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 : 140 ( 8 unt; 78 typ; 0 def)
% Number of atoms : 247 ( 168 equ; 10 cnn)
% Maximal formula atoms : 22 ( 1 avg)
% Number of connectives : 3178 ( 10 ~; 11 |; 104 &;2981 @)
% ( 15 <=>; 57 =>; 0 <=; 0 <~>)
% Maximal formula depth : 39 ( 10 avg;2981 nst)
% Number of types : 5 ( 4 usr)
% Number of type conns : 304 ( 304 >; 0 *; 0 +; 0 <<)
% Number of symbols : 76 ( 74 usr; 6 con; 0-7 aty)
% Number of variables : 688 ( 35 ^ 571 !; 27 ?; 688 :)
% ( 55 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tyop_2Ecanonical_2Ecanonical__sum,type,
tyop_2Ecanonical_2Ecanonical__sum: $tType > $tType ).
thf(tyop_2Ecanonical_2Espolynom,type,
tyop_2Ecanonical_2Espolynom: $tType > $tType ).
thf(tyop_2Eind__type_2Erecspace,type,
tyop_2Eind__type_2Erecspace: $tType > $tType ).
thf(tyop_2Elist_2Elist,type,
tyop_2Elist_2Elist: $tType > $tType ).
thf(tyop_2Emin_2Ebool,type,
tyop_2Emin_2Ebool: $tType ).
thf(tyop_2Emin_2Efun,type,
tyop_2Emin_2Efun: $tType > $tType > $tType ).
thf(tyop_2Enum_2Enum,type,
tyop_2Enum_2Enum: $tType ).
thf(tyop_2Epair_2Eprod,type,
tyop_2Epair_2Eprod: $tType > $tType > $tType ).
thf(tyop_2Equote_2Eindex,type,
tyop_2Equote_2Eindex: $tType ).
thf(tyop_2Equote_2Evarmap,type,
tyop_2Equote_2Evarmap: $tType > $tType ).
thf(tyop_2Esemi__ring_2Esemi__ring,type,
tyop_2Esemi__ring_2Esemi__ring: $tType > $tType ).
thf(tyop_2EternaryComparisons_2Eordering,type,
tyop_2EternaryComparisons_2Eordering: $tType ).
thf(c_2Ebool_2E_21,type,
c_2Ebool_2E_21:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Earithmetic_2E_2B,type,
c_2Earithmetic_2E_2B: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Epair_2E_2C,type,
c_2Epair_2E_2C:
!>[A_27a: $tType,A_27b: $tType] : ( A_27a > A_27b > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) ).
thf(c_2Ebool_2E_2F_5C,type,
c_2Ebool_2E_2F_5C: $o > $o > $o ).
thf(c_2Enum_2E0,type,
c_2Enum_2E0: tyop_2Enum_2Enum ).
thf(c_2Emin_2E_3D,type,
c_2Emin_2E_3D:
!>[A_27a: $tType] : ( A_27a > A_27a > $o ) ).
thf(c_2Emin_2E_3D_3D_3E,type,
c_2Emin_2E_3D_3D_3E: $o > $o > $o ).
thf(c_2Ebool_2E_3F,type,
c_2Ebool_2E_3F:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Ebool_2EARB,type,
c_2Ebool_2EARB:
!>[A_27a: $tType] : A_27a ).
thf(c_2Earithmetic_2EBIT1,type,
c_2Earithmetic_2EBIT1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Eind__type_2EBOTTOM,type,
c_2Eind__type_2EBOTTOM:
!>[A_27a: $tType] : ( tyop_2Eind__type_2Erecspace @ A_27a ) ).
thf(c_2Ebool_2ECOND,type,
c_2Ebool_2ECOND:
!>[A_27a: $tType] : ( $o > A_27a > A_27a > A_27a ) ).
thf(c_2Elist_2ECONS,type,
c_2Elist_2ECONS:
!>[A_27a: $tType] : ( A_27a > ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Elist_2Elist @ A_27a ) ) ).
thf(c_2Eind__type_2ECONSTR,type,
c_2Eind__type_2ECONSTR:
!>[A_27a: $tType] : ( tyop_2Enum_2Enum > A_27a > ( tyop_2Enum_2Enum > ( tyop_2Eind__type_2Erecspace @ A_27a ) ) > ( tyop_2Eind__type_2Erecspace @ A_27a ) ) ).
thf(c_2Ecanonical_2ECons__monom,type,
c_2Ecanonical_2ECons__monom:
!>[A_27a: $tType] : ( A_27a > ( tyop_2Elist_2Elist @ tyop_2Equote_2Eindex ) > ( tyop_2Ecanonical_2Ecanonical__sum @ A_27a ) > ( tyop_2Ecanonical_2Ecanonical__sum @ A_27a ) ) ).
thf(c_2Ecanonical_2ECons__varlist,type,
c_2Ecanonical_2ECons__varlist:
!>[A_27a: $tType] : ( ( tyop_2Elist_2Elist @ tyop_2Equote_2Eindex ) > ( tyop_2Ecanonical_2Ecanonical__sum @ A_27a ) > ( tyop_2Ecanonical_2Ecanonical__sum @ A_27a ) ) ).
thf(c_2Ebool_2EDATATYPE,type,
c_2Ebool_2EDATATYPE:
!>[A_27a: $tType] : ( A_27a > $o ) ).
thf(c_2EternaryComparisons_2EEQUAL,type,
c_2EternaryComparisons_2EEQUAL: tyop_2EternaryComparisons_2Eordering ).
thf(c_2Eind__type_2EFCONS,type,
c_2Eind__type_2EFCONS:
!>[A_27a: $tType] : ( A_27a > ( tyop_2Enum_2Enum > A_27a ) > tyop_2Enum_2Enum > A_27a ) ).
thf(c_2EternaryComparisons_2EGREATER,type,
c_2EternaryComparisons_2EGREATER: tyop_2EternaryComparisons_2Eordering ).
thf(c_2EternaryComparisons_2ELESS,type,
c_2EternaryComparisons_2ELESS: tyop_2EternaryComparisons_2Eordering ).
thf(c_2Elist_2ENIL,type,
c_2Elist_2ENIL:
!>[A_27a: $tType] : ( tyop_2Elist_2Elist @ A_27a ) ).
thf(c_2Earithmetic_2ENUMERAL,type,
c_2Earithmetic_2ENUMERAL: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Ecanonical_2ENil__monom,type,
c_2Ecanonical_2ENil__monom:
!>[A_27a: $tType] : ( tyop_2Ecanonical_2Ecanonical__sum @ A_27a ) ).
thf(c_2Ecanonical_2ESPconst,type,
c_2Ecanonical_2ESPconst:
!>[A_27a: $tType] : ( A_27a > ( tyop_2Ecanonical_2Espolynom @ A_27a ) ) ).
thf(c_2Ecanonical_2ESPmult,type,
c_2Ecanonical_2ESPmult:
!>[A_27a: $tType] : ( ( tyop_2Ecanonical_2Espolynom @ A_27a ) > ( tyop_2Ecanonical_2Espolynom @ A_27a ) > ( tyop_2Ecanonical_2Espolynom @ A_27a ) ) ).
thf(c_2Ecanonical_2ESPplus,type,
c_2Ecanonical_2ESPplus:
!>[A_27a: $tType] : ( ( tyop_2Ecanonical_2Espolynom @ A_27a ) > ( tyop_2Ecanonical_2Espolynom @ A_27a ) > ( tyop_2Ecanonical_2Espolynom @ A_27a ) ) ).
thf(c_2Ecanonical_2ESPvar,type,
c_2Ecanonical_2ESPvar:
!>[A_27a: $tType] : ( tyop_2Equote_2Eindex > ( tyop_2Ecanonical_2Espolynom @ A_27a ) ) ).
thf(c_2Enum_2ESUC,type,
c_2Enum_2ESUC: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Ebool_2ETYPE__DEFINITION,type,
c_2Ebool_2ETYPE__DEFINITION:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > $o ) > ( A_27b > A_27a ) > $o ) ).
thf(c_2Earithmetic_2EZERO,type,
c_2Earithmetic_2EZERO: tyop_2Enum_2Enum ).
thf(c_2Ebool_2E_5C_2F,type,
c_2Ebool_2E_5C_2F: $o > $o > $o ).
thf(c_2Ecanonical_2Ecanonical__sum__CASE,type,
c_2Ecanonical_2Ecanonical__sum__CASE:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Ecanonical_2Ecanonical__sum @ A_27a ) > A_27b > ( A_27a > ( tyop_2Elist_2Elist @ tyop_2Equote_2Eindex ) > ( tyop_2Ecanonical_2Ecanonical__sum @ A_27a ) > A_27b ) > ( ( tyop_2Elist_2Elist @ tyop_2Equote_2Eindex ) > ( tyop_2Ecanonical_2Ecanonical__sum @ A_27a ) > A_27b ) > A_27b ) ).
thf(c_2Ecanonical_2Ecanonical__sum__merge,type,
c_2Ecanonical_2Ecanonical__sum__merge:
!>[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 ) ) ).
thf(c_2Ecanonical_2Ecanonical__sum__prod,type,
c_2Ecanonical_2Ecanonical__sum__prod:
!>[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 ) ) ).
thf(c_2Ecanonical_2Ecanonical__sum__scalar,type,
c_2Ecanonical_2Ecanonical__sum__scalar:
!>[A_27a: $tType] : ( ( tyop_2Esemi__ring_2Esemi__ring @ A_27a ) > A_27a > ( tyop_2Ecanonical_2Ecanonical__sum @ A_27a ) > ( tyop_2Ecanonical_2Ecanonical__sum @ A_27a ) ) ).
thf(c_2Ecanonical_2Ecanonical__sum__scalar2,type,
c_2Ecanonical_2Ecanonical__sum__scalar2:
!>[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 ) ) ).
thf(c_2Ecanonical_2Ecanonical__sum__scalar3,type,
c_2Ecanonical_2Ecanonical__sum__scalar3:
!>[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 ) ) ).
thf(c_2Ecanonical_2Ecanonical__sum__simplify,type,
c_2Ecanonical_2Ecanonical__sum__simplify:
!>[A_27a: $tType] : ( ( tyop_2Esemi__ring_2Esemi__ring @ A_27a ) > ( tyop_2Ecanonical_2Ecanonical__sum @ A_27a ) > ( tyop_2Ecanonical_2Ecanonical__sum @ A_27a ) ) ).
thf(c_2Ecanonical_2Ecanonical__sum__size,type,
c_2Ecanonical_2Ecanonical__sum__size:
!>[A_27a: $tType] : ( ( A_27a > tyop_2Enum_2Enum ) > ( tyop_2Ecanonical_2Ecanonical__sum @ A_27a ) > tyop_2Enum_2Enum ) ).
thf(c_2Ecanonical_2Eics__aux,type,
c_2Ecanonical_2Eics__aux:
!>[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 ) ).
thf(c_2Equote_2Eindex__compare,type,
c_2Equote_2Eindex__compare: tyop_2Equote_2Eindex > tyop_2Equote_2Eindex > tyop_2EternaryComparisons_2Eordering ).
thf(c_2Equote_2Eindex__lt,type,
c_2Equote_2Eindex__lt: tyop_2Equote_2Eindex > tyop_2Equote_2Eindex > $o ).
thf(c_2Equote_2Eindex__size,type,
c_2Equote_2Eindex__size: tyop_2Equote_2Eindex > tyop_2Enum_2Enum ).
thf(c_2Ecanonical_2Einterp__cs,type,
c_2Ecanonical_2Einterp__cs:
!>[A_27a: $tType] : ( ( tyop_2Esemi__ring_2Esemi__ring @ A_27a ) > ( tyop_2Equote_2Evarmap @ A_27a ) > ( tyop_2Ecanonical_2Ecanonical__sum @ A_27a ) > A_27a ) ).
thf(c_2Ecanonical_2Einterp__m,type,
c_2Ecanonical_2Einterp__m:
!>[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 ) ).
thf(c_2Ecanonical_2Einterp__sp,type,
c_2Ecanonical_2Einterp__sp:
!>[A_27a: $tType] : ( ( tyop_2Esemi__ring_2Esemi__ring @ A_27a ) > ( tyop_2Equote_2Evarmap @ A_27a ) > ( tyop_2Ecanonical_2Espolynom @ A_27a ) > A_27a ) ).
thf(c_2Ecanonical_2Einterp__vl,type,
c_2Ecanonical_2Einterp__vl:
!>[A_27a: $tType] : ( ( tyop_2Esemi__ring_2Esemi__ring @ A_27a ) > ( tyop_2Equote_2Evarmap @ A_27a ) > ( tyop_2Elist_2Elist @ tyop_2Equote_2Eindex ) > A_27a ) ).
thf(c_2Esemi__ring_2Eis__semi__ring,type,
c_2Esemi__ring_2Eis__semi__ring:
!>[A_27a: $tType] : ( ( tyop_2Esemi__ring_2Esemi__ring @ A_27a ) > $o ) ).
thf(c_2Ecanonical_2Eivl__aux,type,
c_2Ecanonical_2Eivl__aux:
!>[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 ) ).
thf(c_2EternaryComparisons_2Elist__compare,type,
c_2EternaryComparisons_2Elist__compare:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b > tyop_2EternaryComparisons_2Eordering ) > ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Elist_2Elist @ A_27b ) > tyop_2EternaryComparisons_2Eordering ) ).
thf(c_2EternaryComparisons_2Elist__merge,type,
c_2EternaryComparisons_2Elist__merge:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Elist_2Elist @ A_27a ) ) ).
thf(c_2Elist_2Elist__size,type,
c_2Elist_2Elist__size:
!>[A_27a: $tType] : ( ( A_27a > tyop_2Enum_2Enum ) > ( tyop_2Elist_2Elist @ A_27a ) > tyop_2Enum_2Enum ) ).
thf(c_2Ecanonical_2Emonom__insert,type,
c_2Ecanonical_2Emonom__insert:
!>[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 ) ) ).
thf(c_2EternaryComparisons_2Eordering__CASE,type,
c_2EternaryComparisons_2Eordering__CASE:
!>[A_27a: $tType] : ( tyop_2EternaryComparisons_2Eordering > A_27a > A_27a > A_27a > A_27a ) ).
thf(c_2Esemi__ring_2Esemi__ring__SR0,type,
c_2Esemi__ring_2Esemi__ring__SR0:
!>[A_27a: $tType] : ( ( tyop_2Esemi__ring_2Esemi__ring @ A_27a ) > A_27a ) ).
thf(c_2Esemi__ring_2Esemi__ring__SR1,type,
c_2Esemi__ring_2Esemi__ring__SR1:
!>[A_27a: $tType] : ( ( tyop_2Esemi__ring_2Esemi__ring @ A_27a ) > A_27a ) ).
thf(c_2Esemi__ring_2Esemi__ring__SRM,type,
c_2Esemi__ring_2Esemi__ring__SRM:
!>[A_27a: $tType] : ( ( tyop_2Esemi__ring_2Esemi__ring @ A_27a ) > A_27a > A_27a > A_27a ) ).
thf(c_2Esemi__ring_2Esemi__ring__SRP,type,
c_2Esemi__ring_2Esemi__ring__SRP:
!>[A_27a: $tType] : ( ( tyop_2Esemi__ring_2Esemi__ring @ A_27a ) > A_27a > A_27a > A_27a ) ).
thf(c_2Ecanonical_2Espolynom__CASE,type,
c_2Ecanonical_2Espolynom__CASE:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Ecanonical_2Espolynom @ A_27a ) > ( tyop_2Equote_2Eindex > A_27b ) > ( A_27a > A_27b ) > ( ( tyop_2Ecanonical_2Espolynom @ A_27a ) > ( tyop_2Ecanonical_2Espolynom @ A_27a ) > A_27b ) > ( ( tyop_2Ecanonical_2Espolynom @ A_27a ) > ( tyop_2Ecanonical_2Espolynom @ A_27a ) > A_27b ) > A_27b ) ).
thf(c_2Ecanonical_2Espolynom__normalize,type,
c_2Ecanonical_2Espolynom__normalize:
!>[A_27a: $tType] : ( ( tyop_2Esemi__ring_2Esemi__ring @ A_27a ) > ( tyop_2Ecanonical_2Espolynom @ A_27a ) > ( tyop_2Ecanonical_2Ecanonical__sum @ A_27a ) ) ).
thf(c_2Ecanonical_2Espolynom__simplify,type,
c_2Ecanonical_2Espolynom__simplify:
!>[A_27a: $tType] : ( ( tyop_2Esemi__ring_2Esemi__ring @ A_27a ) > ( tyop_2Ecanonical_2Espolynom @ A_27a ) > ( tyop_2Ecanonical_2Ecanonical__sum @ A_27a ) ) ).
thf(c_2Ecanonical_2Espolynom__size,type,
c_2Ecanonical_2Espolynom__size:
!>[A_27a: $tType] : ( ( A_27a > tyop_2Enum_2Enum ) > ( tyop_2Ecanonical_2Espolynom @ A_27a ) > tyop_2Enum_2Enum ) ).
thf(c_2Ecanonical_2Evarlist__insert,type,
c_2Ecanonical_2Evarlist__insert:
!>[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 ) ) ).
thf(c_2Equote_2Evarmap__find,type,
c_2Equote_2Evarmap__find:
!>[A_27a: $tType] : ( tyop_2Equote_2Eindex > ( tyop_2Equote_2Evarmap @ A_27a ) > A_27a ) ).
thf(c_2Ebool_2E_7E,type,
c_2Ebool_2E_7E: $o > $o ).
thf(logicdef_2E_2F_5C,axiom,
! [V0: $o,V1: $o] :
( ( c_2Ebool_2E_2F_5C @ V0 @ V1 )
<=> ( V0
& V1 ) ) ).
thf(logicdef_2E_5C_2F,axiom,
! [V0: $o,V1: $o] :
( ( c_2Ebool_2E_5C_2F @ V0 @ V1 )
<=> ( V0
| V1 ) ) ).
thf(logicdef_2E_7E,axiom,
! [V0: $o] :
( ( c_2Ebool_2E_7E @ V0 )
<=> ( (~) @ V0 ) ) ).
thf(logicdef_2E_3D_3D_3E,axiom,
! [V0: $o,V1: $o] :
( ( c_2Emin_2E_3D_3D_3E @ V0 @ V1 )
<=> ( V0
=> V1 ) ) ).
thf(logicdef_2E_3D,axiom,
! [A_27a: $tType,V0: A_27a,V1: A_27a] :
( ( c_2Emin_2E_3D @ A_27a @ V0 @ V1 )
<=> ( V0 = V1 ) ) ).
thf(quantdef_2E_21,axiom,
! [A_27a: $tType,V0f: A_27a > $o] :
( ( c_2Ebool_2E_21 @ A_27a @ V0f )
<=> ! [V1x: A_27a] : ( V0f @ V1x ) ) ).
thf(quantdef_2E_3F,axiom,
! [A_27a: $tType,V0f: A_27a > $o] :
( ( c_2Ebool_2E_3F @ A_27a @ V0f )
<=> ? [V1x: A_27a] : ( V0f @ V1x ) ) ).
thf(thm_2Ecanonical_2Ecanonical__sum__TY__DEF,axiom,
! [A_27a: $tType] :
? [V0rep: ( 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 @ ( tyop_2Eind__type_2Erecspace @ ( tyop_2Epair_2Eprod @ A_27a @ ( tyop_2Elist_2Elist @ tyop_2Equote_2Eindex ) ) ) @ ( tyop_2Ecanonical_2Ecanonical__sum @ A_27a )
@ ^ [V1a0_27: tyop_2Eind__type_2Erecspace @ ( tyop_2Epair_2Eprod @ A_27a @ ( tyop_2Elist_2Elist @ tyop_2Equote_2Eindex ) )] :
( c_2Ebool_2E_21 @ ( ( tyop_2Eind__type_2Erecspace @ ( tyop_2Epair_2Eprod @ A_27a @ ( tyop_2Elist_2Elist @ tyop_2Equote_2Eindex ) ) ) > $o )
@ ^ [V2_27canonical__sum_27: ( tyop_2Eind__type_2Erecspace @ ( tyop_2Epair_2Eprod @ A_27a @ ( tyop_2Elist_2Elist @ tyop_2Equote_2Eindex ) ) ) > $o] :
( c_2Emin_2E_3D_3D_3E
@ ( c_2Ebool_2E_21 @ ( tyop_2Eind__type_2Erecspace @ ( tyop_2Epair_2Eprod @ A_27a @ ( tyop_2Elist_2Elist @ tyop_2Equote_2Eindex ) ) )
@ ^ [V3a0_27: tyop_2Eind__type_2Erecspace @ ( tyop_2Epair_2Eprod @ A_27a @ ( tyop_2Elist_2Elist @ tyop_2Equote_2Eindex ) )] :
( c_2Emin_2E_3D_3D_3E
@ ( c_2Ebool_2E_5C_2F
@ ( c_2Emin_2E_3D @ ( tyop_2Eind__type_2Erecspace @ ( tyop_2Epair_2Eprod @ A_27a @ ( tyop_2Elist_2Elist @ tyop_2Equote_2Eindex ) ) ) @ V3a0_27
@ ( c_2Eind__type_2ECONSTR @ ( tyop_2Epair_2Eprod @ A_27a @ ( tyop_2Elist_2Elist @ tyop_2Equote_2Eindex ) ) @ c_2Enum_2E0 @ ( c_2Epair_2E_2C @ A_27a @ ( tyop_2Elist_2Elist @ tyop_2Equote_2Eindex ) @ ( c_2Ebool_2EARB @ A_27a ) @ ( c_2Ebool_2EARB @ ( tyop_2Elist_2Elist @ tyop_2Equote_2Eindex ) ) )
@ ^ [V4n: tyop_2Enum_2Enum] : ( c_2Eind__type_2EBOTTOM @ ( tyop_2Epair_2Eprod @ A_27a @ ( tyop_2Elist_2Elist @ tyop_2Equote_2Eindex ) ) ) ) )
@ ( c_2Ebool_2E_5C_2F
@ ( c_2Ebool_2E_3F @ A_27a
@ ^ [V5a0: A_27a] :
( c_2Ebool_2E_3F @ ( tyop_2Elist_2Elist @ tyop_2Equote_2Eindex )
@ ^ [V6a1: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex] :
( c_2Ebool_2E_3F @ ( tyop_2Eind__type_2Erecspace @ ( tyop_2Epair_2Eprod @ A_27a @ ( tyop_2Elist_2Elist @ tyop_2Equote_2Eindex ) ) )
@ ^ [V7a2: tyop_2Eind__type_2Erecspace @ ( tyop_2Epair_2Eprod @ A_27a @ ( tyop_2Elist_2Elist @ tyop_2Equote_2Eindex ) )] :
( c_2Ebool_2E_2F_5C
@ ( c_2Emin_2E_3D @ ( tyop_2Eind__type_2Erecspace @ ( tyop_2Epair_2Eprod @ A_27a @ ( tyop_2Elist_2Elist @ tyop_2Equote_2Eindex ) ) ) @ V3a0_27
@ ( ^ [V8a0: A_27a,V9a1: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V10a2: tyop_2Eind__type_2Erecspace @ ( tyop_2Epair_2Eprod @ A_27a @ ( tyop_2Elist_2Elist @ tyop_2Equote_2Eindex ) )] :
( c_2Eind__type_2ECONSTR @ ( tyop_2Epair_2Eprod @ A_27a @ ( tyop_2Elist_2Elist @ tyop_2Equote_2Eindex ) ) @ ( c_2Enum_2ESUC @ c_2Enum_2E0 ) @ ( c_2Epair_2E_2C @ A_27a @ ( tyop_2Elist_2Elist @ tyop_2Equote_2Eindex ) @ V8a0 @ V9a1 )
@ ( c_2Eind__type_2EFCONS @ ( tyop_2Eind__type_2Erecspace @ ( tyop_2Epair_2Eprod @ A_27a @ ( tyop_2Elist_2Elist @ tyop_2Equote_2Eindex ) ) ) @ V10a2
@ ^ [V11n: tyop_2Enum_2Enum] : ( c_2Eind__type_2EBOTTOM @ ( tyop_2Epair_2Eprod @ A_27a @ ( tyop_2Elist_2Elist @ tyop_2Equote_2Eindex ) ) ) ) )
@ V5a0
@ V6a1
@ V7a2 ) )
@ ( V2_27canonical__sum_27 @ V7a2 ) ) ) ) )
@ ( c_2Ebool_2E_3F @ ( tyop_2Elist_2Elist @ tyop_2Equote_2Eindex )
@ ^ [V12a0: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex] :
( c_2Ebool_2E_3F @ ( tyop_2Eind__type_2Erecspace @ ( tyop_2Epair_2Eprod @ A_27a @ ( tyop_2Elist_2Elist @ tyop_2Equote_2Eindex ) ) )
@ ^ [V13a1: tyop_2Eind__type_2Erecspace @ ( tyop_2Epair_2Eprod @ A_27a @ ( tyop_2Elist_2Elist @ tyop_2Equote_2Eindex ) )] :
( c_2Ebool_2E_2F_5C
@ ( c_2Emin_2E_3D @ ( tyop_2Eind__type_2Erecspace @ ( tyop_2Epair_2Eprod @ A_27a @ ( tyop_2Elist_2Elist @ tyop_2Equote_2Eindex ) ) ) @ V3a0_27
@ ( ^ [V14a0: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V15a1: tyop_2Eind__type_2Erecspace @ ( tyop_2Epair_2Eprod @ A_27a @ ( tyop_2Elist_2Elist @ tyop_2Equote_2Eindex ) )] :
( c_2Eind__type_2ECONSTR @ ( tyop_2Epair_2Eprod @ A_27a @ ( tyop_2Elist_2Elist @ tyop_2Equote_2Eindex ) ) @ ( c_2Enum_2ESUC @ ( c_2Enum_2ESUC @ c_2Enum_2E0 ) ) @ ( c_2Epair_2E_2C @ A_27a @ ( tyop_2Elist_2Elist @ tyop_2Equote_2Eindex ) @ ( c_2Ebool_2EARB @ A_27a ) @ V14a0 )
@ ( c_2Eind__type_2EFCONS @ ( tyop_2Eind__type_2Erecspace @ ( tyop_2Epair_2Eprod @ A_27a @ ( tyop_2Elist_2Elist @ tyop_2Equote_2Eindex ) ) ) @ V15a1
@ ^ [V16n: tyop_2Enum_2Enum] : ( c_2Eind__type_2EBOTTOM @ ( tyop_2Epair_2Eprod @ A_27a @ ( tyop_2Elist_2Elist @ tyop_2Equote_2Eindex ) ) ) ) )
@ V12a0
@ V13a1 ) )
@ ( V2_27canonical__sum_27 @ V13a1 ) ) ) ) ) )
@ ( V2_27canonical__sum_27 @ V3a0_27 ) ) )
@ ( V2_27canonical__sum_27 @ V1a0_27 ) ) )
@ V0rep ) ).
thf(thm_2Ecanonical_2Ecanonical__sum__case__def,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ! [V0v: A_27b,V1f: A_27a > ( tyop_2Elist_2Elist @ tyop_2Equote_2Eindex ) > ( tyop_2Ecanonical_2Ecanonical__sum @ A_27a ) > A_27b,V2f1: ( tyop_2Elist_2Elist @ tyop_2Equote_2Eindex ) > ( tyop_2Ecanonical_2Ecanonical__sum @ A_27a ) > A_27b] :
( ( c_2Ecanonical_2Ecanonical__sum__CASE @ A_27a @ A_27b @ ( c_2Ecanonical_2ENil__monom @ A_27a ) @ V0v @ V1f @ V2f1 )
= V0v )
& ! [V3a0: A_27a,V4a1: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V5a2: tyop_2Ecanonical_2Ecanonical__sum @ A_27a,V6v: A_27b,V7f: A_27a > ( tyop_2Elist_2Elist @ tyop_2Equote_2Eindex ) > ( tyop_2Ecanonical_2Ecanonical__sum @ A_27a ) > A_27b,V8f1: ( tyop_2Elist_2Elist @ tyop_2Equote_2Eindex ) > ( tyop_2Ecanonical_2Ecanonical__sum @ A_27a ) > A_27b] :
( ( c_2Ecanonical_2Ecanonical__sum__CASE @ A_27a @ A_27b @ ( c_2Ecanonical_2ECons__monom @ A_27a @ V3a0 @ V4a1 @ V5a2 ) @ V6v @ V7f @ V8f1 )
= ( V7f @ V3a0 @ V4a1 @ V5a2 ) )
& ! [V9a0: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V10a1: tyop_2Ecanonical_2Ecanonical__sum @ A_27a,V11v: A_27b,V12f: A_27a > ( tyop_2Elist_2Elist @ tyop_2Equote_2Eindex ) > ( tyop_2Ecanonical_2Ecanonical__sum @ A_27a ) > A_27b,V13f1: ( tyop_2Elist_2Elist @ tyop_2Equote_2Eindex ) > ( tyop_2Ecanonical_2Ecanonical__sum @ A_27a ) > A_27b] :
( ( c_2Ecanonical_2Ecanonical__sum__CASE @ A_27a @ A_27b @ ( c_2Ecanonical_2ECons__varlist @ A_27a @ V9a0 @ V10a1 ) @ V11v @ V12f @ V13f1 )
= ( V13f1 @ V9a0 @ V10a1 ) ) ) ).
thf(thm_2Ecanonical_2Ecanonical__sum__size__def,axiom,
! [A_27a: $tType] :
( ! [V0f: A_27a > tyop_2Enum_2Enum] :
( ( c_2Ecanonical_2Ecanonical__sum__size @ A_27a @ V0f @ ( c_2Ecanonical_2ENil__monom @ A_27a ) )
= c_2Enum_2E0 )
& ! [V1f: A_27a > tyop_2Enum_2Enum,V2a0: A_27a,V3a1: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V4a2: tyop_2Ecanonical_2Ecanonical__sum @ A_27a] :
( ( c_2Ecanonical_2Ecanonical__sum__size @ A_27a @ V1f @ ( c_2Ecanonical_2ECons__monom @ A_27a @ V2a0 @ V3a1 @ V4a2 ) )
= ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ ( c_2Earithmetic_2E_2B @ ( V1f @ V2a0 ) @ ( c_2Earithmetic_2E_2B @ ( c_2Elist_2Elist__size @ tyop_2Equote_2Eindex @ c_2Equote_2Eindex__size @ V3a1 ) @ ( c_2Ecanonical_2Ecanonical__sum__size @ A_27a @ V1f @ V4a2 ) ) ) ) )
& ! [V5f: A_27a > tyop_2Enum_2Enum,V6a0: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V7a1: tyop_2Ecanonical_2Ecanonical__sum @ A_27a] :
( ( c_2Ecanonical_2Ecanonical__sum__size @ A_27a @ V5f @ ( c_2Ecanonical_2ECons__varlist @ A_27a @ V6a0 @ V7a1 ) )
= ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ ( c_2Earithmetic_2E_2B @ ( c_2Elist_2Elist__size @ tyop_2Equote_2Eindex @ c_2Equote_2Eindex__size @ V6a0 ) @ ( c_2Ecanonical_2Ecanonical__sum__size @ A_27a @ V5f @ V7a1 ) ) ) ) ) ).
thf(thm_2Ecanonical_2Ecanonical__sum__scalar__def,axiom,
! [A_27a: $tType] :
( ! [V0sr: tyop_2Esemi__ring_2Esemi__ring @ A_27a,V1c0: A_27a,V2c: A_27a,V3l: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V4t: tyop_2Ecanonical_2Ecanonical__sum @ A_27a] :
( ( c_2Ecanonical_2Ecanonical__sum__scalar @ A_27a @ V0sr @ V1c0 @ ( c_2Ecanonical_2ECons__monom @ A_27a @ V2c @ V3l @ V4t ) )
= ( c_2Ecanonical_2ECons__monom @ A_27a @ ( c_2Esemi__ring_2Esemi__ring__SRM @ A_27a @ V0sr @ V1c0 @ V2c ) @ V3l @ ( c_2Ecanonical_2Ecanonical__sum__scalar @ A_27a @ V0sr @ V1c0 @ V4t ) ) )
& ! [V5sr: tyop_2Esemi__ring_2Esemi__ring @ A_27a,V6c0: A_27a,V7l: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V8t: tyop_2Ecanonical_2Ecanonical__sum @ A_27a] :
( ( c_2Ecanonical_2Ecanonical__sum__scalar @ A_27a @ V5sr @ V6c0 @ ( c_2Ecanonical_2ECons__varlist @ A_27a @ V7l @ V8t ) )
= ( c_2Ecanonical_2ECons__monom @ A_27a @ V6c0 @ V7l @ ( c_2Ecanonical_2Ecanonical__sum__scalar @ A_27a @ V5sr @ V6c0 @ V8t ) ) )
& ! [V9sr: tyop_2Esemi__ring_2Esemi__ring @ A_27a,V10c0: A_27a] :
( ( c_2Ecanonical_2Ecanonical__sum__scalar @ A_27a @ V9sr @ V10c0 @ ( c_2Ecanonical_2ENil__monom @ A_27a ) )
= ( c_2Ecanonical_2ENil__monom @ A_27a ) ) ) ).
thf(thm_2Ecanonical_2Ecanonical__sum__scalar2__def,axiom,
! [A_27a: $tType] :
( ! [V0sr: tyop_2Esemi__ring_2Esemi__ring @ A_27a,V1l0: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V2c: A_27a,V3l: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V4t: tyop_2Ecanonical_2Ecanonical__sum @ A_27a] :
( ( c_2Ecanonical_2Ecanonical__sum__scalar2 @ A_27a @ V0sr @ V1l0 @ ( c_2Ecanonical_2ECons__monom @ A_27a @ V2c @ V3l @ V4t ) )
= ( c_2Ecanonical_2Emonom__insert @ A_27a @ V0sr @ V2c @ ( c_2EternaryComparisons_2Elist__merge @ tyop_2Equote_2Eindex @ c_2Equote_2Eindex__lt @ V1l0 @ V3l ) @ ( c_2Ecanonical_2Ecanonical__sum__scalar2 @ A_27a @ V0sr @ V1l0 @ V4t ) ) )
& ! [V5sr: tyop_2Esemi__ring_2Esemi__ring @ A_27a,V6l0: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V7l: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V8t: tyop_2Ecanonical_2Ecanonical__sum @ A_27a] :
( ( c_2Ecanonical_2Ecanonical__sum__scalar2 @ A_27a @ V5sr @ V6l0 @ ( c_2Ecanonical_2ECons__varlist @ A_27a @ V7l @ V8t ) )
= ( c_2Ecanonical_2Evarlist__insert @ A_27a @ V5sr @ ( c_2EternaryComparisons_2Elist__merge @ tyop_2Equote_2Eindex @ c_2Equote_2Eindex__lt @ V6l0 @ V7l ) @ ( c_2Ecanonical_2Ecanonical__sum__scalar2 @ A_27a @ V5sr @ V6l0 @ V8t ) ) )
& ! [V9sr: tyop_2Esemi__ring_2Esemi__ring @ A_27a,V10l0: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex] :
( ( c_2Ecanonical_2Ecanonical__sum__scalar2 @ A_27a @ V9sr @ V10l0 @ ( c_2Ecanonical_2ENil__monom @ A_27a ) )
= ( c_2Ecanonical_2ENil__monom @ A_27a ) ) ) ).
thf(thm_2Ecanonical_2Ecanonical__sum__scalar3__def,axiom,
! [A_27a: $tType] :
( ! [V0sr: tyop_2Esemi__ring_2Esemi__ring @ A_27a,V1c0: A_27a,V2l0: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V3c: A_27a,V4l: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V5t: tyop_2Ecanonical_2Ecanonical__sum @ A_27a] :
( ( c_2Ecanonical_2Ecanonical__sum__scalar3 @ A_27a @ V0sr @ V1c0 @ V2l0 @ ( c_2Ecanonical_2ECons__monom @ A_27a @ V3c @ V4l @ V5t ) )
= ( c_2Ecanonical_2Emonom__insert @ A_27a @ V0sr @ ( c_2Esemi__ring_2Esemi__ring__SRM @ A_27a @ V0sr @ V1c0 @ V3c ) @ ( c_2EternaryComparisons_2Elist__merge @ tyop_2Equote_2Eindex @ c_2Equote_2Eindex__lt @ V2l0 @ V4l ) @ ( c_2Ecanonical_2Ecanonical__sum__scalar3 @ A_27a @ V0sr @ V1c0 @ V2l0 @ V5t ) ) )
& ! [V6sr: tyop_2Esemi__ring_2Esemi__ring @ A_27a,V7c0: A_27a,V8l0: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V9l: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V10t: tyop_2Ecanonical_2Ecanonical__sum @ A_27a] :
( ( c_2Ecanonical_2Ecanonical__sum__scalar3 @ A_27a @ V6sr @ V7c0 @ V8l0 @ ( c_2Ecanonical_2ECons__varlist @ A_27a @ V9l @ V10t ) )
= ( c_2Ecanonical_2Emonom__insert @ A_27a @ V6sr @ V7c0 @ ( c_2EternaryComparisons_2Elist__merge @ tyop_2Equote_2Eindex @ c_2Equote_2Eindex__lt @ V8l0 @ V9l ) @ ( c_2Ecanonical_2Ecanonical__sum__scalar3 @ A_27a @ V6sr @ V7c0 @ V8l0 @ V10t ) ) )
& ! [V11sr: tyop_2Esemi__ring_2Esemi__ring @ A_27a,V12c0: A_27a,V13l0: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex] :
( ( c_2Ecanonical_2Ecanonical__sum__scalar3 @ A_27a @ V11sr @ V12c0 @ V13l0 @ ( c_2Ecanonical_2ENil__monom @ A_27a ) )
= ( c_2Ecanonical_2ENil__monom @ A_27a ) ) ) ).
thf(thm_2Ecanonical_2Ecanonical__sum__prod__def,axiom,
! [A_27a: $tType] :
( ! [V0sr: tyop_2Esemi__ring_2Esemi__ring @ A_27a,V1c1: A_27a,V2l1: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V3t1: tyop_2Ecanonical_2Ecanonical__sum @ A_27a,V4s2: tyop_2Ecanonical_2Ecanonical__sum @ A_27a] :
( ( c_2Ecanonical_2Ecanonical__sum__prod @ A_27a @ V0sr @ ( c_2Ecanonical_2ECons__monom @ A_27a @ V1c1 @ V2l1 @ V3t1 ) @ V4s2 )
= ( c_2Ecanonical_2Ecanonical__sum__merge @ A_27a @ V0sr @ ( c_2Ecanonical_2Ecanonical__sum__scalar3 @ A_27a @ V0sr @ V1c1 @ V2l1 @ V4s2 ) @ ( c_2Ecanonical_2Ecanonical__sum__prod @ A_27a @ V0sr @ V3t1 @ V4s2 ) ) )
& ! [V5sr: tyop_2Esemi__ring_2Esemi__ring @ A_27a,V6l1: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V7t1: tyop_2Ecanonical_2Ecanonical__sum @ A_27a,V8s2: tyop_2Ecanonical_2Ecanonical__sum @ A_27a] :
( ( c_2Ecanonical_2Ecanonical__sum__prod @ A_27a @ V5sr @ ( c_2Ecanonical_2ECons__varlist @ A_27a @ V6l1 @ V7t1 ) @ V8s2 )
= ( c_2Ecanonical_2Ecanonical__sum__merge @ A_27a @ V5sr @ ( c_2Ecanonical_2Ecanonical__sum__scalar2 @ A_27a @ V5sr @ V6l1 @ V8s2 ) @ ( c_2Ecanonical_2Ecanonical__sum__prod @ A_27a @ V5sr @ V7t1 @ V8s2 ) ) )
& ! [V9sr: tyop_2Esemi__ring_2Esemi__ring @ A_27a,V10s2: tyop_2Ecanonical_2Ecanonical__sum @ A_27a] :
( ( c_2Ecanonical_2Ecanonical__sum__prod @ A_27a @ V9sr @ ( c_2Ecanonical_2ENil__monom @ A_27a ) @ V10s2 )
= ( c_2Ecanonical_2ENil__monom @ A_27a ) ) ) ).
thf(thm_2Ecanonical_2Ecanonical__sum__simplify__def,axiom,
! [A_27a: $tType] :
( ! [V0sr: tyop_2Esemi__ring_2Esemi__ring @ A_27a,V1c: A_27a,V2l: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V3t: tyop_2Ecanonical_2Ecanonical__sum @ A_27a] :
( ( c_2Ecanonical_2Ecanonical__sum__simplify @ A_27a @ V0sr @ ( c_2Ecanonical_2ECons__monom @ A_27a @ V1c @ V2l @ V3t ) )
= ( c_2Ebool_2ECOND @ ( tyop_2Ecanonical_2Ecanonical__sum @ A_27a ) @ ( c_2Emin_2E_3D @ A_27a @ V1c @ ( c_2Esemi__ring_2Esemi__ring__SR0 @ A_27a @ V0sr ) ) @ ( c_2Ecanonical_2Ecanonical__sum__simplify @ A_27a @ V0sr @ V3t ) @ ( c_2Ebool_2ECOND @ ( tyop_2Ecanonical_2Ecanonical__sum @ A_27a ) @ ( c_2Emin_2E_3D @ A_27a @ V1c @ ( c_2Esemi__ring_2Esemi__ring__SR1 @ A_27a @ V0sr ) ) @ ( c_2Ecanonical_2ECons__varlist @ A_27a @ V2l @ ( c_2Ecanonical_2Ecanonical__sum__simplify @ A_27a @ V0sr @ V3t ) ) @ ( c_2Ecanonical_2ECons__monom @ A_27a @ V1c @ V2l @ ( c_2Ecanonical_2Ecanonical__sum__simplify @ A_27a @ V0sr @ V3t ) ) ) ) )
& ! [V4sr: tyop_2Esemi__ring_2Esemi__ring @ A_27a,V5l: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V6t: tyop_2Ecanonical_2Ecanonical__sum @ A_27a] :
( ( c_2Ecanonical_2Ecanonical__sum__simplify @ A_27a @ V4sr @ ( c_2Ecanonical_2ECons__varlist @ A_27a @ V5l @ V6t ) )
= ( c_2Ecanonical_2ECons__varlist @ A_27a @ V5l @ ( c_2Ecanonical_2Ecanonical__sum__simplify @ A_27a @ V4sr @ V6t ) ) )
& ! [V7sr: tyop_2Esemi__ring_2Esemi__ring @ A_27a] :
( ( c_2Ecanonical_2Ecanonical__sum__simplify @ A_27a @ V7sr @ ( c_2Ecanonical_2ENil__monom @ A_27a ) )
= ( c_2Ecanonical_2ENil__monom @ A_27a ) ) ) ).
thf(thm_2Ecanonical_2Eivl__aux__def,axiom,
! [A_27a: $tType] :
( ! [V0sr: tyop_2Esemi__ring_2Esemi__ring @ A_27a,V1vm: tyop_2Equote_2Evarmap @ A_27a,V2x: tyop_2Equote_2Eindex] :
( ( c_2Ecanonical_2Eivl__aux @ A_27a @ V0sr @ V1vm @ V2x @ ( c_2Elist_2ENIL @ tyop_2Equote_2Eindex ) )
= ( c_2Equote_2Evarmap__find @ A_27a @ V2x @ V1vm ) )
& ! [V3sr: tyop_2Esemi__ring_2Esemi__ring @ A_27a,V4vm: tyop_2Equote_2Evarmap @ A_27a,V5x: tyop_2Equote_2Eindex,V6x_27: tyop_2Equote_2Eindex,V7t_27: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex] :
( ( c_2Ecanonical_2Eivl__aux @ A_27a @ V3sr @ V4vm @ V5x @ ( c_2Elist_2ECONS @ tyop_2Equote_2Eindex @ V6x_27 @ V7t_27 ) )
= ( c_2Esemi__ring_2Esemi__ring__SRM @ A_27a @ V3sr @ ( c_2Equote_2Evarmap__find @ A_27a @ V5x @ V4vm ) @ ( c_2Ecanonical_2Eivl__aux @ A_27a @ V3sr @ V4vm @ V6x_27 @ V7t_27 ) ) ) ) ).
thf(thm_2Ecanonical_2Einterp__vl__def,axiom,
! [A_27a: $tType] :
( ! [V0sr: tyop_2Esemi__ring_2Esemi__ring @ A_27a,V1vm: tyop_2Equote_2Evarmap @ A_27a] :
( ( c_2Ecanonical_2Einterp__vl @ A_27a @ V0sr @ V1vm @ ( c_2Elist_2ENIL @ tyop_2Equote_2Eindex ) )
= ( c_2Esemi__ring_2Esemi__ring__SR1 @ A_27a @ V0sr ) )
& ! [V2sr: tyop_2Esemi__ring_2Esemi__ring @ A_27a,V3vm: tyop_2Equote_2Evarmap @ A_27a,V4x: tyop_2Equote_2Eindex,V5t: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex] :
( ( c_2Ecanonical_2Einterp__vl @ A_27a @ V2sr @ V3vm @ ( c_2Elist_2ECONS @ tyop_2Equote_2Eindex @ V4x @ V5t ) )
= ( c_2Ecanonical_2Eivl__aux @ A_27a @ V2sr @ V3vm @ V4x @ V5t ) ) ) ).
thf(thm_2Ecanonical_2Einterp__m__def,axiom,
! [A_27a: $tType] :
( ! [V0sr: tyop_2Esemi__ring_2Esemi__ring @ A_27a,V1vm: tyop_2Equote_2Evarmap @ A_27a,V2c: A_27a] :
( ( c_2Ecanonical_2Einterp__m @ A_27a @ V0sr @ V1vm @ V2c @ ( c_2Elist_2ENIL @ tyop_2Equote_2Eindex ) )
= V2c )
& ! [V3sr: tyop_2Esemi__ring_2Esemi__ring @ A_27a,V4vm: tyop_2Equote_2Evarmap @ A_27a,V5c: A_27a,V6x: tyop_2Equote_2Eindex,V7t: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex] :
( ( c_2Ecanonical_2Einterp__m @ A_27a @ V3sr @ V4vm @ V5c @ ( c_2Elist_2ECONS @ tyop_2Equote_2Eindex @ V6x @ V7t ) )
= ( c_2Esemi__ring_2Esemi__ring__SRM @ A_27a @ V3sr @ V5c @ ( c_2Ecanonical_2Eivl__aux @ A_27a @ V3sr @ V4vm @ V6x @ V7t ) ) ) ) ).
thf(thm_2Ecanonical_2Eics__aux__def,axiom,
! [A_27a: $tType] :
( ! [V0sr: tyop_2Esemi__ring_2Esemi__ring @ A_27a,V1vm: tyop_2Equote_2Evarmap @ A_27a,V2a: A_27a] :
( ( c_2Ecanonical_2Eics__aux @ A_27a @ V0sr @ V1vm @ V2a @ ( c_2Ecanonical_2ENil__monom @ A_27a ) )
= V2a )
& ! [V3sr: tyop_2Esemi__ring_2Esemi__ring @ A_27a,V4vm: tyop_2Equote_2Evarmap @ A_27a,V5a: A_27a,V6l: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V7t: tyop_2Ecanonical_2Ecanonical__sum @ A_27a] :
( ( c_2Ecanonical_2Eics__aux @ A_27a @ V3sr @ V4vm @ V5a @ ( c_2Ecanonical_2ECons__varlist @ A_27a @ V6l @ V7t ) )
= ( c_2Esemi__ring_2Esemi__ring__SRP @ A_27a @ V3sr @ V5a @ ( c_2Ecanonical_2Eics__aux @ A_27a @ V3sr @ V4vm @ ( c_2Ecanonical_2Einterp__vl @ A_27a @ V3sr @ V4vm @ V6l ) @ V7t ) ) )
& ! [V8sr: tyop_2Esemi__ring_2Esemi__ring @ A_27a,V9vm: tyop_2Equote_2Evarmap @ A_27a,V10a: A_27a,V11c: A_27a,V12l: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V13t: tyop_2Ecanonical_2Ecanonical__sum @ A_27a] :
( ( c_2Ecanonical_2Eics__aux @ A_27a @ V8sr @ V9vm @ V10a @ ( c_2Ecanonical_2ECons__monom @ A_27a @ V11c @ V12l @ V13t ) )
= ( c_2Esemi__ring_2Esemi__ring__SRP @ A_27a @ V8sr @ V10a @ ( c_2Ecanonical_2Eics__aux @ A_27a @ V8sr @ V9vm @ ( c_2Ecanonical_2Einterp__m @ A_27a @ V8sr @ V9vm @ V11c @ V12l ) @ V13t ) ) ) ) ).
thf(thm_2Ecanonical_2Einterp__cs__def,axiom,
! [A_27a: $tType] :
( ! [V0sr: tyop_2Esemi__ring_2Esemi__ring @ A_27a,V1vm: tyop_2Equote_2Evarmap @ A_27a] :
( ( c_2Ecanonical_2Einterp__cs @ A_27a @ V0sr @ V1vm @ ( c_2Ecanonical_2ENil__monom @ A_27a ) )
= ( c_2Esemi__ring_2Esemi__ring__SR0 @ A_27a @ V0sr ) )
& ! [V2sr: tyop_2Esemi__ring_2Esemi__ring @ A_27a,V3vm: tyop_2Equote_2Evarmap @ A_27a,V4l: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V5t: tyop_2Ecanonical_2Ecanonical__sum @ A_27a] :
( ( c_2Ecanonical_2Einterp__cs @ A_27a @ V2sr @ V3vm @ ( c_2Ecanonical_2ECons__varlist @ A_27a @ V4l @ V5t ) )
= ( c_2Ecanonical_2Eics__aux @ A_27a @ V2sr @ V3vm @ ( c_2Ecanonical_2Einterp__vl @ A_27a @ V2sr @ V3vm @ V4l ) @ V5t ) )
& ! [V6sr: tyop_2Esemi__ring_2Esemi__ring @ A_27a,V7vm: tyop_2Equote_2Evarmap @ A_27a,V8c: A_27a,V9l: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V10t: tyop_2Ecanonical_2Ecanonical__sum @ A_27a] :
( ( c_2Ecanonical_2Einterp__cs @ A_27a @ V6sr @ V7vm @ ( c_2Ecanonical_2ECons__monom @ A_27a @ V8c @ V9l @ V10t ) )
= ( c_2Ecanonical_2Eics__aux @ A_27a @ V6sr @ V7vm @ ( c_2Ecanonical_2Einterp__m @ A_27a @ V6sr @ V7vm @ V8c @ V9l ) @ V10t ) ) ) ).
thf(thm_2Ecanonical_2Espolynom__TY__DEF,axiom,
! [A_27a: $tType] :
? [V0rep: ( tyop_2Ecanonical_2Espolynom @ A_27a ) > ( tyop_2Eind__type_2Erecspace @ ( tyop_2Epair_2Eprod @ tyop_2Equote_2Eindex @ A_27a ) )] :
( c_2Ebool_2ETYPE__DEFINITION @ ( tyop_2Eind__type_2Erecspace @ ( tyop_2Epair_2Eprod @ tyop_2Equote_2Eindex @ A_27a ) ) @ ( tyop_2Ecanonical_2Espolynom @ A_27a )
@ ^ [V1a0_27: tyop_2Eind__type_2Erecspace @ ( tyop_2Epair_2Eprod @ tyop_2Equote_2Eindex @ A_27a )] :
( c_2Ebool_2E_21 @ ( ( tyop_2Eind__type_2Erecspace @ ( tyop_2Epair_2Eprod @ tyop_2Equote_2Eindex @ A_27a ) ) > $o )
@ ^ [V2_27spolynom_27: ( tyop_2Eind__type_2Erecspace @ ( tyop_2Epair_2Eprod @ tyop_2Equote_2Eindex @ A_27a ) ) > $o] :
( c_2Emin_2E_3D_3D_3E
@ ( c_2Ebool_2E_21 @ ( tyop_2Eind__type_2Erecspace @ ( tyop_2Epair_2Eprod @ tyop_2Equote_2Eindex @ A_27a ) )
@ ^ [V3a0_27: tyop_2Eind__type_2Erecspace @ ( tyop_2Epair_2Eprod @ tyop_2Equote_2Eindex @ A_27a )] :
( c_2Emin_2E_3D_3D_3E
@ ( c_2Ebool_2E_5C_2F
@ ( c_2Ebool_2E_3F @ tyop_2Equote_2Eindex
@ ^ [V4a: tyop_2Equote_2Eindex] :
( c_2Emin_2E_3D @ ( tyop_2Eind__type_2Erecspace @ ( tyop_2Epair_2Eprod @ tyop_2Equote_2Eindex @ A_27a ) ) @ V3a0_27
@ ( ^ [V5a: tyop_2Equote_2Eindex] :
( c_2Eind__type_2ECONSTR @ ( tyop_2Epair_2Eprod @ tyop_2Equote_2Eindex @ A_27a ) @ c_2Enum_2E0 @ ( c_2Epair_2E_2C @ tyop_2Equote_2Eindex @ A_27a @ V5a @ ( c_2Ebool_2EARB @ A_27a ) )
@ ^ [V6n: tyop_2Enum_2Enum] : ( c_2Eind__type_2EBOTTOM @ ( tyop_2Epair_2Eprod @ tyop_2Equote_2Eindex @ A_27a ) ) )
@ V4a ) ) )
@ ( c_2Ebool_2E_5C_2F
@ ( c_2Ebool_2E_3F @ A_27a
@ ^ [V7a: A_27a] :
( c_2Emin_2E_3D @ ( tyop_2Eind__type_2Erecspace @ ( tyop_2Epair_2Eprod @ tyop_2Equote_2Eindex @ A_27a ) ) @ V3a0_27
@ ( ^ [V8a: A_27a] :
( c_2Eind__type_2ECONSTR @ ( tyop_2Epair_2Eprod @ tyop_2Equote_2Eindex @ A_27a ) @ ( c_2Enum_2ESUC @ c_2Enum_2E0 ) @ ( c_2Epair_2E_2C @ tyop_2Equote_2Eindex @ A_27a @ ( c_2Ebool_2EARB @ tyop_2Equote_2Eindex ) @ V8a )
@ ^ [V9n: tyop_2Enum_2Enum] : ( c_2Eind__type_2EBOTTOM @ ( tyop_2Epair_2Eprod @ tyop_2Equote_2Eindex @ A_27a ) ) )
@ V7a ) ) )
@ ( c_2Ebool_2E_5C_2F
@ ( c_2Ebool_2E_3F @ ( tyop_2Eind__type_2Erecspace @ ( tyop_2Epair_2Eprod @ tyop_2Equote_2Eindex @ A_27a ) )
@ ^ [V10a0: tyop_2Eind__type_2Erecspace @ ( tyop_2Epair_2Eprod @ tyop_2Equote_2Eindex @ A_27a )] :
( c_2Ebool_2E_3F @ ( tyop_2Eind__type_2Erecspace @ ( tyop_2Epair_2Eprod @ tyop_2Equote_2Eindex @ A_27a ) )
@ ^ [V11a1: tyop_2Eind__type_2Erecspace @ ( tyop_2Epair_2Eprod @ tyop_2Equote_2Eindex @ A_27a )] :
( c_2Ebool_2E_2F_5C
@ ( c_2Emin_2E_3D @ ( tyop_2Eind__type_2Erecspace @ ( tyop_2Epair_2Eprod @ tyop_2Equote_2Eindex @ A_27a ) ) @ V3a0_27
@ ( ^ [V12a0: tyop_2Eind__type_2Erecspace @ ( tyop_2Epair_2Eprod @ tyop_2Equote_2Eindex @ A_27a ),V13a1: tyop_2Eind__type_2Erecspace @ ( tyop_2Epair_2Eprod @ tyop_2Equote_2Eindex @ A_27a )] :
( c_2Eind__type_2ECONSTR @ ( tyop_2Epair_2Eprod @ tyop_2Equote_2Eindex @ A_27a ) @ ( c_2Enum_2ESUC @ ( c_2Enum_2ESUC @ c_2Enum_2E0 ) ) @ ( c_2Epair_2E_2C @ tyop_2Equote_2Eindex @ A_27a @ ( c_2Ebool_2EARB @ tyop_2Equote_2Eindex ) @ ( c_2Ebool_2EARB @ A_27a ) )
@ ( c_2Eind__type_2EFCONS @ ( tyop_2Eind__type_2Erecspace @ ( tyop_2Epair_2Eprod @ tyop_2Equote_2Eindex @ A_27a ) ) @ V12a0
@ ( c_2Eind__type_2EFCONS @ ( tyop_2Eind__type_2Erecspace @ ( tyop_2Epair_2Eprod @ tyop_2Equote_2Eindex @ A_27a ) ) @ V13a1
@ ^ [V14n: tyop_2Enum_2Enum] : ( c_2Eind__type_2EBOTTOM @ ( tyop_2Epair_2Eprod @ tyop_2Equote_2Eindex @ A_27a ) ) ) ) )
@ V10a0
@ V11a1 ) )
@ ( c_2Ebool_2E_2F_5C @ ( V2_27spolynom_27 @ V10a0 ) @ ( V2_27spolynom_27 @ V11a1 ) ) ) ) )
@ ( c_2Ebool_2E_3F @ ( tyop_2Eind__type_2Erecspace @ ( tyop_2Epair_2Eprod @ tyop_2Equote_2Eindex @ A_27a ) )
@ ^ [V15a0: tyop_2Eind__type_2Erecspace @ ( tyop_2Epair_2Eprod @ tyop_2Equote_2Eindex @ A_27a )] :
( c_2Ebool_2E_3F @ ( tyop_2Eind__type_2Erecspace @ ( tyop_2Epair_2Eprod @ tyop_2Equote_2Eindex @ A_27a ) )
@ ^ [V16a1: tyop_2Eind__type_2Erecspace @ ( tyop_2Epair_2Eprod @ tyop_2Equote_2Eindex @ A_27a )] :
( c_2Ebool_2E_2F_5C
@ ( c_2Emin_2E_3D @ ( tyop_2Eind__type_2Erecspace @ ( tyop_2Epair_2Eprod @ tyop_2Equote_2Eindex @ A_27a ) ) @ V3a0_27
@ ( ^ [V17a0: tyop_2Eind__type_2Erecspace @ ( tyop_2Epair_2Eprod @ tyop_2Equote_2Eindex @ A_27a ),V18a1: tyop_2Eind__type_2Erecspace @ ( tyop_2Epair_2Eprod @ tyop_2Equote_2Eindex @ A_27a )] :
( c_2Eind__type_2ECONSTR @ ( tyop_2Epair_2Eprod @ tyop_2Equote_2Eindex @ A_27a ) @ ( c_2Enum_2ESUC @ ( c_2Enum_2ESUC @ ( c_2Enum_2ESUC @ c_2Enum_2E0 ) ) ) @ ( c_2Epair_2E_2C @ tyop_2Equote_2Eindex @ A_27a @ ( c_2Ebool_2EARB @ tyop_2Equote_2Eindex ) @ ( c_2Ebool_2EARB @ A_27a ) )
@ ( c_2Eind__type_2EFCONS @ ( tyop_2Eind__type_2Erecspace @ ( tyop_2Epair_2Eprod @ tyop_2Equote_2Eindex @ A_27a ) ) @ V17a0
@ ( c_2Eind__type_2EFCONS @ ( tyop_2Eind__type_2Erecspace @ ( tyop_2Epair_2Eprod @ tyop_2Equote_2Eindex @ A_27a ) ) @ V18a1
@ ^ [V19n: tyop_2Enum_2Enum] : ( c_2Eind__type_2EBOTTOM @ ( tyop_2Epair_2Eprod @ tyop_2Equote_2Eindex @ A_27a ) ) ) ) )
@ V15a0
@ V16a1 ) )
@ ( c_2Ebool_2E_2F_5C @ ( V2_27spolynom_27 @ V15a0 ) @ ( V2_27spolynom_27 @ V16a1 ) ) ) ) ) ) ) )
@ ( V2_27spolynom_27 @ V3a0_27 ) ) )
@ ( V2_27spolynom_27 @ V1a0_27 ) ) )
@ V0rep ) ).
thf(thm_2Ecanonical_2Espolynom__case__def,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ! [V0a: tyop_2Equote_2Eindex,V1f: tyop_2Equote_2Eindex > A_27b,V2f1: A_27a > A_27b,V3f2: ( tyop_2Ecanonical_2Espolynom @ A_27a ) > ( tyop_2Ecanonical_2Espolynom @ A_27a ) > A_27b,V4f3: ( tyop_2Ecanonical_2Espolynom @ A_27a ) > ( tyop_2Ecanonical_2Espolynom @ A_27a ) > A_27b] :
( ( c_2Ecanonical_2Espolynom__CASE @ A_27a @ A_27b @ ( c_2Ecanonical_2ESPvar @ A_27a @ V0a ) @ V1f @ V2f1 @ V3f2 @ V4f3 )
= ( V1f @ V0a ) )
& ! [V5a: A_27a,V6f: tyop_2Equote_2Eindex > A_27b,V7f1: A_27a > A_27b,V8f2: ( tyop_2Ecanonical_2Espolynom @ A_27a ) > ( tyop_2Ecanonical_2Espolynom @ A_27a ) > A_27b,V9f3: ( tyop_2Ecanonical_2Espolynom @ A_27a ) > ( tyop_2Ecanonical_2Espolynom @ A_27a ) > A_27b] :
( ( c_2Ecanonical_2Espolynom__CASE @ A_27a @ A_27b @ ( c_2Ecanonical_2ESPconst @ A_27a @ V5a ) @ V6f @ V7f1 @ V8f2 @ V9f3 )
= ( V7f1 @ V5a ) )
& ! [V10a0: tyop_2Ecanonical_2Espolynom @ A_27a,V11a1: tyop_2Ecanonical_2Espolynom @ A_27a,V12f: tyop_2Equote_2Eindex > A_27b,V13f1: A_27a > A_27b,V14f2: ( tyop_2Ecanonical_2Espolynom @ A_27a ) > ( tyop_2Ecanonical_2Espolynom @ A_27a ) > A_27b,V15f3: ( tyop_2Ecanonical_2Espolynom @ A_27a ) > ( tyop_2Ecanonical_2Espolynom @ A_27a ) > A_27b] :
( ( c_2Ecanonical_2Espolynom__CASE @ A_27a @ A_27b @ ( c_2Ecanonical_2ESPplus @ A_27a @ V10a0 @ V11a1 ) @ V12f @ V13f1 @ V14f2 @ V15f3 )
= ( V14f2 @ V10a0 @ V11a1 ) )
& ! [V16a0: tyop_2Ecanonical_2Espolynom @ A_27a,V17a1: tyop_2Ecanonical_2Espolynom @ A_27a,V18f: tyop_2Equote_2Eindex > A_27b,V19f1: A_27a > A_27b,V20f2: ( tyop_2Ecanonical_2Espolynom @ A_27a ) > ( tyop_2Ecanonical_2Espolynom @ A_27a ) > A_27b,V21f3: ( tyop_2Ecanonical_2Espolynom @ A_27a ) > ( tyop_2Ecanonical_2Espolynom @ A_27a ) > A_27b] :
( ( c_2Ecanonical_2Espolynom__CASE @ A_27a @ A_27b @ ( c_2Ecanonical_2ESPmult @ A_27a @ V16a0 @ V17a1 ) @ V18f @ V19f1 @ V20f2 @ V21f3 )
= ( V21f3 @ V16a0 @ V17a1 ) ) ) ).
thf(thm_2Ecanonical_2Espolynom__size__def,axiom,
! [A_27a: $tType] :
( ! [V0f: A_27a > tyop_2Enum_2Enum,V1a: tyop_2Equote_2Eindex] :
( ( c_2Ecanonical_2Espolynom__size @ A_27a @ V0f @ ( c_2Ecanonical_2ESPvar @ A_27a @ V1a ) )
= ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ ( c_2Equote_2Eindex__size @ V1a ) ) )
& ! [V2f: A_27a > tyop_2Enum_2Enum,V3a: A_27a] :
( ( c_2Ecanonical_2Espolynom__size @ A_27a @ V2f @ ( c_2Ecanonical_2ESPconst @ A_27a @ V3a ) )
= ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ ( V2f @ V3a ) ) )
& ! [V4f: A_27a > tyop_2Enum_2Enum,V5a0: tyop_2Ecanonical_2Espolynom @ A_27a,V6a1: tyop_2Ecanonical_2Espolynom @ A_27a] :
( ( c_2Ecanonical_2Espolynom__size @ A_27a @ V4f @ ( c_2Ecanonical_2ESPplus @ A_27a @ V5a0 @ V6a1 ) )
= ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ ( c_2Earithmetic_2E_2B @ ( c_2Ecanonical_2Espolynom__size @ A_27a @ V4f @ V5a0 ) @ ( c_2Ecanonical_2Espolynom__size @ A_27a @ V4f @ V6a1 ) ) ) )
& ! [V7f: A_27a > tyop_2Enum_2Enum,V8a0: tyop_2Ecanonical_2Espolynom @ A_27a,V9a1: tyop_2Ecanonical_2Espolynom @ A_27a] :
( ( c_2Ecanonical_2Espolynom__size @ A_27a @ V7f @ ( c_2Ecanonical_2ESPmult @ A_27a @ V8a0 @ V9a1 ) )
= ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ ( c_2Earithmetic_2E_2B @ ( c_2Ecanonical_2Espolynom__size @ A_27a @ V7f @ V8a0 ) @ ( c_2Ecanonical_2Espolynom__size @ A_27a @ V7f @ V9a1 ) ) ) ) ) ).
thf(thm_2Ecanonical_2Espolynom__normalize__def,axiom,
! [A_27a: $tType] :
( ! [V0sr: tyop_2Esemi__ring_2Esemi__ring @ A_27a,V1i: tyop_2Equote_2Eindex] :
( ( c_2Ecanonical_2Espolynom__normalize @ A_27a @ V0sr @ ( c_2Ecanonical_2ESPvar @ A_27a @ V1i ) )
= ( c_2Ecanonical_2ECons__varlist @ A_27a @ ( c_2Elist_2ECONS @ tyop_2Equote_2Eindex @ V1i @ ( c_2Elist_2ENIL @ tyop_2Equote_2Eindex ) ) @ ( c_2Ecanonical_2ENil__monom @ A_27a ) ) )
& ! [V2sr: tyop_2Esemi__ring_2Esemi__ring @ A_27a,V3c: A_27a] :
( ( c_2Ecanonical_2Espolynom__normalize @ A_27a @ V2sr @ ( c_2Ecanonical_2ESPconst @ A_27a @ V3c ) )
= ( c_2Ecanonical_2ECons__monom @ A_27a @ V3c @ ( c_2Elist_2ENIL @ tyop_2Equote_2Eindex ) @ ( c_2Ecanonical_2ENil__monom @ A_27a ) ) )
& ! [V4sr: tyop_2Esemi__ring_2Esemi__ring @ A_27a,V5l: tyop_2Ecanonical_2Espolynom @ A_27a,V6r: tyop_2Ecanonical_2Espolynom @ A_27a] :
( ( c_2Ecanonical_2Espolynom__normalize @ A_27a @ V4sr @ ( c_2Ecanonical_2ESPplus @ A_27a @ V5l @ V6r ) )
= ( c_2Ecanonical_2Ecanonical__sum__merge @ A_27a @ V4sr @ ( c_2Ecanonical_2Espolynom__normalize @ A_27a @ V4sr @ V5l ) @ ( c_2Ecanonical_2Espolynom__normalize @ A_27a @ V4sr @ V6r ) ) )
& ! [V7sr: tyop_2Esemi__ring_2Esemi__ring @ A_27a,V8l: tyop_2Ecanonical_2Espolynom @ A_27a,V9r: tyop_2Ecanonical_2Espolynom @ A_27a] :
( ( c_2Ecanonical_2Espolynom__normalize @ A_27a @ V7sr @ ( c_2Ecanonical_2ESPmult @ A_27a @ V8l @ V9r ) )
= ( c_2Ecanonical_2Ecanonical__sum__prod @ A_27a @ V7sr @ ( c_2Ecanonical_2Espolynom__normalize @ A_27a @ V7sr @ V8l ) @ ( c_2Ecanonical_2Espolynom__normalize @ A_27a @ V7sr @ V9r ) ) ) ) ).
thf(thm_2Ecanonical_2Espolynom__simplify__def,axiom,
! [A_27a: $tType,V0sr: tyop_2Esemi__ring_2Esemi__ring @ A_27a,V1x: tyop_2Ecanonical_2Espolynom @ A_27a] :
( ( c_2Ecanonical_2Espolynom__simplify @ A_27a @ V0sr @ V1x )
= ( c_2Ecanonical_2Ecanonical__sum__simplify @ A_27a @ V0sr @ ( c_2Ecanonical_2Espolynom__normalize @ A_27a @ V0sr @ V1x ) ) ) ).
thf(thm_2Ecanonical_2Einterp__sp__def,axiom,
! [A_27a: $tType] :
( ! [V0sr: tyop_2Esemi__ring_2Esemi__ring @ A_27a,V1vm: tyop_2Equote_2Evarmap @ A_27a,V2c: A_27a] :
( ( c_2Ecanonical_2Einterp__sp @ A_27a @ V0sr @ V1vm @ ( c_2Ecanonical_2ESPconst @ A_27a @ V2c ) )
= V2c )
& ! [V3sr: tyop_2Esemi__ring_2Esemi__ring @ A_27a,V4vm: tyop_2Equote_2Evarmap @ A_27a,V5i: tyop_2Equote_2Eindex] :
( ( c_2Ecanonical_2Einterp__sp @ A_27a @ V3sr @ V4vm @ ( c_2Ecanonical_2ESPvar @ A_27a @ V5i ) )
= ( c_2Equote_2Evarmap__find @ A_27a @ V5i @ V4vm ) )
& ! [V6sr: tyop_2Esemi__ring_2Esemi__ring @ A_27a,V7vm: tyop_2Equote_2Evarmap @ A_27a,V8p1: tyop_2Ecanonical_2Espolynom @ A_27a,V9p2: tyop_2Ecanonical_2Espolynom @ A_27a] :
( ( c_2Ecanonical_2Einterp__sp @ A_27a @ V6sr @ V7vm @ ( c_2Ecanonical_2ESPplus @ A_27a @ V8p1 @ V9p2 ) )
= ( c_2Esemi__ring_2Esemi__ring__SRP @ A_27a @ V6sr @ ( c_2Ecanonical_2Einterp__sp @ A_27a @ V6sr @ V7vm @ V8p1 ) @ ( c_2Ecanonical_2Einterp__sp @ A_27a @ V6sr @ V7vm @ V9p2 ) ) )
& ! [V10sr: tyop_2Esemi__ring_2Esemi__ring @ A_27a,V11vm: tyop_2Equote_2Evarmap @ A_27a,V12p1: tyop_2Ecanonical_2Espolynom @ A_27a,V13p2: tyop_2Ecanonical_2Espolynom @ A_27a] :
( ( c_2Ecanonical_2Einterp__sp @ A_27a @ V10sr @ V11vm @ ( c_2Ecanonical_2ESPmult @ A_27a @ V12p1 @ V13p2 ) )
= ( c_2Esemi__ring_2Esemi__ring__SRM @ A_27a @ V10sr @ ( c_2Ecanonical_2Einterp__sp @ A_27a @ V10sr @ V11vm @ V12p1 ) @ ( c_2Ecanonical_2Einterp__sp @ A_27a @ V10sr @ V11vm @ V13p2 ) ) ) ) ).
thf(thm_2Ecanonical_2Edatatype__canonical__sum,axiom,
! [A_27a: $tType,V0canonical__sum: ( tyop_2Ecanonical_2Ecanonical__sum @ A_27a ) > ( A_27a > ( tyop_2Elist_2Elist @ tyop_2Equote_2Eindex ) > ( tyop_2Ecanonical_2Ecanonical__sum @ A_27a ) > ( tyop_2Ecanonical_2Ecanonical__sum @ A_27a ) ) > ( ( tyop_2Elist_2Elist @ tyop_2Equote_2Eindex ) > ( tyop_2Ecanonical_2Ecanonical__sum @ A_27a ) > ( tyop_2Ecanonical_2Ecanonical__sum @ A_27a ) ) > $o] : ( c_2Ebool_2EDATATYPE @ $o @ ( V0canonical__sum @ ( c_2Ecanonical_2ENil__monom @ A_27a ) @ ( c_2Ecanonical_2ECons__monom @ A_27a ) @ ( c_2Ecanonical_2ECons__varlist @ A_27a ) ) ) ).
thf(thm_2Ecanonical_2Ecanonical__sum__11,axiom,
! [A_27a: $tType] :
( ! [V0a0: A_27a,V1a1: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V2a2: tyop_2Ecanonical_2Ecanonical__sum @ A_27a,V3a0_27: A_27a,V4a1_27: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V5a2_27: tyop_2Ecanonical_2Ecanonical__sum @ A_27a] :
( ( ( c_2Ecanonical_2ECons__monom @ A_27a @ V0a0 @ V1a1 @ V2a2 )
= ( c_2Ecanonical_2ECons__monom @ A_27a @ V3a0_27 @ V4a1_27 @ V5a2_27 ) )
<=> ( ( V0a0 = V3a0_27 )
& ( V1a1 = V4a1_27 )
& ( V2a2 = V5a2_27 ) ) )
& ! [V6a0: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V7a1: tyop_2Ecanonical_2Ecanonical__sum @ A_27a,V8a0_27: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V9a1_27: tyop_2Ecanonical_2Ecanonical__sum @ A_27a] :
( ( ( c_2Ecanonical_2ECons__varlist @ A_27a @ V6a0 @ V7a1 )
= ( c_2Ecanonical_2ECons__varlist @ A_27a @ V8a0_27 @ V9a1_27 ) )
<=> ( ( V6a0 = V8a0_27 )
& ( V7a1 = V9a1_27 ) ) ) ) ).
thf(thm_2Ecanonical_2Ecanonical__sum__distinct,axiom,
! [A_27a: $tType] :
( ! [V0a2: tyop_2Ecanonical_2Ecanonical__sum @ A_27a,V1a1: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V2a0: A_27a] :
( (~)
@ ( ( c_2Ecanonical_2ENil__monom @ A_27a )
= ( c_2Ecanonical_2ECons__monom @ A_27a @ V2a0 @ V1a1 @ V0a2 ) ) )
& ! [V3a1: tyop_2Ecanonical_2Ecanonical__sum @ A_27a,V4a0: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex] :
( (~)
@ ( ( c_2Ecanonical_2ENil__monom @ A_27a )
= ( c_2Ecanonical_2ECons__varlist @ A_27a @ V4a0 @ V3a1 ) ) )
& ! [V5a2: tyop_2Ecanonical_2Ecanonical__sum @ A_27a,V6a1_27: tyop_2Ecanonical_2Ecanonical__sum @ A_27a,V7a1: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V8a0_27: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V9a0: A_27a] :
( (~)
@ ( ( c_2Ecanonical_2ECons__monom @ A_27a @ V9a0 @ V7a1 @ V5a2 )
= ( c_2Ecanonical_2ECons__varlist @ A_27a @ V8a0_27 @ V6a1_27 ) ) ) ) ).
thf(thm_2Ecanonical_2Ecanonical__sum__nchotomy,axiom,
! [A_27a: $tType,V0cc: tyop_2Ecanonical_2Ecanonical__sum @ A_27a] :
( ( V0cc
= ( c_2Ecanonical_2ENil__monom @ A_27a ) )
| ? [V1a: A_27a,V2l: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V3c: tyop_2Ecanonical_2Ecanonical__sum @ A_27a] :
( V0cc
= ( c_2Ecanonical_2ECons__monom @ A_27a @ V1a @ V2l @ V3c ) )
| ? [V4l: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V5c: tyop_2Ecanonical_2Ecanonical__sum @ A_27a] :
( V0cc
= ( c_2Ecanonical_2ECons__varlist @ A_27a @ V4l @ V5c ) ) ) ).
thf(thm_2Ecanonical_2Ecanonical__sum__Axiom,axiom,
! [A_27a: $tType,A_27b: $tType,V0f0: A_27b,V1f1: A_27a > ( tyop_2Elist_2Elist @ tyop_2Equote_2Eindex ) > ( tyop_2Ecanonical_2Ecanonical__sum @ A_27a ) > A_27b > A_27b,V2f2: ( tyop_2Elist_2Elist @ tyop_2Equote_2Eindex ) > ( tyop_2Ecanonical_2Ecanonical__sum @ A_27a ) > A_27b > A_27b] :
? [V3fn: ( tyop_2Ecanonical_2Ecanonical__sum @ A_27a ) > A_27b] :
( ( ( V3fn @ ( c_2Ecanonical_2ENil__monom @ A_27a ) )
= V0f0 )
& ! [V4a0: A_27a,V5a1: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V6a2: tyop_2Ecanonical_2Ecanonical__sum @ A_27a] :
( ( V3fn @ ( c_2Ecanonical_2ECons__monom @ A_27a @ V4a0 @ V5a1 @ V6a2 ) )
= ( V1f1 @ V4a0 @ V5a1 @ V6a2 @ ( V3fn @ V6a2 ) ) )
& ! [V7a0: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V8a1: tyop_2Ecanonical_2Ecanonical__sum @ A_27a] :
( ( V3fn @ ( c_2Ecanonical_2ECons__varlist @ A_27a @ V7a0 @ V8a1 ) )
= ( V2f2 @ V7a0 @ V8a1 @ ( V3fn @ V8a1 ) ) ) ) ).
thf(thm_2Ecanonical_2Ecanonical__sum__induction,axiom,
! [A_27a: $tType,V0P: ( tyop_2Ecanonical_2Ecanonical__sum @ A_27a ) > $o] :
( ( ( V0P @ ( c_2Ecanonical_2ENil__monom @ A_27a ) )
& ! [V1c: tyop_2Ecanonical_2Ecanonical__sum @ A_27a] :
( ( V0P @ V1c )
=> ! [V2l: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V3a: A_27a] : ( V0P @ ( c_2Ecanonical_2ECons__monom @ A_27a @ V3a @ V2l @ V1c ) ) )
& ! [V4c: tyop_2Ecanonical_2Ecanonical__sum @ A_27a] :
( ( V0P @ V4c )
=> ! [V5l: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex] : ( V0P @ ( c_2Ecanonical_2ECons__varlist @ A_27a @ V5l @ V4c ) ) ) )
=> ! [V6c: tyop_2Ecanonical_2Ecanonical__sum @ A_27a] : ( V0P @ V6c ) ) ).
thf(thm_2Ecanonical_2Ecanonical__sum__case__cong,axiom,
! [A_27a: $tType,A_27b: $tType,V0v_27: A_27b,V1f1_27: ( tyop_2Elist_2Elist @ tyop_2Equote_2Eindex ) > ( tyop_2Ecanonical_2Ecanonical__sum @ A_27a ) > A_27b,V2f_27: A_27a > ( tyop_2Elist_2Elist @ tyop_2Equote_2Eindex ) > ( tyop_2Ecanonical_2Ecanonical__sum @ A_27a ) > A_27b,V3M: tyop_2Ecanonical_2Ecanonical__sum @ A_27a,V4M_27: tyop_2Ecanonical_2Ecanonical__sum @ A_27a,V5v: A_27b,V6f: A_27a > ( tyop_2Elist_2Elist @ tyop_2Equote_2Eindex ) > ( tyop_2Ecanonical_2Ecanonical__sum @ A_27a ) > A_27b,V7f1: ( tyop_2Elist_2Elist @ tyop_2Equote_2Eindex ) > ( tyop_2Ecanonical_2Ecanonical__sum @ A_27a ) > A_27b] :
( ( ( V3M = V4M_27 )
& ( ( V4M_27
= ( c_2Ecanonical_2ENil__monom @ A_27a ) )
=> ( V5v = V0v_27 ) )
& ! [V8a0: A_27a,V9a1: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V10a2: tyop_2Ecanonical_2Ecanonical__sum @ A_27a] :
( ( V4M_27
= ( c_2Ecanonical_2ECons__monom @ A_27a @ V8a0 @ V9a1 @ V10a2 ) )
=> ( ( V6f @ V8a0 @ V9a1 @ V10a2 )
= ( V2f_27 @ V8a0 @ V9a1 @ V10a2 ) ) )
& ! [V11a0: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V12a1: tyop_2Ecanonical_2Ecanonical__sum @ A_27a] :
( ( V4M_27
= ( c_2Ecanonical_2ECons__varlist @ A_27a @ V11a0 @ V12a1 ) )
=> ( ( V7f1 @ V11a0 @ V12a1 )
= ( V1f1_27 @ V11a0 @ V12a1 ) ) ) )
=> ( ( c_2Ecanonical_2Ecanonical__sum__CASE @ A_27a @ A_27b @ V3M @ V5v @ V6f @ V7f1 )
= ( c_2Ecanonical_2Ecanonical__sum__CASE @ A_27a @ A_27b @ V4M_27 @ V0v_27 @ V2f_27 @ V1f1_27 ) ) ) ).
thf(thm_2Ecanonical_2Ecanonical__sum__case__eq,axiom,
! [A_27a: $tType,A_27b: $tType,V0x: tyop_2Ecanonical_2Ecanonical__sum @ A_27a,V1v_27: A_27b,V2v: A_27b,V3f1: ( tyop_2Elist_2Elist @ tyop_2Equote_2Eindex ) > ( tyop_2Ecanonical_2Ecanonical__sum @ A_27a ) > A_27b,V4f: A_27a > ( tyop_2Elist_2Elist @ tyop_2Equote_2Eindex ) > ( tyop_2Ecanonical_2Ecanonical__sum @ A_27a ) > A_27b] :
( ( ( c_2Ecanonical_2Ecanonical__sum__CASE @ A_27a @ A_27b @ V0x @ V2v @ V4f @ V3f1 )
= V1v_27 )
<=> ( ( ( V0x
= ( c_2Ecanonical_2ENil__monom @ A_27a ) )
& ( V2v = V1v_27 ) )
| ? [V5a: A_27a,V6l: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V7c: tyop_2Ecanonical_2Ecanonical__sum @ A_27a] :
( ( V0x
= ( c_2Ecanonical_2ECons__monom @ A_27a @ V5a @ V6l @ V7c ) )
& ( ( V4f @ V5a @ V6l @ V7c )
= V1v_27 ) )
| ? [V8l: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V9c: tyop_2Ecanonical_2Ecanonical__sum @ A_27a] :
( ( V0x
= ( c_2Ecanonical_2ECons__varlist @ A_27a @ V8l @ V9c ) )
& ( ( V3f1 @ V8l @ V9c )
= V1v_27 ) ) ) ) ).
thf(thm_2Ecanonical_2Ecanonical__sum__merge__ind,axiom,
! [A_27a: $tType,V0P: ( tyop_2Esemi__ring_2Esemi__ring @ A_27a ) > ( tyop_2Ecanonical_2Ecanonical__sum @ A_27a ) > ( tyop_2Ecanonical_2Ecanonical__sum @ A_27a ) > $o] :
( ( ! [V1sr: tyop_2Esemi__ring_2Esemi__ring @ A_27a,V2c1: A_27a,V3l1: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V4t1: tyop_2Ecanonical_2Ecanonical__sum @ A_27a,V5c2: A_27a,V6l2: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V7t2: tyop_2Ecanonical_2Ecanonical__sum @ A_27a] :
( ( ( ( ( c_2EternaryComparisons_2Elist__compare @ tyop_2Equote_2Eindex @ tyop_2Equote_2Eindex @ c_2Equote_2Eindex__compare @ V3l1 @ V6l2 )
= c_2EternaryComparisons_2EEQUAL )
=> ( V0P @ V1sr @ V4t1 @ V7t2 ) )
& ( ( ( c_2EternaryComparisons_2Elist__compare @ tyop_2Equote_2Eindex @ tyop_2Equote_2Eindex @ c_2Equote_2Eindex__compare @ V3l1 @ V6l2 )
= c_2EternaryComparisons_2EGREATER )
=> ( V0P @ V1sr @ ( c_2Ecanonical_2ECons__monom @ A_27a @ V2c1 @ V3l1 @ V4t1 ) @ V7t2 ) )
& ( ( ( c_2EternaryComparisons_2Elist__compare @ tyop_2Equote_2Eindex @ tyop_2Equote_2Eindex @ c_2Equote_2Eindex__compare @ V3l1 @ V6l2 )
= c_2EternaryComparisons_2ELESS )
=> ( V0P @ V1sr @ V4t1 @ ( c_2Ecanonical_2ECons__monom @ A_27a @ V5c2 @ V6l2 @ V7t2 ) ) ) )
=> ( V0P @ V1sr @ ( c_2Ecanonical_2ECons__monom @ A_27a @ V2c1 @ V3l1 @ V4t1 ) @ ( c_2Ecanonical_2ECons__monom @ A_27a @ V5c2 @ V6l2 @ V7t2 ) ) )
& ! [V8sr: tyop_2Esemi__ring_2Esemi__ring @ A_27a,V9c1: A_27a,V10l1: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V11t1: tyop_2Ecanonical_2Ecanonical__sum @ A_27a,V12l2: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V13t2: tyop_2Ecanonical_2Ecanonical__sum @ A_27a] :
( ( ( ( ( c_2EternaryComparisons_2Elist__compare @ tyop_2Equote_2Eindex @ tyop_2Equote_2Eindex @ c_2Equote_2Eindex__compare @ V10l1 @ V12l2 )
= c_2EternaryComparisons_2EEQUAL )
=> ( V0P @ V8sr @ V11t1 @ V13t2 ) )
& ( ( ( c_2EternaryComparisons_2Elist__compare @ tyop_2Equote_2Eindex @ tyop_2Equote_2Eindex @ c_2Equote_2Eindex__compare @ V10l1 @ V12l2 )
= c_2EternaryComparisons_2EGREATER )
=> ( V0P @ V8sr @ ( c_2Ecanonical_2ECons__monom @ A_27a @ V9c1 @ V10l1 @ V11t1 ) @ V13t2 ) )
& ( ( ( c_2EternaryComparisons_2Elist__compare @ tyop_2Equote_2Eindex @ tyop_2Equote_2Eindex @ c_2Equote_2Eindex__compare @ V10l1 @ V12l2 )
= c_2EternaryComparisons_2ELESS )
=> ( V0P @ V8sr @ V11t1 @ ( c_2Ecanonical_2ECons__varlist @ A_27a @ V12l2 @ V13t2 ) ) ) )
=> ( V0P @ V8sr @ ( c_2Ecanonical_2ECons__monom @ A_27a @ V9c1 @ V10l1 @ V11t1 ) @ ( c_2Ecanonical_2ECons__varlist @ A_27a @ V12l2 @ V13t2 ) ) )
& ! [V14sr: tyop_2Esemi__ring_2Esemi__ring @ A_27a,V15l1: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V16t1: tyop_2Ecanonical_2Ecanonical__sum @ A_27a,V17c2: A_27a,V18l2: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V19t2: tyop_2Ecanonical_2Ecanonical__sum @ A_27a] :
( ( ( ( ( c_2EternaryComparisons_2Elist__compare @ tyop_2Equote_2Eindex @ tyop_2Equote_2Eindex @ c_2Equote_2Eindex__compare @ V15l1 @ V18l2 )
= c_2EternaryComparisons_2EEQUAL )
=> ( V0P @ V14sr @ V16t1 @ V19t2 ) )
& ( ( ( c_2EternaryComparisons_2Elist__compare @ tyop_2Equote_2Eindex @ tyop_2Equote_2Eindex @ c_2Equote_2Eindex__compare @ V15l1 @ V18l2 )
= c_2EternaryComparisons_2EGREATER )
=> ( V0P @ V14sr @ ( c_2Ecanonical_2ECons__varlist @ A_27a @ V15l1 @ V16t1 ) @ V19t2 ) )
& ( ( ( c_2EternaryComparisons_2Elist__compare @ tyop_2Equote_2Eindex @ tyop_2Equote_2Eindex @ c_2Equote_2Eindex__compare @ V15l1 @ V18l2 )
= c_2EternaryComparisons_2ELESS )
=> ( V0P @ V14sr @ V16t1 @ ( c_2Ecanonical_2ECons__monom @ A_27a @ V17c2 @ V18l2 @ V19t2 ) ) ) )
=> ( V0P @ V14sr @ ( c_2Ecanonical_2ECons__varlist @ A_27a @ V15l1 @ V16t1 ) @ ( c_2Ecanonical_2ECons__monom @ A_27a @ V17c2 @ V18l2 @ V19t2 ) ) )
& ! [V20sr: tyop_2Esemi__ring_2Esemi__ring @ A_27a,V21l1: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V22t1: tyop_2Ecanonical_2Ecanonical__sum @ A_27a,V23l2: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V24t2: tyop_2Ecanonical_2Ecanonical__sum @ A_27a] :
( ( ( ( ( c_2EternaryComparisons_2Elist__compare @ tyop_2Equote_2Eindex @ tyop_2Equote_2Eindex @ c_2Equote_2Eindex__compare @ V21l1 @ V23l2 )
= c_2EternaryComparisons_2EEQUAL )
=> ( V0P @ V20sr @ V22t1 @ V24t2 ) )
& ( ( ( c_2EternaryComparisons_2Elist__compare @ tyop_2Equote_2Eindex @ tyop_2Equote_2Eindex @ c_2Equote_2Eindex__compare @ V21l1 @ V23l2 )
= c_2EternaryComparisons_2EGREATER )
=> ( V0P @ V20sr @ ( c_2Ecanonical_2ECons__varlist @ A_27a @ V21l1 @ V22t1 ) @ V24t2 ) )
& ( ( ( c_2EternaryComparisons_2Elist__compare @ tyop_2Equote_2Eindex @ tyop_2Equote_2Eindex @ c_2Equote_2Eindex__compare @ V21l1 @ V23l2 )
= c_2EternaryComparisons_2ELESS )
=> ( V0P @ V20sr @ V22t1 @ ( c_2Ecanonical_2ECons__varlist @ A_27a @ V23l2 @ V24t2 ) ) ) )
=> ( V0P @ V20sr @ ( c_2Ecanonical_2ECons__varlist @ A_27a @ V21l1 @ V22t1 ) @ ( c_2Ecanonical_2ECons__varlist @ A_27a @ V23l2 @ V24t2 ) ) )
& ! [V25sr: tyop_2Esemi__ring_2Esemi__ring @ A_27a,V26s1: tyop_2Ecanonical_2Ecanonical__sum @ A_27a] : ( V0P @ V25sr @ V26s1 @ ( c_2Ecanonical_2ENil__monom @ A_27a ) )
& ! [V27sr: tyop_2Esemi__ring_2Esemi__ring @ A_27a,V28v4: A_27a,V29v5: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V30v6: tyop_2Ecanonical_2Ecanonical__sum @ A_27a] : ( V0P @ V27sr @ ( c_2Ecanonical_2ENil__monom @ A_27a ) @ ( c_2Ecanonical_2ECons__monom @ A_27a @ V28v4 @ V29v5 @ V30v6 ) )
& ! [V31sr: tyop_2Esemi__ring_2Esemi__ring @ A_27a,V32v7: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V33v8: tyop_2Ecanonical_2Ecanonical__sum @ A_27a] : ( V0P @ V31sr @ ( c_2Ecanonical_2ENil__monom @ A_27a ) @ ( c_2Ecanonical_2ECons__varlist @ A_27a @ V32v7 @ V33v8 ) ) )
=> ! [V34v: tyop_2Esemi__ring_2Esemi__ring @ A_27a,V35v1: tyop_2Ecanonical_2Ecanonical__sum @ A_27a,V36v2: tyop_2Ecanonical_2Ecanonical__sum @ A_27a] : ( V0P @ V34v @ V35v1 @ V36v2 ) ) ).
thf(thm_2Ecanonical_2Ecanonical__sum__merge__def,axiom,
! [A_27a: $tType] :
( ! [V0t2: tyop_2Ecanonical_2Ecanonical__sum @ A_27a,V1t1: tyop_2Ecanonical_2Ecanonical__sum @ A_27a,V2sr: tyop_2Esemi__ring_2Esemi__ring @ A_27a,V3l2: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V4l1: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V5c2: A_27a,V6c1: A_27a] :
( ( c_2Ecanonical_2Ecanonical__sum__merge @ A_27a @ V2sr @ ( c_2Ecanonical_2ECons__monom @ A_27a @ V6c1 @ V4l1 @ V1t1 ) @ ( c_2Ecanonical_2ECons__monom @ A_27a @ V5c2 @ V3l2 @ V0t2 ) )
= ( c_2EternaryComparisons_2Eordering__CASE @ ( tyop_2Ecanonical_2Ecanonical__sum @ A_27a ) @ ( c_2EternaryComparisons_2Elist__compare @ tyop_2Equote_2Eindex @ tyop_2Equote_2Eindex @ c_2Equote_2Eindex__compare @ V4l1 @ V3l2 ) @ ( c_2Ecanonical_2ECons__monom @ A_27a @ V6c1 @ V4l1 @ ( c_2Ecanonical_2Ecanonical__sum__merge @ A_27a @ V2sr @ V1t1 @ ( c_2Ecanonical_2ECons__monom @ A_27a @ V5c2 @ V3l2 @ V0t2 ) ) ) @ ( c_2Ecanonical_2ECons__monom @ A_27a @ ( c_2Esemi__ring_2Esemi__ring__SRP @ A_27a @ V2sr @ V6c1 @ V5c2 ) @ V4l1 @ ( c_2Ecanonical_2Ecanonical__sum__merge @ A_27a @ V2sr @ V1t1 @ V0t2 ) ) @ ( c_2Ecanonical_2ECons__monom @ A_27a @ V5c2 @ V3l2 @ ( c_2Ecanonical_2Ecanonical__sum__merge @ A_27a @ V2sr @ ( c_2Ecanonical_2ECons__monom @ A_27a @ V6c1 @ V4l1 @ V1t1 ) @ V0t2 ) ) ) )
& ! [V7t2: tyop_2Ecanonical_2Ecanonical__sum @ A_27a,V8t1: tyop_2Ecanonical_2Ecanonical__sum @ A_27a,V9sr: tyop_2Esemi__ring_2Esemi__ring @ A_27a,V10l2: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V11l1: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V12c1: A_27a] :
( ( c_2Ecanonical_2Ecanonical__sum__merge @ A_27a @ V9sr @ ( c_2Ecanonical_2ECons__monom @ A_27a @ V12c1 @ V11l1 @ V8t1 ) @ ( c_2Ecanonical_2ECons__varlist @ A_27a @ V10l2 @ V7t2 ) )
= ( c_2EternaryComparisons_2Eordering__CASE @ ( tyop_2Ecanonical_2Ecanonical__sum @ A_27a ) @ ( c_2EternaryComparisons_2Elist__compare @ tyop_2Equote_2Eindex @ tyop_2Equote_2Eindex @ c_2Equote_2Eindex__compare @ V11l1 @ V10l2 ) @ ( c_2Ecanonical_2ECons__monom @ A_27a @ V12c1 @ V11l1 @ ( c_2Ecanonical_2Ecanonical__sum__merge @ A_27a @ V9sr @ V8t1 @ ( c_2Ecanonical_2ECons__varlist @ A_27a @ V10l2 @ V7t2 ) ) ) @ ( c_2Ecanonical_2ECons__monom @ A_27a @ ( c_2Esemi__ring_2Esemi__ring__SRP @ A_27a @ V9sr @ V12c1 @ ( c_2Esemi__ring_2Esemi__ring__SR1 @ A_27a @ V9sr ) ) @ V11l1 @ ( c_2Ecanonical_2Ecanonical__sum__merge @ A_27a @ V9sr @ V8t1 @ V7t2 ) ) @ ( c_2Ecanonical_2ECons__varlist @ A_27a @ V10l2 @ ( c_2Ecanonical_2Ecanonical__sum__merge @ A_27a @ V9sr @ ( c_2Ecanonical_2ECons__monom @ A_27a @ V12c1 @ V11l1 @ V8t1 ) @ V7t2 ) ) ) )
& ! [V13t2: tyop_2Ecanonical_2Ecanonical__sum @ A_27a,V14t1: tyop_2Ecanonical_2Ecanonical__sum @ A_27a,V15sr: tyop_2Esemi__ring_2Esemi__ring @ A_27a,V16l2: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V17l1: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V18c2: A_27a] :
( ( c_2Ecanonical_2Ecanonical__sum__merge @ A_27a @ V15sr @ ( c_2Ecanonical_2ECons__varlist @ A_27a @ V17l1 @ V14t1 ) @ ( c_2Ecanonical_2ECons__monom @ A_27a @ V18c2 @ V16l2 @ V13t2 ) )
= ( c_2EternaryComparisons_2Eordering__CASE @ ( tyop_2Ecanonical_2Ecanonical__sum @ A_27a ) @ ( c_2EternaryComparisons_2Elist__compare @ tyop_2Equote_2Eindex @ tyop_2Equote_2Eindex @ c_2Equote_2Eindex__compare @ V17l1 @ V16l2 ) @ ( c_2Ecanonical_2ECons__varlist @ A_27a @ V17l1 @ ( c_2Ecanonical_2Ecanonical__sum__merge @ A_27a @ V15sr @ V14t1 @ ( c_2Ecanonical_2ECons__monom @ A_27a @ V18c2 @ V16l2 @ V13t2 ) ) ) @ ( c_2Ecanonical_2ECons__monom @ A_27a @ ( c_2Esemi__ring_2Esemi__ring__SRP @ A_27a @ V15sr @ ( c_2Esemi__ring_2Esemi__ring__SR1 @ A_27a @ V15sr ) @ V18c2 ) @ V17l1 @ ( c_2Ecanonical_2Ecanonical__sum__merge @ A_27a @ V15sr @ V14t1 @ V13t2 ) ) @ ( c_2Ecanonical_2ECons__monom @ A_27a @ V18c2 @ V16l2 @ ( c_2Ecanonical_2Ecanonical__sum__merge @ A_27a @ V15sr @ ( c_2Ecanonical_2ECons__varlist @ A_27a @ V17l1 @ V14t1 ) @ V13t2 ) ) ) )
& ! [V19t2: tyop_2Ecanonical_2Ecanonical__sum @ A_27a,V20t1: tyop_2Ecanonical_2Ecanonical__sum @ A_27a,V21sr: tyop_2Esemi__ring_2Esemi__ring @ A_27a,V22l2: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V23l1: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex] :
( ( c_2Ecanonical_2Ecanonical__sum__merge @ A_27a @ V21sr @ ( c_2Ecanonical_2ECons__varlist @ A_27a @ V23l1 @ V20t1 ) @ ( c_2Ecanonical_2ECons__varlist @ A_27a @ V22l2 @ V19t2 ) )
= ( c_2EternaryComparisons_2Eordering__CASE @ ( tyop_2Ecanonical_2Ecanonical__sum @ A_27a ) @ ( c_2EternaryComparisons_2Elist__compare @ tyop_2Equote_2Eindex @ tyop_2Equote_2Eindex @ c_2Equote_2Eindex__compare @ V23l1 @ V22l2 ) @ ( c_2Ecanonical_2ECons__varlist @ A_27a @ V23l1 @ ( c_2Ecanonical_2Ecanonical__sum__merge @ A_27a @ V21sr @ V20t1 @ ( c_2Ecanonical_2ECons__varlist @ A_27a @ V22l2 @ V19t2 ) ) ) @ ( c_2Ecanonical_2ECons__monom @ A_27a @ ( c_2Esemi__ring_2Esemi__ring__SRP @ A_27a @ V21sr @ ( c_2Esemi__ring_2Esemi__ring__SR1 @ A_27a @ V21sr ) @ ( c_2Esemi__ring_2Esemi__ring__SR1 @ A_27a @ V21sr ) ) @ V23l1 @ ( c_2Ecanonical_2Ecanonical__sum__merge @ A_27a @ V21sr @ V20t1 @ V19t2 ) ) @ ( c_2Ecanonical_2ECons__varlist @ A_27a @ V22l2 @ ( c_2Ecanonical_2Ecanonical__sum__merge @ A_27a @ V21sr @ ( c_2Ecanonical_2ECons__varlist @ A_27a @ V23l1 @ V20t1 ) @ V19t2 ) ) ) )
& ! [V24sr: tyop_2Esemi__ring_2Esemi__ring @ A_27a,V25s1: tyop_2Ecanonical_2Ecanonical__sum @ A_27a] :
( ( c_2Ecanonical_2Ecanonical__sum__merge @ A_27a @ V24sr @ V25s1 @ ( c_2Ecanonical_2ENil__monom @ A_27a ) )
= V25s1 )
& ! [V26v6: tyop_2Ecanonical_2Ecanonical__sum @ A_27a,V27v5: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V28v4: A_27a,V29sr: tyop_2Esemi__ring_2Esemi__ring @ A_27a] :
( ( c_2Ecanonical_2Ecanonical__sum__merge @ A_27a @ V29sr @ ( c_2Ecanonical_2ENil__monom @ A_27a ) @ ( c_2Ecanonical_2ECons__monom @ A_27a @ V28v4 @ V27v5 @ V26v6 ) )
= ( c_2Ecanonical_2ECons__monom @ A_27a @ V28v4 @ V27v5 @ V26v6 ) )
& ! [V30v8: tyop_2Ecanonical_2Ecanonical__sum @ A_27a,V31v7: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V32sr: tyop_2Esemi__ring_2Esemi__ring @ A_27a] :
( ( c_2Ecanonical_2Ecanonical__sum__merge @ A_27a @ V32sr @ ( c_2Ecanonical_2ENil__monom @ A_27a ) @ ( c_2Ecanonical_2ECons__varlist @ A_27a @ V31v7 @ V30v8 ) )
= ( c_2Ecanonical_2ECons__varlist @ A_27a @ V31v7 @ V30v8 ) ) ) ).
thf(thm_2Ecanonical_2Emonom__insert__ind,axiom,
! [A_27a: $tType,V0P: ( tyop_2Esemi__ring_2Esemi__ring @ A_27a ) > A_27a > ( tyop_2Elist_2Elist @ tyop_2Equote_2Eindex ) > ( tyop_2Ecanonical_2Ecanonical__sum @ A_27a ) > $o] :
( ( ! [V1sr: tyop_2Esemi__ring_2Esemi__ring @ A_27a,V2c1: A_27a,V3l1: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V4c2: A_27a,V5l2: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V6t2: tyop_2Ecanonical_2Ecanonical__sum @ A_27a] :
( ( ( ( c_2EternaryComparisons_2Elist__compare @ tyop_2Equote_2Eindex @ tyop_2Equote_2Eindex @ c_2Equote_2Eindex__compare @ V3l1 @ V5l2 )
= c_2EternaryComparisons_2EGREATER )
=> ( V0P @ V1sr @ V2c1 @ V3l1 @ V6t2 ) )
=> ( V0P @ V1sr @ V2c1 @ V3l1 @ ( c_2Ecanonical_2ECons__monom @ A_27a @ V4c2 @ V5l2 @ V6t2 ) ) )
& ! [V7sr: tyop_2Esemi__ring_2Esemi__ring @ A_27a,V8c1: A_27a,V9l1: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V10l2: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V11t2: tyop_2Ecanonical_2Ecanonical__sum @ A_27a] :
( ( ( ( c_2EternaryComparisons_2Elist__compare @ tyop_2Equote_2Eindex @ tyop_2Equote_2Eindex @ c_2Equote_2Eindex__compare @ V9l1 @ V10l2 )
= c_2EternaryComparisons_2EGREATER )
=> ( V0P @ V7sr @ V8c1 @ V9l1 @ V11t2 ) )
=> ( V0P @ V7sr @ V8c1 @ V9l1 @ ( c_2Ecanonical_2ECons__varlist @ A_27a @ V10l2 @ V11t2 ) ) )
& ! [V12sr: tyop_2Esemi__ring_2Esemi__ring @ A_27a,V13c1: A_27a,V14l1: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex] : ( V0P @ V12sr @ V13c1 @ V14l1 @ ( c_2Ecanonical_2ENil__monom @ A_27a ) ) )
=> ! [V15v: tyop_2Esemi__ring_2Esemi__ring @ A_27a,V16v1: A_27a,V17v2: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V18v3: tyop_2Ecanonical_2Ecanonical__sum @ A_27a] : ( V0P @ V15v @ V16v1 @ V17v2 @ V18v3 ) ) ).
thf(thm_2Ecanonical_2Emonom__insert__def,axiom,
! [A_27a: $tType] :
( ! [V0t2: tyop_2Ecanonical_2Ecanonical__sum @ A_27a,V1sr: tyop_2Esemi__ring_2Esemi__ring @ A_27a,V2l2: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V3l1: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V4c2: A_27a,V5c1: A_27a] :
( ( c_2Ecanonical_2Emonom__insert @ A_27a @ V1sr @ V5c1 @ V3l1 @ ( c_2Ecanonical_2ECons__monom @ A_27a @ V4c2 @ V2l2 @ V0t2 ) )
= ( c_2EternaryComparisons_2Eordering__CASE @ ( tyop_2Ecanonical_2Ecanonical__sum @ A_27a ) @ ( c_2EternaryComparisons_2Elist__compare @ tyop_2Equote_2Eindex @ tyop_2Equote_2Eindex @ c_2Equote_2Eindex__compare @ V3l1 @ V2l2 ) @ ( c_2Ecanonical_2ECons__monom @ A_27a @ V5c1 @ V3l1 @ ( c_2Ecanonical_2ECons__monom @ A_27a @ V4c2 @ V2l2 @ V0t2 ) ) @ ( c_2Ecanonical_2ECons__monom @ A_27a @ ( c_2Esemi__ring_2Esemi__ring__SRP @ A_27a @ V1sr @ V5c1 @ V4c2 ) @ V3l1 @ V0t2 ) @ ( c_2Ecanonical_2ECons__monom @ A_27a @ V4c2 @ V2l2 @ ( c_2Ecanonical_2Emonom__insert @ A_27a @ V1sr @ V5c1 @ V3l1 @ V0t2 ) ) ) )
& ! [V6t2: tyop_2Ecanonical_2Ecanonical__sum @ A_27a,V7sr: tyop_2Esemi__ring_2Esemi__ring @ A_27a,V8l2: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V9l1: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V10c1: A_27a] :
( ( c_2Ecanonical_2Emonom__insert @ A_27a @ V7sr @ V10c1 @ V9l1 @ ( c_2Ecanonical_2ECons__varlist @ A_27a @ V8l2 @ V6t2 ) )
= ( c_2EternaryComparisons_2Eordering__CASE @ ( tyop_2Ecanonical_2Ecanonical__sum @ A_27a ) @ ( c_2EternaryComparisons_2Elist__compare @ tyop_2Equote_2Eindex @ tyop_2Equote_2Eindex @ c_2Equote_2Eindex__compare @ V9l1 @ V8l2 ) @ ( c_2Ecanonical_2ECons__monom @ A_27a @ V10c1 @ V9l1 @ ( c_2Ecanonical_2ECons__varlist @ A_27a @ V8l2 @ V6t2 ) ) @ ( c_2Ecanonical_2ECons__monom @ A_27a @ ( c_2Esemi__ring_2Esemi__ring__SRP @ A_27a @ V7sr @ V10c1 @ ( c_2Esemi__ring_2Esemi__ring__SR1 @ A_27a @ V7sr ) ) @ V9l1 @ V6t2 ) @ ( c_2Ecanonical_2ECons__varlist @ A_27a @ V8l2 @ ( c_2Ecanonical_2Emonom__insert @ A_27a @ V7sr @ V10c1 @ V9l1 @ V6t2 ) ) ) )
& ! [V11sr: tyop_2Esemi__ring_2Esemi__ring @ A_27a,V12l1: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V13c1: A_27a] :
( ( c_2Ecanonical_2Emonom__insert @ A_27a @ V11sr @ V13c1 @ V12l1 @ ( c_2Ecanonical_2ENil__monom @ A_27a ) )
= ( c_2Ecanonical_2ECons__monom @ A_27a @ V13c1 @ V12l1 @ ( c_2Ecanonical_2ENil__monom @ A_27a ) ) ) ) ).
thf(thm_2Ecanonical_2Evarlist__insert__ind,axiom,
! [A_27a: $tType,V0P: ( tyop_2Esemi__ring_2Esemi__ring @ A_27a ) > ( tyop_2Elist_2Elist @ tyop_2Equote_2Eindex ) > ( tyop_2Ecanonical_2Ecanonical__sum @ A_27a ) > $o] :
( ( ! [V1sr: tyop_2Esemi__ring_2Esemi__ring @ A_27a,V2l1: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V3c2: A_27a,V4l2: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V5t2: tyop_2Ecanonical_2Ecanonical__sum @ A_27a] :
( ( ( ( c_2EternaryComparisons_2Elist__compare @ tyop_2Equote_2Eindex @ tyop_2Equote_2Eindex @ c_2Equote_2Eindex__compare @ V2l1 @ V4l2 )
= c_2EternaryComparisons_2EGREATER )
=> ( V0P @ V1sr @ V2l1 @ V5t2 ) )
=> ( V0P @ V1sr @ V2l1 @ ( c_2Ecanonical_2ECons__monom @ A_27a @ V3c2 @ V4l2 @ V5t2 ) ) )
& ! [V6sr: tyop_2Esemi__ring_2Esemi__ring @ A_27a,V7l1: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V8l2: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V9t2: tyop_2Ecanonical_2Ecanonical__sum @ A_27a] :
( ( ( ( c_2EternaryComparisons_2Elist__compare @ tyop_2Equote_2Eindex @ tyop_2Equote_2Eindex @ c_2Equote_2Eindex__compare @ V7l1 @ V8l2 )
= c_2EternaryComparisons_2EGREATER )
=> ( V0P @ V6sr @ V7l1 @ V9t2 ) )
=> ( V0P @ V6sr @ V7l1 @ ( c_2Ecanonical_2ECons__varlist @ A_27a @ V8l2 @ V9t2 ) ) )
& ! [V10sr: tyop_2Esemi__ring_2Esemi__ring @ A_27a,V11l1: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex] : ( V0P @ V10sr @ V11l1 @ ( c_2Ecanonical_2ENil__monom @ A_27a ) ) )
=> ! [V12v: tyop_2Esemi__ring_2Esemi__ring @ A_27a,V13v1: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V14v2: tyop_2Ecanonical_2Ecanonical__sum @ A_27a] : ( V0P @ V12v @ V13v1 @ V14v2 ) ) ).
thf(thm_2Ecanonical_2Evarlist__insert__def,axiom,
! [A_27a: $tType] :
( ! [V0t2: tyop_2Ecanonical_2Ecanonical__sum @ A_27a,V1sr: tyop_2Esemi__ring_2Esemi__ring @ A_27a,V2l2: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V3l1: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V4c2: A_27a] :
( ( c_2Ecanonical_2Evarlist__insert @ A_27a @ V1sr @ V3l1 @ ( c_2Ecanonical_2ECons__monom @ A_27a @ V4c2 @ V2l2 @ V0t2 ) )
= ( c_2EternaryComparisons_2Eordering__CASE @ ( tyop_2Ecanonical_2Ecanonical__sum @ A_27a ) @ ( c_2EternaryComparisons_2Elist__compare @ tyop_2Equote_2Eindex @ tyop_2Equote_2Eindex @ c_2Equote_2Eindex__compare @ V3l1 @ V2l2 ) @ ( c_2Ecanonical_2ECons__varlist @ A_27a @ V3l1 @ ( c_2Ecanonical_2ECons__monom @ A_27a @ V4c2 @ V2l2 @ V0t2 ) ) @ ( c_2Ecanonical_2ECons__monom @ A_27a @ ( c_2Esemi__ring_2Esemi__ring__SRP @ A_27a @ V1sr @ ( c_2Esemi__ring_2Esemi__ring__SR1 @ A_27a @ V1sr ) @ V4c2 ) @ V3l1 @ V0t2 ) @ ( c_2Ecanonical_2ECons__monom @ A_27a @ V4c2 @ V2l2 @ ( c_2Ecanonical_2Evarlist__insert @ A_27a @ V1sr @ V3l1 @ V0t2 ) ) ) )
& ! [V5t2: tyop_2Ecanonical_2Ecanonical__sum @ A_27a,V6sr: tyop_2Esemi__ring_2Esemi__ring @ A_27a,V7l2: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V8l1: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex] :
( ( c_2Ecanonical_2Evarlist__insert @ A_27a @ V6sr @ V8l1 @ ( c_2Ecanonical_2ECons__varlist @ A_27a @ V7l2 @ V5t2 ) )
= ( c_2EternaryComparisons_2Eordering__CASE @ ( tyop_2Ecanonical_2Ecanonical__sum @ A_27a ) @ ( c_2EternaryComparisons_2Elist__compare @ tyop_2Equote_2Eindex @ tyop_2Equote_2Eindex @ c_2Equote_2Eindex__compare @ V8l1 @ V7l2 ) @ ( c_2Ecanonical_2ECons__varlist @ A_27a @ V8l1 @ ( c_2Ecanonical_2ECons__varlist @ A_27a @ V7l2 @ V5t2 ) ) @ ( c_2Ecanonical_2ECons__monom @ A_27a @ ( c_2Esemi__ring_2Esemi__ring__SRP @ A_27a @ V6sr @ ( c_2Esemi__ring_2Esemi__ring__SR1 @ A_27a @ V6sr ) @ ( c_2Esemi__ring_2Esemi__ring__SR1 @ A_27a @ V6sr ) ) @ V8l1 @ V5t2 ) @ ( c_2Ecanonical_2ECons__varlist @ A_27a @ V7l2 @ ( c_2Ecanonical_2Evarlist__insert @ A_27a @ V6sr @ V8l1 @ V5t2 ) ) ) )
& ! [V9sr: tyop_2Esemi__ring_2Esemi__ring @ A_27a,V10l1: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex] :
( ( c_2Ecanonical_2Evarlist__insert @ A_27a @ V9sr @ V10l1 @ ( c_2Ecanonical_2ENil__monom @ A_27a ) )
= ( c_2Ecanonical_2ECons__varlist @ A_27a @ V10l1 @ ( c_2Ecanonical_2ENil__monom @ A_27a ) ) ) ) ).
thf(thm_2Ecanonical_2Eivl__aux__ok,axiom,
! [A_27a: $tType,V0sr: tyop_2Esemi__ring_2Esemi__ring @ A_27a] :
( ( c_2Esemi__ring_2Eis__semi__ring @ A_27a @ V0sr )
=> ! [V1vm: tyop_2Equote_2Evarmap @ A_27a,V2v: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V3i: tyop_2Equote_2Eindex] :
( ( c_2Ecanonical_2Eivl__aux @ A_27a @ V0sr @ V1vm @ V3i @ V2v )
= ( c_2Esemi__ring_2Esemi__ring__SRM @ A_27a @ V0sr @ ( c_2Equote_2Evarmap__find @ A_27a @ V3i @ V1vm ) @ ( c_2Ecanonical_2Einterp__vl @ A_27a @ V0sr @ V1vm @ V2v ) ) ) ) ).
thf(thm_2Ecanonical_2Evarlist__merge__ok,axiom,
! [A_27a: $tType,V0sr: tyop_2Esemi__ring_2Esemi__ring @ A_27a] :
( ( c_2Esemi__ring_2Eis__semi__ring @ A_27a @ V0sr )
=> ! [V1vm: tyop_2Equote_2Evarmap @ A_27a,V2x: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V3y: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex] :
( ( c_2Ecanonical_2Einterp__vl @ A_27a @ V0sr @ V1vm @ ( c_2EternaryComparisons_2Elist__merge @ tyop_2Equote_2Eindex @ c_2Equote_2Eindex__lt @ V2x @ V3y ) )
= ( c_2Esemi__ring_2Esemi__ring__SRM @ A_27a @ V0sr @ ( c_2Ecanonical_2Einterp__vl @ A_27a @ V0sr @ V1vm @ V2x ) @ ( c_2Ecanonical_2Einterp__vl @ A_27a @ V0sr @ V1vm @ V3y ) ) ) ) ).
thf(thm_2Ecanonical_2Eics__aux__ok,axiom,
! [A_27a: $tType,V0sr: tyop_2Esemi__ring_2Esemi__ring @ A_27a] :
( ( c_2Esemi__ring_2Eis__semi__ring @ A_27a @ V0sr )
=> ! [V1vm: tyop_2Equote_2Evarmap @ A_27a,V2x: A_27a,V3s: tyop_2Ecanonical_2Ecanonical__sum @ A_27a] :
( ( c_2Ecanonical_2Eics__aux @ A_27a @ V0sr @ V1vm @ V2x @ V3s )
= ( c_2Esemi__ring_2Esemi__ring__SRP @ A_27a @ V0sr @ V2x @ ( c_2Ecanonical_2Einterp__cs @ A_27a @ V0sr @ V1vm @ V3s ) ) ) ) ).
thf(thm_2Ecanonical_2Einterp__m__ok,axiom,
! [A_27a: $tType,V0sr: tyop_2Esemi__ring_2Esemi__ring @ A_27a] :
( ( c_2Esemi__ring_2Eis__semi__ring @ A_27a @ V0sr )
=> ! [V1vm: tyop_2Equote_2Evarmap @ A_27a,V2x: A_27a,V3l: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex] :
( ( c_2Ecanonical_2Einterp__m @ A_27a @ V0sr @ V1vm @ V2x @ V3l )
= ( c_2Esemi__ring_2Esemi__ring__SRM @ A_27a @ V0sr @ V2x @ ( c_2Ecanonical_2Einterp__vl @ A_27a @ V0sr @ V1vm @ V3l ) ) ) ) ).
thf(thm_2Ecanonical_2Ecanonical__sum__merge__ok,axiom,
! [A_27a: $tType,V0sr: tyop_2Esemi__ring_2Esemi__ring @ A_27a] :
( ( c_2Esemi__ring_2Eis__semi__ring @ A_27a @ V0sr )
=> ! [V1vm: tyop_2Equote_2Evarmap @ A_27a,V2x: tyop_2Ecanonical_2Ecanonical__sum @ A_27a,V3y: tyop_2Ecanonical_2Ecanonical__sum @ A_27a] :
( ( c_2Ecanonical_2Einterp__cs @ A_27a @ V0sr @ V1vm @ ( c_2Ecanonical_2Ecanonical__sum__merge @ A_27a @ V0sr @ V2x @ V3y ) )
= ( c_2Esemi__ring_2Esemi__ring__SRP @ A_27a @ V0sr @ ( c_2Ecanonical_2Einterp__cs @ A_27a @ V0sr @ V1vm @ V2x ) @ ( c_2Ecanonical_2Einterp__cs @ A_27a @ V0sr @ V1vm @ V3y ) ) ) ) ).
thf(thm_2Ecanonical_2Emonom__insert__ok,axiom,
! [A_27a: $tType,V0sr: tyop_2Esemi__ring_2Esemi__ring @ A_27a] :
( ( c_2Esemi__ring_2Eis__semi__ring @ A_27a @ V0sr )
=> ! [V1vm: tyop_2Equote_2Evarmap @ A_27a,V2a: A_27a,V3l: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V4s: tyop_2Ecanonical_2Ecanonical__sum @ A_27a] :
( ( c_2Ecanonical_2Einterp__cs @ A_27a @ V0sr @ V1vm @ ( c_2Ecanonical_2Emonom__insert @ A_27a @ V0sr @ V2a @ V3l @ V4s ) )
= ( c_2Esemi__ring_2Esemi__ring__SRP @ A_27a @ V0sr @ ( c_2Esemi__ring_2Esemi__ring__SRM @ A_27a @ V0sr @ V2a @ ( c_2Ecanonical_2Einterp__vl @ A_27a @ V0sr @ V1vm @ V3l ) ) @ ( c_2Ecanonical_2Einterp__cs @ A_27a @ V0sr @ V1vm @ V4s ) ) ) ) ).
thf(thm_2Ecanonical_2Evarlist__insert__ok,axiom,
! [A_27a: $tType,V0sr: tyop_2Esemi__ring_2Esemi__ring @ A_27a] :
( ( c_2Esemi__ring_2Eis__semi__ring @ A_27a @ V0sr )
=> ! [V1vm: tyop_2Equote_2Evarmap @ A_27a,V2l: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V3s: tyop_2Ecanonical_2Ecanonical__sum @ A_27a] :
( ( c_2Ecanonical_2Einterp__cs @ A_27a @ V0sr @ V1vm @ ( c_2Ecanonical_2Evarlist__insert @ A_27a @ V0sr @ V2l @ V3s ) )
= ( c_2Esemi__ring_2Esemi__ring__SRP @ A_27a @ V0sr @ ( c_2Ecanonical_2Einterp__vl @ A_27a @ V0sr @ V1vm @ V2l ) @ ( c_2Ecanonical_2Einterp__cs @ A_27a @ V0sr @ V1vm @ V3s ) ) ) ) ).
thf(thm_2Ecanonical_2Ecanonical__sum__scalar__ok,axiom,
! [A_27a: $tType,V0sr: tyop_2Esemi__ring_2Esemi__ring @ A_27a] :
( ( c_2Esemi__ring_2Eis__semi__ring @ A_27a @ V0sr )
=> ! [V1vm: tyop_2Equote_2Evarmap @ A_27a,V2a: A_27a,V3s: tyop_2Ecanonical_2Ecanonical__sum @ A_27a] :
( ( c_2Ecanonical_2Einterp__cs @ A_27a @ V0sr @ V1vm @ ( c_2Ecanonical_2Ecanonical__sum__scalar @ A_27a @ V0sr @ V2a @ V3s ) )
= ( c_2Esemi__ring_2Esemi__ring__SRM @ A_27a @ V0sr @ V2a @ ( c_2Ecanonical_2Einterp__cs @ A_27a @ V0sr @ V1vm @ V3s ) ) ) ) ).
thf(thm_2Ecanonical_2Ecanonical__sum__scalar2__ok,axiom,
! [A_27a: $tType,V0sr: tyop_2Esemi__ring_2Esemi__ring @ A_27a] :
( ( c_2Esemi__ring_2Eis__semi__ring @ A_27a @ V0sr )
=> ! [V1vm: tyop_2Equote_2Evarmap @ A_27a,V2l: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V3s: tyop_2Ecanonical_2Ecanonical__sum @ A_27a] :
( ( c_2Ecanonical_2Einterp__cs @ A_27a @ V0sr @ V1vm @ ( c_2Ecanonical_2Ecanonical__sum__scalar2 @ A_27a @ V0sr @ V2l @ V3s ) )
= ( c_2Esemi__ring_2Esemi__ring__SRM @ A_27a @ V0sr @ ( c_2Ecanonical_2Einterp__vl @ A_27a @ V0sr @ V1vm @ V2l ) @ ( c_2Ecanonical_2Einterp__cs @ A_27a @ V0sr @ V1vm @ V3s ) ) ) ) ).
thf(thm_2Ecanonical_2Ecanonical__sum__scalar3__ok,axiom,
! [A_27a: $tType,V0sr: tyop_2Esemi__ring_2Esemi__ring @ A_27a] :
( ( c_2Esemi__ring_2Eis__semi__ring @ A_27a @ V0sr )
=> ! [V1vm: tyop_2Equote_2Evarmap @ A_27a,V2c: A_27a,V3l: tyop_2Elist_2Elist @ tyop_2Equote_2Eindex,V4s: tyop_2Ecanonical_2Ecanonical__sum @ A_27a] :
( ( c_2Ecanonical_2Einterp__cs @ A_27a @ V0sr @ V1vm @ ( c_2Ecanonical_2Ecanonical__sum__scalar3 @ A_27a @ V0sr @ V2c @ V3l @ V4s ) )
= ( c_2Esemi__ring_2Esemi__ring__SRM @ A_27a @ V0sr @ ( c_2Esemi__ring_2Esemi__ring__SRM @ A_27a @ V0sr @ V2c @ ( c_2Ecanonical_2Einterp__vl @ A_27a @ V0sr @ V1vm @ V3l ) ) @ ( c_2Ecanonical_2Einterp__cs @ A_27a @ V0sr @ V1vm @ V4s ) ) ) ) ).
thf(thm_2Ecanonical_2Ecanonical__sum__prod__ok,axiom,
! [A_27a: $tType,V0sr: tyop_2Esemi__ring_2Esemi__ring @ A_27a] :
( ( c_2Esemi__ring_2Eis__semi__ring @ A_27a @ V0sr )
=> ! [V1vm: tyop_2Equote_2Evarmap @ A_27a,V2x: tyop_2Ecanonical_2Ecanonical__sum @ A_27a,V3y: tyop_2Ecanonical_2Ecanonical__sum @ A_27a] :
( ( c_2Ecanonical_2Einterp__cs @ A_27a @ V0sr @ V1vm @ ( c_2Ecanonical_2Ecanonical__sum__prod @ A_27a @ V0sr @ V2x @ V3y ) )
= ( c_2Esemi__ring_2Esemi__ring__SRM @ A_27a @ V0sr @ ( c_2Ecanonical_2Einterp__cs @ A_27a @ V0sr @ V1vm @ V2x ) @ ( c_2Ecanonical_2Einterp__cs @ A_27a @ V0sr @ V1vm @ V3y ) ) ) ) ).
thf(thm_2Ecanonical_2Ecanonical__sum__simplify__ok,axiom,
! [A_27a: $tType,V0sr: tyop_2Esemi__ring_2Esemi__ring @ A_27a] :
( ( c_2Esemi__ring_2Eis__semi__ring @ A_27a @ V0sr )
=> ! [V1vm: tyop_2Equote_2Evarmap @ A_27a,V2s: tyop_2Ecanonical_2Ecanonical__sum @ A_27a] :
( ( c_2Ecanonical_2Einterp__cs @ A_27a @ V0sr @ V1vm @ ( c_2Ecanonical_2Ecanonical__sum__simplify @ A_27a @ V0sr @ V2s ) )
= ( c_2Ecanonical_2Einterp__cs @ A_27a @ V0sr @ V1vm @ V2s ) ) ) ).
thf(thm_2Ecanonical_2Edatatype__spolynom,axiom,
! [A_27a: $tType,V0spolynom: ( tyop_2Equote_2Eindex > ( tyop_2Ecanonical_2Espolynom @ A_27a ) ) > ( A_27a > ( tyop_2Ecanonical_2Espolynom @ A_27a ) ) > ( ( tyop_2Ecanonical_2Espolynom @ A_27a ) > ( tyop_2Ecanonical_2Espolynom @ A_27a ) > ( tyop_2Ecanonical_2Espolynom @ A_27a ) ) > ( ( tyop_2Ecanonical_2Espolynom @ A_27a ) > ( tyop_2Ecanonical_2Espolynom @ A_27a ) > ( tyop_2Ecanonical_2Espolynom @ A_27a ) ) > $o] : ( c_2Ebool_2EDATATYPE @ $o @ ( V0spolynom @ ( c_2Ecanonical_2ESPvar @ A_27a ) @ ( c_2Ecanonical_2ESPconst @ A_27a ) @ ( c_2Ecanonical_2ESPplus @ A_27a ) @ ( c_2Ecanonical_2ESPmult @ A_27a ) ) ) ).
thf(thm_2Ecanonical_2Espolynom__11,axiom,
! [A_27a: $tType] :
( ! [V0a: tyop_2Equote_2Eindex,V1a_27: tyop_2Equote_2Eindex] :
( ( ( c_2Ecanonical_2ESPvar @ A_27a @ V0a )
= ( c_2Ecanonical_2ESPvar @ A_27a @ V1a_27 ) )
<=> ( V0a = V1a_27 ) )
& ! [V2a: A_27a,V3a_27: A_27a] :
( ( ( c_2Ecanonical_2ESPconst @ A_27a @ V2a )
= ( c_2Ecanonical_2ESPconst @ A_27a @ V3a_27 ) )
<=> ( V2a = V3a_27 ) )
& ! [V4a0: tyop_2Ecanonical_2Espolynom @ A_27a,V5a1: tyop_2Ecanonical_2Espolynom @ A_27a,V6a0_27: tyop_2Ecanonical_2Espolynom @ A_27a,V7a1_27: tyop_2Ecanonical_2Espolynom @ A_27a] :
( ( ( c_2Ecanonical_2ESPplus @ A_27a @ V4a0 @ V5a1 )
= ( c_2Ecanonical_2ESPplus @ A_27a @ V6a0_27 @ V7a1_27 ) )
<=> ( ( V4a0 = V6a0_27 )
& ( V5a1 = V7a1_27 ) ) )
& ! [V8a0: tyop_2Ecanonical_2Espolynom @ A_27a,V9a1: tyop_2Ecanonical_2Espolynom @ A_27a,V10a0_27: tyop_2Ecanonical_2Espolynom @ A_27a,V11a1_27: tyop_2Ecanonical_2Espolynom @ A_27a] :
( ( ( c_2Ecanonical_2ESPmult @ A_27a @ V8a0 @ V9a1 )
= ( c_2Ecanonical_2ESPmult @ A_27a @ V10a0_27 @ V11a1_27 ) )
<=> ( ( V8a0 = V10a0_27 )
& ( V9a1 = V11a1_27 ) ) ) ) ).
thf(thm_2Ecanonical_2Espolynom__distinct,axiom,
! [A_27a: $tType] :
( ! [V0a_27: A_27a,V1a: tyop_2Equote_2Eindex] :
( (~)
@ ( ( c_2Ecanonical_2ESPvar @ A_27a @ V1a )
= ( c_2Ecanonical_2ESPconst @ A_27a @ V0a_27 ) ) )
& ! [V2a1: tyop_2Ecanonical_2Espolynom @ A_27a,V3a0: tyop_2Ecanonical_2Espolynom @ A_27a,V4a: tyop_2Equote_2Eindex] :
( (~)
@ ( ( c_2Ecanonical_2ESPvar @ A_27a @ V4a )
= ( c_2Ecanonical_2ESPplus @ A_27a @ V3a0 @ V2a1 ) ) )
& ! [V5a1: tyop_2Ecanonical_2Espolynom @ A_27a,V6a0: tyop_2Ecanonical_2Espolynom @ A_27a,V7a: tyop_2Equote_2Eindex] :
( (~)
@ ( ( c_2Ecanonical_2ESPvar @ A_27a @ V7a )
= ( c_2Ecanonical_2ESPmult @ A_27a @ V6a0 @ V5a1 ) ) )
& ! [V8a1: tyop_2Ecanonical_2Espolynom @ A_27a,V9a0: tyop_2Ecanonical_2Espolynom @ A_27a,V10a: A_27a] :
( (~)
@ ( ( c_2Ecanonical_2ESPconst @ A_27a @ V10a )
= ( c_2Ecanonical_2ESPplus @ A_27a @ V9a0 @ V8a1 ) ) )
& ! [V11a1: tyop_2Ecanonical_2Espolynom @ A_27a,V12a0: tyop_2Ecanonical_2Espolynom @ A_27a,V13a: A_27a] :
( (~)
@ ( ( c_2Ecanonical_2ESPconst @ A_27a @ V13a )
= ( c_2Ecanonical_2ESPmult @ A_27a @ V12a0 @ V11a1 ) ) )
& ! [V14a1_27: tyop_2Ecanonical_2Espolynom @ A_27a,V15a1: tyop_2Ecanonical_2Espolynom @ A_27a,V16a0_27: tyop_2Ecanonical_2Espolynom @ A_27a,V17a0: tyop_2Ecanonical_2Espolynom @ A_27a] :
( (~)
@ ( ( c_2Ecanonical_2ESPplus @ A_27a @ V17a0 @ V15a1 )
= ( c_2Ecanonical_2ESPmult @ A_27a @ V16a0_27 @ V14a1_27 ) ) ) ) ).
thf(thm_2Ecanonical_2Espolynom__nchotomy,axiom,
! [A_27a: $tType,V0ss: tyop_2Ecanonical_2Espolynom @ A_27a] :
( ? [V1i: tyop_2Equote_2Eindex] :
( V0ss
= ( c_2Ecanonical_2ESPvar @ A_27a @ V1i ) )
| ? [V2a: A_27a] :
( V0ss
= ( c_2Ecanonical_2ESPconst @ A_27a @ V2a ) )
| ? [V3s: tyop_2Ecanonical_2Espolynom @ A_27a,V4s0: tyop_2Ecanonical_2Espolynom @ A_27a] :
( V0ss
= ( c_2Ecanonical_2ESPplus @ A_27a @ V3s @ V4s0 ) )
| ? [V5s: tyop_2Ecanonical_2Espolynom @ A_27a,V6s0: tyop_2Ecanonical_2Espolynom @ A_27a] :
( V0ss
= ( c_2Ecanonical_2ESPmult @ A_27a @ V5s @ V6s0 ) ) ) ).
thf(thm_2Ecanonical_2Espolynom__Axiom,axiom,
! [A_27a: $tType,A_27b: $tType,V0f0: tyop_2Equote_2Eindex > A_27b,V1f1: A_27a > A_27b,V2f2: ( tyop_2Ecanonical_2Espolynom @ A_27a ) > ( tyop_2Ecanonical_2Espolynom @ A_27a ) > A_27b > A_27b > A_27b,V3f3: ( tyop_2Ecanonical_2Espolynom @ A_27a ) > ( tyop_2Ecanonical_2Espolynom @ A_27a ) > A_27b > A_27b > A_27b] :
? [V4fn: ( tyop_2Ecanonical_2Espolynom @ A_27a ) > A_27b] :
( ! [V5a: tyop_2Equote_2Eindex] :
( ( V4fn @ ( c_2Ecanonical_2ESPvar @ A_27a @ V5a ) )
= ( V0f0 @ V5a ) )
& ! [V6a: A_27a] :
( ( V4fn @ ( c_2Ecanonical_2ESPconst @ A_27a @ V6a ) )
= ( V1f1 @ V6a ) )
& ! [V7a0: tyop_2Ecanonical_2Espolynom @ A_27a,V8a1: tyop_2Ecanonical_2Espolynom @ A_27a] :
( ( V4fn @ ( c_2Ecanonical_2ESPplus @ A_27a @ V7a0 @ V8a1 ) )
= ( V2f2 @ V7a0 @ V8a1 @ ( V4fn @ V7a0 ) @ ( V4fn @ V8a1 ) ) )
& ! [V9a0: tyop_2Ecanonical_2Espolynom @ A_27a,V10a1: tyop_2Ecanonical_2Espolynom @ A_27a] :
( ( V4fn @ ( c_2Ecanonical_2ESPmult @ A_27a @ V9a0 @ V10a1 ) )
= ( V3f3 @ V9a0 @ V10a1 @ ( V4fn @ V9a0 ) @ ( V4fn @ V10a1 ) ) ) ) ).
thf(thm_2Ecanonical_2Espolynom__induction,axiom,
! [A_27a: $tType,V0P: ( tyop_2Ecanonical_2Espolynom @ A_27a ) > $o] :
( ( ! [V1i: tyop_2Equote_2Eindex] : ( V0P @ ( c_2Ecanonical_2ESPvar @ A_27a @ V1i ) )
& ! [V2a: A_27a] : ( V0P @ ( c_2Ecanonical_2ESPconst @ A_27a @ V2a ) )
& ! [V3s: tyop_2Ecanonical_2Espolynom @ A_27a,V4s0: tyop_2Ecanonical_2Espolynom @ A_27a] :
( ( ( V0P @ V3s )
& ( V0P @ V4s0 ) )
=> ( V0P @ ( c_2Ecanonical_2ESPplus @ A_27a @ V3s @ V4s0 ) ) )
& ! [V5s: tyop_2Ecanonical_2Espolynom @ A_27a,V6s0: tyop_2Ecanonical_2Espolynom @ A_27a] :
( ( ( V0P @ V5s )
& ( V0P @ V6s0 ) )
=> ( V0P @ ( c_2Ecanonical_2ESPmult @ A_27a @ V5s @ V6s0 ) ) ) )
=> ! [V7s: tyop_2Ecanonical_2Espolynom @ A_27a] : ( V0P @ V7s ) ) ).
thf(thm_2Ecanonical_2Espolynom__case__cong,axiom,
! [A_27a: $tType,A_27b: $tType,V0f3_27: ( tyop_2Ecanonical_2Espolynom @ A_27a ) > ( tyop_2Ecanonical_2Espolynom @ A_27a ) > A_27b,V1f2_27: ( tyop_2Ecanonical_2Espolynom @ A_27a ) > ( tyop_2Ecanonical_2Espolynom @ A_27a ) > A_27b,V2f1_27: A_27a > A_27b,V3f_27: tyop_2Equote_2Eindex > A_27b,V4M: tyop_2Ecanonical_2Espolynom @ A_27a,V5M_27: tyop_2Ecanonical_2Espolynom @ A_27a,V6f: tyop_2Equote_2Eindex > A_27b,V7f1: A_27a > A_27b,V8f2: ( tyop_2Ecanonical_2Espolynom @ A_27a ) > ( tyop_2Ecanonical_2Espolynom @ A_27a ) > A_27b,V9f3: ( tyop_2Ecanonical_2Espolynom @ A_27a ) > ( tyop_2Ecanonical_2Espolynom @ A_27a ) > A_27b] :
( ( ( V4M = V5M_27 )
& ! [V10a: tyop_2Equote_2Eindex] :
( ( V5M_27
= ( c_2Ecanonical_2ESPvar @ A_27a @ V10a ) )
=> ( ( V6f @ V10a )
= ( V3f_27 @ V10a ) ) )
& ! [V11a: A_27a] :
( ( V5M_27
= ( c_2Ecanonical_2ESPconst @ A_27a @ V11a ) )
=> ( ( V7f1 @ V11a )
= ( V2f1_27 @ V11a ) ) )
& ! [V12a0: tyop_2Ecanonical_2Espolynom @ A_27a,V13a1: tyop_2Ecanonical_2Espolynom @ A_27a] :
( ( V5M_27
= ( c_2Ecanonical_2ESPplus @ A_27a @ V12a0 @ V13a1 ) )
=> ( ( V8f2 @ V12a0 @ V13a1 )
= ( V1f2_27 @ V12a0 @ V13a1 ) ) )
& ! [V14a0: tyop_2Ecanonical_2Espolynom @ A_27a,V15a1: tyop_2Ecanonical_2Espolynom @ A_27a] :
( ( V5M_27
= ( c_2Ecanonical_2ESPmult @ A_27a @ V14a0 @ V15a1 ) )
=> ( ( V9f3 @ V14a0 @ V15a1 )
= ( V0f3_27 @ V14a0 @ V15a1 ) ) ) )
=> ( ( c_2Ecanonical_2Espolynom__CASE @ A_27a @ A_27b @ V4M @ V6f @ V7f1 @ V8f2 @ V9f3 )
= ( c_2Ecanonical_2Espolynom__CASE @ A_27a @ A_27b @ V5M_27 @ V3f_27 @ V2f1_27 @ V1f2_27 @ V0f3_27 ) ) ) ).
thf(thm_2Ecanonical_2Espolynom__case__eq,axiom,
! [A_27a: $tType,A_27b: $tType,V0x: tyop_2Ecanonical_2Espolynom @ A_27a,V1v: A_27b,V2f3: ( tyop_2Ecanonical_2Espolynom @ A_27a ) > ( tyop_2Ecanonical_2Espolynom @ A_27a ) > A_27b,V3f2: ( tyop_2Ecanonical_2Espolynom @ A_27a ) > ( tyop_2Ecanonical_2Espolynom @ A_27a ) > A_27b,V4f1: A_27a > A_27b,V5f: tyop_2Equote_2Eindex > A_27b] :
( ( ( c_2Ecanonical_2Espolynom__CASE @ A_27a @ A_27b @ V0x @ V5f @ V4f1 @ V3f2 @ V2f3 )
= V1v )
<=> ( ? [V6i: tyop_2Equote_2Eindex] :
( ( V0x
= ( c_2Ecanonical_2ESPvar @ A_27a @ V6i ) )
& ( ( V5f @ V6i )
= V1v ) )
| ? [V7a: A_27a] :
( ( V0x
= ( c_2Ecanonical_2ESPconst @ A_27a @ V7a ) )
& ( ( V4f1 @ V7a )
= V1v ) )
| ? [V8s: tyop_2Ecanonical_2Espolynom @ A_27a,V9s0: tyop_2Ecanonical_2Espolynom @ A_27a] :
( ( V0x
= ( c_2Ecanonical_2ESPplus @ A_27a @ V8s @ V9s0 ) )
& ( ( V3f2 @ V8s @ V9s0 )
= V1v ) )
| ? [V10s: tyop_2Ecanonical_2Espolynom @ A_27a,V11s0: tyop_2Ecanonical_2Espolynom @ A_27a] :
( ( V0x
= ( c_2Ecanonical_2ESPmult @ A_27a @ V10s @ V11s0 ) )
& ( ( V2f3 @ V10s @ V11s0 )
= V1v ) ) ) ) ).
thf(thm_2Ecanonical_2Espolynomial__normalize__ok,axiom,
! [A_27a: $tType,V0sr: tyop_2Esemi__ring_2Esemi__ring @ A_27a] :
( ( c_2Esemi__ring_2Eis__semi__ring @ A_27a @ V0sr )
=> ! [V1vm: tyop_2Equote_2Evarmap @ A_27a,V2p: tyop_2Ecanonical_2Espolynom @ A_27a] :
( ( c_2Ecanonical_2Einterp__cs @ A_27a @ V0sr @ V1vm @ ( c_2Ecanonical_2Espolynom__normalize @ A_27a @ V0sr @ V2p ) )
= ( c_2Ecanonical_2Einterp__sp @ A_27a @ V0sr @ V1vm @ V2p ) ) ) ).
thf(thm_2Ecanonical_2Espolynomial__simplify__ok,axiom,
! [A_27a: $tType,V0sr: tyop_2Esemi__ring_2Esemi__ring @ A_27a] :
( ( c_2Esemi__ring_2Eis__semi__ring @ A_27a @ V0sr )
=> ! [V1vm: tyop_2Equote_2Evarmap @ A_27a,V2p: tyop_2Ecanonical_2Espolynom @ A_27a] :
( ( c_2Ecanonical_2Einterp__cs @ A_27a @ V0sr @ V1vm @ ( c_2Ecanonical_2Espolynom__simplify @ A_27a @ V0sr @ V2p ) )
= ( c_2Ecanonical_2Einterp__sp @ A_27a @ V0sr @ V1vm @ V2p ) ) ) ).
%------------------------------------------------------------------------------