ITP001 Axioms: ITP096+4.ax
%------------------------------------------------------------------------------
% File : ITP096+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 : canonical.ax [Gau19]
% : HL4096+4.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 129 ( 62 unt; 0 def)
% Number of atoms : 358 ( 247 equ)
% Maximal formula atoms : 32 ( 2 avg)
% Number of connectives : 241 ( 12 ~; 18 |; 111 &)
% ( 25 <=>; 75 =>; 0 <=; 0 <~>)
% Maximal formula depth : 25 ( 7 avg)
% Maximal term depth : 20 ( 3 avg)
% Number of predicates : 2 ( 1 usr; 0 prp; 1-2 aty)
% Number of functors : 140 ( 140 usr; 75 con; 0-5 aty)
% Number of variables : 842 ( 804 !; 38 ?)
% 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_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(arityeq1_2Ec_2Esemi__ring_2Eis__semi__ring_2E1_2Emono_2EA_27a,axiom,
! [A_27a,X0_2E0] : s(tyop_2Emin_2Ebool,c_2Esemi__ring_2Eis__semi__ring_2E1(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),X0_2E0))) = s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Ebool),c_2Esemi__ring_2Eis__semi__ring_2E0),s(tyop_2Esemi__ring_2Esemi__ring(A_27a),X0_2E0))) ).
fof(arityeq1_2Ec_2Esemi__ring_2Esemi__ring__SR0_2E1_2Emono_2EA_27a,axiom,
! [A_27a,X0_2E0] : s(A_27a,c_2Esemi__ring_2Esemi__ring__SR0_2E1(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),X0_2E0))) = s(A_27a,app_2E2(s(tyop_2Emin_2Efun(tyop_2Esemi__ring_2Esemi__ring(A_27a),A_27a),c_2Esemi__ring_2Esemi__ring__SR0_2E0),s(tyop_2Esemi__ring_2Esemi__ring(A_27a),X0_2E0))) ).
fof(arityeq1_2Ec_2Esemi__ring_2Esemi__ring__SR1_2E1_2Emono_2EA_27a,axiom,
! [A_27a,X0_2E0] : s(A_27a,c_2Esemi__ring_2Esemi__ring__SR1_2E1(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),X0_2E0))) = s(A_27a,app_2E2(s(tyop_2Emin_2Efun(tyop_2Esemi__ring_2Esemi__ring(A_27a),A_27a),c_2Esemi__ring_2Esemi__ring__SR1_2E0),s(tyop_2Esemi__ring_2Esemi__ring(A_27a),X0_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(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(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_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_2Ecanonical_2Ecanonical__sum__size_2E2_2Emono_2EA_27a,axiom,
! [A_27a,X0_2E0,X1_2E0] : s(tyop_2Enum_2Enum,c_2Ecanonical_2Ecanonical__sum__size_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),X0_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),X1_2E0))) = s(tyop_2Enum_2Enum,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Enum_2Enum),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Enum_2Enum)),c_2Ecanonical_2Ecanonical__sum__size_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),X0_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),X1_2E0))) ).
fof(arityeq2_2Ec_2Elist_2Elist__size_2E2_2Emono_2Etyop_2Equote_2Eindex,axiom,
! [X0_2E0,X1_2E0] : s(tyop_2Enum_2Enum,c_2Elist_2Elist__size_2E2(s(tyop_2Emin_2Efun(tyop_2Equote_2Eindex,tyop_2Enum_2Enum),X0_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),X1_2E0))) = s(tyop_2Enum_2Enum,app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Enum_2Enum),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Equote_2Eindex,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Enum_2Enum)),c_2Elist_2Elist__size_2E0),s(tyop_2Emin_2Efun(tyop_2Equote_2Eindex,tyop_2Enum_2Enum),X0_2E0))),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),X1_2E0))) ).
fof(arityeq2_2Ec_2Ecanonical_2Espolynom__size_2E2_2Emono_2EA_27a,axiom,
! [A_27a,X0_2E0,X1_2E0] : s(tyop_2Enum_2Enum,c_2Ecanonical_2Espolynom__size_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),X0_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),X1_2E0))) = s(tyop_2Enum_2Enum,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Enum_2Enum),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Enum_2Enum)),c_2Ecanonical_2Espolynom__size_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),X0_2E0))),s(tyop_2Ecanonical_2Espolynom(A_27a),X1_2E0))) ).
fof(arityeq2_2Ec_2Ecanonical_2Ecanonical__sum__simplify_2E2_2Emono_2EA_27a,axiom,
! [A_27a,X0_2E0,X1_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Ecanonical__sum__simplify_2E2(s(tyop_2Esemi__ring_2Esemi__ring(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_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))),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_28A_27a_2Ctyop_2Elist_2Elist_28tyop_2Equote_2Eindex_29_29_29_20mono_2Etyop_2Ecanonical_2Ecanonical__sum_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(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex))),tyop_2Emin_2Ebool),X0_2E0),s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex)))),X1_2E0))) = s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex)))),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex))),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex)))),tyop_2Emin_2Ebool)),c_2Ebool_2ETYPE__DEFINITION_2E0),s(tyop_2Emin_2Efun(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex))),tyop_2Emin_2Ebool),X0_2E0))),s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,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_2Ecanonical_2Espolynom_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_2Ecanonical_2Espolynom(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_2Ecanonical_2Espolynom(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_2Ecanonical_2Espolynom(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_2Ecanonical_2Espolynom(A_27a),tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a))),X1_2E0))) ).
fof(arityeq2_2Ec_2Ecanonical_2Espolynom__normalize_2E2_2Emono_2EA_27a,axiom,
! [A_27a,X0_2E0,X1_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Espolynom__normalize_2E2(s(tyop_2Esemi__ring_2Esemi__ring(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_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))),s(tyop_2Ecanonical_2Espolynom(A_27a),X1_2E0))) ).
fof(arityeq2_2Ec_2Ecanonical_2Espolynom__simplify_2E2_2Emono_2EA_27a,axiom,
! [A_27a,X0_2E0,X1_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Espolynom__simplify_2E2(s(tyop_2Esemi__ring_2Esemi__ring(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_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))),s(tyop_2Ecanonical_2Espolynom(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_2EA_27a_20mono_2Etyop_2Elist_2Elist_28tyop_2Equote_2Eindex_29,axiom,
! [A_27a,X0_2E0,X1_2E0] : s(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex)),c_2Epair_2E_2C_2E2(s(A_27a,X0_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),X1_2E0))) = s(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex))),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex)))),c_2Epair_2E_2C_2E0),s(A_27a,X0_2E0))),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),X1_2E0))) ).
fof(arityeq3_2Ec_2Eind__type_2ECONSTR_2E3_2Emono_2Etyop_2Epair_2Eprod_28A_27a_2Ctyop_2Elist_2Elist_28tyop_2Equote_2Eindex_29_29,axiom,
! [A_27a,X0_2E0,X1_2E0,X2_2E0] : s(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex))),c_2Eind__type_2ECONSTR_2E3(s(tyop_2Enum_2Enum,X0_2E0),s(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex)),X1_2E0),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex)))),X2_2E0))) = s(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex)))),tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex)))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex)))),tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex))))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex)))),tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex)))))),c_2Eind__type_2ECONSTR_2E0),s(tyop_2Enum_2Enum,X0_2E0))),s(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex)),X1_2E0))),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex)))),X2_2E0))) ).
fof(arityeq2_2Ec_2Eind__type_2EFCONS_2E2_2Emono_2Etyop_2Eind__type_2Erecspace_28tyop_2Epair_2Eprod_28A_27a_2Ctyop_2Elist_2Elist_28tyop_2Equote_2Eindex_29_29_29,axiom,
! [A_27a,X0_2E0,X1_2E0] : s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex)))),c_2Eind__type_2EFCONS_2E2(s(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex))),X0_2E0),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex)))),X1_2E0))) = s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex)))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex)))),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex))))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex))),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex)))),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex)))))),c_2Eind__type_2EFCONS_2E0),s(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex))),X0_2E0))),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex)))),X1_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_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(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_2Ecanonical_2Ecanonical__sum__scalar_2E3_2Emono_2EA_27a,axiom,
! [A_27a,X0_2E0,X1_2E0,X2_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Ecanonical__sum__scalar_2E3(s(tyop_2Esemi__ring_2Esemi__ring(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_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))),s(A_27a,X1_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),X2_2E0))) ).
fof(arityeq3_2Ec_2Esemi__ring_2Esemi__ring__SRM_2E3_2Emono_2EA_27a,axiom,
! [A_27a,X0_2E0,X1_2E0,X2_2E0] : s(A_27a,c_2Esemi__ring_2Esemi__ring__SRM_2E3(s(tyop_2Esemi__ring_2Esemi__ring(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_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a))),c_2Esemi__ring_2Esemi__ring__SRM_2E0),s(tyop_2Esemi__ring_2Esemi__ring(A_27a),X0_2E0))),s(A_27a,X1_2E0))),s(A_27a,X2_2E0))) ).
fof(arityeq3_2Ec_2Ecanonical_2Ecanonical__sum__merge_2E3_2Emono_2EA_27a,axiom,
! [A_27a,X0_2E0,X1_2E0,X2_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Ecanonical__sum__merge_2E3(s(tyop_2Esemi__ring_2Esemi__ring(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_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))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),X1_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),X2_2E0))) ).
fof(arityeq3_2Ec_2Ecanonical_2Ecanonical__sum__prod_2E3_2Emono_2EA_27a,axiom,
! [A_27a,X0_2E0,X1_2E0,X2_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Ecanonical__sum__prod_2E3(s(tyop_2Esemi__ring_2Esemi__ring(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_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))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),X1_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),X2_2E0))) ).
fof(arityeq4_2Ec_2Ecanonical_2Ecanonical__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_2Ecanonical_2Ecanonical__sum__scalar3_2E4(s(tyop_2Esemi__ring_2Esemi__ring(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_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))),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_2Ecanonical_2Ecanonical__sum__scalar2_2E3_2Emono_2EA_27a,axiom,
! [A_27a,X0_2E0,X1_2E0,X2_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Ecanonical__sum__scalar2_2E3(s(tyop_2Esemi__ring_2Esemi__ring(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_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))),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),X1_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),X2_2E0))) ).
fof(arityeq3_2Ec_2Ecanonical_2Einterp__cs_2E3_2Emono_2EA_27a,axiom,
! [A_27a,X0_2E0,X1_2E0,X2_2E0] : s(A_27a,c_2Ecanonical_2Einterp__cs_2E3(s(tyop_2Esemi__ring_2Esemi__ring(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_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))),s(tyop_2Equote_2Evarmap(A_27a),X1_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),X2_2E0))) ).
fof(arityeq3_2Ec_2Ecanonical_2Evarlist__insert_2E3_2Emono_2EA_27a,axiom,
! [A_27a,X0_2E0,X1_2E0,X2_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Evarlist__insert_2E3(s(tyop_2Esemi__ring_2Esemi__ring(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_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))),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),X1_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),X2_2E0))) ).
fof(arityeq4_2Ec_2Ecanonical_2Emonom__insert_2E4_2Emono_2EA_27a,axiom,
! [A_27a,X0_2E0,X1_2E0,X2_2E0,X3_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Emonom__insert_2E4(s(tyop_2Esemi__ring_2Esemi__ring(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_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))),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_2Ecanonical_2Einterp__sp_2E3_2Emono_2EA_27a,axiom,
! [A_27a,X0_2E0,X1_2E0,X2_2E0] : s(A_27a,c_2Ecanonical_2Einterp__sp_2E3(s(tyop_2Esemi__ring_2Esemi__ring(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_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))),s(tyop_2Equote_2Evarmap(A_27a),X1_2E0))),s(tyop_2Ecanonical_2Espolynom(A_27a),X2_2E0))) ).
fof(arityeq3_2Ec_2Ecanonical_2Einterp__vl_2E3_2Emono_2EA_27a,axiom,
! [A_27a,X0_2E0,X1_2E0,X2_2E0] : s(A_27a,c_2Ecanonical_2Einterp__vl_2E3(s(tyop_2Esemi__ring_2Esemi__ring(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_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))),s(tyop_2Equote_2Evarmap(A_27a),X1_2E0))),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),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(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_2Ecanonical_2Eivl__aux_2E4_2Emono_2EA_27a,axiom,
! [A_27a,X0_2E0,X1_2E0,X2_2E0,X3_2E0] : s(A_27a,c_2Ecanonical_2Eivl__aux_2E4(s(tyop_2Esemi__ring_2Esemi__ring(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_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))),s(tyop_2Equote_2Evarmap(A_27a),X1_2E0))),s(tyop_2Equote_2Eindex,X2_2E0))),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),X3_2E0))) ).
fof(arityeq3_2Ec_2Esemi__ring_2Esemi__ring__SRP_2E3_2Emono_2EA_27a,axiom,
! [A_27a,X0_2E0,X1_2E0,X2_2E0] : s(A_27a,c_2Esemi__ring_2Esemi__ring__SRP_2E3(s(tyop_2Esemi__ring_2Esemi__ring(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_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a))),c_2Esemi__ring_2Esemi__ring__SRP_2E0),s(tyop_2Esemi__ring_2Esemi__ring(A_27a),X0_2E0))),s(A_27a,X1_2E0))),s(A_27a,X2_2E0))) ).
fof(arityeq4_2Ec_2Ecanonical_2Eics__aux_2E4_2Emono_2EA_27a,axiom,
! [A_27a,X0_2E0,X1_2E0,X2_2E0,X3_2E0] : s(A_27a,c_2Ecanonical_2Eics__aux_2E4(s(tyop_2Esemi__ring_2Esemi__ring(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_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))),s(tyop_2Equote_2Evarmap(A_27a),X1_2E0))),s(A_27a,X2_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),X3_2E0))) ).
fof(arityeq4_2Ec_2Ecanonical_2Einterp__m_2E4_2Emono_2EA_27a,axiom,
! [A_27a,X0_2E0,X1_2E0,X2_2E0,X3_2E0] : s(A_27a,c_2Ecanonical_2Einterp__m_2E4(s(tyop_2Esemi__ring_2Esemi__ring(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_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))),s(tyop_2Equote_2Evarmap(A_27a),X1_2E0))),s(A_27a,X2_2E0))),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),X3_2E0))) ).
fof(arityeq4_2Ec_2Ecanonical_2Ecanonical__sum__CASE_2E4_2Emono_2EA_27a_20mono_2EA_27b,axiom,
! [A_27a,A_27b,X0_2E0,X1_2E0,X2_2E0,X3_2E0] : s(A_27b,c_2Ecanonical_2Ecanonical__sum__CASE_2E4(s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),X0_2E0),s(A_27b,X1_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),A_27b))),X2_2E0),s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27b)),X3_2E0))) = s(A_27b,app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27b)),A_27b),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27b))),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27b)),A_27b)),app_2E2(s(tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27b))),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27b)),A_27b))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27b))),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27b)),A_27b)))),c_2Ecanonical_2Ecanonical__sum__CASE_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),X0_2E0))),s(A_27b,X1_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),A_27b))),X2_2E0))),s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27b)),X3_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(arityeq5_2Ec_2Ecanonical_2Espolynom__CASE_2E5_2Emono_2EA_27a_20mono_2EA_27b,axiom,
! [A_27a,A_27b,X0_2E0,X1_2E0,X2_2E0,X3_2E0,X4_2E0] : s(A_27b,c_2Ecanonical_2Espolynom__CASE_2E5(s(tyop_2Ecanonical_2Espolynom(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_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27b)),X3_2E0),s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27b)),X4_2E0))) = s(A_27b,app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27b)),A_27b),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27b)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(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_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27b)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(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_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27b)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27b)),A_27b)))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(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_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27b)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27b)),A_27b))))),c_2Ecanonical_2Espolynom__CASE_2E0),s(tyop_2Ecanonical_2Espolynom(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_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27b)),X3_2E0))),s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27b)),X4_2E0))) ).
fof(thm_2Ecanonical_2Ecanonical__sum__TY__DEF,axiom,
! [A_27a,F3_2E0,F2_2E0,F1_2E0,F0_2E0] :
( ! [V11n_2E0] : s(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex)))),F3_2E0),s(tyop_2Enum_2Enum,V11n_2E0))) = s(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex))),c_2Eind__type_2EBOTTOM_2E0)
=> ( ! [V8n_2E0] : s(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex)))),F2_2E0),s(tyop_2Enum_2Enum,V8n_2E0))) = s(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex))),c_2Eind__type_2EBOTTOM_2E0)
=> ( ! [V4n_2E0] : s(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex)))),F1_2E0),s(tyop_2Enum_2Enum,V4n_2E0))) = s(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex))),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(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex))),tyop_2Emin_2Ebool),F0_2E0),s(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex))),V1a0_27_2E0))))
<=> ! [V2_27canonical__sum_27_2E0] :
( ! [V3a0_27_2E0] :
( ( s(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex))),V3a0_27_2E0) = s(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex))),c_2Eind__type_2ECONSTR_2E3(s(tyop_2Enum_2Enum,c_2Enum_2E0_2E0),s(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex)),c_2Epair_2E_2C_2E2(s(A_27a,c_2Ebool_2EARB_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),c_2Ebool_2EARB_2E0))),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex)))),F1_2E0)))
| ? [V5a0_2E0,V6a1_2E0,V7a2_2E0] :
( s(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex))),V3a0_27_2E0) = s(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex))),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(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex)),c_2Epair_2E_2C_2E2(s(A_27a,V5a0_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V6a1_2E0))),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex)))),c_2Eind__type_2EFCONS_2E2(s(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex))),V7a2_2E0),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex)))),F2_2E0)))))
& p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex))),tyop_2Emin_2Ebool),V2_27canonical__sum_27_2E0),s(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex))),V7a2_2E0)))) )
| ? [V9a0_2E0,V10a1_2E0] :
( s(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex))),V3a0_27_2E0) = s(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex))),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(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex)),c_2Epair_2E_2C_2E2(s(A_27a,c_2Ebool_2EARB_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V9a0_2E0))),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex)))),c_2Eind__type_2EFCONS_2E2(s(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex))),V10a1_2E0),s(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex)))),F3_2E0)))))
& p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex))),tyop_2Emin_2Ebool),V2_27canonical__sum_27_2E0),s(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex))),V10a1_2E0)))) ) )
=> p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex))),tyop_2Emin_2Ebool),V2_27canonical__sum_27_2E0),s(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex))),V3a0_27_2E0)))) )
=> p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex))),tyop_2Emin_2Ebool),V2_27canonical__sum_27_2E0),s(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex))),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(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex))),tyop_2Emin_2Ebool),F0_2E0),s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex)))),V0rep_2E0)))) ) ) ) ) ).
fof(thm_2Ecanonical_2Ecanonical__sum__case__def,axiom,
! [A_27a,A_27b] :
( ! [V0v_2E0,V1f_2E0,V2f1_2E0] : s(A_27b,c_2Ecanonical_2Ecanonical__sum__CASE_2E4(s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ENil__monom_2E0),s(A_27b,V0v_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),A_27b))),V1f_2E0),s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27b)),V2f1_2E0))) = s(A_27b,V0v_2E0)
& ! [V3a0_2E0,V4a1_2E0,V5a2_2E0,V6v_2E0,V7f_2E0,V8f1_2E0] : s(A_27b,c_2Ecanonical_2Ecanonical__sum__CASE_2E4(s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__monom_2E3(s(A_27a,V3a0_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V4a1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V5a2_2E0))),s(A_27b,V6v_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),A_27b))),V7f_2E0),s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27b)),V8f1_2E0))) = s(A_27b,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27b),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27b)),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),A_27b))),V7f_2E0),s(A_27a,V3a0_2E0))),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V4a1_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V5a2_2E0)))
& ! [V9a0_2E0,V10a1_2E0,V11v_2E0,V12f_2E0,V13f1_2E0] : s(A_27b,c_2Ecanonical_2Ecanonical__sum__CASE_2E4(s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__varlist_2E2(s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V9a0_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V10a1_2E0))),s(A_27b,V11v_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),A_27b))),V12f_2E0),s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27b)),V13f1_2E0))) = s(A_27b,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27b),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27b)),V13f1_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V9a0_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V10a1_2E0))) ) ).
fof(thm_2Ecanonical_2Ecanonical__sum__size__def,axiom,
! [A_27a] :
( ! [V0f_2E0] : s(tyop_2Enum_2Enum,c_2Ecanonical_2Ecanonical__sum__size_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V0f_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ENil__monom_2E0))) = s(tyop_2Enum_2Enum,c_2Enum_2E0_2E0)
& ! [V1f_2E0,V2a0_2E0,V3a1_2E0,V4a2_2E0] : s(tyop_2Enum_2Enum,c_2Ecanonical_2Ecanonical__sum__size_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V1f_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__monom_2E3(s(A_27a,V2a0_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V3a1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V4a2_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,app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V1f_2E0),s(A_27a,V2a0_2E0))),s(tyop_2Enum_2Enum,c_2Earithmetic_2E_2B_2E2(s(tyop_2Enum_2Enum,c_2Elist_2Elist__size_2E2(s(tyop_2Emin_2Efun(tyop_2Equote_2Eindex,tyop_2Enum_2Enum),c_2Equote_2Eindex__size_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V3a1_2E0))),s(tyop_2Enum_2Enum,c_2Ecanonical_2Ecanonical__sum__size_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V1f_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V4a2_2E0)))))))))
& ! [V5f_2E0,V6a0_2E0,V7a1_2E0] : s(tyop_2Enum_2Enum,c_2Ecanonical_2Ecanonical__sum__size_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V5f_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__varlist_2E2(s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V6a0_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V7a1_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_2Elist_2Elist__size_2E2(s(tyop_2Emin_2Efun(tyop_2Equote_2Eindex,tyop_2Enum_2Enum),c_2Equote_2Eindex__size_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V6a0_2E0))),s(tyop_2Enum_2Enum,c_2Ecanonical_2Ecanonical__sum__size_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V5f_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V7a1_2E0))))))) ) ).
fof(thm_2Ecanonical_2Ecanonical__sum__scalar__def,axiom,
! [A_27a] :
( ! [V0sr_2E0,V1c0_2E0,V2c_2E0,V3l_2E0,V4t_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Ecanonical__sum__scalar_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_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_2Esemi__ring_2Esemi__ring__SRM_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_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_2Ecanonical_2Ecanonical__sum__scalar_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0),s(A_27a,V1c0_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V4t_2E0)))))
& ! [V5sr_2E0,V6c0_2E0,V7l_2E0,V8t_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Ecanonical__sum__scalar_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V5sr_2E0),s(A_27a,V6c0_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__varlist_2E2(s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V7l_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V8t_2E0))))) = s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__monom_2E3(s(A_27a,V6c0_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V7l_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Ecanonical__sum__scalar_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V5sr_2E0),s(A_27a,V6c0_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V8t_2E0)))))
& ! [V9sr_2E0,V10c0_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Ecanonical__sum__scalar_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V9sr_2E0),s(A_27a,V10c0_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_2Ecanonical_2Ecanonical__sum__scalar2__def,axiom,
! [A_27a] :
( ! [V0sr_2E0,V1l0_2E0,V2c_2E0,V3l_2E0,V4t_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Ecanonical__sum__scalar2_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_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_2Ecanonical_2Emonom__insert_2E4(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_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_2Ecanonical_2Ecanonical__sum__scalar2_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V1l0_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V4t_2E0)))))
& ! [V5sr_2E0,V6l0_2E0,V7l_2E0,V8t_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Ecanonical__sum__scalar2_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V5sr_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V6l0_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__varlist_2E2(s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V7l_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V8t_2E0))))) = s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Evarlist__insert_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V5sr_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),V6l0_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V7l_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Ecanonical__sum__scalar2_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V5sr_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V6l0_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V8t_2E0)))))
& ! [V9sr_2E0,V10l0_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Ecanonical__sum__scalar2_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V9sr_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V10l0_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_2Ecanonical_2Ecanonical__sum__scalar3__def,axiom,
! [A_27a] :
( ! [V0sr_2E0,V1c0_2E0,V2l0_2E0,V3c_2E0,V4l_2E0,V5t_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Ecanonical__sum__scalar3_2E4(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_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_2Ecanonical_2Emonom__insert_2E4(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0),s(A_27a,c_2Esemi__ring_2Esemi__ring__SRM_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_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_2Ecanonical_2Ecanonical__sum__scalar3_2E4(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0),s(A_27a,V1c0_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V2l0_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V5t_2E0)))))
& ! [V6sr_2E0,V7c0_2E0,V8l0_2E0,V9l_2E0,V10t_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Ecanonical__sum__scalar3_2E4(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V6sr_2E0),s(A_27a,V7c0_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V8l0_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__varlist_2E2(s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V9l_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V10t_2E0))))) = s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Emonom__insert_2E4(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V6sr_2E0),s(A_27a,V7c0_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),V8l0_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V9l_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Ecanonical__sum__scalar3_2E4(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V6sr_2E0),s(A_27a,V7c0_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V8l0_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V10t_2E0)))))
& ! [V11sr_2E0,V12c0_2E0,V13l0_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Ecanonical__sum__scalar3_2E4(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V11sr_2E0),s(A_27a,V12c0_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V13l0_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_2Ecanonical_2Ecanonical__sum__prod__def,axiom,
! [A_27a] :
( ! [V0sr_2E0,V1c1_2E0,V2l1_2E0,V3t1_2E0,V4s2_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Ecanonical__sum__prod_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_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_2Ecanonical_2Ecanonical__sum__merge_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Ecanonical__sum__scalar3_2E4(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_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_2Ecanonical_2Ecanonical__sum__prod_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V3t1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V4s2_2E0)))))
& ! [V5sr_2E0,V6l1_2E0,V7t1_2E0,V8s2_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Ecanonical__sum__prod_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V5sr_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__varlist_2E2(s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V6l1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V7t1_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V8s2_2E0))) = s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Ecanonical__sum__merge_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V5sr_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Ecanonical__sum__scalar2_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V5sr_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V6l1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V8s2_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Ecanonical__sum__prod_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V5sr_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V7t1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V8s2_2E0)))))
& ! [V9sr_2E0,V10s2_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Ecanonical__sum__prod_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V9sr_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ENil__monom_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V10s2_2E0))) = s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ENil__monom_2E0) ) ).
fof(thm_2Ecanonical_2Ecanonical__sum__simplify__def,axiom,
! [A_27a,F1_2E0,F0_2E0] :
( ! [V1c_2E0,V0sr_2E0] :
( p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Ebool)),F1_2E0),s(A_27a,V1c_2E0))),s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0))))
<=> s(A_27a,V1c_2E0) = s(A_27a,c_2Esemi__ring_2Esemi__ring__SR1_2E1(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0))) )
=> ( ! [V1c_2E0,V0sr_2E0] :
( p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Ebool)),F0_2E0),s(A_27a,V1c_2E0))),s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0))))
<=> s(A_27a,V1c_2E0) = s(A_27a,c_2Esemi__ring_2Esemi__ring__SR0_2E1(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0))) )
=> ( ! [V0sr_2E0,V1c_2E0,V2l_2E0,V3t_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Ecanonical__sum__simplify_2E2(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_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_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Ebool)),F0_2E0),s(A_27a,V1c_2E0))),s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Ecanonical__sum__simplify_2E2(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_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_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Ebool)),F1_2E0),s(A_27a,V1c_2E0))),s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_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_2Ecanonical_2Ecanonical__sum__simplify_2E2(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_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_2Ecanonical_2Ecanonical__sum__simplify_2E2(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V3t_2E0)))))))))
& ! [V4sr_2E0,V5l_2E0,V6t_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Ecanonical__sum__simplify_2E2(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V4sr_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(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),c_2Ecanonical_2Ecanonical__sum__simplify_2E2(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V4sr_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V6t_2E0)))))
& ! [V7sr_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Ecanonical__sum__simplify_2E2(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V7sr_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_2Ecanonical_2Eivl__aux__def,axiom,
! [A_27a] :
( ! [V0sr_2E0,V1vm_2E0,V2x_2E0] : s(A_27a,c_2Ecanonical_2Eivl__aux_2E4(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_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)))
& ! [V3sr_2E0,V4vm_2E0,V5x_2E0,V6x_27_2E0,V7t_27_2E0] : s(A_27a,c_2Ecanonical_2Eivl__aux_2E4(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V3sr_2E0),s(tyop_2Equote_2Evarmap(A_27a),V4vm_2E0),s(tyop_2Equote_2Eindex,V5x_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),c_2Elist_2ECONS_2E2(s(tyop_2Equote_2Eindex,V6x_27_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V7t_27_2E0))))) = s(A_27a,c_2Esemi__ring_2Esemi__ring__SRM_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V3sr_2E0),s(A_27a,c_2Equote_2Evarmap__find_2E2(s(tyop_2Equote_2Eindex,V5x_2E0),s(tyop_2Equote_2Evarmap(A_27a),V4vm_2E0))),s(A_27a,c_2Ecanonical_2Eivl__aux_2E4(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V3sr_2E0),s(tyop_2Equote_2Evarmap(A_27a),V4vm_2E0),s(tyop_2Equote_2Eindex,V6x_27_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V7t_27_2E0))))) ) ).
fof(thm_2Ecanonical_2Einterp__vl__def,axiom,
! [A_27a] :
( ! [V0sr_2E0,V1vm_2E0] : s(A_27a,c_2Ecanonical_2Einterp__vl_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0),s(tyop_2Equote_2Evarmap(A_27a),V1vm_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),c_2Elist_2ENIL_2E0))) = s(A_27a,c_2Esemi__ring_2Esemi__ring__SR1_2E1(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0)))
& ! [V2sr_2E0,V3vm_2E0,V4x_2E0,V5t_2E0] : s(A_27a,c_2Ecanonical_2Einterp__vl_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V2sr_2E0),s(tyop_2Equote_2Evarmap(A_27a),V3vm_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),c_2Elist_2ECONS_2E2(s(tyop_2Equote_2Eindex,V4x_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V5t_2E0))))) = s(A_27a,c_2Ecanonical_2Eivl__aux_2E4(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V2sr_2E0),s(tyop_2Equote_2Evarmap(A_27a),V3vm_2E0),s(tyop_2Equote_2Eindex,V4x_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V5t_2E0))) ) ).
fof(thm_2Ecanonical_2Einterp__m__def,axiom,
! [A_27a] :
( ! [V0sr_2E0,V1vm_2E0,V2c_2E0] : s(A_27a,c_2Ecanonical_2Einterp__m_2E4(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_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)
& ! [V3sr_2E0,V4vm_2E0,V5c_2E0,V6x_2E0,V7t_2E0] : s(A_27a,c_2Ecanonical_2Einterp__m_2E4(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V3sr_2E0),s(tyop_2Equote_2Evarmap(A_27a),V4vm_2E0),s(A_27a,V5c_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),c_2Elist_2ECONS_2E2(s(tyop_2Equote_2Eindex,V6x_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V7t_2E0))))) = s(A_27a,c_2Esemi__ring_2Esemi__ring__SRM_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V3sr_2E0),s(A_27a,V5c_2E0),s(A_27a,c_2Ecanonical_2Eivl__aux_2E4(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V3sr_2E0),s(tyop_2Equote_2Evarmap(A_27a),V4vm_2E0),s(tyop_2Equote_2Eindex,V6x_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V7t_2E0))))) ) ).
fof(thm_2Ecanonical_2Eics__aux__def,axiom,
! [A_27a] :
( ! [V0sr_2E0,V1vm_2E0,V2a_2E0] : s(A_27a,c_2Ecanonical_2Eics__aux_2E4(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_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)
& ! [V3sr_2E0,V4vm_2E0,V5a_2E0,V6l_2E0,V7t_2E0] : s(A_27a,c_2Ecanonical_2Eics__aux_2E4(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V3sr_2E0),s(tyop_2Equote_2Evarmap(A_27a),V4vm_2E0),s(A_27a,V5a_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(A_27a,c_2Esemi__ring_2Esemi__ring__SRP_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V3sr_2E0),s(A_27a,V5a_2E0),s(A_27a,c_2Ecanonical_2Eics__aux_2E4(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V3sr_2E0),s(tyop_2Equote_2Evarmap(A_27a),V4vm_2E0),s(A_27a,c_2Ecanonical_2Einterp__vl_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V3sr_2E0),s(tyop_2Equote_2Evarmap(A_27a),V4vm_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V6l_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V7t_2E0)))))
& ! [V8sr_2E0,V9vm_2E0,V10a_2E0,V11c_2E0,V12l_2E0,V13t_2E0] : s(A_27a,c_2Ecanonical_2Eics__aux_2E4(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V8sr_2E0),s(tyop_2Equote_2Evarmap(A_27a),V9vm_2E0),s(A_27a,V10a_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__monom_2E3(s(A_27a,V11c_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V12l_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V13t_2E0))))) = s(A_27a,c_2Esemi__ring_2Esemi__ring__SRP_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V8sr_2E0),s(A_27a,V10a_2E0),s(A_27a,c_2Ecanonical_2Eics__aux_2E4(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V8sr_2E0),s(tyop_2Equote_2Evarmap(A_27a),V9vm_2E0),s(A_27a,c_2Ecanonical_2Einterp__m_2E4(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V8sr_2E0),s(tyop_2Equote_2Evarmap(A_27a),V9vm_2E0),s(A_27a,V11c_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V12l_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V13t_2E0))))) ) ).
fof(thm_2Ecanonical_2Einterp__cs__def,axiom,
! [A_27a] :
( ! [V0sr_2E0,V1vm_2E0] : s(A_27a,c_2Ecanonical_2Einterp__cs_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_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_2Esemi__ring_2Esemi__ring__SR0_2E1(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0)))
& ! [V2sr_2E0,V3vm_2E0,V4l_2E0,V5t_2E0] : s(A_27a,c_2Ecanonical_2Einterp__cs_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V2sr_2E0),s(tyop_2Equote_2Evarmap(A_27a),V3vm_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(A_27a,c_2Ecanonical_2Eics__aux_2E4(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V2sr_2E0),s(tyop_2Equote_2Evarmap(A_27a),V3vm_2E0),s(A_27a,c_2Ecanonical_2Einterp__vl_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V2sr_2E0),s(tyop_2Equote_2Evarmap(A_27a),V3vm_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V4l_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V5t_2E0)))
& ! [V6sr_2E0,V7vm_2E0,V8c_2E0,V9l_2E0,V10t_2E0] : s(A_27a,c_2Ecanonical_2Einterp__cs_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V6sr_2E0),s(tyop_2Equote_2Evarmap(A_27a),V7vm_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__monom_2E3(s(A_27a,V8c_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V9l_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V10t_2E0))))) = s(A_27a,c_2Ecanonical_2Eics__aux_2E4(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V6sr_2E0),s(tyop_2Equote_2Evarmap(A_27a),V7vm_2E0),s(A_27a,c_2Ecanonical_2Einterp__m_2E4(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V6sr_2E0),s(tyop_2Equote_2Evarmap(A_27a),V7vm_2E0),s(A_27a,V8c_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V9l_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V10t_2E0))) ) ).
fof(thm_2Ecanonical_2Espolynom__TY__DEF,axiom,
! [A_27a,F4_2E0,F3_2E0,F2_2E0,F1_2E0,F0_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_27spolynom_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_27spolynom_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_27spolynom_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_27spolynom_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_27spolynom_27_2E0),s(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)),V12a1_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_27spolynom_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_27spolynom_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_2Ecanonical_2Espolynom(A_27a),tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a))),V0rep_2E0)))) ) ) ) ) ) ).
fof(thm_2Ecanonical_2Espolynom__case__def,axiom,
! [A_27a,A_27b] :
( ! [V0a_2E0,V1f_2E0,V2f1_2E0,V3f2_2E0,V4f3_2E0] : s(A_27b,c_2Ecanonical_2Espolynom__CASE_2E5(s(tyop_2Ecanonical_2Espolynom(A_27a),c_2Ecanonical_2ESPvar_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_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27b)),V3f2_2E0),s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27b)),V4f3_2E0))) = s(A_27b,app_2E2(s(tyop_2Emin_2Efun(tyop_2Equote_2Eindex,A_27b),V1f_2E0),s(tyop_2Equote_2Eindex,V0a_2E0)))
& ! [V5a_2E0,V6f_2E0,V7f1_2E0,V8f2_2E0,V9f3_2E0] : s(A_27b,c_2Ecanonical_2Espolynom__CASE_2E5(s(tyop_2Ecanonical_2Espolynom(A_27a),c_2Ecanonical_2ESPconst_2E1(s(A_27a,V5a_2E0))),s(tyop_2Emin_2Efun(tyop_2Equote_2Eindex,A_27b),V6f_2E0),s(tyop_2Emin_2Efun(A_27a,A_27b),V7f1_2E0),s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27b)),V8f2_2E0),s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27b)),V9f3_2E0))) = s(A_27b,app_2E2(s(tyop_2Emin_2Efun(A_27a,A_27b),V7f1_2E0),s(A_27a,V5a_2E0)))
& ! [V10a0_2E0,V11a1_2E0,V12f_2E0,V13f1_2E0,V14f2_2E0,V15f3_2E0] : s(A_27b,c_2Ecanonical_2Espolynom__CASE_2E5(s(tyop_2Ecanonical_2Espolynom(A_27a),c_2Ecanonical_2ESPplus_2E2(s(tyop_2Ecanonical_2Espolynom(A_27a),V10a0_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),V11a1_2E0))),s(tyop_2Emin_2Efun(tyop_2Equote_2Eindex,A_27b),V12f_2E0),s(tyop_2Emin_2Efun(A_27a,A_27b),V13f1_2E0),s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27b)),V14f2_2E0),s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27b)),V15f3_2E0))) = s(A_27b,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27b),app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27b)),V14f2_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),V10a0_2E0))),s(tyop_2Ecanonical_2Espolynom(A_27a),V11a1_2E0)))
& ! [V16a0_2E0,V17a1_2E0,V18f_2E0,V19f1_2E0,V20f2_2E0,V21f3_2E0] : s(A_27b,c_2Ecanonical_2Espolynom__CASE_2E5(s(tyop_2Ecanonical_2Espolynom(A_27a),c_2Ecanonical_2ESPmult_2E2(s(tyop_2Ecanonical_2Espolynom(A_27a),V16a0_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),V17a1_2E0))),s(tyop_2Emin_2Efun(tyop_2Equote_2Eindex,A_27b),V18f_2E0),s(tyop_2Emin_2Efun(A_27a,A_27b),V19f1_2E0),s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27b)),V20f2_2E0),s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27b)),V21f3_2E0))) = s(A_27b,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27b),app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27b)),V21f3_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),V16a0_2E0))),s(tyop_2Ecanonical_2Espolynom(A_27a),V17a1_2E0))) ) ).
fof(thm_2Ecanonical_2Espolynom__size__def,axiom,
! [A_27a] :
( ! [V0f_2E0,V1a_2E0] : s(tyop_2Enum_2Enum,c_2Ecanonical_2Espolynom__size_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V0f_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),c_2Ecanonical_2ESPvar_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_2Ecanonical_2Espolynom__size_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V2f_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),c_2Ecanonical_2ESPconst_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_2Ecanonical_2Espolynom__size_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V4f_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),c_2Ecanonical_2ESPplus_2E2(s(tyop_2Ecanonical_2Espolynom(A_27a),V5a0_2E0),s(tyop_2Ecanonical_2Espolynom(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_2Ecanonical_2Espolynom__size_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V4f_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),V5a0_2E0))),s(tyop_2Enum_2Enum,c_2Ecanonical_2Espolynom__size_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V4f_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),V6a1_2E0)))))))
& ! [V7f_2E0,V8a0_2E0,V9a1_2E0] : s(tyop_2Enum_2Enum,c_2Ecanonical_2Espolynom__size_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V7f_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),c_2Ecanonical_2ESPmult_2E2(s(tyop_2Ecanonical_2Espolynom(A_27a),V8a0_2E0),s(tyop_2Ecanonical_2Espolynom(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_2Ecanonical_2Espolynom__size_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V7f_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),V8a0_2E0))),s(tyop_2Enum_2Enum,c_2Ecanonical_2Espolynom__size_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V7f_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),V9a1_2E0))))))) ) ).
fof(thm_2Ecanonical_2Espolynom__normalize__def,axiom,
! [A_27a] :
( ! [V0sr_2E0,V1i_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Espolynom__normalize_2E2(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_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)))
& ! [V2sr_2E0,V3c_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Espolynom__normalize_2E2(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V2sr_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),c_2Ecanonical_2ESPconst_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)))
& ! [V4sr_2E0,V5l_2E0,V6r_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Espolynom__normalize_2E2(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V4sr_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),c_2Ecanonical_2ESPplus_2E2(s(tyop_2Ecanonical_2Espolynom(A_27a),V5l_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),V6r_2E0))))) = s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Ecanonical__sum__merge_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V4sr_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Espolynom__normalize_2E2(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V4sr_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),V5l_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Espolynom__normalize_2E2(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V4sr_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),V6r_2E0)))))
& ! [V7sr_2E0,V8l_2E0,V9r_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Espolynom__normalize_2E2(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V7sr_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),c_2Ecanonical_2ESPmult_2E2(s(tyop_2Ecanonical_2Espolynom(A_27a),V8l_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),V9r_2E0))))) = s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Ecanonical__sum__prod_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V7sr_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Espolynom__normalize_2E2(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V7sr_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),V8l_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Espolynom__normalize_2E2(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V7sr_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),V9r_2E0))))) ) ).
fof(thm_2Ecanonical_2Espolynom__simplify__def,axiom,
! [A_27a,V0sr_2E0,V1x_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Espolynom__simplify_2E2(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),V1x_2E0))) = s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Ecanonical__sum__simplify_2E2(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Espolynom__normalize_2E2(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),V1x_2E0))))) ).
fof(thm_2Ecanonical_2Einterp__sp__def,axiom,
! [A_27a] :
( ! [V0sr_2E0,V1vm_2E0,V2c_2E0] : s(A_27a,c_2Ecanonical_2Einterp__sp_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_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)
& ! [V3sr_2E0,V4vm_2E0,V5i_2E0] : s(A_27a,c_2Ecanonical_2Einterp__sp_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V3sr_2E0),s(tyop_2Equote_2Evarmap(A_27a),V4vm_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),c_2Ecanonical_2ESPvar_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)))
& ! [V6sr_2E0,V7vm_2E0,V8p1_2E0,V9p2_2E0] : s(A_27a,c_2Ecanonical_2Einterp__sp_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V6sr_2E0),s(tyop_2Equote_2Evarmap(A_27a),V7vm_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),c_2Ecanonical_2ESPplus_2E2(s(tyop_2Ecanonical_2Espolynom(A_27a),V8p1_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),V9p2_2E0))))) = s(A_27a,c_2Esemi__ring_2Esemi__ring__SRP_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V6sr_2E0),s(A_27a,c_2Ecanonical_2Einterp__sp_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V6sr_2E0),s(tyop_2Equote_2Evarmap(A_27a),V7vm_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),V8p1_2E0))),s(A_27a,c_2Ecanonical_2Einterp__sp_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V6sr_2E0),s(tyop_2Equote_2Evarmap(A_27a),V7vm_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),V9p2_2E0)))))
& ! [V10sr_2E0,V11vm_2E0,V12p1_2E0,V13p2_2E0] : s(A_27a,c_2Ecanonical_2Einterp__sp_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V10sr_2E0),s(tyop_2Equote_2Evarmap(A_27a),V11vm_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),c_2Ecanonical_2ESPmult_2E2(s(tyop_2Ecanonical_2Espolynom(A_27a),V12p1_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),V13p2_2E0))))) = s(A_27a,c_2Esemi__ring_2Esemi__ring__SRM_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V10sr_2E0),s(A_27a,c_2Ecanonical_2Einterp__sp_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V10sr_2E0),s(tyop_2Equote_2Evarmap(A_27a),V11vm_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),V12p1_2E0))),s(A_27a,c_2Ecanonical_2Einterp__sp_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V10sr_2E0),s(tyop_2Equote_2Evarmap(A_27a),V11vm_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),V13p2_2E0))))) ) ).
fof(thm_2Ecanonical_2Edatatype__canonical__sum,axiom,
! [A_27a,V0canonical__sum_2E0] : p(s(tyop_2Emin_2Ebool,c_2Ebool_2EDATATYPE_2E1(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a))),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(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)))),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a))),tyop_2Emin_2Ebool)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(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)))),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a))),tyop_2Emin_2Ebool))),V0canonical__sum_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ENil__monom_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_2ECons__monom_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_2ECons__varlist_2E0)))))) ).
fof(thm_2Ecanonical_2Ecanonical__sum__11,axiom,
! [A_27a] :
( ! [V0a0_2E0,V1a1_2E0,V2a2_2E0,V3a0_27_2E0,V4a1_27_2E0,V5a2_27_2E0] :
( s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__monom_2E3(s(A_27a,V0a0_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V1a1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V2a2_2E0))) = s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__monom_2E3(s(A_27a,V3a0_27_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V4a1_27_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V5a2_27_2E0)))
<=> ( s(A_27a,V0a0_2E0) = s(A_27a,V3a0_27_2E0)
& s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V1a1_2E0) = s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V4a1_27_2E0)
& s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V2a2_2E0) = s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V5a2_27_2E0) ) )
& ! [V6a0_2E0,V7a1_2E0,V8a0_27_2E0,V9a1_27_2E0] :
( s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__varlist_2E2(s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V6a0_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V7a1_2E0))) = s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__varlist_2E2(s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V8a0_27_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V9a1_27_2E0)))
<=> ( s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V6a0_2E0) = s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V8a0_27_2E0)
& s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V7a1_2E0) = s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V9a1_27_2E0) ) ) ) ).
fof(thm_2Ecanonical_2Ecanonical__sum__distinct,axiom,
! [A_27a] :
( ! [V0a2_2E0,V1a1_2E0,V2a0_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,V2a0_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V1a1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V0a2_2E0)))
& ! [V3a1_2E0,V4a0_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),V4a0_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V3a1_2E0)))
& ! [V5a2_2E0,V6a1_27_2E0,V7a1_2E0,V8a0_27_2E0,V9a0_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__monom_2E3(s(A_27a,V9a0_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V7a1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V5a2_2E0))) != s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__varlist_2E2(s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V8a0_27_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V6a1_27_2E0))) ) ).
fof(thm_2Ecanonical_2Ecanonical__sum__nchotomy,axiom,
! [A_27a,V0cc_2E0] :
( s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V0cc_2E0) = s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ENil__monom_2E0)
| ? [V1a_2E0,V2l_2E0,V3c_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V0cc_2E0) = s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__monom_2E3(s(A_27a,V1a_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V2l_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V3c_2E0)))
| ? [V4l_2E0,V5c_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V0cc_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),V5c_2E0))) ) ).
fof(thm_2Ecanonical_2Ecanonical__sum__Axiom,axiom,
! [A_27a,A_27b,V0f0_2E0,V1f1_2E0,V2f2_2E0] :
? [V3fn_2E0] :
( s(A_27b,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27b),V3fn_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ENil__monom_2E0))) = s(A_27b,V0f0_2E0)
& ! [V4a0_2E0,V5a1_2E0,V6a2_2E0] : s(A_27b,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27b),V3fn_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__monom_2E3(s(A_27a,V4a0_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V5a1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V6a2_2E0))))) = s(A_27b,app_2E2(s(tyop_2Emin_2Efun(A_27b,A_27b),app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(A_27b,A_27b)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(A_27b,A_27b))),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_2Emin_2Efun(A_27b,A_27b)))),V1f1_2E0),s(A_27a,V4a0_2E0))),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V5a1_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V6a2_2E0))),s(A_27b,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27b),V3fn_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V6a2_2E0)))))
& ! [V7a0_2E0,V8a1_2E0] : s(A_27b,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27b),V3fn_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__varlist_2E2(s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V7a0_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V8a1_2E0))))) = s(A_27b,app_2E2(s(tyop_2Emin_2Efun(A_27b,A_27b),app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(A_27b,A_27b)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(A_27b,A_27b))),V2f2_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V7a0_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V8a1_2E0))),s(A_27b,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27b),V3fn_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V8a1_2E0))))) ) ).
fof(thm_2Ecanonical_2Ecanonical__sum__induction,axiom,
! [A_27a,V0P_2E0] :
( ( p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ENil__monom_2E0))))
& ! [V1c_2E0] :
( p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V1c_2E0))))
=> ! [V2l_2E0,V3a_2E0] : p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__monom_2E3(s(A_27a,V3a_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V2l_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V1c_2E0)))))) )
& ! [V4c_2E0] :
( p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V4c_2E0))))
=> ! [V5l_2E0] : p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool),V0P_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),V4c_2E0)))))) ) )
=> ! [V6c_2E0] : p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V6c_2E0)))) ) ).
fof(thm_2Ecanonical_2Ecanonical__sum__case__cong,axiom,
! [A_27a,A_27b,V0v_27_2E0,V1f1_27_2E0,V2f_27_2E0,V3M_2E0,V4M_27_2E0,V5v_2E0,V6f_2E0,V7f1_2E0] :
( ( s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V3M_2E0) = s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V4M_27_2E0)
& ( s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V4M_27_2E0) = s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ENil__monom_2E0)
=> s(A_27b,V5v_2E0) = s(A_27b,V0v_27_2E0) )
& ! [V8a0_2E0,V9a1_2E0,V10a2_2E0] :
( s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V4M_27_2E0) = s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__monom_2E3(s(A_27a,V8a0_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V9a1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V10a2_2E0)))
=> s(A_27b,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27b),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27b)),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),A_27b))),V6f_2E0),s(A_27a,V8a0_2E0))),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V9a1_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V10a2_2E0))) = s(A_27b,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27b),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27b)),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),A_27b))),V2f_27_2E0),s(A_27a,V8a0_2E0))),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V9a1_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V10a2_2E0))) )
& ! [V11a0_2E0,V12a1_2E0] :
( s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V4M_27_2E0) = s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__varlist_2E2(s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V11a0_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V12a1_2E0)))
=> s(A_27b,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27b),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27b)),V7f1_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V11a0_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V12a1_2E0))) = s(A_27b,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27b),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27b)),V1f1_27_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V11a0_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V12a1_2E0))) ) )
=> s(A_27b,c_2Ecanonical_2Ecanonical__sum__CASE_2E4(s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V3M_2E0),s(A_27b,V5v_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),A_27b))),V6f_2E0),s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27b)),V7f1_2E0))) = s(A_27b,c_2Ecanonical_2Ecanonical__sum__CASE_2E4(s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V4M_27_2E0),s(A_27b,V0v_27_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),A_27b))),V2f_27_2E0),s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27b)),V1f1_27_2E0))) ) ).
fof(thm_2Ecanonical_2Ecanonical__sum__case__eq,axiom,
! [A_27a,A_27b,V0x_2E0,V1v_27_2E0,V2v_2E0,V3f1_2E0,V4f_2E0] :
( s(A_27b,c_2Ecanonical_2Ecanonical__sum__CASE_2E4(s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V0x_2E0),s(A_27b,V2v_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),A_27b))),V4f_2E0),s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27b)),V3f1_2E0))) = s(A_27b,V1v_27_2E0)
<=> ( ( s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V0x_2E0) = s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ENil__monom_2E0)
& s(A_27b,V2v_2E0) = s(A_27b,V1v_27_2E0) )
| ? [V5a_2E0,V6l_2E0,V7c_2E0] :
( s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V0x_2E0) = s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__monom_2E3(s(A_27a,V5a_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V6l_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V7c_2E0)))
& s(A_27b,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27b),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27b)),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),A_27b))),V4f_2E0),s(A_27a,V5a_2E0))),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V6l_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V7c_2E0))) = s(A_27b,V1v_27_2E0) )
| ? [V8l_2E0,V9c_2E0] :
( s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V0x_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),V9c_2E0)))
& s(A_27b,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27b),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27b)),V3f1_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V8l_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V9c_2E0))) = s(A_27b,V1v_27_2E0) ) ) ) ).
fof(thm_2Ecanonical_2Ecanonical__sum__merge__ind,axiom,
! [A_27a,V0P_2E0] :
( ( ! [V1sr_2E0,V2c1_2E0,V3l1_2E0,V4t1_2E0,V5c2_2E0,V6l2_2E0,V7t2_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)),c_2Equote_2Eindex__compare_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V3l1_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V6l2_2E0))) = s(tyop_2EternaryComparisons_2Eordering,c_2EternaryComparisons_2EEQUAL_2E0)
=> p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool)),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_2Emin_2Ebool))),V0P_2E0),s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V1sr_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V4t1_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V7t2_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)),c_2Equote_2Eindex__compare_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V3l1_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V6l2_2E0))) = s(tyop_2EternaryComparisons_2Eordering,c_2EternaryComparisons_2EGREATER_2E0)
=> p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool)),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_2Emin_2Ebool))),V0P_2E0),s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V1sr_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__monom_2E3(s(A_27a,V2c1_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V3l1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V4t1_2E0))))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V7t2_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)),c_2Equote_2Eindex__compare_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V3l1_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V6l2_2E0))) = s(tyop_2EternaryComparisons_2Eordering,c_2EternaryComparisons_2ELESS_2E0)
=> p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool)),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_2Emin_2Ebool))),V0P_2E0),s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V1sr_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V4t1_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),V6l2_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V7t2_2E0)))))) ) )
=> p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool)),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_2Emin_2Ebool))),V0P_2E0),s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V1sr_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__monom_2E3(s(A_27a,V2c1_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V3l1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V4t1_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),V6l2_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V7t2_2E0)))))) )
& ! [V8sr_2E0,V9c1_2E0,V10l1_2E0,V11t1_2E0,V12l2_2E0,V13t2_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)),c_2Equote_2Eindex__compare_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V10l1_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V12l2_2E0))) = s(tyop_2EternaryComparisons_2Eordering,c_2EternaryComparisons_2EEQUAL_2E0)
=> p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool)),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_2Emin_2Ebool))),V0P_2E0),s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V8sr_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V11t1_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V13t2_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)),c_2Equote_2Eindex__compare_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V10l1_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V12l2_2E0))) = s(tyop_2EternaryComparisons_2Eordering,c_2EternaryComparisons_2EGREATER_2E0)
=> p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool)),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_2Emin_2Ebool))),V0P_2E0),s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V8sr_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),V10l1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V11t1_2E0))))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V13t2_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)),c_2Equote_2Eindex__compare_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V10l1_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V12l2_2E0))) = s(tyop_2EternaryComparisons_2Eordering,c_2EternaryComparisons_2ELESS_2E0)
=> p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool)),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_2Emin_2Ebool))),V0P_2E0),s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V8sr_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V11t1_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__varlist_2E2(s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V12l2_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V13t2_2E0)))))) ) )
=> p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool)),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_2Emin_2Ebool))),V0P_2E0),s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V8sr_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),V10l1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V11t1_2E0))))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__varlist_2E2(s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V12l2_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V13t2_2E0)))))) )
& ! [V14sr_2E0,V15l1_2E0,V16t1_2E0,V17c2_2E0,V18l2_2E0,V19t2_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)),c_2Equote_2Eindex__compare_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V15l1_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V18l2_2E0))) = s(tyop_2EternaryComparisons_2Eordering,c_2EternaryComparisons_2EEQUAL_2E0)
=> p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool)),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_2Emin_2Ebool))),V0P_2E0),s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V14sr_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V16t1_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V19t2_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)),c_2Equote_2Eindex__compare_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V15l1_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V18l2_2E0))) = s(tyop_2EternaryComparisons_2Eordering,c_2EternaryComparisons_2EGREATER_2E0)
=> p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool)),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_2Emin_2Ebool))),V0P_2E0),s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V14sr_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),V16t1_2E0))))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V19t2_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)),c_2Equote_2Eindex__compare_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V15l1_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V18l2_2E0))) = s(tyop_2EternaryComparisons_2Eordering,c_2EternaryComparisons_2ELESS_2E0)
=> p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool)),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_2Emin_2Ebool))),V0P_2E0),s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V14sr_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V16t1_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__monom_2E3(s(A_27a,V17c2_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V18l2_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V19t2_2E0)))))) ) )
=> p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool)),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_2Emin_2Ebool))),V0P_2E0),s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V14sr_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),V16t1_2E0))))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__monom_2E3(s(A_27a,V17c2_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V18l2_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V19t2_2E0)))))) )
& ! [V20sr_2E0,V21l1_2E0,V22t1_2E0,V23l2_2E0,V24t2_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)),c_2Equote_2Eindex__compare_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V21l1_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V23l2_2E0))) = s(tyop_2EternaryComparisons_2Eordering,c_2EternaryComparisons_2EEQUAL_2E0)
=> p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool)),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_2Emin_2Ebool))),V0P_2E0),s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V20sr_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V22t1_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V24t2_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)),c_2Equote_2Eindex__compare_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V21l1_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V23l2_2E0))) = s(tyop_2EternaryComparisons_2Eordering,c_2EternaryComparisons_2EGREATER_2E0)
=> p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool)),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_2Emin_2Ebool))),V0P_2E0),s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V20sr_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__varlist_2E2(s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V21l1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V22t1_2E0))))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V24t2_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)),c_2Equote_2Eindex__compare_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V21l1_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V23l2_2E0))) = s(tyop_2EternaryComparisons_2Eordering,c_2EternaryComparisons_2ELESS_2E0)
=> p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool)),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_2Emin_2Ebool))),V0P_2E0),s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V20sr_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V22t1_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__varlist_2E2(s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V23l2_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V24t2_2E0)))))) ) )
=> p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool)),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_2Emin_2Ebool))),V0P_2E0),s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V20sr_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__varlist_2E2(s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V21l1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V22t1_2E0))))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__varlist_2E2(s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V23l2_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V24t2_2E0)))))) )
& ! [V25sr_2E0,V26s1_2E0] : p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool)),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_2Emin_2Ebool))),V0P_2E0),s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V25sr_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V26s1_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ENil__monom_2E0))))
& ! [V27sr_2E0,V28v4_2E0,V29v5_2E0,V30v6_2E0] : p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool)),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_2Emin_2Ebool))),V0P_2E0),s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V27sr_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,V28v4_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V29v5_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V30v6_2E0))))))
& ! [V31sr_2E0,V32v7_2E0,V33v8_2E0] : p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool)),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_2Emin_2Ebool))),V0P_2E0),s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V31sr_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),V32v7_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V33v8_2E0)))))) )
=> ! [V34v_2E0,V35v1_2E0,V36v2_2E0] : p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool)),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_2Emin_2Ebool))),V0P_2E0),s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V34v_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V35v1_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V36v2_2E0)))) ) ).
fof(thm_2Ecanonical_2Ecanonical__sum__merge__def,axiom,
! [A_27a] :
( ! [V0t2_2E0,V1t1_2E0,V2sr_2E0,V3l2_2E0,V4l1_2E0,V5c2_2E0,V6c1_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Ecanonical__sum__merge_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V2sr_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),V1t1_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),V0t2_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_2Ecanonical_2Ecanonical__sum__merge_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V2sr_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V1t1_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),V0t2_2E0))))))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__monom_2E3(s(A_27a,c_2Esemi__ring_2Esemi__ring__SRP_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V2sr_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_2Ecanonical_2Ecanonical__sum__merge_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V2sr_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V1t1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V0t2_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_2Ecanonical_2Ecanonical__sum__merge_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V2sr_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),V1t1_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V0t2_2E0)))))))
& ! [V7t2_2E0,V8t1_2E0,V9sr_2E0,V10l2_2E0,V11l1_2E0,V12c1_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Ecanonical__sum__merge_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V9sr_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__monom_2E3(s(A_27a,V12c1_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V11l1_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),V10l2_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),V11l1_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V10l2_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__monom_2E3(s(A_27a,V12c1_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V11l1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Ecanonical__sum__merge_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V9sr_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),V10l2_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_2Esemi__ring_2Esemi__ring__SRP_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V9sr_2E0),s(A_27a,V12c1_2E0),s(A_27a,c_2Esemi__ring_2Esemi__ring__SR1_2E1(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V9sr_2E0))))),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V11l1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Ecanonical__sum__merge_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V9sr_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),V10l2_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Ecanonical__sum__merge_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V9sr_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__monom_2E3(s(A_27a,V12c1_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V11l1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V8t1_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V7t2_2E0)))))))
& ! [V13t2_2E0,V14t1_2E0,V15sr_2E0,V16l2_2E0,V17l1_2E0,V18c2_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Ecanonical__sum__merge_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V15sr_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__varlist_2E2(s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V17l1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V14t1_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__monom_2E3(s(A_27a,V18c2_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V16l2_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V13t2_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),V17l1_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V16l2_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__varlist_2E2(s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V17l1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Ecanonical__sum__merge_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V15sr_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V14t1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__monom_2E3(s(A_27a,V18c2_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V16l2_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V13t2_2E0))))))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__monom_2E3(s(A_27a,c_2Esemi__ring_2Esemi__ring__SRP_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V15sr_2E0),s(A_27a,c_2Esemi__ring_2Esemi__ring__SR1_2E1(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V15sr_2E0))),s(A_27a,V18c2_2E0))),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V17l1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Ecanonical__sum__merge_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V15sr_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V14t1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V13t2_2E0))))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__monom_2E3(s(A_27a,V18c2_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V16l2_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Ecanonical__sum__merge_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V15sr_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__varlist_2E2(s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V17l1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V14t1_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V13t2_2E0)))))))
& ! [V19t2_2E0,V20t1_2E0,V21sr_2E0,V22l2_2E0,V23l1_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Ecanonical__sum__merge_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V21sr_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__varlist_2E2(s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V23l1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V20t1_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__varlist_2E2(s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V22l2_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V19t2_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),V23l1_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V22l2_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__varlist_2E2(s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V23l1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Ecanonical__sum__merge_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V21sr_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V20t1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__varlist_2E2(s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V22l2_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V19t2_2E0))))))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__monom_2E3(s(A_27a,c_2Esemi__ring_2Esemi__ring__SRP_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V21sr_2E0),s(A_27a,c_2Esemi__ring_2Esemi__ring__SR1_2E1(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V21sr_2E0))),s(A_27a,c_2Esemi__ring_2Esemi__ring__SR1_2E1(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V21sr_2E0))))),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V23l1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Ecanonical__sum__merge_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V21sr_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V20t1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V19t2_2E0))))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__varlist_2E2(s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V22l2_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Ecanonical__sum__merge_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V21sr_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__varlist_2E2(s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V23l1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V20t1_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V19t2_2E0)))))))
& ! [V24sr_2E0,V25s1_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Ecanonical__sum__merge_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V24sr_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V25s1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ENil__monom_2E0))) = s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V25s1_2E0)
& ! [V26v6_2E0,V27v5_2E0,V28v4_2E0,V29sr_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Ecanonical__sum__merge_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V29sr_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,V28v4_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V27v5_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V26v6_2E0))))) = s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__monom_2E3(s(A_27a,V28v4_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V27v5_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V26v6_2E0)))
& ! [V30v8_2E0,V31v7_2E0,V32sr_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Ecanonical__sum__merge_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V32sr_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),V31v7_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V30v8_2E0))))) = s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__varlist_2E2(s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V31v7_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V30v8_2E0))) ) ).
fof(thm_2Ecanonical_2Emonom__insert__ind,axiom,
! [A_27a,V0P_2E0] :
( ( ! [V1sr_2E0,V2c1_2E0,V3l1_2E0,V4c2_2E0,V5l2_2E0,V6t2_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)),c_2Equote_2Eindex__compare_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V3l1_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V5l2_2E0))) = s(tyop_2EternaryComparisons_2Eordering,c_2EternaryComparisons_2EGREATER_2E0)
=> p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool)),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_2Emin_2Ebool))),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_2Emin_2Ebool)))),V0P_2E0),s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V1sr_2E0))),s(A_27a,V2c1_2E0))),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V3l1_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V6t2_2E0)))) )
=> p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool)),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_2Emin_2Ebool))),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_2Emin_2Ebool)))),V0P_2E0),s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V1sr_2E0))),s(A_27a,V2c1_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),V5l2_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V6t2_2E0)))))) )
& ! [V7sr_2E0,V8c1_2E0,V9l1_2E0,V10l2_2E0,V11t2_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)),c_2Equote_2Eindex__compare_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V9l1_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V10l2_2E0))) = s(tyop_2EternaryComparisons_2Eordering,c_2EternaryComparisons_2EGREATER_2E0)
=> p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool)),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_2Emin_2Ebool))),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_2Emin_2Ebool)))),V0P_2E0),s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V7sr_2E0))),s(A_27a,V8c1_2E0))),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V9l1_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V11t2_2E0)))) )
=> p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool)),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_2Emin_2Ebool))),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_2Emin_2Ebool)))),V0P_2E0),s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V7sr_2E0))),s(A_27a,V8c1_2E0))),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V9l1_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__varlist_2E2(s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V10l2_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V11t2_2E0)))))) )
& ! [V12sr_2E0,V13c1_2E0,V14l1_2E0] : p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool)),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_2Emin_2Ebool))),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_2Emin_2Ebool)))),V0P_2E0),s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V12sr_2E0))),s(A_27a,V13c1_2E0))),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V14l1_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ENil__monom_2E0)))) )
=> ! [V15v_2E0,V16v1_2E0,V17v2_2E0,V18v3_2E0] : p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool)),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_2Emin_2Ebool))),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_2Emin_2Ebool)))),V0P_2E0),s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V15v_2E0))),s(A_27a,V16v1_2E0))),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V17v2_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V18v3_2E0)))) ) ).
fof(thm_2Ecanonical_2Emonom__insert__def,axiom,
! [A_27a] :
( ! [V0t2_2E0,V1sr_2E0,V2l2_2E0,V3l1_2E0,V4c2_2E0,V5c1_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Emonom__insert_2E4(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V1sr_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),V0t2_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),V0t2_2E0))))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__monom_2E3(s(A_27a,c_2Esemi__ring_2Esemi__ring__SRP_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V1sr_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),V0t2_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_2Ecanonical_2Emonom__insert_2E4(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V1sr_2E0),s(A_27a,V5c1_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V3l1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V0t2_2E0)))))))
& ! [V6t2_2E0,V7sr_2E0,V8l2_2E0,V9l1_2E0,V10c1_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Emonom__insert_2E4(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V7sr_2E0),s(A_27a,V10c1_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V9l1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__varlist_2E2(s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V8l2_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),V9l1_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V8l2_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__monom_2E3(s(A_27a,V10c1_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V9l1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__varlist_2E2(s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V8l2_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_2Esemi__ring_2Esemi__ring__SRP_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V7sr_2E0),s(A_27a,V10c1_2E0),s(A_27a,c_2Esemi__ring_2Esemi__ring__SR1_2E1(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V7sr_2E0))))),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V9l1_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),V8l2_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Emonom__insert_2E4(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V7sr_2E0),s(A_27a,V10c1_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V9l1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V6t2_2E0)))))))
& ! [V11sr_2E0,V12l1_2E0,V13c1_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Emonom__insert_2E4(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V11sr_2E0),s(A_27a,V13c1_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V12l1_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,V13c1_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V12l1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ENil__monom_2E0))) ) ).
fof(thm_2Ecanonical_2Evarlist__insert__ind,axiom,
! [A_27a,V0P_2E0] :
( ( ! [V1sr_2E0,V2l1_2E0,V3c2_2E0,V4l2_2E0,V5t2_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)),c_2Equote_2Eindex__compare_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V2l1_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V4l2_2E0))) = s(tyop_2EternaryComparisons_2Eordering,c_2EternaryComparisons_2EGREATER_2E0)
=> p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool)),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_2Emin_2Ebool))),V0P_2E0),s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V1sr_2E0))),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V2l1_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V5t2_2E0)))) )
=> p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool)),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_2Emin_2Ebool))),V0P_2E0),s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V1sr_2E0))),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V2l1_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__monom_2E3(s(A_27a,V3c2_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V4l2_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V5t2_2E0)))))) )
& ! [V6sr_2E0,V7l1_2E0,V8l2_2E0,V9t2_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)),c_2Equote_2Eindex__compare_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V7l1_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V8l2_2E0))) = s(tyop_2EternaryComparisons_2Eordering,c_2EternaryComparisons_2EGREATER_2E0)
=> p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool)),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_2Emin_2Ebool))),V0P_2E0),s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V6sr_2E0))),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V7l1_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V9t2_2E0)))) )
=> p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool)),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_2Emin_2Ebool))),V0P_2E0),s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V6sr_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),V8l2_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V9t2_2E0)))))) )
& ! [V10sr_2E0,V11l1_2E0] : p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool)),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_2Emin_2Ebool))),V0P_2E0),s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V10sr_2E0))),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V11l1_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ENil__monom_2E0)))) )
=> ! [V12v_2E0,V13v1_2E0,V14v2_2E0] : p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool)),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_2Emin_2Ebool))),V0P_2E0),s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V12v_2E0))),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V13v1_2E0))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V14v2_2E0)))) ) ).
fof(thm_2Ecanonical_2Evarlist__insert__def,axiom,
! [A_27a] :
( ! [V0t2_2E0,V1sr_2E0,V2l2_2E0,V3l1_2E0,V4c2_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Evarlist__insert_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V1sr_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),V0t2_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),V0t2_2E0))))),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ECons__monom_2E3(s(A_27a,c_2Esemi__ring_2Esemi__ring__SRP_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V1sr_2E0),s(A_27a,c_2Esemi__ring_2Esemi__ring__SR1_2E1(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V1sr_2E0))),s(A_27a,V4c2_2E0))),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V3l1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V0t2_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_2Ecanonical_2Evarlist__insert_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V1sr_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V3l1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V0t2_2E0)))))))
& ! [V5t2_2E0,V6sr_2E0,V7l2_2E0,V8l1_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Evarlist__insert_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V6sr_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),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),V8l1_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V7l2_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_2ECons__varlist_2E2(s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V7l2_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_2Esemi__ring_2Esemi__ring__SRP_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V6sr_2E0),s(A_27a,c_2Esemi__ring_2Esemi__ring__SR1_2E1(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V6sr_2E0))),s(A_27a,c_2Esemi__ring_2Esemi__ring__SR1_2E1(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V6sr_2E0))))),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V8l1_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),V7l2_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Evarlist__insert_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V6sr_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V8l1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V5t2_2E0)))))))
& ! [V9sr_2E0,V10l1_2E0] : s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Evarlist__insert_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V9sr_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__varlist_2E2(s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V10l1_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2ENil__monom_2E0))) ) ).
fof(thm_2Ecanonical_2Eivl__aux__ok,axiom,
! [A_27a,V0sr_2E0] :
( p(s(tyop_2Emin_2Ebool,c_2Esemi__ring_2Eis__semi__ring_2E1(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0))))
=> ! [V1vm_2E0,V2v_2E0,V3i_2E0] : s(A_27a,c_2Ecanonical_2Eivl__aux_2E4(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0),s(tyop_2Equote_2Evarmap(A_27a),V1vm_2E0),s(tyop_2Equote_2Eindex,V3i_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V2v_2E0))) = s(A_27a,c_2Esemi__ring_2Esemi__ring__SRM_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0),s(A_27a,c_2Equote_2Evarmap__find_2E2(s(tyop_2Equote_2Eindex,V3i_2E0),s(tyop_2Equote_2Evarmap(A_27a),V1vm_2E0))),s(A_27a,c_2Ecanonical_2Einterp__vl_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0),s(tyop_2Equote_2Evarmap(A_27a),V1vm_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V2v_2E0))))) ) ).
fof(thm_2Ecanonical_2Evarlist__merge__ok,axiom,
! [A_27a,V0sr_2E0] :
( p(s(tyop_2Emin_2Ebool,c_2Esemi__ring_2Eis__semi__ring_2E1(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0))))
=> ! [V1vm_2E0,V2x_2E0,V3y_2E0] : s(A_27a,c_2Ecanonical_2Einterp__vl_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0),s(tyop_2Equote_2Evarmap(A_27a),V1vm_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),V2x_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V3y_2E0))))) = s(A_27a,c_2Esemi__ring_2Esemi__ring__SRM_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0),s(A_27a,c_2Ecanonical_2Einterp__vl_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0),s(tyop_2Equote_2Evarmap(A_27a),V1vm_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V2x_2E0))),s(A_27a,c_2Ecanonical_2Einterp__vl_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0),s(tyop_2Equote_2Evarmap(A_27a),V1vm_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V3y_2E0))))) ) ).
fof(thm_2Ecanonical_2Eics__aux__ok,axiom,
! [A_27a,V0sr_2E0] :
( p(s(tyop_2Emin_2Ebool,c_2Esemi__ring_2Eis__semi__ring_2E1(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0))))
=> ! [V1vm_2E0,V2x_2E0,V3s_2E0] : s(A_27a,c_2Ecanonical_2Eics__aux_2E4(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0),s(tyop_2Equote_2Evarmap(A_27a),V1vm_2E0),s(A_27a,V2x_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V3s_2E0))) = s(A_27a,c_2Esemi__ring_2Esemi__ring__SRP_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0),s(A_27a,V2x_2E0),s(A_27a,c_2Ecanonical_2Einterp__cs_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0),s(tyop_2Equote_2Evarmap(A_27a),V1vm_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V3s_2E0))))) ) ).
fof(thm_2Ecanonical_2Einterp__m__ok,axiom,
! [A_27a,V0sr_2E0] :
( p(s(tyop_2Emin_2Ebool,c_2Esemi__ring_2Eis__semi__ring_2E1(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0))))
=> ! [V1vm_2E0,V2x_2E0,V3l_2E0] : s(A_27a,c_2Ecanonical_2Einterp__m_2E4(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0),s(tyop_2Equote_2Evarmap(A_27a),V1vm_2E0),s(A_27a,V2x_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V3l_2E0))) = s(A_27a,c_2Esemi__ring_2Esemi__ring__SRM_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0),s(A_27a,V2x_2E0),s(A_27a,c_2Ecanonical_2Einterp__vl_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0),s(tyop_2Equote_2Evarmap(A_27a),V1vm_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V3l_2E0))))) ) ).
fof(thm_2Ecanonical_2Ecanonical__sum__merge__ok,axiom,
! [A_27a,V0sr_2E0] :
( p(s(tyop_2Emin_2Ebool,c_2Esemi__ring_2Eis__semi__ring_2E1(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0))))
=> ! [V1vm_2E0,V2x_2E0,V3y_2E0] : s(A_27a,c_2Ecanonical_2Einterp__cs_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0),s(tyop_2Equote_2Evarmap(A_27a),V1vm_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Ecanonical__sum__merge_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V2x_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V3y_2E0))))) = s(A_27a,c_2Esemi__ring_2Esemi__ring__SRP_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0),s(A_27a,c_2Ecanonical_2Einterp__cs_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0),s(tyop_2Equote_2Evarmap(A_27a),V1vm_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V2x_2E0))),s(A_27a,c_2Ecanonical_2Einterp__cs_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0),s(tyop_2Equote_2Evarmap(A_27a),V1vm_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V3y_2E0))))) ) ).
fof(thm_2Ecanonical_2Emonom__insert__ok,axiom,
! [A_27a,V0sr_2E0] :
( p(s(tyop_2Emin_2Ebool,c_2Esemi__ring_2Eis__semi__ring_2E1(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0))))
=> ! [V1vm_2E0,V2a_2E0,V3l_2E0,V4s_2E0] : s(A_27a,c_2Ecanonical_2Einterp__cs_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0),s(tyop_2Equote_2Evarmap(A_27a),V1vm_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Emonom__insert_2E4(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0),s(A_27a,V2a_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V3l_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V4s_2E0))))) = s(A_27a,c_2Esemi__ring_2Esemi__ring__SRP_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0),s(A_27a,c_2Esemi__ring_2Esemi__ring__SRM_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0),s(A_27a,V2a_2E0),s(A_27a,c_2Ecanonical_2Einterp__vl_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0),s(tyop_2Equote_2Evarmap(A_27a),V1vm_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V3l_2E0))))),s(A_27a,c_2Ecanonical_2Einterp__cs_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0),s(tyop_2Equote_2Evarmap(A_27a),V1vm_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V4s_2E0))))) ) ).
fof(thm_2Ecanonical_2Evarlist__insert__ok,axiom,
! [A_27a,V0sr_2E0] :
( p(s(tyop_2Emin_2Ebool,c_2Esemi__ring_2Eis__semi__ring_2E1(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0))))
=> ! [V1vm_2E0,V2l_2E0,V3s_2E0] : s(A_27a,c_2Ecanonical_2Einterp__cs_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0),s(tyop_2Equote_2Evarmap(A_27a),V1vm_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Evarlist__insert_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V2l_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V3s_2E0))))) = s(A_27a,c_2Esemi__ring_2Esemi__ring__SRP_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0),s(A_27a,c_2Ecanonical_2Einterp__vl_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0),s(tyop_2Equote_2Evarmap(A_27a),V1vm_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V2l_2E0))),s(A_27a,c_2Ecanonical_2Einterp__cs_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0),s(tyop_2Equote_2Evarmap(A_27a),V1vm_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V3s_2E0))))) ) ).
fof(thm_2Ecanonical_2Ecanonical__sum__scalar__ok,axiom,
! [A_27a,V0sr_2E0] :
( p(s(tyop_2Emin_2Ebool,c_2Esemi__ring_2Eis__semi__ring_2E1(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0))))
=> ! [V1vm_2E0,V2a_2E0,V3s_2E0] : s(A_27a,c_2Ecanonical_2Einterp__cs_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0),s(tyop_2Equote_2Evarmap(A_27a),V1vm_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Ecanonical__sum__scalar_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0),s(A_27a,V2a_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V3s_2E0))))) = s(A_27a,c_2Esemi__ring_2Esemi__ring__SRM_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0),s(A_27a,V2a_2E0),s(A_27a,c_2Ecanonical_2Einterp__cs_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0),s(tyop_2Equote_2Evarmap(A_27a),V1vm_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V3s_2E0))))) ) ).
fof(thm_2Ecanonical_2Ecanonical__sum__scalar2__ok,axiom,
! [A_27a,V0sr_2E0] :
( p(s(tyop_2Emin_2Ebool,c_2Esemi__ring_2Eis__semi__ring_2E1(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0))))
=> ! [V1vm_2E0,V2l_2E0,V3s_2E0] : s(A_27a,c_2Ecanonical_2Einterp__cs_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0),s(tyop_2Equote_2Evarmap(A_27a),V1vm_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Ecanonical__sum__scalar2_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V2l_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V3s_2E0))))) = s(A_27a,c_2Esemi__ring_2Esemi__ring__SRM_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0),s(A_27a,c_2Ecanonical_2Einterp__vl_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0),s(tyop_2Equote_2Evarmap(A_27a),V1vm_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V2l_2E0))),s(A_27a,c_2Ecanonical_2Einterp__cs_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0),s(tyop_2Equote_2Evarmap(A_27a),V1vm_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V3s_2E0))))) ) ).
fof(thm_2Ecanonical_2Ecanonical__sum__scalar3__ok,axiom,
! [A_27a,V0sr_2E0] :
( p(s(tyop_2Emin_2Ebool,c_2Esemi__ring_2Eis__semi__ring_2E1(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0))))
=> ! [V1vm_2E0,V2c_2E0,V3l_2E0,V4s_2E0] : s(A_27a,c_2Ecanonical_2Einterp__cs_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0),s(tyop_2Equote_2Evarmap(A_27a),V1vm_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Ecanonical__sum__scalar3_2E4(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0),s(A_27a,V2c_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V3l_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V4s_2E0))))) = s(A_27a,c_2Esemi__ring_2Esemi__ring__SRM_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0),s(A_27a,c_2Esemi__ring_2Esemi__ring__SRM_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0),s(A_27a,V2c_2E0),s(A_27a,c_2Ecanonical_2Einterp__vl_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0),s(tyop_2Equote_2Evarmap(A_27a),V1vm_2E0),s(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V3l_2E0))))),s(A_27a,c_2Ecanonical_2Einterp__cs_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0),s(tyop_2Equote_2Evarmap(A_27a),V1vm_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V4s_2E0))))) ) ).
fof(thm_2Ecanonical_2Ecanonical__sum__prod__ok,axiom,
! [A_27a,V0sr_2E0] :
( p(s(tyop_2Emin_2Ebool,c_2Esemi__ring_2Eis__semi__ring_2E1(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0))))
=> ! [V1vm_2E0,V2x_2E0,V3y_2E0] : s(A_27a,c_2Ecanonical_2Einterp__cs_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0),s(tyop_2Equote_2Evarmap(A_27a),V1vm_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Ecanonical__sum__prod_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V2x_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V3y_2E0))))) = s(A_27a,c_2Esemi__ring_2Esemi__ring__SRM_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0),s(A_27a,c_2Ecanonical_2Einterp__cs_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0),s(tyop_2Equote_2Evarmap(A_27a),V1vm_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V2x_2E0))),s(A_27a,c_2Ecanonical_2Einterp__cs_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0),s(tyop_2Equote_2Evarmap(A_27a),V1vm_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V3y_2E0))))) ) ).
fof(thm_2Ecanonical_2Ecanonical__sum__simplify__ok,axiom,
! [A_27a,V0sr_2E0] :
( p(s(tyop_2Emin_2Ebool,c_2Esemi__ring_2Eis__semi__ring_2E1(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0))))
=> ! [V1vm_2E0,V2s_2E0] : s(A_27a,c_2Ecanonical_2Einterp__cs_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0),s(tyop_2Equote_2Evarmap(A_27a),V1vm_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Ecanonical__sum__simplify_2E2(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V2s_2E0))))) = s(A_27a,c_2Ecanonical_2Einterp__cs_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0),s(tyop_2Equote_2Evarmap(A_27a),V1vm_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),V2s_2E0))) ) ).
fof(thm_2Ecanonical_2Edatatype__spolynom,axiom,
! [A_27a,V0spolynom_2E0] : p(s(tyop_2Emin_2Ebool,c_2Ebool_2EDATATYPE_2E1(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Ecanonical_2Espolynom(A_27a))),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Ecanonical_2Espolynom(A_27a))),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Ecanonical_2Espolynom(A_27a))),tyop_2Emin_2Ebool)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Ecanonical_2Espolynom(A_27a)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Ecanonical_2Espolynom(A_27a))),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Ecanonical_2Espolynom(A_27a))),tyop_2Emin_2Ebool))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Equote_2Eindex,tyop_2Ecanonical_2Espolynom(A_27a)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Ecanonical_2Espolynom(A_27a)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Ecanonical_2Espolynom(A_27a))),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Ecanonical_2Espolynom(A_27a))),tyop_2Emin_2Ebool)))),V0spolynom_2E0),s(tyop_2Emin_2Efun(tyop_2Equote_2Eindex,tyop_2Ecanonical_2Espolynom(A_27a)),c_2Ecanonical_2ESPvar_2E0))),s(tyop_2Emin_2Efun(A_27a,tyop_2Ecanonical_2Espolynom(A_27a)),c_2Ecanonical_2ESPconst_2E0))),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_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Ecanonical_2Espolynom(A_27a))),c_2Ecanonical_2ESPmult_2E0)))))) ).
fof(thm_2Ecanonical_2Espolynom__11,axiom,
! [A_27a] :
( ! [V0a_2E0,V1a_27_2E0] :
( s(tyop_2Ecanonical_2Espolynom(A_27a),c_2Ecanonical_2ESPvar_2E1(s(tyop_2Equote_2Eindex,V0a_2E0))) = s(tyop_2Ecanonical_2Espolynom(A_27a),c_2Ecanonical_2ESPvar_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_2Ecanonical_2Espolynom(A_27a),c_2Ecanonical_2ESPconst_2E1(s(A_27a,V2a_2E0))) = s(tyop_2Ecanonical_2Espolynom(A_27a),c_2Ecanonical_2ESPconst_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_2Ecanonical_2Espolynom(A_27a),c_2Ecanonical_2ESPplus_2E2(s(tyop_2Ecanonical_2Espolynom(A_27a),V4a0_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),V5a1_2E0))) = s(tyop_2Ecanonical_2Espolynom(A_27a),c_2Ecanonical_2ESPplus_2E2(s(tyop_2Ecanonical_2Espolynom(A_27a),V6a0_27_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),V7a1_27_2E0)))
<=> ( s(tyop_2Ecanonical_2Espolynom(A_27a),V4a0_2E0) = s(tyop_2Ecanonical_2Espolynom(A_27a),V6a0_27_2E0)
& s(tyop_2Ecanonical_2Espolynom(A_27a),V5a1_2E0) = s(tyop_2Ecanonical_2Espolynom(A_27a),V7a1_27_2E0) ) )
& ! [V8a0_2E0,V9a1_2E0,V10a0_27_2E0,V11a1_27_2E0] :
( s(tyop_2Ecanonical_2Espolynom(A_27a),c_2Ecanonical_2ESPmult_2E2(s(tyop_2Ecanonical_2Espolynom(A_27a),V8a0_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),V9a1_2E0))) = s(tyop_2Ecanonical_2Espolynom(A_27a),c_2Ecanonical_2ESPmult_2E2(s(tyop_2Ecanonical_2Espolynom(A_27a),V10a0_27_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),V11a1_27_2E0)))
<=> ( s(tyop_2Ecanonical_2Espolynom(A_27a),V8a0_2E0) = s(tyop_2Ecanonical_2Espolynom(A_27a),V10a0_27_2E0)
& s(tyop_2Ecanonical_2Espolynom(A_27a),V9a1_2E0) = s(tyop_2Ecanonical_2Espolynom(A_27a),V11a1_27_2E0) ) ) ) ).
fof(thm_2Ecanonical_2Espolynom__distinct,axiom,
! [A_27a] :
( ! [V0a_27_2E0,V1a_2E0] : s(tyop_2Ecanonical_2Espolynom(A_27a),c_2Ecanonical_2ESPvar_2E1(s(tyop_2Equote_2Eindex,V1a_2E0))) != s(tyop_2Ecanonical_2Espolynom(A_27a),c_2Ecanonical_2ESPconst_2E1(s(A_27a,V0a_27_2E0)))
& ! [V2a1_2E0,V3a0_2E0,V4a_2E0] : s(tyop_2Ecanonical_2Espolynom(A_27a),c_2Ecanonical_2ESPvar_2E1(s(tyop_2Equote_2Eindex,V4a_2E0))) != s(tyop_2Ecanonical_2Espolynom(A_27a),c_2Ecanonical_2ESPplus_2E2(s(tyop_2Ecanonical_2Espolynom(A_27a),V3a0_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),V2a1_2E0)))
& ! [V5a1_2E0,V6a0_2E0,V7a_2E0] : s(tyop_2Ecanonical_2Espolynom(A_27a),c_2Ecanonical_2ESPvar_2E1(s(tyop_2Equote_2Eindex,V7a_2E0))) != s(tyop_2Ecanonical_2Espolynom(A_27a),c_2Ecanonical_2ESPmult_2E2(s(tyop_2Ecanonical_2Espolynom(A_27a),V6a0_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),V5a1_2E0)))
& ! [V8a1_2E0,V9a0_2E0,V10a_2E0] : s(tyop_2Ecanonical_2Espolynom(A_27a),c_2Ecanonical_2ESPconst_2E1(s(A_27a,V10a_2E0))) != s(tyop_2Ecanonical_2Espolynom(A_27a),c_2Ecanonical_2ESPplus_2E2(s(tyop_2Ecanonical_2Espolynom(A_27a),V9a0_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),V8a1_2E0)))
& ! [V11a1_2E0,V12a0_2E0,V13a_2E0] : s(tyop_2Ecanonical_2Espolynom(A_27a),c_2Ecanonical_2ESPconst_2E1(s(A_27a,V13a_2E0))) != s(tyop_2Ecanonical_2Espolynom(A_27a),c_2Ecanonical_2ESPmult_2E2(s(tyop_2Ecanonical_2Espolynom(A_27a),V12a0_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),V11a1_2E0)))
& ! [V14a1_27_2E0,V15a1_2E0,V16a0_27_2E0,V17a0_2E0] : s(tyop_2Ecanonical_2Espolynom(A_27a),c_2Ecanonical_2ESPplus_2E2(s(tyop_2Ecanonical_2Espolynom(A_27a),V17a0_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),V15a1_2E0))) != s(tyop_2Ecanonical_2Espolynom(A_27a),c_2Ecanonical_2ESPmult_2E2(s(tyop_2Ecanonical_2Espolynom(A_27a),V16a0_27_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),V14a1_27_2E0))) ) ).
fof(thm_2Ecanonical_2Espolynom__nchotomy,axiom,
! [A_27a,V0ss_2E0] :
( ? [V1i_2E0] : s(tyop_2Ecanonical_2Espolynom(A_27a),V0ss_2E0) = s(tyop_2Ecanonical_2Espolynom(A_27a),c_2Ecanonical_2ESPvar_2E1(s(tyop_2Equote_2Eindex,V1i_2E0)))
| ? [V2a_2E0] : s(tyop_2Ecanonical_2Espolynom(A_27a),V0ss_2E0) = s(tyop_2Ecanonical_2Espolynom(A_27a),c_2Ecanonical_2ESPconst_2E1(s(A_27a,V2a_2E0)))
| ? [V3s_2E0,V4s0_2E0] : s(tyop_2Ecanonical_2Espolynom(A_27a),V0ss_2E0) = s(tyop_2Ecanonical_2Espolynom(A_27a),c_2Ecanonical_2ESPplus_2E2(s(tyop_2Ecanonical_2Espolynom(A_27a),V3s_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),V4s0_2E0)))
| ? [V5s_2E0,V6s0_2E0] : s(tyop_2Ecanonical_2Espolynom(A_27a),V0ss_2E0) = s(tyop_2Ecanonical_2Espolynom(A_27a),c_2Ecanonical_2ESPmult_2E2(s(tyop_2Ecanonical_2Espolynom(A_27a),V5s_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),V6s0_2E0))) ) ).
fof(thm_2Ecanonical_2Espolynom__Axiom,axiom,
! [A_27a,A_27b,V0f0_2E0,V1f1_2E0,V2f2_2E0,V3f3_2E0] :
? [V4fn_2E0] :
( ! [V5a_2E0] : s(A_27b,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27b),V4fn_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),c_2Ecanonical_2ESPvar_2E1(s(tyop_2Equote_2Eindex,V5a_2E0))))) = s(A_27b,app_2E2(s(tyop_2Emin_2Efun(tyop_2Equote_2Eindex,A_27b),V0f0_2E0),s(tyop_2Equote_2Eindex,V5a_2E0)))
& ! [V6a_2E0] : s(A_27b,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27b),V4fn_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),c_2Ecanonical_2ESPconst_2E1(s(A_27a,V6a_2E0))))) = s(A_27b,app_2E2(s(tyop_2Emin_2Efun(A_27a,A_27b),V1f1_2E0),s(A_27a,V6a_2E0)))
& ! [V7a0_2E0,V8a1_2E0] : s(A_27b,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27b),V4fn_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),c_2Ecanonical_2ESPplus_2E2(s(tyop_2Ecanonical_2Espolynom(A_27a),V7a0_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),V8a1_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_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b)))),V2f2_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),V7a0_2E0))),s(tyop_2Ecanonical_2Espolynom(A_27a),V8a1_2E0))),s(A_27b,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27b),V4fn_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),V7a0_2E0))))),s(A_27b,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27b),V4fn_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),V8a1_2E0)))))
& ! [V9a0_2E0,V10a1_2E0] : s(A_27b,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27b),V4fn_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),c_2Ecanonical_2ESPmult_2E2(s(tyop_2Ecanonical_2Espolynom(A_27a),V9a0_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),V10a1_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_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b)))),V3f3_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),V9a0_2E0))),s(tyop_2Ecanonical_2Espolynom(A_27a),V10a1_2E0))),s(A_27b,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27b),V4fn_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),V9a0_2E0))))),s(A_27b,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27b),V4fn_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),V10a1_2E0))))) ) ).
fof(thm_2Ecanonical_2Espolynom__induction,axiom,
! [A_27a,V0P_2E0] :
( ( ! [V1i_2E0] : p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),c_2Ecanonical_2ESPvar_2E1(s(tyop_2Equote_2Eindex,V1i_2E0))))))
& ! [V2a_2E0] : p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),c_2Ecanonical_2ESPconst_2E1(s(A_27a,V2a_2E0))))))
& ! [V3s_2E0,V4s0_2E0] :
( ( p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),V3s_2E0))))
& p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),V4s0_2E0)))) )
=> p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),c_2Ecanonical_2ESPplus_2E2(s(tyop_2Ecanonical_2Espolynom(A_27a),V3s_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),V4s0_2E0)))))) )
& ! [V5s_2E0,V6s0_2E0] :
( ( p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),V5s_2E0))))
& p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),V6s0_2E0)))) )
=> p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),c_2Ecanonical_2ESPmult_2E2(s(tyop_2Ecanonical_2Espolynom(A_27a),V5s_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),V6s0_2E0)))))) ) )
=> ! [V7s_2E0] : p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),V7s_2E0)))) ) ).
fof(thm_2Ecanonical_2Espolynom__case__cong,axiom,
! [A_27a,A_27b,V0f3_27_2E0,V1f2_27_2E0,V2f1_27_2E0,V3f_27_2E0,V4M_2E0,V5M_27_2E0,V6f_2E0,V7f1_2E0,V8f2_2E0,V9f3_2E0] :
( ( s(tyop_2Ecanonical_2Espolynom(A_27a),V4M_2E0) = s(tyop_2Ecanonical_2Espolynom(A_27a),V5M_27_2E0)
& ! [V10a_2E0] :
( s(tyop_2Ecanonical_2Espolynom(A_27a),V5M_27_2E0) = s(tyop_2Ecanonical_2Espolynom(A_27a),c_2Ecanonical_2ESPvar_2E1(s(tyop_2Equote_2Eindex,V10a_2E0)))
=> s(A_27b,app_2E2(s(tyop_2Emin_2Efun(tyop_2Equote_2Eindex,A_27b),V6f_2E0),s(tyop_2Equote_2Eindex,V10a_2E0))) = s(A_27b,app_2E2(s(tyop_2Emin_2Efun(tyop_2Equote_2Eindex,A_27b),V3f_27_2E0),s(tyop_2Equote_2Eindex,V10a_2E0))) )
& ! [V11a_2E0] :
( s(tyop_2Ecanonical_2Espolynom(A_27a),V5M_27_2E0) = s(tyop_2Ecanonical_2Espolynom(A_27a),c_2Ecanonical_2ESPconst_2E1(s(A_27a,V11a_2E0)))
=> s(A_27b,app_2E2(s(tyop_2Emin_2Efun(A_27a,A_27b),V7f1_2E0),s(A_27a,V11a_2E0))) = s(A_27b,app_2E2(s(tyop_2Emin_2Efun(A_27a,A_27b),V2f1_27_2E0),s(A_27a,V11a_2E0))) )
& ! [V12a0_2E0,V13a1_2E0] :
( s(tyop_2Ecanonical_2Espolynom(A_27a),V5M_27_2E0) = s(tyop_2Ecanonical_2Espolynom(A_27a),c_2Ecanonical_2ESPplus_2E2(s(tyop_2Ecanonical_2Espolynom(A_27a),V12a0_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),V13a1_2E0)))
=> s(A_27b,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27b),app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27b)),V8f2_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),V12a0_2E0))),s(tyop_2Ecanonical_2Espolynom(A_27a),V13a1_2E0))) = s(A_27b,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27b),app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27b)),V1f2_27_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),V12a0_2E0))),s(tyop_2Ecanonical_2Espolynom(A_27a),V13a1_2E0))) )
& ! [V14a0_2E0,V15a1_2E0] :
( s(tyop_2Ecanonical_2Espolynom(A_27a),V5M_27_2E0) = s(tyop_2Ecanonical_2Espolynom(A_27a),c_2Ecanonical_2ESPmult_2E2(s(tyop_2Ecanonical_2Espolynom(A_27a),V14a0_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),V15a1_2E0)))
=> s(A_27b,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27b),app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27b)),V9f3_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),V14a0_2E0))),s(tyop_2Ecanonical_2Espolynom(A_27a),V15a1_2E0))) = s(A_27b,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27b),app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27b)),V0f3_27_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),V14a0_2E0))),s(tyop_2Ecanonical_2Espolynom(A_27a),V15a1_2E0))) ) )
=> s(A_27b,c_2Ecanonical_2Espolynom__CASE_2E5(s(tyop_2Ecanonical_2Espolynom(A_27a),V4M_2E0),s(tyop_2Emin_2Efun(tyop_2Equote_2Eindex,A_27b),V6f_2E0),s(tyop_2Emin_2Efun(A_27a,A_27b),V7f1_2E0),s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27b)),V8f2_2E0),s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27b)),V9f3_2E0))) = s(A_27b,c_2Ecanonical_2Espolynom__CASE_2E5(s(tyop_2Ecanonical_2Espolynom(A_27a),V5M_27_2E0),s(tyop_2Emin_2Efun(tyop_2Equote_2Eindex,A_27b),V3f_27_2E0),s(tyop_2Emin_2Efun(A_27a,A_27b),V2f1_27_2E0),s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27b)),V1f2_27_2E0),s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27b)),V0f3_27_2E0))) ) ).
fof(thm_2Ecanonical_2Espolynom__case__eq,axiom,
! [A_27a,A_27b,V0x_2E0,V1v_2E0,V2f3_2E0,V3f2_2E0,V4f1_2E0,V5f_2E0] :
( s(A_27b,c_2Ecanonical_2Espolynom__CASE_2E5(s(tyop_2Ecanonical_2Espolynom(A_27a),V0x_2E0),s(tyop_2Emin_2Efun(tyop_2Equote_2Eindex,A_27b),V5f_2E0),s(tyop_2Emin_2Efun(A_27a,A_27b),V4f1_2E0),s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27b)),V3f2_2E0),s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27b)),V2f3_2E0))) = s(A_27b,V1v_2E0)
<=> ( ? [V6i_2E0] :
( s(tyop_2Ecanonical_2Espolynom(A_27a),V0x_2E0) = s(tyop_2Ecanonical_2Espolynom(A_27a),c_2Ecanonical_2ESPvar_2E1(s(tyop_2Equote_2Eindex,V6i_2E0)))
& s(A_27b,app_2E2(s(tyop_2Emin_2Efun(tyop_2Equote_2Eindex,A_27b),V5f_2E0),s(tyop_2Equote_2Eindex,V6i_2E0))) = s(A_27b,V1v_2E0) )
| ? [V7a_2E0] :
( s(tyop_2Ecanonical_2Espolynom(A_27a),V0x_2E0) = s(tyop_2Ecanonical_2Espolynom(A_27a),c_2Ecanonical_2ESPconst_2E1(s(A_27a,V7a_2E0)))
& s(A_27b,app_2E2(s(tyop_2Emin_2Efun(A_27a,A_27b),V4f1_2E0),s(A_27a,V7a_2E0))) = s(A_27b,V1v_2E0) )
| ? [V8s_2E0,V9s0_2E0] :
( s(tyop_2Ecanonical_2Espolynom(A_27a),V0x_2E0) = s(tyop_2Ecanonical_2Espolynom(A_27a),c_2Ecanonical_2ESPplus_2E2(s(tyop_2Ecanonical_2Espolynom(A_27a),V8s_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),V9s0_2E0)))
& s(A_27b,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27b),app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27b)),V3f2_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),V8s_2E0))),s(tyop_2Ecanonical_2Espolynom(A_27a),V9s0_2E0))) = s(A_27b,V1v_2E0) )
| ? [V10s_2E0,V11s0_2E0] :
( s(tyop_2Ecanonical_2Espolynom(A_27a),V0x_2E0) = s(tyop_2Ecanonical_2Espolynom(A_27a),c_2Ecanonical_2ESPmult_2E2(s(tyop_2Ecanonical_2Espolynom(A_27a),V10s_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),V11s0_2E0)))
& s(A_27b,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27b),app_2E2(s(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27b)),V2f3_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),V10s_2E0))),s(tyop_2Ecanonical_2Espolynom(A_27a),V11s0_2E0))) = s(A_27b,V1v_2E0) ) ) ) ).
fof(thm_2Ecanonical_2Espolynomial__normalize__ok,axiom,
! [A_27a,V0sr_2E0] :
( p(s(tyop_2Emin_2Ebool,c_2Esemi__ring_2Eis__semi__ring_2E1(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0))))
=> ! [V1vm_2E0,V2p_2E0] : s(A_27a,c_2Ecanonical_2Einterp__cs_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0),s(tyop_2Equote_2Evarmap(A_27a),V1vm_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Espolynom__normalize_2E2(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),V2p_2E0))))) = s(A_27a,c_2Ecanonical_2Einterp__sp_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0),s(tyop_2Equote_2Evarmap(A_27a),V1vm_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),V2p_2E0))) ) ).
fof(thm_2Ecanonical_2Espolynomial__simplify__ok,axiom,
! [A_27a,V0sr_2E0] :
( p(s(tyop_2Emin_2Ebool,c_2Esemi__ring_2Eis__semi__ring_2E1(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0))))
=> ! [V1vm_2E0,V2p_2E0] : s(A_27a,c_2Ecanonical_2Einterp__cs_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0),s(tyop_2Equote_2Evarmap(A_27a),V1vm_2E0),s(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2Ecanonical_2Espolynom__simplify_2E2(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0),s(tyop_2Ecanonical_2Espolynom(A_27a),V2p_2E0))))) = s(A_27a,c_2Ecanonical_2Einterp__sp_2E3(s(tyop_2Esemi__ring_2Esemi__ring(A_27a),V0sr_2E0),s(tyop_2Equote_2Evarmap(A_27a),V1vm_2E0),s(tyop_2Ecanonical_2Espolynom(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) ) ).
%------------------------------------------------------------------------------