiom,
! [A_27a: del,V0polynom: $i] :
( ( mem @ V0polynom @ ( arr @ ( arr @ ty_2Equote_2Eindex @ ( ty_2EringNorm_2Epolynom @ A_27a ) ) @ ( arr @ ( arr @ A_27a @ ( ty_2EringNorm_2Epolynom @ A_27a ) ) @ ( arr @ ( arr @ ( ty_2EringNorm_2Epolynom @ A_27a ) @ ( arr @ ( ty_2EringNorm_2Epolynom @ A_27a ) @ ( ty_2EringNorm_2Epolynom @ A_27a ) ) ) @ ( arr @ ( arr @ ( ty_2EringNorm_2Epolynom @ A_27a ) @ ( arr @ ( ty_2EringNorm_2Epolynom @ A_27a ) @ ( ty_2EringNorm_2Epolynom @ A_27a ) ) ) @ ( arr @ ( arr @ ( ty_2EringNorm_2Epolynom @ A_27a ) @ ( ty_2EringNorm_2Epolynom @ A_27a ) ) @ bool ) ) ) ) ) )
=> ( p @ ( ap @ ( c_2Ebool_2EDATATYPE @ bool ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ V0polynom @ ( c_2EringNorm_2EPvar @ A_27a ) ) @ ( c_2EringNorm_2EPconst @ A_27a ) ) @ ( c_2EringNorm_2EPplus @ A_27a ) ) @ ( c_2EringNorm_2EPmult @ A_27a ) ) @ ( c_2EringNorm_2EPopp @ A_27a ) ) ) ) ) ).
thf(conj_thm_2EringNorm_2Epolynom__11,axiom,
! [A_27a: del] :
( ! [V0a: tp__ty_2Equote_2Eindex,V1a_27: tp__ty_2Equote_2Eindex] :
( ( ( ap @ ( c_2EringNorm_2EPvar @ A_27a ) @ ( inj__ty_2Equote_2Eindex @ V0a ) )
= ( ap @ ( c_2EringNorm_2EPvar @ 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_2EringNorm_2EPconst @ A_27a ) @ V2a )
= ( ap @ ( c_2EringNorm_2EPconst @ A_27a ) @ V3a_27 ) )
<=> ( V2a = V3a_27 ) ) ) )
& ! [V4a0: $i] :
( ( mem @ V4a0 @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
=> ! [V5a1: $i] :
( ( mem @ V5a1 @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
=> ! [V6a0_27: $i] :
( ( mem @ V6a0_27 @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
=> ! [V7a1_27: $i] :
( ( mem @ V7a1_27 @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
=> ( ( ( ap @ ( ap @ ( c_2EringNorm_2EPplus @ A_27a ) @ V4a0 ) @ V5a1 )
= ( ap @ ( ap @ ( c_2EringNorm_2EPplus @ A_27a ) @ V6a0_27 ) @ V7a1_27 ) )
<=> ( ( V4a0 = V6a0_27 )
& ( V5a1 = V7a1_27 ) ) ) ) ) ) )
& ! [V8a0: $i] :
( ( mem @ V8a0 @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
=> ! [V9a1: $i] :
( ( mem @ V9a1 @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
=> ! [V10a0_27: $i] :
( ( mem @ V10a0_27 @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
=> ! [V11a1_27: $i] :
( ( mem @ V11a1_27 @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
=> ( ( ( ap @ ( ap @ ( c_2EringNorm_2EPmult @ A_27a ) @ V8a0 ) @ V9a1 )
= ( ap @ ( ap @ ( c_2EringNorm_2EPmult @ A_27a ) @ V10a0_27 ) @ V11a1_27 ) )
<=> ( ( V8a0 = V10a0_27 )
& ( V9a1 = V11a1_27 ) ) ) ) ) ) )
& ! [V12a: $i] :
( ( mem @ V12a @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
=> ! [V13a_27: $i] :
( ( mem @ V13a_27 @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
=> ( ( ( ap @ ( c_2EringNorm_2EPopp @ A_27a ) @ V12a )
= ( ap @ ( c_2EringNorm_2EPopp @ A_27a ) @ V13a_27 ) )
<=> ( V12a = V13a_27 ) ) ) ) ) ).
thf(conj_thm_2EringNorm_2Epolynom__distinct,axiom,
! [A_27a: del] :
( ! [V0a_27: $i] :
( ( mem @ V0a_27 @ A_27a )
=> ! [V1a: tp__ty_2Equote_2Eindex] :
( ( ap @ ( c_2EringNorm_2EPvar @ A_27a ) @ ( inj__ty_2Equote_2Eindex @ V1a ) )
!= ( ap @ ( c_2EringNorm_2EPconst @ A_27a ) @ V0a_27 ) ) )
& ! [V2a1: $i] :
( ( mem @ V2a1 @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
=> ! [V3a0: $i] :
( ( mem @ V3a0 @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
=> ! [V4a: tp__ty_2Equote_2Eindex] :
( ( ap @ ( c_2EringNorm_2EPvar @ A_27a ) @ ( inj__ty_2Equote_2Eindex @ V4a ) )
!= ( ap @ ( ap @ ( c_2EringNorm_2EPplus @ A_27a ) @ V3a0 ) @ V2a1 ) ) ) )
& ! [V5a1: $i] :
( ( mem @ V5a1 @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
=> ! [V6a0: $i] :
( ( mem @ V6a0 @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
=> ! [V7a: tp__ty_2Equote_2Eindex] :
( ( ap @ ( c_2EringNorm_2EPvar @ A_27a ) @ ( inj__ty_2Equote_2Eindex @ V7a ) )
!= ( ap @ ( ap @ ( c_2EringNorm_2EPmult @ A_27a ) @ V6a0 ) @ V5a1 ) ) ) )
& ! [V8a_27: $i] :
( ( mem @ V8a_27 @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
=> ! [V9a: tp__ty_2Equote_2Eindex] :
( ( ap @ ( c_2EringNorm_2EPvar @ A_27a ) @ ( inj__ty_2Equote_2Eindex @ V9a ) )
!= ( ap @ ( c_2EringNorm_2EPopp @ A_27a ) @ V8a_27 ) ) )
& ! [V10a1: $i] :
( ( mem @ V10a1 @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
=> ! [V11a0: $i] :
( ( mem @ V11a0 @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
=> ! [V12a: $i] :
( ( mem @ V12a @ A_27a )
=> ( ( ap @ ( c_2EringNorm_2EPconst @ A_27a ) @ V12a )
!= ( ap @ ( ap @ ( c_2EringNorm_2EPplus @ A_27a ) @ V11a0 ) @ V10a1 ) ) ) ) )
& ! [V13a1: $i] :
( ( mem @ V13a1 @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
=> ! [V14a0: $i] :
( ( mem @ V14a0 @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
=> ! [V15a: $i] :
( ( mem @ V15a @ A_27a )
=> ( ( ap @ ( c_2EringNorm_2EPconst @ A_27a ) @ V15a )
!= ( ap @ ( ap @ ( c_2EringNorm_2EPmult @ A_27a ) @ V14a0 ) @ V13a1 ) ) ) ) )
& ! [V16a_27: $i] :
( ( mem @ V16a_27 @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
=> ! [V17a: $i] :
( ( mem @ V17a @ A_27a )
=> ( ( ap @ ( c_2EringNorm_2EPconst @ A_27a ) @ V17a )
!= ( ap @ ( c_2EringNorm_2EPopp @ A_27a ) @ V16a_27 ) ) ) )
& ! [V18a1_27: $i] :
( ( mem @ V18a1_27 @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
=> ! [V19a1: $i] :
( ( mem @ V19a1 @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
=> ! [V20a0_27: $i] :
( ( mem @ V20a0_27 @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
=> ! [V21a0: $i] :
( ( mem @ V21a0 @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2EringNorm_2EPplus @ A_27a ) @ V21a0 ) @ V19a1 )
!= ( ap @ ( ap @ ( c_2EringNorm_2EPmult @ A_27a ) @ V20a0_27 ) @ V18a1_27 ) ) ) ) ) )
& ! [V22a1: $i] :
( ( mem @ V22a1 @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
=> ! [V23a0: $i] :
( ( mem @ V23a0 @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
=> ! [V24a: $i] :
( ( mem @ V24a @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2EringNorm_2EPplus @ A_27a ) @ V23a0 ) @ V22a1 )
!= ( ap @ ( c_2EringNorm_2EPopp @ A_27a ) @ V24a ) ) ) ) )
& ! [V25a1: $i] :
( ( mem @ V25a1 @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
=> ! [V26a0: $i] :
( ( mem @ V26a0 @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
=> ! [V27a: $i] :
( ( mem @ V27a @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2EringNorm_2EPmult @ A_27a ) @ V26a0 ) @ V25a1 )
!= ( ap @ ( c_2EringNorm_2EPopp @ A_27a ) @ V27a ) ) ) ) ) ) ).
thf(conj_thm_2EringNorm_2Epolynom__nchotomy,axiom,
! [A_27a: del,V0pp: $i] :
( ( mem @ V0pp @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
=> ( ? [V1i: tp__ty_2Equote_2Eindex] :
( V0pp
= ( ap @ ( c_2EringNorm_2EPvar @ A_27a ) @ ( inj__ty_2Equote_2Eindex @ V1i ) ) )
| ? [V2a: $i] :
( ( mem @ V2a @ A_27a )
& ( V0pp
= ( ap @ ( c_2EringNorm_2EPconst @ A_27a ) @ V2a ) ) )
| ? [V3p: $i] :
( ( mem @ V3p @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
& ? [V4p0: $i] :
( ( mem @ V4p0 @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
& ( V0pp
= ( ap @ ( ap @ ( c_2EringNorm_2EPplus @ A_27a ) @ V3p ) @ V4p0 ) ) ) )
| ? [V5p: $i] :
( ( mem @ V5p @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
& ? [V6p0: $i] :
( ( mem @ V6p0 @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
& ( V0pp
= ( ap @ ( ap @ ( c_2EringNorm_2EPmult @ A_27a ) @ V5p ) @ V6p0 ) ) ) )
| ? [V7p: $i] :
( ( mem @ V7p @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
& ( V0pp
= ( ap @ ( c_2EringNorm_2EPopp @ A_27a ) @ V7p ) ) ) ) ) ).
thf(conj_thm_2EringNorm_2Epolynom__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_2EringNorm_2Epolynom @ A_27a ) @ ( arr @ ( ty_2EringNorm_2Epolynom @ A_27a ) @ ( arr @ A_27b @ ( arr @ A_27b @ A_27b ) ) ) ) )
=> ! [V3f3: $i] :
( ( mem @ V3f3 @ ( arr @ ( ty_2EringNorm_2Epolynom @ A_27a ) @ ( arr @ ( ty_2EringNorm_2Epolynom @ A_27a ) @ ( arr @ A_27b @ ( arr @ A_27b @ A_27b ) ) ) ) )
=> ! [V4f4: $i] :
( ( mem @ V4f4 @ ( arr @ ( ty_2EringNorm_2Epolynom @ A_27a ) @ ( arr @ A_27b @ A_27b ) ) )
=> ? [V5fn: $i] :
( ( mem @ V5fn @ ( arr @ ( ty_2EringNorm_2Epolynom @ A_27a ) @ A_27b ) )
& ! [V6a: tp__ty_2Equote_2Eindex] :
( ( ap @ V5fn @ ( ap @ ( c_2EringNorm_2EPvar @ A_27a ) @ ( inj__ty_2Equote_2Eindex @ V6a ) ) )
= ( ap @ V0f0 @ ( inj__ty_2Equote_2Eindex @ V6a ) ) )
& ! [V7a: $i] :
( ( mem @ V7a @ A_27a )
=> ( ( ap @ V5fn @ ( ap @ ( c_2EringNorm_2EPconst @ A_27a ) @ V7a ) )
= ( ap @ V1f1 @ V7a ) ) )
& ! [V8a0: $i] :
( ( mem @ V8a0 @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
=> ! [V9a1: $i] :
( ( mem @ V9a1 @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
=> ( ( ap @ V5fn @ ( ap @ ( ap @ ( c_2EringNorm_2EPplus @ A_27a ) @ V8a0 ) @ V9a1 ) )
= ( ap @ ( ap @ ( ap @ ( ap @ V2f2 @ V8a0 ) @ V9a1 ) @ ( ap @ V5fn @ V8a0 ) ) @ ( ap @ V5fn @ V9a1 ) ) ) ) )
& ! [V10a0: $i] :
( ( mem @ V10a0 @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
=> ! [V11a1: $i] :
( ( mem @ V11a1 @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
=> ( ( ap @ V5fn @ ( ap @ ( ap @ ( c_2EringNorm_2EPmult @ A_27a ) @ V10a0 ) @ V11a1 ) )
= ( ap @ ( ap @ ( ap @ ( ap @ V3f3 @ V10a0 ) @ V11a1 ) @ ( ap @ V5fn @ V10a0 ) ) @ ( ap @ V5fn @ V11a1 ) ) ) ) )
& ! [V12a: $i] :
( ( mem @ V12a @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
=> ( ( ap @ V5fn @ ( ap @ ( c_2EringNorm_2EPopp @ A_27a ) @ V12a ) )
= ( ap @ ( ap @ V4f4 @ V12a ) @ ( ap @ V5fn @ V12a ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2EringNorm_2Epolynom__induction,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2EringNorm_2Epolynom @ A_27a ) @ bool ) )
=> ( ( ! [V1i: tp__ty_2Equote_2Eindex] : ( p @ ( ap @ V0P @ ( ap @ ( c_2EringNorm_2EPvar @ A_27a ) @ ( inj__ty_2Equote_2Eindex @ V1i ) ) ) )
& ! [V2a: $i] :
( ( mem @ V2a @ A_27a )
=> ( p @ ( ap @ V0P @ ( ap @ ( c_2EringNorm_2EPconst @ A_27a ) @ V2a ) ) ) )
& ! [V3p: $i] :
( ( mem @ V3p @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
=> ! [V4p0: $i] :
( ( mem @ V4p0 @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
=> ( ( ( p @ ( ap @ V0P @ V3p ) )
& ( p @ ( ap @ V0P @ V4p0 ) ) )
=> ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2EringNorm_2EPplus @ A_27a ) @ V3p ) @ V4p0 ) ) ) ) ) )
& ! [V5p: $i] :
( ( mem @ V5p @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
=> ! [V6p0: $i] :
( ( mem @ V6p0 @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
=> ( ( ( p @ ( ap @ V0P @ V5p ) )
& ( p @ ( ap @ V0P @ V6p0 ) ) )
=> ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2EringNorm_2EPmult @ A_27a ) @ V5p ) @ V6p0 ) ) ) ) ) )
& ! [V7p: $i] :
( ( mem @ V7p @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
=> ( ( p @ ( ap @ V0P @ V7p ) )
=> ( p @ ( ap @ V0P @ ( ap @ ( c_2EringNorm_2EPopp @ A_27a ) @ V7p ) ) ) ) ) )
=> ! [V8p: $i] :
( ( mem @ V8p @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
=> ( p @ ( ap @ V0P @ V8p ) ) ) ) ) ).
thf(conj_thm_2EringNorm_2Epolynom__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_2EringNorm_2Epolynom @ A_27a ) @ ( arr @ ( ty_2EringNorm_2Epolynom @ A_27a ) @ A_27b ) ) )
=> ! [V3f3_27: $i] :
( ( mem @ V3f3_27 @ ( arr @ ( ty_2EringNorm_2Epolynom @ A_27a ) @ ( arr @ ( ty_2EringNorm_2Epolynom @ A_27a ) @ A_27b ) ) )
=> ! [V4f4_27: $i] :
( ( mem @ V4f4_27 @ ( arr @ ( ty_2EringNorm_2Epolynom @ A_27a ) @ A_27b ) )
=> ! [V5M: $i] :
( ( mem @ V5M @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
=> ! [V6M_27: $i] :
( ( mem @ V6M_27 @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
=> ! [V7f: $i] :
( ( mem @ V7f @ ( arr @ ty_2Equote_2Eindex @ A_27b ) )
=> ! [V8f1: $i] :
( ( mem @ V8f1 @ ( arr @ A_27a @ A_27b ) )
=> ! [V9f2: $i] :
( ( mem @ V9f2 @ ( arr @ ( ty_2EringNorm_2Epolynom @ A_27a ) @ ( arr @ ( ty_2EringNorm_2Epolynom @ A_27a ) @ A_27b ) ) )
=> ! [V10f3: $i] :
( ( mem @ V10f3 @ ( arr @ ( ty_2EringNorm_2Epolynom @ A_27a ) @ ( arr @ ( ty_2EringNorm_2Epolynom @ A_27a ) @ A_27b ) ) )
=> ! [V11f4: $i] :
( ( mem @ V11f4 @ ( arr @ ( ty_2EringNorm_2Epolynom @ A_27a ) @ A_27b ) )
=> ( ( ( V5M = V6M_27 )
& ! [V12a: tp__ty_2Equote_2Eindex] :
( ( V6M_27
= ( ap @ ( c_2EringNorm_2EPvar @ A_27a ) @ ( inj__ty_2Equote_2Eindex @ V12a ) ) )
=> ( ( ap @ V7f @ ( inj__ty_2Equote_2Eindex @ V12a ) )
= ( ap @ V0f_27 @ ( inj__ty_2Equote_2Eindex @ V12a ) ) ) )
& ! [V13a: $i] :
( ( mem @ V13a @ A_27a )
=> ( ( V6M_27
= ( ap @ ( c_2EringNorm_2EPconst @ A_27a ) @ V13a ) )
=> ( ( ap @ V8f1 @ V13a )
= ( ap @ V1f1_27 @ V13a ) ) ) )
& ! [V14a0: $i] :
( ( mem @ V14a0 @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
=> ! [V15a1: $i] :
( ( mem @ V15a1 @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
=> ( ( V6M_27
= ( ap @ ( ap @ ( c_2EringNorm_2EPplus @ A_27a ) @ V14a0 ) @ V15a1 ) )
=> ( ( ap @ ( ap @ V9f2 @ V14a0 ) @ V15a1 )
= ( ap @ ( ap @ V2f2_27 @ V14a0 ) @ V15a1 ) ) ) ) )
& ! [V16a0: $i] :
( ( mem @ V16a0 @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
=> ! [V17a1: $i] :
( ( mem @ V17a1 @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
=> ( ( V6M_27
= ( ap @ ( ap @ ( c_2EringNorm_2EPmult @ A_27a ) @ V16a0 ) @ V17a1 ) )
=> ( ( ap @ ( ap @ V10f3 @ V16a0 ) @ V17a1 )
= ( ap @ ( ap @ V3f3_27 @ V16a0 ) @ V17a1 ) ) ) ) )
& ! [V18a: $i] :
( ( mem @ V18a @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
=> ( ( V6M_27
= ( ap @ ( c_2EringNorm_2EPopp @ A_27a ) @ V18a ) )
=> ( ( ap @ V11f4 @ V18a )
= ( ap @ V4f4_27 @ V18a ) ) ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Epolynom__CASE @ A_27a @ A_27b ) @ V5M ) @ V7f ) @ V8f1 ) @ V9f2 ) @ V10f3 ) @ V11f4 )
= ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Epolynom__CASE @ A_27a @ A_27b ) @ V6M_27 ) @ V0f_27 ) @ V1f1_27 ) @ V2f2_27 ) @ V3f3_27 ) @ V4f4_27 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2EringNorm_2Epolynom__case__eq,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ ( ty_2EringNorm_2Epolynom @ 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_2EringNorm_2Epolynom @ A_27a ) @ ( arr @ ( ty_2EringNorm_2Epolynom @ A_27a ) @ A_27b ) ) )
=> ! [V4f3: $i] :
( ( mem @ V4f3 @ ( arr @ ( ty_2EringNorm_2Epolynom @ A_27a ) @ ( arr @ ( ty_2EringNorm_2Epolynom @ A_27a ) @ A_27b ) ) )
=> ! [V5f4: $i] :
( ( mem @ V5f4 @ ( arr @ ( ty_2EringNorm_2Epolynom @ A_27a ) @ A_27b ) )
=> ! [V6v: $i] :
( ( mem @ V6v @ A_27b )
=> ( ( ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Epolynom__CASE @ A_27a @ A_27b ) @ V0x ) @ V1f ) @ V2f1 ) @ V3f2 ) @ V4f3 ) @ V5f4 )
= V6v )
<=> ( ? [V7i: tp__ty_2Equote_2Eindex] :
( ( V0x
= ( ap @ ( c_2EringNorm_2EPvar @ A_27a ) @ ( inj__ty_2Equote_2Eindex @ V7i ) ) )
& ( ( ap @ V1f @ ( inj__ty_2Equote_2Eindex @ V7i ) )
= V6v ) )
| ? [V8a: $i] :
( ( mem @ V8a @ A_27a )
& ( V0x
= ( ap @ ( c_2EringNorm_2EPconst @ A_27a ) @ V8a ) )
& ( ( ap @ V2f1 @ V8a )
= V6v ) )
| ? [V9p: $i] :
( ( mem @ V9p @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
& ? [V10p0: $i] :
( ( mem @ V10p0 @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
& ( V0x
= ( ap @ ( ap @ ( c_2EringNorm_2EPplus @ A_27a ) @ V9p ) @ V10p0 ) )
& ( ( ap @ ( ap @ V3f2 @ V9p ) @ V10p0 )
= V6v ) ) )
| ? [V11p: $i] :
( ( mem @ V11p @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
& ? [V12p0: $i] :
( ( mem @ V12p0 @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
& ( V0x
= ( ap @ ( ap @ ( c_2EringNorm_2EPmult @ A_27a ) @ V11p ) @ V12p0 ) )
& ( ( ap @ ( ap @ V4f3 @ V11p ) @ V12p0 )
= V6v ) ) )
| ? [V13p: $i] :
( ( mem @ V13p @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
& ( V0x
= ( ap @ ( c_2EringNorm_2EPopp @ A_27a ) @ V13p ) )
& ( ( ap @ V5f4 @ V13p )
= V6v ) ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2EringNorm_2Epolynom__normalize__def,axiom,
! [A_27a: del] :
( ! [V0r: $i] :
( ( mem @ V0r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ! [V1i: tp__ty_2Equote_2Eindex] :
( ( ap @ ( ap @ ( c_2EringNorm_2Epolynom__normalize @ A_27a ) @ V0r ) @ ( ap @ ( c_2EringNorm_2EPvar @ 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 ) ) ) )
& ! [V2r: $i] :
( ( mem @ V2r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ! [V3c: $i] :
( ( mem @ V3c @ A_27a )
=> ( ( ap @ ( ap @ ( c_2EringNorm_2Epolynom__normalize @ A_27a ) @ V2r ) @ ( ap @ ( c_2EringNorm_2EPconst @ 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 ) ) ) ) )
& ! [V4r: $i] :
( ( mem @ V4r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ! [V5pl: $i] :
( ( mem @ V5pl @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
=> ! [V6pr: $i] :
( ( mem @ V6pr @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2EringNorm_2Epolynom__normalize @ A_27a ) @ V4r ) @ ( ap @ ( ap @ ( c_2EringNorm_2EPplus @ A_27a ) @ V5pl ) @ V6pr ) )
= ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Er__canonical__sum__merge @ A_27a ) @ V4r ) @ ( ap @ ( ap @ ( c_2EringNorm_2Epolynom__normalize @ A_27a ) @ V4r ) @ V5pl ) ) @ ( ap @ ( ap @ ( c_2EringNorm_2Epolynom__normalize @ A_27a ) @ V4r ) @ V6pr ) ) ) ) ) )
& ! [V7r: $i] :
( ( mem @ V7r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ! [V8pl: $i] :
( ( mem @ V8pl @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
=> ! [V9pr: $i] :
( ( mem @ V9pr @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2EringNorm_2Epolynom__normalize @ A_27a ) @ V7r ) @ ( ap @ ( ap @ ( c_2EringNorm_2EPmult @ A_27a ) @ V8pl ) @ V9pr ) )
= ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Er__canonical__sum__prod @ A_27a ) @ V7r ) @ ( ap @ ( ap @ ( c_2EringNorm_2Epolynom__normalize @ A_27a ) @ V7r ) @ V8pl ) ) @ ( ap @ ( ap @ ( c_2EringNorm_2Epolynom__normalize @ A_27a ) @ V7r ) @ V9pr ) ) ) ) ) )
& ! [V10r: $i] :
( ( mem @ V10r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ! [V11p: $i] :
( ( mem @ V11p @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2EringNorm_2Epolynom__normalize @ A_27a ) @ V10r ) @ ( ap @ ( c_2EringNorm_2EPopp @ A_27a ) @ V11p ) )
= ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Er__canonical__sum__scalar3 @ A_27a ) @ V10r ) @ ( ap @ ( ap @ ( c_2Ering_2Ering__RN @ A_27a ) @ V10r ) @ ( ap @ ( c_2Ering_2Ering__R1 @ A_27a ) @ V10r ) ) ) @ ( c_2Elist_2ENIL @ ty_2Equote_2Eindex ) ) @ ( ap @ ( ap @ ( c_2EringNorm_2Epolynom__normalize @ A_27a ) @ V10r ) @ V11p ) ) ) ) ) ) ).
thf(ax_thm_2EringNorm_2Epolynom__simplify__def,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2EringNorm_2Epolynom__simplify @ A_27a ) @ V0r ) @ V1x )
= ( ap @ ( ap @ ( c_2EringNorm_2Er__canonical__sum__simplify @ A_27a ) @ V0r ) @ ( ap @ ( ap @ ( c_2EringNorm_2Epolynom__normalize @ A_27a ) @ V0r ) @ V1x ) ) ) ) ) ).
thf(ax_thm_2EringNorm_2Einterp__p__def,axiom,
! [A_27a: del] :
( ! [V0r: $i] :
( ( mem @ V0r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ! [V1vm: $i] :
( ( mem @ V1vm @ ( ty_2Equote_2Evarmap @ A_27a ) )
=> ! [V2c: $i] :
( ( mem @ V2c @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Einterp__p @ A_27a ) @ V0r ) @ V1vm ) @ ( ap @ ( c_2EringNorm_2EPconst @ A_27a ) @ V2c ) )
= V2c ) ) ) )
& ! [V3r: $i] :
( ( mem @ V3r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ! [V4vm: $i] :
( ( mem @ V4vm @ ( ty_2Equote_2Evarmap @ A_27a ) )
=> ! [V5i: tp__ty_2Equote_2Eindex] :
( ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Einterp__p @ A_27a ) @ V3r ) @ V4vm ) @ ( ap @ ( c_2EringNorm_2EPvar @ A_27a ) @ ( inj__ty_2Equote_2Eindex @ V5i ) ) )
= ( ap @ ( ap @ ( c_2Equote_2Evarmap__find @ A_27a ) @ ( inj__ty_2Equote_2Eindex @ V5i ) ) @ V4vm ) ) ) )
& ! [V6r: $i] :
( ( mem @ V6r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ! [V7vm: $i] :
( ( mem @ V7vm @ ( ty_2Equote_2Evarmap @ A_27a ) )
=> ! [V8p1: $i] :
( ( mem @ V8p1 @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
=> ! [V9p2: $i] :
( ( mem @ V9p2 @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Einterp__p @ A_27a ) @ V6r ) @ V7vm ) @ ( ap @ ( ap @ ( c_2EringNorm_2EPplus @ A_27a ) @ V8p1 ) @ V9p2 ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ering_2Ering__RP @ A_27a ) @ V6r ) @ ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Einterp__p @ A_27a ) @ V6r ) @ V7vm ) @ V8p1 ) ) @ ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Einterp__p @ A_27a ) @ V6r ) @ V7vm ) @ V9p2 ) ) ) ) ) ) )
& ! [V10r: $i] :
( ( mem @ V10r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ! [V11vm: $i] :
( ( mem @ V11vm @ ( ty_2Equote_2Evarmap @ A_27a ) )
=> ! [V12p1: $i] :
( ( mem @ V12p1 @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
=> ! [V13p2: $i] :
( ( mem @ V13p2 @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Einterp__p @ A_27a ) @ V10r ) @ V11vm ) @ ( ap @ ( ap @ ( c_2EringNorm_2EPmult @ A_27a ) @ V12p1 ) @ V13p2 ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ering_2Ering__RM @ A_27a ) @ V10r ) @ ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Einterp__p @ A_27a ) @ V10r ) @ V11vm ) @ V12p1 ) ) @ ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Einterp__p @ A_27a ) @ V10r ) @ V11vm ) @ V13p2 ) ) ) ) ) ) )
& ! [V14r: $i] :
( ( mem @ V14r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ! [V15vm: $i] :
( ( mem @ V15vm @ ( ty_2Equote_2Evarmap @ A_27a ) )
=> ! [V16p1: $i] :
( ( mem @ V16p1 @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Einterp__p @ A_27a ) @ V14r ) @ V15vm ) @ ( ap @ ( c_2EringNorm_2EPopp @ A_27a ) @ V16p1 ) )
= ( ap @ ( ap @ ( c_2Ering_2Ering__RN @ A_27a ) @ V14r ) @ ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Einterp__p @ A_27a ) @ V14r ) @ V15vm ) @ V16p1 ) ) ) ) ) ) ) ).
thf(conj_thm_2EringNorm_2Epolynom__normalize__ok,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Ering_2Eis__ring @ A_27a ) @ V0r ) )
=> ! [V1vm: $i] :
( ( mem @ V1vm @ ( ty_2Equote_2Evarmap @ A_27a ) )
=> ! [V2p: $i] :
( ( mem @ V2p @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Er__interp__cs @ A_27a ) @ V0r ) @ V1vm ) @ ( ap @ ( ap @ ( c_2EringNorm_2Epolynom__normalize @ A_27a ) @ V0r ) @ V2p ) )
= ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Einterp__p @ A_27a ) @ V0r ) @ V1vm ) @ V2p ) ) ) ) ) ) ).
thf(conj_thm_2EringNorm_2Epolynom__simplify__ok,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( ty_2Ering_2Ering @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Ering_2Eis__ring @ A_27a ) @ V0r ) )
=> ! [V1vm: $i] :
( ( mem @ V1vm @ ( ty_2Equote_2Evarmap @ A_27a ) )
=> ! [V2p: $i] :
( ( mem @ V2p @ ( ty_2EringNorm_2Epolynom @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Er__interp__cs @ A_27a ) @ V0r ) @ V1vm ) @ ( ap @ ( ap @ ( c_2EringNorm_2Epolynom__simplify @ A_27a ) @ V0r ) @ V2p ) )
= ( ap @ ( ap @ ( ap @ ( c_2EringNorm_2Einterp__p @ A_27a ) @ V0r ) @ V1vm ) @ V2p ) ) ) ) ) ) ).
%------------------------------------------------------------------------------