ITP001 Axioms: ITP096_7.ax
%------------------------------------------------------------------------------
% File : ITP096_7 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 syntactic export, chainy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau19] Gauthier (2019), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : canonical.ax [Gau19]
% : HL4096_7.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 269 ( 62 unt; 140 typ; 0 def)
% Number of atoms : 358 ( 247 equ)
% Maximal formula atoms : 32 ( 1 avg)
% Number of connectives : 241 ( 12 ~; 18 |; 111 &)
% ( 25 <=>; 75 =>; 0 <=; 0 <~>)
% Maximal formula depth : 25 ( 7 avg)
% Maximal term depth : 13 ( 2 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 142 ( 65 >; 77 *; 0 +; 0 <<)
% Number of predicates : 2 ( 1 usr; 0 prp; 1-2 aty)
% Number of functors : 135 ( 135 usr; 18 con; 0-7 aty)
% Number of variables : 956 ( 804 !; 38 ?; 956 :)
% ( 114 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
tff(tyop_2Ecanonical_2Ecanonical__sum,type,
tyop_2Ecanonical_2Ecanonical__sum: $tType > $tType ).
tff(tyop_2Ecanonical_2Espolynom,type,
tyop_2Ecanonical_2Espolynom: $tType > $tType ).
tff(tyop_2Eind__type_2Erecspace,type,
tyop_2Eind__type_2Erecspace: $tType > $tType ).
tff(tyop_2Elist_2Elist,type,
tyop_2Elist_2Elist: $tType > $tType ).
tff(tyop_2Emin_2Ebool,type,
tyop_2Emin_2Ebool: $tType ).
tff(tyop_2Emin_2Efun,type,
tyop_2Emin_2Efun: ( $tType * $tType ) > $tType ).
tff(tyop_2Enum_2Enum,type,
tyop_2Enum_2Enum: $tType ).
tff(tyop_2Epair_2Eprod,type,
tyop_2Epair_2Eprod: ( $tType * $tType ) > $tType ).
tff(tyop_2Equote_2Eindex,type,
tyop_2Equote_2Eindex: $tType ).
tff(tyop_2Equote_2Evarmap,type,
tyop_2Equote_2Evarmap: $tType > $tType ).
tff(tyop_2Esemi__ring_2Esemi__ring,type,
tyop_2Esemi__ring_2Esemi__ring: $tType > $tType ).
tff(tyop_2EternaryComparisons_2Eordering,type,
tyop_2EternaryComparisons_2Eordering: $tType ).
tff(app_2E2,type,
app_2E2:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Emin_2Efun(A_27a,A_27b) * A_27a ) > A_27b ) ).
tff(p,type,
p: tyop_2Emin_2Ebool > $o ).
tff(combin_i_2E0,type,
combin_i_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,A_27a) ).
tff(combin_k_2E0,type,
combin_k_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27a)) ).
tff(combin_s_2E0,type,
combin_s_2E0:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27c))) ).
tff(c_2Ebool_2E_21_2E0,type,
c_2Ebool_2E_21_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool) ).
tff(c_2Ebool_2E_21_2E1,type,
c_2Ebool_2E_21_2E1:
!>[A_27a: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) > tyop_2Emin_2Ebool ) ).
tff(c_2Earithmetic_2E_2B_2E0,type,
c_2Earithmetic_2E_2B_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum)) ).
tff(c_2Earithmetic_2E_2B_2E2,type,
c_2Earithmetic_2E_2B_2E2: ( tyop_2Enum_2Enum * tyop_2Enum_2Enum ) > tyop_2Enum_2Enum ).
tff(c_2Epair_2E_2C_2E0,type,
c_2Epair_2E_2C_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2Epair_2Eprod(A_27a,A_27b))) ).
tff(c_2Epair_2E_2C_2E2,type,
c_2Epair_2E_2C_2E2:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a * A_27b ) > tyop_2Epair_2Eprod(A_27a,A_27b) ) ).
tff(c_2Ebool_2E_2F_5C_2E0,type,
c_2Ebool_2E_2F_5C_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool)) ).
tff(c_2Ebool_2E_2F_5C_2E2,type,
c_2Ebool_2E_2F_5C_2E2: ( tyop_2Emin_2Ebool * tyop_2Emin_2Ebool ) > tyop_2Emin_2Ebool ).
tff(c_2Enum_2E0_2E0,type,
c_2Enum_2E0_2E0: tyop_2Enum_2Enum ).
tff(c_2Emin_2E_3D_2E0,type,
c_2Emin_2E_3D_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)) ).
tff(c_2Emin_2E_3D_2E2,type,
c_2Emin_2E_3D_2E2:
!>[A_27a: $tType] : ( ( A_27a * A_27a ) > tyop_2Emin_2Ebool ) ).
tff(c_2Emin_2E_3D_3D_3E_2E0,type,
c_2Emin_2E_3D_3D_3E_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool)) ).
tff(c_2Emin_2E_3D_3D_3E_2E2,type,
c_2Emin_2E_3D_3D_3E_2E2: ( tyop_2Emin_2Ebool * tyop_2Emin_2Ebool ) > tyop_2Emin_2Ebool ).
tff(c_2Ebool_2E_3F_2E0,type,
c_2Ebool_2E_3F_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool) ).
tff(c_2Ebool_2E_3F_2E1,type,
c_2Ebool_2E_3F_2E1:
!>[A_27a: $tType] : ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) > tyop_2Emin_2Ebool ) ).
tff(c_2Ebool_2EARB_2E0,type,
c_2Ebool_2EARB_2E0:
!>[A_27a: $tType] : A_27a ).
tff(c_2Earithmetic_2EBIT1_2E0,type,
c_2Earithmetic_2EBIT1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum) ).
tff(c_2Earithmetic_2EBIT1_2E1,type,
c_2Earithmetic_2EBIT1_2E1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
tff(c_2Eind__type_2EBOTTOM_2E0,type,
c_2Eind__type_2EBOTTOM_2E0:
!>[A_27a: $tType] : tyop_2Eind__type_2Erecspace(A_27a) ).
tff(c_2Ebool_2ECOND_2E0,type,
c_2Ebool_2ECOND_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a))) ).
tff(c_2Ebool_2ECOND_2E3,type,
c_2Ebool_2ECOND_2E3:
!>[A_27a: $tType] : ( ( tyop_2Emin_2Ebool * A_27a * A_27a ) > A_27a ) ).
tff(c_2Elist_2ECONS_2E0,type,
c_2Elist_2ECONS_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Elist_2Elist(A_27a))) ).
tff(c_2Elist_2ECONS_2E2,type,
c_2Elist_2ECONS_2E2:
!>[A_27a: $tType] : ( ( A_27a * tyop_2Elist_2Elist(A_27a) ) > tyop_2Elist_2Elist(A_27a) ) ).
tff(c_2Eind__type_2ECONSTR_2E0,type,
c_2Eind__type_2ECONSTR_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(A_27a)),tyop_2Eind__type_2Erecspace(A_27a)))) ).
tff(c_2Eind__type_2ECONSTR_2E3,type,
c_2Eind__type_2ECONSTR_2E3:
!>[A_27a: $tType] : ( ( tyop_2Enum_2Enum * A_27a * tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(A_27a)) ) > tyop_2Eind__type_2Erecspace(A_27a) ) ).
tff(c_2Ecanonical_2ECons__monom_2E0,type,
c_2Ecanonical_2ECons__monom_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)))) ).
tff(c_2Ecanonical_2ECons__monom_2E3,type,
c_2Ecanonical_2ECons__monom_2E3:
!>[A_27a: $tType] : ( ( A_27a * tyop_2Elist_2Elist(tyop_2Equote_2Eindex) * tyop_2Ecanonical_2Ecanonical__sum(A_27a) ) > tyop_2Ecanonical_2Ecanonical__sum(A_27a) ) ).
tff(c_2Ecanonical_2ECons__varlist_2E0,type,
c_2Ecanonical_2ECons__varlist_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a))) ).
tff(c_2Ecanonical_2ECons__varlist_2E2,type,
c_2Ecanonical_2ECons__varlist_2E2:
!>[A_27a: $tType] : ( ( tyop_2Elist_2Elist(tyop_2Equote_2Eindex) * tyop_2Ecanonical_2Ecanonical__sum(A_27a) ) > tyop_2Ecanonical_2Ecanonical__sum(A_27a) ) ).
tff(c_2Ebool_2EDATATYPE_2E0,type,
c_2Ebool_2EDATATYPE_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) ).
tff(c_2Ebool_2EDATATYPE_2E1,type,
c_2Ebool_2EDATATYPE_2E1:
!>[A_27a: $tType] : ( A_27a > tyop_2Emin_2Ebool ) ).
tff(c_2EternaryComparisons_2EEQUAL_2E0,type,
c_2EternaryComparisons_2EEQUAL_2E0: tyop_2EternaryComparisons_2Eordering ).
tff(c_2Ebool_2EF_2E0,type,
c_2Ebool_2EF_2E0: tyop_2Emin_2Ebool ).
tff(c_2Eind__type_2EFCONS_2E0,type,
c_2Eind__type_2EFCONS_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a),tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a))) ).
tff(c_2Eind__type_2EFCONS_2E2,type,
c_2Eind__type_2EFCONS_2E2:
!>[A_27a: $tType] : ( ( A_27a * tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a) ) > tyop_2Emin_2Efun(tyop_2Enum_2Enum,A_27a) ) ).
tff(c_2EternaryComparisons_2EGREATER_2E0,type,
c_2EternaryComparisons_2EGREATER_2E0: tyop_2EternaryComparisons_2Eordering ).
tff(c_2EternaryComparisons_2ELESS_2E0,type,
c_2EternaryComparisons_2ELESS_2E0: tyop_2EternaryComparisons_2Eordering ).
tff(c_2Elist_2ENIL_2E0,type,
c_2Elist_2ENIL_2E0:
!>[A_27a: $tType] : tyop_2Elist_2Elist(A_27a) ).
tff(c_2Earithmetic_2ENUMERAL_2E0,type,
c_2Earithmetic_2ENUMERAL_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum) ).
tff(c_2Earithmetic_2ENUMERAL_2E1,type,
c_2Earithmetic_2ENUMERAL_2E1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
tff(c_2Ecanonical_2ENil__monom_2E0,type,
c_2Ecanonical_2ENil__monom_2E0:
!>[A_27a: $tType] : tyop_2Ecanonical_2Ecanonical__sum(A_27a) ).
tff(c_2Ecanonical_2ESPconst_2E0,type,
c_2Ecanonical_2ESPconst_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(A_27a,tyop_2Ecanonical_2Espolynom(A_27a)) ).
tff(c_2Ecanonical_2ESPconst_2E1,type,
c_2Ecanonical_2ESPconst_2E1:
!>[A_27a: $tType] : ( A_27a > tyop_2Ecanonical_2Espolynom(A_27a) ) ).
tff(c_2Ecanonical_2ESPmult_2E0,type,
c_2Ecanonical_2ESPmult_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Ecanonical_2Espolynom(A_27a))) ).
tff(c_2Ecanonical_2ESPmult_2E2,type,
c_2Ecanonical_2ESPmult_2E2:
!>[A_27a: $tType] : ( ( tyop_2Ecanonical_2Espolynom(A_27a) * tyop_2Ecanonical_2Espolynom(A_27a) ) > tyop_2Ecanonical_2Espolynom(A_27a) ) ).
tff(c_2Ecanonical_2ESPplus_2E0,type,
c_2Ecanonical_2ESPplus_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Ecanonical_2Espolynom(A_27a))) ).
tff(c_2Ecanonical_2ESPplus_2E2,type,
c_2Ecanonical_2ESPplus_2E2:
!>[A_27a: $tType] : ( ( tyop_2Ecanonical_2Espolynom(A_27a) * tyop_2Ecanonical_2Espolynom(A_27a) ) > tyop_2Ecanonical_2Espolynom(A_27a) ) ).
tff(c_2Ecanonical_2ESPvar_2E0,type,
c_2Ecanonical_2ESPvar_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Equote_2Eindex,tyop_2Ecanonical_2Espolynom(A_27a)) ).
tff(c_2Ecanonical_2ESPvar_2E1,type,
c_2Ecanonical_2ESPvar_2E1:
!>[A_27a: $tType] : ( tyop_2Equote_2Eindex > tyop_2Ecanonical_2Espolynom(A_27a) ) ).
tff(c_2Enum_2ESUC_2E0,type,
c_2Enum_2ESUC_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum) ).
tff(c_2Enum_2ESUC_2E1,type,
c_2Enum_2ESUC_2E1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
tff(c_2Ebool_2ET_2E0,type,
c_2Ebool_2ET_2E0: tyop_2Emin_2Ebool ).
tff(c_2Ebool_2ETYPE__DEFINITION_2E0,type,
c_2Ebool_2ETYPE__DEFINITION_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27b,A_27a),tyop_2Emin_2Ebool)) ).
tff(c_2Ebool_2ETYPE__DEFINITION_2E2,type,
c_2Ebool_2ETYPE__DEFINITION_2E2:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool) * tyop_2Emin_2Efun(A_27b,A_27a) ) > tyop_2Emin_2Ebool ) ).
tff(c_2Earithmetic_2EZERO_2E0,type,
c_2Earithmetic_2EZERO_2E0: tyop_2Enum_2Enum ).
tff(c_2Ebool_2E_5C_2F_2E0,type,
c_2Ebool_2E_5C_2F_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool)) ).
tff(c_2Ebool_2E_5C_2F_2E2,type,
c_2Ebool_2E_5C_2F_2E2: ( tyop_2Emin_2Ebool * tyop_2Emin_2Ebool ) > tyop_2Emin_2Ebool ).
tff(c_2Ecanonical_2Ecanonical__sum__CASE_2E0,type,
c_2Ecanonical_2Ecanonical__sum__CASE_2E0:
!>[A_27a: $tType,A_27b: $tType] : 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)))) ).
tff(c_2Ecanonical_2Ecanonical__sum__CASE_2E4,type,
c_2Ecanonical_2Ecanonical__sum__CASE_2E4:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Ecanonical_2Ecanonical__sum(A_27a) * A_27b * 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_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27b)) ) > A_27b ) ).
tff(c_2Ecanonical_2Ecanonical__sum__merge_2E0,type,
c_2Ecanonical_2Ecanonical__sum__merge_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)))) ).
tff(c_2Ecanonical_2Ecanonical__sum__merge_2E3,type,
c_2Ecanonical_2Ecanonical__sum__merge_2E3:
!>[A_27a: $tType] : ( ( tyop_2Esemi__ring_2Esemi__ring(A_27a) * tyop_2Ecanonical_2Ecanonical__sum(A_27a) * tyop_2Ecanonical_2Ecanonical__sum(A_27a) ) > tyop_2Ecanonical_2Ecanonical__sum(A_27a) ) ).
tff(c_2Ecanonical_2Ecanonical__sum__prod_2E0,type,
c_2Ecanonical_2Ecanonical__sum__prod_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)))) ).
tff(c_2Ecanonical_2Ecanonical__sum__prod_2E3,type,
c_2Ecanonical_2Ecanonical__sum__prod_2E3:
!>[A_27a: $tType] : ( ( tyop_2Esemi__ring_2Esemi__ring(A_27a) * tyop_2Ecanonical_2Ecanonical__sum(A_27a) * tyop_2Ecanonical_2Ecanonical__sum(A_27a) ) > tyop_2Ecanonical_2Ecanonical__sum(A_27a) ) ).
tff(c_2Ecanonical_2Ecanonical__sum__scalar_2E0,type,
c_2Ecanonical_2Ecanonical__sum__scalar_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)))) ).
tff(c_2Ecanonical_2Ecanonical__sum__scalar_2E3,type,
c_2Ecanonical_2Ecanonical__sum__scalar_2E3:
!>[A_27a: $tType] : ( ( tyop_2Esemi__ring_2Esemi__ring(A_27a) * A_27a * tyop_2Ecanonical_2Ecanonical__sum(A_27a) ) > tyop_2Ecanonical_2Ecanonical__sum(A_27a) ) ).
tff(c_2Ecanonical_2Ecanonical__sum__scalar2_2E0,type,
c_2Ecanonical_2Ecanonical__sum__scalar2_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)))) ).
tff(c_2Ecanonical_2Ecanonical__sum__scalar2_2E3,type,
c_2Ecanonical_2Ecanonical__sum__scalar2_2E3:
!>[A_27a: $tType] : ( ( tyop_2Esemi__ring_2Esemi__ring(A_27a) * tyop_2Elist_2Elist(tyop_2Equote_2Eindex) * tyop_2Ecanonical_2Ecanonical__sum(A_27a) ) > tyop_2Ecanonical_2Ecanonical__sum(A_27a) ) ).
tff(c_2Ecanonical_2Ecanonical__sum__scalar3_2E0,type,
c_2Ecanonical_2Ecanonical__sum__scalar3_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a))))) ).
tff(c_2Ecanonical_2Ecanonical__sum__scalar3_2E4,type,
c_2Ecanonical_2Ecanonical__sum__scalar3_2E4:
!>[A_27a: $tType] : ( ( tyop_2Esemi__ring_2Esemi__ring(A_27a) * A_27a * tyop_2Elist_2Elist(tyop_2Equote_2Eindex) * tyop_2Ecanonical_2Ecanonical__sum(A_27a) ) > tyop_2Ecanonical_2Ecanonical__sum(A_27a) ) ).
tff(c_2Ecanonical_2Ecanonical__sum__simplify_2E0,type,
c_2Ecanonical_2Ecanonical__sum__simplify_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a))) ).
tff(c_2Ecanonical_2Ecanonical__sum__simplify_2E2,type,
c_2Ecanonical_2Ecanonical__sum__simplify_2E2:
!>[A_27a: $tType] : ( ( tyop_2Esemi__ring_2Esemi__ring(A_27a) * tyop_2Ecanonical_2Ecanonical__sum(A_27a) ) > tyop_2Ecanonical_2Ecanonical__sum(A_27a) ) ).
tff(c_2Ecanonical_2Ecanonical__sum__size_2E0,type,
c_2Ecanonical_2Ecanonical__sum__size_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Enum_2Enum)) ).
tff(c_2Ecanonical_2Ecanonical__sum__size_2E2,type,
c_2Ecanonical_2Ecanonical__sum__size_2E2:
!>[A_27a: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum) * tyop_2Ecanonical_2Ecanonical__sum(A_27a) ) > tyop_2Enum_2Enum ) ).
tff(c_2Ecanonical_2Eics__aux_2E0,type,
c_2Ecanonical_2Eics__aux_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(tyop_2Equote_2Evarmap(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27a)))) ).
tff(c_2Ecanonical_2Eics__aux_2E4,type,
c_2Ecanonical_2Eics__aux_2E4:
!>[A_27a: $tType] : ( ( tyop_2Esemi__ring_2Esemi__ring(A_27a) * tyop_2Equote_2Evarmap(A_27a) * A_27a * tyop_2Ecanonical_2Ecanonical__sum(A_27a) ) > A_27a ) ).
tff(c_2Equote_2Eindex__compare_2E0,type,
c_2Equote_2Eindex__compare_2E0: tyop_2Emin_2Efun(tyop_2Equote_2Eindex,tyop_2Emin_2Efun(tyop_2Equote_2Eindex,tyop_2EternaryComparisons_2Eordering)) ).
tff(c_2Equote_2Eindex__lt_2E0,type,
c_2Equote_2Eindex__lt_2E0: tyop_2Emin_2Efun(tyop_2Equote_2Eindex,tyop_2Emin_2Efun(tyop_2Equote_2Eindex,tyop_2Emin_2Ebool)) ).
tff(c_2Equote_2Eindex__size_2E0,type,
c_2Equote_2Eindex__size_2E0: tyop_2Emin_2Efun(tyop_2Equote_2Eindex,tyop_2Enum_2Enum) ).
tff(c_2Equote_2Eindex__size_2E1,type,
c_2Equote_2Eindex__size_2E1: tyop_2Equote_2Eindex > tyop_2Enum_2Enum ).
tff(c_2Ecanonical_2Einterp__cs_2E0,type,
c_2Ecanonical_2Einterp__cs_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(tyop_2Equote_2Evarmap(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27a))) ).
tff(c_2Ecanonical_2Einterp__cs_2E3,type,
c_2Ecanonical_2Einterp__cs_2E3:
!>[A_27a: $tType] : ( ( tyop_2Esemi__ring_2Esemi__ring(A_27a) * tyop_2Equote_2Evarmap(A_27a) * tyop_2Ecanonical_2Ecanonical__sum(A_27a) ) > A_27a ) ).
tff(c_2Ecanonical_2Einterp__m_2E0,type,
c_2Ecanonical_2Einterp__m_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(tyop_2Equote_2Evarmap(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),A_27a)))) ).
tff(c_2Ecanonical_2Einterp__m_2E4,type,
c_2Ecanonical_2Einterp__m_2E4:
!>[A_27a: $tType] : ( ( tyop_2Esemi__ring_2Esemi__ring(A_27a) * tyop_2Equote_2Evarmap(A_27a) * A_27a * tyop_2Elist_2Elist(tyop_2Equote_2Eindex) ) > A_27a ) ).
tff(c_2Ecanonical_2Einterp__sp_2E0,type,
c_2Ecanonical_2Einterp__sp_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(tyop_2Equote_2Evarmap(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27a))) ).
tff(c_2Ecanonical_2Einterp__sp_2E3,type,
c_2Ecanonical_2Einterp__sp_2E3:
!>[A_27a: $tType] : ( ( tyop_2Esemi__ring_2Esemi__ring(A_27a) * tyop_2Equote_2Evarmap(A_27a) * tyop_2Ecanonical_2Espolynom(A_27a) ) > A_27a ) ).
tff(c_2Ecanonical_2Einterp__vl_2E0,type,
c_2Ecanonical_2Einterp__vl_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(tyop_2Equote_2Evarmap(A_27a),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),A_27a))) ).
tff(c_2Ecanonical_2Einterp__vl_2E3,type,
c_2Ecanonical_2Einterp__vl_2E3:
!>[A_27a: $tType] : ( ( tyop_2Esemi__ring_2Esemi__ring(A_27a) * tyop_2Equote_2Evarmap(A_27a) * tyop_2Elist_2Elist(tyop_2Equote_2Eindex) ) > A_27a ) ).
tff(c_2Esemi__ring_2Eis__semi__ring_2E0,type,
c_2Esemi__ring_2Eis__semi__ring_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Ebool) ).
tff(c_2Esemi__ring_2Eis__semi__ring_2E1,type,
c_2Esemi__ring_2Eis__semi__ring_2E1:
!>[A_27a: $tType] : ( tyop_2Esemi__ring_2Esemi__ring(A_27a) > tyop_2Emin_2Ebool ) ).
tff(c_2Ecanonical_2Eivl__aux_2E0,type,
c_2Ecanonical_2Eivl__aux_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(tyop_2Equote_2Evarmap(A_27a),tyop_2Emin_2Efun(tyop_2Equote_2Eindex,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),A_27a)))) ).
tff(c_2Ecanonical_2Eivl__aux_2E4,type,
c_2Ecanonical_2Eivl__aux_2E4:
!>[A_27a: $tType] : ( ( tyop_2Esemi__ring_2Esemi__ring(A_27a) * tyop_2Equote_2Evarmap(A_27a) * tyop_2Equote_2Eindex * tyop_2Elist_2Elist(tyop_2Equote_2Eindex) ) > A_27a ) ).
tff(c_2EternaryComparisons_2Elist__compare_2E0,type,
c_2EternaryComparisons_2Elist__compare_2E0:
!>[A_27a: $tType,A_27b: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2EternaryComparisons_2Eordering)),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27b),tyop_2EternaryComparisons_2Eordering))) ).
tff(c_2EternaryComparisons_2Elist__compare_2E3,type,
c_2EternaryComparisons_2Elist__compare_2E3:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,tyop_2EternaryComparisons_2Eordering)) * tyop_2Elist_2Elist(A_27a) * tyop_2Elist_2Elist(A_27b) ) > tyop_2EternaryComparisons_2Eordering ) ).
tff(c_2EternaryComparisons_2Elist__merge_2E0,type,
c_2EternaryComparisons_2Elist__merge_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Elist_2Elist(A_27a)))) ).
tff(c_2EternaryComparisons_2Elist__merge_2E3,type,
c_2EternaryComparisons_2Elist__merge_2E3:
!>[A_27a: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)) * tyop_2Elist_2Elist(A_27a) * tyop_2Elist_2Elist(A_27a) ) > tyop_2Elist_2Elist(A_27a) ) ).
tff(c_2Elist_2Elist__size_2E0,type,
c_2Elist_2Elist__size_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Elist_2Elist(A_27a),tyop_2Enum_2Enum)) ).
tff(c_2Elist_2Elist__size_2E2,type,
c_2Elist_2Elist__size_2E2:
!>[A_27a: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum) * tyop_2Elist_2Elist(A_27a) ) > tyop_2Enum_2Enum ) ).
tff(c_2Ecanonical_2Emonom__insert_2E0,type,
c_2Ecanonical_2Emonom__insert_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a))))) ).
tff(c_2Ecanonical_2Emonom__insert_2E4,type,
c_2Ecanonical_2Emonom__insert_2E4:
!>[A_27a: $tType] : ( ( tyop_2Esemi__ring_2Esemi__ring(A_27a) * A_27a * tyop_2Elist_2Elist(tyop_2Equote_2Eindex) * tyop_2Ecanonical_2Ecanonical__sum(A_27a) ) > tyop_2Ecanonical_2Ecanonical__sum(A_27a) ) ).
tff(c_2EternaryComparisons_2Eordering__CASE_2E0,type,
c_2EternaryComparisons_2Eordering__CASE_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2EternaryComparisons_2Eordering,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a)))) ).
tff(c_2EternaryComparisons_2Eordering__CASE_2E4,type,
c_2EternaryComparisons_2Eordering__CASE_2E4:
!>[A_27a: $tType] : ( ( tyop_2EternaryComparisons_2Eordering * A_27a * A_27a * A_27a ) > A_27a ) ).
tff(c_2Esemi__ring_2Esemi__ring__SR0_2E0,type,
c_2Esemi__ring_2Esemi__ring__SR0_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Esemi__ring_2Esemi__ring(A_27a),A_27a) ).
tff(c_2Esemi__ring_2Esemi__ring__SR0_2E1,type,
c_2Esemi__ring_2Esemi__ring__SR0_2E1:
!>[A_27a: $tType] : ( tyop_2Esemi__ring_2Esemi__ring(A_27a) > A_27a ) ).
tff(c_2Esemi__ring_2Esemi__ring__SR1_2E0,type,
c_2Esemi__ring_2Esemi__ring__SR1_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Esemi__ring_2Esemi__ring(A_27a),A_27a) ).
tff(c_2Esemi__ring_2Esemi__ring__SR1_2E1,type,
c_2Esemi__ring_2Esemi__ring__SR1_2E1:
!>[A_27a: $tType] : ( tyop_2Esemi__ring_2Esemi__ring(A_27a) > A_27a ) ).
tff(c_2Esemi__ring_2Esemi__ring__SRM_2E0,type,
c_2Esemi__ring_2Esemi__ring__SRM_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a))) ).
tff(c_2Esemi__ring_2Esemi__ring__SRM_2E3,type,
c_2Esemi__ring_2Esemi__ring__SRM_2E3:
!>[A_27a: $tType] : ( ( tyop_2Esemi__ring_2Esemi__ring(A_27a) * A_27a * A_27a ) > A_27a ) ).
tff(c_2Esemi__ring_2Esemi__ring__SRP_2E0,type,
c_2Esemi__ring_2Esemi__ring__SRP_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,A_27a))) ).
tff(c_2Esemi__ring_2Esemi__ring__SRP_2E3,type,
c_2Esemi__ring_2Esemi__ring__SRP_2E3:
!>[A_27a: $tType] : ( ( tyop_2Esemi__ring_2Esemi__ring(A_27a) * A_27a * A_27a ) > A_27a ) ).
tff(c_2Ecanonical_2Espolynom__CASE_2E0,type,
c_2Ecanonical_2Espolynom__CASE_2E0:
!>[A_27a: $tType,A_27b: $tType] : 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))))) ).
tff(c_2Ecanonical_2Espolynom__CASE_2E5,type,
c_2Ecanonical_2Espolynom__CASE_2E5:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Ecanonical_2Espolynom(A_27a) * tyop_2Emin_2Efun(tyop_2Equote_2Eindex,A_27b) * tyop_2Emin_2Efun(A_27a,A_27b) * tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27b)) * tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27b)) ) > A_27b ) ).
tff(c_2Ecanonical_2Espolynom__normalize_2E0,type,
c_2Ecanonical_2Espolynom__normalize_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a))) ).
tff(c_2Ecanonical_2Espolynom__normalize_2E2,type,
c_2Ecanonical_2Espolynom__normalize_2E2:
!>[A_27a: $tType] : ( ( tyop_2Esemi__ring_2Esemi__ring(A_27a) * tyop_2Ecanonical_2Espolynom(A_27a) ) > tyop_2Ecanonical_2Ecanonical__sum(A_27a) ) ).
tff(c_2Ecanonical_2Espolynom__simplify_2E0,type,
c_2Ecanonical_2Espolynom__simplify_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a))) ).
tff(c_2Ecanonical_2Espolynom__simplify_2E2,type,
c_2Ecanonical_2Espolynom__simplify_2E2:
!>[A_27a: $tType] : ( ( tyop_2Esemi__ring_2Esemi__ring(A_27a) * tyop_2Ecanonical_2Espolynom(A_27a) ) > tyop_2Ecanonical_2Ecanonical__sum(A_27a) ) ).
tff(c_2Ecanonical_2Espolynom__size_2E0,type,
c_2Ecanonical_2Espolynom__size_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Enum_2Enum)) ).
tff(c_2Ecanonical_2Espolynom__size_2E2,type,
c_2Ecanonical_2Espolynom__size_2E2:
!>[A_27a: $tType] : ( ( tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum) * tyop_2Ecanonical_2Espolynom(A_27a) ) > tyop_2Enum_2Enum ) ).
tff(c_2Ecanonical_2Evarlist__insert_2E0,type,
c_2Ecanonical_2Evarlist__insert_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)))) ).
tff(c_2Ecanonical_2Evarlist__insert_2E3,type,
c_2Ecanonical_2Evarlist__insert_2E3:
!>[A_27a: $tType] : ( ( tyop_2Esemi__ring_2Esemi__ring(A_27a) * tyop_2Elist_2Elist(tyop_2Equote_2Eindex) * tyop_2Ecanonical_2Ecanonical__sum(A_27a) ) > tyop_2Ecanonical_2Ecanonical__sum(A_27a) ) ).
tff(c_2Equote_2Evarmap__find_2E0,type,
c_2Equote_2Evarmap__find_2E0:
!>[A_27a: $tType] : tyop_2Emin_2Efun(tyop_2Equote_2Eindex,tyop_2Emin_2Efun(tyop_2Equote_2Evarmap(A_27a),A_27a)) ).
tff(c_2Equote_2Evarmap__find_2E2,type,
c_2Equote_2Evarmap__find_2E2:
!>[A_27a: $tType] : ( ( tyop_2Equote_2Eindex * tyop_2Equote_2Evarmap(A_27a) ) > A_27a ) ).
tff(c_2Ebool_2E_7E_2E0,type,
c_2Ebool_2E_7E_2E0: tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool) ).
tff(c_2Ebool_2E_7E_2E1,type,
c_2Ebool_2E_7E_2E1: tyop_2Emin_2Ebool > tyop_2Emin_2Ebool ).
tff(thm_2Eextra_2Dho_2Eeq__ext,axiom,
! [A_27a: $tType,A_27b: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V1g_2E0: tyop_2Emin_2Efun(A_27a,A_27b)] :
( ! [V2x_2E0: A_27a] : ( app_2E2(A_27a,A_27b,V0f_2E0,V2x_2E0) = app_2E2(A_27a,A_27b,V1g_2E0,V2x_2E0) )
=> ( V0f_2E0 = V1g_2E0 ) ) ).
tff(thm_2Eextra_2Dho_2Eboolext,axiom,
! [V0_2E0: tyop_2Emin_2Ebool,V1_2E0: tyop_2Emin_2Ebool] :
( ( p(V0_2E0)
<=> p(V1_2E0) )
=> ( V0_2E0 = V1_2E0 ) ) ).
tff(thm_2Eextra_2Dho_2Etruth,axiom,
p(c_2Ebool_2ET_2E0) ).
tff(thm_2Eextra_2Dho_2Enotfalse,axiom,
~ p(c_2Ebool_2EF_2E0) ).
tff(thm_2Eextra_2Dho_2Ebool__cases__ax,axiom,
! [V0t_2E0: tyop_2Emin_2Ebool] :
( ( V0t_2E0 = c_2Ebool_2ET_2E0 )
| ( V0t_2E0 = c_2Ebool_2EF_2E0 ) ) ).
tff(thm_2Eextra_2Dho_2Ei__thm,axiom,
! [A_27a: $tType,V0x_2E0: A_27a] : ( app_2E2(A_27a,A_27a,combin_i_2E0(A_27a),V0x_2E0) = V0x_2E0 ) ).
tff(thm_2Eextra_2Dho_2Ek__thm,axiom,
! [A_27a: $tType,A_27b: $tType,V0x_2E0: A_27a,V1y_2E0: A_27b] : ( app_2E2(A_27b,A_27a,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27a),combin_k_2E0(A_27a,A_27b),V0x_2E0),V1y_2E0) = V0x_2E0 ) ).
tff(thm_2Eextra_2Dho_2Es__thm,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),V1g_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V2x_2E0: A_27a] : ( app_2E2(A_27a,A_27c,app_2E2(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27c),app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27c)),combin_s_2E0(A_27a,A_27b,A_27c),V0f_2E0),V1g_2E0),V2x_2E0) = app_2E2(A_27b,A_27c,app_2E2(A_27a,tyop_2Emin_2Efun(A_27b,A_27c),V0f_2E0,V2x_2E0),app_2E2(A_27a,A_27b,V1g_2E0,V2x_2E0)) ) ).
tff(logicdef_2E_2F_5C,axiom,
! [V0_2E0: tyop_2Emin_2Ebool,V1_2E0: tyop_2Emin_2Ebool] :
( p(c_2Ebool_2E_2F_5C_2E2(V0_2E0,V1_2E0))
<=> ( p(V0_2E0)
& p(V1_2E0) ) ) ).
tff(logicdef_2E_5C_2F,axiom,
! [V0_2E0: tyop_2Emin_2Ebool,V1_2E0: tyop_2Emin_2Ebool] :
( p(c_2Ebool_2E_5C_2F_2E2(V0_2E0,V1_2E0))
<=> ( p(V0_2E0)
| p(V1_2E0) ) ) ).
tff(logicdef_2E_7E,axiom,
! [V0_2E0: tyop_2Emin_2Ebool] :
( p(c_2Ebool_2E_7E_2E1(V0_2E0))
<=> ~ p(V0_2E0) ) ).
tff(logicdef_2E_3D_3D_3E,axiom,
! [V0_2E0: tyop_2Emin_2Ebool,V1_2E0: tyop_2Emin_2Ebool] :
( p(c_2Emin_2E_3D_3D_3E_2E2(V0_2E0,V1_2E0))
<=> ( p(V0_2E0)
=> p(V1_2E0) ) ) ).
tff(logicdef_2E_3D,axiom,
! [A_27a: $tType,V0_2E0: A_27a,V1_2E0: A_27a] :
( p(c_2Emin_2E_3D_2E2(A_27a,V0_2E0,V1_2E0))
<=> ( V0_2E0 = V1_2E0 ) ) ).
tff(quantdef_2E_21,axiom,
! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2E_21_2E1(A_27a,V0f_2E0))
<=> ! [V1x_2E0: A_27a] : p(app_2E2(A_27a,tyop_2Emin_2Ebool,V0f_2E0,V1x_2E0)) ) ).
tff(quantdef_2E_3F,axiom,
! [A_27a: $tType,V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] :
( p(c_2Ebool_2E_3F_2E1(A_27a,V0f_2E0))
<=> ? [V1x_2E0: A_27a] : p(app_2E2(A_27a,tyop_2Emin_2Ebool,V0f_2E0,V1x_2E0)) ) ).
tff(arityeq1_2Ec_2Ebool_2E_21_2E1_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Ebool_2E_21_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Ebool_2E_21_2E0(A_27a),X0_2E0) ) ).
tff(arityeq2_2Ec_2Earithmetic_2E_2B_2E2,axiom,
! [X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_2B_2E2(X0_2E0,X1_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Enum_2Enum),c_2Earithmetic_2E_2B_2E0,X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2EA_27a_20mono_2Etyop_2Elist_2Elist_28tyop_2Equote_2Eindex_29,axiom,
! [A_27a: $tType,X0_2E0: A_27a,X1_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex)] : ( c_2Epair_2E_2C_2E2(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex),X0_2E0,X1_2E0) = app_2E2(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex)),app_2E2(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(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex)),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2Etyop_2Equote_2Eindex_20mono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Equote_2Eindex,X1_2E0: A_27a] : ( c_2Epair_2E_2C_2E2(tyop_2Equote_2Eindex,A_27a,X0_2E0,X1_2E0) = app_2E2(A_27a,tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a),app_2E2(tyop_2Equote_2Eindex,tyop_2Emin_2Efun(A_27a,tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)),c_2Epair_2E_2C_2E0(tyop_2Equote_2Eindex,A_27a),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ebool_2E_2F_5C_2E2,axiom,
! [X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Emin_2Ebool] :
( ( p(X0_2E0)
& p(X1_2E0) )
<=> p(app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),c_2Ebool_2E_2F_5C_2E0,X0_2E0),X1_2E0)) ) ).
tff(arityeq2_2Ec_2Emin_2E_3D_2E2_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: A_27a,X1_2E0: A_27a] :
( ( X0_2E0 = X1_2E0 )
<=> p(app_2E2(A_27a,tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),c_2Emin_2E_3D_2E0(A_27a),X0_2E0),X1_2E0)) ) ).
tff(arityeq2_2Ec_2Emin_2E_3D_3D_3E_2E2,axiom,
! [X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Emin_2Ebool] :
( ( p(X0_2E0)
=> p(X1_2E0) )
<=> p(app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),c_2Emin_2E_3D_3D_3E_2E0,X0_2E0),X1_2E0)) ) ).
tff(arityeq1_2Ec_2Ebool_2E_3F_2E1_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)] : ( c_2Ebool_2E_3F_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool,c_2Ebool_2E_3F_2E0(A_27a),X0_2E0) ) ).
tff(arityeq1_2Ec_2Earithmetic_2EBIT1_2E1,axiom,
! [X0_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2EBIT1_2E1(X0_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Earithmetic_2EBIT1_2E0,X0_2E0) ) ).
tff(arityeq3_2Ec_2Ebool_2ECOND_2E3_2Emono_2Etyop_2Ecanonical_2Ecanonical__sum_28A_27a_29,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a),X2_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : ( c_2Ebool_2ECOND_2E3(tyop_2Ecanonical_2Ecanonical__sum(A_27a),X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a),app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)),app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a))),c_2Ebool_2ECOND_2E0(tyop_2Ecanonical_2Ecanonical__sum(A_27a)),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq2_2Ec_2Elist_2ECONS_2E2_2Emono_2Etyop_2Equote_2Eindex,axiom,
! [X0_2E0: tyop_2Equote_2Eindex,X1_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex)] : ( c_2Elist_2ECONS_2E2(tyop_2Equote_2Eindex,X0_2E0,X1_2E0) = app_2E2(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Elist_2Elist(tyop_2Equote_2Eindex),app_2E2(tyop_2Equote_2Eindex,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Elist_2Elist(tyop_2Equote_2Eindex)),c_2Elist_2ECONS_2E0(tyop_2Equote_2Eindex),X0_2E0),X1_2E0) ) ).
tff(arityeq3_2Ec_2Eind__type_2ECONSTR_2E3_2Emono_2Etyop_2Epair_2Eprod_28A_27a_2Ctyop_2Elist_2Elist_28tyop_2Equote_2Eindex_29_29,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex)),X2_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex))))] : ( c_2Eind__type_2ECONSTR_2E3(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex)),X0_2E0,X1_2E0,X2_2E0) = app_2E2(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(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(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(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex))),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq3_2Ec_2Eind__type_2ECONSTR_2E3_2Emono_2Etyop_2Epair_2Eprod_28tyop_2Equote_2Eindex_2CA_27a_29,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Enum_2Enum,X1_2E0: tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a),X2_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)))] : ( c_2Eind__type_2ECONSTR_2E3(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a),X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a))),tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)),app_2E2(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a))),tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a))),app_2E2(tyop_2Enum_2Enum,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a))),tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)))),c_2Eind__type_2ECONSTR_2E0(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq3_2Ec_2Ecanonical_2ECons__monom_2E3_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: A_27a,X1_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),X2_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : ( c_2Ecanonical_2ECons__monom_2E3(A_27a,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a),app_2E2(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a))),c_2Ecanonical_2ECons__monom_2E0(A_27a),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq2_2Ec_2Ecanonical_2ECons__varlist_2E2_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),X1_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : ( c_2Ecanonical_2ECons__varlist_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a),app_2E2(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)),c_2Ecanonical_2ECons__varlist_2E0(A_27a),X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Ebool_2EDATATYPE_2E1_2Emono_2Etyop_2Emin_2Ebool,axiom,
! [X0_2E0: tyop_2Emin_2Ebool] : ( c_2Ebool_2EDATATYPE_2E1(tyop_2Emin_2Ebool,X0_2E0) = app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool,c_2Ebool_2EDATATYPE_2E0(tyop_2Emin_2Ebool),X0_2E0) ) ).
tff(arityeq2_2Ec_2Eind__type_2EFCONS_2E2_2Emono_2Etyop_2Eind__type_2Erecspace_28tyop_2Epair_2Eprod_28A_27a_2Ctyop_2Elist_2Elist_28tyop_2Equote_2Eindex_29_29_29,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex))),X1_2E0: 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(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex))),X0_2E0,X1_2E0) = app_2E2(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(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(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex)))),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Eind__type_2EFCONS_2E2_2Emono_2Etyop_2Eind__type_2Erecspace_28tyop_2Epair_2Eprod_28tyop_2Equote_2Eindex_2CA_27a_29_29,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)),X1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)))] : ( c_2Eind__type_2EFCONS_2E2(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)),X0_2E0,X1_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a))),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a))),app_2E2(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a))),tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)))),c_2Eind__type_2EFCONS_2E0(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a))),X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Earithmetic_2ENUMERAL_2E1,axiom,
! [X0_2E0: tyop_2Enum_2Enum] : ( c_2Earithmetic_2ENUMERAL_2E1(X0_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Earithmetic_2ENUMERAL_2E0,X0_2E0) ) ).
tff(arityeq1_2Ec_2Ecanonical_2ESPconst_2E1_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: A_27a] : ( c_2Ecanonical_2ESPconst_2E1(A_27a,X0_2E0) = app_2E2(A_27a,tyop_2Ecanonical_2Espolynom(A_27a),c_2Ecanonical_2ESPconst_2E0(A_27a),X0_2E0) ) ).
tff(arityeq2_2Ec_2Ecanonical_2ESPmult_2E2_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Ecanonical_2Espolynom(A_27a),X1_2E0: tyop_2Ecanonical_2Espolynom(A_27a)] : ( c_2Ecanonical_2ESPmult_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Ecanonical_2Espolynom(A_27a),app_2E2(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Ecanonical_2Espolynom(A_27a)),c_2Ecanonical_2ESPmult_2E0(A_27a),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ecanonical_2ESPplus_2E2_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Ecanonical_2Espolynom(A_27a),X1_2E0: tyop_2Ecanonical_2Espolynom(A_27a)] : ( c_2Ecanonical_2ESPplus_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Ecanonical_2Espolynom(A_27a),app_2E2(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Ecanonical_2Espolynom(A_27a)),c_2Ecanonical_2ESPplus_2E0(A_27a),X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Ecanonical_2ESPvar_2E1_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Equote_2Eindex] : ( c_2Ecanonical_2ESPvar_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Equote_2Eindex,tyop_2Ecanonical_2Espolynom(A_27a),c_2Ecanonical_2ESPvar_2E0(A_27a),X0_2E0) ) ).
tff(arityeq1_2Ec_2Enum_2ESUC_2E1,axiom,
! [X0_2E0: tyop_2Enum_2Enum] : ( c_2Enum_2ESUC_2E1(X0_2E0) = app_2E2(tyop_2Enum_2Enum,tyop_2Enum_2Enum,c_2Enum_2ESUC_2E0,X0_2E0) ) ).
tff(arityeq2_2Ec_2Ebool_2ETYPE__DEFINITION_2E2_2Emono_2Etyop_2Eind__type_2Erecspace_28tyop_2Epair_2Eprod_28A_27a_2Ctyop_2Elist_2Elist_28tyop_2Equote_2Eindex_29_29_29_20mono_2Etyop_2Ecanonical_2Ecanonical__sum_28A_27a_29,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex))),tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex))))] : ( c_2Ebool_2ETYPE__DEFINITION_2E2(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex))),tyop_2Ecanonical_2Ecanonical__sum(A_27a),X0_2E0,X1_2E0) = app_2E2(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(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(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex))),tyop_2Ecanonical_2Ecanonical__sum(A_27a)),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ebool_2ETYPE__DEFINITION_2E2_2Emono_2Etyop_2Eind__type_2Erecspace_28tyop_2Epair_2Eprod_28tyop_2Equote_2Eindex_2CA_27a_29_29_20mono_2Etyop_2Ecanonical_2Espolynom_28A_27a_29,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)),tyop_2Emin_2Ebool),X1_2E0: tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)))] : ( c_2Ebool_2ETYPE__DEFINITION_2E2(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)),tyop_2Ecanonical_2Espolynom(A_27a),X0_2E0,X1_2E0) = app_2E2(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(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(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)),tyop_2Ecanonical_2Espolynom(A_27a)),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ebool_2E_5C_2F_2E2,axiom,
! [X0_2E0: tyop_2Emin_2Ebool,X1_2E0: tyop_2Emin_2Ebool] :
( ( p(X0_2E0)
| p(X1_2E0) )
<=> p(app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool,app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),c_2Ebool_2E_5C_2F_2E0,X0_2E0),X1_2E0)) ) ).
tff(arityeq4_2Ec_2Ecanonical_2Ecanonical__sum__CASE_2E4_2Emono_2EA_27a_20mono_2EA_27b,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a),X1_2E0: A_27b,X2_2E0: 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))),X3_2E0: tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27b))] : ( c_2Ecanonical_2Ecanonical__sum__CASE_2E4(A_27a,A_27b,X0_2E0,X1_2E0,X2_2E0,X3_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27b)),A_27b,app_2E2(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(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(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(A_27a,A_27b),X0_2E0),X1_2E0),X2_2E0),X3_2E0) ) ).
tff(arityeq3_2Ec_2Ecanonical_2Ecanonical__sum__merge_2E3_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a),X1_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a),X2_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : ( c_2Ecanonical_2Ecanonical__sum__merge_2E3(A_27a,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a),app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)),app_2E2(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a))),c_2Ecanonical_2Ecanonical__sum__merge_2E0(A_27a),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq3_2Ec_2Ecanonical_2Ecanonical__sum__prod_2E3_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a),X1_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a),X2_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : ( c_2Ecanonical_2Ecanonical__sum__prod_2E3(A_27a,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a),app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)),app_2E2(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a))),c_2Ecanonical_2Ecanonical__sum__prod_2E0(A_27a),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq3_2Ec_2Ecanonical_2Ecanonical__sum__scalar_2E3_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a),X1_2E0: A_27a,X2_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : ( c_2Ecanonical_2Ecanonical__sum__scalar_2E3(A_27a,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)),app_2E2(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a))),c_2Ecanonical_2Ecanonical__sum__scalar_2E0(A_27a),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq3_2Ec_2Ecanonical_2Ecanonical__sum__scalar2_2E3_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a),X1_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),X2_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : ( c_2Ecanonical_2Ecanonical__sum__scalar2_2E3(A_27a,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a),app_2E2(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)),app_2E2(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a))),c_2Ecanonical_2Ecanonical__sum__scalar2_2E0(A_27a),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq4_2Ec_2Ecanonical_2Ecanonical__sum__scalar3_2E4_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a),X1_2E0: A_27a,X2_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),X3_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : ( c_2Ecanonical_2Ecanonical__sum__scalar3_2E4(A_27a,X0_2E0,X1_2E0,X2_2E0,X3_2E0) = app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a),app_2E2(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a))),app_2E2(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)))),c_2Ecanonical_2Ecanonical__sum__scalar3_2E0(A_27a),X0_2E0),X1_2E0),X2_2E0),X3_2E0) ) ).
tff(arityeq2_2Ec_2Ecanonical_2Ecanonical__sum__simplify_2E2_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a),X1_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : ( c_2Ecanonical_2Ecanonical__sum__simplify_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a),app_2E2(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)),c_2Ecanonical_2Ecanonical__sum__simplify_2E0(A_27a),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ecanonical_2Ecanonical__sum__size_2E2_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),X1_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : ( c_2Ecanonical_2Ecanonical__sum__size_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Enum_2Enum,app_2E2(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(A_27a),X0_2E0),X1_2E0) ) ).
tff(arityeq4_2Ec_2Ecanonical_2Eics__aux_2E4_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a),X1_2E0: tyop_2Equote_2Evarmap(A_27a),X2_2E0: A_27a,X3_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : ( c_2Ecanonical_2Eics__aux_2E4(A_27a,X0_2E0,X1_2E0,X2_2E0,X3_2E0) = app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27a,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27a),app_2E2(tyop_2Equote_2Evarmap(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27a)),app_2E2(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(tyop_2Equote_2Evarmap(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27a))),c_2Ecanonical_2Eics__aux_2E0(A_27a),X0_2E0),X1_2E0),X2_2E0),X3_2E0) ) ).
tff(arityeq1_2Ec_2Equote_2Eindex__size_2E1,axiom,
! [X0_2E0: tyop_2Equote_2Eindex] : ( c_2Equote_2Eindex__size_2E1(X0_2E0) = app_2E2(tyop_2Equote_2Eindex,tyop_2Enum_2Enum,c_2Equote_2Eindex__size_2E0,X0_2E0) ) ).
tff(arityeq3_2Ec_2Ecanonical_2Einterp__cs_2E3_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a),X1_2E0: tyop_2Equote_2Evarmap(A_27a),X2_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : ( c_2Ecanonical_2Einterp__cs_2E3(A_27a,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27a,app_2E2(tyop_2Equote_2Evarmap(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27a),app_2E2(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(tyop_2Equote_2Evarmap(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27a)),c_2Ecanonical_2Einterp__cs_2E0(A_27a),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq4_2Ec_2Ecanonical_2Einterp__m_2E4_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a),X1_2E0: tyop_2Equote_2Evarmap(A_27a),X2_2E0: A_27a,X3_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex)] : ( c_2Ecanonical_2Einterp__m_2E4(A_27a,X0_2E0,X1_2E0,X2_2E0,X3_2E0) = app_2E2(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),A_27a,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),A_27a),app_2E2(tyop_2Equote_2Evarmap(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),A_27a)),app_2E2(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(tyop_2Equote_2Evarmap(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),A_27a))),c_2Ecanonical_2Einterp__m_2E0(A_27a),X0_2E0),X1_2E0),X2_2E0),X3_2E0) ) ).
tff(arityeq3_2Ec_2Ecanonical_2Einterp__sp_2E3_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a),X1_2E0: tyop_2Equote_2Evarmap(A_27a),X2_2E0: tyop_2Ecanonical_2Espolynom(A_27a)] : ( c_2Ecanonical_2Einterp__sp_2E3(A_27a,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Ecanonical_2Espolynom(A_27a),A_27a,app_2E2(tyop_2Equote_2Evarmap(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27a),app_2E2(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(tyop_2Equote_2Evarmap(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27a)),c_2Ecanonical_2Einterp__sp_2E0(A_27a),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq3_2Ec_2Ecanonical_2Einterp__vl_2E3_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a),X1_2E0: tyop_2Equote_2Evarmap(A_27a),X2_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex)] : ( c_2Ecanonical_2Einterp__vl_2E3(A_27a,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),A_27a,app_2E2(tyop_2Equote_2Evarmap(A_27a),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),A_27a),app_2E2(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(tyop_2Equote_2Evarmap(A_27a),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),A_27a)),c_2Ecanonical_2Einterp__vl_2E0(A_27a),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq1_2Ec_2Esemi__ring_2Eis__semi__ring_2E1_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a)] : ( c_2Esemi__ring_2Eis__semi__ring_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Ebool,c_2Esemi__ring_2Eis__semi__ring_2E0(A_27a),X0_2E0) ) ).
tff(arityeq4_2Ec_2Ecanonical_2Eivl__aux_2E4_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a),X1_2E0: tyop_2Equote_2Evarmap(A_27a),X2_2E0: tyop_2Equote_2Eindex,X3_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex)] : ( c_2Ecanonical_2Eivl__aux_2E4(A_27a,X0_2E0,X1_2E0,X2_2E0,X3_2E0) = app_2E2(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),A_27a,app_2E2(tyop_2Equote_2Eindex,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),A_27a),app_2E2(tyop_2Equote_2Evarmap(A_27a),tyop_2Emin_2Efun(tyop_2Equote_2Eindex,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),A_27a)),app_2E2(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(tyop_2Equote_2Evarmap(A_27a),tyop_2Emin_2Efun(tyop_2Equote_2Eindex,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),A_27a))),c_2Ecanonical_2Eivl__aux_2E0(A_27a),X0_2E0),X1_2E0),X2_2E0),X3_2E0) ) ).
tff(arityeq3_2Ec_2EternaryComparisons_2Elist__compare_2E3_2Emono_2Etyop_2Equote_2Eindex_20mono_2Etyop_2Equote_2Eindex,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Equote_2Eindex,tyop_2Emin_2Efun(tyop_2Equote_2Eindex,tyop_2EternaryComparisons_2Eordering)),X1_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),X2_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex)] : ( c_2EternaryComparisons_2Elist__compare_2E3(tyop_2Equote_2Eindex,tyop_2Equote_2Eindex,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2EternaryComparisons_2Eordering,app_2E2(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2EternaryComparisons_2Eordering),app_2E2(tyop_2Emin_2Efun(tyop_2Equote_2Eindex,tyop_2Emin_2Efun(tyop_2Equote_2Eindex,tyop_2EternaryComparisons_2Eordering)),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2EternaryComparisons_2Eordering)),c_2EternaryComparisons_2Elist__compare_2E0(tyop_2Equote_2Eindex,tyop_2Equote_2Eindex),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq3_2Ec_2EternaryComparisons_2Elist__merge_2E3_2Emono_2Etyop_2Equote_2Eindex,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Equote_2Eindex,tyop_2Emin_2Efun(tyop_2Equote_2Eindex,tyop_2Emin_2Ebool)),X1_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),X2_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex)] : ( c_2EternaryComparisons_2Elist__merge_2E3(tyop_2Equote_2Eindex,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Elist_2Elist(tyop_2Equote_2Eindex),app_2E2(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Elist_2Elist(tyop_2Equote_2Eindex)),app_2E2(tyop_2Emin_2Efun(tyop_2Equote_2Eindex,tyop_2Emin_2Efun(tyop_2Equote_2Eindex,tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Elist_2Elist(tyop_2Equote_2Eindex))),c_2EternaryComparisons_2Elist__merge_2E0(tyop_2Equote_2Eindex),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq2_2Ec_2Elist_2Elist__size_2E2_2Emono_2Etyop_2Equote_2Eindex,axiom,
! [X0_2E0: tyop_2Emin_2Efun(tyop_2Equote_2Eindex,tyop_2Enum_2Enum),X1_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex)] : ( c_2Elist_2Elist__size_2E2(tyop_2Equote_2Eindex,X0_2E0,X1_2E0) = app_2E2(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Enum_2Enum,app_2E2(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(tyop_2Equote_2Eindex),X0_2E0),X1_2E0) ) ).
tff(arityeq4_2Ec_2Ecanonical_2Emonom__insert_2E4_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a),X1_2E0: A_27a,X2_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),X3_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : ( c_2Ecanonical_2Emonom__insert_2E4(A_27a,X0_2E0,X1_2E0,X2_2E0,X3_2E0) = app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a),app_2E2(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a))),app_2E2(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)))),c_2Ecanonical_2Emonom__insert_2E0(A_27a),X0_2E0),X1_2E0),X2_2E0),X3_2E0) ) ).
tff(arityeq4_2Ec_2EternaryComparisons_2Eordering__CASE_2E4_2Emono_2Etyop_2Ecanonical_2Ecanonical__sum_28A_27a_29,axiom,
! [A_27a: $tType,X0_2E0: tyop_2EternaryComparisons_2Eordering,X1_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a),X2_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a),X3_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : ( c_2EternaryComparisons_2Eordering__CASE_2E4(tyop_2Ecanonical_2Ecanonical__sum(A_27a),X0_2E0,X1_2E0,X2_2E0,X3_2E0) = app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a),app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)),app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a))),app_2E2(tyop_2EternaryComparisons_2Eordering,tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)))),c_2EternaryComparisons_2Eordering__CASE_2E0(tyop_2Ecanonical_2Ecanonical__sum(A_27a)),X0_2E0),X1_2E0),X2_2E0),X3_2E0) ) ).
tff(arityeq1_2Ec_2Esemi__ring_2Esemi__ring__SR0_2E1_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a)] : ( c_2Esemi__ring_2Esemi__ring__SR0_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Esemi__ring_2Esemi__ring(A_27a),A_27a,c_2Esemi__ring_2Esemi__ring__SR0_2E0(A_27a),X0_2E0) ) ).
tff(arityeq1_2Ec_2Esemi__ring_2Esemi__ring__SR1_2E1_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a)] : ( c_2Esemi__ring_2Esemi__ring__SR1_2E1(A_27a,X0_2E0) = app_2E2(tyop_2Esemi__ring_2Esemi__ring(A_27a),A_27a,c_2Esemi__ring_2Esemi__ring__SR1_2E0(A_27a),X0_2E0) ) ).
tff(arityeq3_2Ec_2Esemi__ring_2Esemi__ring__SRM_2E3_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a),X1_2E0: A_27a,X2_2E0: A_27a] : ( c_2Esemi__ring_2Esemi__ring__SRM_2E3(A_27a,X0_2E0,X1_2E0,X2_2E0) = app_2E2(A_27a,A_27a,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,A_27a),app_2E2(tyop_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(A_27a),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq3_2Ec_2Esemi__ring_2Esemi__ring__SRP_2E3_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a),X1_2E0: A_27a,X2_2E0: A_27a] : ( c_2Esemi__ring_2Esemi__ring__SRP_2E3(A_27a,X0_2E0,X1_2E0,X2_2E0) = app_2E2(A_27a,A_27a,app_2E2(A_27a,tyop_2Emin_2Efun(A_27a,A_27a),app_2E2(tyop_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(A_27a),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq5_2Ec_2Ecanonical_2Espolynom__CASE_2E5_2Emono_2EA_27a_20mono_2EA_27b,axiom,
! [A_27a: $tType,A_27b: $tType,X0_2E0: tyop_2Ecanonical_2Espolynom(A_27a),X1_2E0: tyop_2Emin_2Efun(tyop_2Equote_2Eindex,A_27b),X2_2E0: tyop_2Emin_2Efun(A_27a,A_27b),X3_2E0: tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27b)),X4_2E0: tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27b))] : ( c_2Ecanonical_2Espolynom__CASE_2E5(A_27a,A_27b,X0_2E0,X1_2E0,X2_2E0,X3_2E0,X4_2E0) = app_2E2(tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27b)),A_27b,app_2E2(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(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(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(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(A_27a,A_27b),X0_2E0),X1_2E0),X2_2E0),X3_2E0),X4_2E0) ) ).
tff(arityeq2_2Ec_2Ecanonical_2Espolynom__normalize_2E2_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a),X1_2E0: tyop_2Ecanonical_2Espolynom(A_27a)] : ( c_2Ecanonical_2Espolynom__normalize_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a),app_2E2(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)),c_2Ecanonical_2Espolynom__normalize_2E0(A_27a),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ecanonical_2Espolynom__simplify_2E2_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a),X1_2E0: tyop_2Ecanonical_2Espolynom(A_27a)] : ( c_2Ecanonical_2Espolynom__simplify_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a),app_2E2(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)),c_2Ecanonical_2Espolynom__simplify_2E0(A_27a),X0_2E0),X1_2E0) ) ).
tff(arityeq2_2Ec_2Ecanonical_2Espolynom__size_2E2_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),X1_2E0: tyop_2Ecanonical_2Espolynom(A_27a)] : ( c_2Ecanonical_2Espolynom__size_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Enum_2Enum,app_2E2(tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Enum_2Enum),c_2Ecanonical_2Espolynom__size_2E0(A_27a),X0_2E0),X1_2E0) ) ).
tff(arityeq3_2Ec_2Ecanonical_2Evarlist__insert_2E3_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a),X1_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),X2_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : ( c_2Ecanonical_2Evarlist__insert_2E3(A_27a,X0_2E0,X1_2E0,X2_2E0) = app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a),app_2E2(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a)),app_2E2(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Ecanonical_2Ecanonical__sum(A_27a))),c_2Ecanonical_2Evarlist__insert_2E0(A_27a),X0_2E0),X1_2E0),X2_2E0) ) ).
tff(arityeq2_2Ec_2Equote_2Evarmap__find_2E2_2Emono_2EA_27a,axiom,
! [A_27a: $tType,X0_2E0: tyop_2Equote_2Eindex,X1_2E0: tyop_2Equote_2Evarmap(A_27a)] : ( c_2Equote_2Evarmap__find_2E2(A_27a,X0_2E0,X1_2E0) = app_2E2(tyop_2Equote_2Evarmap(A_27a),A_27a,app_2E2(tyop_2Equote_2Eindex,tyop_2Emin_2Efun(tyop_2Equote_2Evarmap(A_27a),A_27a),c_2Equote_2Evarmap__find_2E0(A_27a),X0_2E0),X1_2E0) ) ).
tff(arityeq1_2Ec_2Ebool_2E_7E_2E1,axiom,
! [X0_2E0: tyop_2Emin_2Ebool] :
( ~ p(X0_2E0)
<=> p(app_2E2(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool,c_2Ebool_2E_7E_2E0,X0_2E0)) ) ).
tff(thm_2Ecanonical_2Ecanonical__sum__TY__DEF,axiom,
! [A_27a: $tType,F3_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex)))),F2_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex)))),F1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex)))),F0_2E0: tyop_2Emin_2Efun(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex))),tyop_2Emin_2Ebool)] :
( ! [V11n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex))),F3_2E0,V11n_2E0) = c_2Eind__type_2EBOTTOM_2E0(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex))) )
=> ( ! [V8n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex))),F2_2E0,V8n_2E0) = c_2Eind__type_2EBOTTOM_2E0(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex))) )
=> ( ! [V4n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex))),F1_2E0,V4n_2E0) = c_2Eind__type_2EBOTTOM_2E0(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex))) )
=> ( ! [V1a0_27_2E0: tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex)))] :
( p(app_2E2(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex))),tyop_2Emin_2Ebool,F0_2E0,V1a0_27_2E0))
<=> ! [V2_27canonical__sum_27_2E0: tyop_2Emin_2Efun(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex))),tyop_2Emin_2Ebool)] :
( ! [V3a0_27_2E0: tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex)))] :
( ( ( V3a0_27_2E0 = c_2Eind__type_2ECONSTR_2E3(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex)),c_2Enum_2E0_2E0,c_2Epair_2E_2C_2E2(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex),c_2Ebool_2EARB_2E0(A_27a),c_2Ebool_2EARB_2E0(tyop_2Elist_2Elist(tyop_2Equote_2Eindex))),F1_2E0) )
| ? [V5a0_2E0: A_27a,V6a1_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V7a2_2E0: tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex)))] :
( ( V3a0_27_2E0 = c_2Eind__type_2ECONSTR_2E3(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex)),c_2Enum_2ESUC_2E1(c_2Enum_2E0_2E0),c_2Epair_2E_2C_2E2(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V5a0_2E0,V6a1_2E0),c_2Eind__type_2EFCONS_2E2(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex))),V7a2_2E0,F2_2E0)) )
& p(app_2E2(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex))),tyop_2Emin_2Ebool,V2_27canonical__sum_27_2E0,V7a2_2E0)) )
| ? [V9a0_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V10a1_2E0: tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex)))] :
( ( V3a0_27_2E0 = c_2Eind__type_2ECONSTR_2E3(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex)),c_2Enum_2ESUC_2E1(c_2Enum_2ESUC_2E1(c_2Enum_2E0_2E0)),c_2Epair_2E_2C_2E2(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex),c_2Ebool_2EARB_2E0(A_27a),V9a0_2E0),c_2Eind__type_2EFCONS_2E2(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex))),V10a1_2E0,F3_2E0)) )
& p(app_2E2(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex))),tyop_2Emin_2Ebool,V2_27canonical__sum_27_2E0,V10a1_2E0)) ) )
=> p(app_2E2(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex))),tyop_2Emin_2Ebool,V2_27canonical__sum_27_2E0,V3a0_27_2E0)) )
=> p(app_2E2(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex))),tyop_2Emin_2Ebool,V2_27canonical__sum_27_2E0,V1a0_27_2E0)) ) )
=> ? [V0rep_2E0: tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex))))] : p(c_2Ebool_2ETYPE__DEFINITION_2E2(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(A_27a,tyop_2Elist_2Elist(tyop_2Equote_2Eindex))),tyop_2Ecanonical_2Ecanonical__sum(A_27a),F0_2E0,V0rep_2E0)) ) ) ) ) ).
tff(thm_2Ecanonical_2Ecanonical__sum__case__def,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ! [V0v_2E0: A_27b,V1f_2E0: 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))),V2f1_2E0: tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27b))] : ( c_2Ecanonical_2Ecanonical__sum__CASE_2E4(A_27a,A_27b,c_2Ecanonical_2ENil__monom_2E0(A_27a),V0v_2E0,V1f_2E0,V2f1_2E0) = V0v_2E0 )
& ! [V3a0_2E0: A_27a,V4a1_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V5a2_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a),V6v_2E0: A_27b,V7f_2E0: 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))),V8f1_2E0: tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27b))] : ( c_2Ecanonical_2Ecanonical__sum__CASE_2E4(A_27a,A_27b,c_2Ecanonical_2ECons__monom_2E3(A_27a,V3a0_2E0,V4a1_2E0,V5a2_2E0),V6v_2E0,V7f_2E0,V8f1_2E0) = app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27b,app_2E2(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27b),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27b)),V7f_2E0,V3a0_2E0),V4a1_2E0),V5a2_2E0) )
& ! [V9a0_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V10a1_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a),V11v_2E0: A_27b,V12f_2E0: 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))),V13f1_2E0: tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27b))] : ( c_2Ecanonical_2Ecanonical__sum__CASE_2E4(A_27a,A_27b,c_2Ecanonical_2ECons__varlist_2E2(A_27a,V9a0_2E0,V10a1_2E0),V11v_2E0,V12f_2E0,V13f1_2E0) = app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27b,app_2E2(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27b),V13f1_2E0,V9a0_2E0),V10a1_2E0) ) ) ).
tff(thm_2Ecanonical_2Ecanonical__sum__size__def,axiom,
! [A_27a: $tType] :
( ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum)] : ( c_2Ecanonical_2Ecanonical__sum__size_2E2(A_27a,V0f_2E0,c_2Ecanonical_2ENil__monom_2E0(A_27a)) = c_2Enum_2E0_2E0 )
& ! [V1f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V2a0_2E0: A_27a,V3a1_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V4a2_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : ( c_2Ecanonical_2Ecanonical__sum__size_2E2(A_27a,V1f_2E0,c_2Ecanonical_2ECons__monom_2E3(A_27a,V2a0_2E0,V3a1_2E0,V4a2_2E0)) = c_2Earithmetic_2E_2B_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)),c_2Earithmetic_2E_2B_2E2(app_2E2(A_27a,tyop_2Enum_2Enum,V1f_2E0,V2a0_2E0),c_2Earithmetic_2E_2B_2E2(c_2Elist_2Elist__size_2E2(tyop_2Equote_2Eindex,c_2Equote_2Eindex__size_2E0,V3a1_2E0),c_2Ecanonical_2Ecanonical__sum__size_2E2(A_27a,V1f_2E0,V4a2_2E0)))) )
& ! [V5f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V6a0_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V7a1_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : ( c_2Ecanonical_2Ecanonical__sum__size_2E2(A_27a,V5f_2E0,c_2Ecanonical_2ECons__varlist_2E2(A_27a,V6a0_2E0,V7a1_2E0)) = c_2Earithmetic_2E_2B_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)),c_2Earithmetic_2E_2B_2E2(c_2Elist_2Elist__size_2E2(tyop_2Equote_2Eindex,c_2Equote_2Eindex__size_2E0,V6a0_2E0),c_2Ecanonical_2Ecanonical__sum__size_2E2(A_27a,V5f_2E0,V7a1_2E0))) ) ) ).
tff(thm_2Ecanonical_2Ecanonical__sum__scalar__def,axiom,
! [A_27a: $tType] :
( ! [V0sr_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a),V1c0_2E0: A_27a,V2c_2E0: A_27a,V3l_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V4t_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : ( c_2Ecanonical_2Ecanonical__sum__scalar_2E3(A_27a,V0sr_2E0,V1c0_2E0,c_2Ecanonical_2ECons__monom_2E3(A_27a,V2c_2E0,V3l_2E0,V4t_2E0)) = c_2Ecanonical_2ECons__monom_2E3(A_27a,c_2Esemi__ring_2Esemi__ring__SRM_2E3(A_27a,V0sr_2E0,V1c0_2E0,V2c_2E0),V3l_2E0,c_2Ecanonical_2Ecanonical__sum__scalar_2E3(A_27a,V0sr_2E0,V1c0_2E0,V4t_2E0)) )
& ! [V5sr_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a),V6c0_2E0: A_27a,V7l_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V8t_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : ( c_2Ecanonical_2Ecanonical__sum__scalar_2E3(A_27a,V5sr_2E0,V6c0_2E0,c_2Ecanonical_2ECons__varlist_2E2(A_27a,V7l_2E0,V8t_2E0)) = c_2Ecanonical_2ECons__monom_2E3(A_27a,V6c0_2E0,V7l_2E0,c_2Ecanonical_2Ecanonical__sum__scalar_2E3(A_27a,V5sr_2E0,V6c0_2E0,V8t_2E0)) )
& ! [V9sr_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a),V10c0_2E0: A_27a] : ( c_2Ecanonical_2Ecanonical__sum__scalar_2E3(A_27a,V9sr_2E0,V10c0_2E0,c_2Ecanonical_2ENil__monom_2E0(A_27a)) = c_2Ecanonical_2ENil__monom_2E0(A_27a) ) ) ).
tff(thm_2Ecanonical_2Ecanonical__sum__scalar2__def,axiom,
! [A_27a: $tType] :
( ! [V0sr_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a),V1l0_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V2c_2E0: A_27a,V3l_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V4t_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : ( c_2Ecanonical_2Ecanonical__sum__scalar2_2E3(A_27a,V0sr_2E0,V1l0_2E0,c_2Ecanonical_2ECons__monom_2E3(A_27a,V2c_2E0,V3l_2E0,V4t_2E0)) = c_2Ecanonical_2Emonom__insert_2E4(A_27a,V0sr_2E0,V2c_2E0,c_2EternaryComparisons_2Elist__merge_2E3(tyop_2Equote_2Eindex,c_2Equote_2Eindex__lt_2E0,V1l0_2E0,V3l_2E0),c_2Ecanonical_2Ecanonical__sum__scalar2_2E3(A_27a,V0sr_2E0,V1l0_2E0,V4t_2E0)) )
& ! [V5sr_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a),V6l0_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V7l_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V8t_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : ( c_2Ecanonical_2Ecanonical__sum__scalar2_2E3(A_27a,V5sr_2E0,V6l0_2E0,c_2Ecanonical_2ECons__varlist_2E2(A_27a,V7l_2E0,V8t_2E0)) = c_2Ecanonical_2Evarlist__insert_2E3(A_27a,V5sr_2E0,c_2EternaryComparisons_2Elist__merge_2E3(tyop_2Equote_2Eindex,c_2Equote_2Eindex__lt_2E0,V6l0_2E0,V7l_2E0),c_2Ecanonical_2Ecanonical__sum__scalar2_2E3(A_27a,V5sr_2E0,V6l0_2E0,V8t_2E0)) )
& ! [V9sr_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a),V10l0_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex)] : ( c_2Ecanonical_2Ecanonical__sum__scalar2_2E3(A_27a,V9sr_2E0,V10l0_2E0,c_2Ecanonical_2ENil__monom_2E0(A_27a)) = c_2Ecanonical_2ENil__monom_2E0(A_27a) ) ) ).
tff(thm_2Ecanonical_2Ecanonical__sum__scalar3__def,axiom,
! [A_27a: $tType] :
( ! [V0sr_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a),V1c0_2E0: A_27a,V2l0_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V3c_2E0: A_27a,V4l_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V5t_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : ( c_2Ecanonical_2Ecanonical__sum__scalar3_2E4(A_27a,V0sr_2E0,V1c0_2E0,V2l0_2E0,c_2Ecanonical_2ECons__monom_2E3(A_27a,V3c_2E0,V4l_2E0,V5t_2E0)) = c_2Ecanonical_2Emonom__insert_2E4(A_27a,V0sr_2E0,c_2Esemi__ring_2Esemi__ring__SRM_2E3(A_27a,V0sr_2E0,V1c0_2E0,V3c_2E0),c_2EternaryComparisons_2Elist__merge_2E3(tyop_2Equote_2Eindex,c_2Equote_2Eindex__lt_2E0,V2l0_2E0,V4l_2E0),c_2Ecanonical_2Ecanonical__sum__scalar3_2E4(A_27a,V0sr_2E0,V1c0_2E0,V2l0_2E0,V5t_2E0)) )
& ! [V6sr_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a),V7c0_2E0: A_27a,V8l0_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V9l_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V10t_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : ( c_2Ecanonical_2Ecanonical__sum__scalar3_2E4(A_27a,V6sr_2E0,V7c0_2E0,V8l0_2E0,c_2Ecanonical_2ECons__varlist_2E2(A_27a,V9l_2E0,V10t_2E0)) = c_2Ecanonical_2Emonom__insert_2E4(A_27a,V6sr_2E0,V7c0_2E0,c_2EternaryComparisons_2Elist__merge_2E3(tyop_2Equote_2Eindex,c_2Equote_2Eindex__lt_2E0,V8l0_2E0,V9l_2E0),c_2Ecanonical_2Ecanonical__sum__scalar3_2E4(A_27a,V6sr_2E0,V7c0_2E0,V8l0_2E0,V10t_2E0)) )
& ! [V11sr_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a),V12c0_2E0: A_27a,V13l0_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex)] : ( c_2Ecanonical_2Ecanonical__sum__scalar3_2E4(A_27a,V11sr_2E0,V12c0_2E0,V13l0_2E0,c_2Ecanonical_2ENil__monom_2E0(A_27a)) = c_2Ecanonical_2ENil__monom_2E0(A_27a) ) ) ).
tff(thm_2Ecanonical_2Ecanonical__sum__prod__def,axiom,
! [A_27a: $tType] :
( ! [V0sr_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a),V1c1_2E0: A_27a,V2l1_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V3t1_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a),V4s2_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : ( c_2Ecanonical_2Ecanonical__sum__prod_2E3(A_27a,V0sr_2E0,c_2Ecanonical_2ECons__monom_2E3(A_27a,V1c1_2E0,V2l1_2E0,V3t1_2E0),V4s2_2E0) = c_2Ecanonical_2Ecanonical__sum__merge_2E3(A_27a,V0sr_2E0,c_2Ecanonical_2Ecanonical__sum__scalar3_2E4(A_27a,V0sr_2E0,V1c1_2E0,V2l1_2E0,V4s2_2E0),c_2Ecanonical_2Ecanonical__sum__prod_2E3(A_27a,V0sr_2E0,V3t1_2E0,V4s2_2E0)) )
& ! [V5sr_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a),V6l1_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V7t1_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a),V8s2_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : ( c_2Ecanonical_2Ecanonical__sum__prod_2E3(A_27a,V5sr_2E0,c_2Ecanonical_2ECons__varlist_2E2(A_27a,V6l1_2E0,V7t1_2E0),V8s2_2E0) = c_2Ecanonical_2Ecanonical__sum__merge_2E3(A_27a,V5sr_2E0,c_2Ecanonical_2Ecanonical__sum__scalar2_2E3(A_27a,V5sr_2E0,V6l1_2E0,V8s2_2E0),c_2Ecanonical_2Ecanonical__sum__prod_2E3(A_27a,V5sr_2E0,V7t1_2E0,V8s2_2E0)) )
& ! [V9sr_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a),V10s2_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : ( c_2Ecanonical_2Ecanonical__sum__prod_2E3(A_27a,V9sr_2E0,c_2Ecanonical_2ENil__monom_2E0(A_27a),V10s2_2E0) = c_2Ecanonical_2ENil__monom_2E0(A_27a) ) ) ).
tff(thm_2Ecanonical_2Ecanonical__sum__simplify__def,axiom,
! [A_27a: $tType,F1_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Ebool)),F0_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Ebool))] :
( ! [V1c_2E0: A_27a,V0sr_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a)] :
( p(app_2E2(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Ebool),F1_2E0,V1c_2E0),V0sr_2E0))
<=> ( V1c_2E0 = c_2Esemi__ring_2Esemi__ring__SR1_2E1(A_27a,V0sr_2E0) ) )
=> ( ! [V1c_2E0: A_27a,V0sr_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a)] :
( p(app_2E2(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Ebool),F0_2E0,V1c_2E0),V0sr_2E0))
<=> ( V1c_2E0 = c_2Esemi__ring_2Esemi__ring__SR0_2E1(A_27a,V0sr_2E0) ) )
=> ( ! [V0sr_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a),V1c_2E0: A_27a,V2l_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V3t_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : ( c_2Ecanonical_2Ecanonical__sum__simplify_2E2(A_27a,V0sr_2E0,c_2Ecanonical_2ECons__monom_2E3(A_27a,V1c_2E0,V2l_2E0,V3t_2E0)) = c_2Ebool_2ECOND_2E3(tyop_2Ecanonical_2Ecanonical__sum(A_27a),app_2E2(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Ebool),F0_2E0,V1c_2E0),V0sr_2E0),c_2Ecanonical_2Ecanonical__sum__simplify_2E2(A_27a,V0sr_2E0,V3t_2E0),c_2Ebool_2ECOND_2E3(tyop_2Ecanonical_2Ecanonical__sum(A_27a),app_2E2(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Ebool,app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Ebool),F1_2E0,V1c_2E0),V0sr_2E0),c_2Ecanonical_2ECons__varlist_2E2(A_27a,V2l_2E0,c_2Ecanonical_2Ecanonical__sum__simplify_2E2(A_27a,V0sr_2E0,V3t_2E0)),c_2Ecanonical_2ECons__monom_2E3(A_27a,V1c_2E0,V2l_2E0,c_2Ecanonical_2Ecanonical__sum__simplify_2E2(A_27a,V0sr_2E0,V3t_2E0)))) )
& ! [V4sr_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a),V5l_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V6t_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : ( c_2Ecanonical_2Ecanonical__sum__simplify_2E2(A_27a,V4sr_2E0,c_2Ecanonical_2ECons__varlist_2E2(A_27a,V5l_2E0,V6t_2E0)) = c_2Ecanonical_2ECons__varlist_2E2(A_27a,V5l_2E0,c_2Ecanonical_2Ecanonical__sum__simplify_2E2(A_27a,V4sr_2E0,V6t_2E0)) )
& ! [V7sr_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a)] : ( c_2Ecanonical_2Ecanonical__sum__simplify_2E2(A_27a,V7sr_2E0,c_2Ecanonical_2ENil__monom_2E0(A_27a)) = c_2Ecanonical_2ENil__monom_2E0(A_27a) ) ) ) ) ).
tff(thm_2Ecanonical_2Eivl__aux__def,axiom,
! [A_27a: $tType] :
( ! [V0sr_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a),V1vm_2E0: tyop_2Equote_2Evarmap(A_27a),V2x_2E0: tyop_2Equote_2Eindex] : ( c_2Ecanonical_2Eivl__aux_2E4(A_27a,V0sr_2E0,V1vm_2E0,V2x_2E0,c_2Elist_2ENIL_2E0(tyop_2Equote_2Eindex)) = c_2Equote_2Evarmap__find_2E2(A_27a,V2x_2E0,V1vm_2E0) )
& ! [V3sr_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a),V4vm_2E0: tyop_2Equote_2Evarmap(A_27a),V5x_2E0: tyop_2Equote_2Eindex,V6x_27_2E0: tyop_2Equote_2Eindex,V7t_27_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex)] : ( c_2Ecanonical_2Eivl__aux_2E4(A_27a,V3sr_2E0,V4vm_2E0,V5x_2E0,c_2Elist_2ECONS_2E2(tyop_2Equote_2Eindex,V6x_27_2E0,V7t_27_2E0)) = c_2Esemi__ring_2Esemi__ring__SRM_2E3(A_27a,V3sr_2E0,c_2Equote_2Evarmap__find_2E2(A_27a,V5x_2E0,V4vm_2E0),c_2Ecanonical_2Eivl__aux_2E4(A_27a,V3sr_2E0,V4vm_2E0,V6x_27_2E0,V7t_27_2E0)) ) ) ).
tff(thm_2Ecanonical_2Einterp__vl__def,axiom,
! [A_27a: $tType] :
( ! [V0sr_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a),V1vm_2E0: tyop_2Equote_2Evarmap(A_27a)] : ( c_2Ecanonical_2Einterp__vl_2E3(A_27a,V0sr_2E0,V1vm_2E0,c_2Elist_2ENIL_2E0(tyop_2Equote_2Eindex)) = c_2Esemi__ring_2Esemi__ring__SR1_2E1(A_27a,V0sr_2E0) )
& ! [V2sr_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a),V3vm_2E0: tyop_2Equote_2Evarmap(A_27a),V4x_2E0: tyop_2Equote_2Eindex,V5t_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex)] : ( c_2Ecanonical_2Einterp__vl_2E3(A_27a,V2sr_2E0,V3vm_2E0,c_2Elist_2ECONS_2E2(tyop_2Equote_2Eindex,V4x_2E0,V5t_2E0)) = c_2Ecanonical_2Eivl__aux_2E4(A_27a,V2sr_2E0,V3vm_2E0,V4x_2E0,V5t_2E0) ) ) ).
tff(thm_2Ecanonical_2Einterp__m__def,axiom,
! [A_27a: $tType] :
( ! [V0sr_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a),V1vm_2E0: tyop_2Equote_2Evarmap(A_27a),V2c_2E0: A_27a] : ( c_2Ecanonical_2Einterp__m_2E4(A_27a,V0sr_2E0,V1vm_2E0,V2c_2E0,c_2Elist_2ENIL_2E0(tyop_2Equote_2Eindex)) = V2c_2E0 )
& ! [V3sr_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a),V4vm_2E0: tyop_2Equote_2Evarmap(A_27a),V5c_2E0: A_27a,V6x_2E0: tyop_2Equote_2Eindex,V7t_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex)] : ( c_2Ecanonical_2Einterp__m_2E4(A_27a,V3sr_2E0,V4vm_2E0,V5c_2E0,c_2Elist_2ECONS_2E2(tyop_2Equote_2Eindex,V6x_2E0,V7t_2E0)) = c_2Esemi__ring_2Esemi__ring__SRM_2E3(A_27a,V3sr_2E0,V5c_2E0,c_2Ecanonical_2Eivl__aux_2E4(A_27a,V3sr_2E0,V4vm_2E0,V6x_2E0,V7t_2E0)) ) ) ).
tff(thm_2Ecanonical_2Eics__aux__def,axiom,
! [A_27a: $tType] :
( ! [V0sr_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a),V1vm_2E0: tyop_2Equote_2Evarmap(A_27a),V2a_2E0: A_27a] : ( c_2Ecanonical_2Eics__aux_2E4(A_27a,V0sr_2E0,V1vm_2E0,V2a_2E0,c_2Ecanonical_2ENil__monom_2E0(A_27a)) = V2a_2E0 )
& ! [V3sr_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a),V4vm_2E0: tyop_2Equote_2Evarmap(A_27a),V5a_2E0: A_27a,V6l_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V7t_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : ( c_2Ecanonical_2Eics__aux_2E4(A_27a,V3sr_2E0,V4vm_2E0,V5a_2E0,c_2Ecanonical_2ECons__varlist_2E2(A_27a,V6l_2E0,V7t_2E0)) = c_2Esemi__ring_2Esemi__ring__SRP_2E3(A_27a,V3sr_2E0,V5a_2E0,c_2Ecanonical_2Eics__aux_2E4(A_27a,V3sr_2E0,V4vm_2E0,c_2Ecanonical_2Einterp__vl_2E3(A_27a,V3sr_2E0,V4vm_2E0,V6l_2E0),V7t_2E0)) )
& ! [V8sr_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a),V9vm_2E0: tyop_2Equote_2Evarmap(A_27a),V10a_2E0: A_27a,V11c_2E0: A_27a,V12l_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V13t_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : ( c_2Ecanonical_2Eics__aux_2E4(A_27a,V8sr_2E0,V9vm_2E0,V10a_2E0,c_2Ecanonical_2ECons__monom_2E3(A_27a,V11c_2E0,V12l_2E0,V13t_2E0)) = c_2Esemi__ring_2Esemi__ring__SRP_2E3(A_27a,V8sr_2E0,V10a_2E0,c_2Ecanonical_2Eics__aux_2E4(A_27a,V8sr_2E0,V9vm_2E0,c_2Ecanonical_2Einterp__m_2E4(A_27a,V8sr_2E0,V9vm_2E0,V11c_2E0,V12l_2E0),V13t_2E0)) ) ) ).
tff(thm_2Ecanonical_2Einterp__cs__def,axiom,
! [A_27a: $tType] :
( ! [V0sr_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a),V1vm_2E0: tyop_2Equote_2Evarmap(A_27a)] : ( c_2Ecanonical_2Einterp__cs_2E3(A_27a,V0sr_2E0,V1vm_2E0,c_2Ecanonical_2ENil__monom_2E0(A_27a)) = c_2Esemi__ring_2Esemi__ring__SR0_2E1(A_27a,V0sr_2E0) )
& ! [V2sr_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a),V3vm_2E0: tyop_2Equote_2Evarmap(A_27a),V4l_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V5t_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : ( c_2Ecanonical_2Einterp__cs_2E3(A_27a,V2sr_2E0,V3vm_2E0,c_2Ecanonical_2ECons__varlist_2E2(A_27a,V4l_2E0,V5t_2E0)) = c_2Ecanonical_2Eics__aux_2E4(A_27a,V2sr_2E0,V3vm_2E0,c_2Ecanonical_2Einterp__vl_2E3(A_27a,V2sr_2E0,V3vm_2E0,V4l_2E0),V5t_2E0) )
& ! [V6sr_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a),V7vm_2E0: tyop_2Equote_2Evarmap(A_27a),V8c_2E0: A_27a,V9l_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V10t_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : ( c_2Ecanonical_2Einterp__cs_2E3(A_27a,V6sr_2E0,V7vm_2E0,c_2Ecanonical_2ECons__monom_2E3(A_27a,V8c_2E0,V9l_2E0,V10t_2E0)) = c_2Ecanonical_2Eics__aux_2E4(A_27a,V6sr_2E0,V7vm_2E0,c_2Ecanonical_2Einterp__m_2E4(A_27a,V6sr_2E0,V7vm_2E0,V8c_2E0,V9l_2E0),V10t_2E0) ) ) ).
tff(thm_2Ecanonical_2Espolynom__TY__DEF,axiom,
! [A_27a: $tType,F4_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a))),F3_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a))),F2_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a))),F1_2E0: tyop_2Emin_2Efun(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a))),F0_2E0: tyop_2Emin_2Efun(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)),tyop_2Emin_2Ebool)] :
( ! [V13n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)),F4_2E0,V13n_2E0) = c_2Eind__type_2EBOTTOM_2E0(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)) )
=> ( ! [V10n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)),F3_2E0,V10n_2E0) = c_2Eind__type_2EBOTTOM_2E0(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)) )
=> ( ! [V7n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)),F2_2E0,V7n_2E0) = c_2Eind__type_2EBOTTOM_2E0(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)) )
=> ( ! [V5n_2E0: tyop_2Enum_2Enum] : ( app_2E2(tyop_2Enum_2Enum,tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)),F1_2E0,V5n_2E0) = c_2Eind__type_2EBOTTOM_2E0(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)) )
=> ( ! [V1a0_27_2E0: tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a))] :
( p(app_2E2(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)),tyop_2Emin_2Ebool,F0_2E0,V1a0_27_2E0))
<=> ! [V2_27spolynom_27_2E0: tyop_2Emin_2Efun(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)),tyop_2Emin_2Ebool)] :
( ! [V3a0_27_2E0: tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a))] :
( ( ? [V4a_2E0: tyop_2Equote_2Eindex] : ( V3a0_27_2E0 = c_2Eind__type_2ECONSTR_2E3(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a),c_2Enum_2E0_2E0,c_2Epair_2E_2C_2E2(tyop_2Equote_2Eindex,A_27a,V4a_2E0,c_2Ebool_2EARB_2E0(A_27a)),F1_2E0) )
| ? [V6a_2E0: A_27a] : ( V3a0_27_2E0 = c_2Eind__type_2ECONSTR_2E3(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a),c_2Enum_2ESUC_2E1(c_2Enum_2E0_2E0),c_2Epair_2E_2C_2E2(tyop_2Equote_2Eindex,A_27a,c_2Ebool_2EARB_2E0(tyop_2Equote_2Eindex),V6a_2E0),F2_2E0) )
| ? [V8a0_2E0: tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)),V9a1_2E0: tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a))] :
( ( V3a0_27_2E0 = c_2Eind__type_2ECONSTR_2E3(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a),c_2Enum_2ESUC_2E1(c_2Enum_2ESUC_2E1(c_2Enum_2E0_2E0)),c_2Epair_2E_2C_2E2(tyop_2Equote_2Eindex,A_27a,c_2Ebool_2EARB_2E0(tyop_2Equote_2Eindex),c_2Ebool_2EARB_2E0(A_27a)),c_2Eind__type_2EFCONS_2E2(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)),V8a0_2E0,c_2Eind__type_2EFCONS_2E2(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)),V9a1_2E0,F3_2E0))) )
& p(app_2E2(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)),tyop_2Emin_2Ebool,V2_27spolynom_27_2E0,V8a0_2E0))
& p(app_2E2(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)),tyop_2Emin_2Ebool,V2_27spolynom_27_2E0,V9a1_2E0)) )
| ? [V11a0_2E0: tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)),V12a1_2E0: tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a))] :
( ( V3a0_27_2E0 = c_2Eind__type_2ECONSTR_2E3(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a),c_2Enum_2ESUC_2E1(c_2Enum_2ESUC_2E1(c_2Enum_2ESUC_2E1(c_2Enum_2E0_2E0))),c_2Epair_2E_2C_2E2(tyop_2Equote_2Eindex,A_27a,c_2Ebool_2EARB_2E0(tyop_2Equote_2Eindex),c_2Ebool_2EARB_2E0(A_27a)),c_2Eind__type_2EFCONS_2E2(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)),V11a0_2E0,c_2Eind__type_2EFCONS_2E2(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)),V12a1_2E0,F4_2E0))) )
& p(app_2E2(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)),tyop_2Emin_2Ebool,V2_27spolynom_27_2E0,V11a0_2E0))
& p(app_2E2(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)),tyop_2Emin_2Ebool,V2_27spolynom_27_2E0,V12a1_2E0)) ) )
=> p(app_2E2(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)),tyop_2Emin_2Ebool,V2_27spolynom_27_2E0,V3a0_27_2E0)) )
=> p(app_2E2(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)),tyop_2Emin_2Ebool,V2_27spolynom_27_2E0,V1a0_27_2E0)) ) )
=> ? [V0rep_2E0: tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)))] : p(c_2Ebool_2ETYPE__DEFINITION_2E2(tyop_2Eind__type_2Erecspace(tyop_2Epair_2Eprod(tyop_2Equote_2Eindex,A_27a)),tyop_2Ecanonical_2Espolynom(A_27a),F0_2E0,V0rep_2E0)) ) ) ) ) ) ).
tff(thm_2Ecanonical_2Espolynom__case__def,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ! [V0a_2E0: tyop_2Equote_2Eindex,V1f_2E0: tyop_2Emin_2Efun(tyop_2Equote_2Eindex,A_27b),V2f1_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V3f2_2E0: tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27b)),V4f3_2E0: tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27b))] : ( c_2Ecanonical_2Espolynom__CASE_2E5(A_27a,A_27b,c_2Ecanonical_2ESPvar_2E1(A_27a,V0a_2E0),V1f_2E0,V2f1_2E0,V3f2_2E0,V4f3_2E0) = app_2E2(tyop_2Equote_2Eindex,A_27b,V1f_2E0,V0a_2E0) )
& ! [V5a_2E0: A_27a,V6f_2E0: tyop_2Emin_2Efun(tyop_2Equote_2Eindex,A_27b),V7f1_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V8f2_2E0: tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27b)),V9f3_2E0: tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27b))] : ( c_2Ecanonical_2Espolynom__CASE_2E5(A_27a,A_27b,c_2Ecanonical_2ESPconst_2E1(A_27a,V5a_2E0),V6f_2E0,V7f1_2E0,V8f2_2E0,V9f3_2E0) = app_2E2(A_27a,A_27b,V7f1_2E0,V5a_2E0) )
& ! [V10a0_2E0: tyop_2Ecanonical_2Espolynom(A_27a),V11a1_2E0: tyop_2Ecanonical_2Espolynom(A_27a),V12f_2E0: tyop_2Emin_2Efun(tyop_2Equote_2Eindex,A_27b),V13f1_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V14f2_2E0: tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27b)),V15f3_2E0: tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27b))] : ( c_2Ecanonical_2Espolynom__CASE_2E5(A_27a,A_27b,c_2Ecanonical_2ESPplus_2E2(A_27a,V10a0_2E0,V11a1_2E0),V12f_2E0,V13f1_2E0,V14f2_2E0,V15f3_2E0) = app_2E2(tyop_2Ecanonical_2Espolynom(A_27a),A_27b,app_2E2(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27b),V14f2_2E0,V10a0_2E0),V11a1_2E0) )
& ! [V16a0_2E0: tyop_2Ecanonical_2Espolynom(A_27a),V17a1_2E0: tyop_2Ecanonical_2Espolynom(A_27a),V18f_2E0: tyop_2Emin_2Efun(tyop_2Equote_2Eindex,A_27b),V19f1_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V20f2_2E0: tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27b)),V21f3_2E0: tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27b))] : ( c_2Ecanonical_2Espolynom__CASE_2E5(A_27a,A_27b,c_2Ecanonical_2ESPmult_2E2(A_27a,V16a0_2E0,V17a1_2E0),V18f_2E0,V19f1_2E0,V20f2_2E0,V21f3_2E0) = app_2E2(tyop_2Ecanonical_2Espolynom(A_27a),A_27b,app_2E2(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27b),V21f3_2E0,V16a0_2E0),V17a1_2E0) ) ) ).
tff(thm_2Ecanonical_2Espolynom__size__def,axiom,
! [A_27a: $tType] :
( ! [V0f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V1a_2E0: tyop_2Equote_2Eindex] : ( c_2Ecanonical_2Espolynom__size_2E2(A_27a,V0f_2E0,c_2Ecanonical_2ESPvar_2E1(A_27a,V1a_2E0)) = c_2Earithmetic_2E_2B_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)),c_2Equote_2Eindex__size_2E1(V1a_2E0)) )
& ! [V2f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V3a_2E0: A_27a] : ( c_2Ecanonical_2Espolynom__size_2E2(A_27a,V2f_2E0,c_2Ecanonical_2ESPconst_2E1(A_27a,V3a_2E0)) = c_2Earithmetic_2E_2B_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)),app_2E2(A_27a,tyop_2Enum_2Enum,V2f_2E0,V3a_2E0)) )
& ! [V4f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V5a0_2E0: tyop_2Ecanonical_2Espolynom(A_27a),V6a1_2E0: tyop_2Ecanonical_2Espolynom(A_27a)] : ( c_2Ecanonical_2Espolynom__size_2E2(A_27a,V4f_2E0,c_2Ecanonical_2ESPplus_2E2(A_27a,V5a0_2E0,V6a1_2E0)) = c_2Earithmetic_2E_2B_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)),c_2Earithmetic_2E_2B_2E2(c_2Ecanonical_2Espolynom__size_2E2(A_27a,V4f_2E0,V5a0_2E0),c_2Ecanonical_2Espolynom__size_2E2(A_27a,V4f_2E0,V6a1_2E0))) )
& ! [V7f_2E0: tyop_2Emin_2Efun(A_27a,tyop_2Enum_2Enum),V8a0_2E0: tyop_2Ecanonical_2Espolynom(A_27a),V9a1_2E0: tyop_2Ecanonical_2Espolynom(A_27a)] : ( c_2Ecanonical_2Espolynom__size_2E2(A_27a,V7f_2E0,c_2Ecanonical_2ESPmult_2E2(A_27a,V8a0_2E0,V9a1_2E0)) = c_2Earithmetic_2E_2B_2E2(c_2Earithmetic_2ENUMERAL_2E1(c_2Earithmetic_2EBIT1_2E1(c_2Earithmetic_2EZERO_2E0)),c_2Earithmetic_2E_2B_2E2(c_2Ecanonical_2Espolynom__size_2E2(A_27a,V7f_2E0,V8a0_2E0),c_2Ecanonical_2Espolynom__size_2E2(A_27a,V7f_2E0,V9a1_2E0))) ) ) ).
tff(thm_2Ecanonical_2Espolynom__normalize__def,axiom,
! [A_27a: $tType] :
( ! [V0sr_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a),V1i_2E0: tyop_2Equote_2Eindex] : ( c_2Ecanonical_2Espolynom__normalize_2E2(A_27a,V0sr_2E0,c_2Ecanonical_2ESPvar_2E1(A_27a,V1i_2E0)) = c_2Ecanonical_2ECons__varlist_2E2(A_27a,c_2Elist_2ECONS_2E2(tyop_2Equote_2Eindex,V1i_2E0,c_2Elist_2ENIL_2E0(tyop_2Equote_2Eindex)),c_2Ecanonical_2ENil__monom_2E0(A_27a)) )
& ! [V2sr_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a),V3c_2E0: A_27a] : ( c_2Ecanonical_2Espolynom__normalize_2E2(A_27a,V2sr_2E0,c_2Ecanonical_2ESPconst_2E1(A_27a,V3c_2E0)) = c_2Ecanonical_2ECons__monom_2E3(A_27a,V3c_2E0,c_2Elist_2ENIL_2E0(tyop_2Equote_2Eindex),c_2Ecanonical_2ENil__monom_2E0(A_27a)) )
& ! [V4sr_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a),V5l_2E0: tyop_2Ecanonical_2Espolynom(A_27a),V6r_2E0: tyop_2Ecanonical_2Espolynom(A_27a)] : ( c_2Ecanonical_2Espolynom__normalize_2E2(A_27a,V4sr_2E0,c_2Ecanonical_2ESPplus_2E2(A_27a,V5l_2E0,V6r_2E0)) = c_2Ecanonical_2Ecanonical__sum__merge_2E3(A_27a,V4sr_2E0,c_2Ecanonical_2Espolynom__normalize_2E2(A_27a,V4sr_2E0,V5l_2E0),c_2Ecanonical_2Espolynom__normalize_2E2(A_27a,V4sr_2E0,V6r_2E0)) )
& ! [V7sr_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a),V8l_2E0: tyop_2Ecanonical_2Espolynom(A_27a),V9r_2E0: tyop_2Ecanonical_2Espolynom(A_27a)] : ( c_2Ecanonical_2Espolynom__normalize_2E2(A_27a,V7sr_2E0,c_2Ecanonical_2ESPmult_2E2(A_27a,V8l_2E0,V9r_2E0)) = c_2Ecanonical_2Ecanonical__sum__prod_2E3(A_27a,V7sr_2E0,c_2Ecanonical_2Espolynom__normalize_2E2(A_27a,V7sr_2E0,V8l_2E0),c_2Ecanonical_2Espolynom__normalize_2E2(A_27a,V7sr_2E0,V9r_2E0)) ) ) ).
tff(thm_2Ecanonical_2Espolynom__simplify__def,axiom,
! [A_27a: $tType,V0sr_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a),V1x_2E0: tyop_2Ecanonical_2Espolynom(A_27a)] : ( c_2Ecanonical_2Espolynom__simplify_2E2(A_27a,V0sr_2E0,V1x_2E0) = c_2Ecanonical_2Ecanonical__sum__simplify_2E2(A_27a,V0sr_2E0,c_2Ecanonical_2Espolynom__normalize_2E2(A_27a,V0sr_2E0,V1x_2E0)) ) ).
tff(thm_2Ecanonical_2Einterp__sp__def,axiom,
! [A_27a: $tType] :
( ! [V0sr_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a),V1vm_2E0: tyop_2Equote_2Evarmap(A_27a),V2c_2E0: A_27a] : ( c_2Ecanonical_2Einterp__sp_2E3(A_27a,V0sr_2E0,V1vm_2E0,c_2Ecanonical_2ESPconst_2E1(A_27a,V2c_2E0)) = V2c_2E0 )
& ! [V3sr_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a),V4vm_2E0: tyop_2Equote_2Evarmap(A_27a),V5i_2E0: tyop_2Equote_2Eindex] : ( c_2Ecanonical_2Einterp__sp_2E3(A_27a,V3sr_2E0,V4vm_2E0,c_2Ecanonical_2ESPvar_2E1(A_27a,V5i_2E0)) = c_2Equote_2Evarmap__find_2E2(A_27a,V5i_2E0,V4vm_2E0) )
& ! [V6sr_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a),V7vm_2E0: tyop_2Equote_2Evarmap(A_27a),V8p1_2E0: tyop_2Ecanonical_2Espolynom(A_27a),V9p2_2E0: tyop_2Ecanonical_2Espolynom(A_27a)] : ( c_2Ecanonical_2Einterp__sp_2E3(A_27a,V6sr_2E0,V7vm_2E0,c_2Ecanonical_2ESPplus_2E2(A_27a,V8p1_2E0,V9p2_2E0)) = c_2Esemi__ring_2Esemi__ring__SRP_2E3(A_27a,V6sr_2E0,c_2Ecanonical_2Einterp__sp_2E3(A_27a,V6sr_2E0,V7vm_2E0,V8p1_2E0),c_2Ecanonical_2Einterp__sp_2E3(A_27a,V6sr_2E0,V7vm_2E0,V9p2_2E0)) )
& ! [V10sr_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a),V11vm_2E0: tyop_2Equote_2Evarmap(A_27a),V12p1_2E0: tyop_2Ecanonical_2Espolynom(A_27a),V13p2_2E0: tyop_2Ecanonical_2Espolynom(A_27a)] : ( c_2Ecanonical_2Einterp__sp_2E3(A_27a,V10sr_2E0,V11vm_2E0,c_2Ecanonical_2ESPmult_2E2(A_27a,V12p1_2E0,V13p2_2E0)) = c_2Esemi__ring_2Esemi__ring__SRM_2E3(A_27a,V10sr_2E0,c_2Ecanonical_2Einterp__sp_2E3(A_27a,V10sr_2E0,V11vm_2E0,V12p1_2E0),c_2Ecanonical_2Einterp__sp_2E3(A_27a,V10sr_2E0,V11vm_2E0,V13p2_2E0)) ) ) ).
tff(thm_2Ecanonical_2Edatatype__canonical__sum,axiom,
! [A_27a: $tType,V0canonical__sum_2E0: 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)))] : p(c_2Ebool_2EDATATYPE_2E1(tyop_2Emin_2Ebool,app_2E2(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(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(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,c_2Ecanonical_2ENil__monom_2E0(A_27a)),c_2Ecanonical_2ECons__monom_2E0(A_27a)),c_2Ecanonical_2ECons__varlist_2E0(A_27a)))) ).
tff(thm_2Ecanonical_2Ecanonical__sum__11,axiom,
! [A_27a: $tType] :
( ! [V0a0_2E0: A_27a,V1a1_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V2a2_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a),V3a0_27_2E0: A_27a,V4a1_27_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V5a2_27_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] :
( ( c_2Ecanonical_2ECons__monom_2E3(A_27a,V0a0_2E0,V1a1_2E0,V2a2_2E0) = c_2Ecanonical_2ECons__monom_2E3(A_27a,V3a0_27_2E0,V4a1_27_2E0,V5a2_27_2E0) )
<=> ( ( V0a0_2E0 = V3a0_27_2E0 )
& ( V1a1_2E0 = V4a1_27_2E0 )
& ( V2a2_2E0 = V5a2_27_2E0 ) ) )
& ! [V6a0_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V7a1_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a),V8a0_27_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V9a1_27_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] :
( ( c_2Ecanonical_2ECons__varlist_2E2(A_27a,V6a0_2E0,V7a1_2E0) = c_2Ecanonical_2ECons__varlist_2E2(A_27a,V8a0_27_2E0,V9a1_27_2E0) )
<=> ( ( V6a0_2E0 = V8a0_27_2E0 )
& ( V7a1_2E0 = V9a1_27_2E0 ) ) ) ) ).
tff(thm_2Ecanonical_2Ecanonical__sum__distinct,axiom,
! [A_27a: $tType] :
( ! [V0a2_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a),V1a1_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V2a0_2E0: A_27a] : ( c_2Ecanonical_2ENil__monom_2E0(A_27a) != c_2Ecanonical_2ECons__monom_2E3(A_27a,V2a0_2E0,V1a1_2E0,V0a2_2E0) )
& ! [V3a1_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a),V4a0_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex)] : ( c_2Ecanonical_2ENil__monom_2E0(A_27a) != c_2Ecanonical_2ECons__varlist_2E2(A_27a,V4a0_2E0,V3a1_2E0) )
& ! [V5a2_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a),V6a1_27_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a),V7a1_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V8a0_27_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V9a0_2E0: A_27a] : ( c_2Ecanonical_2ECons__monom_2E3(A_27a,V9a0_2E0,V7a1_2E0,V5a2_2E0) != c_2Ecanonical_2ECons__varlist_2E2(A_27a,V8a0_27_2E0,V6a1_27_2E0) ) ) ).
tff(thm_2Ecanonical_2Ecanonical__sum__nchotomy,axiom,
! [A_27a: $tType,V0cc_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] :
( ( V0cc_2E0 = c_2Ecanonical_2ENil__monom_2E0(A_27a) )
| ? [V1a_2E0: A_27a,V2l_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V3c_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : ( V0cc_2E0 = c_2Ecanonical_2ECons__monom_2E3(A_27a,V1a_2E0,V2l_2E0,V3c_2E0) )
| ? [V4l_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V5c_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : ( V0cc_2E0 = c_2Ecanonical_2ECons__varlist_2E2(A_27a,V4l_2E0,V5c_2E0) ) ) ).
tff(thm_2Ecanonical_2Ecanonical__sum__Axiom,axiom,
! [A_27a: $tType,A_27b: $tType,V0f0_2E0: A_27b,V1f1_2E0: 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)))),V2f2_2E0: 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)))] :
? [V3fn_2E0: tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27b)] :
( ( app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27b,V3fn_2E0,c_2Ecanonical_2ENil__monom_2E0(A_27a)) = V0f0_2E0 )
& ! [V4a0_2E0: A_27a,V5a1_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V6a2_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : ( app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27b,V3fn_2E0,c_2Ecanonical_2ECons__monom_2E3(A_27a,V4a0_2E0,V5a1_2E0,V6a2_2E0)) = app_2E2(A_27b,A_27b,app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(A_27b,A_27b),app_2E2(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(A_27b,A_27b)),app_2E2(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,V4a0_2E0),V5a1_2E0),V6a2_2E0),app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27b,V3fn_2E0,V6a2_2E0)) )
& ! [V7a0_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V8a1_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : ( app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27b,V3fn_2E0,c_2Ecanonical_2ECons__varlist_2E2(A_27a,V7a0_2E0,V8a1_2E0)) = app_2E2(A_27b,A_27b,app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(A_27b,A_27b),app_2E2(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(A_27b,A_27b)),V2f2_2E0,V7a0_2E0),V8a1_2E0),app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27b,V3fn_2E0,V8a1_2E0)) ) ) ).
tff(thm_2Ecanonical_2Ecanonical__sum__induction,axiom,
! [A_27a: $tType,V0P_2E0: tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool)] :
( ( p(app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool,V0P_2E0,c_2Ecanonical_2ENil__monom_2E0(A_27a)))
& ! [V1c_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] :
( p(app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool,V0P_2E0,V1c_2E0))
=> ! [V2l_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V3a_2E0: A_27a] : p(app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool,V0P_2E0,c_2Ecanonical_2ECons__monom_2E3(A_27a,V3a_2E0,V2l_2E0,V1c_2E0))) )
& ! [V4c_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] :
( p(app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool,V0P_2E0,V4c_2E0))
=> ! [V5l_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex)] : p(app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool,V0P_2E0,c_2Ecanonical_2ECons__varlist_2E2(A_27a,V5l_2E0,V4c_2E0))) ) )
=> ! [V6c_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : p(app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool,V0P_2E0,V6c_2E0)) ) ).
tff(thm_2Ecanonical_2Ecanonical__sum__case__cong,axiom,
! [A_27a: $tType,A_27b: $tType,V0v_27_2E0: A_27b,V1f1_27_2E0: tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27b)),V2f_27_2E0: 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))),V3M_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a),V4M_27_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a),V5v_2E0: A_27b,V6f_2E0: 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))),V7f1_2E0: tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27b))] :
( ( ( V3M_2E0 = V4M_27_2E0 )
& ( ( V4M_27_2E0 = c_2Ecanonical_2ENil__monom_2E0(A_27a) )
=> ( V5v_2E0 = V0v_27_2E0 ) )
& ! [V8a0_2E0: A_27a,V9a1_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V10a2_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] :
( ( V4M_27_2E0 = c_2Ecanonical_2ECons__monom_2E3(A_27a,V8a0_2E0,V9a1_2E0,V10a2_2E0) )
=> ( app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27b,app_2E2(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27b),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27b)),V6f_2E0,V8a0_2E0),V9a1_2E0),V10a2_2E0) = app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27b,app_2E2(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27b),app_2E2(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,V8a0_2E0),V9a1_2E0),V10a2_2E0) ) )
& ! [V11a0_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V12a1_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] :
( ( V4M_27_2E0 = c_2Ecanonical_2ECons__varlist_2E2(A_27a,V11a0_2E0,V12a1_2E0) )
=> ( app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27b,app_2E2(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27b),V7f1_2E0,V11a0_2E0),V12a1_2E0) = app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27b,app_2E2(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27b),V1f1_27_2E0,V11a0_2E0),V12a1_2E0) ) ) )
=> ( c_2Ecanonical_2Ecanonical__sum__CASE_2E4(A_27a,A_27b,V3M_2E0,V5v_2E0,V6f_2E0,V7f1_2E0) = c_2Ecanonical_2Ecanonical__sum__CASE_2E4(A_27a,A_27b,V4M_27_2E0,V0v_27_2E0,V2f_27_2E0,V1f1_27_2E0) ) ) ).
tff(thm_2Ecanonical_2Ecanonical__sum__case__eq,axiom,
! [A_27a: $tType,A_27b: $tType,V0x_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a),V1v_27_2E0: A_27b,V2v_2E0: A_27b,V3f1_2E0: tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27b)),V4f_2E0: 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)))] :
( ( c_2Ecanonical_2Ecanonical__sum__CASE_2E4(A_27a,A_27b,V0x_2E0,V2v_2E0,V4f_2E0,V3f1_2E0) = V1v_27_2E0 )
<=> ( ( ( V0x_2E0 = c_2Ecanonical_2ENil__monom_2E0(A_27a) )
& ( V2v_2E0 = V1v_27_2E0 ) )
| ? [V5a_2E0: A_27a,V6l_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V7c_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] :
( ( V0x_2E0 = c_2Ecanonical_2ECons__monom_2E3(A_27a,V5a_2E0,V6l_2E0,V7c_2E0) )
& ( app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27b,app_2E2(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27b),app_2E2(A_27a,tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27b)),V4f_2E0,V5a_2E0),V6l_2E0),V7c_2E0) = V1v_27_2E0 ) )
| ? [V8l_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V9c_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] :
( ( V0x_2E0 = c_2Ecanonical_2ECons__varlist_2E2(A_27a,V8l_2E0,V9c_2E0) )
& ( app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27b,app_2E2(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),A_27b),V3f1_2E0,V8l_2E0),V9c_2E0) = V1v_27_2E0 ) ) ) ) ).
tff(thm_2Ecanonical_2Ecanonical__sum__merge__ind,axiom,
! [A_27a: $tType,V0P_2E0: 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)))] :
( ( ! [V1sr_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a),V2c1_2E0: A_27a,V3l1_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V4t1_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a),V5c2_2E0: A_27a,V6l2_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V7t2_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] :
( ( ( ( c_2EternaryComparisons_2Elist__compare_2E3(tyop_2Equote_2Eindex,tyop_2Equote_2Eindex,c_2Equote_2Eindex__compare_2E0,V3l1_2E0,V6l2_2E0) = c_2EternaryComparisons_2EEQUAL_2E0 )
=> p(app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool,app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool),app_2E2(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool)),V0P_2E0,V1sr_2E0),V4t1_2E0),V7t2_2E0)) )
& ( ( c_2EternaryComparisons_2Elist__compare_2E3(tyop_2Equote_2Eindex,tyop_2Equote_2Eindex,c_2Equote_2Eindex__compare_2E0,V3l1_2E0,V6l2_2E0) = c_2EternaryComparisons_2EGREATER_2E0 )
=> p(app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool,app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool),app_2E2(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool)),V0P_2E0,V1sr_2E0),c_2Ecanonical_2ECons__monom_2E3(A_27a,V2c1_2E0,V3l1_2E0,V4t1_2E0)),V7t2_2E0)) )
& ( ( c_2EternaryComparisons_2Elist__compare_2E3(tyop_2Equote_2Eindex,tyop_2Equote_2Eindex,c_2Equote_2Eindex__compare_2E0,V3l1_2E0,V6l2_2E0) = c_2EternaryComparisons_2ELESS_2E0 )
=> p(app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool,app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool),app_2E2(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool)),V0P_2E0,V1sr_2E0),V4t1_2E0),c_2Ecanonical_2ECons__monom_2E3(A_27a,V5c2_2E0,V6l2_2E0,V7t2_2E0))) ) )
=> p(app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool,app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool),app_2E2(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool)),V0P_2E0,V1sr_2E0),c_2Ecanonical_2ECons__monom_2E3(A_27a,V2c1_2E0,V3l1_2E0,V4t1_2E0)),c_2Ecanonical_2ECons__monom_2E3(A_27a,V5c2_2E0,V6l2_2E0,V7t2_2E0))) )
& ! [V8sr_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a),V9c1_2E0: A_27a,V10l1_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V11t1_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a),V12l2_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V13t2_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] :
( ( ( ( c_2EternaryComparisons_2Elist__compare_2E3(tyop_2Equote_2Eindex,tyop_2Equote_2Eindex,c_2Equote_2Eindex__compare_2E0,V10l1_2E0,V12l2_2E0) = c_2EternaryComparisons_2EEQUAL_2E0 )
=> p(app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool,app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool),app_2E2(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool)),V0P_2E0,V8sr_2E0),V11t1_2E0),V13t2_2E0)) )
& ( ( c_2EternaryComparisons_2Elist__compare_2E3(tyop_2Equote_2Eindex,tyop_2Equote_2Eindex,c_2Equote_2Eindex__compare_2E0,V10l1_2E0,V12l2_2E0) = c_2EternaryComparisons_2EGREATER_2E0 )
=> p(app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool,app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool),app_2E2(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool)),V0P_2E0,V8sr_2E0),c_2Ecanonical_2ECons__monom_2E3(A_27a,V9c1_2E0,V10l1_2E0,V11t1_2E0)),V13t2_2E0)) )
& ( ( c_2EternaryComparisons_2Elist__compare_2E3(tyop_2Equote_2Eindex,tyop_2Equote_2Eindex,c_2Equote_2Eindex__compare_2E0,V10l1_2E0,V12l2_2E0) = c_2EternaryComparisons_2ELESS_2E0 )
=> p(app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool,app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool),app_2E2(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool)),V0P_2E0,V8sr_2E0),V11t1_2E0),c_2Ecanonical_2ECons__varlist_2E2(A_27a,V12l2_2E0,V13t2_2E0))) ) )
=> p(app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool,app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool),app_2E2(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool)),V0P_2E0,V8sr_2E0),c_2Ecanonical_2ECons__monom_2E3(A_27a,V9c1_2E0,V10l1_2E0,V11t1_2E0)),c_2Ecanonical_2ECons__varlist_2E2(A_27a,V12l2_2E0,V13t2_2E0))) )
& ! [V14sr_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a),V15l1_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V16t1_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a),V17c2_2E0: A_27a,V18l2_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V19t2_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] :
( ( ( ( c_2EternaryComparisons_2Elist__compare_2E3(tyop_2Equote_2Eindex,tyop_2Equote_2Eindex,c_2Equote_2Eindex__compare_2E0,V15l1_2E0,V18l2_2E0) = c_2EternaryComparisons_2EEQUAL_2E0 )
=> p(app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool,app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool),app_2E2(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool)),V0P_2E0,V14sr_2E0),V16t1_2E0),V19t2_2E0)) )
& ( ( c_2EternaryComparisons_2Elist__compare_2E3(tyop_2Equote_2Eindex,tyop_2Equote_2Eindex,c_2Equote_2Eindex__compare_2E0,V15l1_2E0,V18l2_2E0) = c_2EternaryComparisons_2EGREATER_2E0 )
=> p(app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool,app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool),app_2E2(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool)),V0P_2E0,V14sr_2E0),c_2Ecanonical_2ECons__varlist_2E2(A_27a,V15l1_2E0,V16t1_2E0)),V19t2_2E0)) )
& ( ( c_2EternaryComparisons_2Elist__compare_2E3(tyop_2Equote_2Eindex,tyop_2Equote_2Eindex,c_2Equote_2Eindex__compare_2E0,V15l1_2E0,V18l2_2E0) = c_2EternaryComparisons_2ELESS_2E0 )
=> p(app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool,app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool),app_2E2(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool)),V0P_2E0,V14sr_2E0),V16t1_2E0),c_2Ecanonical_2ECons__monom_2E3(A_27a,V17c2_2E0,V18l2_2E0,V19t2_2E0))) ) )
=> p(app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool,app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool),app_2E2(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool)),V0P_2E0,V14sr_2E0),c_2Ecanonical_2ECons__varlist_2E2(A_27a,V15l1_2E0,V16t1_2E0)),c_2Ecanonical_2ECons__monom_2E3(A_27a,V17c2_2E0,V18l2_2E0,V19t2_2E0))) )
& ! [V20sr_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a),V21l1_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V22t1_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a),V23l2_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V24t2_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] :
( ( ( ( c_2EternaryComparisons_2Elist__compare_2E3(tyop_2Equote_2Eindex,tyop_2Equote_2Eindex,c_2Equote_2Eindex__compare_2E0,V21l1_2E0,V23l2_2E0) = c_2EternaryComparisons_2EEQUAL_2E0 )
=> p(app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool,app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool),app_2E2(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool)),V0P_2E0,V20sr_2E0),V22t1_2E0),V24t2_2E0)) )
& ( ( c_2EternaryComparisons_2Elist__compare_2E3(tyop_2Equote_2Eindex,tyop_2Equote_2Eindex,c_2Equote_2Eindex__compare_2E0,V21l1_2E0,V23l2_2E0) = c_2EternaryComparisons_2EGREATER_2E0 )
=> p(app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool,app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool),app_2E2(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool)),V0P_2E0,V20sr_2E0),c_2Ecanonical_2ECons__varlist_2E2(A_27a,V21l1_2E0,V22t1_2E0)),V24t2_2E0)) )
& ( ( c_2EternaryComparisons_2Elist__compare_2E3(tyop_2Equote_2Eindex,tyop_2Equote_2Eindex,c_2Equote_2Eindex__compare_2E0,V21l1_2E0,V23l2_2E0) = c_2EternaryComparisons_2ELESS_2E0 )
=> p(app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool,app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool),app_2E2(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool)),V0P_2E0,V20sr_2E0),V22t1_2E0),c_2Ecanonical_2ECons__varlist_2E2(A_27a,V23l2_2E0,V24t2_2E0))) ) )
=> p(app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool,app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool),app_2E2(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool)),V0P_2E0,V20sr_2E0),c_2Ecanonical_2ECons__varlist_2E2(A_27a,V21l1_2E0,V22t1_2E0)),c_2Ecanonical_2ECons__varlist_2E2(A_27a,V23l2_2E0,V24t2_2E0))) )
& ! [V25sr_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a),V26s1_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : p(app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool,app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool),app_2E2(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool)),V0P_2E0,V25sr_2E0),V26s1_2E0),c_2Ecanonical_2ENil__monom_2E0(A_27a)))
& ! [V27sr_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a),V28v4_2E0: A_27a,V29v5_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V30v6_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : p(app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool,app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool),app_2E2(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool)),V0P_2E0,V27sr_2E0),c_2Ecanonical_2ENil__monom_2E0(A_27a)),c_2Ecanonical_2ECons__monom_2E3(A_27a,V28v4_2E0,V29v5_2E0,V30v6_2E0)))
& ! [V31sr_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a),V32v7_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V33v8_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : p(app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool,app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool),app_2E2(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool)),V0P_2E0,V31sr_2E0),c_2Ecanonical_2ENil__monom_2E0(A_27a)),c_2Ecanonical_2ECons__varlist_2E2(A_27a,V32v7_2E0,V33v8_2E0))) )
=> ! [V34v_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a),V35v1_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a),V36v2_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : p(app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool,app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool),app_2E2(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool)),V0P_2E0,V34v_2E0),V35v1_2E0),V36v2_2E0)) ) ).
tff(thm_2Ecanonical_2Ecanonical__sum__merge__def,axiom,
! [A_27a: $tType] :
( ! [V0t2_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a),V1t1_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a),V2sr_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a),V3l2_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V4l1_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V5c2_2E0: A_27a,V6c1_2E0: A_27a] : ( c_2Ecanonical_2Ecanonical__sum__merge_2E3(A_27a,V2sr_2E0,c_2Ecanonical_2ECons__monom_2E3(A_27a,V6c1_2E0,V4l1_2E0,V1t1_2E0),c_2Ecanonical_2ECons__monom_2E3(A_27a,V5c2_2E0,V3l2_2E0,V0t2_2E0)) = c_2EternaryComparisons_2Eordering__CASE_2E4(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EternaryComparisons_2Elist__compare_2E3(tyop_2Equote_2Eindex,tyop_2Equote_2Eindex,c_2Equote_2Eindex__compare_2E0,V4l1_2E0,V3l2_2E0),c_2Ecanonical_2ECons__monom_2E3(A_27a,V6c1_2E0,V4l1_2E0,c_2Ecanonical_2Ecanonical__sum__merge_2E3(A_27a,V2sr_2E0,V1t1_2E0,c_2Ecanonical_2ECons__monom_2E3(A_27a,V5c2_2E0,V3l2_2E0,V0t2_2E0))),c_2Ecanonical_2ECons__monom_2E3(A_27a,c_2Esemi__ring_2Esemi__ring__SRP_2E3(A_27a,V2sr_2E0,V6c1_2E0,V5c2_2E0),V4l1_2E0,c_2Ecanonical_2Ecanonical__sum__merge_2E3(A_27a,V2sr_2E0,V1t1_2E0,V0t2_2E0)),c_2Ecanonical_2ECons__monom_2E3(A_27a,V5c2_2E0,V3l2_2E0,c_2Ecanonical_2Ecanonical__sum__merge_2E3(A_27a,V2sr_2E0,c_2Ecanonical_2ECons__monom_2E3(A_27a,V6c1_2E0,V4l1_2E0,V1t1_2E0),V0t2_2E0))) )
& ! [V7t2_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a),V8t1_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a),V9sr_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a),V10l2_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V11l1_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V12c1_2E0: A_27a] : ( c_2Ecanonical_2Ecanonical__sum__merge_2E3(A_27a,V9sr_2E0,c_2Ecanonical_2ECons__monom_2E3(A_27a,V12c1_2E0,V11l1_2E0,V8t1_2E0),c_2Ecanonical_2ECons__varlist_2E2(A_27a,V10l2_2E0,V7t2_2E0)) = c_2EternaryComparisons_2Eordering__CASE_2E4(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EternaryComparisons_2Elist__compare_2E3(tyop_2Equote_2Eindex,tyop_2Equote_2Eindex,c_2Equote_2Eindex__compare_2E0,V11l1_2E0,V10l2_2E0),c_2Ecanonical_2ECons__monom_2E3(A_27a,V12c1_2E0,V11l1_2E0,c_2Ecanonical_2Ecanonical__sum__merge_2E3(A_27a,V9sr_2E0,V8t1_2E0,c_2Ecanonical_2ECons__varlist_2E2(A_27a,V10l2_2E0,V7t2_2E0))),c_2Ecanonical_2ECons__monom_2E3(A_27a,c_2Esemi__ring_2Esemi__ring__SRP_2E3(A_27a,V9sr_2E0,V12c1_2E0,c_2Esemi__ring_2Esemi__ring__SR1_2E1(A_27a,V9sr_2E0)),V11l1_2E0,c_2Ecanonical_2Ecanonical__sum__merge_2E3(A_27a,V9sr_2E0,V8t1_2E0,V7t2_2E0)),c_2Ecanonical_2ECons__varlist_2E2(A_27a,V10l2_2E0,c_2Ecanonical_2Ecanonical__sum__merge_2E3(A_27a,V9sr_2E0,c_2Ecanonical_2ECons__monom_2E3(A_27a,V12c1_2E0,V11l1_2E0,V8t1_2E0),V7t2_2E0))) )
& ! [V13t2_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a),V14t1_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a),V15sr_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a),V16l2_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V17l1_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V18c2_2E0: A_27a] : ( c_2Ecanonical_2Ecanonical__sum__merge_2E3(A_27a,V15sr_2E0,c_2Ecanonical_2ECons__varlist_2E2(A_27a,V17l1_2E0,V14t1_2E0),c_2Ecanonical_2ECons__monom_2E3(A_27a,V18c2_2E0,V16l2_2E0,V13t2_2E0)) = c_2EternaryComparisons_2Eordering__CASE_2E4(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EternaryComparisons_2Elist__compare_2E3(tyop_2Equote_2Eindex,tyop_2Equote_2Eindex,c_2Equote_2Eindex__compare_2E0,V17l1_2E0,V16l2_2E0),c_2Ecanonical_2ECons__varlist_2E2(A_27a,V17l1_2E0,c_2Ecanonical_2Ecanonical__sum__merge_2E3(A_27a,V15sr_2E0,V14t1_2E0,c_2Ecanonical_2ECons__monom_2E3(A_27a,V18c2_2E0,V16l2_2E0,V13t2_2E0))),c_2Ecanonical_2ECons__monom_2E3(A_27a,c_2Esemi__ring_2Esemi__ring__SRP_2E3(A_27a,V15sr_2E0,c_2Esemi__ring_2Esemi__ring__SR1_2E1(A_27a,V15sr_2E0),V18c2_2E0),V17l1_2E0,c_2Ecanonical_2Ecanonical__sum__merge_2E3(A_27a,V15sr_2E0,V14t1_2E0,V13t2_2E0)),c_2Ecanonical_2ECons__monom_2E3(A_27a,V18c2_2E0,V16l2_2E0,c_2Ecanonical_2Ecanonical__sum__merge_2E3(A_27a,V15sr_2E0,c_2Ecanonical_2ECons__varlist_2E2(A_27a,V17l1_2E0,V14t1_2E0),V13t2_2E0))) )
& ! [V19t2_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a),V20t1_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a),V21sr_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a),V22l2_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V23l1_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex)] : ( c_2Ecanonical_2Ecanonical__sum__merge_2E3(A_27a,V21sr_2E0,c_2Ecanonical_2ECons__varlist_2E2(A_27a,V23l1_2E0,V20t1_2E0),c_2Ecanonical_2ECons__varlist_2E2(A_27a,V22l2_2E0,V19t2_2E0)) = c_2EternaryComparisons_2Eordering__CASE_2E4(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EternaryComparisons_2Elist__compare_2E3(tyop_2Equote_2Eindex,tyop_2Equote_2Eindex,c_2Equote_2Eindex__compare_2E0,V23l1_2E0,V22l2_2E0),c_2Ecanonical_2ECons__varlist_2E2(A_27a,V23l1_2E0,c_2Ecanonical_2Ecanonical__sum__merge_2E3(A_27a,V21sr_2E0,V20t1_2E0,c_2Ecanonical_2ECons__varlist_2E2(A_27a,V22l2_2E0,V19t2_2E0))),c_2Ecanonical_2ECons__monom_2E3(A_27a,c_2Esemi__ring_2Esemi__ring__SRP_2E3(A_27a,V21sr_2E0,c_2Esemi__ring_2Esemi__ring__SR1_2E1(A_27a,V21sr_2E0),c_2Esemi__ring_2Esemi__ring__SR1_2E1(A_27a,V21sr_2E0)),V23l1_2E0,c_2Ecanonical_2Ecanonical__sum__merge_2E3(A_27a,V21sr_2E0,V20t1_2E0,V19t2_2E0)),c_2Ecanonical_2ECons__varlist_2E2(A_27a,V22l2_2E0,c_2Ecanonical_2Ecanonical__sum__merge_2E3(A_27a,V21sr_2E0,c_2Ecanonical_2ECons__varlist_2E2(A_27a,V23l1_2E0,V20t1_2E0),V19t2_2E0))) )
& ! [V24sr_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a),V25s1_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : ( c_2Ecanonical_2Ecanonical__sum__merge_2E3(A_27a,V24sr_2E0,V25s1_2E0,c_2Ecanonical_2ENil__monom_2E0(A_27a)) = V25s1_2E0 )
& ! [V26v6_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a),V27v5_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V28v4_2E0: A_27a,V29sr_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a)] : ( c_2Ecanonical_2Ecanonical__sum__merge_2E3(A_27a,V29sr_2E0,c_2Ecanonical_2ENil__monom_2E0(A_27a),c_2Ecanonical_2ECons__monom_2E3(A_27a,V28v4_2E0,V27v5_2E0,V26v6_2E0)) = c_2Ecanonical_2ECons__monom_2E3(A_27a,V28v4_2E0,V27v5_2E0,V26v6_2E0) )
& ! [V30v8_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a),V31v7_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V32sr_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a)] : ( c_2Ecanonical_2Ecanonical__sum__merge_2E3(A_27a,V32sr_2E0,c_2Ecanonical_2ENil__monom_2E0(A_27a),c_2Ecanonical_2ECons__varlist_2E2(A_27a,V31v7_2E0,V30v8_2E0)) = c_2Ecanonical_2ECons__varlist_2E2(A_27a,V31v7_2E0,V30v8_2E0) ) ) ).
tff(thm_2Ecanonical_2Emonom__insert__ind,axiom,
! [A_27a: $tType,V0P_2E0: 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))))] :
( ( ! [V1sr_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a),V2c1_2E0: A_27a,V3l1_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V4c2_2E0: A_27a,V5l2_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V6t2_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] :
( ( ( c_2EternaryComparisons_2Elist__compare_2E3(tyop_2Equote_2Eindex,tyop_2Equote_2Eindex,c_2Equote_2Eindex__compare_2E0,V3l1_2E0,V5l2_2E0) = c_2EternaryComparisons_2EGREATER_2E0 )
=> p(app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool,app_2E2(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool),app_2E2(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(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,V1sr_2E0),V2c1_2E0),V3l1_2E0),V6t2_2E0)) )
=> p(app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool,app_2E2(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool),app_2E2(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(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,V1sr_2E0),V2c1_2E0),V3l1_2E0),c_2Ecanonical_2ECons__monom_2E3(A_27a,V4c2_2E0,V5l2_2E0,V6t2_2E0))) )
& ! [V7sr_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a),V8c1_2E0: A_27a,V9l1_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V10l2_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V11t2_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] :
( ( ( c_2EternaryComparisons_2Elist__compare_2E3(tyop_2Equote_2Eindex,tyop_2Equote_2Eindex,c_2Equote_2Eindex__compare_2E0,V9l1_2E0,V10l2_2E0) = c_2EternaryComparisons_2EGREATER_2E0 )
=> p(app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool,app_2E2(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool),app_2E2(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(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,V7sr_2E0),V8c1_2E0),V9l1_2E0),V11t2_2E0)) )
=> p(app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool,app_2E2(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool),app_2E2(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(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,V7sr_2E0),V8c1_2E0),V9l1_2E0),c_2Ecanonical_2ECons__varlist_2E2(A_27a,V10l2_2E0,V11t2_2E0))) )
& ! [V12sr_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a),V13c1_2E0: A_27a,V14l1_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex)] : p(app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool,app_2E2(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool),app_2E2(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(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,V12sr_2E0),V13c1_2E0),V14l1_2E0),c_2Ecanonical_2ENil__monom_2E0(A_27a))) )
=> ! [V15v_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a),V16v1_2E0: A_27a,V17v2_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V18v3_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : p(app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool,app_2E2(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool),app_2E2(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(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,V15v_2E0),V16v1_2E0),V17v2_2E0),V18v3_2E0)) ) ).
tff(thm_2Ecanonical_2Emonom__insert__def,axiom,
! [A_27a: $tType] :
( ! [V0t2_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a),V1sr_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a),V2l2_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V3l1_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V4c2_2E0: A_27a,V5c1_2E0: A_27a] : ( c_2Ecanonical_2Emonom__insert_2E4(A_27a,V1sr_2E0,V5c1_2E0,V3l1_2E0,c_2Ecanonical_2ECons__monom_2E3(A_27a,V4c2_2E0,V2l2_2E0,V0t2_2E0)) = c_2EternaryComparisons_2Eordering__CASE_2E4(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EternaryComparisons_2Elist__compare_2E3(tyop_2Equote_2Eindex,tyop_2Equote_2Eindex,c_2Equote_2Eindex__compare_2E0,V3l1_2E0,V2l2_2E0),c_2Ecanonical_2ECons__monom_2E3(A_27a,V5c1_2E0,V3l1_2E0,c_2Ecanonical_2ECons__monom_2E3(A_27a,V4c2_2E0,V2l2_2E0,V0t2_2E0)),c_2Ecanonical_2ECons__monom_2E3(A_27a,c_2Esemi__ring_2Esemi__ring__SRP_2E3(A_27a,V1sr_2E0,V5c1_2E0,V4c2_2E0),V3l1_2E0,V0t2_2E0),c_2Ecanonical_2ECons__monom_2E3(A_27a,V4c2_2E0,V2l2_2E0,c_2Ecanonical_2Emonom__insert_2E4(A_27a,V1sr_2E0,V5c1_2E0,V3l1_2E0,V0t2_2E0))) )
& ! [V6t2_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a),V7sr_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a),V8l2_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V9l1_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V10c1_2E0: A_27a] : ( c_2Ecanonical_2Emonom__insert_2E4(A_27a,V7sr_2E0,V10c1_2E0,V9l1_2E0,c_2Ecanonical_2ECons__varlist_2E2(A_27a,V8l2_2E0,V6t2_2E0)) = c_2EternaryComparisons_2Eordering__CASE_2E4(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EternaryComparisons_2Elist__compare_2E3(tyop_2Equote_2Eindex,tyop_2Equote_2Eindex,c_2Equote_2Eindex__compare_2E0,V9l1_2E0,V8l2_2E0),c_2Ecanonical_2ECons__monom_2E3(A_27a,V10c1_2E0,V9l1_2E0,c_2Ecanonical_2ECons__varlist_2E2(A_27a,V8l2_2E0,V6t2_2E0)),c_2Ecanonical_2ECons__monom_2E3(A_27a,c_2Esemi__ring_2Esemi__ring__SRP_2E3(A_27a,V7sr_2E0,V10c1_2E0,c_2Esemi__ring_2Esemi__ring__SR1_2E1(A_27a,V7sr_2E0)),V9l1_2E0,V6t2_2E0),c_2Ecanonical_2ECons__varlist_2E2(A_27a,V8l2_2E0,c_2Ecanonical_2Emonom__insert_2E4(A_27a,V7sr_2E0,V10c1_2E0,V9l1_2E0,V6t2_2E0))) )
& ! [V11sr_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a),V12l1_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V13c1_2E0: A_27a] : ( c_2Ecanonical_2Emonom__insert_2E4(A_27a,V11sr_2E0,V13c1_2E0,V12l1_2E0,c_2Ecanonical_2ENil__monom_2E0(A_27a)) = c_2Ecanonical_2ECons__monom_2E3(A_27a,V13c1_2E0,V12l1_2E0,c_2Ecanonical_2ENil__monom_2E0(A_27a)) ) ) ).
tff(thm_2Ecanonical_2Evarlist__insert__ind,axiom,
! [A_27a: $tType,V0P_2E0: 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)))] :
( ( ! [V1sr_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a),V2l1_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V3c2_2E0: A_27a,V4l2_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V5t2_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] :
( ( ( c_2EternaryComparisons_2Elist__compare_2E3(tyop_2Equote_2Eindex,tyop_2Equote_2Eindex,c_2Equote_2Eindex__compare_2E0,V2l1_2E0,V4l2_2E0) = c_2EternaryComparisons_2EGREATER_2E0 )
=> p(app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool,app_2E2(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool),app_2E2(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool)),V0P_2E0,V1sr_2E0),V2l1_2E0),V5t2_2E0)) )
=> p(app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool,app_2E2(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool),app_2E2(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool)),V0P_2E0,V1sr_2E0),V2l1_2E0),c_2Ecanonical_2ECons__monom_2E3(A_27a,V3c2_2E0,V4l2_2E0,V5t2_2E0))) )
& ! [V6sr_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a),V7l1_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V8l2_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V9t2_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] :
( ( ( c_2EternaryComparisons_2Elist__compare_2E3(tyop_2Equote_2Eindex,tyop_2Equote_2Eindex,c_2Equote_2Eindex__compare_2E0,V7l1_2E0,V8l2_2E0) = c_2EternaryComparisons_2EGREATER_2E0 )
=> p(app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool,app_2E2(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool),app_2E2(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool)),V0P_2E0,V6sr_2E0),V7l1_2E0),V9t2_2E0)) )
=> p(app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool,app_2E2(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool),app_2E2(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool)),V0P_2E0,V6sr_2E0),V7l1_2E0),c_2Ecanonical_2ECons__varlist_2E2(A_27a,V8l2_2E0,V9t2_2E0))) )
& ! [V10sr_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a),V11l1_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex)] : p(app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool,app_2E2(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool),app_2E2(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool)),V0P_2E0,V10sr_2E0),V11l1_2E0),c_2Ecanonical_2ENil__monom_2E0(A_27a))) )
=> ! [V12v_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a),V13v1_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V14v2_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : p(app_2E2(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool,app_2E2(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool),app_2E2(tyop_2Esemi__ring_2Esemi__ring(A_27a),tyop_2Emin_2Efun(tyop_2Elist_2Elist(tyop_2Equote_2Eindex),tyop_2Emin_2Efun(tyop_2Ecanonical_2Ecanonical__sum(A_27a),tyop_2Emin_2Ebool)),V0P_2E0,V12v_2E0),V13v1_2E0),V14v2_2E0)) ) ).
tff(thm_2Ecanonical_2Evarlist__insert__def,axiom,
! [A_27a: $tType] :
( ! [V0t2_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a),V1sr_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a),V2l2_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V3l1_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V4c2_2E0: A_27a] : ( c_2Ecanonical_2Evarlist__insert_2E3(A_27a,V1sr_2E0,V3l1_2E0,c_2Ecanonical_2ECons__monom_2E3(A_27a,V4c2_2E0,V2l2_2E0,V0t2_2E0)) = c_2EternaryComparisons_2Eordering__CASE_2E4(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EternaryComparisons_2Elist__compare_2E3(tyop_2Equote_2Eindex,tyop_2Equote_2Eindex,c_2Equote_2Eindex__compare_2E0,V3l1_2E0,V2l2_2E0),c_2Ecanonical_2ECons__varlist_2E2(A_27a,V3l1_2E0,c_2Ecanonical_2ECons__monom_2E3(A_27a,V4c2_2E0,V2l2_2E0,V0t2_2E0)),c_2Ecanonical_2ECons__monom_2E3(A_27a,c_2Esemi__ring_2Esemi__ring__SRP_2E3(A_27a,V1sr_2E0,c_2Esemi__ring_2Esemi__ring__SR1_2E1(A_27a,V1sr_2E0),V4c2_2E0),V3l1_2E0,V0t2_2E0),c_2Ecanonical_2ECons__monom_2E3(A_27a,V4c2_2E0,V2l2_2E0,c_2Ecanonical_2Evarlist__insert_2E3(A_27a,V1sr_2E0,V3l1_2E0,V0t2_2E0))) )
& ! [V5t2_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a),V6sr_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a),V7l2_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V8l1_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex)] : ( c_2Ecanonical_2Evarlist__insert_2E3(A_27a,V6sr_2E0,V8l1_2E0,c_2Ecanonical_2ECons__varlist_2E2(A_27a,V7l2_2E0,V5t2_2E0)) = c_2EternaryComparisons_2Eordering__CASE_2E4(tyop_2Ecanonical_2Ecanonical__sum(A_27a),c_2EternaryComparisons_2Elist__compare_2E3(tyop_2Equote_2Eindex,tyop_2Equote_2Eindex,c_2Equote_2Eindex__compare_2E0,V8l1_2E0,V7l2_2E0),c_2Ecanonical_2ECons__varlist_2E2(A_27a,V8l1_2E0,c_2Ecanonical_2ECons__varlist_2E2(A_27a,V7l2_2E0,V5t2_2E0)),c_2Ecanonical_2ECons__monom_2E3(A_27a,c_2Esemi__ring_2Esemi__ring__SRP_2E3(A_27a,V6sr_2E0,c_2Esemi__ring_2Esemi__ring__SR1_2E1(A_27a,V6sr_2E0),c_2Esemi__ring_2Esemi__ring__SR1_2E1(A_27a,V6sr_2E0)),V8l1_2E0,V5t2_2E0),c_2Ecanonical_2ECons__varlist_2E2(A_27a,V7l2_2E0,c_2Ecanonical_2Evarlist__insert_2E3(A_27a,V6sr_2E0,V8l1_2E0,V5t2_2E0))) )
& ! [V9sr_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a),V10l1_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex)] : ( c_2Ecanonical_2Evarlist__insert_2E3(A_27a,V9sr_2E0,V10l1_2E0,c_2Ecanonical_2ENil__monom_2E0(A_27a)) = c_2Ecanonical_2ECons__varlist_2E2(A_27a,V10l1_2E0,c_2Ecanonical_2ENil__monom_2E0(A_27a)) ) ) ).
tff(thm_2Ecanonical_2Eivl__aux__ok,axiom,
! [A_27a: $tType,V0sr_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a)] :
( p(c_2Esemi__ring_2Eis__semi__ring_2E1(A_27a,V0sr_2E0))
=> ! [V1vm_2E0: tyop_2Equote_2Evarmap(A_27a),V2v_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V3i_2E0: tyop_2Equote_2Eindex] : ( c_2Ecanonical_2Eivl__aux_2E4(A_27a,V0sr_2E0,V1vm_2E0,V3i_2E0,V2v_2E0) = c_2Esemi__ring_2Esemi__ring__SRM_2E3(A_27a,V0sr_2E0,c_2Equote_2Evarmap__find_2E2(A_27a,V3i_2E0,V1vm_2E0),c_2Ecanonical_2Einterp__vl_2E3(A_27a,V0sr_2E0,V1vm_2E0,V2v_2E0)) ) ) ).
tff(thm_2Ecanonical_2Evarlist__merge__ok,axiom,
! [A_27a: $tType,V0sr_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a)] :
( p(c_2Esemi__ring_2Eis__semi__ring_2E1(A_27a,V0sr_2E0))
=> ! [V1vm_2E0: tyop_2Equote_2Evarmap(A_27a),V2x_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V3y_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex)] : ( c_2Ecanonical_2Einterp__vl_2E3(A_27a,V0sr_2E0,V1vm_2E0,c_2EternaryComparisons_2Elist__merge_2E3(tyop_2Equote_2Eindex,c_2Equote_2Eindex__lt_2E0,V2x_2E0,V3y_2E0)) = c_2Esemi__ring_2Esemi__ring__SRM_2E3(A_27a,V0sr_2E0,c_2Ecanonical_2Einterp__vl_2E3(A_27a,V0sr_2E0,V1vm_2E0,V2x_2E0),c_2Ecanonical_2Einterp__vl_2E3(A_27a,V0sr_2E0,V1vm_2E0,V3y_2E0)) ) ) ).
tff(thm_2Ecanonical_2Eics__aux__ok,axiom,
! [A_27a: $tType,V0sr_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a)] :
( p(c_2Esemi__ring_2Eis__semi__ring_2E1(A_27a,V0sr_2E0))
=> ! [V1vm_2E0: tyop_2Equote_2Evarmap(A_27a),V2x_2E0: A_27a,V3s_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : ( c_2Ecanonical_2Eics__aux_2E4(A_27a,V0sr_2E0,V1vm_2E0,V2x_2E0,V3s_2E0) = c_2Esemi__ring_2Esemi__ring__SRP_2E3(A_27a,V0sr_2E0,V2x_2E0,c_2Ecanonical_2Einterp__cs_2E3(A_27a,V0sr_2E0,V1vm_2E0,V3s_2E0)) ) ) ).
tff(thm_2Ecanonical_2Einterp__m__ok,axiom,
! [A_27a: $tType,V0sr_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a)] :
( p(c_2Esemi__ring_2Eis__semi__ring_2E1(A_27a,V0sr_2E0))
=> ! [V1vm_2E0: tyop_2Equote_2Evarmap(A_27a),V2x_2E0: A_27a,V3l_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex)] : ( c_2Ecanonical_2Einterp__m_2E4(A_27a,V0sr_2E0,V1vm_2E0,V2x_2E0,V3l_2E0) = c_2Esemi__ring_2Esemi__ring__SRM_2E3(A_27a,V0sr_2E0,V2x_2E0,c_2Ecanonical_2Einterp__vl_2E3(A_27a,V0sr_2E0,V1vm_2E0,V3l_2E0)) ) ) ).
tff(thm_2Ecanonical_2Ecanonical__sum__merge__ok,axiom,
! [A_27a: $tType,V0sr_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a)] :
( p(c_2Esemi__ring_2Eis__semi__ring_2E1(A_27a,V0sr_2E0))
=> ! [V1vm_2E0: tyop_2Equote_2Evarmap(A_27a),V2x_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a),V3y_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : ( c_2Ecanonical_2Einterp__cs_2E3(A_27a,V0sr_2E0,V1vm_2E0,c_2Ecanonical_2Ecanonical__sum__merge_2E3(A_27a,V0sr_2E0,V2x_2E0,V3y_2E0)) = c_2Esemi__ring_2Esemi__ring__SRP_2E3(A_27a,V0sr_2E0,c_2Ecanonical_2Einterp__cs_2E3(A_27a,V0sr_2E0,V1vm_2E0,V2x_2E0),c_2Ecanonical_2Einterp__cs_2E3(A_27a,V0sr_2E0,V1vm_2E0,V3y_2E0)) ) ) ).
tff(thm_2Ecanonical_2Emonom__insert__ok,axiom,
! [A_27a: $tType,V0sr_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a)] :
( p(c_2Esemi__ring_2Eis__semi__ring_2E1(A_27a,V0sr_2E0))
=> ! [V1vm_2E0: tyop_2Equote_2Evarmap(A_27a),V2a_2E0: A_27a,V3l_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V4s_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : ( c_2Ecanonical_2Einterp__cs_2E3(A_27a,V0sr_2E0,V1vm_2E0,c_2Ecanonical_2Emonom__insert_2E4(A_27a,V0sr_2E0,V2a_2E0,V3l_2E0,V4s_2E0)) = c_2Esemi__ring_2Esemi__ring__SRP_2E3(A_27a,V0sr_2E0,c_2Esemi__ring_2Esemi__ring__SRM_2E3(A_27a,V0sr_2E0,V2a_2E0,c_2Ecanonical_2Einterp__vl_2E3(A_27a,V0sr_2E0,V1vm_2E0,V3l_2E0)),c_2Ecanonical_2Einterp__cs_2E3(A_27a,V0sr_2E0,V1vm_2E0,V4s_2E0)) ) ) ).
tff(thm_2Ecanonical_2Evarlist__insert__ok,axiom,
! [A_27a: $tType,V0sr_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a)] :
( p(c_2Esemi__ring_2Eis__semi__ring_2E1(A_27a,V0sr_2E0))
=> ! [V1vm_2E0: tyop_2Equote_2Evarmap(A_27a),V2l_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V3s_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : ( c_2Ecanonical_2Einterp__cs_2E3(A_27a,V0sr_2E0,V1vm_2E0,c_2Ecanonical_2Evarlist__insert_2E3(A_27a,V0sr_2E0,V2l_2E0,V3s_2E0)) = c_2Esemi__ring_2Esemi__ring__SRP_2E3(A_27a,V0sr_2E0,c_2Ecanonical_2Einterp__vl_2E3(A_27a,V0sr_2E0,V1vm_2E0,V2l_2E0),c_2Ecanonical_2Einterp__cs_2E3(A_27a,V0sr_2E0,V1vm_2E0,V3s_2E0)) ) ) ).
tff(thm_2Ecanonical_2Ecanonical__sum__scalar__ok,axiom,
! [A_27a: $tType,V0sr_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a)] :
( p(c_2Esemi__ring_2Eis__semi__ring_2E1(A_27a,V0sr_2E0))
=> ! [V1vm_2E0: tyop_2Equote_2Evarmap(A_27a),V2a_2E0: A_27a,V3s_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : ( c_2Ecanonical_2Einterp__cs_2E3(A_27a,V0sr_2E0,V1vm_2E0,c_2Ecanonical_2Ecanonical__sum__scalar_2E3(A_27a,V0sr_2E0,V2a_2E0,V3s_2E0)) = c_2Esemi__ring_2Esemi__ring__SRM_2E3(A_27a,V0sr_2E0,V2a_2E0,c_2Ecanonical_2Einterp__cs_2E3(A_27a,V0sr_2E0,V1vm_2E0,V3s_2E0)) ) ) ).
tff(thm_2Ecanonical_2Ecanonical__sum__scalar2__ok,axiom,
! [A_27a: $tType,V0sr_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a)] :
( p(c_2Esemi__ring_2Eis__semi__ring_2E1(A_27a,V0sr_2E0))
=> ! [V1vm_2E0: tyop_2Equote_2Evarmap(A_27a),V2l_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V3s_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : ( c_2Ecanonical_2Einterp__cs_2E3(A_27a,V0sr_2E0,V1vm_2E0,c_2Ecanonical_2Ecanonical__sum__scalar2_2E3(A_27a,V0sr_2E0,V2l_2E0,V3s_2E0)) = c_2Esemi__ring_2Esemi__ring__SRM_2E3(A_27a,V0sr_2E0,c_2Ecanonical_2Einterp__vl_2E3(A_27a,V0sr_2E0,V1vm_2E0,V2l_2E0),c_2Ecanonical_2Einterp__cs_2E3(A_27a,V0sr_2E0,V1vm_2E0,V3s_2E0)) ) ) ).
tff(thm_2Ecanonical_2Ecanonical__sum__scalar3__ok,axiom,
! [A_27a: $tType,V0sr_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a)] :
( p(c_2Esemi__ring_2Eis__semi__ring_2E1(A_27a,V0sr_2E0))
=> ! [V1vm_2E0: tyop_2Equote_2Evarmap(A_27a),V2c_2E0: A_27a,V3l_2E0: tyop_2Elist_2Elist(tyop_2Equote_2Eindex),V4s_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : ( c_2Ecanonical_2Einterp__cs_2E3(A_27a,V0sr_2E0,V1vm_2E0,c_2Ecanonical_2Ecanonical__sum__scalar3_2E4(A_27a,V0sr_2E0,V2c_2E0,V3l_2E0,V4s_2E0)) = c_2Esemi__ring_2Esemi__ring__SRM_2E3(A_27a,V0sr_2E0,c_2Esemi__ring_2Esemi__ring__SRM_2E3(A_27a,V0sr_2E0,V2c_2E0,c_2Ecanonical_2Einterp__vl_2E3(A_27a,V0sr_2E0,V1vm_2E0,V3l_2E0)),c_2Ecanonical_2Einterp__cs_2E3(A_27a,V0sr_2E0,V1vm_2E0,V4s_2E0)) ) ) ).
tff(thm_2Ecanonical_2Ecanonical__sum__prod__ok,axiom,
! [A_27a: $tType,V0sr_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a)] :
( p(c_2Esemi__ring_2Eis__semi__ring_2E1(A_27a,V0sr_2E0))
=> ! [V1vm_2E0: tyop_2Equote_2Evarmap(A_27a),V2x_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a),V3y_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : ( c_2Ecanonical_2Einterp__cs_2E3(A_27a,V0sr_2E0,V1vm_2E0,c_2Ecanonical_2Ecanonical__sum__prod_2E3(A_27a,V0sr_2E0,V2x_2E0,V3y_2E0)) = c_2Esemi__ring_2Esemi__ring__SRM_2E3(A_27a,V0sr_2E0,c_2Ecanonical_2Einterp__cs_2E3(A_27a,V0sr_2E0,V1vm_2E0,V2x_2E0),c_2Ecanonical_2Einterp__cs_2E3(A_27a,V0sr_2E0,V1vm_2E0,V3y_2E0)) ) ) ).
tff(thm_2Ecanonical_2Ecanonical__sum__simplify__ok,axiom,
! [A_27a: $tType,V0sr_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a)] :
( p(c_2Esemi__ring_2Eis__semi__ring_2E1(A_27a,V0sr_2E0))
=> ! [V1vm_2E0: tyop_2Equote_2Evarmap(A_27a),V2s_2E0: tyop_2Ecanonical_2Ecanonical__sum(A_27a)] : ( c_2Ecanonical_2Einterp__cs_2E3(A_27a,V0sr_2E0,V1vm_2E0,c_2Ecanonical_2Ecanonical__sum__simplify_2E2(A_27a,V0sr_2E0,V2s_2E0)) = c_2Ecanonical_2Einterp__cs_2E3(A_27a,V0sr_2E0,V1vm_2E0,V2s_2E0) ) ) ).
tff(thm_2Ecanonical_2Edatatype__spolynom,axiom,
! [A_27a: $tType,V0spolynom_2E0: 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))))] : p(c_2Ebool_2EDATATYPE_2E1(tyop_2Emin_2Ebool,app_2E2(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(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(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(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,c_2Ecanonical_2ESPvar_2E0(A_27a)),c_2Ecanonical_2ESPconst_2E0(A_27a)),c_2Ecanonical_2ESPplus_2E0(A_27a)),c_2Ecanonical_2ESPmult_2E0(A_27a)))) ).
tff(thm_2Ecanonical_2Espolynom__11,axiom,
! [A_27a: $tType] :
( ! [V0a_2E0: tyop_2Equote_2Eindex,V1a_27_2E0: tyop_2Equote_2Eindex] :
( ( c_2Ecanonical_2ESPvar_2E1(A_27a,V0a_2E0) = c_2Ecanonical_2ESPvar_2E1(A_27a,V1a_27_2E0) )
<=> ( V0a_2E0 = V1a_27_2E0 ) )
& ! [V2a_2E0: A_27a,V3a_27_2E0: A_27a] :
( ( c_2Ecanonical_2ESPconst_2E1(A_27a,V2a_2E0) = c_2Ecanonical_2ESPconst_2E1(A_27a,V3a_27_2E0) )
<=> ( V2a_2E0 = V3a_27_2E0 ) )
& ! [V4a0_2E0: tyop_2Ecanonical_2Espolynom(A_27a),V5a1_2E0: tyop_2Ecanonical_2Espolynom(A_27a),V6a0_27_2E0: tyop_2Ecanonical_2Espolynom(A_27a),V7a1_27_2E0: tyop_2Ecanonical_2Espolynom(A_27a)] :
( ( c_2Ecanonical_2ESPplus_2E2(A_27a,V4a0_2E0,V5a1_2E0) = c_2Ecanonical_2ESPplus_2E2(A_27a,V6a0_27_2E0,V7a1_27_2E0) )
<=> ( ( V4a0_2E0 = V6a0_27_2E0 )
& ( V5a1_2E0 = V7a1_27_2E0 ) ) )
& ! [V8a0_2E0: tyop_2Ecanonical_2Espolynom(A_27a),V9a1_2E0: tyop_2Ecanonical_2Espolynom(A_27a),V10a0_27_2E0: tyop_2Ecanonical_2Espolynom(A_27a),V11a1_27_2E0: tyop_2Ecanonical_2Espolynom(A_27a)] :
( ( c_2Ecanonical_2ESPmult_2E2(A_27a,V8a0_2E0,V9a1_2E0) = c_2Ecanonical_2ESPmult_2E2(A_27a,V10a0_27_2E0,V11a1_27_2E0) )
<=> ( ( V8a0_2E0 = V10a0_27_2E0 )
& ( V9a1_2E0 = V11a1_27_2E0 ) ) ) ) ).
tff(thm_2Ecanonical_2Espolynom__distinct,axiom,
! [A_27a: $tType] :
( ! [V0a_27_2E0: A_27a,V1a_2E0: tyop_2Equote_2Eindex] : ( c_2Ecanonical_2ESPvar_2E1(A_27a,V1a_2E0) != c_2Ecanonical_2ESPconst_2E1(A_27a,V0a_27_2E0) )
& ! [V2a1_2E0: tyop_2Ecanonical_2Espolynom(A_27a),V3a0_2E0: tyop_2Ecanonical_2Espolynom(A_27a),V4a_2E0: tyop_2Equote_2Eindex] : ( c_2Ecanonical_2ESPvar_2E1(A_27a,V4a_2E0) != c_2Ecanonical_2ESPplus_2E2(A_27a,V3a0_2E0,V2a1_2E0) )
& ! [V5a1_2E0: tyop_2Ecanonical_2Espolynom(A_27a),V6a0_2E0: tyop_2Ecanonical_2Espolynom(A_27a),V7a_2E0: tyop_2Equote_2Eindex] : ( c_2Ecanonical_2ESPvar_2E1(A_27a,V7a_2E0) != c_2Ecanonical_2ESPmult_2E2(A_27a,V6a0_2E0,V5a1_2E0) )
& ! [V8a1_2E0: tyop_2Ecanonical_2Espolynom(A_27a),V9a0_2E0: tyop_2Ecanonical_2Espolynom(A_27a),V10a_2E0: A_27a] : ( c_2Ecanonical_2ESPconst_2E1(A_27a,V10a_2E0) != c_2Ecanonical_2ESPplus_2E2(A_27a,V9a0_2E0,V8a1_2E0) )
& ! [V11a1_2E0: tyop_2Ecanonical_2Espolynom(A_27a),V12a0_2E0: tyop_2Ecanonical_2Espolynom(A_27a),V13a_2E0: A_27a] : ( c_2Ecanonical_2ESPconst_2E1(A_27a,V13a_2E0) != c_2Ecanonical_2ESPmult_2E2(A_27a,V12a0_2E0,V11a1_2E0) )
& ! [V14a1_27_2E0: tyop_2Ecanonical_2Espolynom(A_27a),V15a1_2E0: tyop_2Ecanonical_2Espolynom(A_27a),V16a0_27_2E0: tyop_2Ecanonical_2Espolynom(A_27a),V17a0_2E0: tyop_2Ecanonical_2Espolynom(A_27a)] : ( c_2Ecanonical_2ESPplus_2E2(A_27a,V17a0_2E0,V15a1_2E0) != c_2Ecanonical_2ESPmult_2E2(A_27a,V16a0_27_2E0,V14a1_27_2E0) ) ) ).
tff(thm_2Ecanonical_2Espolynom__nchotomy,axiom,
! [A_27a: $tType,V0ss_2E0: tyop_2Ecanonical_2Espolynom(A_27a)] :
( ? [V1i_2E0: tyop_2Equote_2Eindex] : ( V0ss_2E0 = c_2Ecanonical_2ESPvar_2E1(A_27a,V1i_2E0) )
| ? [V2a_2E0: A_27a] : ( V0ss_2E0 = c_2Ecanonical_2ESPconst_2E1(A_27a,V2a_2E0) )
| ? [V3s_2E0: tyop_2Ecanonical_2Espolynom(A_27a),V4s0_2E0: tyop_2Ecanonical_2Espolynom(A_27a)] : ( V0ss_2E0 = c_2Ecanonical_2ESPplus_2E2(A_27a,V3s_2E0,V4s0_2E0) )
| ? [V5s_2E0: tyop_2Ecanonical_2Espolynom(A_27a),V6s0_2E0: tyop_2Ecanonical_2Espolynom(A_27a)] : ( V0ss_2E0 = c_2Ecanonical_2ESPmult_2E2(A_27a,V5s_2E0,V6s0_2E0) ) ) ).
tff(thm_2Ecanonical_2Espolynom__Axiom,axiom,
! [A_27a: $tType,A_27b: $tType,V0f0_2E0: tyop_2Emin_2Efun(tyop_2Equote_2Eindex,A_27b),V1f1_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V2f2_2E0: tyop_2Emin_2Efun(tyop_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: 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))))] :
? [V4fn_2E0: tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27b)] :
( ! [V5a_2E0: tyop_2Equote_2Eindex] : ( app_2E2(tyop_2Ecanonical_2Espolynom(A_27a),A_27b,V4fn_2E0,c_2Ecanonical_2ESPvar_2E1(A_27a,V5a_2E0)) = app_2E2(tyop_2Equote_2Eindex,A_27b,V0f0_2E0,V5a_2E0) )
& ! [V6a_2E0: A_27a] : ( app_2E2(tyop_2Ecanonical_2Espolynom(A_27a),A_27b,V4fn_2E0,c_2Ecanonical_2ESPconst_2E1(A_27a,V6a_2E0)) = app_2E2(A_27a,A_27b,V1f1_2E0,V6a_2E0) )
& ! [V7a0_2E0: tyop_2Ecanonical_2Espolynom(A_27a),V8a1_2E0: tyop_2Ecanonical_2Espolynom(A_27a)] : ( app_2E2(tyop_2Ecanonical_2Espolynom(A_27a),A_27b,V4fn_2E0,c_2Ecanonical_2ESPplus_2E2(A_27a,V7a0_2E0,V8a1_2E0)) = app_2E2(A_27b,A_27b,app_2E2(A_27b,tyop_2Emin_2Efun(A_27b,A_27b),app_2E2(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b)),app_2E2(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,V7a0_2E0),V8a1_2E0),app_2E2(tyop_2Ecanonical_2Espolynom(A_27a),A_27b,V4fn_2E0,V7a0_2E0)),app_2E2(tyop_2Ecanonical_2Espolynom(A_27a),A_27b,V4fn_2E0,V8a1_2E0)) )
& ! [V9a0_2E0: tyop_2Ecanonical_2Espolynom(A_27a),V10a1_2E0: tyop_2Ecanonical_2Espolynom(A_27a)] : ( app_2E2(tyop_2Ecanonical_2Espolynom(A_27a),A_27b,V4fn_2E0,c_2Ecanonical_2ESPmult_2E2(A_27a,V9a0_2E0,V10a1_2E0)) = app_2E2(A_27b,A_27b,app_2E2(A_27b,tyop_2Emin_2Efun(A_27b,A_27b),app_2E2(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(A_27b,tyop_2Emin_2Efun(A_27b,A_27b)),app_2E2(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,V9a0_2E0),V10a1_2E0),app_2E2(tyop_2Ecanonical_2Espolynom(A_27a),A_27b,V4fn_2E0,V9a0_2E0)),app_2E2(tyop_2Ecanonical_2Espolynom(A_27a),A_27b,V4fn_2E0,V10a1_2E0)) ) ) ).
tff(thm_2Ecanonical_2Espolynom__induction,axiom,
! [A_27a: $tType,V0P_2E0: tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Ebool)] :
( ( ! [V1i_2E0: tyop_2Equote_2Eindex] : p(app_2E2(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Ebool,V0P_2E0,c_2Ecanonical_2ESPvar_2E1(A_27a,V1i_2E0)))
& ! [V2a_2E0: A_27a] : p(app_2E2(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Ebool,V0P_2E0,c_2Ecanonical_2ESPconst_2E1(A_27a,V2a_2E0)))
& ! [V3s_2E0: tyop_2Ecanonical_2Espolynom(A_27a),V4s0_2E0: tyop_2Ecanonical_2Espolynom(A_27a)] :
( ( p(app_2E2(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Ebool,V0P_2E0,V3s_2E0))
& p(app_2E2(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Ebool,V0P_2E0,V4s0_2E0)) )
=> p(app_2E2(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Ebool,V0P_2E0,c_2Ecanonical_2ESPplus_2E2(A_27a,V3s_2E0,V4s0_2E0))) )
& ! [V5s_2E0: tyop_2Ecanonical_2Espolynom(A_27a),V6s0_2E0: tyop_2Ecanonical_2Espolynom(A_27a)] :
( ( p(app_2E2(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Ebool,V0P_2E0,V5s_2E0))
& p(app_2E2(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Ebool,V0P_2E0,V6s0_2E0)) )
=> p(app_2E2(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Ebool,V0P_2E0,c_2Ecanonical_2ESPmult_2E2(A_27a,V5s_2E0,V6s0_2E0))) ) )
=> ! [V7s_2E0: tyop_2Ecanonical_2Espolynom(A_27a)] : p(app_2E2(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Ebool,V0P_2E0,V7s_2E0)) ) ).
tff(thm_2Ecanonical_2Espolynom__case__cong,axiom,
! [A_27a: $tType,A_27b: $tType,V0f3_27_2E0: tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27b)),V1f2_27_2E0: tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27b)),V2f1_27_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V3f_27_2E0: tyop_2Emin_2Efun(tyop_2Equote_2Eindex,A_27b),V4M_2E0: tyop_2Ecanonical_2Espolynom(A_27a),V5M_27_2E0: tyop_2Ecanonical_2Espolynom(A_27a),V6f_2E0: tyop_2Emin_2Efun(tyop_2Equote_2Eindex,A_27b),V7f1_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V8f2_2E0: tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27b)),V9f3_2E0: tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27b))] :
( ( ( V4M_2E0 = V5M_27_2E0 )
& ! [V10a_2E0: tyop_2Equote_2Eindex] :
( ( V5M_27_2E0 = c_2Ecanonical_2ESPvar_2E1(A_27a,V10a_2E0) )
=> ( app_2E2(tyop_2Equote_2Eindex,A_27b,V6f_2E0,V10a_2E0) = app_2E2(tyop_2Equote_2Eindex,A_27b,V3f_27_2E0,V10a_2E0) ) )
& ! [V11a_2E0: A_27a] :
( ( V5M_27_2E0 = c_2Ecanonical_2ESPconst_2E1(A_27a,V11a_2E0) )
=> ( app_2E2(A_27a,A_27b,V7f1_2E0,V11a_2E0) = app_2E2(A_27a,A_27b,V2f1_27_2E0,V11a_2E0) ) )
& ! [V12a0_2E0: tyop_2Ecanonical_2Espolynom(A_27a),V13a1_2E0: tyop_2Ecanonical_2Espolynom(A_27a)] :
( ( V5M_27_2E0 = c_2Ecanonical_2ESPplus_2E2(A_27a,V12a0_2E0,V13a1_2E0) )
=> ( app_2E2(tyop_2Ecanonical_2Espolynom(A_27a),A_27b,app_2E2(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27b),V8f2_2E0,V12a0_2E0),V13a1_2E0) = app_2E2(tyop_2Ecanonical_2Espolynom(A_27a),A_27b,app_2E2(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27b),V1f2_27_2E0,V12a0_2E0),V13a1_2E0) ) )
& ! [V14a0_2E0: tyop_2Ecanonical_2Espolynom(A_27a),V15a1_2E0: tyop_2Ecanonical_2Espolynom(A_27a)] :
( ( V5M_27_2E0 = c_2Ecanonical_2ESPmult_2E2(A_27a,V14a0_2E0,V15a1_2E0) )
=> ( app_2E2(tyop_2Ecanonical_2Espolynom(A_27a),A_27b,app_2E2(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27b),V9f3_2E0,V14a0_2E0),V15a1_2E0) = app_2E2(tyop_2Ecanonical_2Espolynom(A_27a),A_27b,app_2E2(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27b),V0f3_27_2E0,V14a0_2E0),V15a1_2E0) ) ) )
=> ( c_2Ecanonical_2Espolynom__CASE_2E5(A_27a,A_27b,V4M_2E0,V6f_2E0,V7f1_2E0,V8f2_2E0,V9f3_2E0) = c_2Ecanonical_2Espolynom__CASE_2E5(A_27a,A_27b,V5M_27_2E0,V3f_27_2E0,V2f1_27_2E0,V1f2_27_2E0,V0f3_27_2E0) ) ) ).
tff(thm_2Ecanonical_2Espolynom__case__eq,axiom,
! [A_27a: $tType,A_27b: $tType,V0x_2E0: tyop_2Ecanonical_2Espolynom(A_27a),V1v_2E0: A_27b,V2f3_2E0: tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27b)),V3f2_2E0: tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27b)),V4f1_2E0: tyop_2Emin_2Efun(A_27a,A_27b),V5f_2E0: tyop_2Emin_2Efun(tyop_2Equote_2Eindex,A_27b)] :
( ( c_2Ecanonical_2Espolynom__CASE_2E5(A_27a,A_27b,V0x_2E0,V5f_2E0,V4f1_2E0,V3f2_2E0,V2f3_2E0) = V1v_2E0 )
<=> ( ? [V6i_2E0: tyop_2Equote_2Eindex] :
( ( V0x_2E0 = c_2Ecanonical_2ESPvar_2E1(A_27a,V6i_2E0) )
& ( app_2E2(tyop_2Equote_2Eindex,A_27b,V5f_2E0,V6i_2E0) = V1v_2E0 ) )
| ? [V7a_2E0: A_27a] :
( ( V0x_2E0 = c_2Ecanonical_2ESPconst_2E1(A_27a,V7a_2E0) )
& ( app_2E2(A_27a,A_27b,V4f1_2E0,V7a_2E0) = V1v_2E0 ) )
| ? [V8s_2E0: tyop_2Ecanonical_2Espolynom(A_27a),V9s0_2E0: tyop_2Ecanonical_2Espolynom(A_27a)] :
( ( V0x_2E0 = c_2Ecanonical_2ESPplus_2E2(A_27a,V8s_2E0,V9s0_2E0) )
& ( app_2E2(tyop_2Ecanonical_2Espolynom(A_27a),A_27b,app_2E2(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27b),V3f2_2E0,V8s_2E0),V9s0_2E0) = V1v_2E0 ) )
| ? [V10s_2E0: tyop_2Ecanonical_2Espolynom(A_27a),V11s0_2E0: tyop_2Ecanonical_2Espolynom(A_27a)] :
( ( V0x_2E0 = c_2Ecanonical_2ESPmult_2E2(A_27a,V10s_2E0,V11s0_2E0) )
& ( app_2E2(tyop_2Ecanonical_2Espolynom(A_27a),A_27b,app_2E2(tyop_2Ecanonical_2Espolynom(A_27a),tyop_2Emin_2Efun(tyop_2Ecanonical_2Espolynom(A_27a),A_27b),V2f3_2E0,V10s_2E0),V11s0_2E0) = V1v_2E0 ) ) ) ) ).
tff(thm_2Ecanonical_2Espolynomial__normalize__ok,axiom,
! [A_27a: $tType,V0sr_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a)] :
( p(c_2Esemi__ring_2Eis__semi__ring_2E1(A_27a,V0sr_2E0))
=> ! [V1vm_2E0: tyop_2Equote_2Evarmap(A_27a),V2p_2E0: tyop_2Ecanonical_2Espolynom(A_27a)] : ( c_2Ecanonical_2Einterp__cs_2E3(A_27a,V0sr_2E0,V1vm_2E0,c_2Ecanonical_2Espolynom__normalize_2E2(A_27a,V0sr_2E0,V2p_2E0)) = c_2Ecanonical_2Einterp__sp_2E3(A_27a,V0sr_2E0,V1vm_2E0,V2p_2E0) ) ) ).
tff(thm_2Ecanonical_2Espolynomial__simplify__ok,axiom,
! [A_27a: $tType,V0sr_2E0: tyop_2Esemi__ring_2Esemi__ring(A_27a)] :
( p(c_2Esemi__ring_2Eis__semi__ring_2E1(A_27a,V0sr_2E0))
=> ! [V1vm_2E0: tyop_2Equote_2Evarmap(A_27a),V2p_2E0: tyop_2Ecanonical_2Espolynom(A_27a)] : ( c_2Ecanonical_2Einterp__cs_2E3(A_27a,V0sr_2E0,V1vm_2E0,c_2Ecanonical_2Espolynom__simplify_2E2(A_27a,V0sr_2E0,V2p_2E0)) = c_2Ecanonical_2Einterp__sp_2E3(A_27a,V0sr_2E0,V1vm_2E0,V2p_2E0) ) ) ).
%------------------------------------------------------------------------------