%------------------------------------------------------------------------------
% File     : ITP114+4 : 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+4.ax [TPAP]

% Status   : Satisfiable
% Syntax   : Number of formulae    :  157 ( 113 unt;   0 def)
%            Number of atoms       :  311 ( 254 equ)
%            Maximal formula atoms :   19 (   1 avg)
%            Number of connectives :  167 (  13   ~;  15   |;  91   &)
%                                         (  22 <=>;  26  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   28 (   5 avg)
%            Maximal term depth    :   22 (   2 avg)
%            Number of predicates  :    2 (   1 usr;   0 prp; 1-2 aty)
%            Number of functors    :  201 ( 201 usr;  95 con; 0-6 aty)
%            Number of variables   :  725 ( 701   !;  24   ?)
% SPC      : FOF_SAT_RFO_SEQ

% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
fof(reserved_2Eho_2Eboolext,axiom,
    ! [V0_2E0,V1_2E0] :
      ( ( p(s(tyop_2Emin_2Ebool,V0_2E0))
      <=> p(s(tyop_2Emin_2Ebool,V1_2E0)) )
     => s(tyop_2Emin_2Ebool,V0_2E0) = s(tyop_2Emin_2Ebool,V1_2E0) ) ).

fof(reserved_2Eho_2Etruth,axiom,
    p(s(tyop_2Emin_2Ebool,c_2Ebool_2ET_2E0)) ).

fof(reserved_2Eho_2Enotfalse,axiom,
    ~ p(s(tyop_2Emin_2Ebool,c_2Ebool_2EF_2E0)) ).

fof(reserved_2Eho_2Ebool__cases__ax,axiom,
    ! [V0t_2E0] :
      ( s(tyop_2Emin_2Ebool,V0t_2E0) = s(tyop_2Emin_2Ebool,c_2Ebool_2ET_2E0)
      | s(tyop_2Emin_2Ebool,V0t_2E0) = s(tyop_2Emin_2Ebool,c_2Ebool_2EF_2E0) ) ).

fof(reserved_2Eho_2Ei__thm,axiom,
    ! [A_27a,V0x_2E0] : s(A_27a,app_2E2(s(tyop_2Emin_2Efun(A_27a,A_27a),combin_i_2E0),s(A_27a,V0x_2E0))) = s(A_27a,V0x_2E0) ).

fof(reserved_2Eho_2Ek__thm,axiom,
    ! [A_27a,A_27b,V0x_2E0,V1y_2E0] : s(A_27a,app_2E2(s(tyop_2Emin_2Efun(A_27b,A_27a),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27a)),combin_k_2E0),s(A_27a,V0x_2E0))),s(A_27b,V1y_2E0))) = s(A_27a,V0x_2E0) ).

fof(reserved_2Eho_2Es__thm,axiom,
    ! [A_27a,A_27b,A_27c,V0f_2E0,V1g_2E0,V2x_2E0] : s(A_27c,app_2E2(s(tyop_2Emin_2Efun(A_27a,A_27c),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27c)),app_2E2(s(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))),combin_s_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),V0f_2E0))),s(tyop_2Emin_2Efun(A_27a,A_27b),V1g_2E0))),s(A_27a,V2x_2E0))) = s(A_27c,app_2E2(s(tyop_2Emin_2Efun(A_27b,A_27c),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),V0f_2E0),s(A_27a,V2x_2E0))),s(A_27b,app_2E2(s(tyop_2Emin_2Efun(A_27a,A_27b),V1g_2E0),s(A_27a,V2x_2E0))))) ).

fof(reserved_2Elogic_2E_2F_5C,axiom,
    ! [V0_2E0,V1_2E0] :
      ( p(s(tyop_2Emin_2Ebool,c_2Ebool_2E_2F_5C_2E2(s(tyop_2Emin_2Ebool,V0_2E0),s(tyop_2Emin_2Ebool,V1_2E0))))
    <=> ( p(s(tyop_2Emin_2Ebool,V0_2E0))
        & p(s(tyop_2Emin_2Ebool,V1_2E0)) ) ) ).

fof(reserved_2Elogic_2E_5C_2F,axiom,
    ! [V0_2E0,V1_2E0] :
      ( p(s(tyop_2Emin_2Ebool,c_2Ebool_2E_5C_2F_2E2(s(tyop_2Emin_2Ebool,V0_2E0),s(tyop_2Emin_2Ebool,V1_2E0))))
    <=> ( p(s(tyop_2Emin_2Ebool,V0_2E0))
        | p(s(tyop_2Emin_2Ebool,V1_2E0)) ) ) ).

fof(reserved_2Elogic_2E_7E,axiom,
    ! [V0_2E0] :
      ( p(s(tyop_2Emin_2Ebool,c_2Ebool_2E_7E_2E1(s(tyop_2Emin_2Ebool,V0_2E0))))
    <=> ~ p(s(tyop_2Emin_2Ebool,V0_2E0)) ) ).

fof(reserved_2Elogic_2E_3D_3D_3E,axiom,
    ! [V0_2E0,V1_2E0] :
      ( p(s(tyop_2Emin_2Ebool,c_2Emin_2E_3D_3D_3E_2E2(s(tyop_2Emin_2Ebool,V0_2E0),s(tyop_2Emin_2Ebool,V1_2E0))))
    <=> ( p(s(tyop_2Emin_2Ebool,V0_2E0))
       => p(s(tyop_2Emin_2Ebool,V1_2E0)) ) ) ).

fof(reserved_2Elogic_2E_3D,axiom,
    ! [A_27a,V0_2E0,V1_2E0] :
      ( p(s(tyop_2Emin_2Ebool,c_2Emin_2E_3D_2E2(s(A_27a,V0_2E0),s(A_27a,V1_2E0))))
    <=> s(A_27a,V0_2E0) = s(A_27a,V1_2E0) ) ).

fof(reserved_2Equant_2E_21,axiom,
    ! [A_27a,V0f_2E0] :
      ( p(s(tyop_2Emin_2Ebool,c_2Ebool_2E_21_2E1(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0f_2E0))))
    <=> ! [V1x_2E0] : p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0f_2E0),s(A_27a,V1x_2E0)))) ) ).

fof(reserved_2Equant_2E_3F,axiom,
    ! [A_27a,V0f_2E0] :
      ( p(s(tyop_2Emin_2Ebool,c_2Ebool_2E_3F_2E1(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0f_2E0))))
    <=> ? [V1x_2E0] : p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0f_2E0),s(A_27a,V1x_2E0)))) ) ).

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

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

fof(arityeq1_2Ec_2Ebool_2E_7E_2E1,axiom,
    ! [X0_2E0] :
      ( ~ p(s(tyop_2Emin_2Ebool,X0_2E0))
    <=> p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),c_2Ebool_2E_7E_2E0),s(tyop_2Emin_2Ebool,X0_2E0)))) ) ).

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

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

fof(arityeq1_2Ec_2Ebool_2E_21_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a,X0_2E0] : s(tyop_2Emin_2Ebool,c_2Ebool_2E_21_2E1(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X0_2E0))) = s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),c_2Ebool_2E_21_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X0_2E0))) ).

fof(arityeq1_2Ec_2Ebool_2E_3F_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a,X0_2E0] : s(tyop_2Emin_2Ebool,c_2Ebool_2E_3F_2E1(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X0_2E0))) = s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),c_2Ebool_2E_3F_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X0_2E0))) ).

fof(arityeq1_2Ec_2Ebool_2EDATATYPE_2E1_2Emono_2Etyop_2Emin_2Ebool,axiom,
    ! [X0_2E0] : s(tyop_2Emin_2Ebool,c_2Ebool_2EDATATYPE_2E1(s(tyop_2Emin_2Ebool,X0_2E0))) = s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),c_2Ebool_2EDATATYPE_2E0),s(tyop_2Emin_2Ebool,X0_2E0))) ).

fof(arityeq1_2Ec_2EringNorm_2EPconst_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a,X0_2E0] : s(tyop_2EringNorm_2Epolynom(A_27a),c_2EringNorm_2EPconst_2E1(s(A_27a,X0_2E0))) = s(tyop_2EringNorm_2Epolynom(A_27a),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2EringNorm_2Epolynom(A_27a)),c_2EringNorm_2EPconst_2E0),s(A_27a,X0_2E0))) ).

fof(arityeq1_2Ec_2EringNorm_2EPopp_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a,X0_2E0] : s(tyop_2EringNorm_2Epolynom(A_27a),c_2EringNorm_2EPopp_2E1(s(tyop_2EringNorm_2Epolynom(A_27a),X0_2E0))) = s(tyop_2EringNorm_2Epolynom(A_27a),app_2E2(s(tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2EringNorm_2Epolynom(A_27a)),c_2EringNorm_2EPopp_2E0),s(tyop_2EringNorm_2Epolynom(A_27a),X0_2E0))) ).

fof(arityeq1_2Ec_2EringNorm_2EPvar_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a,X0_2E0] : s(tyop_2EringNorm_2Epolynom(A_27a),c_2EringNorm_2EPvar_2E1(s(tyop_2Equote_2Eindex,X0_2E0))) = s(tyop_2EringNorm_2Epolynom(A_27a),app_2E2(s(tyop_2Emin_2Efun(tyop_2Equote_2Eindex,tyop_2EringNorm_2Epolynom(A_27a)),c_2EringNorm_2EPvar_2E0),s(tyop_2Equote_2Eindex,X0_2E0))) ).

fof(arityeq1_2Ec_2Ecanonical_2Ecanonical__sum__merge_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a,X0_2E0] : s(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_2E1(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),X0_2E0))) = s(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(s(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)))),c_2Ecanonical_2Ecanonical__sum__merge_2E0),s(tyop_2Esemi__ring_2Esemi__ring(A_27a),X0_2E0))) ).

fof(arityeq1_2Ec_2Ering_2Esemi__ring__of_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a,X0_2E0] : s(tyop_2Esemi__ring_2Esemi__ring(A_27a),c_2Ering_2Esemi__ring__of_2E1(s(tyop_2Ering_2Ering(A_27a),X0_2E0))) = s(tyop_2Esemi__ring_2Esemi__ring(A_27a),app_2E2(s(tyop_2Emin_2Efun(tyop_2Ering_2Ering(A_27a),tyop_2Esemi__ring_2Esemi__ring(A_27a)),c_2Ering_2Esemi__ring__of_2E0),s(tyop_2Ering_2Ering(A_27a),X0_2E0))) ).

fof(arityeq1_2Ec_2Ecanonical_2Ecanonical__sum__prod_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a,X0_2E0] : s(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_2E1(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),X0_2E0))) = s(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(s(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)))),c_2Ecanonical_2Ecanonical__sum__prod_2E0),s(tyop_2Esemi__ring_2Esemi__ring(A_27a),X0_2E0))) ).

fof(arityeq1_2Ec_2Ecanonical_2Ecanonical__sum__scalar_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a,X0_2E0] : s(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_2E1(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),X0_2E0))) = s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a))),app_2E2(s(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)))),c_2Ecanonical_2Ecanonical__sum__scalar_2E0),s(tyop_2Esemi__ring_2Esemi__ring(A_27a),X0_2E0))) ).

fof(arityeq1_2Ec_2Ecanonical_2Ecanonical__sum__scalar2_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a,X0_2E0] : s(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_2E1(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),X0_2E0))) = s(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(s(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)))),c_2Ecanonical_2Ecanonical__sum__scalar2_2E0),s(tyop_2Esemi__ring_2Esemi__ring(A_27a),X0_2E0))) ).

fof(arityeq1_2Ec_2Ecanonical_2Ecanonical__sum__scalar3_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a,X0_2E0] : s(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_2E1(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),X0_2E0))) = s(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)))),app_2E2(s(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))))),c_2Ecanonical_2Ecanonical__sum__scalar3_2E0),s(tyop_2Esemi__ring_2Esemi__ring(A_27a),X0_2E0))) ).

fof(arityeq1_2Ec_2Ecanonical_2Ecanonical__sum__simplify_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a,X0_2E0] : s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)),c_2Ecanonical_2Ecanonical__sum__simplify_2E1(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),X0_2E0))) = s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)),app_2E2(s(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))),c_2Ecanonical_2Ecanonical__sum__simplify_2E0),s(tyop_2Esemi__ring_2Esemi__ring(A_27a),X0_2E0))) ).

fof(arityeq1_2Ec_2Ecanonical_2Eics__aux_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a,X0_2E0] : s(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_2E1(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),X0_2E0))) = s(tyop_2Emin_2Efun(tyop_2Equote_2Evarmap(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27a))),app_2E2(s(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)))),c_2Ecanonical_2Eics__aux_2E0),s(tyop_2Esemi__ring_2Esemi__ring(A_27a),X0_2E0))) ).

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

fof(arityeq1_2Ec_2Ecanonical_2Einterp__m_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a,X0_2E0] : s(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_2E1(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),X0_2E0))) = s(tyop_2Emin_2Efun(tyop_2Equote_2Evarmap(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),A_27a))),app_2E2(s(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)))),c_2Ecanonical_2Einterp__m_2E0),s(tyop_2Esemi__ring_2Esemi__ring(A_27a),X0_2E0))) ).

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

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

fof(arityeq1_2Ec_2Ering_2Eis__ring_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a,X0_2E0] : s(tyop_2Emin_2Ebool,c_2Ering_2Eis__ring_2E1(s(tyop_2Ering_2Ering(A_27a),X0_2E0))) = s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ering_2Ering(A_27a),tyop_2Emin_2Ebool),c_2Ering_2Eis__ring_2E0),s(tyop_2Ering_2Ering(A_27a),X0_2E0))) ).

fof(arityeq1_2Ec_2Ecanonical_2Eivl__aux_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a,X0_2E0] : s(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_2E1(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),X0_2E0))) = s(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))),app_2E2(s(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)))),c_2Ecanonical_2Eivl__aux_2E0),s(tyop_2Esemi__ring_2Esemi__ring(A_27a),X0_2E0))) ).

fof(arityeq1_2Ec_2Ecanonical_2Emonom__insert_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a,X0_2E0] : s(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_2E1(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),X0_2E0))) = s(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)))),app_2E2(s(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))))),c_2Ecanonical_2Emonom__insert_2E0),s(tyop_2Esemi__ring_2Esemi__ring(A_27a),X0_2E0))) ).

fof(arityeq1_2Ec_2EringNorm_2Er__canonical__sum__merge_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a,X0_2E0] : s(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_2E1(s(tyop_2Ering_2Ering(A_27a),X0_2E0))) = s(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(s(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)))),c_2EringNorm_2Er__canonical__sum__merge_2E0),s(tyop_2Ering_2Ering(A_27a),X0_2E0))) ).

fof(arityeq1_2Ec_2EringNorm_2Er__canonical__sum__prod_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a,X0_2E0] : s(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_2E1(s(tyop_2Ering_2Ering(A_27a),X0_2E0))) = s(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(s(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)))),c_2EringNorm_2Er__canonical__sum__prod_2E0),s(tyop_2Ering_2Ering(A_27a),X0_2E0))) ).

fof(arityeq1_2Ec_2EringNorm_2Er__canonical__sum__scalar_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a,X0_2E0] : s(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_2E1(s(tyop_2Ering_2Ering(A_27a),X0_2E0))) = s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a))),app_2E2(s(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)))),c_2EringNorm_2Er__canonical__sum__scalar_2E0),s(tyop_2Ering_2Ering(A_27a),X0_2E0))) ).

fof(arityeq1_2Ec_2EringNorm_2Er__canonical__sum__scalar2_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a,X0_2E0] : s(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_2E1(s(tyop_2Ering_2Ering(A_27a),X0_2E0))) = s(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(s(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)))),c_2EringNorm_2Er__canonical__sum__scalar2_2E0),s(tyop_2Ering_2Ering(A_27a),X0_2E0))) ).

fof(arityeq1_2Ec_2EringNorm_2Er__canonical__sum__scalar3_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a,X0_2E0] : s(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_2E1(s(tyop_2Ering_2Ering(A_27a),X0_2E0))) = s(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)))),app_2E2(s(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))))),c_2EringNorm_2Er__canonical__sum__scalar3_2E0),s(tyop_2Ering_2Ering(A_27a),X0_2E0))) ).

fof(arityeq1_2Ec_2EringNorm_2Er__canonical__sum__simplify_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a,X0_2E0] : s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)),c_2EringNorm_2Er__canonical__sum__simplify_2E1(s(tyop_2Ering_2Ering(A_27a),X0_2E0))) = s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)),app_2E2(s(tyop_2Emin_2Efun(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),s(tyop_2Ering_2Ering(A_27a),X0_2E0))) ).

fof(arityeq1_2Ec_2EringNorm_2Er__ics__aux_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a,X0_2E0] : s(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_2E1(s(tyop_2Ering_2Ering(A_27a),X0_2E0))) = s(tyop_2Emin_2Efun(tyop_2Equote_2Evarmap(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27a))),app_2E2(s(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)))),c_2EringNorm_2Er__ics__aux_2E0),s(tyop_2Ering_2Ering(A_27a),X0_2E0))) ).

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

fof(arityeq1_2Ec_2EringNorm_2Er__interp__m_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a,X0_2E0] : s(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_2E1(s(tyop_2Ering_2Ering(A_27a),X0_2E0))) = s(tyop_2Emin_2Efun(tyop_2Equote_2Evarmap(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),A_27a))),app_2E2(s(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)))),c_2EringNorm_2Er__interp__m_2E0),s(tyop_2Ering_2Ering(A_27a),X0_2E0))) ).

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

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

fof(arityeq1_2Ec_2EringNorm_2Er__ivl__aux_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a,X0_2E0] : s(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_2E1(s(tyop_2Ering_2Ering(A_27a),X0_2E0))) = s(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))),app_2E2(s(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)))),c_2EringNorm_2Er__ivl__aux_2E0),s(tyop_2Ering_2Ering(A_27a),X0_2E0))) ).

fof(arityeq1_2Ec_2EringNorm_2Er__monom__insert_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a,X0_2E0] : s(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_2E1(s(tyop_2Ering_2Ering(A_27a),X0_2E0))) = s(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)))),app_2E2(s(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))))),c_2EringNorm_2Er__monom__insert_2E0),s(tyop_2Ering_2Ering(A_27a),X0_2E0))) ).

fof(arityeq1_2Ec_2EringNorm_2Er__spolynom__normalize_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a,X0_2E0] : s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)),c_2EringNorm_2Er__spolynom__normalize_2E1(s(tyop_2Ering_2Ering(A_27a),X0_2E0))) = s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)),app_2E2(s(tyop_2Emin_2Efun(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),s(tyop_2Ering_2Ering(A_27a),X0_2E0))) ).

fof(arityeq1_2Ec_2EringNorm_2Er__spolynom__simplify_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a,X0_2E0] : s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)),c_2EringNorm_2Er__spolynom__simplify_2E1(s(tyop_2Ering_2Ering(A_27a),X0_2E0))) = s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)),app_2E2(s(tyop_2Emin_2Efun(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),s(tyop_2Ering_2Ering(A_27a),X0_2E0))) ).

fof(arityeq1_2Ec_2EringNorm_2Er__varlist__insert_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a,X0_2E0] : s(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_2E1(s(tyop_2Ering_2Ering(A_27a),X0_2E0))) = s(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(s(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)))),c_2EringNorm_2Er__varlist__insert_2E0),s(tyop_2Ering_2Ering(A_27a),X0_2E0))) ).

fof(arityeq1_2Ec_2Ering_2Ering__R0_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a,X0_2E0] : s(A_27a,c_2Ering_2Ering__R0_2E1(s(tyop_2Ering_2Ering(A_27a),X0_2E0))) = s(A_27a,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ering_2Ering(A_27a),A_27a),c_2Ering_2Ering__R0_2E0),s(tyop_2Ering_2Ering(A_27a),X0_2E0))) ).

fof(arityeq1_2Ec_2Ering_2Ering__R1_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a,X0_2E0] : s(A_27a,c_2Ering_2Ering__R1_2E1(s(tyop_2Ering_2Ering(A_27a),X0_2E0))) = s(A_27a,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ering_2Ering(A_27a),A_27a),c_2Ering_2Ering__R1_2E0),s(tyop_2Ering_2Ering(A_27a),X0_2E0))) ).

fof(arityeq1_2Ec_2Ecanonical_2Espolynom__normalize_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a,X0_2E0] : s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)),c_2Ecanonical_2Espolynom__normalize_2E1(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),X0_2E0))) = s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)),app_2E2(s(tyop_2Emin_2Efun(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),s(tyop_2Esemi__ring_2Esemi__ring(A_27a),X0_2E0))) ).

fof(arityeq1_2Ec_2Ecanonical_2Espolynom__simplify_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a,X0_2E0] : s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)),c_2Ecanonical_2Espolynom__simplify_2E1(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),X0_2E0))) = s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)),app_2E2(s(tyop_2Emin_2Efun(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),s(tyop_2Esemi__ring_2Esemi__ring(A_27a),X0_2E0))) ).

fof(arityeq1_2Ec_2Ecanonical_2Evarlist__insert_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a,X0_2E0] : s(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_2E1(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),X0_2E0))) = s(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(s(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)))),c_2Ecanonical_2Evarlist__insert_2E0),s(tyop_2Esemi__ring_2Esemi__ring(A_27a),X0_2E0))) ).

fof(arityeq2_2Ec_2EringNorm_2EPmult_2E2_2Emono_2EA_27a,axiom,
    ! [A_27a,X0_2E0,X1_2E0] : s(tyop_2EringNorm_2Epolynom(A_27a),c_2EringNorm_2EPmult_2E2(s(tyop_2EringNorm_2Epolynom(A_27a),X0_2E0),s(tyop_2EringNorm_2Epolynom(A_27a),X1_2E0))) = s(tyop_2EringNorm_2Epolynom(A_27a),app_2E2(s(tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2EringNorm_2Epolynom(A_27a)),app_2E2(s(tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2EringNorm_2Epolynom(A_27a))),c_2EringNorm_2EPmult_2E0),s(tyop_2EringNorm_2Epolynom(A_27a),X0_2E0))),s(tyop_2EringNorm_2Epolynom(A_27a),X1_2E0))) ).

fof(arityeq2_2Ec_2EringNorm_2EPplus_2E2_2Emono_2EA_27a,axiom,
    ! [A_27a,X0_2E0,X1_2E0] : s(tyop_2EringNorm_2Epolynom(A_27a),c_2EringNorm_2EPplus_2E2(s(tyop_2EringNorm_2Epolynom(A_27a),X0_2E0),s(tyop_2EringNorm_2Epolynom(A_27a),X1_2E0))) = s(tyop_2EringNorm_2Epolynom(A_27a),app_2E2(s(tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2EringNorm_2Epolynom(A_27a)),app_2E2(s(tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2EringNorm_2Epolynom(A_27a))),c_2EringNorm_2EPplus_2E0),s(tyop_2EringNorm_2Epolynom(A_27a),X0_2E0))),s(tyop_2EringNorm_2Epolynom(A_27a),X1_2E0))) ).

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

fof(arityeq1_2Ec_2Earithmetic_2EBIT1_2E1,axiom,
    ! [X0_2E0] : s(tyop_2Enum_2Enum,c_2Earithmetic_2EBIT1_2E1(s(tyop_2Enum_2Enum,X0_2E0))) = s(tyop_2Enum_2Enum,app_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),c_2Earithmetic_2EBIT1_2E0),s(tyop_2Enum_2Enum,X0_2E0))) ).

fof(arityeq1_2Ec_2Earithmetic_2ENUMERAL_2E1,axiom,
    ! [X0_2E0] : s(tyop_2Enum_2Enum,c_2Earithmetic_2ENUMERAL_2E1(s(tyop_2Enum_2Enum,X0_2E0))) = s(tyop_2Enum_2Enum,app_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),c_2Earithmetic_2ENUMERAL_2E0),s(tyop_2Enum_2Enum,X0_2E0))) ).

fof(arityeq1_2Ec_2Equote_2Eindex__size_2E1,axiom,
    ! [X0_2E0] : s(tyop_2Enum_2Enum,c_2Equote_2Eindex__size_2E1(s(tyop_2Equote_2Eindex,X0_2E0))) = s(tyop_2Enum_2Enum,app_2E2(s(tyop_2Emin_2Efun(tyop_2Equote_2Eindex,tyop_2Enum_2Enum),c_2Equote_2Eindex__size_2E0),s(tyop_2Equote_2Eindex,X0_2E0))) ).

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

fof(arityeq2_2Ec_2Ecanonical_2ECons__varlist_2E2_2Emono_2EA_27a,axiom,
    ! [A_27a,X0_2E0,X1_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__varlist_2E2(s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),X0_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),X1_2E0))) = s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)),app_2E2(s(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__varlist_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),X0_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),X1_2E0))) ).

fof(arityeq2_2Ec_2EringNorm_2Er__canonical__sum__simplify_2E2_2Emono_2EA_27a,axiom,
    ! [A_27a,X0_2E0,X1_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EringNorm_2Er__canonical__sum__simplify_2E2(s(tyop_2Ering_2Ering(A_27a),X0_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),X1_2E0))) = s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)),app_2E2(s(tyop_2Emin_2Efun(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),s(tyop_2Ering_2Ering(A_27a),X0_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),X1_2E0))) ).

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

fof(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,X0_2E0,X1_2E0] : s(tyop_2Emin_2Ebool,c_2Ebool_2ETYPE__DEFINITION_2E2(s(tyop_2Emin_2Efun(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)),tyop_2Emin_2Ebool),X0_2E0),s(tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a))),X1_2E0))) = s(tyop_2Emin_2Ebool,app_2E2(s(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),app_2E2(s(tyop_2Emin_2Efun(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),s(tyop_2Emin_2Efun(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)),tyop_2Emin_2Ebool),X0_2E0))),s(tyop_2Emin_2Efun(tyop_2EringNorm_2Epolynom(A_27a),tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a))),X1_2E0))) ).

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

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

fof(arityeq2_2Ec_2EringNorm_2Er__spolynom__normalize_2E2_2Emono_2EA_27a,axiom,
    ! [A_27a,X0_2E0,X1_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EringNorm_2Er__spolynom__normalize_2E2(s(tyop_2Ering_2Ering(A_27a),X0_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),X1_2E0))) = s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)),app_2E2(s(tyop_2Emin_2Efun(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),s(tyop_2Ering_2Ering(A_27a),X0_2E0))),s(tyop_2Ecanonical_2Espolynom(A_27a),X1_2E0))) ).

fof(arityeq3_2Ec_2Ecanonical_2ECons__monom_2E3_2Emono_2EA_27a,axiom,
    ! [A_27a,X0_2E0,X1_2E0,X2_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__monom_2E3(s(A_27a,X0_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),X1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),X2_2E0))) = s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)),app_2E2(s(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(s(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_2ECons__monom_2E0),s(A_27a,X0_2E0))),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),X1_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),X2_2E0))) ).

fof(arityeq1_2Ec_2Ecanonical_2ESPconst_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a,X0_2E0] : s(tyop_2Ecanonical_2Espolynom(A_27a),c_2Ecanonical_2ESPconst_2E1(s(A_27a,X0_2E0))) = s(tyop_2Ecanonical_2Espolynom(A_27a),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Ecanonical_2Espolynom(A_27a)),c_2Ecanonical_2ESPconst_2E0),s(A_27a,X0_2E0))) ).

fof(arityeq1_2Ec_2Ecanonical_2ESPvar_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a,X0_2E0] : s(tyop_2Ecanonical_2Espolynom(A_27a),c_2Ecanonical_2ESPvar_2E1(s(tyop_2Equote_2Eindex,X0_2E0))) = s(tyop_2Ecanonical_2Espolynom(A_27a),app_2E2(s(tyop_2Emin_2Efun(tyop_2Equote_2Eindex,tyop_2Ecanonical_2Espolynom(A_27a)),c_2Ecanonical_2ESPvar_2E0),s(tyop_2Equote_2Eindex,X0_2E0))) ).

fof(arityeq2_2Ec_2Ecanonical_2ESPmult_2E2_2Emono_2EA_27a,axiom,
    ! [A_27a,X0_2E0,X1_2E0] : s(tyop_2Ecanonical_2Espolynom(A_27a),c_2Ecanonical_2ESPmult_2E2(s(tyop_2Ecanonical_2Espolynom(A_27a),X0_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),X1_2E0))) = s(tyop_2Ecanonical_2Espolynom(A_27a),app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Ecanonical_2Espolynom(A_27a)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Ecanonical_2Espolynom(A_27a))),c_2Ecanonical_2ESPmult_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),X0_2E0))),s(tyop_2Ecanonical_2Espolynom(A_27a),X1_2E0))) ).

fof(arityeq2_2Ec_2Ecanonical_2ESPplus_2E2_2Emono_2EA_27a,axiom,
    ! [A_27a,X0_2E0,X1_2E0] : s(tyop_2Ecanonical_2Espolynom(A_27a),c_2Ecanonical_2ESPplus_2E2(s(tyop_2Ecanonical_2Espolynom(A_27a),X0_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),X1_2E0))) = s(tyop_2Ecanonical_2Espolynom(A_27a),app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Ecanonical_2Espolynom(A_27a)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Ecanonical_2Espolynom(A_27a))),c_2Ecanonical_2ESPplus_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),X0_2E0))),s(tyop_2Ecanonical_2Espolynom(A_27a),X1_2E0))) ).

fof(arityeq2_2Ec_2EringNorm_2Er__spolynom__simplify_2E2_2Emono_2EA_27a,axiom,
    ! [A_27a,X0_2E0,X1_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EringNorm_2Er__spolynom__simplify_2E2(s(tyop_2Ering_2Ering(A_27a),X0_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),X1_2E0))) = s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)),app_2E2(s(tyop_2Emin_2Efun(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),s(tyop_2Ering_2Ering(A_27a),X0_2E0))),s(tyop_2Ecanonical_2Espolynom(A_27a),X1_2E0))) ).

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

fof(arityeq2_2Ec_2Ering_2Ering__RN_2E2_2Emono_2EA_27a,axiom,
    ! [A_27a,X0_2E0,X1_2E0] : s(A_27a,c_2Ering_2Ering__RN_2E2(s(tyop_2Ering_2Ering(A_27a),X0_2E0),s(A_27a,X1_2E0))) = s(A_27a,app_2E2(s(tyop_2Emin_2Efun(A_27a,A_27a),app_2E2(s(tyop_2Emin_2Efun(tyop_2Ering_2Ering(A_27a),tyop_2Emin_2Efun(A_27a,A_27a)),c_2Ering_2Ering__RN_2E0),s(tyop_2Ering_2Ering(A_27a),X0_2E0))),s(A_27a,X1_2E0))) ).

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

fof(arityeq3_2Ec_2Ebool_2ECOND_2E3_2Emono_2Etyop_2Ecanonical_2Ecanonical__sum_28A_27a_29,axiom,
    ! [A_27a,X0_2E0,X1_2E0,X2_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ebool_2ECOND_2E3(s(tyop_2Emin_2Ebool,X0_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),X1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),X2_2E0))) = s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)),app_2E2(s(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(s(tyop_2Emin_2Efun(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),s(tyop_2Emin_2Ebool,X0_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),X1_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),X2_2E0))) ).

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

fof(arityeq3_2Ec_2Eind__type_2ECONSTR_2E3_2Emono_2Etyop_2Epair_2Eprod_28tyop_2Equote_2Eindex_2CA_27a_29,axiom,
    ! [A_27a,X0_2E0,X1_2E0,X2_2E0] : s(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)),c_2Eind__type_2ECONSTR_2E3(s(tyop_2Enum_2Enum,X0_2E0),s(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a),X1_2E0),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a))),X2_2E0))) = s(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)),app_2E2(s(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(s(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)))),app_2E2(s(tyop_2Emin_2Efun(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),s(tyop_2Enum_2Enum,X0_2E0))),s(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a),X1_2E0))),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a))),X2_2E0))) ).

fof(arityeq1_2Ec_2Enum_2ESUC_2E1,axiom,
    ! [X0_2E0] : s(tyop_2Enum_2Enum,c_2Enum_2ESUC_2E1(s(tyop_2Enum_2Enum,X0_2E0))) = s(tyop_2Enum_2Enum,app_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),c_2Enum_2ESUC_2E0),s(tyop_2Enum_2Enum,X0_2E0))) ).

fof(arityeq2_2Ec_2Eind__type_2EFCONS_2E2_2Emono_2Etyop_2Eind__type_2Erecspace_28tyop_2Epair_2Eprod_28tyop_2Equote_2Eindex_2CA_27a_29_29,axiom,
    ! [A_27a,X0_2E0,X1_2E0] : s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a))),c_2Eind__type_2EFCONS_2E2(s(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)),X0_2E0),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a))),X1_2E0))) = s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a))),app_2E2(s(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)))),app_2E2(s(tyop_2Emin_2Efun(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),s(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)),X0_2E0))),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a))),X1_2E0))) ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

fof(arityeq4_2Ec_2EternaryComparisons_2Eordering__CASE_2E4_2Emono_2Etyop_2Ecanonical_2Ecanonical__sum_28A_27a_29,axiom,
    ! [A_27a,X0_2E0,X1_2E0,X2_2E0,X3_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EternaryComparisons_2Eordering__CASE_2E4(s(tyop_2EternaryComparisons_2Eordering,X0_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),X1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),X2_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),X3_2E0))) = s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)),app_2E2(s(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(s(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)))),app_2E2(s(tyop_2Emin_2Efun(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),s(tyop_2EternaryComparisons_2Eordering,X0_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),X1_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),X2_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),X3_2E0))) ).

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

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

fof(thm_2EringNorm_2Er__interp__sp__def,axiom,
    ! [A_27a,V0r_2E0] : s(tyop_2Emin_2Efun(tyop_2Equote_2Evarmap(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27a)),c_2EringNorm_2Er__interp__sp_2E1(s(tyop_2Ering_2Ering(A_27a),V0r_2E0))) = s(tyop_2Emin_2Efun(tyop_2Equote_2Evarmap(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27a)),c_2Ecanonical_2Einterp__sp_2E1(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),c_2Ering_2Esemi__ring__of_2E1(s(tyop_2Ering_2Ering(A_27a),V0r_2E0))))) ).

fof(thm_2EringNorm_2Er__spolynom__simplify__def,axiom,
    ! [A_27a,V0r_2E0] : s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)),c_2EringNorm_2Er__spolynom__simplify_2E1(s(tyop_2Ering_2Ering(A_27a),V0r_2E0))) = s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)),c_2Ecanonical_2Espolynom__simplify_2E1(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),c_2Ering_2Esemi__ring__of_2E1(s(tyop_2Ering_2Ering(A_27a),V0r_2E0))))) ).

fof(thm_2EringNorm_2Er__spolynom__normalize__def,axiom,
    ! [A_27a,V0r_2E0] : s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)),c_2EringNorm_2Er__spolynom__normalize_2E1(s(tyop_2Ering_2Ering(A_27a),V0r_2E0))) = s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)),c_2Ecanonical_2Espolynom__normalize_2E1(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),c_2Ering_2Esemi__ring__of_2E1(s(tyop_2Ering_2Ering(A_27a),V0r_2E0))))) ).

fof(thm_2EringNorm_2Er__interp__cs__def,axiom,
    ! [A_27a,V0r_2E0] : s(tyop_2Emin_2Efun(tyop_2Equote_2Evarmap(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27a)),c_2EringNorm_2Er__interp__cs_2E1(s(tyop_2Ering_2Ering(A_27a),V0r_2E0))) = s(tyop_2Emin_2Efun(tyop_2Equote_2Evarmap(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27a)),c_2Ecanonical_2Einterp__cs_2E1(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),c_2Ering_2Esemi__ring__of_2E1(s(tyop_2Ering_2Ering(A_27a),V0r_2E0))))) ).

fof(thm_2EringNorm_2Er__ics__aux__def,axiom,
    ! [A_27a,V0r_2E0] : s(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_2E1(s(tyop_2Ering_2Ering(A_27a),V0r_2E0))) = s(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_2E1(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),c_2Ering_2Esemi__ring__of_2E1(s(tyop_2Ering_2Ering(A_27a),V0r_2E0))))) ).

fof(thm_2EringNorm_2Er__interp__m__def,axiom,
    ! [A_27a,V0r_2E0] : s(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_2E1(s(tyop_2Ering_2Ering(A_27a),V0r_2E0))) = s(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_2E1(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),c_2Ering_2Esemi__ring__of_2E1(s(tyop_2Ering_2Ering(A_27a),V0r_2E0))))) ).

fof(thm_2EringNorm_2Er__interp__vl__def,axiom,
    ! [A_27a,V0r_2E0] : s(tyop_2Emin_2Efun(tyop_2Equote_2Evarmap(A_27a),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),A_27a)),c_2EringNorm_2Er__interp__vl_2E1(s(tyop_2Ering_2Ering(A_27a),V0r_2E0))) = s(tyop_2Emin_2Efun(tyop_2Equote_2Evarmap(A_27a),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),A_27a)),c_2Ecanonical_2Einterp__vl_2E1(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),c_2Ering_2Esemi__ring__of_2E1(s(tyop_2Ering_2Ering(A_27a),V0r_2E0))))) ).

fof(thm_2EringNorm_2Er__ivl__aux__def,axiom,
    ! [A_27a,V0r_2E0] : s(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_2E1(s(tyop_2Ering_2Ering(A_27a),V0r_2E0))) = s(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_2E1(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),c_2Ering_2Esemi__ring__of_2E1(s(tyop_2Ering_2Ering(A_27a),V0r_2E0))))) ).

fof(thm_2EringNorm_2Er__canonical__sum__simplify__def,axiom,
    ! [A_27a,V0r_2E0] : s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)),c_2EringNorm_2Er__canonical__sum__simplify_2E1(s(tyop_2Ering_2Ering(A_27a),V0r_2E0))) = s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)),c_2Ecanonical_2Ecanonical__sum__simplify_2E1(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),c_2Ering_2Esemi__ring__of_2E1(s(tyop_2Ering_2Ering(A_27a),V0r_2E0))))) ).

fof(thm_2EringNorm_2Er__canonical__sum__prod__def,axiom,
    ! [A_27a,V0r_2E0] : s(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_2E1(s(tyop_2Ering_2Ering(A_27a),V0r_2E0))) = s(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_2E1(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),c_2Ering_2Esemi__ring__of_2E1(s(tyop_2Ering_2Ering(A_27a),V0r_2E0))))) ).

fof(thm_2EringNorm_2Er__canonical__sum__scalar3__def,axiom,
    ! [A_27a,V0r_2E0] : s(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_2E1(s(tyop_2Ering_2Ering(A_27a),V0r_2E0))) = s(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_2E1(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),c_2Ering_2Esemi__ring__of_2E1(s(tyop_2Ering_2Ering(A_27a),V0r_2E0))))) ).

fof(thm_2EringNorm_2Er__canonical__sum__scalar2__def,axiom,
    ! [A_27a,V0r_2E0] : s(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_2E1(s(tyop_2Ering_2Ering(A_27a),V0r_2E0))) = s(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_2E1(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),c_2Ering_2Esemi__ring__of_2E1(s(tyop_2Ering_2Ering(A_27a),V0r_2E0))))) ).

fof(thm_2EringNorm_2Er__canonical__sum__scalar__def,axiom,
    ! [A_27a,V0r_2E0] : s(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_2E1(s(tyop_2Ering_2Ering(A_27a),V0r_2E0))) = s(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_2E1(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),c_2Ering_2Esemi__ring__of_2E1(s(tyop_2Ering_2Ering(A_27a),V0r_2E0))))) ).

fof(thm_2EringNorm_2Er__varlist__insert__def,axiom,
    ! [A_27a,V0r_2E0] : s(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_2E1(s(tyop_2Ering_2Ering(A_27a),V0r_2E0))) = s(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_2E1(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),c_2Ering_2Esemi__ring__of_2E1(s(tyop_2Ering_2Ering(A_27a),V0r_2E0))))) ).

fof(thm_2EringNorm_2Er__monom__insert__def,axiom,
    ! [A_27a,V0r_2E0] : s(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_2E1(s(tyop_2Ering_2Ering(A_27a),V0r_2E0))) = s(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_2E1(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),c_2Ering_2Esemi__ring__of_2E1(s(tyop_2Ering_2Ering(A_27a),V0r_2E0))))) ).

fof(thm_2EringNorm_2Er__canonical__sum__merge__def,axiom,
    ! [A_27a,V0r_2E0] : s(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_2E1(s(tyop_2Ering_2Ering(A_27a),V0r_2E0))) = s(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_2E1(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),c_2Ering_2Esemi__ring__of_2E1(s(tyop_2Ering_2Ering(A_27a),V0r_2E0))))) ).

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

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

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

fof(thm_2EringNorm_2Epolynom__normalize__def,axiom,
    ! [A_27a] :
      ( ! [V0r_2E0,V1i_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EringNorm_2Epolynom__normalize_2E2(s(tyop_2Ering_2Ering(A_27a),V0r_2E0),s(tyop_2EringNorm_2Epolynom(A_27a),c_2EringNorm_2EPvar_2E1(s(tyop_2Equote_2Eindex,V1i_2E0))))) = s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__varlist_2E2(s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),c_2Elist_2ECONS_2E2(s(tyop_2Equote_2Eindex,V1i_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),c_2Elist_2ENIL_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ENil__monom_2E0)))
      & ! [V2r_2E0,V3c_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EringNorm_2Epolynom__normalize_2E2(s(tyop_2Ering_2Ering(A_27a),V2r_2E0),s(tyop_2EringNorm_2Epolynom(A_27a),c_2EringNorm_2EPconst_2E1(s(A_27a,V3c_2E0))))) = s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__monom_2E3(s(A_27a,V3c_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),c_2Elist_2ENIL_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ENil__monom_2E0)))
      & ! [V4r_2E0,V5pl_2E0,V6pr_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EringNorm_2Epolynom__normalize_2E2(s(tyop_2Ering_2Ering(A_27a),V4r_2E0),s(tyop_2EringNorm_2Epolynom(A_27a),c_2EringNorm_2EPplus_2E2(s(tyop_2EringNorm_2Epolynom(A_27a),V5pl_2E0),s(tyop_2EringNorm_2Epolynom(A_27a),V6pr_2E0))))) = s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EringNorm_2Er__canonical__sum__merge_2E3(s(tyop_2Ering_2Ering(A_27a),V4r_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EringNorm_2Epolynom__normalize_2E2(s(tyop_2Ering_2Ering(A_27a),V4r_2E0),s(tyop_2EringNorm_2Epolynom(A_27a),V5pl_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EringNorm_2Epolynom__normalize_2E2(s(tyop_2Ering_2Ering(A_27a),V4r_2E0),s(tyop_2EringNorm_2Epolynom(A_27a),V6pr_2E0)))))
      & ! [V7r_2E0,V8pl_2E0,V9pr_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EringNorm_2Epolynom__normalize_2E2(s(tyop_2Ering_2Ering(A_27a),V7r_2E0),s(tyop_2EringNorm_2Epolynom(A_27a),c_2EringNorm_2EPmult_2E2(s(tyop_2EringNorm_2Epolynom(A_27a),V8pl_2E0),s(tyop_2EringNorm_2Epolynom(A_27a),V9pr_2E0))))) = s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EringNorm_2Er__canonical__sum__prod_2E3(s(tyop_2Ering_2Ering(A_27a),V7r_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EringNorm_2Epolynom__normalize_2E2(s(tyop_2Ering_2Ering(A_27a),V7r_2E0),s(tyop_2EringNorm_2Epolynom(A_27a),V8pl_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EringNorm_2Epolynom__normalize_2E2(s(tyop_2Ering_2Ering(A_27a),V7r_2E0),s(tyop_2EringNorm_2Epolynom(A_27a),V9pr_2E0)))))
      & ! [V10r_2E0,V11p_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EringNorm_2Epolynom__normalize_2E2(s(tyop_2Ering_2Ering(A_27a),V10r_2E0),s(tyop_2EringNorm_2Epolynom(A_27a),c_2EringNorm_2EPopp_2E1(s(tyop_2EringNorm_2Epolynom(A_27a),V11p_2E0))))) = s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EringNorm_2Er__canonical__sum__scalar3_2E4(s(tyop_2Ering_2Ering(A_27a),V10r_2E0),s(A_27a,c_2Ering_2Ering__RN_2E2(s(tyop_2Ering_2Ering(A_27a),V10r_2E0),s(A_27a,c_2Ering_2Ering__R1_2E1(s(tyop_2Ering_2Ering(A_27a),V10r_2E0))))),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),c_2Elist_2ENIL_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EringNorm_2Epolynom__normalize_2E2(s(tyop_2Ering_2Ering(A_27a),V10r_2E0),s(tyop_2EringNorm_2Epolynom(A_27a),V11p_2E0))))) ) ).

fof(thm_2EringNorm_2Epolynom__simplify__def,axiom,
    ! [A_27a,V0r_2E0,V1x_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EringNorm_2Epolynom__simplify_2E2(s(tyop_2Ering_2Ering(A_27a),V0r_2E0),s(tyop_2EringNorm_2Epolynom(A_27a),V1x_2E0))) = s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EringNorm_2Er__canonical__sum__simplify_2E2(s(tyop_2Ering_2Ering(A_27a),V0r_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EringNorm_2Epolynom__normalize_2E2(s(tyop_2Ering_2Ering(A_27a),V0r_2E0),s(tyop_2EringNorm_2Epolynom(A_27a),V1x_2E0))))) ).

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

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

fof(thm_2EringNorm_2Ecanonical__sum__merge__def,axiom,
    ! [A_27a,V0r_2E0] :
      ( ! [V1t2_2E0,V2t1_2E0,V3l2_2E0,V4l1_2E0,V5c2_2E0,V6c1_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EringNorm_2Er__canonical__sum__merge_2E3(s(tyop_2Ering_2Ering(A_27a),V0r_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__monom_2E3(s(A_27a,V6c1_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V4l1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V2t1_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__monom_2E3(s(A_27a,V5c2_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V3l2_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V1t2_2E0))))) = s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EternaryComparisons_2Eordering__CASE_2E4(s(tyop_2EternaryComparisons_2Eordering,c_2EternaryComparisons_2Elist__compare_2E3(s(tyop_2Emin_2Efun(tyop_2Equote_2Eindex,tyop_2Emin_2Efun(tyop_2Equote_2Eindex,tyop_2EternaryComparisons_2Eordering)),c_2Equote_2Eindex__compare_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V4l1_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V3l2_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__monom_2E3(s(A_27a,V6c1_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V4l1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EringNorm_2Er__canonical__sum__merge_2E3(s(tyop_2Ering_2Ering(A_27a),V0r_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V2t1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__monom_2E3(s(A_27a,V5c2_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V3l2_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V1t2_2E0))))))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__monom_2E3(s(A_27a,c_2Ering_2Ering__RP_2E3(s(tyop_2Ering_2Ering(A_27a),V0r_2E0),s(A_27a,V6c1_2E0),s(A_27a,V5c2_2E0))),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V4l1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EringNorm_2Er__canonical__sum__merge_2E3(s(tyop_2Ering_2Ering(A_27a),V0r_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V2t1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V1t2_2E0))))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__monom_2E3(s(A_27a,V5c2_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V3l2_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EringNorm_2Er__canonical__sum__merge_2E3(s(tyop_2Ering_2Ering(A_27a),V0r_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__monom_2E3(s(A_27a,V6c1_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V4l1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V2t1_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V1t2_2E0)))))))
      & ! [V7t2_2E0,V8t1_2E0,V9l2_2E0,V10l1_2E0,V11c1_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EringNorm_2Er__canonical__sum__merge_2E3(s(tyop_2Ering_2Ering(A_27a),V0r_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__monom_2E3(s(A_27a,V11c1_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V10l1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V8t1_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__varlist_2E2(s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V9l2_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V7t2_2E0))))) = s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EternaryComparisons_2Eordering__CASE_2E4(s(tyop_2EternaryComparisons_2Eordering,c_2EternaryComparisons_2Elist__compare_2E3(s(tyop_2Emin_2Efun(tyop_2Equote_2Eindex,tyop_2Emin_2Efun(tyop_2Equote_2Eindex,tyop_2EternaryComparisons_2Eordering)),c_2Equote_2Eindex__compare_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V10l1_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V9l2_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__monom_2E3(s(A_27a,V11c1_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V10l1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EringNorm_2Er__canonical__sum__merge_2E3(s(tyop_2Ering_2Ering(A_27a),V0r_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V8t1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__varlist_2E2(s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V9l2_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V7t2_2E0))))))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__monom_2E3(s(A_27a,c_2Ering_2Ering__RP_2E3(s(tyop_2Ering_2Ering(A_27a),V0r_2E0),s(A_27a,V11c1_2E0),s(A_27a,c_2Ering_2Ering__R1_2E1(s(tyop_2Ering_2Ering(A_27a),V0r_2E0))))),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V10l1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EringNorm_2Er__canonical__sum__merge_2E3(s(tyop_2Ering_2Ering(A_27a),V0r_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V8t1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V7t2_2E0))))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__varlist_2E2(s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V9l2_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EringNorm_2Er__canonical__sum__merge_2E3(s(tyop_2Ering_2Ering(A_27a),V0r_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__monom_2E3(s(A_27a,V11c1_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V10l1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V8t1_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V7t2_2E0)))))))
      & ! [V12t2_2E0,V13t1_2E0,V14l2_2E0,V15l1_2E0,V16c2_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EringNorm_2Er__canonical__sum__merge_2E3(s(tyop_2Ering_2Ering(A_27a),V0r_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__varlist_2E2(s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V15l1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V13t1_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__monom_2E3(s(A_27a,V16c2_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V14l2_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V12t2_2E0))))) = s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EternaryComparisons_2Eordering__CASE_2E4(s(tyop_2EternaryComparisons_2Eordering,c_2EternaryComparisons_2Elist__compare_2E3(s(tyop_2Emin_2Efun(tyop_2Equote_2Eindex,tyop_2Emin_2Efun(tyop_2Equote_2Eindex,tyop_2EternaryComparisons_2Eordering)),c_2Equote_2Eindex__compare_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V15l1_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V14l2_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__varlist_2E2(s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V15l1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EringNorm_2Er__canonical__sum__merge_2E3(s(tyop_2Ering_2Ering(A_27a),V0r_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V13t1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__monom_2E3(s(A_27a,V16c2_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V14l2_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V12t2_2E0))))))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__monom_2E3(s(A_27a,c_2Ering_2Ering__RP_2E3(s(tyop_2Ering_2Ering(A_27a),V0r_2E0),s(A_27a,c_2Ering_2Ering__R1_2E1(s(tyop_2Ering_2Ering(A_27a),V0r_2E0))),s(A_27a,V16c2_2E0))),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V15l1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EringNorm_2Er__canonical__sum__merge_2E3(s(tyop_2Ering_2Ering(A_27a),V0r_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V13t1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V12t2_2E0))))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__monom_2E3(s(A_27a,V16c2_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V14l2_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EringNorm_2Er__canonical__sum__merge_2E3(s(tyop_2Ering_2Ering(A_27a),V0r_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__varlist_2E2(s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V15l1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V13t1_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V12t2_2E0)))))))
      & ! [V17t2_2E0,V18t1_2E0,V19l2_2E0,V20l1_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EringNorm_2Er__canonical__sum__merge_2E3(s(tyop_2Ering_2Ering(A_27a),V0r_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__varlist_2E2(s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V20l1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V18t1_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__varlist_2E2(s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V19l2_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V17t2_2E0))))) = s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EternaryComparisons_2Eordering__CASE_2E4(s(tyop_2EternaryComparisons_2Eordering,c_2EternaryComparisons_2Elist__compare_2E3(s(tyop_2Emin_2Efun(tyop_2Equote_2Eindex,tyop_2Emin_2Efun(tyop_2Equote_2Eindex,tyop_2EternaryComparisons_2Eordering)),c_2Equote_2Eindex__compare_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V20l1_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V19l2_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__varlist_2E2(s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V20l1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EringNorm_2Er__canonical__sum__merge_2E3(s(tyop_2Ering_2Ering(A_27a),V0r_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V18t1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__varlist_2E2(s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V19l2_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V17t2_2E0))))))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__monom_2E3(s(A_27a,c_2Ering_2Ering__RP_2E3(s(tyop_2Ering_2Ering(A_27a),V0r_2E0),s(A_27a,c_2Ering_2Ering__R1_2E1(s(tyop_2Ering_2Ering(A_27a),V0r_2E0))),s(A_27a,c_2Ering_2Ering__R1_2E1(s(tyop_2Ering_2Ering(A_27a),V0r_2E0))))),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V20l1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EringNorm_2Er__canonical__sum__merge_2E3(s(tyop_2Ering_2Ering(A_27a),V0r_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V18t1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V17t2_2E0))))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__varlist_2E2(s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V19l2_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EringNorm_2Er__canonical__sum__merge_2E3(s(tyop_2Ering_2Ering(A_27a),V0r_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__varlist_2E2(s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V20l1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V18t1_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V17t2_2E0)))))))
      & ! [V21s1_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EringNorm_2Er__canonical__sum__merge_2E3(s(tyop_2Ering_2Ering(A_27a),V0r_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V21s1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ENil__monom_2E0))) = s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V21s1_2E0)
      & ! [V22v6_2E0,V23v5_2E0,V24v4_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EringNorm_2Er__canonical__sum__merge_2E3(s(tyop_2Ering_2Ering(A_27a),V0r_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ENil__monom_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__monom_2E3(s(A_27a,V24v4_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V23v5_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V22v6_2E0))))) = s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__monom_2E3(s(A_27a,V24v4_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V23v5_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V22v6_2E0)))
      & ! [V25v8_2E0,V26v7_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EringNorm_2Er__canonical__sum__merge_2E3(s(tyop_2Ering_2Ering(A_27a),V0r_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ENil__monom_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__varlist_2E2(s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V26v7_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V25v8_2E0))))) = s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__varlist_2E2(s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V26v7_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V25v8_2E0))) ) ).

fof(thm_2EringNorm_2Emonom__insert__def,axiom,
    ! [A_27a,V0r_2E0] :
      ( ! [V1t2_2E0,V2l2_2E0,V3l1_2E0,V4c2_2E0,V5c1_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EringNorm_2Er__monom__insert_2E4(s(tyop_2Ering_2Ering(A_27a),V0r_2E0),s(A_27a,V5c1_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V3l1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__monom_2E3(s(A_27a,V4c2_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V2l2_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V1t2_2E0))))) = s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EternaryComparisons_2Eordering__CASE_2E4(s(tyop_2EternaryComparisons_2Eordering,c_2EternaryComparisons_2Elist__compare_2E3(s(tyop_2Emin_2Efun(tyop_2Equote_2Eindex,tyop_2Emin_2Efun(tyop_2Equote_2Eindex,tyop_2EternaryComparisons_2Eordering)),c_2Equote_2Eindex__compare_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V3l1_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V2l2_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__monom_2E3(s(A_27a,V5c1_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V3l1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__monom_2E3(s(A_27a,V4c2_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V2l2_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V1t2_2E0))))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__monom_2E3(s(A_27a,c_2Ering_2Ering__RP_2E3(s(tyop_2Ering_2Ering(A_27a),V0r_2E0),s(A_27a,V5c1_2E0),s(A_27a,V4c2_2E0))),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V3l1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V1t2_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__monom_2E3(s(A_27a,V4c2_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V2l2_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EringNorm_2Er__monom__insert_2E4(s(tyop_2Ering_2Ering(A_27a),V0r_2E0),s(A_27a,V5c1_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V3l1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V1t2_2E0)))))))
      & ! [V6t2_2E0,V7l2_2E0,V8l1_2E0,V9c1_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EringNorm_2Er__monom__insert_2E4(s(tyop_2Ering_2Ering(A_27a),V0r_2E0),s(A_27a,V9c1_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V8l1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__varlist_2E2(s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V7l2_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V6t2_2E0))))) = s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EternaryComparisons_2Eordering__CASE_2E4(s(tyop_2EternaryComparisons_2Eordering,c_2EternaryComparisons_2Elist__compare_2E3(s(tyop_2Emin_2Efun(tyop_2Equote_2Eindex,tyop_2Emin_2Efun(tyop_2Equote_2Eindex,tyop_2EternaryComparisons_2Eordering)),c_2Equote_2Eindex__compare_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V8l1_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V7l2_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__monom_2E3(s(A_27a,V9c1_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V8l1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__varlist_2E2(s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V7l2_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V6t2_2E0))))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__monom_2E3(s(A_27a,c_2Ering_2Ering__RP_2E3(s(tyop_2Ering_2Ering(A_27a),V0r_2E0),s(A_27a,V9c1_2E0),s(A_27a,c_2Ering_2Ering__R1_2E1(s(tyop_2Ering_2Ering(A_27a),V0r_2E0))))),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V8l1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V6t2_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__varlist_2E2(s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V7l2_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EringNorm_2Er__monom__insert_2E4(s(tyop_2Ering_2Ering(A_27a),V0r_2E0),s(A_27a,V9c1_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V8l1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V6t2_2E0)))))))
      & ! [V10l1_2E0,V11c1_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EringNorm_2Er__monom__insert_2E4(s(tyop_2Ering_2Ering(A_27a),V0r_2E0),s(A_27a,V11c1_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V10l1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ENil__monom_2E0))) = s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__monom_2E3(s(A_27a,V11c1_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V10l1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ENil__monom_2E0))) ) ).

fof(thm_2EringNorm_2Evarlist__insert__def,axiom,
    ! [A_27a,V0r_2E0] :
      ( ! [V1t2_2E0,V2l2_2E0,V3l1_2E0,V4c2_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EringNorm_2Er__varlist__insert_2E3(s(tyop_2Ering_2Ering(A_27a),V0r_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V3l1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__monom_2E3(s(A_27a,V4c2_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V2l2_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V1t2_2E0))))) = s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EternaryComparisons_2Eordering__CASE_2E4(s(tyop_2EternaryComparisons_2Eordering,c_2EternaryComparisons_2Elist__compare_2E3(s(tyop_2Emin_2Efun(tyop_2Equote_2Eindex,tyop_2Emin_2Efun(tyop_2Equote_2Eindex,tyop_2EternaryComparisons_2Eordering)),c_2Equote_2Eindex__compare_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V3l1_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V2l2_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__varlist_2E2(s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V3l1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__monom_2E3(s(A_27a,V4c2_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V2l2_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V1t2_2E0))))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__monom_2E3(s(A_27a,c_2Ering_2Ering__RP_2E3(s(tyop_2Ering_2Ering(A_27a),V0r_2E0),s(A_27a,c_2Ering_2Ering__R1_2E1(s(tyop_2Ering_2Ering(A_27a),V0r_2E0))),s(A_27a,V4c2_2E0))),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V3l1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V1t2_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__monom_2E3(s(A_27a,V4c2_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V2l2_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EringNorm_2Er__varlist__insert_2E3(s(tyop_2Ering_2Ering(A_27a),V0r_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V3l1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V1t2_2E0)))))))
      & ! [V5t2_2E0,V6l2_2E0,V7l1_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EringNorm_2Er__varlist__insert_2E3(s(tyop_2Ering_2Ering(A_27a),V0r_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V7l1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__varlist_2E2(s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V6l2_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V5t2_2E0))))) = s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EternaryComparisons_2Eordering__CASE_2E4(s(tyop_2EternaryComparisons_2Eordering,c_2EternaryComparisons_2Elist__compare_2E3(s(tyop_2Emin_2Efun(tyop_2Equote_2Eindex,tyop_2Emin_2Efun(tyop_2Equote_2Eindex,tyop_2EternaryComparisons_2Eordering)),c_2Equote_2Eindex__compare_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V7l1_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V6l2_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__varlist_2E2(s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V7l1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__varlist_2E2(s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V6l2_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V5t2_2E0))))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__monom_2E3(s(A_27a,c_2Ering_2Ering__RP_2E3(s(tyop_2Ering_2Ering(A_27a),V0r_2E0),s(A_27a,c_2Ering_2Ering__R1_2E1(s(tyop_2Ering_2Ering(A_27a),V0r_2E0))),s(A_27a,c_2Ering_2Ering__R1_2E1(s(tyop_2Ering_2Ering(A_27a),V0r_2E0))))),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V7l1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V5t2_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__varlist_2E2(s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V6l2_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EringNorm_2Er__varlist__insert_2E3(s(tyop_2Ering_2Ering(A_27a),V0r_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V7l1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V5t2_2E0)))))))
      & ! [V8l1_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EringNorm_2Er__varlist__insert_2E3(s(tyop_2Ering_2Ering(A_27a),V0r_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V8l1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ENil__monom_2E0))) = s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__varlist_2E2(s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V8l1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ENil__monom_2E0))) ) ).

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

fof(thm_2EringNorm_2Ecanonical__sum__scalar2__def,axiom,
    ! [A_27a,V0r_2E0] :
      ( ! [V1l0_2E0,V2c_2E0,V3l_2E0,V4t_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EringNorm_2Er__canonical__sum__scalar2_2E3(s(tyop_2Ering_2Ering(A_27a),V0r_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V1l0_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__monom_2E3(s(A_27a,V2c_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V3l_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V4t_2E0))))) = s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EringNorm_2Er__monom__insert_2E4(s(tyop_2Ering_2Ering(A_27a),V0r_2E0),s(A_27a,V2c_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),c_2EternaryComparisons_2Elist__merge_2E3(s(tyop_2Emin_2Efun(tyop_2Equote_2Eindex,tyop_2Emin_2Efun(tyop_2Equote_2Eindex,tyop_2Emin_2Ebool)),c_2Equote_2Eindex__lt_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V1l0_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V3l_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EringNorm_2Er__canonical__sum__scalar2_2E3(s(tyop_2Ering_2Ering(A_27a),V0r_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V1l0_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V4t_2E0)))))
      & ! [V5l0_2E0,V6l_2E0,V7t_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EringNorm_2Er__canonical__sum__scalar2_2E3(s(tyop_2Ering_2Ering(A_27a),V0r_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V5l0_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__varlist_2E2(s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V6l_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V7t_2E0))))) = s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EringNorm_2Er__varlist__insert_2E3(s(tyop_2Ering_2Ering(A_27a),V0r_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),c_2EternaryComparisons_2Elist__merge_2E3(s(tyop_2Emin_2Efun(tyop_2Equote_2Eindex,tyop_2Emin_2Efun(tyop_2Equote_2Eindex,tyop_2Emin_2Ebool)),c_2Equote_2Eindex__lt_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V5l0_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V6l_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EringNorm_2Er__canonical__sum__scalar2_2E3(s(tyop_2Ering_2Ering(A_27a),V0r_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V5l0_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V7t_2E0)))))
      & ! [V8l0_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EringNorm_2Er__canonical__sum__scalar2_2E3(s(tyop_2Ering_2Ering(A_27a),V0r_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V8l0_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ENil__monom_2E0))) = s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ENil__monom_2E0) ) ).

fof(thm_2EringNorm_2Ecanonical__sum__scalar3__def,axiom,
    ! [A_27a,V0r_2E0] :
      ( ! [V1c0_2E0,V2l0_2E0,V3c_2E0,V4l_2E0,V5t_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EringNorm_2Er__canonical__sum__scalar3_2E4(s(tyop_2Ering_2Ering(A_27a),V0r_2E0),s(A_27a,V1c0_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V2l0_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__monom_2E3(s(A_27a,V3c_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V4l_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V5t_2E0))))) = s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EringNorm_2Er__monom__insert_2E4(s(tyop_2Ering_2Ering(A_27a),V0r_2E0),s(A_27a,c_2Ering_2Ering__RM_2E3(s(tyop_2Ering_2Ering(A_27a),V0r_2E0),s(A_27a,V1c0_2E0),s(A_27a,V3c_2E0))),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),c_2EternaryComparisons_2Elist__merge_2E3(s(tyop_2Emin_2Efun(tyop_2Equote_2Eindex,tyop_2Emin_2Efun(tyop_2Equote_2Eindex,tyop_2Emin_2Ebool)),c_2Equote_2Eindex__lt_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V2l0_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V4l_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EringNorm_2Er__canonical__sum__scalar3_2E4(s(tyop_2Ering_2Ering(A_27a),V0r_2E0),s(A_27a,V1c0_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V2l0_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V5t_2E0)))))
      & ! [V6c0_2E0,V7l0_2E0,V8l_2E0,V9t_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EringNorm_2Er__canonical__sum__scalar3_2E4(s(tyop_2Ering_2Ering(A_27a),V0r_2E0),s(A_27a,V6c0_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V7l0_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__varlist_2E2(s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V8l_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V9t_2E0))))) = s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EringNorm_2Er__monom__insert_2E4(s(tyop_2Ering_2Ering(A_27a),V0r_2E0),s(A_27a,V6c0_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),c_2EternaryComparisons_2Elist__merge_2E3(s(tyop_2Emin_2Efun(tyop_2Equote_2Eindex,tyop_2Emin_2Efun(tyop_2Equote_2Eindex,tyop_2Emin_2Ebool)),c_2Equote_2Eindex__lt_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V7l0_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V8l_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EringNorm_2Er__canonical__sum__scalar3_2E4(s(tyop_2Ering_2Ering(A_27a),V0r_2E0),s(A_27a,V6c0_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V7l0_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V9t_2E0)))))
      & ! [V10c0_2E0,V11l0_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EringNorm_2Er__canonical__sum__scalar3_2E4(s(tyop_2Ering_2Ering(A_27a),V0r_2E0),s(A_27a,V10c0_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V11l0_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ENil__monom_2E0))) = s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ENil__monom_2E0) ) ).

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

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

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

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

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

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

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

fof(thm_2EringNorm_2Espolynom__normalize__def,axiom,
    ! [A_27a,V0r_2E0] :
      ( ! [V1i_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EringNorm_2Er__spolynom__normalize_2E2(s(tyop_2Ering_2Ering(A_27a),V0r_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),c_2Ecanonical_2ESPvar_2E1(s(tyop_2Equote_2Eindex,V1i_2E0))))) = s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__varlist_2E2(s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),c_2Elist_2ECONS_2E2(s(tyop_2Equote_2Eindex,V1i_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),c_2Elist_2ENIL_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ENil__monom_2E0)))
      & ! [V2c_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EringNorm_2Er__spolynom__normalize_2E2(s(tyop_2Ering_2Ering(A_27a),V0r_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),c_2Ecanonical_2ESPconst_2E1(s(A_27a,V2c_2E0))))) = s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__monom_2E3(s(A_27a,V2c_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),c_2Elist_2ENIL_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ENil__monom_2E0)))
      & ! [V3l_2E0,V4r_27_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EringNorm_2Er__spolynom__normalize_2E2(s(tyop_2Ering_2Ering(A_27a),V0r_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),c_2Ecanonical_2ESPplus_2E2(s(tyop_2Ecanonical_2Espolynom(A_27a),V3l_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),V4r_27_2E0))))) = s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EringNorm_2Er__canonical__sum__merge_2E3(s(tyop_2Ering_2Ering(A_27a),V0r_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EringNorm_2Er__spolynom__normalize_2E2(s(tyop_2Ering_2Ering(A_27a),V0r_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),V3l_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EringNorm_2Er__spolynom__normalize_2E2(s(tyop_2Ering_2Ering(A_27a),V0r_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),V4r_27_2E0)))))
      & ! [V5l_2E0,V6r_27_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EringNorm_2Er__spolynom__normalize_2E2(s(tyop_2Ering_2Ering(A_27a),V0r_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),c_2Ecanonical_2ESPmult_2E2(s(tyop_2Ecanonical_2Espolynom(A_27a),V5l_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),V6r_27_2E0))))) = s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EringNorm_2Er__canonical__sum__prod_2E3(s(tyop_2Ering_2Ering(A_27a),V0r_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EringNorm_2Er__spolynom__normalize_2E2(s(tyop_2Ering_2Ering(A_27a),V0r_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),V5l_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EringNorm_2Er__spolynom__normalize_2E2(s(tyop_2Ering_2Ering(A_27a),V0r_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),V6r_27_2E0))))) ) ).

fof(thm_2EringNorm_2Espolynom__simplify__def,axiom,
    ! [A_27a,V0r_2E0,V1x_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EringNorm_2Er__spolynom__simplify_2E2(s(tyop_2Ering_2Ering(A_27a),V0r_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),V1x_2E0))) = s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EringNorm_2Er__canonical__sum__simplify_2E2(s(tyop_2Ering_2Ering(A_27a),V0r_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EringNorm_2Er__spolynom__normalize_2E2(s(tyop_2Ering_2Ering(A_27a),V0r_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),V1x_2E0))))) ).

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

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

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

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

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

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

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

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

fof(thm_2EringNorm_2Epolynom__normalize__ok,axiom,
    ! [A_27a,V0r_2E0] :
      ( p(s(tyop_2Emin_2Ebool,c_2Ering_2Eis__ring_2E1(s(tyop_2Ering_2Ering(A_27a),V0r_2E0))))
     => ! [V1vm_2E0,V2p_2E0] : s(A_27a,c_2EringNorm_2Er__interp__cs_2E3(s(tyop_2Ering_2Ering(A_27a),V0r_2E0),s(tyop_2Equote_2Evarmap(A_27a),V1vm_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EringNorm_2Epolynom__normalize_2E2(s(tyop_2Ering_2Ering(A_27a),V0r_2E0),s(tyop_2EringNorm_2Epolynom(A_27a),V2p_2E0))))) = s(A_27a,c_2EringNorm_2Einterp__p_2E3(s(tyop_2Ering_2Ering(A_27a),V0r_2E0),s(tyop_2Equote_2Evarmap(A_27a),V1vm_2E0),s(tyop_2EringNorm_2Epolynom(A_27a),V2p_2E0))) ) ).

fof(thm_2EringNorm_2Epolynom__simplify__ok,axiom,
    ! [A_27a,V0r_2E0] :
      ( p(s(tyop_2Emin_2Ebool,c_2Ering_2Eis__ring_2E1(s(tyop_2Ering_2Ering(A_27a),V0r_2E0))))
     => ! [V1vm_2E0,V2p_2E0] : s(A_27a,c_2EringNorm_2Er__interp__cs_2E3(s(tyop_2Ering_2Ering(A_27a),V0r_2E0),s(tyop_2Equote_2Evarmap(A_27a),V1vm_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EringNorm_2Epolynom__simplify_2E2(s(tyop_2Ering_2Ering(A_27a),V0r_2E0),s(tyop_2EringNorm_2Epolynom(A_27a),V2p_2E0))))) = s(A_27a,c_2EringNorm_2Einterp__p_2E3(s(tyop_2Ering_2Ering(A_27a),V0r_2E0),s(tyop_2Equote_2Evarmap(A_27a),V1vm_2E0),s(tyop_2EringNorm_2Epolynom(A_27a),V2p_2E0))) ) ).

fof(reserved_2Eho_2Eeq__ext,axiom,
    ! [A_27a,A_27b,V0f_2E0,V1g_2E0] :
      ( ! [V2x_2E0] : s(A_27b,app_2E2(s(tyop_2Emin_2Efun(A_27a,A_27b),V0f_2E0),s(A_27a,V2x_2E0))) = s(A_27b,app_2E2(s(tyop_2Emin_2Efun(A_27a,A_27b),V1g_2E0),s(A_27a,V2x_2E0)))
     => s(tyop_2Emin_2Efun(A_27a,A_27b),V0f_2E0) = s(tyop_2Emin_2Efun(A_27a,A_27b),V1g_2E0) ) ).

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