%------------------------------------------------------------------------------
% File     : ITP114_7 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain   : Interactive Theorem Proving
% Axioms   : HOL4 syntactic export, chainy mode
% Version  : [BG+19] axioms.
% English  :

% Refs     : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
%          : [Gau19] Gauthier (2019), Email to Geoff Sutcliffe
% Source   : [BG+19]
% Names    : ringNorm.ax [Gau19]
%          : HL4114_7.ax [TPAP]

% Status   : Satisfiable
% Syntax   : Number of formulae    :  358 ( 113 unt; 201 typ;   0 def)
%            Number of atoms       :  311 ( 254 equ)
%            Maximal formula atoms :   19 (   0 avg)
%            Number of connectives :  167 (  13   ~;  15   |;  91   &)
%                                         (  22 <=>;  26  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   28 (   5 avg)
%            Maximal term depth    :   14 (   2 avg)
%            Number of types       :    5 (   4 usr)
%            Number of type conns  :  186 ( 106   >;  80   *;   0   +;   0  <<)
%            Number of predicates  :    2 (   1 usr;   0 prp; 1-2 aty)
%            Number of functors    :  196 ( 196 usr;  15 con; 0-8 aty)
%            Number of variables   :  899 ( 701   !;  24   ?; 899   :)
%                                         ( 174  !>;   0  ?*;   0  @-;   0  @+)
% SPC      : TF1_SAT_EQU_NAR

% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
tff(tyop_2Ecanonical_2Ecanonical__sum,type,
    tyop_2Ecanonical_2Ecanonical__sum: $tType > $tType ).

tff(tyop_2Ecanonical_2Espolynom,type,
    tyop_2Ecanonical_2Espolynom: $tType > $tType ).

tff(tyop_2Eind__type_2Erecspace,type,
    tyop_2Eind__type_2Erecspace: $tType > $tType ).

tff(tyop_2Elist_2Elist,type,
    tyop_2Elist_2Elist: $tType > $tType ).

tff(tyop_2Emin_2Ebool,type,
    tyop_2Emin_2Ebool: $tType ).

tff(tyop_2Emin_2Efun,type,
    tyop_2Emin_2Efun: ( $tType * $tType ) > $tType ).

tff(tyop_2Enum_2Enum,type,
    tyop_2Enum_2Enum: $tType ).

tff(tyop_2Epair_2Eprod,type,
    tyop_2Epair_2Eprod: ( $tType * $tType ) > $tType ).

tff(tyop_2Equote_2Eindex,type,
    tyop_2Equote_2Eindex: $tType ).

tff(tyop_2Equote_2Evarmap,type,
    tyop_2Equote_2Evarmap: $tType > $tType ).

tff(tyop_2Ering_2Ering,type,
    tyop_2Ering_2Ering: $tType > $tType ).

tff(tyop_2EringNorm_2Epolynom,type,
    tyop_2EringNorm_2Epolynom: $tType > $tType ).

tff(tyop_2Esemi__ring_2Esemi__ring,type,
    tyop_2Esemi__ring_2Esemi__ring: $tType > $tType ).

tff(tyop_2EternaryComparisons_2Eordering,type,
    tyop_2EternaryComparisons_2Eordering: $tType ).

tff(app_2E2,type,
    app_2E2: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Emin_2Efun(A_27a,A_27b) * A_27a ) > A_27b ) ).

tff(p,type,
    p: tyop_2Emin_2Ebool > $o ).

tff(combin_i_2E0,type,
    combin_i_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,A_27a) ).

tff(combin_k_2E0,type,
    combin_k_2E0: 
      !>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27a)) ).

tff(combin_s_2E0,type,
    combin_s_2E0: 
      !>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27c))) ).

tff(c_2Ebool_2E_21_2E0,type,
    c_2Ebool_2E_21_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool) ).

tff(c_2Ebool_2E_21_2E1,type,
    c_2Ebool_2E_21_2E1: 
      !>[A_27a: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) > tyop_2Emin_2Ebool ) ).

tff(c_2Earithmetic_2E_2B_2E0,type,
    c_2Earithmetic_2E_2B_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)) ).

tff(c_2Earithmetic_2E_2B_2E2,type,
    c_2Earithmetic_2E_2B_2E2: ( tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > tyop_2Enum_2Enum ).

tff(c_2Epair_2E_2C_2E0,type,
    c_2Epair_2E_2C_2E0: 
      !>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27a,A_27b))) ).

tff(c_2Epair_2E_2C_2E2,type,
    c_2Epair_2E_2C_2E2: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( A_27a * A_27b ) > tyop_2Epair_2Eprod(A_27a,A_27b) ) ).

tff(c_2Ebool_2E_2F_5C_2E0,type,
    c_2Ebool_2E_2F_5C_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool)) ).

tff(c_2Ebool_2E_2F_5C_2E2,type,
    c_2Ebool_2E_2F_5C_2E2: ( tyop_2Emin_2Ebool * tyop_2Emin_2Ebool ) > tyop_2Emin_2Ebool ).

tff(c_2Enum_2E0_2E0,type,
    c_2Enum_2E0_2E0: tyop_2Enum_2Enum ).

tff(c_2Emin_2E_3D_2E0,type,
    c_2Emin_2E_3D_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)) ).

tff(c_2Emin_2E_3D_2E2,type,
    c_2Emin_2E_3D_2E2: 
      !>[A_27a: $tType] : ( ( A_27a * A_27a ) > tyop_2Emin_2Ebool ) ).

tff(c_2Emin_2E_3D_3D_3E_2E0,type,
    c_2Emin_2E_3D_3D_3E_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool)) ).

tff(c_2Emin_2E_3D_3D_3E_2E2,type,
    c_2Emin_2E_3D_3D_3E_2E2: ( tyop_2Emin_2Ebool * tyop_2Emin_2Ebool ) > tyop_2Emin_2Ebool ).

tff(c_2Ebool_2E_3F_2E0,type,
    c_2Ebool_2E_3F_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool) ).

tff(c_2Ebool_2E_3F_2E1,type,
    c_2Ebool_2E_3F_2E1: 
      !>[A_27a: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) > tyop_2Emin_2Ebool ) ).

tff(c_2Ebool_2EARB_2E0,type,
    c_2Ebool_2EARB_2E0: 
      !>[A_27a: $tType] : A_27a ).

tff(c_2Earithmetic_2EBIT1_2E0,type,
    c_2Earithmetic_2EBIT1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum) ).

tff(c_2Earithmetic_2EBIT1_2E1,type,
    c_2Earithmetic_2EBIT1_2E1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).

tff(c_2Eind__type_2EBOTTOM_2E0,type,
    c_2Eind__type_2EBOTTOM_2E0: 
      !>[A_27a: $tType] : tyop_2Eind__type_2Erecspace(A_27a) ).

tff(c_2Ebool_2ECOND_2E0,type,
    c_2Ebool_2ECOND_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a))) ).

tff(c_2Ebool_2ECOND_2E3,type,
    c_2Ebool_2ECOND_2E3: 
      !>[A_27a: $tType] : ( ( tyop_2Emin_2Ebool * A_27a * A_27a ) > A_27a ) ).

tff(c_2Elist_2ECONS_2E0,type,
    c_2Elist_2ECONS_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Elist_2Elist(A_27a))) ).

tff(c_2Elist_2ECONS_2E2,type,
    c_2Elist_2ECONS_2E2: 
      !>[A_27a: $tType] : ( ( A_27a * tyop_2Elist_2Elist(A_27a) ) > tyop_2Elist_2Elist(A_27a) ) ).

tff(c_2Eind__type_2ECONSTR_2E0,type,
    c_2Eind__type_2ECONSTR_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(A_27a)),tyop_2Eind__type_2Erecspace(A_27a)))) ).

tff(c_2Eind__type_2ECONSTR_2E3,type,
    c_2Eind__type_2ECONSTR_2E3: 
      !>[A_27a: $tType] : ( ( tyop_2Enum_2Enum * A_27a * tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(A_27a)) ) > tyop_2Eind__type_2Erecspace(A_27a) ) ).

tff(c_2Ecanonical_2ECons__monom_2E0,type,
    c_2Ecanonical_2ECons__monom_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)))) ).

tff(c_2Ecanonical_2ECons__monom_2E3,type,
    c_2Ecanonical_2ECons__monom_2E3: 
      !>[A_27a: $tType] : ( ( A_27a * tyop_2Elist_2Elist(tyop_2Equote_2Eindex) * tyop_2Ecanonical_2Ecanonical__sum(A_27a) ) > tyop_2Ecanonical_2Ecanonical__sum(A_27a) ) ).

tff(c_2Ecanonical_2ECons__varlist_2E0,type,
    c_2Ecanonical_2ECons__varlist_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a))) ).

tff(c_2Ecanonical_2ECons__varlist_2E2,type,
    c_2Ecanonical_2ECons__varlist_2E2: 
      !>[A_27a: $tType] : ( ( tyop_2Elist_2Elist(tyop_2Equote_2Eindex) * tyop_2Ecanonical_2Ecanonical__sum(A_27a) ) > tyop_2Ecanonical_2Ecanonical__sum(A_27a) ) ).

tff(c_2Ebool_2EDATATYPE_2E0,type,
    c_2Ebool_2EDATATYPE_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ).

tff(c_2Ebool_2EDATATYPE_2E1,type,
    c_2Ebool_2EDATATYPE_2E1: 
      !>[A_27a: $tType] : ( A_27a > tyop_2Emin_2Ebool ) ).

tff(c_2Ebool_2EF_2E0,type,
    c_2Ebool_2EF_2E0: tyop_2Emin_2Ebool ).

tff(c_2Eind__type_2EFCONS_2E0,type,
    c_2Eind__type_2EFCONS_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a))) ).

tff(c_2Eind__type_2EFCONS_2E2,type,
    c_2Eind__type_2EFCONS_2E2: 
      !>[A_27a: $tType] : ( ( A_27a * tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a) ) > tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a) ) ).

tff(c_2Elist_2ENIL_2E0,type,
    c_2Elist_2ENIL_2E0: 
      !>[A_27a: $tType] : tyop_2Elist_2Elist(A_27a) ).

tff(c_2Earithmetic_2ENUMERAL_2E0,type,
    c_2Earithmetic_2ENUMERAL_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum) ).

tff(c_2Earithmetic_2ENUMERAL_2E1,type,
    c_2Earithmetic_2ENUMERAL_2E1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).

tff(c_2Ecanonical_2ENil__monom_2E0,type,
    c_2Ecanonical_2ENil__monom_2E0: 
      !>[A_27a: $tType] : tyop_2Ecanonical_2Ecanonical__sum(A_27a) ).

tff(c_2EringNorm_2EPconst_2E0,type,
    c_2EringNorm_2EPconst_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2EringNorm_2Epolynom(A_27a)) ).

tff(c_2EringNorm_2EPconst_2E1,type,
    c_2EringNorm_2EPconst_2E1: 
      !>[A_27a: $tType] : ( A_27a > tyop_2EringNorm_2Epolynom(A_27a) ) ).

tff(c_2EringNorm_2EPmult_2E0,type,
    c_2EringNorm_2EPmult_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2EringNorm_2Epolynom(A_27a))) ).

tff(c_2EringNorm_2EPmult_2E2,type,
    c_2EringNorm_2EPmult_2E2: 
      !>[A_27a: $tType] : ( ( tyop_2EringNorm_2Epolynom(A_27a) * tyop_2EringNorm_2Epolynom(A_27a) ) > tyop_2EringNorm_2Epolynom(A_27a) ) ).

tff(c_2EringNorm_2EPopp_2E0,type,
    c_2EringNorm_2EPopp_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2EringNorm_2Epolynom(A_27a)) ).

tff(c_2EringNorm_2EPopp_2E1,type,
    c_2EringNorm_2EPopp_2E1: 
      !>[A_27a: $tType] : ( tyop_2EringNorm_2Epolynom(A_27a) > tyop_2EringNorm_2Epolynom(A_27a) ) ).

tff(c_2EringNorm_2EPplus_2E0,type,
    c_2EringNorm_2EPplus_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2EringNorm_2Epolynom(A_27a))) ).

tff(c_2EringNorm_2EPplus_2E2,type,
    c_2EringNorm_2EPplus_2E2: 
      !>[A_27a: $tType] : ( ( tyop_2EringNorm_2Epolynom(A_27a) * tyop_2EringNorm_2Epolynom(A_27a) ) > tyop_2EringNorm_2Epolynom(A_27a) ) ).

tff(c_2EringNorm_2EPvar_2E0,type,
    c_2EringNorm_2EPvar_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Equote_2Eindex,tyop_2EringNorm_2Epolynom(A_27a)) ).

tff(c_2EringNorm_2EPvar_2E1,type,
    c_2EringNorm_2EPvar_2E1: 
      !>[A_27a: $tType] : ( tyop_2Equote_2Eindex > tyop_2EringNorm_2Epolynom(A_27a) ) ).

tff(c_2Ecanonical_2ESPconst_2E0,type,
    c_2Ecanonical_2ESPconst_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Ecanonical_2Espolynom(A_27a)) ).

tff(c_2Ecanonical_2ESPconst_2E1,type,
    c_2Ecanonical_2ESPconst_2E1: 
      !>[A_27a: $tType] : ( A_27a > tyop_2Ecanonical_2Espolynom(A_27a) ) ).

tff(c_2Ecanonical_2ESPmult_2E0,type,
    c_2Ecanonical_2ESPmult_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Ecanonical_2Espolynom(A_27a))) ).

tff(c_2Ecanonical_2ESPmult_2E2,type,
    c_2Ecanonical_2ESPmult_2E2: 
      !>[A_27a: $tType] : ( ( tyop_2Ecanonical_2Espolynom(A_27a) * tyop_2Ecanonical_2Espolynom(A_27a) ) > tyop_2Ecanonical_2Espolynom(A_27a) ) ).

tff(c_2Ecanonical_2ESPplus_2E0,type,
    c_2Ecanonical_2ESPplus_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Ecanonical_2Espolynom(A_27a))) ).

tff(c_2Ecanonical_2ESPplus_2E2,type,
    c_2Ecanonical_2ESPplus_2E2: 
      !>[A_27a: $tType] : ( ( tyop_2Ecanonical_2Espolynom(A_27a) * tyop_2Ecanonical_2Espolynom(A_27a) ) > tyop_2Ecanonical_2Espolynom(A_27a) ) ).

tff(c_2Ecanonical_2ESPvar_2E0,type,
    c_2Ecanonical_2ESPvar_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Equote_2Eindex,tyop_2Ecanonical_2Espolynom(A_27a)) ).

tff(c_2Ecanonical_2ESPvar_2E1,type,
    c_2Ecanonical_2ESPvar_2E1: 
      !>[A_27a: $tType] : ( tyop_2Equote_2Eindex > tyop_2Ecanonical_2Espolynom(A_27a) ) ).

tff(c_2Enum_2ESUC_2E0,type,
    c_2Enum_2ESUC_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum) ).

tff(c_2Enum_2ESUC_2E1,type,
    c_2Enum_2ESUC_2E1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).

tff(c_2Ebool_2ET_2E0,type,
    c_2Ebool_2ET_2E0: tyop_2Emin_2Ebool ).

tff(c_2Ebool_2ETYPE__DEFINITION_2E0,type,
    c_2Ebool_2ETYPE__DEFINITION_2E0: 
      !>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,A_27a),tyop_2Emin_2Ebool)) ).

tff(c_2Ebool_2ETYPE__DEFINITION_2E2,type,
    c_2Ebool_2ETYPE__DEFINITION_2E2: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) * tyop_2Emin_2Efun(A_27b,A_27a) ) > tyop_2Emin_2Ebool ) ).

tff(c_2Earithmetic_2EZERO_2E0,type,
    c_2Earithmetic_2EZERO_2E0: tyop_2Enum_2Enum ).

tff(c_2Ebool_2E_5C_2F_2E0,type,
    c_2Ebool_2E_5C_2F_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool)) ).

tff(c_2Ebool_2E_5C_2F_2E2,type,
    c_2Ebool_2E_5C_2F_2E2: ( tyop_2Emin_2Ebool * tyop_2Emin_2Ebool ) > tyop_2Emin_2Ebool ).

tff(c_2Ecanonical_2Ecanonical__sum__merge_2E0,type,
    c_2Ecanonical_2Ecanonical__sum__merge_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)))) ).

tff(c_2Ecanonical_2Ecanonical__sum__merge_2E1,type,
    c_2Ecanonical_2Ecanonical__sum__merge_2E1: 
      !>[A_27a: $tType] : ( tyop_2Esemi__ring_2Esemi__ring(A_27a) > tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a))) ) ).

tff(c_2Ecanonical_2Ecanonical__sum__prod_2E0,type,
    c_2Ecanonical_2Ecanonical__sum__prod_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)))) ).

tff(c_2Ecanonical_2Ecanonical__sum__prod_2E1,type,
    c_2Ecanonical_2Ecanonical__sum__prod_2E1: 
      !>[A_27a: $tType] : ( tyop_2Esemi__ring_2Esemi__ring(A_27a) > tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a))) ) ).

tff(c_2Ecanonical_2Ecanonical__sum__scalar_2E0,type,
    c_2Ecanonical_2Ecanonical__sum__scalar_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)))) ).

tff(c_2Ecanonical_2Ecanonical__sum__scalar_2E1,type,
    c_2Ecanonical_2Ecanonical__sum__scalar_2E1: 
      !>[A_27a: $tType] : ( tyop_2Esemi__ring_2Esemi__ring(A_27a) > tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a))) ) ).

tff(c_2Ecanonical_2Ecanonical__sum__scalar2_2E0,type,
    c_2Ecanonical_2Ecanonical__sum__scalar2_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)))) ).

tff(c_2Ecanonical_2Ecanonical__sum__scalar2_2E1,type,
    c_2Ecanonical_2Ecanonical__sum__scalar2_2E1: 
      !>[A_27a: $tType] : ( tyop_2Esemi__ring_2Esemi__ring(A_27a) > tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a))) ) ).

tff(c_2Ecanonical_2Ecanonical__sum__scalar3_2E0,type,
    c_2Ecanonical_2Ecanonical__sum__scalar3_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a))))) ).

tff(c_2Ecanonical_2Ecanonical__sum__scalar3_2E1,type,
    c_2Ecanonical_2Ecanonical__sum__scalar3_2E1: 
      !>[A_27a: $tType] : ( tyop_2Esemi__ring_2Esemi__ring(A_27a) > tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)))) ) ).

tff(c_2Ecanonical_2Ecanonical__sum__simplify_2E0,type,
    c_2Ecanonical_2Ecanonical__sum__simplify_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a))) ).

tff(c_2Ecanonical_2Ecanonical__sum__simplify_2E1,type,
    c_2Ecanonical_2Ecanonical__sum__simplify_2E1: 
      !>[A_27a: $tType] : ( tyop_2Esemi__ring_2Esemi__ring(A_27a) > tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)) ) ).

tff(c_2Ecanonical_2Eics__aux_2E0,type,
    c_2Ecanonical_2Eics__aux_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(tyop_2Equote_2Evarmap(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27a)))) ).

tff(c_2Ecanonical_2Eics__aux_2E1,type,
    c_2Ecanonical_2Eics__aux_2E1: 
      !>[A_27a: $tType] : ( tyop_2Esemi__ring_2Esemi__ring(A_27a) > tyop_2Emin_2Efun(tyop_2Equote_2Evarmap(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27a))) ) ).

tff(c_2Equote_2Eindex__compare_2E0,type,
    c_2Equote_2Eindex__compare_2E0: tyop_2Emin_2Efun(tyop_2Equote_2Eindex,tyop_2Emin_2Efun(tyop_2Equote_2Eindex,tyop_2EternaryComparisons_2Eordering)) ).

tff(c_2Equote_2Eindex__lt_2E0,type,
    c_2Equote_2Eindex__lt_2E0: tyop_2Emin_2Efun(tyop_2Equote_2Eindex,tyop_2Emin_2Efun(tyop_2Equote_2Eindex,tyop_2Emin_2Ebool)) ).

tff(c_2Equote_2Eindex__size_2E0,type,
    c_2Equote_2Eindex__size_2E0: tyop_2Emin_2Efun(tyop_2Equote_2Eindex,tyop_2Enum_2Enum) ).

tff(c_2Equote_2Eindex__size_2E1,type,
    c_2Equote_2Eindex__size_2E1: tyop_2Equote_2Eindex > tyop_2Enum_2Enum ).

tff(c_2Ecanonical_2Einterp__cs_2E0,type,
    c_2Ecanonical_2Einterp__cs_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(tyop_2Equote_2Evarmap(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27a))) ).

tff(c_2Ecanonical_2Einterp__cs_2E1,type,
    c_2Ecanonical_2Einterp__cs_2E1: 
      !>[A_27a: $tType] : ( tyop_2Esemi__ring_2Esemi__ring(A_27a) > tyop_2Emin_2Efun(tyop_2Equote_2Evarmap(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27a)) ) ).

tff(c_2Ecanonical_2Einterp__m_2E0,type,
    c_2Ecanonical_2Einterp__m_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(tyop_2Equote_2Evarmap(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),A_27a)))) ).

tff(c_2Ecanonical_2Einterp__m_2E1,type,
    c_2Ecanonical_2Einterp__m_2E1: 
      !>[A_27a: $tType] : ( tyop_2Esemi__ring_2Esemi__ring(A_27a) > tyop_2Emin_2Efun(tyop_2Equote_2Evarmap(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),A_27a))) ) ).

tff(c_2EringNorm_2Einterp__p_2E0,type,
    c_2EringNorm_2Einterp__p_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Ering_2Ering(A_27a),tyop_2Emin_2Efun(tyop_2Equote_2Evarmap(A_27a),tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),A_27a))) ).

tff(c_2EringNorm_2Einterp__p_2E3,type,
    c_2EringNorm_2Einterp__p_2E3: 
      !>[A_27a: $tType] : ( ( tyop_2Ering_2Ering(A_27a) * tyop_2Equote_2Evarmap(A_27a) * tyop_2EringNorm_2Epolynom(A_27a) ) > A_27a ) ).

tff(c_2Ecanonical_2Einterp__sp_2E0,type,
    c_2Ecanonical_2Einterp__sp_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(tyop_2Equote_2Evarmap(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27a))) ).

tff(c_2Ecanonical_2Einterp__sp_2E1,type,
    c_2Ecanonical_2Einterp__sp_2E1: 
      !>[A_27a: $tType] : ( tyop_2Esemi__ring_2Esemi__ring(A_27a) > tyop_2Emin_2Efun(tyop_2Equote_2Evarmap(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27a)) ) ).

tff(c_2Ecanonical_2Einterp__vl_2E0,type,
    c_2Ecanonical_2Einterp__vl_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(tyop_2Equote_2Evarmap(A_27a),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),A_27a))) ).

tff(c_2Ecanonical_2Einterp__vl_2E1,type,
    c_2Ecanonical_2Einterp__vl_2E1: 
      !>[A_27a: $tType] : ( tyop_2Esemi__ring_2Esemi__ring(A_27a) > tyop_2Emin_2Efun(tyop_2Equote_2Evarmap(A_27a),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),A_27a)) ) ).

tff(c_2Ering_2Eis__ring_2E0,type,
    c_2Ering_2Eis__ring_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Ering_2Ering(A_27a),tyop_2Emin_2Ebool) ).

tff(c_2Ering_2Eis__ring_2E1,type,
    c_2Ering_2Eis__ring_2E1: 
      !>[A_27a: $tType] : ( tyop_2Ering_2Ering(A_27a) > tyop_2Emin_2Ebool ) ).

tff(c_2Ecanonical_2Eivl__aux_2E0,type,
    c_2Ecanonical_2Eivl__aux_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(tyop_2Equote_2Evarmap(A_27a),tyop_2Emin_2Efun(tyop_2Equote_2Eindex,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),A_27a)))) ).

tff(c_2Ecanonical_2Eivl__aux_2E1,type,
    c_2Ecanonical_2Eivl__aux_2E1: 
      !>[A_27a: $tType] : ( tyop_2Esemi__ring_2Esemi__ring(A_27a) > tyop_2Emin_2Efun(tyop_2Equote_2Evarmap(A_27a),tyop_2Emin_2Efun(tyop_2Equote_2Eindex,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),A_27a))) ) ).

tff(c_2EternaryComparisons_2Elist__compare_2E0,type,
    c_2EternaryComparisons_2Elist__compare_2E0: 
      !>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2EternaryComparisons_2Eordering)),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2EternaryComparisons_2Eordering))) ).

tff(c_2EternaryComparisons_2Elist__compare_2E3,type,
    c_2EternaryComparisons_2Elist__compare_2E3: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2EternaryComparisons_2Eordering)) * tyop_2Elist_2Elist(A_27a) * tyop_2Elist_2Elist(A_27b) ) > tyop_2EternaryComparisons_2Eordering ) ).

tff(c_2EternaryComparisons_2Elist__merge_2E0,type,
    c_2EternaryComparisons_2Elist__merge_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Elist_2Elist(A_27a)))) ).

tff(c_2EternaryComparisons_2Elist__merge_2E3,type,
    c_2EternaryComparisons_2Elist__merge_2E3: 
      !>[A_27a: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)) * tyop_2Elist_2Elist(A_27a) * tyop_2Elist_2Elist(A_27a) ) > tyop_2Elist_2Elist(A_27a) ) ).

tff(c_2Ecanonical_2Emonom__insert_2E0,type,
    c_2Ecanonical_2Emonom__insert_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a))))) ).

tff(c_2Ecanonical_2Emonom__insert_2E1,type,
    c_2Ecanonical_2Emonom__insert_2E1: 
      !>[A_27a: $tType] : ( tyop_2Esemi__ring_2Esemi__ring(A_27a) > tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)))) ) ).

tff(c_2EternaryComparisons_2Eordering__CASE_2E0,type,
    c_2EternaryComparisons_2Eordering__CASE_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2EternaryComparisons_2Eordering,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a)))) ).

tff(c_2EternaryComparisons_2Eordering__CASE_2E4,type,
    c_2EternaryComparisons_2Eordering__CASE_2E4: 
      !>[A_27a: $tType] : ( ( tyop_2EternaryComparisons_2Eordering * A_27a * A_27a * A_27a ) > A_27a ) ).

tff(c_2EringNorm_2Epolynom__CASE_2E0,type,
    c_2EringNorm_2Epolynom__CASE_2E0: 
      !>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Equote_2Eindex,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),A_27b)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),A_27b)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),A_27b),A_27b)))))) ).

tff(c_2EringNorm_2Epolynom__CASE_2E6,type,
    c_2EringNorm_2Epolynom__CASE_2E6: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2EringNorm_2Epolynom(A_27a) * tyop_2Emin_2Efun(tyop_2Equote_2Eindex,A_27b) * tyop_2Emin_2Efun(A_27a,A_27b) * tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),A_27b)) * tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),A_27b)) * tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),A_27b) ) > A_27b ) ).

tff(c_2EringNorm_2Epolynom__normalize_2E0,type,
    c_2EringNorm_2Epolynom__normalize_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Ering_2Ering(A_27a),tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a))) ).

tff(c_2EringNorm_2Epolynom__normalize_2E2,type,
    c_2EringNorm_2Epolynom__normalize_2E2: 
      !>[A_27a: $tType] : ( ( tyop_2Ering_2Ering(A_27a) * tyop_2EringNorm_2Epolynom(A_27a) ) > tyop_2Ecanonical_2Ecanonical__sum(A_27a) ) ).

tff(c_2EringNorm_2Epolynom__simplify_2E0,type,
    c_2EringNorm_2Epolynom__simplify_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Ering_2Ering(A_27a),tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a))) ).

tff(c_2EringNorm_2Epolynom__simplify_2E2,type,
    c_2EringNorm_2Epolynom__simplify_2E2: 
      !>[A_27a: $tType] : ( ( tyop_2Ering_2Ering(A_27a) * tyop_2EringNorm_2Epolynom(A_27a) ) > tyop_2Ecanonical_2Ecanonical__sum(A_27a) ) ).

tff(c_2EringNorm_2Epolynom__size_2E0,type,
    c_2EringNorm_2Epolynom__size_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Enum_2Enum)) ).

tff(c_2EringNorm_2Epolynom__size_2E2,type,
    c_2EringNorm_2Epolynom__size_2E2: 
      !>[A_27a: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum) * tyop_2EringNorm_2Epolynom(A_27a) ) > tyop_2Enum_2Enum ) ).

tff(c_2EringNorm_2Er__canonical__sum__merge_2E0,type,
    c_2EringNorm_2Er__canonical__sum__merge_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Ering_2Ering(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)))) ).

tff(c_2EringNorm_2Er__canonical__sum__merge_2E1,type,
    c_2EringNorm_2Er__canonical__sum__merge_2E1: 
      !>[A_27a: $tType] : ( tyop_2Ering_2Ering(A_27a) > tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a))) ) ).

tff(c_2EringNorm_2Er__canonical__sum__merge_2E3,type,
    c_2EringNorm_2Er__canonical__sum__merge_2E3: 
      !>[A_27a: $tType] : ( ( tyop_2Ering_2Ering(A_27a) * tyop_2Ecanonical_2Ecanonical__sum(A_27a) * tyop_2Ecanonical_2Ecanonical__sum(A_27a) ) > tyop_2Ecanonical_2Ecanonical__sum(A_27a) ) ).

tff(c_2EringNorm_2Er__canonical__sum__prod_2E0,type,
    c_2EringNorm_2Er__canonical__sum__prod_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Ering_2Ering(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)))) ).

tff(c_2EringNorm_2Er__canonical__sum__prod_2E1,type,
    c_2EringNorm_2Er__canonical__sum__prod_2E1: 
      !>[A_27a: $tType] : ( tyop_2Ering_2Ering(A_27a) > tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a))) ) ).

tff(c_2EringNorm_2Er__canonical__sum__prod_2E3,type,
    c_2EringNorm_2Er__canonical__sum__prod_2E3: 
      !>[A_27a: $tType] : ( ( tyop_2Ering_2Ering(A_27a) * tyop_2Ecanonical_2Ecanonical__sum(A_27a) * tyop_2Ecanonical_2Ecanonical__sum(A_27a) ) > tyop_2Ecanonical_2Ecanonical__sum(A_27a) ) ).

tff(c_2EringNorm_2Er__canonical__sum__scalar_2E0,type,
    c_2EringNorm_2Er__canonical__sum__scalar_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Ering_2Ering(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)))) ).

tff(c_2EringNorm_2Er__canonical__sum__scalar_2E1,type,
    c_2EringNorm_2Er__canonical__sum__scalar_2E1: 
      !>[A_27a: $tType] : ( tyop_2Ering_2Ering(A_27a) > tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a))) ) ).

tff(c_2EringNorm_2Er__canonical__sum__scalar_2E3,type,
    c_2EringNorm_2Er__canonical__sum__scalar_2E3: 
      !>[A_27a: $tType] : ( ( tyop_2Ering_2Ering(A_27a) * A_27a * tyop_2Ecanonical_2Ecanonical__sum(A_27a) ) > tyop_2Ecanonical_2Ecanonical__sum(A_27a) ) ).

tff(c_2EringNorm_2Er__canonical__sum__scalar2_2E0,type,
    c_2EringNorm_2Er__canonical__sum__scalar2_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Ering_2Ering(A_27a),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)))) ).

tff(c_2EringNorm_2Er__canonical__sum__scalar2_2E1,type,
    c_2EringNorm_2Er__canonical__sum__scalar2_2E1: 
      !>[A_27a: $tType] : ( tyop_2Ering_2Ering(A_27a) > tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a))) ) ).

tff(c_2EringNorm_2Er__canonical__sum__scalar2_2E3,type,
    c_2EringNorm_2Er__canonical__sum__scalar2_2E3: 
      !>[A_27a: $tType] : ( ( tyop_2Ering_2Ering(A_27a) * tyop_2Elist_2Elist(tyop_2Equote_2Eindex) * tyop_2Ecanonical_2Ecanonical__sum(A_27a) ) > tyop_2Ecanonical_2Ecanonical__sum(A_27a) ) ).

tff(c_2EringNorm_2Er__canonical__sum__scalar3_2E0,type,
    c_2EringNorm_2Er__canonical__sum__scalar3_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Ering_2Ering(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a))))) ).

tff(c_2EringNorm_2Er__canonical__sum__scalar3_2E1,type,
    c_2EringNorm_2Er__canonical__sum__scalar3_2E1: 
      !>[A_27a: $tType] : ( tyop_2Ering_2Ering(A_27a) > tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)))) ) ).

tff(c_2EringNorm_2Er__canonical__sum__scalar3_2E4,type,
    c_2EringNorm_2Er__canonical__sum__scalar3_2E4: 
      !>[A_27a: $tType] : ( ( tyop_2Ering_2Ering(A_27a) * A_27a * tyop_2Elist_2Elist(tyop_2Equote_2Eindex) * tyop_2Ecanonical_2Ecanonical__sum(A_27a) ) > tyop_2Ecanonical_2Ecanonical__sum(A_27a) ) ).

tff(c_2EringNorm_2Er__canonical__sum__simplify_2E0,type,
    c_2EringNorm_2Er__canonical__sum__simplify_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Ering_2Ering(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a))) ).

tff(c_2EringNorm_2Er__canonical__sum__simplify_2E1,type,
    c_2EringNorm_2Er__canonical__sum__simplify_2E1: 
      !>[A_27a: $tType] : ( tyop_2Ering_2Ering(A_27a) > tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)) ) ).

tff(c_2EringNorm_2Er__canonical__sum__simplify_2E2,type,
    c_2EringNorm_2Er__canonical__sum__simplify_2E2: 
      !>[A_27a: $tType] : ( ( tyop_2Ering_2Ering(A_27a) * tyop_2Ecanonical_2Ecanonical__sum(A_27a) ) > tyop_2Ecanonical_2Ecanonical__sum(A_27a) ) ).

tff(c_2EringNorm_2Er__ics__aux_2E0,type,
    c_2EringNorm_2Er__ics__aux_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Ering_2Ering(A_27a),tyop_2Emin_2Efun(tyop_2Equote_2Evarmap(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27a)))) ).

tff(c_2EringNorm_2Er__ics__aux_2E1,type,
    c_2EringNorm_2Er__ics__aux_2E1: 
      !>[A_27a: $tType] : ( tyop_2Ering_2Ering(A_27a) > tyop_2Emin_2Efun(tyop_2Equote_2Evarmap(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27a))) ) ).

tff(c_2EringNorm_2Er__ics__aux_2E4,type,
    c_2EringNorm_2Er__ics__aux_2E4: 
      !>[A_27a: $tType] : ( ( tyop_2Ering_2Ering(A_27a) * tyop_2Equote_2Evarmap(A_27a) * A_27a * tyop_2Ecanonical_2Ecanonical__sum(A_27a) ) > A_27a ) ).

tff(c_2EringNorm_2Er__interp__cs_2E0,type,
    c_2EringNorm_2Er__interp__cs_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Ering_2Ering(A_27a),tyop_2Emin_2Efun(tyop_2Equote_2Evarmap(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27a))) ).

tff(c_2EringNorm_2Er__interp__cs_2E1,type,
    c_2EringNorm_2Er__interp__cs_2E1: 
      !>[A_27a: $tType] : ( tyop_2Ering_2Ering(A_27a) > tyop_2Emin_2Efun(tyop_2Equote_2Evarmap(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27a)) ) ).

tff(c_2EringNorm_2Er__interp__cs_2E3,type,
    c_2EringNorm_2Er__interp__cs_2E3: 
      !>[A_27a: $tType] : ( ( tyop_2Ering_2Ering(A_27a) * tyop_2Equote_2Evarmap(A_27a) * tyop_2Ecanonical_2Ecanonical__sum(A_27a) ) > A_27a ) ).

tff(c_2EringNorm_2Er__interp__m_2E0,type,
    c_2EringNorm_2Er__interp__m_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Ering_2Ering(A_27a),tyop_2Emin_2Efun(tyop_2Equote_2Evarmap(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),A_27a)))) ).

tff(c_2EringNorm_2Er__interp__m_2E1,type,
    c_2EringNorm_2Er__interp__m_2E1: 
      !>[A_27a: $tType] : ( tyop_2Ering_2Ering(A_27a) > tyop_2Emin_2Efun(tyop_2Equote_2Evarmap(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),A_27a))) ) ).

tff(c_2EringNorm_2Er__interp__m_2E4,type,
    c_2EringNorm_2Er__interp__m_2E4: 
      !>[A_27a: $tType] : ( ( tyop_2Ering_2Ering(A_27a) * tyop_2Equote_2Evarmap(A_27a) * A_27a * tyop_2Elist_2Elist(tyop_2Equote_2Eindex) ) > A_27a ) ).

tff(c_2EringNorm_2Er__interp__sp_2E0,type,
    c_2EringNorm_2Er__interp__sp_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Ering_2Ering(A_27a),tyop_2Emin_2Efun(tyop_2Equote_2Evarmap(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27a))) ).

tff(c_2EringNorm_2Er__interp__sp_2E1,type,
    c_2EringNorm_2Er__interp__sp_2E1: 
      !>[A_27a: $tType] : ( tyop_2Ering_2Ering(A_27a) > tyop_2Emin_2Efun(tyop_2Equote_2Evarmap(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27a)) ) ).

tff(c_2EringNorm_2Er__interp__sp_2E3,type,
    c_2EringNorm_2Er__interp__sp_2E3: 
      !>[A_27a: $tType] : ( ( tyop_2Ering_2Ering(A_27a) * tyop_2Equote_2Evarmap(A_27a) * tyop_2Ecanonical_2Espolynom(A_27a) ) > A_27a ) ).

tff(c_2EringNorm_2Er__interp__vl_2E0,type,
    c_2EringNorm_2Er__interp__vl_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Ering_2Ering(A_27a),tyop_2Emin_2Efun(tyop_2Equote_2Evarmap(A_27a),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),A_27a))) ).

tff(c_2EringNorm_2Er__interp__vl_2E1,type,
    c_2EringNorm_2Er__interp__vl_2E1: 
      !>[A_27a: $tType] : ( tyop_2Ering_2Ering(A_27a) > tyop_2Emin_2Efun(tyop_2Equote_2Evarmap(A_27a),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),A_27a)) ) ).

tff(c_2EringNorm_2Er__interp__vl_2E3,type,
    c_2EringNorm_2Er__interp__vl_2E3: 
      !>[A_27a: $tType] : ( ( tyop_2Ering_2Ering(A_27a) * tyop_2Equote_2Evarmap(A_27a) * tyop_2Elist_2Elist(tyop_2Equote_2Eindex) ) > A_27a ) ).

tff(c_2EringNorm_2Er__ivl__aux_2E0,type,
    c_2EringNorm_2Er__ivl__aux_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Ering_2Ering(A_27a),tyop_2Emin_2Efun(tyop_2Equote_2Evarmap(A_27a),tyop_2Emin_2Efun(tyop_2Equote_2Eindex,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),A_27a)))) ).

tff(c_2EringNorm_2Er__ivl__aux_2E1,type,
    c_2EringNorm_2Er__ivl__aux_2E1: 
      !>[A_27a: $tType] : ( tyop_2Ering_2Ering(A_27a) > tyop_2Emin_2Efun(tyop_2Equote_2Evarmap(A_27a),tyop_2Emin_2Efun(tyop_2Equote_2Eindex,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),A_27a))) ) ).

tff(c_2EringNorm_2Er__ivl__aux_2E4,type,
    c_2EringNorm_2Er__ivl__aux_2E4: 
      !>[A_27a: $tType] : ( ( tyop_2Ering_2Ering(A_27a) * tyop_2Equote_2Evarmap(A_27a) * tyop_2Equote_2Eindex * tyop_2Elist_2Elist(tyop_2Equote_2Eindex) ) > A_27a ) ).

tff(c_2EringNorm_2Er__monom__insert_2E0,type,
    c_2EringNorm_2Er__monom__insert_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Ering_2Ering(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a))))) ).

tff(c_2EringNorm_2Er__monom__insert_2E1,type,
    c_2EringNorm_2Er__monom__insert_2E1: 
      !>[A_27a: $tType] : ( tyop_2Ering_2Ering(A_27a) > tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)))) ) ).

tff(c_2EringNorm_2Er__monom__insert_2E4,type,
    c_2EringNorm_2Er__monom__insert_2E4: 
      !>[A_27a: $tType] : ( ( tyop_2Ering_2Ering(A_27a) * A_27a * tyop_2Elist_2Elist(tyop_2Equote_2Eindex) * tyop_2Ecanonical_2Ecanonical__sum(A_27a) ) > tyop_2Ecanonical_2Ecanonical__sum(A_27a) ) ).

tff(c_2EringNorm_2Er__spolynom__normalize_2E0,type,
    c_2EringNorm_2Er__spolynom__normalize_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Ering_2Ering(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a))) ).

tff(c_2EringNorm_2Er__spolynom__normalize_2E1,type,
    c_2EringNorm_2Er__spolynom__normalize_2E1: 
      !>[A_27a: $tType] : ( tyop_2Ering_2Ering(A_27a) > tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)) ) ).

tff(c_2EringNorm_2Er__spolynom__normalize_2E2,type,
    c_2EringNorm_2Er__spolynom__normalize_2E2: 
      !>[A_27a: $tType] : ( ( tyop_2Ering_2Ering(A_27a) * tyop_2Ecanonical_2Espolynom(A_27a) ) > tyop_2Ecanonical_2Ecanonical__sum(A_27a) ) ).

tff(c_2EringNorm_2Er__spolynom__simplify_2E0,type,
    c_2EringNorm_2Er__spolynom__simplify_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Ering_2Ering(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a))) ).

tff(c_2EringNorm_2Er__spolynom__simplify_2E1,type,
    c_2EringNorm_2Er__spolynom__simplify_2E1: 
      !>[A_27a: $tType] : ( tyop_2Ering_2Ering(A_27a) > tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)) ) ).

tff(c_2EringNorm_2Er__spolynom__simplify_2E2,type,
    c_2EringNorm_2Er__spolynom__simplify_2E2: 
      !>[A_27a: $tType] : ( ( tyop_2Ering_2Ering(A_27a) * tyop_2Ecanonical_2Espolynom(A_27a) ) > tyop_2Ecanonical_2Ecanonical__sum(A_27a) ) ).

tff(c_2EringNorm_2Er__varlist__insert_2E0,type,
    c_2EringNorm_2Er__varlist__insert_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Ering_2Ering(A_27a),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)))) ).

tff(c_2EringNorm_2Er__varlist__insert_2E1,type,
    c_2EringNorm_2Er__varlist__insert_2E1: 
      !>[A_27a: $tType] : ( tyop_2Ering_2Ering(A_27a) > tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a))) ) ).

tff(c_2EringNorm_2Er__varlist__insert_2E3,type,
    c_2EringNorm_2Er__varlist__insert_2E3: 
      !>[A_27a: $tType] : ( ( tyop_2Ering_2Ering(A_27a) * tyop_2Elist_2Elist(tyop_2Equote_2Eindex) * tyop_2Ecanonical_2Ecanonical__sum(A_27a) ) > tyop_2Ecanonical_2Ecanonical__sum(A_27a) ) ).

tff(c_2Ering_2Ering__R0_2E0,type,
    c_2Ering_2Ering__R0_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Ering_2Ering(A_27a),A_27a) ).

tff(c_2Ering_2Ering__R0_2E1,type,
    c_2Ering_2Ering__R0_2E1: 
      !>[A_27a: $tType] : ( tyop_2Ering_2Ering(A_27a) > A_27a ) ).

tff(c_2Ering_2Ering__R1_2E0,type,
    c_2Ering_2Ering__R1_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Ering_2Ering(A_27a),A_27a) ).

tff(c_2Ering_2Ering__R1_2E1,type,
    c_2Ering_2Ering__R1_2E1: 
      !>[A_27a: $tType] : ( tyop_2Ering_2Ering(A_27a) > A_27a ) ).

tff(c_2Ering_2Ering__RM_2E0,type,
    c_2Ering_2Ering__RM_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Ering_2Ering(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a))) ).

tff(c_2Ering_2Ering__RM_2E3,type,
    c_2Ering_2Ering__RM_2E3: 
      !>[A_27a: $tType] : ( ( tyop_2Ering_2Ering(A_27a) * A_27a * A_27a ) > A_27a ) ).

tff(c_2Ering_2Ering__RN_2E0,type,
    c_2Ering_2Ering__RN_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Ering_2Ering(A_27a),tyop_2Emin_2Efun(A_27a,A_27a)) ).

tff(c_2Ering_2Ering__RN_2E2,type,
    c_2Ering_2Ering__RN_2E2: 
      !>[A_27a: $tType] : ( ( tyop_2Ering_2Ering(A_27a) * A_27a ) > A_27a ) ).

tff(c_2Ering_2Ering__RP_2E0,type,
    c_2Ering_2Ering__RP_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Ering_2Ering(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a))) ).

tff(c_2Ering_2Ering__RP_2E3,type,
    c_2Ering_2Ering__RP_2E3: 
      !>[A_27a: $tType] : ( ( tyop_2Ering_2Ering(A_27a) * A_27a * A_27a ) > A_27a ) ).

tff(c_2Ering_2Esemi__ring__of_2E0,type,
    c_2Ering_2Esemi__ring__of_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Ering_2Ering(A_27a),tyop_2Esemi__ring_2Esemi__ring(A_27a)) ).

tff(c_2Ering_2Esemi__ring__of_2E1,type,
    c_2Ering_2Esemi__ring__of_2E1: 
      !>[A_27a: $tType] : ( tyop_2Ering_2Ering(A_27a) > tyop_2Esemi__ring_2Esemi__ring(A_27a) ) ).

tff(c_2Ecanonical_2Espolynom__normalize_2E0,type,
    c_2Ecanonical_2Espolynom__normalize_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a))) ).

tff(c_2Ecanonical_2Espolynom__normalize_2E1,type,
    c_2Ecanonical_2Espolynom__normalize_2E1: 
      !>[A_27a: $tType] : ( tyop_2Esemi__ring_2Esemi__ring(A_27a) > tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)) ) ).

tff(c_2Ecanonical_2Espolynom__simplify_2E0,type,
    c_2Ecanonical_2Espolynom__simplify_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a))) ).

tff(c_2Ecanonical_2Espolynom__simplify_2E1,type,
    c_2Ecanonical_2Espolynom__simplify_2E1: 
      !>[A_27a: $tType] : ( tyop_2Esemi__ring_2Esemi__ring(A_27a) > tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)) ) ).

tff(c_2Ecanonical_2Evarlist__insert_2E0,type,
    c_2Ecanonical_2Evarlist__insert_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)))) ).

tff(c_2Ecanonical_2Evarlist__insert_2E1,type,
    c_2Ecanonical_2Evarlist__insert_2E1: 
      !>[A_27a: $tType] : ( tyop_2Esemi__ring_2Esemi__ring(A_27a) > tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a))) ) ).

tff(c_2Equote_2Evarmap__find_2E0,type,
    c_2Equote_2Evarmap__find_2E0: 
      !>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Equote_2Eindex,tyop_2Emin_2Efun(tyop_2Equote_2Evarmap(A_27a),A_27a)) ).

tff(c_2Equote_2Evarmap__find_2E2,type,
    c_2Equote_2Evarmap__find_2E2: 
      !>[A_27a: $tType] : ( ( tyop_2Equote_2Eindex * tyop_2Equote_2Evarmap(A_27a) ) > A_27a ) ).

tff(c_2Ebool_2E_7E_2E0,type,
    c_2Ebool_2E_7E_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool) ).

tff(c_2Ebool_2E_7E_2E1,type,
    c_2Ebool_2E_7E_2E1: tyop_2Emin_2Ebool > tyop_2Emin_2Ebool ).

tff(thm_2Eextra_2Dho_2Eeq__ext,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V1g_2E0: tyop_2Emin_2Efun(A_27a,A_27b)] :
      ( ! [V2x_2E0: A_27a] : ( app_2E2(A_27a,A_27b,V0f_2E0,V2x_2E0) = app_2E2(A_27a,A_27b,V1g_2E0,V2x_2E0) )
     => ( V0f_2E0 = V1g_2E0 ) ) ).

tff(thm_2Eextra_2Dho_2Eboolext,axiom,
    ! [V0_2E0: tyop_2Emin_2Ebool,V1_2E0: tyop_2Emin_2Ebool] :
      ( ( p(V0_2E0)
      <=> p(V1_2E0) )
     => ( V0_2E0 = V1_2E0 ) ) ).

tff(thm_2Eextra_2Dho_2Etruth,axiom,
    p(c_2Ebool_2ET_2E0) ).

tff(thm_2Eextra_2Dho_2Enotfalse,axiom,
    ~ p(c_2Ebool_2EF_2E0) ).

tff(thm_2Eextra_2Dho_2Ebool__cases__ax,axiom,
    ! [V0t_2E0: tyop_2Emin_2Ebool] :
      ( ( V0t_2E0 = c_2Ebool_2ET_2E0 )
      | ( V0t_2E0 = c_2Ebool_2EF_2E0 ) ) ).

tff(thm_2Eextra_2Dho_2Ei__thm,axiom,
    ! [A_27a: $tType,V0x_2E0: A_27a] : ( app_2E2(A_27a,A_27a,combin_i_2E0(A_27a),V0x_2E0) = V0x_2E0 ) ).

tff(thm_2Eextra_2Dho_2Ek__thm,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0x_2E0: A_27a,V1y_2E0: A_27b] : ( app_2E2(A_27b,A_27a,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27a),combin_k_2E0(A_27a,A_27b),V0x_2E0),V1y_2E0) = V0x_2E0 ) ).

tff(thm_2Eextra_2Dho_2Es__thm,axiom,
    ! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),V1g_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V2x_2E0: A_27a] : ( app_2E2(A_27a,A_27c,app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27c),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27c)),combin_s_2E0(A_27a,A_27b,A_27c),V0f_2E0),V1g_2E0),V2x_2E0) = app_2E2(A_27b,A_27c,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27c),V0f_2E0,V2x_2E0),app_2E2(A_27a,A_27b,V1g_2E0,V2x_2E0)) ) ).

tff(logicdef_2E_2F_5C,axiom,
    ! [V0_2E0: tyop_2Emin_2Ebool,V1_2E0: tyop_2Emin_2Ebool] :
      ( p(c_2Ebool_2E_2F_5C_2E2(V0_2E0,V1_2E0))
    <=> ( p(V0_2E0)
        & p(V1_2E0) ) ) ).

tff(logicdef_2E_5C_2F,axiom,
    ! [V0_2E0: tyop_2Emin_2Ebool,V1_2E0: tyop_2Emin_2Ebool] :
      ( p(c_2Ebool_2E_5C_2F_2E2(V0_2E0,V1_2E0))
    <=> ( p(V0_2E0)
        | p(V1_2E0) ) ) ).

tff(logicdef_2E_7E,axiom,
    ! [V0_2E0: tyop_2Emin_2Ebool] :
      ( p(c_2Ebool_2E_7E_2E1(V0_2E0))
    <=> ~ p(V0_2E0) ) ).

tff(logicdef_2E_3D_3D_3E,axiom,
    ! [V0_2E0: tyop_2Emin_2Ebool,V1_2E0: tyop_2Emin_2Ebool] :
      ( p(c_2Emin_2E_3D_3D_3E_2E2(V0_2E0,V1_2E0))
    <=> ( p(V0_2E0)
       => p(V1_2E0) ) ) ).

tff(logicdef_2E_3D,axiom,
    ! [A_27a: $tType,V0_2E0: A_27a,V1_2E0: A_27a] :
      ( p(c_2Emin_2E_3D_2E2(A_27a,V0_2E0,V1_2E0))
    <=> ( V0_2E0 = V1_2E0 ) ) ).

tff(quantdef_2E_21,axiom,
    ! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
      ( p(c_2Ebool_2E_21_2E1(A_27a,V0f_2E0))
    <=> ! [V1x_2E0: A_27a] : p(app_2E2(A_27a,tyop_2Emin_2Ebool,V0f_2E0,V1x_2E0)) ) ).

tff(quantdef_2E_3F,axiom,
    ! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
      ( p(c_2Ebool_2E_3F_2E1(A_27a,V0f_2E0))
    <=> ? [V1x_2E0: A_27a] : p(app_2E2(A_27a,tyop_2Emin_2Ebool,V0f_2E0,V1x_2E0)) ) ).

tff(arityeq1_2Ec_2Ebool_2E_21_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Ebool_2E_21_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Ebool_2E_21_2E0(A_27a),X0_2E0) ) ).

tff(arityeq2_2Ec_2Earithmetic_2E_2B_2E2,axiom,
    ! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_2B_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),c_2Earithmetic_2E_2B_2E0,X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2Etyop_2Equote_2Eindex_20mono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Equote_2Eindex,X1_2E0: A_27a] : ( c_2Epair_2E_2C_2E2(tyop_2Equote_2Eindex,A_27a,X0_2E0,X1_2E0) = app_2E2(A_27a,tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a),app_2E2(tyop_2Equote_2Eindex,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)),c_2Epair_2E_2C_2E0(tyop_2Equote_2Eindex,A_27a),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Ebool_2E_2F_5C_2E2,axiom,
    ! [X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Emin_2Ebool] :
      ( ( p(X0_2E0)
        & p(X1_2E0) )
    <=> p(app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),c_2Ebool_2E_2F_5C_2E0,X0_2E0),X1_2E0)) ) ).

tff(arityeq2_2Ec_2Emin_2E_3D_2E2_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: A_27a,X1_2E0: A_27a] :
      ( ( X0_2E0 = X1_2E0 )
    <=> p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),c_2Emin_2E_3D_2E0(A_27a),X0_2E0),X1_2E0)) ) ).

tff(arityeq2_2Ec_2Emin_2E_3D_3D_3E_2E2,axiom,
    ! [X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Emin_2Ebool] :
      ( ( p(X0_2E0)
       => p(X1_2E0) )
    <=> p(app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),c_2Emin_2E_3D_3D_3E_2E0,X0_2E0),X1_2E0)) ) ).

tff(arityeq1_2Ec_2Ebool_2E_3F_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Ebool_2E_3F_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Ebool_2E_3F_2E0(A_27a),X0_2E0) ) ).

tff(arityeq1_2Ec_2Earithmetic_2EBIT1_2E1,axiom,
    ! [X0_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2EBIT1_2E1(X0_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Earithmetic_2EBIT1_2E0,X0_2E0) ) ).

tff(arityeq3_2Ec_2Ebool_2ECOND_2E3_2Emono_2Etyop_2Ecanonical_2Ecanonical__sum_28A_27a_29,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a),X2_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : ( c_2Ebool_2ECOND_2E3(tyop_2Ecanonical_2Ecanonical__sum(A_27a),X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a),app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)),app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a))),c_2Ebool_2ECOND_2E0(tyop_2Ecanonical_2Ecanonical__sum(A_27a)),X0_2E0),X1_2E0),X2_2E0) ) ).

tff(arityeq2_2Ec_2Elist_2ECONS_2E2_2Emono_2Etyop_2Equote_2Eindex,axiom,
    ! [X0_2E0: tyop_2Equote_2Eindex,X1_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex)] : ( c_2Elist_2ECONS_2E2(tyop_2Equote_2Eindex,X0_2E0,X1_2E0) = app_2E2(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Elist_2Elist(tyop_2Equote_2Eindex),app_2E2(tyop_2Equote_2Eindex,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Elist_2Elist(tyop_2Equote_2Eindex)),c_2Elist_2ECONS_2E0(tyop_2Equote_2Eindex),X0_2E0),X1_2E0) ) ).

tff(arityeq3_2Ec_2Eind__type_2ECONSTR_2E3_2Emono_2Etyop_2Epair_2Eprod_28tyop_2Equote_2Eindex_2CA_27a_29,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a),X2_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)))] : ( c_2Eind__type_2ECONSTR_2E3(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a),X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a))),tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)),app_2E2(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a))),tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a))),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a))),tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)))),c_2Eind__type_2ECONSTR_2E0(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)),X0_2E0),X1_2E0),X2_2E0) ) ).

tff(arityeq3_2Ec_2Ecanonical_2ECons__monom_2E3_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: A_27a,X1_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),X2_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : ( c_2Ecanonical_2ECons__monom_2E3(A_27a,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a),app_2E2(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a))),c_2Ecanonical_2ECons__monom_2E0(A_27a),X0_2E0),X1_2E0),X2_2E0) ) ).

tff(arityeq2_2Ec_2Ecanonical_2ECons__varlist_2E2_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),X1_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : ( c_2Ecanonical_2ECons__varlist_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a),app_2E2(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)),c_2Ecanonical_2ECons__varlist_2E0(A_27a),X0_2E0),X1_2E0) ) ).

tff(arityeq1_2Ec_2Ebool_2EDATATYPE_2E1_2Emono_2Etyop_2Emin_2Ebool,axiom,
    ! [X0_2E0: tyop_2Emin_2Ebool] : ( c_2Ebool_2EDATATYPE_2E1(tyop_2Emin_2Ebool,X0_2E0) = app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool,c_2Ebool_2EDATATYPE_2E0(tyop_2Emin_2Ebool),X0_2E0) ) ).

tff(arityeq2_2Ec_2Eind__type_2EFCONS_2E2_2Emono_2Etyop_2Eind__type_2Erecspace_28tyop_2Epair_2Eprod_28tyop_2Equote_2Eindex_2CA_27a_29_29,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)),X1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)))] : ( c_2Eind__type_2EFCONS_2E2(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)),X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a))),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a))),app_2E2(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a))),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)))),c_2Eind__type_2EFCONS_2E0(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a))),X0_2E0),X1_2E0) ) ).

tff(arityeq1_2Ec_2Earithmetic_2ENUMERAL_2E1,axiom,
    ! [X0_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2ENUMERAL_2E1(X0_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Earithmetic_2ENUMERAL_2E0,X0_2E0) ) ).

tff(arityeq1_2Ec_2EringNorm_2EPconst_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: A_27a] : ( c_2EringNorm_2EPconst_2E1(A_27a,X0_2E0) = app_2E2(A_27a,tyop_2EringNorm_2Epolynom(A_27a),c_2EringNorm_2EPconst_2E0(A_27a),X0_2E0) ) ).

tff(arityeq2_2Ec_2EringNorm_2EPmult_2E2_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2EringNorm_2Epolynom(A_27a),X1_2E0: tyop_2EringNorm_2Epolynom(A_27a)] : ( c_2EringNorm_2EPmult_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2EringNorm_2Epolynom(A_27a),tyop_2EringNorm_2Epolynom(A_27a),app_2E2(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2EringNorm_2Epolynom(A_27a)),c_2EringNorm_2EPmult_2E0(A_27a),X0_2E0),X1_2E0) ) ).

tff(arityeq1_2Ec_2EringNorm_2EPopp_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2EringNorm_2Epolynom(A_27a)] : ( c_2EringNorm_2EPopp_2E1(A_27a,X0_2E0) = app_2E2(tyop_2EringNorm_2Epolynom(A_27a),tyop_2EringNorm_2Epolynom(A_27a),c_2EringNorm_2EPopp_2E0(A_27a),X0_2E0) ) ).

tff(arityeq2_2Ec_2EringNorm_2EPplus_2E2_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2EringNorm_2Epolynom(A_27a),X1_2E0: tyop_2EringNorm_2Epolynom(A_27a)] : ( c_2EringNorm_2EPplus_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2EringNorm_2Epolynom(A_27a),tyop_2EringNorm_2Epolynom(A_27a),app_2E2(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2EringNorm_2Epolynom(A_27a)),c_2EringNorm_2EPplus_2E0(A_27a),X0_2E0),X1_2E0) ) ).

tff(arityeq1_2Ec_2EringNorm_2EPvar_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Equote_2Eindex] : ( c_2EringNorm_2EPvar_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Equote_2Eindex,tyop_2EringNorm_2Epolynom(A_27a),c_2EringNorm_2EPvar_2E0(A_27a),X0_2E0) ) ).

tff(arityeq1_2Ec_2Ecanonical_2ESPconst_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: A_27a] : ( c_2Ecanonical_2ESPconst_2E1(A_27a,X0_2E0) = app_2E2(A_27a,tyop_2Ecanonical_2Espolynom(A_27a),c_2Ecanonical_2ESPconst_2E0(A_27a),X0_2E0) ) ).

tff(arityeq2_2Ec_2Ecanonical_2ESPmult_2E2_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Ecanonical_2Espolynom(A_27a),X1_2E0: tyop_2Ecanonical_2Espolynom(A_27a)] : ( c_2Ecanonical_2ESPmult_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Ecanonical_2Espolynom(A_27a),app_2E2(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Ecanonical_2Espolynom(A_27a)),c_2Ecanonical_2ESPmult_2E0(A_27a),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Ecanonical_2ESPplus_2E2_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Ecanonical_2Espolynom(A_27a),X1_2E0: tyop_2Ecanonical_2Espolynom(A_27a)] : ( c_2Ecanonical_2ESPplus_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Ecanonical_2Espolynom(A_27a),app_2E2(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Ecanonical_2Espolynom(A_27a)),c_2Ecanonical_2ESPplus_2E0(A_27a),X0_2E0),X1_2E0) ) ).

tff(arityeq1_2Ec_2Ecanonical_2ESPvar_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Equote_2Eindex] : ( c_2Ecanonical_2ESPvar_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Equote_2Eindex,tyop_2Ecanonical_2Espolynom(A_27a),c_2Ecanonical_2ESPvar_2E0(A_27a),X0_2E0) ) ).

tff(arityeq1_2Ec_2Enum_2ESUC_2E1,axiom,
    ! [X0_2E0: tyop_2Enum_2Enum] : ( c_2Enum_2ESUC_2E1(X0_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Enum_2ESUC_2E0,X0_2E0) ) ).

tff(arityeq2_2Ec_2Ebool_2ETYPE__DEFINITION_2E2_2Emono_2Etyop_2Eind__type_2Erecspace_28tyop_2Epair_2Eprod_28tyop_2Equote_2Eindex_2CA_27a_29_29_20mono_2Etyop_2EringNorm_2Epolynom_28A_27a_29,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)),tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)))] : ( c_2Ebool_2ETYPE__DEFINITION_2E2(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)),tyop_2EringNorm_2Epolynom(A_27a),X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a))),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a))),tyop_2Emin_2Ebool),c_2Ebool_2ETYPE__DEFINITION_2E0(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)),tyop_2EringNorm_2Epolynom(A_27a)),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2Ebool_2E_5C_2F_2E2,axiom,
    ! [X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Emin_2Ebool] :
      ( ( p(X0_2E0)
        | p(X1_2E0) )
    <=> p(app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),c_2Ebool_2E_5C_2F_2E0,X0_2E0),X1_2E0)) ) ).

tff(arityeq1_2Ec_2Ecanonical_2Ecanonical__sum__merge_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a)] : ( c_2Ecanonical_2Ecanonical__sum__merge_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a))),c_2Ecanonical_2Ecanonical__sum__merge_2E0(A_27a),X0_2E0) ) ).

tff(arityeq1_2Ec_2Ecanonical_2Ecanonical__sum__prod_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a)] : ( c_2Ecanonical_2Ecanonical__sum__prod_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a))),c_2Ecanonical_2Ecanonical__sum__prod_2E0(A_27a),X0_2E0) ) ).

tff(arityeq1_2Ec_2Ecanonical_2Ecanonical__sum__scalar_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a)] : ( c_2Ecanonical_2Ecanonical__sum__scalar_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a))),c_2Ecanonical_2Ecanonical__sum__scalar_2E0(A_27a),X0_2E0) ) ).

tff(arityeq1_2Ec_2Ecanonical_2Ecanonical__sum__scalar2_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a)] : ( c_2Ecanonical_2Ecanonical__sum__scalar2_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a))),c_2Ecanonical_2Ecanonical__sum__scalar2_2E0(A_27a),X0_2E0) ) ).

tff(arityeq1_2Ec_2Ecanonical_2Ecanonical__sum__scalar3_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a)] : ( c_2Ecanonical_2Ecanonical__sum__scalar3_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)))),c_2Ecanonical_2Ecanonical__sum__scalar3_2E0(A_27a),X0_2E0) ) ).

tff(arityeq1_2Ec_2Ecanonical_2Ecanonical__sum__simplify_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a)] : ( c_2Ecanonical_2Ecanonical__sum__simplify_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)),c_2Ecanonical_2Ecanonical__sum__simplify_2E0(A_27a),X0_2E0) ) ).

tff(arityeq1_2Ec_2Ecanonical_2Eics__aux_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a)] : ( c_2Ecanonical_2Eics__aux_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(tyop_2Equote_2Evarmap(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27a))),c_2Ecanonical_2Eics__aux_2E0(A_27a),X0_2E0) ) ).

tff(arityeq1_2Ec_2Equote_2Eindex__size_2E1,axiom,
    ! [X0_2E0: tyop_2Equote_2Eindex] : ( c_2Equote_2Eindex__size_2E1(X0_2E0) = app_2E2(tyop_2Equote_2Eindex,tyop_2Enum_2Enum,c_2Equote_2Eindex__size_2E0,X0_2E0) ) ).

tff(arityeq1_2Ec_2Ecanonical_2Einterp__cs_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a)] : ( c_2Ecanonical_2Einterp__cs_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(tyop_2Equote_2Evarmap(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27a)),c_2Ecanonical_2Einterp__cs_2E0(A_27a),X0_2E0) ) ).

tff(arityeq1_2Ec_2Ecanonical_2Einterp__m_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a)] : ( c_2Ecanonical_2Einterp__m_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(tyop_2Equote_2Evarmap(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),A_27a))),c_2Ecanonical_2Einterp__m_2E0(A_27a),X0_2E0) ) ).

tff(arityeq3_2Ec_2EringNorm_2Einterp__p_2E3_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Ering_2Ering(A_27a),X1_2E0: tyop_2Equote_2Evarmap(A_27a),X2_2E0: tyop_2EringNorm_2Epolynom(A_27a)] : ( c_2EringNorm_2Einterp__p_2E3(A_27a,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2EringNorm_2Epolynom(A_27a),A_27a,app_2E2(tyop_2Equote_2Evarmap(A_27a),tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),A_27a),app_2E2(tyop_2Ering_2Ering(A_27a),tyop_2Emin_2Efun(tyop_2Equote_2Evarmap(A_27a),tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),A_27a)),c_2EringNorm_2Einterp__p_2E0(A_27a),X0_2E0),X1_2E0),X2_2E0) ) ).

tff(arityeq1_2Ec_2Ecanonical_2Einterp__sp_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a)] : ( c_2Ecanonical_2Einterp__sp_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(tyop_2Equote_2Evarmap(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27a)),c_2Ecanonical_2Einterp__sp_2E0(A_27a),X0_2E0) ) ).

tff(arityeq1_2Ec_2Ecanonical_2Einterp__vl_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a)] : ( c_2Ecanonical_2Einterp__vl_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(tyop_2Equote_2Evarmap(A_27a),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),A_27a)),c_2Ecanonical_2Einterp__vl_2E0(A_27a),X0_2E0) ) ).

tff(arityeq1_2Ec_2Ering_2Eis__ring_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Ering_2Ering(A_27a)] : ( c_2Ering_2Eis__ring_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Ering_2Ering(A_27a),tyop_2Emin_2Ebool,c_2Ering_2Eis__ring_2E0(A_27a),X0_2E0) ) ).

tff(arityeq1_2Ec_2Ecanonical_2Eivl__aux_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a)] : ( c_2Ecanonical_2Eivl__aux_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(tyop_2Equote_2Evarmap(A_27a),tyop_2Emin_2Efun(tyop_2Equote_2Eindex,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),A_27a))),c_2Ecanonical_2Eivl__aux_2E0(A_27a),X0_2E0) ) ).

tff(arityeq3_2Ec_2EternaryComparisons_2Elist__compare_2E3_2Emono_2Etyop_2Equote_2Eindex_20mono_2Etyop_2Equote_2Eindex,axiom,
    ! [X0_2E0: tyop_2Emin_2Efun(tyop_2Equote_2Eindex,tyop_2Emin_2Efun(tyop_2Equote_2Eindex,tyop_2EternaryComparisons_2Eordering)),X1_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),X2_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex)] : ( c_2EternaryComparisons_2Elist__compare_2E3(tyop_2Equote_2Eindex,tyop_2Equote_2Eindex,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2EternaryComparisons_2Eordering,app_2E2(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2EternaryComparisons_2Eordering),app_2E2(tyop_2Emin_2Efun(tyop_2Equote_2Eindex,tyop_2Emin_2Efun(tyop_2Equote_2Eindex,tyop_2EternaryComparisons_2Eordering)),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2EternaryComparisons_2Eordering)),c_2EternaryComparisons_2Elist__compare_2E0(tyop_2Equote_2Eindex,tyop_2Equote_2Eindex),X0_2E0),X1_2E0),X2_2E0) ) ).

tff(arityeq3_2Ec_2EternaryComparisons_2Elist__merge_2E3_2Emono_2Etyop_2Equote_2Eindex,axiom,
    ! [X0_2E0: tyop_2Emin_2Efun(tyop_2Equote_2Eindex,tyop_2Emin_2Efun(tyop_2Equote_2Eindex,tyop_2Emin_2Ebool)),X1_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),X2_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex)] : ( c_2EternaryComparisons_2Elist__merge_2E3(tyop_2Equote_2Eindex,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Elist_2Elist(tyop_2Equote_2Eindex),app_2E2(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Elist_2Elist(tyop_2Equote_2Eindex)),app_2E2(tyop_2Emin_2Efun(tyop_2Equote_2Eindex,tyop_2Emin_2Efun(tyop_2Equote_2Eindex,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Elist_2Elist(tyop_2Equote_2Eindex))),c_2EternaryComparisons_2Elist__merge_2E0(tyop_2Equote_2Eindex),X0_2E0),X1_2E0),X2_2E0) ) ).

tff(arityeq1_2Ec_2Ecanonical_2Emonom__insert_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a)] : ( c_2Ecanonical_2Emonom__insert_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)))),c_2Ecanonical_2Emonom__insert_2E0(A_27a),X0_2E0) ) ).

tff(arityeq4_2Ec_2EternaryComparisons_2Eordering__CASE_2E4_2Emono_2Etyop_2Ecanonical_2Ecanonical__sum_28A_27a_29,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2EternaryComparisons_2Eordering,X1_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a),X2_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a),X3_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : ( c_2EternaryComparisons_2Eordering__CASE_2E4(tyop_2Ecanonical_2Ecanonical__sum(A_27a),X0_2E0,X1_2E0,X2_2E0,X3_2E0) = app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a),app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)),app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a))),app_2E2(tyop_2EternaryComparisons_2Eordering,tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)))),c_2EternaryComparisons_2Eordering__CASE_2E0(tyop_2Ecanonical_2Ecanonical__sum(A_27a)),X0_2E0),X1_2E0),X2_2E0),X3_2E0) ) ).

tff(arityeq6_2Ec_2EringNorm_2Epolynom__CASE_2E6_2Emono_2EA_27a_20mono_2EA_27b,axiom,
    ! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2EringNorm_2Epolynom(A_27a),X1_2E0: tyop_2Emin_2Efun(tyop_2Equote_2Eindex,A_27b),X2_2E0: tyop_2Emin_2Efun(A_27a,A_27b),X3_2E0: tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),A_27b)),X4_2E0: tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),A_27b)),X5_2E0: tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),A_27b)] : ( c_2EringNorm_2Epolynom__CASE_2E6(A_27a,A_27b,X0_2E0,X1_2E0,X2_2E0,X3_2E0,X4_2E0,X5_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),A_27b),A_27b,app_2E2(tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),A_27b)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),A_27b),A_27b),app_2E2(tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),A_27b)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),A_27b)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),A_27b),A_27b)),app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),A_27b)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),A_27b)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),A_27b),A_27b))),app_2E2(tyop_2Emin_2Efun(tyop_2Equote_2Eindex,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),A_27b)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),A_27b)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),A_27b),A_27b)))),app_2E2(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Equote_2Eindex,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),A_27b)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),A_27b)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),A_27b),A_27b))))),c_2EringNorm_2Epolynom__CASE_2E0(A_27a,A_27b),X0_2E0),X1_2E0),X2_2E0),X3_2E0),X4_2E0),X5_2E0) ) ).

tff(arityeq2_2Ec_2EringNorm_2Epolynom__normalize_2E2_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Ering_2Ering(A_27a),X1_2E0: tyop_2EringNorm_2Epolynom(A_27a)] : ( c_2EringNorm_2Epolynom__normalize_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a),app_2E2(tyop_2Ering_2Ering(A_27a),tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)),c_2EringNorm_2Epolynom__normalize_2E0(A_27a),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2EringNorm_2Epolynom__simplify_2E2_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Ering_2Ering(A_27a),X1_2E0: tyop_2EringNorm_2Epolynom(A_27a)] : ( c_2EringNorm_2Epolynom__simplify_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a),app_2E2(tyop_2Ering_2Ering(A_27a),tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)),c_2EringNorm_2Epolynom__simplify_2E0(A_27a),X0_2E0),X1_2E0) ) ).

tff(arityeq2_2Ec_2EringNorm_2Epolynom__size_2E2_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),X1_2E0: tyop_2EringNorm_2Epolynom(A_27a)] : ( c_2EringNorm_2Epolynom__size_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Enum_2Enum,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Enum_2Enum),c_2EringNorm_2Epolynom__size_2E0(A_27a),X0_2E0),X1_2E0) ) ).

tff(arityeq1_2Ec_2EringNorm_2Er__canonical__sum__merge_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Ering_2Ering(A_27a)] : ( c_2EringNorm_2Er__canonical__sum__merge_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Ering_2Ering(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a))),c_2EringNorm_2Er__canonical__sum__merge_2E0(A_27a),X0_2E0) ) ).

tff(arityeq3_2Ec_2EringNorm_2Er__canonical__sum__merge_2E3_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Ering_2Ering(A_27a),X1_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a),X2_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : ( c_2EringNorm_2Er__canonical__sum__merge_2E3(A_27a,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a),app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)),app_2E2(tyop_2Ering_2Ering(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a))),c_2EringNorm_2Er__canonical__sum__merge_2E0(A_27a),X0_2E0),X1_2E0),X2_2E0) ) ).

tff(arityeq1_2Ec_2EringNorm_2Er__canonical__sum__prod_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Ering_2Ering(A_27a)] : ( c_2EringNorm_2Er__canonical__sum__prod_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Ering_2Ering(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a))),c_2EringNorm_2Er__canonical__sum__prod_2E0(A_27a),X0_2E0) ) ).

tff(arityeq3_2Ec_2EringNorm_2Er__canonical__sum__prod_2E3_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Ering_2Ering(A_27a),X1_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a),X2_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : ( c_2EringNorm_2Er__canonical__sum__prod_2E3(A_27a,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a),app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)),app_2E2(tyop_2Ering_2Ering(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a))),c_2EringNorm_2Er__canonical__sum__prod_2E0(A_27a),X0_2E0),X1_2E0),X2_2E0) ) ).

tff(arityeq1_2Ec_2EringNorm_2Er__canonical__sum__scalar_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Ering_2Ering(A_27a)] : ( c_2EringNorm_2Er__canonical__sum__scalar_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Ering_2Ering(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a))),c_2EringNorm_2Er__canonical__sum__scalar_2E0(A_27a),X0_2E0) ) ).

tff(arityeq3_2Ec_2EringNorm_2Er__canonical__sum__scalar_2E3_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Ering_2Ering(A_27a),X1_2E0: A_27a,X2_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : ( c_2EringNorm_2Er__canonical__sum__scalar_2E3(A_27a,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)),app_2E2(tyop_2Ering_2Ering(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a))),c_2EringNorm_2Er__canonical__sum__scalar_2E0(A_27a),X0_2E0),X1_2E0),X2_2E0) ) ).

tff(arityeq1_2Ec_2EringNorm_2Er__canonical__sum__scalar2_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Ering_2Ering(A_27a)] : ( c_2EringNorm_2Er__canonical__sum__scalar2_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Ering_2Ering(A_27a),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a))),c_2EringNorm_2Er__canonical__sum__scalar2_2E0(A_27a),X0_2E0) ) ).

tff(arityeq3_2Ec_2EringNorm_2Er__canonical__sum__scalar2_2E3_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Ering_2Ering(A_27a),X1_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),X2_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : ( c_2EringNorm_2Er__canonical__sum__scalar2_2E3(A_27a,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a),app_2E2(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)),app_2E2(tyop_2Ering_2Ering(A_27a),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a))),c_2EringNorm_2Er__canonical__sum__scalar2_2E0(A_27a),X0_2E0),X1_2E0),X2_2E0) ) ).

tff(arityeq1_2Ec_2EringNorm_2Er__canonical__sum__scalar3_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Ering_2Ering(A_27a)] : ( c_2EringNorm_2Er__canonical__sum__scalar3_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Ering_2Ering(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)))),c_2EringNorm_2Er__canonical__sum__scalar3_2E0(A_27a),X0_2E0) ) ).

tff(arityeq4_2Ec_2EringNorm_2Er__canonical__sum__scalar3_2E4_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Ering_2Ering(A_27a),X1_2E0: A_27a,X2_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),X3_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : ( c_2EringNorm_2Er__canonical__sum__scalar3_2E4(A_27a,X0_2E0,X1_2E0,X2_2E0,X3_2E0) = app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a),app_2E2(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a))),app_2E2(tyop_2Ering_2Ering(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)))),c_2EringNorm_2Er__canonical__sum__scalar3_2E0(A_27a),X0_2E0),X1_2E0),X2_2E0),X3_2E0) ) ).

tff(arityeq1_2Ec_2EringNorm_2Er__canonical__sum__simplify_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Ering_2Ering(A_27a)] : ( c_2EringNorm_2Er__canonical__sum__simplify_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Ering_2Ering(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)),c_2EringNorm_2Er__canonical__sum__simplify_2E0(A_27a),X0_2E0) ) ).

tff(arityeq2_2Ec_2EringNorm_2Er__canonical__sum__simplify_2E2_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Ering_2Ering(A_27a),X1_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : ( c_2EringNorm_2Er__canonical__sum__simplify_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a),app_2E2(tyop_2Ering_2Ering(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)),c_2EringNorm_2Er__canonical__sum__simplify_2E0(A_27a),X0_2E0),X1_2E0) ) ).

tff(arityeq1_2Ec_2EringNorm_2Er__ics__aux_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Ering_2Ering(A_27a)] : ( c_2EringNorm_2Er__ics__aux_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Ering_2Ering(A_27a),tyop_2Emin_2Efun(tyop_2Equote_2Evarmap(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27a))),c_2EringNorm_2Er__ics__aux_2E0(A_27a),X0_2E0) ) ).

tff(arityeq4_2Ec_2EringNorm_2Er__ics__aux_2E4_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Ering_2Ering(A_27a),X1_2E0: tyop_2Equote_2Evarmap(A_27a),X2_2E0: A_27a,X3_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : ( c_2EringNorm_2Er__ics__aux_2E4(A_27a,X0_2E0,X1_2E0,X2_2E0,X3_2E0) = app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27a,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27a),app_2E2(tyop_2Equote_2Evarmap(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27a)),app_2E2(tyop_2Ering_2Ering(A_27a),tyop_2Emin_2Efun(tyop_2Equote_2Evarmap(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27a))),c_2EringNorm_2Er__ics__aux_2E0(A_27a),X0_2E0),X1_2E0),X2_2E0),X3_2E0) ) ).

tff(arityeq1_2Ec_2EringNorm_2Er__interp__cs_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Ering_2Ering(A_27a)] : ( c_2EringNorm_2Er__interp__cs_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Ering_2Ering(A_27a),tyop_2Emin_2Efun(tyop_2Equote_2Evarmap(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27a)),c_2EringNorm_2Er__interp__cs_2E0(A_27a),X0_2E0) ) ).

tff(arityeq3_2Ec_2EringNorm_2Er__interp__cs_2E3_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Ering_2Ering(A_27a),X1_2E0: tyop_2Equote_2Evarmap(A_27a),X2_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : ( c_2EringNorm_2Er__interp__cs_2E3(A_27a,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27a,app_2E2(tyop_2Equote_2Evarmap(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27a),app_2E2(tyop_2Ering_2Ering(A_27a),tyop_2Emin_2Efun(tyop_2Equote_2Evarmap(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27a)),c_2EringNorm_2Er__interp__cs_2E0(A_27a),X0_2E0),X1_2E0),X2_2E0) ) ).

tff(arityeq1_2Ec_2EringNorm_2Er__interp__m_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Ering_2Ering(A_27a)] : ( c_2EringNorm_2Er__interp__m_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Ering_2Ering(A_27a),tyop_2Emin_2Efun(tyop_2Equote_2Evarmap(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),A_27a))),c_2EringNorm_2Er__interp__m_2E0(A_27a),X0_2E0) ) ).

tff(arityeq4_2Ec_2EringNorm_2Er__interp__m_2E4_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Ering_2Ering(A_27a),X1_2E0: tyop_2Equote_2Evarmap(A_27a),X2_2E0: A_27a,X3_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex)] : ( c_2EringNorm_2Er__interp__m_2E4(A_27a,X0_2E0,X1_2E0,X2_2E0,X3_2E0) = app_2E2(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),A_27a,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),A_27a),app_2E2(tyop_2Equote_2Evarmap(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),A_27a)),app_2E2(tyop_2Ering_2Ering(A_27a),tyop_2Emin_2Efun(tyop_2Equote_2Evarmap(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),A_27a))),c_2EringNorm_2Er__interp__m_2E0(A_27a),X0_2E0),X1_2E0),X2_2E0),X3_2E0) ) ).

tff(arityeq1_2Ec_2EringNorm_2Er__interp__sp_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Ering_2Ering(A_27a)] : ( c_2EringNorm_2Er__interp__sp_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Ering_2Ering(A_27a),tyop_2Emin_2Efun(tyop_2Equote_2Evarmap(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27a)),c_2EringNorm_2Er__interp__sp_2E0(A_27a),X0_2E0) ) ).

tff(arityeq3_2Ec_2EringNorm_2Er__interp__sp_2E3_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Ering_2Ering(A_27a),X1_2E0: tyop_2Equote_2Evarmap(A_27a),X2_2E0: tyop_2Ecanonical_2Espolynom(A_27a)] : ( c_2EringNorm_2Er__interp__sp_2E3(A_27a,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Ecanonical_2Espolynom(A_27a),A_27a,app_2E2(tyop_2Equote_2Evarmap(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27a),app_2E2(tyop_2Ering_2Ering(A_27a),tyop_2Emin_2Efun(tyop_2Equote_2Evarmap(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27a)),c_2EringNorm_2Er__interp__sp_2E0(A_27a),X0_2E0),X1_2E0),X2_2E0) ) ).

tff(arityeq1_2Ec_2EringNorm_2Er__interp__vl_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Ering_2Ering(A_27a)] : ( c_2EringNorm_2Er__interp__vl_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Ering_2Ering(A_27a),tyop_2Emin_2Efun(tyop_2Equote_2Evarmap(A_27a),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),A_27a)),c_2EringNorm_2Er__interp__vl_2E0(A_27a),X0_2E0) ) ).

tff(arityeq3_2Ec_2EringNorm_2Er__interp__vl_2E3_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Ering_2Ering(A_27a),X1_2E0: tyop_2Equote_2Evarmap(A_27a),X2_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex)] : ( c_2EringNorm_2Er__interp__vl_2E3(A_27a,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),A_27a,app_2E2(tyop_2Equote_2Evarmap(A_27a),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),A_27a),app_2E2(tyop_2Ering_2Ering(A_27a),tyop_2Emin_2Efun(tyop_2Equote_2Evarmap(A_27a),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),A_27a)),c_2EringNorm_2Er__interp__vl_2E0(A_27a),X0_2E0),X1_2E0),X2_2E0) ) ).

tff(arityeq1_2Ec_2EringNorm_2Er__ivl__aux_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Ering_2Ering(A_27a)] : ( c_2EringNorm_2Er__ivl__aux_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Ering_2Ering(A_27a),tyop_2Emin_2Efun(tyop_2Equote_2Evarmap(A_27a),tyop_2Emin_2Efun(tyop_2Equote_2Eindex,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),A_27a))),c_2EringNorm_2Er__ivl__aux_2E0(A_27a),X0_2E0) ) ).

tff(arityeq4_2Ec_2EringNorm_2Er__ivl__aux_2E4_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Ering_2Ering(A_27a),X1_2E0: tyop_2Equote_2Evarmap(A_27a),X2_2E0: tyop_2Equote_2Eindex,X3_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex)] : ( c_2EringNorm_2Er__ivl__aux_2E4(A_27a,X0_2E0,X1_2E0,X2_2E0,X3_2E0) = app_2E2(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),A_27a,app_2E2(tyop_2Equote_2Eindex,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),A_27a),app_2E2(tyop_2Equote_2Evarmap(A_27a),tyop_2Emin_2Efun(tyop_2Equote_2Eindex,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),A_27a)),app_2E2(tyop_2Ering_2Ering(A_27a),tyop_2Emin_2Efun(tyop_2Equote_2Evarmap(A_27a),tyop_2Emin_2Efun(tyop_2Equote_2Eindex,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),A_27a))),c_2EringNorm_2Er__ivl__aux_2E0(A_27a),X0_2E0),X1_2E0),X2_2E0),X3_2E0) ) ).

tff(arityeq1_2Ec_2EringNorm_2Er__monom__insert_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Ering_2Ering(A_27a)] : ( c_2EringNorm_2Er__monom__insert_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Ering_2Ering(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)))),c_2EringNorm_2Er__monom__insert_2E0(A_27a),X0_2E0) ) ).

tff(arityeq4_2Ec_2EringNorm_2Er__monom__insert_2E4_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Ering_2Ering(A_27a),X1_2E0: A_27a,X2_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),X3_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : ( c_2EringNorm_2Er__monom__insert_2E4(A_27a,X0_2E0,X1_2E0,X2_2E0,X3_2E0) = app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a),app_2E2(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a))),app_2E2(tyop_2Ering_2Ering(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)))),c_2EringNorm_2Er__monom__insert_2E0(A_27a),X0_2E0),X1_2E0),X2_2E0),X3_2E0) ) ).

tff(arityeq1_2Ec_2EringNorm_2Er__spolynom__normalize_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Ering_2Ering(A_27a)] : ( c_2EringNorm_2Er__spolynom__normalize_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Ering_2Ering(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)),c_2EringNorm_2Er__spolynom__normalize_2E0(A_27a),X0_2E0) ) ).

tff(arityeq2_2Ec_2EringNorm_2Er__spolynom__normalize_2E2_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Ering_2Ering(A_27a),X1_2E0: tyop_2Ecanonical_2Espolynom(A_27a)] : ( c_2EringNorm_2Er__spolynom__normalize_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a),app_2E2(tyop_2Ering_2Ering(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)),c_2EringNorm_2Er__spolynom__normalize_2E0(A_27a),X0_2E0),X1_2E0) ) ).

tff(arityeq1_2Ec_2EringNorm_2Er__spolynom__simplify_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Ering_2Ering(A_27a)] : ( c_2EringNorm_2Er__spolynom__simplify_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Ering_2Ering(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)),c_2EringNorm_2Er__spolynom__simplify_2E0(A_27a),X0_2E0) ) ).

tff(arityeq2_2Ec_2EringNorm_2Er__spolynom__simplify_2E2_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Ering_2Ering(A_27a),X1_2E0: tyop_2Ecanonical_2Espolynom(A_27a)] : ( c_2EringNorm_2Er__spolynom__simplify_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a),app_2E2(tyop_2Ering_2Ering(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)),c_2EringNorm_2Er__spolynom__simplify_2E0(A_27a),X0_2E0),X1_2E0) ) ).

tff(arityeq1_2Ec_2EringNorm_2Er__varlist__insert_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Ering_2Ering(A_27a)] : ( c_2EringNorm_2Er__varlist__insert_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Ering_2Ering(A_27a),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a))),c_2EringNorm_2Er__varlist__insert_2E0(A_27a),X0_2E0) ) ).

tff(arityeq3_2Ec_2EringNorm_2Er__varlist__insert_2E3_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Ering_2Ering(A_27a),X1_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),X2_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : ( c_2EringNorm_2Er__varlist__insert_2E3(A_27a,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a),app_2E2(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)),app_2E2(tyop_2Ering_2Ering(A_27a),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a))),c_2EringNorm_2Er__varlist__insert_2E0(A_27a),X0_2E0),X1_2E0),X2_2E0) ) ).

tff(arityeq1_2Ec_2Ering_2Ering__R0_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Ering_2Ering(A_27a)] : ( c_2Ering_2Ering__R0_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Ering_2Ering(A_27a),A_27a,c_2Ering_2Ering__R0_2E0(A_27a),X0_2E0) ) ).

tff(arityeq1_2Ec_2Ering_2Ering__R1_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Ering_2Ering(A_27a)] : ( c_2Ering_2Ering__R1_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Ering_2Ering(A_27a),A_27a,c_2Ering_2Ering__R1_2E0(A_27a),X0_2E0) ) ).

tff(arityeq3_2Ec_2Ering_2Ering__RM_2E3_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Ering_2Ering(A_27a),X1_2E0: A_27a,X2_2E0: A_27a] : ( c_2Ering_2Ering__RM_2E3(A_27a,X0_2E0,X1_2E0,X2_2E0) = app_2E2(A_27a,A_27a,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,A_27a),app_2E2(tyop_2Ering_2Ering(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a)),c_2Ering_2Ering__RM_2E0(A_27a),X0_2E0),X1_2E0),X2_2E0) ) ).

tff(arityeq2_2Ec_2Ering_2Ering__RN_2E2_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Ering_2Ering(A_27a),X1_2E0: A_27a] : ( c_2Ering_2Ering__RN_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(A_27a,A_27a,app_2E2(tyop_2Ering_2Ering(A_27a),tyop_2Emin_2Efun(A_27a,A_27a),c_2Ering_2Ering__RN_2E0(A_27a),X0_2E0),X1_2E0) ) ).

tff(arityeq3_2Ec_2Ering_2Ering__RP_2E3_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Ering_2Ering(A_27a),X1_2E0: A_27a,X2_2E0: A_27a] : ( c_2Ering_2Ering__RP_2E3(A_27a,X0_2E0,X1_2E0,X2_2E0) = app_2E2(A_27a,A_27a,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,A_27a),app_2E2(tyop_2Ering_2Ering(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a)),c_2Ering_2Ering__RP_2E0(A_27a),X0_2E0),X1_2E0),X2_2E0) ) ).

tff(arityeq1_2Ec_2Ering_2Esemi__ring__of_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Ering_2Ering(A_27a)] : ( c_2Ering_2Esemi__ring__of_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Ering_2Ering(A_27a),tyop_2Esemi__ring_2Esemi__ring(A_27a),c_2Ering_2Esemi__ring__of_2E0(A_27a),X0_2E0) ) ).

tff(arityeq1_2Ec_2Ecanonical_2Espolynom__normalize_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a)] : ( c_2Ecanonical_2Espolynom__normalize_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)),c_2Ecanonical_2Espolynom__normalize_2E0(A_27a),X0_2E0) ) ).

tff(arityeq1_2Ec_2Ecanonical_2Espolynom__simplify_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a)] : ( c_2Ecanonical_2Espolynom__simplify_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)),c_2Ecanonical_2Espolynom__simplify_2E0(A_27a),X0_2E0) ) ).

tff(arityeq1_2Ec_2Ecanonical_2Evarlist__insert_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a)] : ( c_2Ecanonical_2Evarlist__insert_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a))),c_2Ecanonical_2Evarlist__insert_2E0(A_27a),X0_2E0) ) ).

tff(arityeq2_2Ec_2Equote_2Evarmap__find_2E2_2Emono_2EA_27a,axiom,
    ! [A_27a: $tType,X0_2E0: tyop_2Equote_2Eindex,X1_2E0: tyop_2Equote_2Evarmap(A_27a)] : ( c_2Equote_2Evarmap__find_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Equote_2Evarmap(A_27a),A_27a,app_2E2(tyop_2Equote_2Eindex,tyop_2Emin_2Efun(tyop_2Equote_2Evarmap(A_27a),A_27a),c_2Equote_2Evarmap__find_2E0(A_27a),X0_2E0),X1_2E0) ) ).

tff(arityeq1_2Ec_2Ebool_2E_7E_2E1,axiom,
    ! [X0_2E0: tyop_2Emin_2Ebool] :
      ( ~ p(X0_2E0)
    <=> p(app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool,c_2Ebool_2E_7E_2E0,X0_2E0)) ) ).

tff(thm_2EringNorm_2Er__interp__sp__def,axiom,
    ! [A_27a: $tType,V0r_2E0: tyop_2Ering_2Ering(A_27a)] : ( c_2EringNorm_2Er__interp__sp_2E1(A_27a,V0r_2E0) = c_2Ecanonical_2Einterp__sp_2E1(A_27a,c_2Ering_2Esemi__ring__of_2E1(A_27a,V0r_2E0)) ) ).

tff(thm_2EringNorm_2Er__spolynom__simplify__def,axiom,
    ! [A_27a: $tType,V0r_2E0: tyop_2Ering_2Ering(A_27a)] : ( c_2EringNorm_2Er__spolynom__simplify_2E1(A_27a,V0r_2E0) = c_2Ecanonical_2Espolynom__simplify_2E1(A_27a,c_2Ering_2Esemi__ring__of_2E1(A_27a,V0r_2E0)) ) ).

tff(thm_2EringNorm_2Er__spolynom__normalize__def,axiom,
    ! [A_27a: $tType,V0r_2E0: tyop_2Ering_2Ering(A_27a)] : ( c_2EringNorm_2Er__spolynom__normalize_2E1(A_27a,V0r_2E0) = c_2Ecanonical_2Espolynom__normalize_2E1(A_27a,c_2Ering_2Esemi__ring__of_2E1(A_27a,V0r_2E0)) ) ).

tff(thm_2EringNorm_2Er__interp__cs__def,axiom,
    ! [A_27a: $tType,V0r_2E0: tyop_2Ering_2Ering(A_27a)] : ( c_2EringNorm_2Er__interp__cs_2E1(A_27a,V0r_2E0) = c_2Ecanonical_2Einterp__cs_2E1(A_27a,c_2Ering_2Esemi__ring__of_2E1(A_27a,V0r_2E0)) ) ).

tff(thm_2EringNorm_2Er__ics__aux__def,axiom,
    ! [A_27a: $tType,V0r_2E0: tyop_2Ering_2Ering(A_27a)] : ( c_2EringNorm_2Er__ics__aux_2E1(A_27a,V0r_2E0) = c_2Ecanonical_2Eics__aux_2E1(A_27a,c_2Ering_2Esemi__ring__of_2E1(A_27a,V0r_2E0)) ) ).

tff(thm_2EringNorm_2Er__interp__m__def,axiom,
    ! [A_27a: $tType,V0r_2E0: tyop_2Ering_2Ering(A_27a)] : ( c_2EringNorm_2Er__interp__m_2E1(A_27a,V0r_2E0) = c_2Ecanonical_2Einterp__m_2E1(A_27a,c_2Ering_2Esemi__ring__of_2E1(A_27a,V0r_2E0)) ) ).

tff(thm_2EringNorm_2Er__interp__vl__def,axiom,
    ! [A_27a: $tType,V0r_2E0: tyop_2Ering_2Ering(A_27a)] : ( c_2EringNorm_2Er__interp__vl_2E1(A_27a,V0r_2E0) = c_2Ecanonical_2Einterp__vl_2E1(A_27a,c_2Ering_2Esemi__ring__of_2E1(A_27a,V0r_2E0)) ) ).

tff(thm_2EringNorm_2Er__ivl__aux__def,axiom,
    ! [A_27a: $tType,V0r_2E0: tyop_2Ering_2Ering(A_27a)] : ( c_2EringNorm_2Er__ivl__aux_2E1(A_27a,V0r_2E0) = c_2Ecanonical_2Eivl__aux_2E1(A_27a,c_2Ering_2Esemi__ring__of_2E1(A_27a,V0r_2E0)) ) ).

tff(thm_2EringNorm_2Er__canonical__sum__simplify__def,axiom,
    ! [A_27a: $tType,V0r_2E0: tyop_2Ering_2Ering(A_27a)] : ( c_2EringNorm_2Er__canonical__sum__simplify_2E1(A_27a,V0r_2E0) = c_2Ecanonical_2Ecanonical__sum__simplify_2E1(A_27a,c_2Ering_2Esemi__ring__of_2E1(A_27a,V0r_2E0)) ) ).

tff(thm_2EringNorm_2Er__canonical__sum__prod__def,axiom,
    ! [A_27a: $tType,V0r_2E0: tyop_2Ering_2Ering(A_27a)] : ( c_2EringNorm_2Er__canonical__sum__prod_2E1(A_27a,V0r_2E0) = c_2Ecanonical_2Ecanonical__sum__prod_2E1(A_27a,c_2Ering_2Esemi__ring__of_2E1(A_27a,V0r_2E0)) ) ).

tff(thm_2EringNorm_2Er__canonical__sum__scalar3__def,axiom,
    ! [A_27a: $tType,V0r_2E0: tyop_2Ering_2Ering(A_27a)] : ( c_2EringNorm_2Er__canonical__sum__scalar3_2E1(A_27a,V0r_2E0) = c_2Ecanonical_2Ecanonical__sum__scalar3_2E1(A_27a,c_2Ering_2Esemi__ring__of_2E1(A_27a,V0r_2E0)) ) ).

tff(thm_2EringNorm_2Er__canonical__sum__scalar2__def,axiom,
    ! [A_27a: $tType,V0r_2E0: tyop_2Ering_2Ering(A_27a)] : ( c_2EringNorm_2Er__canonical__sum__scalar2_2E1(A_27a,V0r_2E0) = c_2Ecanonical_2Ecanonical__sum__scalar2_2E1(A_27a,c_2Ering_2Esemi__ring__of_2E1(A_27a,V0r_2E0)) ) ).

tff(thm_2EringNorm_2Er__canonical__sum__scalar__def,axiom,
    ! [A_27a: $tType,V0r_2E0: tyop_2Ering_2Ering(A_27a)] : ( c_2EringNorm_2Er__canonical__sum__scalar_2E1(A_27a,V0r_2E0) = c_2Ecanonical_2Ecanonical__sum__scalar_2E1(A_27a,c_2Ering_2Esemi__ring__of_2E1(A_27a,V0r_2E0)) ) ).

tff(thm_2EringNorm_2Er__varlist__insert__def,axiom,
    ! [A_27a: $tType,V0r_2E0: tyop_2Ering_2Ering(A_27a)] : ( c_2EringNorm_2Er__varlist__insert_2E1(A_27a,V0r_2E0) = c_2Ecanonical_2Evarlist__insert_2E1(A_27a,c_2Ering_2Esemi__ring__of_2E1(A_27a,V0r_2E0)) ) ).

tff(thm_2EringNorm_2Er__monom__insert__def,axiom,
    ! [A_27a: $tType,V0r_2E0: tyop_2Ering_2Ering(A_27a)] : ( c_2EringNorm_2Er__monom__insert_2E1(A_27a,V0r_2E0) = c_2Ecanonical_2Emonom__insert_2E1(A_27a,c_2Ering_2Esemi__ring__of_2E1(A_27a,V0r_2E0)) ) ).

tff(thm_2EringNorm_2Er__canonical__sum__merge__def,axiom,
    ! [A_27a: $tType,V0r_2E0: tyop_2Ering_2Ering(A_27a)] : ( c_2EringNorm_2Er__canonical__sum__merge_2E1(A_27a,V0r_2E0) = c_2Ecanonical_2Ecanonical__sum__merge_2E1(A_27a,c_2Ering_2Esemi__ring__of_2E1(A_27a,V0r_2E0)) ) ).

tff(thm_2EringNorm_2Epolynom__TY__DEF,axiom,
    ! [A_27a: $tType,F5_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a))),F4_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a))),F3_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a))),F2_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a))),F1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a))),F0_2E0: tyop_2Emin_2Efun(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)),tyop_2Emin_2Ebool)] :
      ( ! [V15n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)),F5_2E0,V15n_2E0) = c_2Eind__type_2EBOTTOM_2E0(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)) )
     => ( ! [V13n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)),F4_2E0,V13n_2E0) = c_2Eind__type_2EBOTTOM_2E0(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)) )
       => ( ! [V10n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)),F3_2E0,V10n_2E0) = c_2Eind__type_2EBOTTOM_2E0(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)) )
         => ( ! [V7n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)),F2_2E0,V7n_2E0) = c_2Eind__type_2EBOTTOM_2E0(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)) )
           => ( ! [V5n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)),F1_2E0,V5n_2E0) = c_2Eind__type_2EBOTTOM_2E0(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)) )
             => ( ! [V1a0_27_2E0: tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a))] :
                    ( p(app_2E2(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)),tyop_2Emin_2Ebool,F0_2E0,V1a0_27_2E0))
                  <=> ! [V2_27polynom_27_2E0: tyop_2Emin_2Efun(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)),tyop_2Emin_2Ebool)] :
                        ( ! [V3a0_27_2E0: tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a))] :
                            ( ( ? [V4a_2E0: tyop_2Equote_2Eindex] : ( V3a0_27_2E0 = c_2Eind__type_2ECONSTR_2E3(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a),c_2Enum_2E0_2E0,c_2Epair_2E_2C_2E2(tyop_2Equote_2Eindex,A_27a,V4a_2E0,c_2Ebool_2EARB_2E0(A_27a)),F1_2E0) )
                              | ? [V6a_2E0: A_27a] : ( V3a0_27_2E0 = c_2Eind__type_2ECONSTR_2E3(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a),c_2Enum_2ESUC_2E1(c_2Enum_2E0_2E0),c_2Epair_2E_2C_2E2(tyop_2Equote_2Eindex,A_27a,c_2Ebool_2EARB_2E0(tyop_2Equote_2Eindex),V6a_2E0),F2_2E0) )
                              | ? [V8a0_2E0: tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)),V9a1_2E0: tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a))] :
                                  ( ( V3a0_27_2E0 = c_2Eind__type_2ECONSTR_2E3(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a),c_2Enum_2ESUC_2E1(c_2Enum_2ESUC_2E1(c_2Enum_2E0_2E0)),c_2Epair_2E_2C_2E2(tyop_2Equote_2Eindex,A_27a,c_2Ebool_2EARB_2E0(tyop_2Equote_2Eindex),c_2Ebool_2EARB_2E0(A_27a)),c_2Eind__type_2EFCONS_2E2(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)),V8a0_2E0,c_2Eind__type_2EFCONS_2E2(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)),V9a1_2E0,F3_2E0))) )
                                  & p(app_2E2(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)),tyop_2Emin_2Ebool,V2_27polynom_27_2E0,V8a0_2E0))
                                  & p(app_2E2(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)),tyop_2Emin_2Ebool,V2_27polynom_27_2E0,V9a1_2E0)) )
                              | ? [V11a0_2E0: tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)),V12a1_2E0: tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a))] :
                                  ( ( V3a0_27_2E0 = c_2Eind__type_2ECONSTR_2E3(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a),c_2Enum_2ESUC_2E1(c_2Enum_2ESUC_2E1(c_2Enum_2ESUC_2E1(c_2Enum_2E0_2E0))),c_2Epair_2E_2C_2E2(tyop_2Equote_2Eindex,A_27a,c_2Ebool_2EARB_2E0(tyop_2Equote_2Eindex),c_2Ebool_2EARB_2E0(A_27a)),c_2Eind__type_2EFCONS_2E2(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)),V11a0_2E0,c_2Eind__type_2EFCONS_2E2(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)),V12a1_2E0,F4_2E0))) )
                                  & p(app_2E2(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)),tyop_2Emin_2Ebool,V2_27polynom_27_2E0,V11a0_2E0))
                                  & p(app_2E2(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)),tyop_2Emin_2Ebool,V2_27polynom_27_2E0,V12a1_2E0)) )
                              | ? [V14a_2E0: tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a))] :
                                  ( ( V3a0_27_2E0 = c_2Eind__type_2ECONSTR_2E3(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a),c_2Enum_2ESUC_2E1(c_2Enum_2ESUC_2E1(c_2Enum_2ESUC_2E1(c_2Enum_2ESUC_2E1(c_2Enum_2E0_2E0)))),c_2Epair_2E_2C_2E2(tyop_2Equote_2Eindex,A_27a,c_2Ebool_2EARB_2E0(tyop_2Equote_2Eindex),c_2Ebool_2EARB_2E0(A_27a)),c_2Eind__type_2EFCONS_2E2(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)),V14a_2E0,F5_2E0)) )
                                  & p(app_2E2(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)),tyop_2Emin_2Ebool,V2_27polynom_27_2E0,V14a_2E0)) ) )
                           => p(app_2E2(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)),tyop_2Emin_2Ebool,V2_27polynom_27_2E0,V3a0_27_2E0)) )
                       => p(app_2E2(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)),tyop_2Emin_2Ebool,V2_27polynom_27_2E0,V1a0_27_2E0)) ) )
               => ? [V0rep_2E0: tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)))] : p(c_2Ebool_2ETYPE__DEFINITION_2E2(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)),tyop_2EringNorm_2Epolynom(A_27a),F0_2E0,V0rep_2E0)) ) ) ) ) ) ) ).

tff(thm_2EringNorm_2Epolynom__case__def,axiom,
    ! [A_27a: $tType,A_27b: $tType] :
      ( ! [V0a_2E0: tyop_2Equote_2Eindex,V1f_2E0: tyop_2Emin_2Efun(tyop_2Equote_2Eindex,A_27b),V2f1_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V3f2_2E0: tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),A_27b)),V4f3_2E0: tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),A_27b)),V5f4_2E0: tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),A_27b)] : ( c_2EringNorm_2Epolynom__CASE_2E6(A_27a,A_27b,c_2EringNorm_2EPvar_2E1(A_27a,V0a_2E0),V1f_2E0,V2f1_2E0,V3f2_2E0,V4f3_2E0,V5f4_2E0) = app_2E2(tyop_2Equote_2Eindex,A_27b,V1f_2E0,V0a_2E0) )
      & ! [V6a_2E0: A_27a,V7f_2E0: tyop_2Emin_2Efun(tyop_2Equote_2Eindex,A_27b),V8f1_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V9f2_2E0: tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),A_27b)),V10f3_2E0: tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),A_27b)),V11f4_2E0: tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),A_27b)] : ( c_2EringNorm_2Epolynom__CASE_2E6(A_27a,A_27b,c_2EringNorm_2EPconst_2E1(A_27a,V6a_2E0),V7f_2E0,V8f1_2E0,V9f2_2E0,V10f3_2E0,V11f4_2E0) = app_2E2(A_27a,A_27b,V8f1_2E0,V6a_2E0) )
      & ! [V12a0_2E0: tyop_2EringNorm_2Epolynom(A_27a),V13a1_2E0: tyop_2EringNorm_2Epolynom(A_27a),V14f_2E0: tyop_2Emin_2Efun(tyop_2Equote_2Eindex,A_27b),V15f1_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V16f2_2E0: tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),A_27b)),V17f3_2E0: tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),A_27b)),V18f4_2E0: tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),A_27b)] : ( c_2EringNorm_2Epolynom__CASE_2E6(A_27a,A_27b,c_2EringNorm_2EPplus_2E2(A_27a,V12a0_2E0,V13a1_2E0),V14f_2E0,V15f1_2E0,V16f2_2E0,V17f3_2E0,V18f4_2E0) = app_2E2(tyop_2EringNorm_2Epolynom(A_27a),A_27b,app_2E2(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),A_27b),V16f2_2E0,V12a0_2E0),V13a1_2E0) )
      & ! [V19a0_2E0: tyop_2EringNorm_2Epolynom(A_27a),V20a1_2E0: tyop_2EringNorm_2Epolynom(A_27a),V21f_2E0: tyop_2Emin_2Efun(tyop_2Equote_2Eindex,A_27b),V22f1_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V23f2_2E0: tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),A_27b)),V24f3_2E0: tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),A_27b)),V25f4_2E0: tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),A_27b)] : ( c_2EringNorm_2Epolynom__CASE_2E6(A_27a,A_27b,c_2EringNorm_2EPmult_2E2(A_27a,V19a0_2E0,V20a1_2E0),V21f_2E0,V22f1_2E0,V23f2_2E0,V24f3_2E0,V25f4_2E0) = app_2E2(tyop_2EringNorm_2Epolynom(A_27a),A_27b,app_2E2(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),A_27b),V24f3_2E0,V19a0_2E0),V20a1_2E0) )
      & ! [V26a_2E0: tyop_2EringNorm_2Epolynom(A_27a),V27f_2E0: tyop_2Emin_2Efun(tyop_2Equote_2Eindex,A_27b),V28f1_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V29f2_2E0: tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),A_27b)),V30f3_2E0: tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),A_27b)),V31f4_2E0: tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),A_27b)] : ( c_2EringNorm_2Epolynom__CASE_2E6(A_27a,A_27b,c_2EringNorm_2EPopp_2E1(A_27a,V26a_2E0),V27f_2E0,V28f1_2E0,V29f2_2E0,V30f3_2E0,V31f4_2E0) = app_2E2(tyop_2EringNorm_2Epolynom(A_27a),A_27b,V31f4_2E0,V26a_2E0) ) ) ).

tff(thm_2EringNorm_2Epolynom__size__def,axiom,
    ! [A_27a: $tType] :
      ( ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V1a_2E0: tyop_2Equote_2Eindex] : ( c_2EringNorm_2Epolynom__size_2E2(A_27a,V0f_2E0,c_2EringNorm_2EPvar_2E1(A_27a,V1a_2E0)) = c_2Earithmetic_2E_2B_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)),c_2Equote_2Eindex__size_2E1(V1a_2E0)) )
      & ! [V2f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V3a_2E0: A_27a] : ( c_2EringNorm_2Epolynom__size_2E2(A_27a,V2f_2E0,c_2EringNorm_2EPconst_2E1(A_27a,V3a_2E0)) = c_2Earithmetic_2E_2B_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)),app_2E2(A_27a,tyop_2Enum_2Enum,V2f_2E0,V3a_2E0)) )
      & ! [V4f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V5a0_2E0: tyop_2EringNorm_2Epolynom(A_27a),V6a1_2E0: tyop_2EringNorm_2Epolynom(A_27a)] : ( c_2EringNorm_2Epolynom__size_2E2(A_27a,V4f_2E0,c_2EringNorm_2EPplus_2E2(A_27a,V5a0_2E0,V6a1_2E0)) = c_2Earithmetic_2E_2B_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)),c_2Earithmetic_2E_2B_2E2(c_2EringNorm_2Epolynom__size_2E2(A_27a,V4f_2E0,V5a0_2E0),c_2EringNorm_2Epolynom__size_2E2(A_27a,V4f_2E0,V6a1_2E0))) )
      & ! [V7f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V8a0_2E0: tyop_2EringNorm_2Epolynom(A_27a),V9a1_2E0: tyop_2EringNorm_2Epolynom(A_27a)] : ( c_2EringNorm_2Epolynom__size_2E2(A_27a,V7f_2E0,c_2EringNorm_2EPmult_2E2(A_27a,V8a0_2E0,V9a1_2E0)) = c_2Earithmetic_2E_2B_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)),c_2Earithmetic_2E_2B_2E2(c_2EringNorm_2Epolynom__size_2E2(A_27a,V7f_2E0,V8a0_2E0),c_2EringNorm_2Epolynom__size_2E2(A_27a,V7f_2E0,V9a1_2E0))) )
      & ! [V10f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V11a_2E0: tyop_2EringNorm_2Epolynom(A_27a)] : ( c_2EringNorm_2Epolynom__size_2E2(A_27a,V10f_2E0,c_2EringNorm_2EPopp_2E1(A_27a,V11a_2E0)) = c_2Earithmetic_2E_2B_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)),c_2EringNorm_2Epolynom__size_2E2(A_27a,V10f_2E0,V11a_2E0)) ) ) ).

tff(thm_2EringNorm_2Epolynom__normalize__def,axiom,
    ! [A_27a: $tType] :
      ( ! [V0r_2E0: tyop_2Ering_2Ering(A_27a),V1i_2E0: tyop_2Equote_2Eindex] : ( c_2EringNorm_2Epolynom__normalize_2E2(A_27a,V0r_2E0,c_2EringNorm_2EPvar_2E1(A_27a,V1i_2E0)) = c_2Ecanonical_2ECons__varlist_2E2(A_27a,c_2Elist_2ECONS_2E2(tyop_2Equote_2Eindex,V1i_2E0,c_2Elist_2ENIL_2E0(tyop_2Equote_2Eindex)),c_2Ecanonical_2ENil__monom_2E0(A_27a)) )
      & ! [V2r_2E0: tyop_2Ering_2Ering(A_27a),V3c_2E0: A_27a] : ( c_2EringNorm_2Epolynom__normalize_2E2(A_27a,V2r_2E0,c_2EringNorm_2EPconst_2E1(A_27a,V3c_2E0)) = c_2Ecanonical_2ECons__monom_2E3(A_27a,V3c_2E0,c_2Elist_2ENIL_2E0(tyop_2Equote_2Eindex),c_2Ecanonical_2ENil__monom_2E0(A_27a)) )
      & ! [V4r_2E0: tyop_2Ering_2Ering(A_27a),V5pl_2E0: tyop_2EringNorm_2Epolynom(A_27a),V6pr_2E0: tyop_2EringNorm_2Epolynom(A_27a)] : ( c_2EringNorm_2Epolynom__normalize_2E2(A_27a,V4r_2E0,c_2EringNorm_2EPplus_2E2(A_27a,V5pl_2E0,V6pr_2E0)) = c_2EringNorm_2Er__canonical__sum__merge_2E3(A_27a,V4r_2E0,c_2EringNorm_2Epolynom__normalize_2E2(A_27a,V4r_2E0,V5pl_2E0),c_2EringNorm_2Epolynom__normalize_2E2(A_27a,V4r_2E0,V6pr_2E0)) )
      & ! [V7r_2E0: tyop_2Ering_2Ering(A_27a),V8pl_2E0: tyop_2EringNorm_2Epolynom(A_27a),V9pr_2E0: tyop_2EringNorm_2Epolynom(A_27a)] : ( c_2EringNorm_2Epolynom__normalize_2E2(A_27a,V7r_2E0,c_2EringNorm_2EPmult_2E2(A_27a,V8pl_2E0,V9pr_2E0)) = c_2EringNorm_2Er__canonical__sum__prod_2E3(A_27a,V7r_2E0,c_2EringNorm_2Epolynom__normalize_2E2(A_27a,V7r_2E0,V8pl_2E0),c_2EringNorm_2Epolynom__normalize_2E2(A_27a,V7r_2E0,V9pr_2E0)) )
      & ! [V10r_2E0: tyop_2Ering_2Ering(A_27a),V11p_2E0: tyop_2EringNorm_2Epolynom(A_27a)] : ( c_2EringNorm_2Epolynom__normalize_2E2(A_27a,V10r_2E0,c_2EringNorm_2EPopp_2E1(A_27a,V11p_2E0)) = c_2EringNorm_2Er__canonical__sum__scalar3_2E4(A_27a,V10r_2E0,c_2Ering_2Ering__RN_2E2(A_27a,V10r_2E0,c_2Ering_2Ering__R1_2E1(A_27a,V10r_2E0)),c_2Elist_2ENIL_2E0(tyop_2Equote_2Eindex),c_2EringNorm_2Epolynom__normalize_2E2(A_27a,V10r_2E0,V11p_2E0)) ) ) ).

tff(thm_2EringNorm_2Epolynom__simplify__def,axiom,
    ! [A_27a: $tType,V0r_2E0: tyop_2Ering_2Ering(A_27a),V1x_2E0: tyop_2EringNorm_2Epolynom(A_27a)] : ( c_2EringNorm_2Epolynom__simplify_2E2(A_27a,V0r_2E0,V1x_2E0) = c_2EringNorm_2Er__canonical__sum__simplify_2E2(A_27a,V0r_2E0,c_2EringNorm_2Epolynom__normalize_2E2(A_27a,V0r_2E0,V1x_2E0)) ) ).

tff(thm_2EringNorm_2Einterp__p__def,axiom,
    ! [A_27a: $tType] :
      ( ! [V0r_2E0: tyop_2Ering_2Ering(A_27a),V1vm_2E0: tyop_2Equote_2Evarmap(A_27a),V2c_2E0: A_27a] : ( c_2EringNorm_2Einterp__p_2E3(A_27a,V0r_2E0,V1vm_2E0,c_2EringNorm_2EPconst_2E1(A_27a,V2c_2E0)) = V2c_2E0 )
      & ! [V3r_2E0: tyop_2Ering_2Ering(A_27a),V4vm_2E0: tyop_2Equote_2Evarmap(A_27a),V5i_2E0: tyop_2Equote_2Eindex] : ( c_2EringNorm_2Einterp__p_2E3(A_27a,V3r_2E0,V4vm_2E0,c_2EringNorm_2EPvar_2E1(A_27a,V5i_2E0)) = c_2Equote_2Evarmap__find_2E2(A_27a,V5i_2E0,V4vm_2E0) )
      & ! [V6r_2E0: tyop_2Ering_2Ering(A_27a),V7vm_2E0: tyop_2Equote_2Evarmap(A_27a),V8p1_2E0: tyop_2EringNorm_2Epolynom(A_27a),V9p2_2E0: tyop_2EringNorm_2Epolynom(A_27a)] : ( c_2EringNorm_2Einterp__p_2E3(A_27a,V6r_2E0,V7vm_2E0,c_2EringNorm_2EPplus_2E2(A_27a,V8p1_2E0,V9p2_2E0)) = c_2Ering_2Ering__RP_2E3(A_27a,V6r_2E0,c_2EringNorm_2Einterp__p_2E3(A_27a,V6r_2E0,V7vm_2E0,V8p1_2E0),c_2EringNorm_2Einterp__p_2E3(A_27a,V6r_2E0,V7vm_2E0,V9p2_2E0)) )
      & ! [V10r_2E0: tyop_2Ering_2Ering(A_27a),V11vm_2E0: tyop_2Equote_2Evarmap(A_27a),V12p1_2E0: tyop_2EringNorm_2Epolynom(A_27a),V13p2_2E0: tyop_2EringNorm_2Epolynom(A_27a)] : ( c_2EringNorm_2Einterp__p_2E3(A_27a,V10r_2E0,V11vm_2E0,c_2EringNorm_2EPmult_2E2(A_27a,V12p1_2E0,V13p2_2E0)) = c_2Ering_2Ering__RM_2E3(A_27a,V10r_2E0,c_2EringNorm_2Einterp__p_2E3(A_27a,V10r_2E0,V11vm_2E0,V12p1_2E0),c_2EringNorm_2Einterp__p_2E3(A_27a,V10r_2E0,V11vm_2E0,V13p2_2E0)) )
      & ! [V14r_2E0: tyop_2Ering_2Ering(A_27a),V15vm_2E0: tyop_2Equote_2Evarmap(A_27a),V16p1_2E0: tyop_2EringNorm_2Epolynom(A_27a)] : ( c_2EringNorm_2Einterp__p_2E3(A_27a,V14r_2E0,V15vm_2E0,c_2EringNorm_2EPopp_2E1(A_27a,V16p1_2E0)) = c_2Ering_2Ering__RN_2E2(A_27a,V14r_2E0,c_2EringNorm_2Einterp__p_2E3(A_27a,V14r_2E0,V15vm_2E0,V16p1_2E0)) ) ) ).

tff(thm_2EringNorm_2Einterp__sp__def,axiom,
    ! [A_27a: $tType,V0r_2E0: tyop_2Ering_2Ering(A_27a)] :
      ( ! [V1vm_2E0: tyop_2Equote_2Evarmap(A_27a),V2c_2E0: A_27a] : ( c_2EringNorm_2Er__interp__sp_2E3(A_27a,V0r_2E0,V1vm_2E0,c_2Ecanonical_2ESPconst_2E1(A_27a,V2c_2E0)) = V2c_2E0 )
      & ! [V3vm_2E0: tyop_2Equote_2Evarmap(A_27a),V4i_2E0: tyop_2Equote_2Eindex] : ( c_2EringNorm_2Er__interp__sp_2E3(A_27a,V0r_2E0,V3vm_2E0,c_2Ecanonical_2ESPvar_2E1(A_27a,V4i_2E0)) = c_2Equote_2Evarmap__find_2E2(A_27a,V4i_2E0,V3vm_2E0) )
      & ! [V5vm_2E0: tyop_2Equote_2Evarmap(A_27a),V6p1_2E0: tyop_2Ecanonical_2Espolynom(A_27a),V7p2_2E0: tyop_2Ecanonical_2Espolynom(A_27a)] : ( c_2EringNorm_2Er__interp__sp_2E3(A_27a,V0r_2E0,V5vm_2E0,c_2Ecanonical_2ESPplus_2E2(A_27a,V6p1_2E0,V7p2_2E0)) = c_2Ering_2Ering__RP_2E3(A_27a,V0r_2E0,c_2EringNorm_2Er__interp__sp_2E3(A_27a,V0r_2E0,V5vm_2E0,V6p1_2E0),c_2EringNorm_2Er__interp__sp_2E3(A_27a,V0r_2E0,V5vm_2E0,V7p2_2E0)) )
      & ! [V8vm_2E0: tyop_2Equote_2Evarmap(A_27a),V9p1_2E0: tyop_2Ecanonical_2Espolynom(A_27a),V10p2_2E0: tyop_2Ecanonical_2Espolynom(A_27a)] : ( c_2EringNorm_2Er__interp__sp_2E3(A_27a,V0r_2E0,V8vm_2E0,c_2Ecanonical_2ESPmult_2E2(A_27a,V9p1_2E0,V10p2_2E0)) = c_2Ering_2Ering__RM_2E3(A_27a,V0r_2E0,c_2EringNorm_2Er__interp__sp_2E3(A_27a,V0r_2E0,V8vm_2E0,V9p1_2E0),c_2EringNorm_2Er__interp__sp_2E3(A_27a,V0r_2E0,V8vm_2E0,V10p2_2E0)) ) ) ).

tff(thm_2EringNorm_2Ecanonical__sum__merge__def,axiom,
    ! [A_27a: $tType,V0r_2E0: tyop_2Ering_2Ering(A_27a)] :
      ( ! [V1t2_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a),V2t1_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a),V3l2_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V4l1_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V5c2_2E0: A_27a,V6c1_2E0: A_27a] : ( c_2EringNorm_2Er__canonical__sum__merge_2E3(A_27a,V0r_2E0,c_2Ecanonical_2ECons__monom_2E3(A_27a,V6c1_2E0,V4l1_2E0,V2t1_2E0),c_2Ecanonical_2ECons__monom_2E3(A_27a,V5c2_2E0,V3l2_2E0,V1t2_2E0)) = c_2EternaryComparisons_2Eordering__CASE_2E4(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EternaryComparisons_2Elist__compare_2E3(tyop_2Equote_2Eindex,tyop_2Equote_2Eindex,c_2Equote_2Eindex__compare_2E0,V4l1_2E0,V3l2_2E0),c_2Ecanonical_2ECons__monom_2E3(A_27a,V6c1_2E0,V4l1_2E0,c_2EringNorm_2Er__canonical__sum__merge_2E3(A_27a,V0r_2E0,V2t1_2E0,c_2Ecanonical_2ECons__monom_2E3(A_27a,V5c2_2E0,V3l2_2E0,V1t2_2E0))),c_2Ecanonical_2ECons__monom_2E3(A_27a,c_2Ering_2Ering__RP_2E3(A_27a,V0r_2E0,V6c1_2E0,V5c2_2E0),V4l1_2E0,c_2EringNorm_2Er__canonical__sum__merge_2E3(A_27a,V0r_2E0,V2t1_2E0,V1t2_2E0)),c_2Ecanonical_2ECons__monom_2E3(A_27a,V5c2_2E0,V3l2_2E0,c_2EringNorm_2Er__canonical__sum__merge_2E3(A_27a,V0r_2E0,c_2Ecanonical_2ECons__monom_2E3(A_27a,V6c1_2E0,V4l1_2E0,V2t1_2E0),V1t2_2E0))) )
      & ! [V7t2_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a),V8t1_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a),V9l2_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V10l1_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V11c1_2E0: A_27a] : ( c_2EringNorm_2Er__canonical__sum__merge_2E3(A_27a,V0r_2E0,c_2Ecanonical_2ECons__monom_2E3(A_27a,V11c1_2E0,V10l1_2E0,V8t1_2E0),c_2Ecanonical_2ECons__varlist_2E2(A_27a,V9l2_2E0,V7t2_2E0)) = c_2EternaryComparisons_2Eordering__CASE_2E4(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EternaryComparisons_2Elist__compare_2E3(tyop_2Equote_2Eindex,tyop_2Equote_2Eindex,c_2Equote_2Eindex__compare_2E0,V10l1_2E0,V9l2_2E0),c_2Ecanonical_2ECons__monom_2E3(A_27a,V11c1_2E0,V10l1_2E0,c_2EringNorm_2Er__canonical__sum__merge_2E3(A_27a,V0r_2E0,V8t1_2E0,c_2Ecanonical_2ECons__varlist_2E2(A_27a,V9l2_2E0,V7t2_2E0))),c_2Ecanonical_2ECons__monom_2E3(A_27a,c_2Ering_2Ering__RP_2E3(A_27a,V0r_2E0,V11c1_2E0,c_2Ering_2Ering__R1_2E1(A_27a,V0r_2E0)),V10l1_2E0,c_2EringNorm_2Er__canonical__sum__merge_2E3(A_27a,V0r_2E0,V8t1_2E0,V7t2_2E0)),c_2Ecanonical_2ECons__varlist_2E2(A_27a,V9l2_2E0,c_2EringNorm_2Er__canonical__sum__merge_2E3(A_27a,V0r_2E0,c_2Ecanonical_2ECons__monom_2E3(A_27a,V11c1_2E0,V10l1_2E0,V8t1_2E0),V7t2_2E0))) )
      & ! [V12t2_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a),V13t1_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a),V14l2_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V15l1_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V16c2_2E0: A_27a] : ( c_2EringNorm_2Er__canonical__sum__merge_2E3(A_27a,V0r_2E0,c_2Ecanonical_2ECons__varlist_2E2(A_27a,V15l1_2E0,V13t1_2E0),c_2Ecanonical_2ECons__monom_2E3(A_27a,V16c2_2E0,V14l2_2E0,V12t2_2E0)) = c_2EternaryComparisons_2Eordering__CASE_2E4(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EternaryComparisons_2Elist__compare_2E3(tyop_2Equote_2Eindex,tyop_2Equote_2Eindex,c_2Equote_2Eindex__compare_2E0,V15l1_2E0,V14l2_2E0),c_2Ecanonical_2ECons__varlist_2E2(A_27a,V15l1_2E0,c_2EringNorm_2Er__canonical__sum__merge_2E3(A_27a,V0r_2E0,V13t1_2E0,c_2Ecanonical_2ECons__monom_2E3(A_27a,V16c2_2E0,V14l2_2E0,V12t2_2E0))),c_2Ecanonical_2ECons__monom_2E3(A_27a,c_2Ering_2Ering__RP_2E3(A_27a,V0r_2E0,c_2Ering_2Ering__R1_2E1(A_27a,V0r_2E0),V16c2_2E0),V15l1_2E0,c_2EringNorm_2Er__canonical__sum__merge_2E3(A_27a,V0r_2E0,V13t1_2E0,V12t2_2E0)),c_2Ecanonical_2ECons__monom_2E3(A_27a,V16c2_2E0,V14l2_2E0,c_2EringNorm_2Er__canonical__sum__merge_2E3(A_27a,V0r_2E0,c_2Ecanonical_2ECons__varlist_2E2(A_27a,V15l1_2E0,V13t1_2E0),V12t2_2E0))) )
      & ! [V17t2_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a),V18t1_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a),V19l2_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V20l1_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex)] : ( c_2EringNorm_2Er__canonical__sum__merge_2E3(A_27a,V0r_2E0,c_2Ecanonical_2ECons__varlist_2E2(A_27a,V20l1_2E0,V18t1_2E0),c_2Ecanonical_2ECons__varlist_2E2(A_27a,V19l2_2E0,V17t2_2E0)) = c_2EternaryComparisons_2Eordering__CASE_2E4(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EternaryComparisons_2Elist__compare_2E3(tyop_2Equote_2Eindex,tyop_2Equote_2Eindex,c_2Equote_2Eindex__compare_2E0,V20l1_2E0,V19l2_2E0),c_2Ecanonical_2ECons__varlist_2E2(A_27a,V20l1_2E0,c_2EringNorm_2Er__canonical__sum__merge_2E3(A_27a,V0r_2E0,V18t1_2E0,c_2Ecanonical_2ECons__varlist_2E2(A_27a,V19l2_2E0,V17t2_2E0))),c_2Ecanonical_2ECons__monom_2E3(A_27a,c_2Ering_2Ering__RP_2E3(A_27a,V0r_2E0,c_2Ering_2Ering__R1_2E1(A_27a,V0r_2E0),c_2Ering_2Ering__R1_2E1(A_27a,V0r_2E0)),V20l1_2E0,c_2EringNorm_2Er__canonical__sum__merge_2E3(A_27a,V0r_2E0,V18t1_2E0,V17t2_2E0)),c_2Ecanonical_2ECons__varlist_2E2(A_27a,V19l2_2E0,c_2EringNorm_2Er__canonical__sum__merge_2E3(A_27a,V0r_2E0,c_2Ecanonical_2ECons__varlist_2E2(A_27a,V20l1_2E0,V18t1_2E0),V17t2_2E0))) )
      & ! [V21s1_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : ( c_2EringNorm_2Er__canonical__sum__merge_2E3(A_27a,V0r_2E0,V21s1_2E0,c_2Ecanonical_2ENil__monom_2E0(A_27a)) = V21s1_2E0 )
      & ! [V22v6_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a),V23v5_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V24v4_2E0: A_27a] : ( c_2EringNorm_2Er__canonical__sum__merge_2E3(A_27a,V0r_2E0,c_2Ecanonical_2ENil__monom_2E0(A_27a),c_2Ecanonical_2ECons__monom_2E3(A_27a,V24v4_2E0,V23v5_2E0,V22v6_2E0)) = c_2Ecanonical_2ECons__monom_2E3(A_27a,V24v4_2E0,V23v5_2E0,V22v6_2E0) )
      & ! [V25v8_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a),V26v7_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex)] : ( c_2EringNorm_2Er__canonical__sum__merge_2E3(A_27a,V0r_2E0,c_2Ecanonical_2ENil__monom_2E0(A_27a),c_2Ecanonical_2ECons__varlist_2E2(A_27a,V26v7_2E0,V25v8_2E0)) = c_2Ecanonical_2ECons__varlist_2E2(A_27a,V26v7_2E0,V25v8_2E0) ) ) ).

tff(thm_2EringNorm_2Emonom__insert__def,axiom,
    ! [A_27a: $tType,V0r_2E0: tyop_2Ering_2Ering(A_27a)] :
      ( ! [V1t2_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a),V2l2_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V3l1_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V4c2_2E0: A_27a,V5c1_2E0: A_27a] : ( c_2EringNorm_2Er__monom__insert_2E4(A_27a,V0r_2E0,V5c1_2E0,V3l1_2E0,c_2Ecanonical_2ECons__monom_2E3(A_27a,V4c2_2E0,V2l2_2E0,V1t2_2E0)) = c_2EternaryComparisons_2Eordering__CASE_2E4(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EternaryComparisons_2Elist__compare_2E3(tyop_2Equote_2Eindex,tyop_2Equote_2Eindex,c_2Equote_2Eindex__compare_2E0,V3l1_2E0,V2l2_2E0),c_2Ecanonical_2ECons__monom_2E3(A_27a,V5c1_2E0,V3l1_2E0,c_2Ecanonical_2ECons__monom_2E3(A_27a,V4c2_2E0,V2l2_2E0,V1t2_2E0)),c_2Ecanonical_2ECons__monom_2E3(A_27a,c_2Ering_2Ering__RP_2E3(A_27a,V0r_2E0,V5c1_2E0,V4c2_2E0),V3l1_2E0,V1t2_2E0),c_2Ecanonical_2ECons__monom_2E3(A_27a,V4c2_2E0,V2l2_2E0,c_2EringNorm_2Er__monom__insert_2E4(A_27a,V0r_2E0,V5c1_2E0,V3l1_2E0,V1t2_2E0))) )
      & ! [V6t2_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a),V7l2_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V8l1_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V9c1_2E0: A_27a] : ( c_2EringNorm_2Er__monom__insert_2E4(A_27a,V0r_2E0,V9c1_2E0,V8l1_2E0,c_2Ecanonical_2ECons__varlist_2E2(A_27a,V7l2_2E0,V6t2_2E0)) = c_2EternaryComparisons_2Eordering__CASE_2E4(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EternaryComparisons_2Elist__compare_2E3(tyop_2Equote_2Eindex,tyop_2Equote_2Eindex,c_2Equote_2Eindex__compare_2E0,V8l1_2E0,V7l2_2E0),c_2Ecanonical_2ECons__monom_2E3(A_27a,V9c1_2E0,V8l1_2E0,c_2Ecanonical_2ECons__varlist_2E2(A_27a,V7l2_2E0,V6t2_2E0)),c_2Ecanonical_2ECons__monom_2E3(A_27a,c_2Ering_2Ering__RP_2E3(A_27a,V0r_2E0,V9c1_2E0,c_2Ering_2Ering__R1_2E1(A_27a,V0r_2E0)),V8l1_2E0,V6t2_2E0),c_2Ecanonical_2ECons__varlist_2E2(A_27a,V7l2_2E0,c_2EringNorm_2Er__monom__insert_2E4(A_27a,V0r_2E0,V9c1_2E0,V8l1_2E0,V6t2_2E0))) )
      & ! [V10l1_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V11c1_2E0: A_27a] : ( c_2EringNorm_2Er__monom__insert_2E4(A_27a,V0r_2E0,V11c1_2E0,V10l1_2E0,c_2Ecanonical_2ENil__monom_2E0(A_27a)) = c_2Ecanonical_2ECons__monom_2E3(A_27a,V11c1_2E0,V10l1_2E0,c_2Ecanonical_2ENil__monom_2E0(A_27a)) ) ) ).

tff(thm_2EringNorm_2Evarlist__insert__def,axiom,
    ! [A_27a: $tType,V0r_2E0: tyop_2Ering_2Ering(A_27a)] :
      ( ! [V1t2_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a),V2l2_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V3l1_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V4c2_2E0: A_27a] : ( c_2EringNorm_2Er__varlist__insert_2E3(A_27a,V0r_2E0,V3l1_2E0,c_2Ecanonical_2ECons__monom_2E3(A_27a,V4c2_2E0,V2l2_2E0,V1t2_2E0)) = c_2EternaryComparisons_2Eordering__CASE_2E4(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EternaryComparisons_2Elist__compare_2E3(tyop_2Equote_2Eindex,tyop_2Equote_2Eindex,c_2Equote_2Eindex__compare_2E0,V3l1_2E0,V2l2_2E0),c_2Ecanonical_2ECons__varlist_2E2(A_27a,V3l1_2E0,c_2Ecanonical_2ECons__monom_2E3(A_27a,V4c2_2E0,V2l2_2E0,V1t2_2E0)),c_2Ecanonical_2ECons__monom_2E3(A_27a,c_2Ering_2Ering__RP_2E3(A_27a,V0r_2E0,c_2Ering_2Ering__R1_2E1(A_27a,V0r_2E0),V4c2_2E0),V3l1_2E0,V1t2_2E0),c_2Ecanonical_2ECons__monom_2E3(A_27a,V4c2_2E0,V2l2_2E0,c_2EringNorm_2Er__varlist__insert_2E3(A_27a,V0r_2E0,V3l1_2E0,V1t2_2E0))) )
      & ! [V5t2_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a),V6l2_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V7l1_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex)] : ( c_2EringNorm_2Er__varlist__insert_2E3(A_27a,V0r_2E0,V7l1_2E0,c_2Ecanonical_2ECons__varlist_2E2(A_27a,V6l2_2E0,V5t2_2E0)) = c_2EternaryComparisons_2Eordering__CASE_2E4(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EternaryComparisons_2Elist__compare_2E3(tyop_2Equote_2Eindex,tyop_2Equote_2Eindex,c_2Equote_2Eindex__compare_2E0,V7l1_2E0,V6l2_2E0),c_2Ecanonical_2ECons__varlist_2E2(A_27a,V7l1_2E0,c_2Ecanonical_2ECons__varlist_2E2(A_27a,V6l2_2E0,V5t2_2E0)),c_2Ecanonical_2ECons__monom_2E3(A_27a,c_2Ering_2Ering__RP_2E3(A_27a,V0r_2E0,c_2Ering_2Ering__R1_2E1(A_27a,V0r_2E0),c_2Ering_2Ering__R1_2E1(A_27a,V0r_2E0)),V7l1_2E0,V5t2_2E0),c_2Ecanonical_2ECons__varlist_2E2(A_27a,V6l2_2E0,c_2EringNorm_2Er__varlist__insert_2E3(A_27a,V0r_2E0,V7l1_2E0,V5t2_2E0))) )
      & ! [V8l1_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex)] : ( c_2EringNorm_2Er__varlist__insert_2E3(A_27a,V0r_2E0,V8l1_2E0,c_2Ecanonical_2ENil__monom_2E0(A_27a)) = c_2Ecanonical_2ECons__varlist_2E2(A_27a,V8l1_2E0,c_2Ecanonical_2ENil__monom_2E0(A_27a)) ) ) ).

tff(thm_2EringNorm_2Ecanonical__sum__scalar__def,axiom,
    ! [A_27a: $tType,V0r_2E0: tyop_2Ering_2Ering(A_27a)] :
      ( ! [V1c0_2E0: A_27a,V2c_2E0: A_27a,V3l_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V4t_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : ( c_2EringNorm_2Er__canonical__sum__scalar_2E3(A_27a,V0r_2E0,V1c0_2E0,c_2Ecanonical_2ECons__monom_2E3(A_27a,V2c_2E0,V3l_2E0,V4t_2E0)) = c_2Ecanonical_2ECons__monom_2E3(A_27a,c_2Ering_2Ering__RM_2E3(A_27a,V0r_2E0,V1c0_2E0,V2c_2E0),V3l_2E0,c_2EringNorm_2Er__canonical__sum__scalar_2E3(A_27a,V0r_2E0,V1c0_2E0,V4t_2E0)) )
      & ! [V5c0_2E0: A_27a,V6l_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V7t_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : ( c_2EringNorm_2Er__canonical__sum__scalar_2E3(A_27a,V0r_2E0,V5c0_2E0,c_2Ecanonical_2ECons__varlist_2E2(A_27a,V6l_2E0,V7t_2E0)) = c_2Ecanonical_2ECons__monom_2E3(A_27a,V5c0_2E0,V6l_2E0,c_2EringNorm_2Er__canonical__sum__scalar_2E3(A_27a,V0r_2E0,V5c0_2E0,V7t_2E0)) )
      & ! [V8c0_2E0: A_27a] : ( c_2EringNorm_2Er__canonical__sum__scalar_2E3(A_27a,V0r_2E0,V8c0_2E0,c_2Ecanonical_2ENil__monom_2E0(A_27a)) = c_2Ecanonical_2ENil__monom_2E0(A_27a) ) ) ).

tff(thm_2EringNorm_2Ecanonical__sum__scalar2__def,axiom,
    ! [A_27a: $tType,V0r_2E0: tyop_2Ering_2Ering(A_27a)] :
      ( ! [V1l0_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V2c_2E0: A_27a,V3l_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V4t_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : ( c_2EringNorm_2Er__canonical__sum__scalar2_2E3(A_27a,V0r_2E0,V1l0_2E0,c_2Ecanonical_2ECons__monom_2E3(A_27a,V2c_2E0,V3l_2E0,V4t_2E0)) = c_2EringNorm_2Er__monom__insert_2E4(A_27a,V0r_2E0,V2c_2E0,c_2EternaryComparisons_2Elist__merge_2E3(tyop_2Equote_2Eindex,c_2Equote_2Eindex__lt_2E0,V1l0_2E0,V3l_2E0),c_2EringNorm_2Er__canonical__sum__scalar2_2E3(A_27a,V0r_2E0,V1l0_2E0,V4t_2E0)) )
      & ! [V5l0_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V6l_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V7t_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : ( c_2EringNorm_2Er__canonical__sum__scalar2_2E3(A_27a,V0r_2E0,V5l0_2E0,c_2Ecanonical_2ECons__varlist_2E2(A_27a,V6l_2E0,V7t_2E0)) = c_2EringNorm_2Er__varlist__insert_2E3(A_27a,V0r_2E0,c_2EternaryComparisons_2Elist__merge_2E3(tyop_2Equote_2Eindex,c_2Equote_2Eindex__lt_2E0,V5l0_2E0,V6l_2E0),c_2EringNorm_2Er__canonical__sum__scalar2_2E3(A_27a,V0r_2E0,V5l0_2E0,V7t_2E0)) )
      & ! [V8l0_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex)] : ( c_2EringNorm_2Er__canonical__sum__scalar2_2E3(A_27a,V0r_2E0,V8l0_2E0,c_2Ecanonical_2ENil__monom_2E0(A_27a)) = c_2Ecanonical_2ENil__monom_2E0(A_27a) ) ) ).

tff(thm_2EringNorm_2Ecanonical__sum__scalar3__def,axiom,
    ! [A_27a: $tType,V0r_2E0: tyop_2Ering_2Ering(A_27a)] :
      ( ! [V1c0_2E0: A_27a,V2l0_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V3c_2E0: A_27a,V4l_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V5t_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : ( c_2EringNorm_2Er__canonical__sum__scalar3_2E4(A_27a,V0r_2E0,V1c0_2E0,V2l0_2E0,c_2Ecanonical_2ECons__monom_2E3(A_27a,V3c_2E0,V4l_2E0,V5t_2E0)) = c_2EringNorm_2Er__monom__insert_2E4(A_27a,V0r_2E0,c_2Ering_2Ering__RM_2E3(A_27a,V0r_2E0,V1c0_2E0,V3c_2E0),c_2EternaryComparisons_2Elist__merge_2E3(tyop_2Equote_2Eindex,c_2Equote_2Eindex__lt_2E0,V2l0_2E0,V4l_2E0),c_2EringNorm_2Er__canonical__sum__scalar3_2E4(A_27a,V0r_2E0,V1c0_2E0,V2l0_2E0,V5t_2E0)) )
      & ! [V6c0_2E0: A_27a,V7l0_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V8l_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V9t_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : ( c_2EringNorm_2Er__canonical__sum__scalar3_2E4(A_27a,V0r_2E0,V6c0_2E0,V7l0_2E0,c_2Ecanonical_2ECons__varlist_2E2(A_27a,V8l_2E0,V9t_2E0)) = c_2EringNorm_2Er__monom__insert_2E4(A_27a,V0r_2E0,V6c0_2E0,c_2EternaryComparisons_2Elist__merge_2E3(tyop_2Equote_2Eindex,c_2Equote_2Eindex__lt_2E0,V7l0_2E0,V8l_2E0),c_2EringNorm_2Er__canonical__sum__scalar3_2E4(A_27a,V0r_2E0,V6c0_2E0,V7l0_2E0,V9t_2E0)) )
      & ! [V10c0_2E0: A_27a,V11l0_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex)] : ( c_2EringNorm_2Er__canonical__sum__scalar3_2E4(A_27a,V0r_2E0,V10c0_2E0,V11l0_2E0,c_2Ecanonical_2ENil__monom_2E0(A_27a)) = c_2Ecanonical_2ENil__monom_2E0(A_27a) ) ) ).

tff(thm_2EringNorm_2Ecanonical__sum__prod__def,axiom,
    ! [A_27a: $tType,V0r_2E0: tyop_2Ering_2Ering(A_27a)] :
      ( ! [V1c1_2E0: A_27a,V2l1_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V3t1_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a),V4s2_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : ( c_2EringNorm_2Er__canonical__sum__prod_2E3(A_27a,V0r_2E0,c_2Ecanonical_2ECons__monom_2E3(A_27a,V1c1_2E0,V2l1_2E0,V3t1_2E0),V4s2_2E0) = c_2EringNorm_2Er__canonical__sum__merge_2E3(A_27a,V0r_2E0,c_2EringNorm_2Er__canonical__sum__scalar3_2E4(A_27a,V0r_2E0,V1c1_2E0,V2l1_2E0,V4s2_2E0),c_2EringNorm_2Er__canonical__sum__prod_2E3(A_27a,V0r_2E0,V3t1_2E0,V4s2_2E0)) )
      & ! [V5l1_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V6t1_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a),V7s2_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : ( c_2EringNorm_2Er__canonical__sum__prod_2E3(A_27a,V0r_2E0,c_2Ecanonical_2ECons__varlist_2E2(A_27a,V5l1_2E0,V6t1_2E0),V7s2_2E0) = c_2EringNorm_2Er__canonical__sum__merge_2E3(A_27a,V0r_2E0,c_2EringNorm_2Er__canonical__sum__scalar2_2E3(A_27a,V0r_2E0,V5l1_2E0,V7s2_2E0),c_2EringNorm_2Er__canonical__sum__prod_2E3(A_27a,V0r_2E0,V6t1_2E0,V7s2_2E0)) )
      & ! [V8s2_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : ( c_2EringNorm_2Er__canonical__sum__prod_2E3(A_27a,V0r_2E0,c_2Ecanonical_2ENil__monom_2E0(A_27a),V8s2_2E0) = c_2Ecanonical_2ENil__monom_2E0(A_27a) ) ) ).

tff(thm_2EringNorm_2Ecanonical__sum__simplify__def,axiom,
    ! [A_27a: $tType,F1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Ering_2Ering(A_27a),tyop_2Emin_2Ebool)),F0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Ering_2Ering(A_27a),tyop_2Emin_2Ebool))] :
      ( ! [V1c_2E0: A_27a,V0r_2E0: tyop_2Ering_2Ering(A_27a)] :
          ( p(app_2E2(tyop_2Ering_2Ering(A_27a),tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Ering_2Ering(A_27a),tyop_2Emin_2Ebool),F1_2E0,V1c_2E0),V0r_2E0))
        <=> ( V1c_2E0 = c_2Ering_2Ering__R1_2E1(A_27a,V0r_2E0) ) )
     => ( ! [V1c_2E0: A_27a,V0r_2E0: tyop_2Ering_2Ering(A_27a)] :
            ( p(app_2E2(tyop_2Ering_2Ering(A_27a),tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Ering_2Ering(A_27a),tyop_2Emin_2Ebool),F0_2E0,V1c_2E0),V0r_2E0))
          <=> ( V1c_2E0 = c_2Ering_2Ering__R0_2E1(A_27a,V0r_2E0) ) )
       => ! [V0r_2E0: tyop_2Ering_2Ering(A_27a)] :
            ( ! [V1c_2E0: A_27a,V2l_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V3t_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : ( c_2EringNorm_2Er__canonical__sum__simplify_2E2(A_27a,V0r_2E0,c_2Ecanonical_2ECons__monom_2E3(A_27a,V1c_2E0,V2l_2E0,V3t_2E0)) = c_2Ebool_2ECOND_2E3(tyop_2Ecanonical_2Ecanonical__sum(A_27a),app_2E2(tyop_2Ering_2Ering(A_27a),tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Ering_2Ering(A_27a),tyop_2Emin_2Ebool),F0_2E0,V1c_2E0),V0r_2E0),c_2EringNorm_2Er__canonical__sum__simplify_2E2(A_27a,V0r_2E0,V3t_2E0),c_2Ebool_2ECOND_2E3(tyop_2Ecanonical_2Ecanonical__sum(A_27a),app_2E2(tyop_2Ering_2Ering(A_27a),tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Ering_2Ering(A_27a),tyop_2Emin_2Ebool),F1_2E0,V1c_2E0),V0r_2E0),c_2Ecanonical_2ECons__varlist_2E2(A_27a,V2l_2E0,c_2EringNorm_2Er__canonical__sum__simplify_2E2(A_27a,V0r_2E0,V3t_2E0)),c_2Ecanonical_2ECons__monom_2E3(A_27a,V1c_2E0,V2l_2E0,c_2EringNorm_2Er__canonical__sum__simplify_2E2(A_27a,V0r_2E0,V3t_2E0)))) )
            & ! [V4l_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V5t_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : ( c_2EringNorm_2Er__canonical__sum__simplify_2E2(A_27a,V0r_2E0,c_2Ecanonical_2ECons__varlist_2E2(A_27a,V4l_2E0,V5t_2E0)) = c_2Ecanonical_2ECons__varlist_2E2(A_27a,V4l_2E0,c_2EringNorm_2Er__canonical__sum__simplify_2E2(A_27a,V0r_2E0,V5t_2E0)) )
            & ( c_2EringNorm_2Er__canonical__sum__simplify_2E2(A_27a,V0r_2E0,c_2Ecanonical_2ENil__monom_2E0(A_27a)) = c_2Ecanonical_2ENil__monom_2E0(A_27a) ) ) ) ) ).

tff(thm_2EringNorm_2Eivl__aux__def,axiom,
    ! [A_27a: $tType,V0r_2E0: tyop_2Ering_2Ering(A_27a)] :
      ( ! [V1vm_2E0: tyop_2Equote_2Evarmap(A_27a),V2x_2E0: tyop_2Equote_2Eindex] : ( c_2EringNorm_2Er__ivl__aux_2E4(A_27a,V0r_2E0,V1vm_2E0,V2x_2E0,c_2Elist_2ENIL_2E0(tyop_2Equote_2Eindex)) = c_2Equote_2Evarmap__find_2E2(A_27a,V2x_2E0,V1vm_2E0) )
      & ! [V3vm_2E0: tyop_2Equote_2Evarmap(A_27a),V4x_2E0: tyop_2Equote_2Eindex,V5x_27_2E0: tyop_2Equote_2Eindex,V6t_27_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex)] : ( c_2EringNorm_2Er__ivl__aux_2E4(A_27a,V0r_2E0,V3vm_2E0,V4x_2E0,c_2Elist_2ECONS_2E2(tyop_2Equote_2Eindex,V5x_27_2E0,V6t_27_2E0)) = c_2Ering_2Ering__RM_2E3(A_27a,V0r_2E0,c_2Equote_2Evarmap__find_2E2(A_27a,V4x_2E0,V3vm_2E0),c_2EringNorm_2Er__ivl__aux_2E4(A_27a,V0r_2E0,V3vm_2E0,V5x_27_2E0,V6t_27_2E0)) ) ) ).

tff(thm_2EringNorm_2Einterp__vl__def,axiom,
    ! [A_27a: $tType,V0r_2E0: tyop_2Ering_2Ering(A_27a)] :
      ( ! [V1vm_2E0: tyop_2Equote_2Evarmap(A_27a)] : ( c_2EringNorm_2Er__interp__vl_2E3(A_27a,V0r_2E0,V1vm_2E0,c_2Elist_2ENIL_2E0(tyop_2Equote_2Eindex)) = c_2Ering_2Ering__R1_2E1(A_27a,V0r_2E0) )
      & ! [V2vm_2E0: tyop_2Equote_2Evarmap(A_27a),V3x_2E0: tyop_2Equote_2Eindex,V4t_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex)] : ( c_2EringNorm_2Er__interp__vl_2E3(A_27a,V0r_2E0,V2vm_2E0,c_2Elist_2ECONS_2E2(tyop_2Equote_2Eindex,V3x_2E0,V4t_2E0)) = c_2EringNorm_2Er__ivl__aux_2E4(A_27a,V0r_2E0,V2vm_2E0,V3x_2E0,V4t_2E0) ) ) ).

tff(thm_2EringNorm_2Einterp__m__def,axiom,
    ! [A_27a: $tType,V0r_2E0: tyop_2Ering_2Ering(A_27a)] :
      ( ! [V1vm_2E0: tyop_2Equote_2Evarmap(A_27a),V2c_2E0: A_27a] : ( c_2EringNorm_2Er__interp__m_2E4(A_27a,V0r_2E0,V1vm_2E0,V2c_2E0,c_2Elist_2ENIL_2E0(tyop_2Equote_2Eindex)) = V2c_2E0 )
      & ! [V3vm_2E0: tyop_2Equote_2Evarmap(A_27a),V4c_2E0: A_27a,V5x_2E0: tyop_2Equote_2Eindex,V6t_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex)] : ( c_2EringNorm_2Er__interp__m_2E4(A_27a,V0r_2E0,V3vm_2E0,V4c_2E0,c_2Elist_2ECONS_2E2(tyop_2Equote_2Eindex,V5x_2E0,V6t_2E0)) = c_2Ering_2Ering__RM_2E3(A_27a,V0r_2E0,V4c_2E0,c_2EringNorm_2Er__ivl__aux_2E4(A_27a,V0r_2E0,V3vm_2E0,V5x_2E0,V6t_2E0)) ) ) ).

tff(thm_2EringNorm_2Eics__aux__def,axiom,
    ! [A_27a: $tType,V0r_2E0: tyop_2Ering_2Ering(A_27a)] :
      ( ! [V1vm_2E0: tyop_2Equote_2Evarmap(A_27a),V2a_2E0: A_27a] : ( c_2EringNorm_2Er__ics__aux_2E4(A_27a,V0r_2E0,V1vm_2E0,V2a_2E0,c_2Ecanonical_2ENil__monom_2E0(A_27a)) = V2a_2E0 )
      & ! [V3vm_2E0: tyop_2Equote_2Evarmap(A_27a),V4a_2E0: A_27a,V5l_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V6t_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : ( c_2EringNorm_2Er__ics__aux_2E4(A_27a,V0r_2E0,V3vm_2E0,V4a_2E0,c_2Ecanonical_2ECons__varlist_2E2(A_27a,V5l_2E0,V6t_2E0)) = c_2Ering_2Ering__RP_2E3(A_27a,V0r_2E0,V4a_2E0,c_2EringNorm_2Er__ics__aux_2E4(A_27a,V0r_2E0,V3vm_2E0,c_2EringNorm_2Er__interp__vl_2E3(A_27a,V0r_2E0,V3vm_2E0,V5l_2E0),V6t_2E0)) )
      & ! [V7vm_2E0: tyop_2Equote_2Evarmap(A_27a),V8a_2E0: A_27a,V9c_2E0: A_27a,V10l_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V11t_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : ( c_2EringNorm_2Er__ics__aux_2E4(A_27a,V0r_2E0,V7vm_2E0,V8a_2E0,c_2Ecanonical_2ECons__monom_2E3(A_27a,V9c_2E0,V10l_2E0,V11t_2E0)) = c_2Ering_2Ering__RP_2E3(A_27a,V0r_2E0,V8a_2E0,c_2EringNorm_2Er__ics__aux_2E4(A_27a,V0r_2E0,V7vm_2E0,c_2EringNorm_2Er__interp__m_2E4(A_27a,V0r_2E0,V7vm_2E0,V9c_2E0,V10l_2E0),V11t_2E0)) ) ) ).

tff(thm_2EringNorm_2Einterp__cs__def,axiom,
    ! [A_27a: $tType,V0r_2E0: tyop_2Ering_2Ering(A_27a)] :
      ( ! [V1vm_2E0: tyop_2Equote_2Evarmap(A_27a)] : ( c_2EringNorm_2Er__interp__cs_2E3(A_27a,V0r_2E0,V1vm_2E0,c_2Ecanonical_2ENil__monom_2E0(A_27a)) = c_2Ering_2Ering__R0_2E1(A_27a,V0r_2E0) )
      & ! [V2vm_2E0: tyop_2Equote_2Evarmap(A_27a),V3l_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V4t_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : ( c_2EringNorm_2Er__interp__cs_2E3(A_27a,V0r_2E0,V2vm_2E0,c_2Ecanonical_2ECons__varlist_2E2(A_27a,V3l_2E0,V4t_2E0)) = c_2EringNorm_2Er__ics__aux_2E4(A_27a,V0r_2E0,V2vm_2E0,c_2EringNorm_2Er__interp__vl_2E3(A_27a,V0r_2E0,V2vm_2E0,V3l_2E0),V4t_2E0) )
      & ! [V5vm_2E0: tyop_2Equote_2Evarmap(A_27a),V6c_2E0: A_27a,V7l_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V8t_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : ( c_2EringNorm_2Er__interp__cs_2E3(A_27a,V0r_2E0,V5vm_2E0,c_2Ecanonical_2ECons__monom_2E3(A_27a,V6c_2E0,V7l_2E0,V8t_2E0)) = c_2EringNorm_2Er__ics__aux_2E4(A_27a,V0r_2E0,V5vm_2E0,c_2EringNorm_2Er__interp__m_2E4(A_27a,V0r_2E0,V5vm_2E0,V6c_2E0,V7l_2E0),V8t_2E0) ) ) ).

tff(thm_2EringNorm_2Espolynom__normalize__def,axiom,
    ! [A_27a: $tType,V0r_2E0: tyop_2Ering_2Ering(A_27a)] :
      ( ! [V1i_2E0: tyop_2Equote_2Eindex] : ( c_2EringNorm_2Er__spolynom__normalize_2E2(A_27a,V0r_2E0,c_2Ecanonical_2ESPvar_2E1(A_27a,V1i_2E0)) = c_2Ecanonical_2ECons__varlist_2E2(A_27a,c_2Elist_2ECONS_2E2(tyop_2Equote_2Eindex,V1i_2E0,c_2Elist_2ENIL_2E0(tyop_2Equote_2Eindex)),c_2Ecanonical_2ENil__monom_2E0(A_27a)) )
      & ! [V2c_2E0: A_27a] : ( c_2EringNorm_2Er__spolynom__normalize_2E2(A_27a,V0r_2E0,c_2Ecanonical_2ESPconst_2E1(A_27a,V2c_2E0)) = c_2Ecanonical_2ECons__monom_2E3(A_27a,V2c_2E0,c_2Elist_2ENIL_2E0(tyop_2Equote_2Eindex),c_2Ecanonical_2ENil__monom_2E0(A_27a)) )
      & ! [V3l_2E0: tyop_2Ecanonical_2Espolynom(A_27a),V4r_27_2E0: tyop_2Ecanonical_2Espolynom(A_27a)] : ( c_2EringNorm_2Er__spolynom__normalize_2E2(A_27a,V0r_2E0,c_2Ecanonical_2ESPplus_2E2(A_27a,V3l_2E0,V4r_27_2E0)) = c_2EringNorm_2Er__canonical__sum__merge_2E3(A_27a,V0r_2E0,c_2EringNorm_2Er__spolynom__normalize_2E2(A_27a,V0r_2E0,V3l_2E0),c_2EringNorm_2Er__spolynom__normalize_2E2(A_27a,V0r_2E0,V4r_27_2E0)) )
      & ! [V5l_2E0: tyop_2Ecanonical_2Espolynom(A_27a),V6r_27_2E0: tyop_2Ecanonical_2Espolynom(A_27a)] : ( c_2EringNorm_2Er__spolynom__normalize_2E2(A_27a,V0r_2E0,c_2Ecanonical_2ESPmult_2E2(A_27a,V5l_2E0,V6r_27_2E0)) = c_2EringNorm_2Er__canonical__sum__prod_2E3(A_27a,V0r_2E0,c_2EringNorm_2Er__spolynom__normalize_2E2(A_27a,V0r_2E0,V5l_2E0),c_2EringNorm_2Er__spolynom__normalize_2E2(A_27a,V0r_2E0,V6r_27_2E0)) ) ) ).

tff(thm_2EringNorm_2Espolynom__simplify__def,axiom,
    ! [A_27a: $tType,V0r_2E0: tyop_2Ering_2Ering(A_27a),V1x_2E0: tyop_2Ecanonical_2Espolynom(A_27a)] : ( c_2EringNorm_2Er__spolynom__simplify_2E2(A_27a,V0r_2E0,V1x_2E0) = c_2EringNorm_2Er__canonical__sum__simplify_2E2(A_27a,V0r_2E0,c_2EringNorm_2Er__spolynom__normalize_2E2(A_27a,V0r_2E0,V1x_2E0)) ) ).

tff(thm_2EringNorm_2Edatatype__polynom,axiom,
    ! [A_27a: $tType,V0polynom_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Equote_2Eindex,tyop_2EringNorm_2Epolynom(A_27a)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2EringNorm_2Epolynom(A_27a)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2EringNorm_2Epolynom(A_27a))),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2EringNorm_2Epolynom(A_27a))),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2EringNorm_2Epolynom(A_27a)),tyop_2Emin_2Ebool)))))] : p(c_2Ebool_2EDATATYPE_2E1(tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2EringNorm_2Epolynom(A_27a)),tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2EringNorm_2Epolynom(A_27a))),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2EringNorm_2Epolynom(A_27a)),tyop_2Emin_2Ebool),app_2E2(tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2EringNorm_2Epolynom(A_27a))),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2EringNorm_2Epolynom(A_27a))),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2EringNorm_2Epolynom(A_27a)),tyop_2Emin_2Ebool)),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2EringNorm_2Epolynom(A_27a)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2EringNorm_2Epolynom(A_27a))),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2EringNorm_2Epolynom(A_27a))),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2EringNorm_2Epolynom(A_27a)),tyop_2Emin_2Ebool))),app_2E2(tyop_2Emin_2Efun(tyop_2Equote_2Eindex,tyop_2EringNorm_2Epolynom(A_27a)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2EringNorm_2Epolynom(A_27a)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2EringNorm_2Epolynom(A_27a))),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2EringNorm_2Epolynom(A_27a))),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2EringNorm_2Epolynom(A_27a)),tyop_2Emin_2Ebool)))),V0polynom_2E0,c_2EringNorm_2EPvar_2E0(A_27a)),c_2EringNorm_2EPconst_2E0(A_27a)),c_2EringNorm_2EPplus_2E0(A_27a)),c_2EringNorm_2EPmult_2E0(A_27a)),c_2EringNorm_2EPopp_2E0(A_27a)))) ).

tff(thm_2EringNorm_2Epolynom__11,axiom,
    ! [A_27a: $tType] :
      ( ! [V0a_2E0: tyop_2Equote_2Eindex,V1a_27_2E0: tyop_2Equote_2Eindex] :
          ( ( c_2EringNorm_2EPvar_2E1(A_27a,V0a_2E0) = c_2EringNorm_2EPvar_2E1(A_27a,V1a_27_2E0) )
        <=> ( V0a_2E0 = V1a_27_2E0 ) )
      & ! [V2a_2E0: A_27a,V3a_27_2E0: A_27a] :
          ( ( c_2EringNorm_2EPconst_2E1(A_27a,V2a_2E0) = c_2EringNorm_2EPconst_2E1(A_27a,V3a_27_2E0) )
        <=> ( V2a_2E0 = V3a_27_2E0 ) )
      & ! [V4a0_2E0: tyop_2EringNorm_2Epolynom(A_27a),V5a1_2E0: tyop_2EringNorm_2Epolynom(A_27a),V6a0_27_2E0: tyop_2EringNorm_2Epolynom(A_27a),V7a1_27_2E0: tyop_2EringNorm_2Epolynom(A_27a)] :
          ( ( c_2EringNorm_2EPplus_2E2(A_27a,V4a0_2E0,V5a1_2E0) = c_2EringNorm_2EPplus_2E2(A_27a,V6a0_27_2E0,V7a1_27_2E0) )
        <=> ( ( V4a0_2E0 = V6a0_27_2E0 )
            & ( V5a1_2E0 = V7a1_27_2E0 ) ) )
      & ! [V8a0_2E0: tyop_2EringNorm_2Epolynom(A_27a),V9a1_2E0: tyop_2EringNorm_2Epolynom(A_27a),V10a0_27_2E0: tyop_2EringNorm_2Epolynom(A_27a),V11a1_27_2E0: tyop_2EringNorm_2Epolynom(A_27a)] :
          ( ( c_2EringNorm_2EPmult_2E2(A_27a,V8a0_2E0,V9a1_2E0) = c_2EringNorm_2EPmult_2E2(A_27a,V10a0_27_2E0,V11a1_27_2E0) )
        <=> ( ( V8a0_2E0 = V10a0_27_2E0 )
            & ( V9a1_2E0 = V11a1_27_2E0 ) ) )
      & ! [V12a_2E0: tyop_2EringNorm_2Epolynom(A_27a),V13a_27_2E0: tyop_2EringNorm_2Epolynom(A_27a)] :
          ( ( c_2EringNorm_2EPopp_2E1(A_27a,V12a_2E0) = c_2EringNorm_2EPopp_2E1(A_27a,V13a_27_2E0) )
        <=> ( V12a_2E0 = V13a_27_2E0 ) ) ) ).

tff(thm_2EringNorm_2Epolynom__distinct,axiom,
    ! [A_27a: $tType] :
      ( ! [V0a_27_2E0: A_27a,V1a_2E0: tyop_2Equote_2Eindex] : ( c_2EringNorm_2EPvar_2E1(A_27a,V1a_2E0) != c_2EringNorm_2EPconst_2E1(A_27a,V0a_27_2E0) )
      & ! [V2a1_2E0: tyop_2EringNorm_2Epolynom(A_27a),V3a0_2E0: tyop_2EringNorm_2Epolynom(A_27a),V4a_2E0: tyop_2Equote_2Eindex] : ( c_2EringNorm_2EPvar_2E1(A_27a,V4a_2E0) != c_2EringNorm_2EPplus_2E2(A_27a,V3a0_2E0,V2a1_2E0) )
      & ! [V5a1_2E0: tyop_2EringNorm_2Epolynom(A_27a),V6a0_2E0: tyop_2EringNorm_2Epolynom(A_27a),V7a_2E0: tyop_2Equote_2Eindex] : ( c_2EringNorm_2EPvar_2E1(A_27a,V7a_2E0) != c_2EringNorm_2EPmult_2E2(A_27a,V6a0_2E0,V5a1_2E0) )
      & ! [V8a_27_2E0: tyop_2EringNorm_2Epolynom(A_27a),V9a_2E0: tyop_2Equote_2Eindex] : ( c_2EringNorm_2EPvar_2E1(A_27a,V9a_2E0) != c_2EringNorm_2EPopp_2E1(A_27a,V8a_27_2E0) )
      & ! [V10a1_2E0: tyop_2EringNorm_2Epolynom(A_27a),V11a0_2E0: tyop_2EringNorm_2Epolynom(A_27a),V12a_2E0: A_27a] : ( c_2EringNorm_2EPconst_2E1(A_27a,V12a_2E0) != c_2EringNorm_2EPplus_2E2(A_27a,V11a0_2E0,V10a1_2E0) )
      & ! [V13a1_2E0: tyop_2EringNorm_2Epolynom(A_27a),V14a0_2E0: tyop_2EringNorm_2Epolynom(A_27a),V15a_2E0: A_27a] : ( c_2EringNorm_2EPconst_2E1(A_27a,V15a_2E0) != c_2EringNorm_2EPmult_2E2(A_27a,V14a0_2E0,V13a1_2E0) )
      & ! [V16a_27_2E0: tyop_2EringNorm_2Epolynom(A_27a),V17a_2E0: A_27a] : ( c_2EringNorm_2EPconst_2E1(A_27a,V17a_2E0) != c_2EringNorm_2EPopp_2E1(A_27a,V16a_27_2E0) )
      & ! [V18a1_27_2E0: tyop_2EringNorm_2Epolynom(A_27a),V19a1_2E0: tyop_2EringNorm_2Epolynom(A_27a),V20a0_27_2E0: tyop_2EringNorm_2Epolynom(A_27a),V21a0_2E0: tyop_2EringNorm_2Epolynom(A_27a)] : ( c_2EringNorm_2EPplus_2E2(A_27a,V21a0_2E0,V19a1_2E0) != c_2EringNorm_2EPmult_2E2(A_27a,V20a0_27_2E0,V18a1_27_2E0) )
      & ! [V22a1_2E0: tyop_2EringNorm_2Epolynom(A_27a),V23a0_2E0: tyop_2EringNorm_2Epolynom(A_27a),V24a_2E0: tyop_2EringNorm_2Epolynom(A_27a)] : ( c_2EringNorm_2EPplus_2E2(A_27a,V23a0_2E0,V22a1_2E0) != c_2EringNorm_2EPopp_2E1(A_27a,V24a_2E0) )
      & ! [V25a1_2E0: tyop_2EringNorm_2Epolynom(A_27a),V26a0_2E0: tyop_2EringNorm_2Epolynom(A_27a),V27a_2E0: tyop_2EringNorm_2Epolynom(A_27a)] : ( c_2EringNorm_2EPmult_2E2(A_27a,V26a0_2E0,V25a1_2E0) != c_2EringNorm_2EPopp_2E1(A_27a,V27a_2E0) ) ) ).

tff(thm_2EringNorm_2Epolynom__nchotomy,axiom,
    ! [A_27a: $tType,V0pp_2E0: tyop_2EringNorm_2Epolynom(A_27a)] :
      ( ? [V1i_2E0: tyop_2Equote_2Eindex] : ( V0pp_2E0 = c_2EringNorm_2EPvar_2E1(A_27a,V1i_2E0) )
      | ? [V2a_2E0: A_27a] : ( V0pp_2E0 = c_2EringNorm_2EPconst_2E1(A_27a,V2a_2E0) )
      | ? [V3p_2E0: tyop_2EringNorm_2Epolynom(A_27a),V4p0_2E0: tyop_2EringNorm_2Epolynom(A_27a)] : ( V0pp_2E0 = c_2EringNorm_2EPplus_2E2(A_27a,V3p_2E0,V4p0_2E0) )
      | ? [V5p_2E0: tyop_2EringNorm_2Epolynom(A_27a),V6p0_2E0: tyop_2EringNorm_2Epolynom(A_27a)] : ( V0pp_2E0 = c_2EringNorm_2EPmult_2E2(A_27a,V5p_2E0,V6p0_2E0) )
      | ? [V7p_2E0: tyop_2EringNorm_2Epolynom(A_27a)] : ( V0pp_2E0 = c_2EringNorm_2EPopp_2E1(A_27a,V7p_2E0) ) ) ).

tff(thm_2EringNorm_2Epolynom__Axiom,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0f0_2E0: tyop_2Emin_2Efun(tyop_2Equote_2Eindex,A_27b),V1f1_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V2f2_2E0: tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b)))),V3f3_2E0: tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b)))),V4f4_2E0: tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Emin_2Efun(A_27b,A_27b))] :
    ? [V5fn_2E0: tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),A_27b)] :
      ( ! [V6a_2E0: tyop_2Equote_2Eindex] : ( app_2E2(tyop_2EringNorm_2Epolynom(A_27a),A_27b,V5fn_2E0,c_2EringNorm_2EPvar_2E1(A_27a,V6a_2E0)) = app_2E2(tyop_2Equote_2Eindex,A_27b,V0f0_2E0,V6a_2E0) )
      & ! [V7a_2E0: A_27a] : ( app_2E2(tyop_2EringNorm_2Epolynom(A_27a),A_27b,V5fn_2E0,c_2EringNorm_2EPconst_2E1(A_27a,V7a_2E0)) = app_2E2(A_27a,A_27b,V1f1_2E0,V7a_2E0) )
      & ! [V8a0_2E0: tyop_2EringNorm_2Epolynom(A_27a),V9a1_2E0: tyop_2EringNorm_2Epolynom(A_27a)] : ( app_2E2(tyop_2EringNorm_2Epolynom(A_27a),A_27b,V5fn_2E0,c_2EringNorm_2EPplus_2E2(A_27a,V8a0_2E0,V9a1_2E0)) = app_2E2(A_27b,A_27b,app_2E2(A_27b,tyop_2Emin_2Efun(A_27b,A_27b),app_2E2(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b)),app_2E2(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b))),V2f2_2E0,V8a0_2E0),V9a1_2E0),app_2E2(tyop_2EringNorm_2Epolynom(A_27a),A_27b,V5fn_2E0,V8a0_2E0)),app_2E2(tyop_2EringNorm_2Epolynom(A_27a),A_27b,V5fn_2E0,V9a1_2E0)) )
      & ! [V10a0_2E0: tyop_2EringNorm_2Epolynom(A_27a),V11a1_2E0: tyop_2EringNorm_2Epolynom(A_27a)] : ( app_2E2(tyop_2EringNorm_2Epolynom(A_27a),A_27b,V5fn_2E0,c_2EringNorm_2EPmult_2E2(A_27a,V10a0_2E0,V11a1_2E0)) = app_2E2(A_27b,A_27b,app_2E2(A_27b,tyop_2Emin_2Efun(A_27b,A_27b),app_2E2(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b)),app_2E2(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b))),V3f3_2E0,V10a0_2E0),V11a1_2E0),app_2E2(tyop_2EringNorm_2Epolynom(A_27a),A_27b,V5fn_2E0,V10a0_2E0)),app_2E2(tyop_2EringNorm_2Epolynom(A_27a),A_27b,V5fn_2E0,V11a1_2E0)) )
      & ! [V12a_2E0: tyop_2EringNorm_2Epolynom(A_27a)] : ( app_2E2(tyop_2EringNorm_2Epolynom(A_27a),A_27b,V5fn_2E0,c_2EringNorm_2EPopp_2E1(A_27a,V12a_2E0)) = app_2E2(A_27b,A_27b,app_2E2(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Emin_2Efun(A_27b,A_27b),V4f4_2E0,V12a_2E0),app_2E2(tyop_2EringNorm_2Epolynom(A_27a),A_27b,V5fn_2E0,V12a_2E0)) ) ) ).

tff(thm_2EringNorm_2Epolynom__induction,axiom,
    ! [A_27a: $tType,V0P_2E0: tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Emin_2Ebool)] :
      ( ( ! [V1i_2E0: tyop_2Equote_2Eindex] : p(app_2E2(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Emin_2Ebool,V0P_2E0,c_2EringNorm_2EPvar_2E1(A_27a,V1i_2E0)))
        & ! [V2a_2E0: A_27a] : p(app_2E2(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Emin_2Ebool,V0P_2E0,c_2EringNorm_2EPconst_2E1(A_27a,V2a_2E0)))
        & ! [V3p_2E0: tyop_2EringNorm_2Epolynom(A_27a),V4p0_2E0: tyop_2EringNorm_2Epolynom(A_27a)] :
            ( ( p(app_2E2(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Emin_2Ebool,V0P_2E0,V3p_2E0))
              & p(app_2E2(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Emin_2Ebool,V0P_2E0,V4p0_2E0)) )
           => p(app_2E2(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Emin_2Ebool,V0P_2E0,c_2EringNorm_2EPplus_2E2(A_27a,V3p_2E0,V4p0_2E0))) )
        & ! [V5p_2E0: tyop_2EringNorm_2Epolynom(A_27a),V6p0_2E0: tyop_2EringNorm_2Epolynom(A_27a)] :
            ( ( p(app_2E2(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Emin_2Ebool,V0P_2E0,V5p_2E0))
              & p(app_2E2(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Emin_2Ebool,V0P_2E0,V6p0_2E0)) )
           => p(app_2E2(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Emin_2Ebool,V0P_2E0,c_2EringNorm_2EPmult_2E2(A_27a,V5p_2E0,V6p0_2E0))) )
        & ! [V7p_2E0: tyop_2EringNorm_2Epolynom(A_27a)] :
            ( p(app_2E2(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Emin_2Ebool,V0P_2E0,V7p_2E0))
           => p(app_2E2(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Emin_2Ebool,V0P_2E0,c_2EringNorm_2EPopp_2E1(A_27a,V7p_2E0))) ) )
     => ! [V8p_2E0: tyop_2EringNorm_2Epolynom(A_27a)] : p(app_2E2(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Emin_2Ebool,V0P_2E0,V8p_2E0)) ) ).

tff(thm_2EringNorm_2Epolynom__case__cong,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0f4_27_2E0: tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),A_27b),V1f3_27_2E0: tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),A_27b)),V2f2_27_2E0: tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),A_27b)),V3f1_27_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V4f_27_2E0: tyop_2Emin_2Efun(tyop_2Equote_2Eindex,A_27b),V5M_2E0: tyop_2EringNorm_2Epolynom(A_27a),V6M_27_2E0: tyop_2EringNorm_2Epolynom(A_27a),V7f_2E0: tyop_2Emin_2Efun(tyop_2Equote_2Eindex,A_27b),V8f1_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V9f2_2E0: tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),A_27b)),V10f3_2E0: tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),A_27b)),V11f4_2E0: tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),A_27b)] :
      ( ( ( V5M_2E0 = V6M_27_2E0 )
        & ! [V12a_2E0: tyop_2Equote_2Eindex] :
            ( ( V6M_27_2E0 = c_2EringNorm_2EPvar_2E1(A_27a,V12a_2E0) )
           => ( app_2E2(tyop_2Equote_2Eindex,A_27b,V7f_2E0,V12a_2E0) = app_2E2(tyop_2Equote_2Eindex,A_27b,V4f_27_2E0,V12a_2E0) ) )
        & ! [V13a_2E0: A_27a] :
            ( ( V6M_27_2E0 = c_2EringNorm_2EPconst_2E1(A_27a,V13a_2E0) )
           => ( app_2E2(A_27a,A_27b,V8f1_2E0,V13a_2E0) = app_2E2(A_27a,A_27b,V3f1_27_2E0,V13a_2E0) ) )
        & ! [V14a0_2E0: tyop_2EringNorm_2Epolynom(A_27a),V15a1_2E0: tyop_2EringNorm_2Epolynom(A_27a)] :
            ( ( V6M_27_2E0 = c_2EringNorm_2EPplus_2E2(A_27a,V14a0_2E0,V15a1_2E0) )
           => ( app_2E2(tyop_2EringNorm_2Epolynom(A_27a),A_27b,app_2E2(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),A_27b),V9f2_2E0,V14a0_2E0),V15a1_2E0) = app_2E2(tyop_2EringNorm_2Epolynom(A_27a),A_27b,app_2E2(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),A_27b),V2f2_27_2E0,V14a0_2E0),V15a1_2E0) ) )
        & ! [V16a0_2E0: tyop_2EringNorm_2Epolynom(A_27a),V17a1_2E0: tyop_2EringNorm_2Epolynom(A_27a)] :
            ( ( V6M_27_2E0 = c_2EringNorm_2EPmult_2E2(A_27a,V16a0_2E0,V17a1_2E0) )
           => ( app_2E2(tyop_2EringNorm_2Epolynom(A_27a),A_27b,app_2E2(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),A_27b),V10f3_2E0,V16a0_2E0),V17a1_2E0) = app_2E2(tyop_2EringNorm_2Epolynom(A_27a),A_27b,app_2E2(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),A_27b),V1f3_27_2E0,V16a0_2E0),V17a1_2E0) ) )
        & ! [V18a_2E0: tyop_2EringNorm_2Epolynom(A_27a)] :
            ( ( V6M_27_2E0 = c_2EringNorm_2EPopp_2E1(A_27a,V18a_2E0) )
           => ( app_2E2(tyop_2EringNorm_2Epolynom(A_27a),A_27b,V11f4_2E0,V18a_2E0) = app_2E2(tyop_2EringNorm_2Epolynom(A_27a),A_27b,V0f4_27_2E0,V18a_2E0) ) ) )
     => ( c_2EringNorm_2Epolynom__CASE_2E6(A_27a,A_27b,V5M_2E0,V7f_2E0,V8f1_2E0,V9f2_2E0,V10f3_2E0,V11f4_2E0) = c_2EringNorm_2Epolynom__CASE_2E6(A_27a,A_27b,V6M_27_2E0,V4f_27_2E0,V3f1_27_2E0,V2f2_27_2E0,V1f3_27_2E0,V0f4_27_2E0) ) ) ).

tff(thm_2EringNorm_2Epolynom__case__eq,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0x_2E0: tyop_2EringNorm_2Epolynom(A_27a),V1v_2E0: A_27b,V2f4_2E0: tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),A_27b),V3f3_2E0: tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),A_27b)),V4f2_2E0: tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),A_27b)),V5f1_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V6f_2E0: tyop_2Emin_2Efun(tyop_2Equote_2Eindex,A_27b)] :
      ( ( c_2EringNorm_2Epolynom__CASE_2E6(A_27a,A_27b,V0x_2E0,V6f_2E0,V5f1_2E0,V4f2_2E0,V3f3_2E0,V2f4_2E0) = V1v_2E0 )
    <=> ( ? [V7i_2E0: tyop_2Equote_2Eindex] :
            ( ( V0x_2E0 = c_2EringNorm_2EPvar_2E1(A_27a,V7i_2E0) )
            & ( app_2E2(tyop_2Equote_2Eindex,A_27b,V6f_2E0,V7i_2E0) = V1v_2E0 ) )
        | ? [V8a_2E0: A_27a] :
            ( ( V0x_2E0 = c_2EringNorm_2EPconst_2E1(A_27a,V8a_2E0) )
            & ( app_2E2(A_27a,A_27b,V5f1_2E0,V8a_2E0) = V1v_2E0 ) )
        | ? [V9p_2E0: tyop_2EringNorm_2Epolynom(A_27a),V10p0_2E0: tyop_2EringNorm_2Epolynom(A_27a)] :
            ( ( V0x_2E0 = c_2EringNorm_2EPplus_2E2(A_27a,V9p_2E0,V10p0_2E0) )
            & ( app_2E2(tyop_2EringNorm_2Epolynom(A_27a),A_27b,app_2E2(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),A_27b),V4f2_2E0,V9p_2E0),V10p0_2E0) = V1v_2E0 ) )
        | ? [V11p_2E0: tyop_2EringNorm_2Epolynom(A_27a),V12p0_2E0: tyop_2EringNorm_2Epolynom(A_27a)] :
            ( ( V0x_2E0 = c_2EringNorm_2EPmult_2E2(A_27a,V11p_2E0,V12p0_2E0) )
            & ( app_2E2(tyop_2EringNorm_2Epolynom(A_27a),A_27b,app_2E2(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),A_27b),V3f3_2E0,V11p_2E0),V12p0_2E0) = V1v_2E0 ) )
        | ? [V13p_2E0: tyop_2EringNorm_2Epolynom(A_27a)] :
            ( ( V0x_2E0 = c_2EringNorm_2EPopp_2E1(A_27a,V13p_2E0) )
            & ( app_2E2(tyop_2EringNorm_2Epolynom(A_27a),A_27b,V2f4_2E0,V13p_2E0) = V1v_2E0 ) ) ) ) ).

tff(thm_2EringNorm_2Epolynom__normalize__ok,axiom,
    ! [A_27a: $tType,V0r_2E0: tyop_2Ering_2Ering(A_27a)] :
      ( p(c_2Ering_2Eis__ring_2E1(A_27a,V0r_2E0))
     => ! [V1vm_2E0: tyop_2Equote_2Evarmap(A_27a),V2p_2E0: tyop_2EringNorm_2Epolynom(A_27a)] : ( c_2EringNorm_2Er__interp__cs_2E3(A_27a,V0r_2E0,V1vm_2E0,c_2EringNorm_2Epolynom__normalize_2E2(A_27a,V0r_2E0,V2p_2E0)) = c_2EringNorm_2Einterp__p_2E3(A_27a,V0r_2E0,V1vm_2E0,V2p_2E0) ) ) ).

tff(thm_2EringNorm_2Epolynom__simplify__ok,axiom,
    ! [A_27a: $tType,V0r_2E0: tyop_2Ering_2Ering(A_27a)] :
      ( p(c_2Ering_2Eis__ring_2E1(A_27a,V0r_2E0))
     => ! [V1vm_2E0: tyop_2Equote_2Evarmap(A_27a),V2p_2E0: tyop_2EringNorm_2Epolynom(A_27a)] : ( c_2EringNorm_2Er__interp__cs_2E3(A_27a,V0r_2E0,V1vm_2E0,c_2EringNorm_2Epolynom__simplify_2E2(A_27a,V0r_2E0,V2p_2E0)) = c_2EringNorm_2Einterp__p_2E3(A_27a,V0r_2E0,V1vm_2E0,V2p_2E0) ) ) ).

%------------------------------------------------------------------------------