ITP001 Axioms: ITP114+5.ax
%------------------------------------------------------------------------------
% File : ITP114+5 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 set theory export, chainy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau20] Gauthier (2020), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : ringNorm+2.ax [Gau20]
% : HL4114+5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 84 ( 2 unt; 0 def)
% Number of atoms : 645 ( 152 equ)
% Maximal formula atoms : 39 ( 7 avg)
% Number of connectives : 571 ( 10 ~; 8 |; 100 &)
% ( 6 <=>; 447 =>; 0 <=; 0 <~>)
% Maximal formula depth : 40 ( 8 avg)
% Maximal term depth : 12 ( 2 avg)
% Number of predicates : 4 ( 3 usr; 0 prp; 1-2 aty)
% Number of functors : 112 ( 112 usr; 15 con; 0-4 aty)
% Number of variables : 469 ( 453 !; 16 ?)
% SPC : FOF_SAT_RFO_SEQ
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
fof(ne_ty_2EringNorm_2Epolynom,axiom,
! [A0] :
( ne(A0)
=> ne(ty_2EringNorm_2Epolynom(A0)) ) ).
fof(mem_c_2EringNorm_2EPconst,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2EringNorm_2EPconst(A_27a),arr(A_27a,ty_2EringNorm_2Epolynom(A_27a))) ) ).
fof(mem_c_2EringNorm_2EPmult,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2EringNorm_2EPmult(A_27a),arr(ty_2EringNorm_2Epolynom(A_27a),arr(ty_2EringNorm_2Epolynom(A_27a),ty_2EringNorm_2Epolynom(A_27a)))) ) ).
fof(mem_c_2EringNorm_2EPopp,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2EringNorm_2EPopp(A_27a),arr(ty_2EringNorm_2Epolynom(A_27a),ty_2EringNorm_2Epolynom(A_27a))) ) ).
fof(mem_c_2EringNorm_2EPplus,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2EringNorm_2EPplus(A_27a),arr(ty_2EringNorm_2Epolynom(A_27a),arr(ty_2EringNorm_2Epolynom(A_27a),ty_2EringNorm_2Epolynom(A_27a)))) ) ).
fof(mem_c_2EringNorm_2EPvar,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2EringNorm_2EPvar(A_27a),arr(ty_2Equote_2Eindex,ty_2EringNorm_2Epolynom(A_27a))) ) ).
fof(mem_c_2EringNorm_2Einterp__p,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2EringNorm_2Einterp__p(A_27a),arr(ty_2Ering_2Ering(A_27a),arr(ty_2Equote_2Evarmap(A_27a),arr(ty_2EringNorm_2Epolynom(A_27a),A_27a)))) ) ).
fof(mem_c_2EringNorm_2Epolynom__CASE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2EringNorm_2Epolynom__CASE(A_27a,A_27b),arr(ty_2EringNorm_2Epolynom(A_27a),arr(arr(ty_2Equote_2Eindex,A_27b),arr(arr(A_27a,A_27b),arr(arr(ty_2EringNorm_2Epolynom(A_27a),arr(ty_2EringNorm_2Epolynom(A_27a),A_27b)),arr(arr(ty_2EringNorm_2Epolynom(A_27a),arr(ty_2EringNorm_2Epolynom(A_27a),A_27b)),arr(arr(ty_2EringNorm_2Epolynom(A_27a),A_27b),A_27b))))))) ) ) ).
fof(mem_c_2EringNorm_2Epolynom__normalize,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2EringNorm_2Epolynom__normalize(A_27a),arr(ty_2Ering_2Ering(A_27a),arr(ty_2EringNorm_2Epolynom(A_27a),ty_2Ecanonical_2Ecanonical__sum(A_27a)))) ) ).
fof(mem_c_2EringNorm_2Epolynom__simplify,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2EringNorm_2Epolynom__simplify(A_27a),arr(ty_2Ering_2Ering(A_27a),arr(ty_2EringNorm_2Epolynom(A_27a),ty_2Ecanonical_2Ecanonical__sum(A_27a)))) ) ).
fof(mem_c_2EringNorm_2Epolynom__size,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2EringNorm_2Epolynom__size(A_27a),arr(arr(A_27a,ty_2Enum_2Enum),arr(ty_2EringNorm_2Epolynom(A_27a),ty_2Enum_2Enum))) ) ).
fof(mem_c_2EringNorm_2Er__canonical__sum__merge,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2EringNorm_2Er__canonical__sum__merge(A_27a),arr(ty_2Ering_2Ering(A_27a),arr(ty_2Ecanonical_2Ecanonical__sum(A_27a),arr(ty_2Ecanonical_2Ecanonical__sum(A_27a),ty_2Ecanonical_2Ecanonical__sum(A_27a))))) ) ).
fof(mem_c_2EringNorm_2Er__canonical__sum__prod,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2EringNorm_2Er__canonical__sum__prod(A_27a),arr(ty_2Ering_2Ering(A_27a),arr(ty_2Ecanonical_2Ecanonical__sum(A_27a),arr(ty_2Ecanonical_2Ecanonical__sum(A_27a),ty_2Ecanonical_2Ecanonical__sum(A_27a))))) ) ).
fof(mem_c_2EringNorm_2Er__canonical__sum__scalar,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2EringNorm_2Er__canonical__sum__scalar(A_27a),arr(ty_2Ering_2Ering(A_27a),arr(A_27a,arr(ty_2Ecanonical_2Ecanonical__sum(A_27a),ty_2Ecanonical_2Ecanonical__sum(A_27a))))) ) ).
fof(mem_c_2EringNorm_2Er__canonical__sum__scalar2,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2EringNorm_2Er__canonical__sum__scalar2(A_27a),arr(ty_2Ering_2Ering(A_27a),arr(ty_2Elist_2Elist(ty_2Equote_2Eindex),arr(ty_2Ecanonical_2Ecanonical__sum(A_27a),ty_2Ecanonical_2Ecanonical__sum(A_27a))))) ) ).
fof(mem_c_2EringNorm_2Er__canonical__sum__scalar3,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2EringNorm_2Er__canonical__sum__scalar3(A_27a),arr(ty_2Ering_2Ering(A_27a),arr(A_27a,arr(ty_2Elist_2Elist(ty_2Equote_2Eindex),arr(ty_2Ecanonical_2Ecanonical__sum(A_27a),ty_2Ecanonical_2Ecanonical__sum(A_27a)))))) ) ).
fof(mem_c_2EringNorm_2Er__canonical__sum__simplify,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2EringNorm_2Er__canonical__sum__simplify(A_27a),arr(ty_2Ering_2Ering(A_27a),arr(ty_2Ecanonical_2Ecanonical__sum(A_27a),ty_2Ecanonical_2Ecanonical__sum(A_27a)))) ) ).
fof(mem_c_2EringNorm_2Er__ics__aux,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2EringNorm_2Er__ics__aux(A_27a),arr(ty_2Ering_2Ering(A_27a),arr(ty_2Equote_2Evarmap(A_27a),arr(A_27a,arr(ty_2Ecanonical_2Ecanonical__sum(A_27a),A_27a))))) ) ).
fof(mem_c_2EringNorm_2Er__interp__cs,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2EringNorm_2Er__interp__cs(A_27a),arr(ty_2Ering_2Ering(A_27a),arr(ty_2Equote_2Evarmap(A_27a),arr(ty_2Ecanonical_2Ecanonical__sum(A_27a),A_27a)))) ) ).
fof(mem_c_2EringNorm_2Er__interp__m,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2EringNorm_2Er__interp__m(A_27a),arr(ty_2Ering_2Ering(A_27a),arr(ty_2Equote_2Evarmap(A_27a),arr(A_27a,arr(ty_2Elist_2Elist(ty_2Equote_2Eindex),A_27a))))) ) ).
fof(mem_c_2EringNorm_2Er__interp__sp,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2EringNorm_2Er__interp__sp(A_27a),arr(ty_2Ering_2Ering(A_27a),arr(ty_2Equote_2Evarmap(A_27a),arr(ty_2Ecanonical_2Espolynom(A_27a),A_27a)))) ) ).
fof(mem_c_2EringNorm_2Er__interp__vl,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2EringNorm_2Er__interp__vl(A_27a),arr(ty_2Ering_2Ering(A_27a),arr(ty_2Equote_2Evarmap(A_27a),arr(ty_2Elist_2Elist(ty_2Equote_2Eindex),A_27a)))) ) ).
fof(mem_c_2EringNorm_2Er__ivl__aux,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2EringNorm_2Er__ivl__aux(A_27a),arr(ty_2Ering_2Ering(A_27a),arr(ty_2Equote_2Evarmap(A_27a),arr(ty_2Equote_2Eindex,arr(ty_2Elist_2Elist(ty_2Equote_2Eindex),A_27a))))) ) ).
fof(mem_c_2EringNorm_2Er__monom__insert,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2EringNorm_2Er__monom__insert(A_27a),arr(ty_2Ering_2Ering(A_27a),arr(A_27a,arr(ty_2Elist_2Elist(ty_2Equote_2Eindex),arr(ty_2Ecanonical_2Ecanonical__sum(A_27a),ty_2Ecanonical_2Ecanonical__sum(A_27a)))))) ) ).
fof(mem_c_2EringNorm_2Er__spolynom__normalize,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2EringNorm_2Er__spolynom__normalize(A_27a),arr(ty_2Ering_2Ering(A_27a),arr(ty_2Ecanonical_2Espolynom(A_27a),ty_2Ecanonical_2Ecanonical__sum(A_27a)))) ) ).
fof(mem_c_2EringNorm_2Er__spolynom__simplify,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2EringNorm_2Er__spolynom__simplify(A_27a),arr(ty_2Ering_2Ering(A_27a),arr(ty_2Ecanonical_2Espolynom(A_27a),ty_2Ecanonical_2Ecanonical__sum(A_27a)))) ) ).
fof(mem_c_2EringNorm_2Er__varlist__insert,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2EringNorm_2Er__varlist__insert(A_27a),arr(ty_2Ering_2Ering(A_27a),arr(ty_2Elist_2Elist(ty_2Equote_2Eindex),arr(ty_2Ecanonical_2Ecanonical__sum(A_27a),ty_2Ecanonical_2Ecanonical__sum(A_27a))))) ) ).
fof(ax_thm_2EringNorm_2Er__interp__sp__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0r] :
( mem(V0r,ty_2Ering_2Ering(A_27a))
=> ap(c_2EringNorm_2Er__interp__sp(A_27a),V0r) = ap(c_2Ecanonical_2Einterp__sp(A_27a),ap(c_2Ering_2Esemi__ring__of(A_27a),V0r)) ) ) ).
fof(ax_thm_2EringNorm_2Er__spolynom__simplify__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0r] :
( mem(V0r,ty_2Ering_2Ering(A_27a))
=> ap(c_2EringNorm_2Er__spolynom__simplify(A_27a),V0r) = ap(c_2Ecanonical_2Espolynom__simplify(A_27a),ap(c_2Ering_2Esemi__ring__of(A_27a),V0r)) ) ) ).
fof(ax_thm_2EringNorm_2Er__spolynom__normalize__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0r] :
( mem(V0r,ty_2Ering_2Ering(A_27a))
=> ap(c_2EringNorm_2Er__spolynom__normalize(A_27a),V0r) = ap(c_2Ecanonical_2Espolynom__normalize(A_27a),ap(c_2Ering_2Esemi__ring__of(A_27a),V0r)) ) ) ).
fof(ax_thm_2EringNorm_2Er__interp__cs__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0r] :
( mem(V0r,ty_2Ering_2Ering(A_27a))
=> ap(c_2EringNorm_2Er__interp__cs(A_27a),V0r) = ap(c_2Ecanonical_2Einterp__cs(A_27a),ap(c_2Ering_2Esemi__ring__of(A_27a),V0r)) ) ) ).
fof(ax_thm_2EringNorm_2Er__ics__aux__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0r] :
( mem(V0r,ty_2Ering_2Ering(A_27a))
=> ap(c_2EringNorm_2Er__ics__aux(A_27a),V0r) = ap(c_2Ecanonical_2Eics__aux(A_27a),ap(c_2Ering_2Esemi__ring__of(A_27a),V0r)) ) ) ).
fof(ax_thm_2EringNorm_2Er__interp__m__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0r] :
( mem(V0r,ty_2Ering_2Ering(A_27a))
=> ap(c_2EringNorm_2Er__interp__m(A_27a),V0r) = ap(c_2Ecanonical_2Einterp__m(A_27a),ap(c_2Ering_2Esemi__ring__of(A_27a),V0r)) ) ) ).
fof(ax_thm_2EringNorm_2Er__interp__vl__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0r] :
( mem(V0r,ty_2Ering_2Ering(A_27a))
=> ap(c_2EringNorm_2Er__interp__vl(A_27a),V0r) = ap(c_2Ecanonical_2Einterp__vl(A_27a),ap(c_2Ering_2Esemi__ring__of(A_27a),V0r)) ) ) ).
fof(ax_thm_2EringNorm_2Er__ivl__aux__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0r] :
( mem(V0r,ty_2Ering_2Ering(A_27a))
=> ap(c_2EringNorm_2Er__ivl__aux(A_27a),V0r) = ap(c_2Ecanonical_2Eivl__aux(A_27a),ap(c_2Ering_2Esemi__ring__of(A_27a),V0r)) ) ) ).
fof(ax_thm_2EringNorm_2Er__canonical__sum__simplify__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0r] :
( mem(V0r,ty_2Ering_2Ering(A_27a))
=> ap(c_2EringNorm_2Er__canonical__sum__simplify(A_27a),V0r) = ap(c_2Ecanonical_2Ecanonical__sum__simplify(A_27a),ap(c_2Ering_2Esemi__ring__of(A_27a),V0r)) ) ) ).
fof(ax_thm_2EringNorm_2Er__canonical__sum__prod__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0r] :
( mem(V0r,ty_2Ering_2Ering(A_27a))
=> ap(c_2EringNorm_2Er__canonical__sum__prod(A_27a),V0r) = ap(c_2Ecanonical_2Ecanonical__sum__prod(A_27a),ap(c_2Ering_2Esemi__ring__of(A_27a),V0r)) ) ) ).
fof(ax_thm_2EringNorm_2Er__canonical__sum__scalar3__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0r] :
( mem(V0r,ty_2Ering_2Ering(A_27a))
=> ap(c_2EringNorm_2Er__canonical__sum__scalar3(A_27a),V0r) = ap(c_2Ecanonical_2Ecanonical__sum__scalar3(A_27a),ap(c_2Ering_2Esemi__ring__of(A_27a),V0r)) ) ) ).
fof(ax_thm_2EringNorm_2Er__canonical__sum__scalar2__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0r] :
( mem(V0r,ty_2Ering_2Ering(A_27a))
=> ap(c_2EringNorm_2Er__canonical__sum__scalar2(A_27a),V0r) = ap(c_2Ecanonical_2Ecanonical__sum__scalar2(A_27a),ap(c_2Ering_2Esemi__ring__of(A_27a),V0r)) ) ) ).
fof(ax_thm_2EringNorm_2Er__canonical__sum__scalar__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0r] :
( mem(V0r,ty_2Ering_2Ering(A_27a))
=> ap(c_2EringNorm_2Er__canonical__sum__scalar(A_27a),V0r) = ap(c_2Ecanonical_2Ecanonical__sum__scalar(A_27a),ap(c_2Ering_2Esemi__ring__of(A_27a),V0r)) ) ) ).
fof(ax_thm_2EringNorm_2Er__varlist__insert__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0r] :
( mem(V0r,ty_2Ering_2Ering(A_27a))
=> ap(c_2EringNorm_2Er__varlist__insert(A_27a),V0r) = ap(c_2Ecanonical_2Evarlist__insert(A_27a),ap(c_2Ering_2Esemi__ring__of(A_27a),V0r)) ) ) ).
fof(ax_thm_2EringNorm_2Er__monom__insert__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0r] :
( mem(V0r,ty_2Ering_2Ering(A_27a))
=> ap(c_2EringNorm_2Er__monom__insert(A_27a),V0r) = ap(c_2Ecanonical_2Emonom__insert(A_27a),ap(c_2Ering_2Esemi__ring__of(A_27a),V0r)) ) ) ).
fof(ax_thm_2EringNorm_2Er__canonical__sum__merge__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0r] :
( mem(V0r,ty_2Ering_2Ering(A_27a))
=> ap(c_2EringNorm_2Er__canonical__sum__merge(A_27a),V0r) = ap(c_2Ecanonical_2Ecanonical__sum__merge(A_27a),ap(c_2Ering_2Esemi__ring__of(A_27a),V0r)) ) ) ).
fof(conj_thm_2EringNorm_2Einterp__sp__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0r] :
( mem(V0r,ty_2Ering_2Ering(A_27a))
=> ( ! [V1vm] :
( mem(V1vm,ty_2Equote_2Evarmap(A_27a))
=> ! [V2c] :
( mem(V2c,A_27a)
=> ap(ap(ap(c_2EringNorm_2Er__interp__sp(A_27a),V0r),V1vm),ap(c_2Ecanonical_2ESPconst(A_27a),V2c)) = V2c ) )
& ! [V3vm] :
( mem(V3vm,ty_2Equote_2Evarmap(A_27a))
=> ! [V4i] :
( mem(V4i,ty_2Equote_2Eindex)
=> ap(ap(ap(c_2EringNorm_2Er__interp__sp(A_27a),V0r),V3vm),ap(c_2Ecanonical_2ESPvar(A_27a),V4i)) = ap(ap(c_2Equote_2Evarmap__find(A_27a),V4i),V3vm) ) )
& ! [V5vm] :
( mem(V5vm,ty_2Equote_2Evarmap(A_27a))
=> ! [V6p1] :
( mem(V6p1,ty_2Ecanonical_2Espolynom(A_27a))
=> ! [V7p2] :
( mem(V7p2,ty_2Ecanonical_2Espolynom(A_27a))
=> ap(ap(ap(c_2EringNorm_2Er__interp__sp(A_27a),V0r),V5vm),ap(ap(c_2Ecanonical_2ESPplus(A_27a),V6p1),V7p2)) = ap(ap(ap(c_2Ering_2Ering__RP(A_27a),V0r),ap(ap(ap(c_2EringNorm_2Er__interp__sp(A_27a),V0r),V5vm),V6p1)),ap(ap(ap(c_2EringNorm_2Er__interp__sp(A_27a),V0r),V5vm),V7p2)) ) ) )
& ! [V8vm] :
( mem(V8vm,ty_2Equote_2Evarmap(A_27a))
=> ! [V9p1] :
( mem(V9p1,ty_2Ecanonical_2Espolynom(A_27a))
=> ! [V10p2] :
( mem(V10p2,ty_2Ecanonical_2Espolynom(A_27a))
=> ap(ap(ap(c_2EringNorm_2Er__interp__sp(A_27a),V0r),V8vm),ap(ap(c_2Ecanonical_2ESPmult(A_27a),V9p1),V10p2)) = ap(ap(ap(c_2Ering_2Ering__RM(A_27a),V0r),ap(ap(ap(c_2EringNorm_2Er__interp__sp(A_27a),V0r),V8vm),V9p1)),ap(ap(ap(c_2EringNorm_2Er__interp__sp(A_27a),V0r),V8vm),V10p2)) ) ) ) ) ) ) ).
fof(conj_thm_2EringNorm_2Ecanonical__sum__merge__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0r] :
( mem(V0r,ty_2Ering_2Ering(A_27a))
=> ( ! [V1t2] :
( mem(V1t2,ty_2Ecanonical_2Ecanonical__sum(A_27a))
=> ! [V2t1] :
( mem(V2t1,ty_2Ecanonical_2Ecanonical__sum(A_27a))
=> ! [V3l2] :
( mem(V3l2,ty_2Elist_2Elist(ty_2Equote_2Eindex))
=> ! [V4l1] :
( mem(V4l1,ty_2Elist_2Elist(ty_2Equote_2Eindex))
=> ! [V5c2] :
( mem(V5c2,A_27a)
=> ! [V6c1] :
( mem(V6c1,A_27a)
=> ap(ap(ap(c_2EringNorm_2Er__canonical__sum__merge(A_27a),V0r),ap(ap(ap(c_2Ecanonical_2ECons__monom(A_27a),V6c1),V4l1),V2t1)),ap(ap(ap(c_2Ecanonical_2ECons__monom(A_27a),V5c2),V3l2),V1t2)) = ap(ap(ap(ap(c_2EternaryComparisons_2Eordering__CASE(ty_2Ecanonical_2Ecanonical__sum(A_27a)),ap(ap(ap(c_2EternaryComparisons_2Elist__compare(ty_2Equote_2Eindex,ty_2Equote_2Eindex),c_2Equote_2Eindex__compare),V4l1),V3l2)),ap(ap(ap(c_2Ecanonical_2ECons__monom(A_27a),V6c1),V4l1),ap(ap(ap(c_2EringNorm_2Er__canonical__sum__merge(A_27a),V0r),V2t1),ap(ap(ap(c_2Ecanonical_2ECons__monom(A_27a),V5c2),V3l2),V1t2)))),ap(ap(ap(c_2Ecanonical_2ECons__monom(A_27a),ap(ap(ap(c_2Ering_2Ering__RP(A_27a),V0r),V6c1),V5c2)),V4l1),ap(ap(ap(c_2EringNorm_2Er__canonical__sum__merge(A_27a),V0r),V2t1),V1t2))),ap(ap(ap(c_2Ecanonical_2ECons__monom(A_27a),V5c2),V3l2),ap(ap(ap(c_2EringNorm_2Er__canonical__sum__merge(A_27a),V0r),ap(ap(ap(c_2Ecanonical_2ECons__monom(A_27a),V6c1),V4l1),V2t1)),V1t2))) ) ) ) ) ) )
& ! [V7t2] :
( mem(V7t2,ty_2Ecanonical_2Ecanonical__sum(A_27a))
=> ! [V8t1] :
( mem(V8t1,ty_2Ecanonical_2Ecanonical__sum(A_27a))
=> ! [V9l2] :
( mem(V9l2,ty_2Elist_2Elist(ty_2Equote_2Eindex))
=> ! [V10l1] :
( mem(V10l1,ty_2Elist_2Elist(ty_2Equote_2Eindex))
=> ! [V11c1] :
( mem(V11c1,A_27a)
=> ap(ap(ap(c_2EringNorm_2Er__canonical__sum__merge(A_27a),V0r),ap(ap(ap(c_2Ecanonical_2ECons__monom(A_27a),V11c1),V10l1),V8t1)),ap(ap(c_2Ecanonical_2ECons__varlist(A_27a),V9l2),V7t2)) = ap(ap(ap(ap(c_2EternaryComparisons_2Eordering__CASE(ty_2Ecanonical_2Ecanonical__sum(A_27a)),ap(ap(ap(c_2EternaryComparisons_2Elist__compare(ty_2Equote_2Eindex,ty_2Equote_2Eindex),c_2Equote_2Eindex__compare),V10l1),V9l2)),ap(ap(ap(c_2Ecanonical_2ECons__monom(A_27a),V11c1),V10l1),ap(ap(ap(c_2EringNorm_2Er__canonical__sum__merge(A_27a),V0r),V8t1),ap(ap(c_2Ecanonical_2ECons__varlist(A_27a),V9l2),V7t2)))),ap(ap(ap(c_2Ecanonical_2ECons__monom(A_27a),ap(ap(ap(c_2Ering_2Ering__RP(A_27a),V0r),V11c1),ap(c_2Ering_2Ering__R1(A_27a),V0r))),V10l1),ap(ap(ap(c_2EringNorm_2Er__canonical__sum__merge(A_27a),V0r),V8t1),V7t2))),ap(ap(c_2Ecanonical_2ECons__varlist(A_27a),V9l2),ap(ap(ap(c_2EringNorm_2Er__canonical__sum__merge(A_27a),V0r),ap(ap(ap(c_2Ecanonical_2ECons__monom(A_27a),V11c1),V10l1),V8t1)),V7t2))) ) ) ) ) )
& ! [V12t2] :
( mem(V12t2,ty_2Ecanonical_2Ecanonical__sum(A_27a))
=> ! [V13t1] :
( mem(V13t1,ty_2Ecanonical_2Ecanonical__sum(A_27a))
=> ! [V14l2] :
( mem(V14l2,ty_2Elist_2Elist(ty_2Equote_2Eindex))
=> ! [V15l1] :
( mem(V15l1,ty_2Elist_2Elist(ty_2Equote_2Eindex))
=> ! [V16c2] :
( mem(V16c2,A_27a)
=> ap(ap(ap(c_2EringNorm_2Er__canonical__sum__merge(A_27a),V0r),ap(ap(c_2Ecanonical_2ECons__varlist(A_27a),V15l1),V13t1)),ap(ap(ap(c_2Ecanonical_2ECons__monom(A_27a),V16c2),V14l2),V12t2)) = ap(ap(ap(ap(c_2EternaryComparisons_2Eordering__CASE(ty_2Ecanonical_2Ecanonical__sum(A_27a)),ap(ap(ap(c_2EternaryComparisons_2Elist__compare(ty_2Equote_2Eindex,ty_2Equote_2Eindex),c_2Equote_2Eindex__compare),V15l1),V14l2)),ap(ap(c_2Ecanonical_2ECons__varlist(A_27a),V15l1),ap(ap(ap(c_2EringNorm_2Er__canonical__sum__merge(A_27a),V0r),V13t1),ap(ap(ap(c_2Ecanonical_2ECons__monom(A_27a),V16c2),V14l2),V12t2)))),ap(ap(ap(c_2Ecanonical_2ECons__monom(A_27a),ap(ap(ap(c_2Ering_2Ering__RP(A_27a),V0r),ap(c_2Ering_2Ering__R1(A_27a),V0r)),V16c2)),V15l1),ap(ap(ap(c_2EringNorm_2Er__canonical__sum__merge(A_27a),V0r),V13t1),V12t2))),ap(ap(ap(c_2Ecanonical_2ECons__monom(A_27a),V16c2),V14l2),ap(ap(ap(c_2EringNorm_2Er__canonical__sum__merge(A_27a),V0r),ap(ap(c_2Ecanonical_2ECons__varlist(A_27a),V15l1),V13t1)),V12t2))) ) ) ) ) )
& ! [V17t2] :
( mem(V17t2,ty_2Ecanonical_2Ecanonical__sum(A_27a))
=> ! [V18t1] :
( mem(V18t1,ty_2Ecanonical_2Ecanonical__sum(A_27a))
=> ! [V19l2] :
( mem(V19l2,ty_2Elist_2Elist(ty_2Equote_2Eindex))
=> ! [V20l1] :
( mem(V20l1,ty_2Elist_2Elist(ty_2Equote_2Eindex))
=> ap(ap(ap(c_2EringNorm_2Er__canonical__sum__merge(A_27a),V0r),ap(ap(c_2Ecanonical_2ECons__varlist(A_27a),V20l1),V18t1)),ap(ap(c_2Ecanonical_2ECons__varlist(A_27a),V19l2),V17t2)) = ap(ap(ap(ap(c_2EternaryComparisons_2Eordering__CASE(ty_2Ecanonical_2Ecanonical__sum(A_27a)),ap(ap(ap(c_2EternaryComparisons_2Elist__compare(ty_2Equote_2Eindex,ty_2Equote_2Eindex),c_2Equote_2Eindex__compare),V20l1),V19l2)),ap(ap(c_2Ecanonical_2ECons__varlist(A_27a),V20l1),ap(ap(ap(c_2EringNorm_2Er__canonical__sum__merge(A_27a),V0r),V18t1),ap(ap(c_2Ecanonical_2ECons__varlist(A_27a),V19l2),V17t2)))),ap(ap(ap(c_2Ecanonical_2ECons__monom(A_27a),ap(ap(ap(c_2Ering_2Ering__RP(A_27a),V0r),ap(c_2Ering_2Ering__R1(A_27a),V0r)),ap(c_2Ering_2Ering__R1(A_27a),V0r))),V20l1),ap(ap(ap(c_2EringNorm_2Er__canonical__sum__merge(A_27a),V0r),V18t1),V17t2))),ap(ap(c_2Ecanonical_2ECons__varlist(A_27a),V19l2),ap(ap(ap(c_2EringNorm_2Er__canonical__sum__merge(A_27a),V0r),ap(ap(c_2Ecanonical_2ECons__varlist(A_27a),V20l1),V18t1)),V17t2))) ) ) ) )
& ! [V21s1] :
( mem(V21s1,ty_2Ecanonical_2Ecanonical__sum(A_27a))
=> ap(ap(ap(c_2EringNorm_2Er__canonical__sum__merge(A_27a),V0r),V21s1),c_2Ecanonical_2ENil__monom(A_27a)) = V21s1 )
& ! [V22v6] :
( mem(V22v6,ty_2Ecanonical_2Ecanonical__sum(A_27a))
=> ! [V23v5] :
( mem(V23v5,ty_2Elist_2Elist(ty_2Equote_2Eindex))
=> ! [V24v4] :
( mem(V24v4,A_27a)
=> ap(ap(ap(c_2EringNorm_2Er__canonical__sum__merge(A_27a),V0r),c_2Ecanonical_2ENil__monom(A_27a)),ap(ap(ap(c_2Ecanonical_2ECons__monom(A_27a),V24v4),V23v5),V22v6)) = ap(ap(ap(c_2Ecanonical_2ECons__monom(A_27a),V24v4),V23v5),V22v6) ) ) )
& ! [V25v8] :
( mem(V25v8,ty_2Ecanonical_2Ecanonical__sum(A_27a))
=> ! [V26v7] :
( mem(V26v7,ty_2Elist_2Elist(ty_2Equote_2Eindex))
=> ap(ap(ap(c_2EringNorm_2Er__canonical__sum__merge(A_27a),V0r),c_2Ecanonical_2ENil__monom(A_27a)),ap(ap(c_2Ecanonical_2ECons__varlist(A_27a),V26v7),V25v8)) = ap(ap(c_2Ecanonical_2ECons__varlist(A_27a),V26v7),V25v8) ) ) ) ) ) ).
fof(conj_thm_2EringNorm_2Emonom__insert__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0r] :
( mem(V0r,ty_2Ering_2Ering(A_27a))
=> ( ! [V1t2] :
( mem(V1t2,ty_2Ecanonical_2Ecanonical__sum(A_27a))
=> ! [V2l2] :
( mem(V2l2,ty_2Elist_2Elist(ty_2Equote_2Eindex))
=> ! [V3l1] :
( mem(V3l1,ty_2Elist_2Elist(ty_2Equote_2Eindex))
=> ! [V4c2] :
( mem(V4c2,A_27a)
=> ! [V5c1] :
( mem(V5c1,A_27a)
=> ap(ap(ap(ap(c_2EringNorm_2Er__monom__insert(A_27a),V0r),V5c1),V3l1),ap(ap(ap(c_2Ecanonical_2ECons__monom(A_27a),V4c2),V2l2),V1t2)) = ap(ap(ap(ap(c_2EternaryComparisons_2Eordering__CASE(ty_2Ecanonical_2Ecanonical__sum(A_27a)),ap(ap(ap(c_2EternaryComparisons_2Elist__compare(ty_2Equote_2Eindex,ty_2Equote_2Eindex),c_2Equote_2Eindex__compare),V3l1),V2l2)),ap(ap(ap(c_2Ecanonical_2ECons__monom(A_27a),V5c1),V3l1),ap(ap(ap(c_2Ecanonical_2ECons__monom(A_27a),V4c2),V2l2),V1t2))),ap(ap(ap(c_2Ecanonical_2ECons__monom(A_27a),ap(ap(ap(c_2Ering_2Ering__RP(A_27a),V0r),V5c1),V4c2)),V3l1),V1t2)),ap(ap(ap(c_2Ecanonical_2ECons__monom(A_27a),V4c2),V2l2),ap(ap(ap(ap(c_2EringNorm_2Er__monom__insert(A_27a),V0r),V5c1),V3l1),V1t2))) ) ) ) ) )
& ! [V6t2] :
( mem(V6t2,ty_2Ecanonical_2Ecanonical__sum(A_27a))
=> ! [V7l2] :
( mem(V7l2,ty_2Elist_2Elist(ty_2Equote_2Eindex))
=> ! [V8l1] :
( mem(V8l1,ty_2Elist_2Elist(ty_2Equote_2Eindex))
=> ! [V9c1] :
( mem(V9c1,A_27a)
=> ap(ap(ap(ap(c_2EringNorm_2Er__monom__insert(A_27a),V0r),V9c1),V8l1),ap(ap(c_2Ecanonical_2ECons__varlist(A_27a),V7l2),V6t2)) = ap(ap(ap(ap(c_2EternaryComparisons_2Eordering__CASE(ty_2Ecanonical_2Ecanonical__sum(A_27a)),ap(ap(ap(c_2EternaryComparisons_2Elist__compare(ty_2Equote_2Eindex,ty_2Equote_2Eindex),c_2Equote_2Eindex__compare),V8l1),V7l2)),ap(ap(ap(c_2Ecanonical_2ECons__monom(A_27a),V9c1),V8l1),ap(ap(c_2Ecanonical_2ECons__varlist(A_27a),V7l2),V6t2))),ap(ap(ap(c_2Ecanonical_2ECons__monom(A_27a),ap(ap(ap(c_2Ering_2Ering__RP(A_27a),V0r),V9c1),ap(c_2Ering_2Ering__R1(A_27a),V0r))),V8l1),V6t2)),ap(ap(c_2Ecanonical_2ECons__varlist(A_27a),V7l2),ap(ap(ap(ap(c_2EringNorm_2Er__monom__insert(A_27a),V0r),V9c1),V8l1),V6t2))) ) ) ) )
& ! [V10l1] :
( mem(V10l1,ty_2Elist_2Elist(ty_2Equote_2Eindex))
=> ! [V11c1] :
( mem(V11c1,A_27a)
=> ap(ap(ap(ap(c_2EringNorm_2Er__monom__insert(A_27a),V0r),V11c1),V10l1),c_2Ecanonical_2ENil__monom(A_27a)) = ap(ap(ap(c_2Ecanonical_2ECons__monom(A_27a),V11c1),V10l1),c_2Ecanonical_2ENil__monom(A_27a)) ) ) ) ) ) ).
fof(conj_thm_2EringNorm_2Evarlist__insert__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0r] :
( mem(V0r,ty_2Ering_2Ering(A_27a))
=> ( ! [V1t2] :
( mem(V1t2,ty_2Ecanonical_2Ecanonical__sum(A_27a))
=> ! [V2l2] :
( mem(V2l2,ty_2Elist_2Elist(ty_2Equote_2Eindex))
=> ! [V3l1] :
( mem(V3l1,ty_2Elist_2Elist(ty_2Equote_2Eindex))
=> ! [V4c2] :
( mem(V4c2,A_27a)
=> ap(ap(ap(c_2EringNorm_2Er__varlist__insert(A_27a),V0r),V3l1),ap(ap(ap(c_2Ecanonical_2ECons__monom(A_27a),V4c2),V2l2),V1t2)) = ap(ap(ap(ap(c_2EternaryComparisons_2Eordering__CASE(ty_2Ecanonical_2Ecanonical__sum(A_27a)),ap(ap(ap(c_2EternaryComparisons_2Elist__compare(ty_2Equote_2Eindex,ty_2Equote_2Eindex),c_2Equote_2Eindex__compare),V3l1),V2l2)),ap(ap(c_2Ecanonical_2ECons__varlist(A_27a),V3l1),ap(ap(ap(c_2Ecanonical_2ECons__monom(A_27a),V4c2),V2l2),V1t2))),ap(ap(ap(c_2Ecanonical_2ECons__monom(A_27a),ap(ap(ap(c_2Ering_2Ering__RP(A_27a),V0r),ap(c_2Ering_2Ering__R1(A_27a),V0r)),V4c2)),V3l1),V1t2)),ap(ap(ap(c_2Ecanonical_2ECons__monom(A_27a),V4c2),V2l2),ap(ap(ap(c_2EringNorm_2Er__varlist__insert(A_27a),V0r),V3l1),V1t2))) ) ) ) )
& ! [V5t2] :
( mem(V5t2,ty_2Ecanonical_2Ecanonical__sum(A_27a))
=> ! [V6l2] :
( mem(V6l2,ty_2Elist_2Elist(ty_2Equote_2Eindex))
=> ! [V7l1] :
( mem(V7l1,ty_2Elist_2Elist(ty_2Equote_2Eindex))
=> ap(ap(ap(c_2EringNorm_2Er__varlist__insert(A_27a),V0r),V7l1),ap(ap(c_2Ecanonical_2ECons__varlist(A_27a),V6l2),V5t2)) = ap(ap(ap(ap(c_2EternaryComparisons_2Eordering__CASE(ty_2Ecanonical_2Ecanonical__sum(A_27a)),ap(ap(ap(c_2EternaryComparisons_2Elist__compare(ty_2Equote_2Eindex,ty_2Equote_2Eindex),c_2Equote_2Eindex__compare),V7l1),V6l2)),ap(ap(c_2Ecanonical_2ECons__varlist(A_27a),V7l1),ap(ap(c_2Ecanonical_2ECons__varlist(A_27a),V6l2),V5t2))),ap(ap(ap(c_2Ecanonical_2ECons__monom(A_27a),ap(ap(ap(c_2Ering_2Ering__RP(A_27a),V0r),ap(c_2Ering_2Ering__R1(A_27a),V0r)),ap(c_2Ering_2Ering__R1(A_27a),V0r))),V7l1),V5t2)),ap(ap(c_2Ecanonical_2ECons__varlist(A_27a),V6l2),ap(ap(ap(c_2EringNorm_2Er__varlist__insert(A_27a),V0r),V7l1),V5t2))) ) ) )
& ! [V8l1] :
( mem(V8l1,ty_2Elist_2Elist(ty_2Equote_2Eindex))
=> ap(ap(ap(c_2EringNorm_2Er__varlist__insert(A_27a),V0r),V8l1),c_2Ecanonical_2ENil__monom(A_27a)) = ap(ap(c_2Ecanonical_2ECons__varlist(A_27a),V8l1),c_2Ecanonical_2ENil__monom(A_27a)) ) ) ) ) ).
fof(conj_thm_2EringNorm_2Ecanonical__sum__scalar__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0r] :
( mem(V0r,ty_2Ering_2Ering(A_27a))
=> ( ! [V1c0] :
( mem(V1c0,A_27a)
=> ! [V2c] :
( mem(V2c,A_27a)
=> ! [V3l] :
( mem(V3l,ty_2Elist_2Elist(ty_2Equote_2Eindex))
=> ! [V4t] :
( mem(V4t,ty_2Ecanonical_2Ecanonical__sum(A_27a))
=> ap(ap(ap(c_2EringNorm_2Er__canonical__sum__scalar(A_27a),V0r),V1c0),ap(ap(ap(c_2Ecanonical_2ECons__monom(A_27a),V2c),V3l),V4t)) = ap(ap(ap(c_2Ecanonical_2ECons__monom(A_27a),ap(ap(ap(c_2Ering_2Ering__RM(A_27a),V0r),V1c0),V2c)),V3l),ap(ap(ap(c_2EringNorm_2Er__canonical__sum__scalar(A_27a),V0r),V1c0),V4t)) ) ) ) )
& ! [V5c0] :
( mem(V5c0,A_27a)
=> ! [V6l] :
( mem(V6l,ty_2Elist_2Elist(ty_2Equote_2Eindex))
=> ! [V7t] :
( mem(V7t,ty_2Ecanonical_2Ecanonical__sum(A_27a))
=> ap(ap(ap(c_2EringNorm_2Er__canonical__sum__scalar(A_27a),V0r),V5c0),ap(ap(c_2Ecanonical_2ECons__varlist(A_27a),V6l),V7t)) = ap(ap(ap(c_2Ecanonical_2ECons__monom(A_27a),V5c0),V6l),ap(ap(ap(c_2EringNorm_2Er__canonical__sum__scalar(A_27a),V0r),V5c0),V7t)) ) ) )
& ! [V8c0] :
( mem(V8c0,A_27a)
=> ap(ap(ap(c_2EringNorm_2Er__canonical__sum__scalar(A_27a),V0r),V8c0),c_2Ecanonical_2ENil__monom(A_27a)) = c_2Ecanonical_2ENil__monom(A_27a) ) ) ) ) ).
fof(conj_thm_2EringNorm_2Ecanonical__sum__scalar2__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0r] :
( mem(V0r,ty_2Ering_2Ering(A_27a))
=> ( ! [V1l0] :
( mem(V1l0,ty_2Elist_2Elist(ty_2Equote_2Eindex))
=> ! [V2c] :
( mem(V2c,A_27a)
=> ! [V3l] :
( mem(V3l,ty_2Elist_2Elist(ty_2Equote_2Eindex))
=> ! [V4t] :
( mem(V4t,ty_2Ecanonical_2Ecanonical__sum(A_27a))
=> ap(ap(ap(c_2EringNorm_2Er__canonical__sum__scalar2(A_27a),V0r),V1l0),ap(ap(ap(c_2Ecanonical_2ECons__monom(A_27a),V2c),V3l),V4t)) = ap(ap(ap(ap(c_2EringNorm_2Er__monom__insert(A_27a),V0r),V2c),ap(ap(ap(c_2EternaryComparisons_2Elist__merge(ty_2Equote_2Eindex),c_2Equote_2Eindex__lt),V1l0),V3l)),ap(ap(ap(c_2EringNorm_2Er__canonical__sum__scalar2(A_27a),V0r),V1l0),V4t)) ) ) ) )
& ! [V5l0] :
( mem(V5l0,ty_2Elist_2Elist(ty_2Equote_2Eindex))
=> ! [V6l] :
( mem(V6l,ty_2Elist_2Elist(ty_2Equote_2Eindex))
=> ! [V7t] :
( mem(V7t,ty_2Ecanonical_2Ecanonical__sum(A_27a))
=> ap(ap(ap(c_2EringNorm_2Er__canonical__sum__scalar2(A_27a),V0r),V5l0),ap(ap(c_2Ecanonical_2ECons__varlist(A_27a),V6l),V7t)) = ap(ap(ap(c_2EringNorm_2Er__varlist__insert(A_27a),V0r),ap(ap(ap(c_2EternaryComparisons_2Elist__merge(ty_2Equote_2Eindex),c_2Equote_2Eindex__lt),V5l0),V6l)),ap(ap(ap(c_2EringNorm_2Er__canonical__sum__scalar2(A_27a),V0r),V5l0),V7t)) ) ) )
& ! [V8l0] :
( mem(V8l0,ty_2Elist_2Elist(ty_2Equote_2Eindex))
=> ap(ap(ap(c_2EringNorm_2Er__canonical__sum__scalar2(A_27a),V0r),V8l0),c_2Ecanonical_2ENil__monom(A_27a)) = c_2Ecanonical_2ENil__monom(A_27a) ) ) ) ) ).
fof(conj_thm_2EringNorm_2Ecanonical__sum__scalar3__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0r] :
( mem(V0r,ty_2Ering_2Ering(A_27a))
=> ( ! [V1c0] :
( mem(V1c0,A_27a)
=> ! [V2l0] :
( mem(V2l0,ty_2Elist_2Elist(ty_2Equote_2Eindex))
=> ! [V3c] :
( mem(V3c,A_27a)
=> ! [V4l] :
( mem(V4l,ty_2Elist_2Elist(ty_2Equote_2Eindex))
=> ! [V5t] :
( mem(V5t,ty_2Ecanonical_2Ecanonical__sum(A_27a))
=> ap(ap(ap(ap(c_2EringNorm_2Er__canonical__sum__scalar3(A_27a),V0r),V1c0),V2l0),ap(ap(ap(c_2Ecanonical_2ECons__monom(A_27a),V3c),V4l),V5t)) = ap(ap(ap(ap(c_2EringNorm_2Er__monom__insert(A_27a),V0r),ap(ap(ap(c_2Ering_2Ering__RM(A_27a),V0r),V1c0),V3c)),ap(ap(ap(c_2EternaryComparisons_2Elist__merge(ty_2Equote_2Eindex),c_2Equote_2Eindex__lt),V2l0),V4l)),ap(ap(ap(ap(c_2EringNorm_2Er__canonical__sum__scalar3(A_27a),V0r),V1c0),V2l0),V5t)) ) ) ) ) )
& ! [V6c0] :
( mem(V6c0,A_27a)
=> ! [V7l0] :
( mem(V7l0,ty_2Elist_2Elist(ty_2Equote_2Eindex))
=> ! [V8l] :
( mem(V8l,ty_2Elist_2Elist(ty_2Equote_2Eindex))
=> ! [V9t] :
( mem(V9t,ty_2Ecanonical_2Ecanonical__sum(A_27a))
=> ap(ap(ap(ap(c_2EringNorm_2Er__canonical__sum__scalar3(A_27a),V0r),V6c0),V7l0),ap(ap(c_2Ecanonical_2ECons__varlist(A_27a),V8l),V9t)) = ap(ap(ap(ap(c_2EringNorm_2Er__monom__insert(A_27a),V0r),V6c0),ap(ap(ap(c_2EternaryComparisons_2Elist__merge(ty_2Equote_2Eindex),c_2Equote_2Eindex__lt),V7l0),V8l)),ap(ap(ap(ap(c_2EringNorm_2Er__canonical__sum__scalar3(A_27a),V0r),V6c0),V7l0),V9t)) ) ) ) )
& ! [V10c0] :
( mem(V10c0,A_27a)
=> ! [V11l0] :
( mem(V11l0,ty_2Elist_2Elist(ty_2Equote_2Eindex))
=> ap(ap(ap(ap(c_2EringNorm_2Er__canonical__sum__scalar3(A_27a),V0r),V10c0),V11l0),c_2Ecanonical_2ENil__monom(A_27a)) = c_2Ecanonical_2ENil__monom(A_27a) ) ) ) ) ) ).
fof(conj_thm_2EringNorm_2Ecanonical__sum__prod__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0r] :
( mem(V0r,ty_2Ering_2Ering(A_27a))
=> ( ! [V1c1] :
( mem(V1c1,A_27a)
=> ! [V2l1] :
( mem(V2l1,ty_2Elist_2Elist(ty_2Equote_2Eindex))
=> ! [V3t1] :
( mem(V3t1,ty_2Ecanonical_2Ecanonical__sum(A_27a))
=> ! [V4s2] :
( mem(V4s2,ty_2Ecanonical_2Ecanonical__sum(A_27a))
=> ap(ap(ap(c_2EringNorm_2Er__canonical__sum__prod(A_27a),V0r),ap(ap(ap(c_2Ecanonical_2ECons__monom(A_27a),V1c1),V2l1),V3t1)),V4s2) = ap(ap(ap(c_2EringNorm_2Er__canonical__sum__merge(A_27a),V0r),ap(ap(ap(ap(c_2EringNorm_2Er__canonical__sum__scalar3(A_27a),V0r),V1c1),V2l1),V4s2)),ap(ap(ap(c_2EringNorm_2Er__canonical__sum__prod(A_27a),V0r),V3t1),V4s2)) ) ) ) )
& ! [V5l1] :
( mem(V5l1,ty_2Elist_2Elist(ty_2Equote_2Eindex))
=> ! [V6t1] :
( mem(V6t1,ty_2Ecanonical_2Ecanonical__sum(A_27a))
=> ! [V7s2] :
( mem(V7s2,ty_2Ecanonical_2Ecanonical__sum(A_27a))
=> ap(ap(ap(c_2EringNorm_2Er__canonical__sum__prod(A_27a),V0r),ap(ap(c_2Ecanonical_2ECons__varlist(A_27a),V5l1),V6t1)),V7s2) = ap(ap(ap(c_2EringNorm_2Er__canonical__sum__merge(A_27a),V0r),ap(ap(ap(c_2EringNorm_2Er__canonical__sum__scalar2(A_27a),V0r),V5l1),V7s2)),ap(ap(ap(c_2EringNorm_2Er__canonical__sum__prod(A_27a),V0r),V6t1),V7s2)) ) ) )
& ! [V8s2] :
( mem(V8s2,ty_2Ecanonical_2Ecanonical__sum(A_27a))
=> ap(ap(ap(c_2EringNorm_2Er__canonical__sum__prod(A_27a),V0r),c_2Ecanonical_2ENil__monom(A_27a)),V8s2) = c_2Ecanonical_2ENil__monom(A_27a) ) ) ) ) ).
fof(conj_thm_2EringNorm_2Ecanonical__sum__simplify__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0r] :
( mem(V0r,ty_2Ering_2Ering(A_27a))
=> ( ! [V1c] :
( mem(V1c,A_27a)
=> ! [V2l] :
( mem(V2l,ty_2Elist_2Elist(ty_2Equote_2Eindex))
=> ! [V3t] :
( mem(V3t,ty_2Ecanonical_2Ecanonical__sum(A_27a))
=> ap(ap(c_2EringNorm_2Er__canonical__sum__simplify(A_27a),V0r),ap(ap(ap(c_2Ecanonical_2ECons__monom(A_27a),V1c),V2l),V3t)) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Ecanonical_2Ecanonical__sum(A_27a)),ap(ap(c_2Emin_2E_3D(A_27a),V1c),ap(c_2Ering_2Ering__R0(A_27a),V0r))),ap(ap(c_2EringNorm_2Er__canonical__sum__simplify(A_27a),V0r),V3t)),ap(ap(ap(c_2Ebool_2ECOND(ty_2Ecanonical_2Ecanonical__sum(A_27a)),ap(ap(c_2Emin_2E_3D(A_27a),V1c),ap(c_2Ering_2Ering__R1(A_27a),V0r))),ap(ap(c_2Ecanonical_2ECons__varlist(A_27a),V2l),ap(ap(c_2EringNorm_2Er__canonical__sum__simplify(A_27a),V0r),V3t))),ap(ap(ap(c_2Ecanonical_2ECons__monom(A_27a),V1c),V2l),ap(ap(c_2EringNorm_2Er__canonical__sum__simplify(A_27a),V0r),V3t)))) ) ) )
& ! [V4l] :
( mem(V4l,ty_2Elist_2Elist(ty_2Equote_2Eindex))
=> ! [V5t] :
( mem(V5t,ty_2Ecanonical_2Ecanonical__sum(A_27a))
=> ap(ap(c_2EringNorm_2Er__canonical__sum__simplify(A_27a),V0r),ap(ap(c_2Ecanonical_2ECons__varlist(A_27a),V4l),V5t)) = ap(ap(c_2Ecanonical_2ECons__varlist(A_27a),V4l),ap(ap(c_2EringNorm_2Er__canonical__sum__simplify(A_27a),V0r),V5t)) ) )
& ap(ap(c_2EringNorm_2Er__canonical__sum__simplify(A_27a),V0r),c_2Ecanonical_2ENil__monom(A_27a)) = c_2Ecanonical_2ENil__monom(A_27a) ) ) ) ).
fof(conj_thm_2EringNorm_2Eivl__aux__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0r] :
( mem(V0r,ty_2Ering_2Ering(A_27a))
=> ( ! [V1vm] :
( mem(V1vm,ty_2Equote_2Evarmap(A_27a))
=> ! [V2x] :
( mem(V2x,ty_2Equote_2Eindex)
=> ap(ap(ap(ap(c_2EringNorm_2Er__ivl__aux(A_27a),V0r),V1vm),V2x),c_2Elist_2ENIL(ty_2Equote_2Eindex)) = ap(ap(c_2Equote_2Evarmap__find(A_27a),V2x),V1vm) ) )
& ! [V3vm] :
( mem(V3vm,ty_2Equote_2Evarmap(A_27a))
=> ! [V4x] :
( mem(V4x,ty_2Equote_2Eindex)
=> ! [V5x_27] :
( mem(V5x_27,ty_2Equote_2Eindex)
=> ! [V6t_27] :
( mem(V6t_27,ty_2Elist_2Elist(ty_2Equote_2Eindex))
=> ap(ap(ap(ap(c_2EringNorm_2Er__ivl__aux(A_27a),V0r),V3vm),V4x),ap(ap(c_2Elist_2ECONS(ty_2Equote_2Eindex),V5x_27),V6t_27)) = ap(ap(ap(c_2Ering_2Ering__RM(A_27a),V0r),ap(ap(c_2Equote_2Evarmap__find(A_27a),V4x),V3vm)),ap(ap(ap(ap(c_2EringNorm_2Er__ivl__aux(A_27a),V0r),V3vm),V5x_27),V6t_27)) ) ) ) ) ) ) ) ).
fof(conj_thm_2EringNorm_2Einterp__vl__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0r] :
( mem(V0r,ty_2Ering_2Ering(A_27a))
=> ( ! [V1vm] :
( mem(V1vm,ty_2Equote_2Evarmap(A_27a))
=> ap(ap(ap(c_2EringNorm_2Er__interp__vl(A_27a),V0r),V1vm),c_2Elist_2ENIL(ty_2Equote_2Eindex)) = ap(c_2Ering_2Ering__R1(A_27a),V0r) )
& ! [V2vm] :
( mem(V2vm,ty_2Equote_2Evarmap(A_27a))
=> ! [V3x] :
( mem(V3x,ty_2Equote_2Eindex)
=> ! [V4t] :
( mem(V4t,ty_2Elist_2Elist(ty_2Equote_2Eindex))
=> ap(ap(ap(c_2EringNorm_2Er__interp__vl(A_27a),V0r),V2vm),ap(ap(c_2Elist_2ECONS(ty_2Equote_2Eindex),V3x),V4t)) = ap(ap(ap(ap(c_2EringNorm_2Er__ivl__aux(A_27a),V0r),V2vm),V3x),V4t) ) ) ) ) ) ) ).
fof(conj_thm_2EringNorm_2Einterp__m__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0r] :
( mem(V0r,ty_2Ering_2Ering(A_27a))
=> ( ! [V1vm] :
( mem(V1vm,ty_2Equote_2Evarmap(A_27a))
=> ! [V2c] :
( mem(V2c,A_27a)
=> ap(ap(ap(ap(c_2EringNorm_2Er__interp__m(A_27a),V0r),V1vm),V2c),c_2Elist_2ENIL(ty_2Equote_2Eindex)) = V2c ) )
& ! [V3vm] :
( mem(V3vm,ty_2Equote_2Evarmap(A_27a))
=> ! [V4c] :
( mem(V4c,A_27a)
=> ! [V5x] :
( mem(V5x,ty_2Equote_2Eindex)
=> ! [V6t] :
( mem(V6t,ty_2Elist_2Elist(ty_2Equote_2Eindex))
=> ap(ap(ap(ap(c_2EringNorm_2Er__interp__m(A_27a),V0r),V3vm),V4c),ap(ap(c_2Elist_2ECONS(ty_2Equote_2Eindex),V5x),V6t)) = ap(ap(ap(c_2Ering_2Ering__RM(A_27a),V0r),V4c),ap(ap(ap(ap(c_2EringNorm_2Er__ivl__aux(A_27a),V0r),V3vm),V5x),V6t)) ) ) ) ) ) ) ) ).
fof(conj_thm_2EringNorm_2Eics__aux__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0r] :
( mem(V0r,ty_2Ering_2Ering(A_27a))
=> ( ! [V1vm] :
( mem(V1vm,ty_2Equote_2Evarmap(A_27a))
=> ! [V2a] :
( mem(V2a,A_27a)
=> ap(ap(ap(ap(c_2EringNorm_2Er__ics__aux(A_27a),V0r),V1vm),V2a),c_2Ecanonical_2ENil__monom(A_27a)) = V2a ) )
& ! [V3vm] :
( mem(V3vm,ty_2Equote_2Evarmap(A_27a))
=> ! [V4a] :
( mem(V4a,A_27a)
=> ! [V5l] :
( mem(V5l,ty_2Elist_2Elist(ty_2Equote_2Eindex))
=> ! [V6t] :
( mem(V6t,ty_2Ecanonical_2Ecanonical__sum(A_27a))
=> ap(ap(ap(ap(c_2EringNorm_2Er__ics__aux(A_27a),V0r),V3vm),V4a),ap(ap(c_2Ecanonical_2ECons__varlist(A_27a),V5l),V6t)) = ap(ap(ap(c_2Ering_2Ering__RP(A_27a),V0r),V4a),ap(ap(ap(ap(c_2EringNorm_2Er__ics__aux(A_27a),V0r),V3vm),ap(ap(ap(c_2EringNorm_2Er__interp__vl(A_27a),V0r),V3vm),V5l)),V6t)) ) ) ) )
& ! [V7vm] :
( mem(V7vm,ty_2Equote_2Evarmap(A_27a))
=> ! [V8a] :
( mem(V8a,A_27a)
=> ! [V9c] :
( mem(V9c,A_27a)
=> ! [V10l] :
( mem(V10l,ty_2Elist_2Elist(ty_2Equote_2Eindex))
=> ! [V11t] :
( mem(V11t,ty_2Ecanonical_2Ecanonical__sum(A_27a))
=> ap(ap(ap(ap(c_2EringNorm_2Er__ics__aux(A_27a),V0r),V7vm),V8a),ap(ap(ap(c_2Ecanonical_2ECons__monom(A_27a),V9c),V10l),V11t)) = ap(ap(ap(c_2Ering_2Ering__RP(A_27a),V0r),V8a),ap(ap(ap(ap(c_2EringNorm_2Er__ics__aux(A_27a),V0r),V7vm),ap(ap(ap(ap(c_2EringNorm_2Er__interp__m(A_27a),V0r),V7vm),V9c),V10l)),V11t)) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2EringNorm_2Einterp__cs__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0r] :
( mem(V0r,ty_2Ering_2Ering(A_27a))
=> ( ! [V1vm] :
( mem(V1vm,ty_2Equote_2Evarmap(A_27a))
=> ap(ap(ap(c_2EringNorm_2Er__interp__cs(A_27a),V0r),V1vm),c_2Ecanonical_2ENil__monom(A_27a)) = ap(c_2Ering_2Ering__R0(A_27a),V0r) )
& ! [V2vm] :
( mem(V2vm,ty_2Equote_2Evarmap(A_27a))
=> ! [V3l] :
( mem(V3l,ty_2Elist_2Elist(ty_2Equote_2Eindex))
=> ! [V4t] :
( mem(V4t,ty_2Ecanonical_2Ecanonical__sum(A_27a))
=> ap(ap(ap(c_2EringNorm_2Er__interp__cs(A_27a),V0r),V2vm),ap(ap(c_2Ecanonical_2ECons__varlist(A_27a),V3l),V4t)) = ap(ap(ap(ap(c_2EringNorm_2Er__ics__aux(A_27a),V0r),V2vm),ap(ap(ap(c_2EringNorm_2Er__interp__vl(A_27a),V0r),V2vm),V3l)),V4t) ) ) )
& ! [V5vm] :
( mem(V5vm,ty_2Equote_2Evarmap(A_27a))
=> ! [V6c] :
( mem(V6c,A_27a)
=> ! [V7l] :
( mem(V7l,ty_2Elist_2Elist(ty_2Equote_2Eindex))
=> ! [V8t] :
( mem(V8t,ty_2Ecanonical_2Ecanonical__sum(A_27a))
=> ap(ap(ap(c_2EringNorm_2Er__interp__cs(A_27a),V0r),V5vm),ap(ap(ap(c_2Ecanonical_2ECons__monom(A_27a),V6c),V7l),V8t)) = ap(ap(ap(ap(c_2EringNorm_2Er__ics__aux(A_27a),V0r),V5vm),ap(ap(ap(ap(c_2EringNorm_2Er__interp__m(A_27a),V0r),V5vm),V6c),V7l)),V8t) ) ) ) ) ) ) ) ).
fof(conj_thm_2EringNorm_2Espolynom__normalize__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0r] :
( mem(V0r,ty_2Ering_2Ering(A_27a))
=> ( ! [V1i] :
( mem(V1i,ty_2Equote_2Eindex)
=> ap(ap(c_2EringNorm_2Er__spolynom__normalize(A_27a),V0r),ap(c_2Ecanonical_2ESPvar(A_27a),V1i)) = ap(ap(c_2Ecanonical_2ECons__varlist(A_27a),ap(ap(c_2Elist_2ECONS(ty_2Equote_2Eindex),V1i),c_2Elist_2ENIL(ty_2Equote_2Eindex))),c_2Ecanonical_2ENil__monom(A_27a)) )
& ! [V2c] :
( mem(V2c,A_27a)
=> ap(ap(c_2EringNorm_2Er__spolynom__normalize(A_27a),V0r),ap(c_2Ecanonical_2ESPconst(A_27a),V2c)) = ap(ap(ap(c_2Ecanonical_2ECons__monom(A_27a),V2c),c_2Elist_2ENIL(ty_2Equote_2Eindex)),c_2Ecanonical_2ENil__monom(A_27a)) )
& ! [V3l] :
( mem(V3l,ty_2Ecanonical_2Espolynom(A_27a))
=> ! [V4r_27] :
( mem(V4r_27,ty_2Ecanonical_2Espolynom(A_27a))
=> ap(ap(c_2EringNorm_2Er__spolynom__normalize(A_27a),V0r),ap(ap(c_2Ecanonical_2ESPplus(A_27a),V3l),V4r_27)) = ap(ap(ap(c_2EringNorm_2Er__canonical__sum__merge(A_27a),V0r),ap(ap(c_2EringNorm_2Er__spolynom__normalize(A_27a),V0r),V3l)),ap(ap(c_2EringNorm_2Er__spolynom__normalize(A_27a),V0r),V4r_27)) ) )
& ! [V5l] :
( mem(V5l,ty_2Ecanonical_2Espolynom(A_27a))
=> ! [V6r_27] :
( mem(V6r_27,ty_2Ecanonical_2Espolynom(A_27a))
=> ap(ap(c_2EringNorm_2Er__spolynom__normalize(A_27a),V0r),ap(ap(c_2Ecanonical_2ESPmult(A_27a),V5l),V6r_27)) = ap(ap(ap(c_2EringNorm_2Er__canonical__sum__prod(A_27a),V0r),ap(ap(c_2EringNorm_2Er__spolynom__normalize(A_27a),V0r),V5l)),ap(ap(c_2EringNorm_2Er__spolynom__normalize(A_27a),V0r),V6r_27)) ) ) ) ) ) ).
fof(conj_thm_2EringNorm_2Espolynom__simplify__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0r] :
( mem(V0r,ty_2Ering_2Ering(A_27a))
=> ! [V1x] :
( mem(V1x,ty_2Ecanonical_2Espolynom(A_27a))
=> ap(ap(c_2EringNorm_2Er__spolynom__simplify(A_27a),V0r),V1x) = ap(ap(c_2EringNorm_2Er__canonical__sum__simplify(A_27a),V0r),ap(ap(c_2EringNorm_2Er__spolynom__normalize(A_27a),V0r),V1x)) ) ) ) ).
fof(lameq_f2659,axiom,
! [A_27a,V3a0_27] :
( mem(V3a0_27,ty_2Eind__type_2Erecspace(ty_2Epair_2Eprod(ty_2Equote_2Eindex,A_27a)))
=> ! [V10a0] :
( mem(V10a0,ty_2Eind__type_2Erecspace(ty_2Epair_2Eprod(ty_2Equote_2Eindex,A_27a)))
=> ! [V2_27polynom_27] :
( mem(V2_27polynom_27,arr(ty_2Eind__type_2Erecspace(ty_2Epair_2Eprod(ty_2Equote_2Eindex,A_27a)),bool))
=> ! [V11a1] : ap(f2659(A_27a,V3a0_27,V10a0,V2_27polynom_27),V11a1) = ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Emin_2E_3D(ty_2Eind__type_2Erecspace(ty_2Epair_2Eprod(ty_2Equote_2Eindex,A_27a))),V3a0_27),ap(ap(f2501(A_27a),V10a0),V11a1))),ap(ap(c_2Ebool_2E_2F_5C,ap(V2_27polynom_27,V10a0)),ap(V2_27polynom_27,V11a1))) ) ) ) ).
fof(lameq_f2660,axiom,
! [A_27a,V2_27polynom_27] :
( mem(V2_27polynom_27,arr(ty_2Eind__type_2Erecspace(ty_2Epair_2Eprod(ty_2Equote_2Eindex,A_27a)),bool))
=> ! [V3a0_27] :
( mem(V3a0_27,ty_2Eind__type_2Erecspace(ty_2Epair_2Eprod(ty_2Equote_2Eindex,A_27a)))
=> ! [V10a0] : ap(f2660(A_27a,V2_27polynom_27,V3a0_27),V10a0) = ap(c_2Ebool_2E_3F(ty_2Eind__type_2Erecspace(ty_2Epair_2Eprod(ty_2Equote_2Eindex,A_27a))),f2659(A_27a,V3a0_27,V10a0,V2_27polynom_27)) ) ) ).
fof(lameq_f2661,axiom,
! [A_27a,V3a0_27] :
( mem(V3a0_27,ty_2Eind__type_2Erecspace(ty_2Epair_2Eprod(ty_2Equote_2Eindex,A_27a)))
=> ! [V15a0] :
( mem(V15a0,ty_2Eind__type_2Erecspace(ty_2Epair_2Eprod(ty_2Equote_2Eindex,A_27a)))
=> ! [V2_27polynom_27] :
( mem(V2_27polynom_27,arr(ty_2Eind__type_2Erecspace(ty_2Epair_2Eprod(ty_2Equote_2Eindex,A_27a)),bool))
=> ! [V16a1] : ap(f2661(A_27a,V3a0_27,V15a0,V2_27polynom_27),V16a1) = ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Emin_2E_3D(ty_2Eind__type_2Erecspace(ty_2Epair_2Eprod(ty_2Equote_2Eindex,A_27a))),V3a0_27),ap(ap(f2505(A_27a),V15a0),V16a1))),ap(ap(c_2Ebool_2E_2F_5C,ap(V2_27polynom_27,V15a0)),ap(V2_27polynom_27,V16a1))) ) ) ) ).
fof(lameq_f2662,axiom,
! [A_27a,V2_27polynom_27] :
( mem(V2_27polynom_27,arr(ty_2Eind__type_2Erecspace(ty_2Epair_2Eprod(ty_2Equote_2Eindex,A_27a)),bool))
=> ! [V3a0_27] :
( mem(V3a0_27,ty_2Eind__type_2Erecspace(ty_2Epair_2Eprod(ty_2Equote_2Eindex,A_27a)))
=> ! [V15a0] : ap(f2662(A_27a,V2_27polynom_27,V3a0_27),V15a0) = ap(c_2Ebool_2E_3F(ty_2Eind__type_2Erecspace(ty_2Epair_2Eprod(ty_2Equote_2Eindex,A_27a))),f2661(A_27a,V3a0_27,V15a0,V2_27polynom_27)) ) ) ).
fof(lameq_f2663,axiom,
! [A_27a,V21a] : ap(f2663(A_27a),V21a) = ap(ap(ap(c_2Eind__type_2ECONSTR(ty_2Epair_2Eprod(ty_2Equote_2Eindex,A_27a)),ap(c_2Enum_2ESUC,ap(c_2Enum_2ESUC,ap(c_2Enum_2ESUC,ap(c_2Enum_2ESUC,c_2Enum_2E0))))),ap(ap(c_2Epair_2E_2C(ty_2Equote_2Eindex,A_27a),c_2Ebool_2EARB(ty_2Equote_2Eindex)),c_2Ebool_2EARB(A_27a))),ap(ap(c_2Eind__type_2EFCONS(ty_2Eind__type_2Erecspace(ty_2Epair_2Eprod(ty_2Equote_2Eindex,A_27a))),V21a),k(ty_2Enum_2Enum,c_2Eind__type_2EBOTTOM(ty_2Epair_2Eprod(ty_2Equote_2Eindex,A_27a))))) ).
fof(lameq_f2664,axiom,
! [A_27a,V3a0_27] :
( mem(V3a0_27,ty_2Eind__type_2Erecspace(ty_2Epair_2Eprod(ty_2Equote_2Eindex,A_27a)))
=> ! [V2_27polynom_27] :
( mem(V2_27polynom_27,arr(ty_2Eind__type_2Erecspace(ty_2Epair_2Eprod(ty_2Equote_2Eindex,A_27a)),bool))
=> ! [V20a] : ap(f2664(A_27a,V3a0_27,V2_27polynom_27),V20a) = ap(ap(c_2Ebool_2E_2F_5C,ap(ap(c_2Emin_2E_3D(ty_2Eind__type_2Erecspace(ty_2Epair_2Eprod(ty_2Equote_2Eindex,A_27a))),V3a0_27),ap(f2663(A_27a),V20a))),ap(V2_27polynom_27,V20a)) ) ) ).
fof(lameq_f2665,axiom,
! [A_27a,V2_27polynom_27] :
( mem(V2_27polynom_27,arr(ty_2Eind__type_2Erecspace(ty_2Epair_2Eprod(ty_2Equote_2Eindex,A_27a)),bool))
=> ! [V3a0_27] : ap(f2665(A_27a,V2_27polynom_27),V3a0_27) = ap(ap(c_2Emin_2E_3D_3D_3E,ap(ap(c_2Ebool_2E_5C_2F,ap(c_2Ebool_2E_3F(ty_2Equote_2Eindex),f2497(A_27a,V3a0_27))),ap(ap(c_2Ebool_2E_5C_2F,ap(c_2Ebool_2E_3F(A_27a),f2499(A_27a,V3a0_27))),ap(ap(c_2Ebool_2E_5C_2F,ap(c_2Ebool_2E_3F(ty_2Eind__type_2Erecspace(ty_2Epair_2Eprod(ty_2Equote_2Eindex,A_27a))),f2660(A_27a,V2_27polynom_27,V3a0_27))),ap(ap(c_2Ebool_2E_5C_2F,ap(c_2Ebool_2E_3F(ty_2Eind__type_2Erecspace(ty_2Epair_2Eprod(ty_2Equote_2Eindex,A_27a))),f2662(A_27a,V2_27polynom_27,V3a0_27))),ap(c_2Ebool_2E_3F(ty_2Eind__type_2Erecspace(ty_2Epair_2Eprod(ty_2Equote_2Eindex,A_27a))),f2664(A_27a,V3a0_27,V2_27polynom_27))))))),ap(V2_27polynom_27,V3a0_27)) ) ).
fof(lameq_f2666,axiom,
! [A_27a,V1a0_27] :
( mem(V1a0_27,ty_2Eind__type_2Erecspace(ty_2Epair_2Eprod(ty_2Equote_2Eindex,A_27a)))
=> ! [V2_27polynom_27] : ap(f2666(A_27a,V1a0_27),V2_27polynom_27) = ap(ap(c_2Emin_2E_3D_3D_3E,ap(c_2Ebool_2E_21(ty_2Eind__type_2Erecspace(ty_2Epair_2Eprod(ty_2Equote_2Eindex,A_27a))),f2665(A_27a,V2_27polynom_27))),ap(V2_27polynom_27,V1a0_27)) ) ).
fof(lameq_f2667,axiom,
! [A_27a,V1a0_27] : ap(f2667(A_27a),V1a0_27) = ap(c_2Ebool_2E_21(arr(ty_2Eind__type_2Erecspace(ty_2Epair_2Eprod(ty_2Equote_2Eindex,A_27a)),bool)),f2666(A_27a,V1a0_27)) ).
fof(ax_thm_2EringNorm_2Epolynom__TY__DEF,axiom,
! [A_27a] :
( ne(A_27a)
=> ? [V0rep] :
( mem(V0rep,arr(ty_2EringNorm_2Epolynom(A_27a),ty_2Eind__type_2Erecspace(ty_2Epair_2Eprod(ty_2Equote_2Eindex,A_27a))))
& p(ap(ap(c_2Ebool_2ETYPE__DEFINITION(ty_2Eind__type_2Erecspace(ty_2Epair_2Eprod(ty_2Equote_2Eindex,A_27a)),ty_2EringNorm_2Epolynom(A_27a)),f2667(A_27a)),V0rep)) ) ) ).
fof(ax_thm_2EringNorm_2Epolynom__case__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ( ! [V0a] :
( mem(V0a,ty_2Equote_2Eindex)
=> ! [V1f] :
( mem(V1f,arr(ty_2Equote_2Eindex,A_27b))
=> ! [V2f1] :
( mem(V2f1,arr(A_27a,A_27b))
=> ! [V3f2] :
( mem(V3f2,arr(ty_2EringNorm_2Epolynom(A_27a),arr(ty_2EringNorm_2Epolynom(A_27a),A_27b)))
=> ! [V4f3] :
( mem(V4f3,arr(ty_2EringNorm_2Epolynom(A_27a),arr(ty_2EringNorm_2Epolynom(A_27a),A_27b)))
=> ! [V5f4] :
( mem(V5f4,arr(ty_2EringNorm_2Epolynom(A_27a),A_27b))
=> ap(ap(ap(ap(ap(ap(c_2EringNorm_2Epolynom__CASE(A_27a,A_27b),ap(c_2EringNorm_2EPvar(A_27a),V0a)),V1f),V2f1),V3f2),V4f3),V5f4) = ap(V1f,V0a) ) ) ) ) ) )
& ! [V6a] :
( mem(V6a,A_27a)
=> ! [V7f] :
( mem(V7f,arr(ty_2Equote_2Eindex,A_27b))
=> ! [V8f1] :
( mem(V8f1,arr(A_27a,A_27b))
=> ! [V9f2] :
( mem(V9f2,arr(ty_2EringNorm_2Epolynom(A_27a),arr(ty_2EringNorm_2Epolynom(A_27a),A_27b)))
=> ! [V10f3] :
( mem(V10f3,arr(ty_2EringNorm_2Epolynom(A_27a),arr(ty_2EringNorm_2Epolynom(A_27a),A_27b)))
=> ! [V11f4] :
( mem(V11f4,arr(ty_2EringNorm_2Epolynom(A_27a),A_27b))
=> ap(ap(ap(ap(ap(ap(c_2EringNorm_2Epolynom__CASE(A_27a,A_27b),ap(c_2EringNorm_2EPconst(A_27a),V6a)),V7f),V8f1),V9f2),V10f3),V11f4) = ap(V8f1,V6a) ) ) ) ) ) )
& ! [V12a0] :
( mem(V12a0,ty_2EringNorm_2Epolynom(A_27a))
=> ! [V13a1] :
( mem(V13a1,ty_2EringNorm_2Epolynom(A_27a))
=> ! [V14f] :
( mem(V14f,arr(ty_2Equote_2Eindex,A_27b))
=> ! [V15f1] :
( mem(V15f1,arr(A_27a,A_27b))
=> ! [V16f2] :
( mem(V16f2,arr(ty_2EringNorm_2Epolynom(A_27a),arr(ty_2EringNorm_2Epolynom(A_27a),A_27b)))
=> ! [V17f3] :
( mem(V17f3,arr(ty_2EringNorm_2Epolynom(A_27a),arr(ty_2EringNorm_2Epolynom(A_27a),A_27b)))
=> ! [V18f4] :
( mem(V18f4,arr(ty_2EringNorm_2Epolynom(A_27a),A_27b))
=> ap(ap(ap(ap(ap(ap(c_2EringNorm_2Epolynom__CASE(A_27a,A_27b),ap(ap(c_2EringNorm_2EPplus(A_27a),V12a0),V13a1)),V14f),V15f1),V16f2),V17f3),V18f4) = ap(ap(V16f2,V12a0),V13a1) ) ) ) ) ) ) )
& ! [V19a0] :
( mem(V19a0,ty_2EringNorm_2Epolynom(A_27a))
=> ! [V20a1] :
( mem(V20a1,ty_2EringNorm_2Epolynom(A_27a))
=> ! [V21f] :
( mem(V21f,arr(ty_2Equote_2Eindex,A_27b))
=> ! [V22f1] :
( mem(V22f1,arr(A_27a,A_27b))
=> ! [V23f2] :
( mem(V23f2,arr(ty_2EringNorm_2Epolynom(A_27a),arr(ty_2EringNorm_2Epolynom(A_27a),A_27b)))
=> ! [V24f3] :
( mem(V24f3,arr(ty_2EringNorm_2Epolynom(A_27a),arr(ty_2EringNorm_2Epolynom(A_27a),A_27b)))
=> ! [V25f4] :
( mem(V25f4,arr(ty_2EringNorm_2Epolynom(A_27a),A_27b))
=> ap(ap(ap(ap(ap(ap(c_2EringNorm_2Epolynom__CASE(A_27a,A_27b),ap(ap(c_2EringNorm_2EPmult(A_27a),V19a0),V20a1)),V21f),V22f1),V23f2),V24f3),V25f4) = ap(ap(V24f3,V19a0),V20a1) ) ) ) ) ) ) )
& ! [V26a] :
( mem(V26a,ty_2EringNorm_2Epolynom(A_27a))
=> ! [V27f] :
( mem(V27f,arr(ty_2Equote_2Eindex,A_27b))
=> ! [V28f1] :
( mem(V28f1,arr(A_27a,A_27b))
=> ! [V29f2] :
( mem(V29f2,arr(ty_2EringNorm_2Epolynom(A_27a),arr(ty_2EringNorm_2Epolynom(A_27a),A_27b)))
=> ! [V30f3] :
( mem(V30f3,arr(ty_2EringNorm_2Epolynom(A_27a),arr(ty_2EringNorm_2Epolynom(A_27a),A_27b)))
=> ! [V31f4] :
( mem(V31f4,arr(ty_2EringNorm_2Epolynom(A_27a),A_27b))
=> ap(ap(ap(ap(ap(ap(c_2EringNorm_2Epolynom__CASE(A_27a,A_27b),ap(c_2EringNorm_2EPopp(A_27a),V26a)),V27f),V28f1),V29f2),V30f3),V31f4) = ap(V31f4,V26a) ) ) ) ) ) ) ) ) ) ).
fof(ax_thm_2EringNorm_2Epolynom__size__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ( ! [V0f] :
( mem(V0f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V1a] :
( mem(V1a,ty_2Equote_2Eindex)
=> ap(ap(c_2EringNorm_2Epolynom__size(A_27a),V0f),ap(c_2EringNorm_2EPvar(A_27a),V1a)) = ap(ap(c_2Earithmetic_2E_2B,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))),ap(c_2Equote_2Eindex__size,V1a)) ) )
& ! [V2f] :
( mem(V2f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V3a] :
( mem(V3a,A_27a)
=> ap(ap(c_2EringNorm_2Epolynom__size(A_27a),V2f),ap(c_2EringNorm_2EPconst(A_27a),V3a)) = ap(ap(c_2Earithmetic_2E_2B,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))),ap(V2f,V3a)) ) )
& ! [V4f] :
( mem(V4f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V5a0] :
( mem(V5a0,ty_2EringNorm_2Epolynom(A_27a))
=> ! [V6a1] :
( mem(V6a1,ty_2EringNorm_2Epolynom(A_27a))
=> ap(ap(c_2EringNorm_2Epolynom__size(A_27a),V4f),ap(ap(c_2EringNorm_2EPplus(A_27a),V5a0),V6a1)) = ap(ap(c_2Earithmetic_2E_2B,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))),ap(ap(c_2Earithmetic_2E_2B,ap(ap(c_2EringNorm_2Epolynom__size(A_27a),V4f),V5a0)),ap(ap(c_2EringNorm_2Epolynom__size(A_27a),V4f),V6a1))) ) ) )
& ! [V7f] :
( mem(V7f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V8a0] :
( mem(V8a0,ty_2EringNorm_2Epolynom(A_27a))
=> ! [V9a1] :
( mem(V9a1,ty_2EringNorm_2Epolynom(A_27a))
=> ap(ap(c_2EringNorm_2Epolynom__size(A_27a),V7f),ap(ap(c_2EringNorm_2EPmult(A_27a),V8a0),V9a1)) = ap(ap(c_2Earithmetic_2E_2B,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))),ap(ap(c_2Earithmetic_2E_2B,ap(ap(c_2EringNorm_2Epolynom__size(A_27a),V7f),V8a0)),ap(ap(c_2EringNorm_2Epolynom__size(A_27a),V7f),V9a1))) ) ) )
& ! [V10f] :
( mem(V10f,arr(A_27a,ty_2Enum_2Enum))
=> ! [V11a] :
( mem(V11a,ty_2EringNorm_2Epolynom(A_27a))
=> ap(ap(c_2EringNorm_2Epolynom__size(A_27a),V10f),ap(c_2EringNorm_2EPopp(A_27a),V11a)) = ap(ap(c_2Earithmetic_2E_2B,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))),ap(ap(c_2EringNorm_2Epolynom__size(A_27a),V10f),V11a)) ) ) ) ) ).
fof(conj_thm_2EringNorm_2Edatatype__polynom,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0polynom] :
( mem(V0polynom,arr(arr(ty_2Equote_2Eindex,ty_2EringNorm_2Epolynom(A_27a)),arr(arr(A_27a,ty_2EringNorm_2Epolynom(A_27a)),arr(arr(ty_2EringNorm_2Epolynom(A_27a),arr(ty_2EringNorm_2Epolynom(A_27a),ty_2EringNorm_2Epolynom(A_27a))),arr(arr(ty_2EringNorm_2Epolynom(A_27a),arr(ty_2EringNorm_2Epolynom(A_27a),ty_2EringNorm_2Epolynom(A_27a))),arr(arr(ty_2EringNorm_2Epolynom(A_27a),ty_2EringNorm_2Epolynom(A_27a)),bool))))))
=> p(ap(c_2Ebool_2EDATATYPE(bool),ap(ap(ap(ap(ap(V0polynom,c_2EringNorm_2EPvar(A_27a)),c_2EringNorm_2EPconst(A_27a)),c_2EringNorm_2EPplus(A_27a)),c_2EringNorm_2EPmult(A_27a)),c_2EringNorm_2EPopp(A_27a)))) ) ) ).
fof(conj_thm_2EringNorm_2Epolynom__11,axiom,
! [A_27a] :
( ne(A_27a)
=> ( ! [V0a] :
( mem(V0a,ty_2Equote_2Eindex)
=> ! [V1a_27] :
( mem(V1a_27,ty_2Equote_2Eindex)
=> ( ap(c_2EringNorm_2EPvar(A_27a),V0a) = ap(c_2EringNorm_2EPvar(A_27a),V1a_27)
<=> V0a = V1a_27 ) ) )
& ! [V2a] :
( mem(V2a,A_27a)
=> ! [V3a_27] :
( mem(V3a_27,A_27a)
=> ( ap(c_2EringNorm_2EPconst(A_27a),V2a) = ap(c_2EringNorm_2EPconst(A_27a),V3a_27)
<=> V2a = V3a_27 ) ) )
& ! [V4a0] :
( mem(V4a0,ty_2EringNorm_2Epolynom(A_27a))
=> ! [V5a1] :
( mem(V5a1,ty_2EringNorm_2Epolynom(A_27a))
=> ! [V6a0_27] :
( mem(V6a0_27,ty_2EringNorm_2Epolynom(A_27a))
=> ! [V7a1_27] :
( mem(V7a1_27,ty_2EringNorm_2Epolynom(A_27a))
=> ( ap(ap(c_2EringNorm_2EPplus(A_27a),V4a0),V5a1) = ap(ap(c_2EringNorm_2EPplus(A_27a),V6a0_27),V7a1_27)
<=> ( V4a0 = V6a0_27
& V5a1 = V7a1_27 ) ) ) ) ) )
& ! [V8a0] :
( mem(V8a0,ty_2EringNorm_2Epolynom(A_27a))
=> ! [V9a1] :
( mem(V9a1,ty_2EringNorm_2Epolynom(A_27a))
=> ! [V10a0_27] :
( mem(V10a0_27,ty_2EringNorm_2Epolynom(A_27a))
=> ! [V11a1_27] :
( mem(V11a1_27,ty_2EringNorm_2Epolynom(A_27a))
=> ( ap(ap(c_2EringNorm_2EPmult(A_27a),V8a0),V9a1) = ap(ap(c_2EringNorm_2EPmult(A_27a),V10a0_27),V11a1_27)
<=> ( V8a0 = V10a0_27
& V9a1 = V11a1_27 ) ) ) ) ) )
& ! [V12a] :
( mem(V12a,ty_2EringNorm_2Epolynom(A_27a))
=> ! [V13a_27] :
( mem(V13a_27,ty_2EringNorm_2Epolynom(A_27a))
=> ( ap(c_2EringNorm_2EPopp(A_27a),V12a) = ap(c_2EringNorm_2EPopp(A_27a),V13a_27)
<=> V12a = V13a_27 ) ) ) ) ) ).
fof(conj_thm_2EringNorm_2Epolynom__distinct,axiom,
! [A_27a] :
( ne(A_27a)
=> ( ! [V0a_27] :
( mem(V0a_27,A_27a)
=> ! [V1a] :
( mem(V1a,ty_2Equote_2Eindex)
=> ap(c_2EringNorm_2EPvar(A_27a),V1a) != ap(c_2EringNorm_2EPconst(A_27a),V0a_27) ) )
& ! [V2a1] :
( mem(V2a1,ty_2EringNorm_2Epolynom(A_27a))
=> ! [V3a0] :
( mem(V3a0,ty_2EringNorm_2Epolynom(A_27a))
=> ! [V4a] :
( mem(V4a,ty_2Equote_2Eindex)
=> ap(c_2EringNorm_2EPvar(A_27a),V4a) != ap(ap(c_2EringNorm_2EPplus(A_27a),V3a0),V2a1) ) ) )
& ! [V5a1] :
( mem(V5a1,ty_2EringNorm_2Epolynom(A_27a))
=> ! [V6a0] :
( mem(V6a0,ty_2EringNorm_2Epolynom(A_27a))
=> ! [V7a] :
( mem(V7a,ty_2Equote_2Eindex)
=> ap(c_2EringNorm_2EPvar(A_27a),V7a) != ap(ap(c_2EringNorm_2EPmult(A_27a),V6a0),V5a1) ) ) )
& ! [V8a_27] :
( mem(V8a_27,ty_2EringNorm_2Epolynom(A_27a))
=> ! [V9a] :
( mem(V9a,ty_2Equote_2Eindex)
=> ap(c_2EringNorm_2EPvar(A_27a),V9a) != ap(c_2EringNorm_2EPopp(A_27a),V8a_27) ) )
& ! [V10a1] :
( mem(V10a1,ty_2EringNorm_2Epolynom(A_27a))
=> ! [V11a0] :
( mem(V11a0,ty_2EringNorm_2Epolynom(A_27a))
=> ! [V12a] :
( mem(V12a,A_27a)
=> ap(c_2EringNorm_2EPconst(A_27a),V12a) != ap(ap(c_2EringNorm_2EPplus(A_27a),V11a0),V10a1) ) ) )
& ! [V13a1] :
( mem(V13a1,ty_2EringNorm_2Epolynom(A_27a))
=> ! [V14a0] :
( mem(V14a0,ty_2EringNorm_2Epolynom(A_27a))
=> ! [V15a] :
( mem(V15a,A_27a)
=> ap(c_2EringNorm_2EPconst(A_27a),V15a) != ap(ap(c_2EringNorm_2EPmult(A_27a),V14a0),V13a1) ) ) )
& ! [V16a_27] :
( mem(V16a_27,ty_2EringNorm_2Epolynom(A_27a))
=> ! [V17a] :
( mem(V17a,A_27a)
=> ap(c_2EringNorm_2EPconst(A_27a),V17a) != ap(c_2EringNorm_2EPopp(A_27a),V16a_27) ) )
& ! [V18a1_27] :
( mem(V18a1_27,ty_2EringNorm_2Epolynom(A_27a))
=> ! [V19a1] :
( mem(V19a1,ty_2EringNorm_2Epolynom(A_27a))
=> ! [V20a0_27] :
( mem(V20a0_27,ty_2EringNorm_2Epolynom(A_27a))
=> ! [V21a0] :
( mem(V21a0,ty_2EringNorm_2Epolynom(A_27a))
=> ap(ap(c_2EringNorm_2EPplus(A_27a),V21a0),V19a1) != ap(ap(c_2EringNorm_2EPmult(A_27a),V20a0_27),V18a1_27) ) ) ) )
& ! [V22a1] :
( mem(V22a1,ty_2EringNorm_2Epolynom(A_27a))
=> ! [V23a0] :
( mem(V23a0,ty_2EringNorm_2Epolynom(A_27a))
=> ! [V24a] :
( mem(V24a,ty_2EringNorm_2Epolynom(A_27a))
=> ap(ap(c_2EringNorm_2EPplus(A_27a),V23a0),V22a1) != ap(c_2EringNorm_2EPopp(A_27a),V24a) ) ) )
& ! [V25a1] :
( mem(V25a1,ty_2EringNorm_2Epolynom(A_27a))
=> ! [V26a0] :
( mem(V26a0,ty_2EringNorm_2Epolynom(A_27a))
=> ! [V27a] :
( mem(V27a,ty_2EringNorm_2Epolynom(A_27a))
=> ap(ap(c_2EringNorm_2EPmult(A_27a),V26a0),V25a1) != ap(c_2EringNorm_2EPopp(A_27a),V27a) ) ) ) ) ) ).
fof(conj_thm_2EringNorm_2Epolynom__nchotomy,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0pp] :
( mem(V0pp,ty_2EringNorm_2Epolynom(A_27a))
=> ( ? [V1i] :
( mem(V1i,ty_2Equote_2Eindex)
& V0pp = ap(c_2EringNorm_2EPvar(A_27a),V1i) )
| ? [V2a] :
( mem(V2a,A_27a)
& V0pp = ap(c_2EringNorm_2EPconst(A_27a),V2a) )
| ? [V3p] :
( mem(V3p,ty_2EringNorm_2Epolynom(A_27a))
& ? [V4p0] :
( mem(V4p0,ty_2EringNorm_2Epolynom(A_27a))
& V0pp = ap(ap(c_2EringNorm_2EPplus(A_27a),V3p),V4p0) ) )
| ? [V5p] :
( mem(V5p,ty_2EringNorm_2Epolynom(A_27a))
& ? [V6p0] :
( mem(V6p0,ty_2EringNorm_2Epolynom(A_27a))
& V0pp = ap(ap(c_2EringNorm_2EPmult(A_27a),V5p),V6p0) ) )
| ? [V7p] :
( mem(V7p,ty_2EringNorm_2Epolynom(A_27a))
& V0pp = ap(c_2EringNorm_2EPopp(A_27a),V7p) ) ) ) ) ).
fof(conj_thm_2EringNorm_2Epolynom__Axiom,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f0] :
( mem(V0f0,arr(ty_2Equote_2Eindex,A_27b))
=> ! [V1f1] :
( mem(V1f1,arr(A_27a,A_27b))
=> ! [V2f2] :
( mem(V2f2,arr(ty_2EringNorm_2Epolynom(A_27a),arr(ty_2EringNorm_2Epolynom(A_27a),arr(A_27b,arr(A_27b,A_27b)))))
=> ! [V3f3] :
( mem(V3f3,arr(ty_2EringNorm_2Epolynom(A_27a),arr(ty_2EringNorm_2Epolynom(A_27a),arr(A_27b,arr(A_27b,A_27b)))))
=> ! [V4f4] :
( mem(V4f4,arr(ty_2EringNorm_2Epolynom(A_27a),arr(A_27b,A_27b)))
=> ? [V5fn] :
( mem(V5fn,arr(ty_2EringNorm_2Epolynom(A_27a),A_27b))
& ! [V6a] :
( mem(V6a,ty_2Equote_2Eindex)
=> ap(V5fn,ap(c_2EringNorm_2EPvar(A_27a),V6a)) = ap(V0f0,V6a) )
& ! [V7a] :
( mem(V7a,A_27a)
=> ap(V5fn,ap(c_2EringNorm_2EPconst(A_27a),V7a)) = ap(V1f1,V7a) )
& ! [V8a0] :
( mem(V8a0,ty_2EringNorm_2Epolynom(A_27a))
=> ! [V9a1] :
( mem(V9a1,ty_2EringNorm_2Epolynom(A_27a))
=> ap(V5fn,ap(ap(c_2EringNorm_2EPplus(A_27a),V8a0),V9a1)) = ap(ap(ap(ap(V2f2,V8a0),V9a1),ap(V5fn,V8a0)),ap(V5fn,V9a1)) ) )
& ! [V10a0] :
( mem(V10a0,ty_2EringNorm_2Epolynom(A_27a))
=> ! [V11a1] :
( mem(V11a1,ty_2EringNorm_2Epolynom(A_27a))
=> ap(V5fn,ap(ap(c_2EringNorm_2EPmult(A_27a),V10a0),V11a1)) = ap(ap(ap(ap(V3f3,V10a0),V11a1),ap(V5fn,V10a0)),ap(V5fn,V11a1)) ) )
& ! [V12a] :
( mem(V12a,ty_2EringNorm_2Epolynom(A_27a))
=> ap(V5fn,ap(c_2EringNorm_2EPopp(A_27a),V12a)) = ap(ap(V4f4,V12a),ap(V5fn,V12a)) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2EringNorm_2Epolynom__induction,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(ty_2EringNorm_2Epolynom(A_27a),bool))
=> ( ( ! [V1i] :
( mem(V1i,ty_2Equote_2Eindex)
=> p(ap(V0P,ap(c_2EringNorm_2EPvar(A_27a),V1i))) )
& ! [V2a] :
( mem(V2a,A_27a)
=> p(ap(V0P,ap(c_2EringNorm_2EPconst(A_27a),V2a))) )
& ! [V3p] :
( mem(V3p,ty_2EringNorm_2Epolynom(A_27a))
=> ! [V4p0] :
( mem(V4p0,ty_2EringNorm_2Epolynom(A_27a))
=> ( ( p(ap(V0P,V3p))
& p(ap(V0P,V4p0)) )
=> p(ap(V0P,ap(ap(c_2EringNorm_2EPplus(A_27a),V3p),V4p0))) ) ) )
& ! [V5p] :
( mem(V5p,ty_2EringNorm_2Epolynom(A_27a))
=> ! [V6p0] :
( mem(V6p0,ty_2EringNorm_2Epolynom(A_27a))
=> ( ( p(ap(V0P,V5p))
& p(ap(V0P,V6p0)) )
=> p(ap(V0P,ap(ap(c_2EringNorm_2EPmult(A_27a),V5p),V6p0))) ) ) )
& ! [V7p] :
( mem(V7p,ty_2EringNorm_2Epolynom(A_27a))
=> ( p(ap(V0P,V7p))
=> p(ap(V0P,ap(c_2EringNorm_2EPopp(A_27a),V7p))) ) ) )
=> ! [V8p] :
( mem(V8p,ty_2EringNorm_2Epolynom(A_27a))
=> p(ap(V0P,V8p)) ) ) ) ) ).
fof(conj_thm_2EringNorm_2Epolynom__case__cong,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f_27] :
( mem(V0f_27,arr(ty_2Equote_2Eindex,A_27b))
=> ! [V1f1_27] :
( mem(V1f1_27,arr(A_27a,A_27b))
=> ! [V2f2_27] :
( mem(V2f2_27,arr(ty_2EringNorm_2Epolynom(A_27a),arr(ty_2EringNorm_2Epolynom(A_27a),A_27b)))
=> ! [V3f3_27] :
( mem(V3f3_27,arr(ty_2EringNorm_2Epolynom(A_27a),arr(ty_2EringNorm_2Epolynom(A_27a),A_27b)))
=> ! [V4f4_27] :
( mem(V4f4_27,arr(ty_2EringNorm_2Epolynom(A_27a),A_27b))
=> ! [V5M] :
( mem(V5M,ty_2EringNorm_2Epolynom(A_27a))
=> ! [V6M_27] :
( mem(V6M_27,ty_2EringNorm_2Epolynom(A_27a))
=> ! [V7f] :
( mem(V7f,arr(ty_2Equote_2Eindex,A_27b))
=> ! [V8f1] :
( mem(V8f1,arr(A_27a,A_27b))
=> ! [V9f2] :
( mem(V9f2,arr(ty_2EringNorm_2Epolynom(A_27a),arr(ty_2EringNorm_2Epolynom(A_27a),A_27b)))
=> ! [V10f3] :
( mem(V10f3,arr(ty_2EringNorm_2Epolynom(A_27a),arr(ty_2EringNorm_2Epolynom(A_27a),A_27b)))
=> ! [V11f4] :
( mem(V11f4,arr(ty_2EringNorm_2Epolynom(A_27a),A_27b))
=> ( ( V5M = V6M_27
& ! [V12a] :
( mem(V12a,ty_2Equote_2Eindex)
=> ( V6M_27 = ap(c_2EringNorm_2EPvar(A_27a),V12a)
=> ap(V7f,V12a) = ap(V0f_27,V12a) ) )
& ! [V13a] :
( mem(V13a,A_27a)
=> ( V6M_27 = ap(c_2EringNorm_2EPconst(A_27a),V13a)
=> ap(V8f1,V13a) = ap(V1f1_27,V13a) ) )
& ! [V14a0] :
( mem(V14a0,ty_2EringNorm_2Epolynom(A_27a))
=> ! [V15a1] :
( mem(V15a1,ty_2EringNorm_2Epolynom(A_27a))
=> ( V6M_27 = ap(ap(c_2EringNorm_2EPplus(A_27a),V14a0),V15a1)
=> ap(ap(V9f2,V14a0),V15a1) = ap(ap(V2f2_27,V14a0),V15a1) ) ) )
& ! [V16a0] :
( mem(V16a0,ty_2EringNorm_2Epolynom(A_27a))
=> ! [V17a1] :
( mem(V17a1,ty_2EringNorm_2Epolynom(A_27a))
=> ( V6M_27 = ap(ap(c_2EringNorm_2EPmult(A_27a),V16a0),V17a1)
=> ap(ap(V10f3,V16a0),V17a1) = ap(ap(V3f3_27,V16a0),V17a1) ) ) )
& ! [V18a] :
( mem(V18a,ty_2EringNorm_2Epolynom(A_27a))
=> ( V6M_27 = ap(c_2EringNorm_2EPopp(A_27a),V18a)
=> ap(V11f4,V18a) = ap(V4f4_27,V18a) ) ) )
=> ap(ap(ap(ap(ap(ap(c_2EringNorm_2Epolynom__CASE(A_27a,A_27b),V5M),V7f),V8f1),V9f2),V10f3),V11f4) = ap(ap(ap(ap(ap(ap(c_2EringNorm_2Epolynom__CASE(A_27a,A_27b),V6M_27),V0f_27),V1f1_27),V2f2_27),V3f3_27),V4f4_27) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2EringNorm_2Epolynom__case__eq,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0x] :
( mem(V0x,ty_2EringNorm_2Epolynom(A_27a))
=> ! [V1f] :
( mem(V1f,arr(ty_2Equote_2Eindex,A_27b))
=> ! [V2f1] :
( mem(V2f1,arr(A_27a,A_27b))
=> ! [V3f2] :
( mem(V3f2,arr(ty_2EringNorm_2Epolynom(A_27a),arr(ty_2EringNorm_2Epolynom(A_27a),A_27b)))
=> ! [V4f3] :
( mem(V4f3,arr(ty_2EringNorm_2Epolynom(A_27a),arr(ty_2EringNorm_2Epolynom(A_27a),A_27b)))
=> ! [V5f4] :
( mem(V5f4,arr(ty_2EringNorm_2Epolynom(A_27a),A_27b))
=> ! [V6v] :
( mem(V6v,A_27b)
=> ( ap(ap(ap(ap(ap(ap(c_2EringNorm_2Epolynom__CASE(A_27a,A_27b),V0x),V1f),V2f1),V3f2),V4f3),V5f4) = V6v
<=> ( ? [V7i] :
( mem(V7i,ty_2Equote_2Eindex)
& V0x = ap(c_2EringNorm_2EPvar(A_27a),V7i)
& ap(V1f,V7i) = V6v )
| ? [V8a] :
( mem(V8a,A_27a)
& V0x = ap(c_2EringNorm_2EPconst(A_27a),V8a)
& ap(V2f1,V8a) = V6v )
| ? [V9p] :
( mem(V9p,ty_2EringNorm_2Epolynom(A_27a))
& ? [V10p0] :
( mem(V10p0,ty_2EringNorm_2Epolynom(A_27a))
& V0x = ap(ap(c_2EringNorm_2EPplus(A_27a),V9p),V10p0)
& ap(ap(V3f2,V9p),V10p0) = V6v ) )
| ? [V11p] :
( mem(V11p,ty_2EringNorm_2Epolynom(A_27a))
& ? [V12p0] :
( mem(V12p0,ty_2EringNorm_2Epolynom(A_27a))
& V0x = ap(ap(c_2EringNorm_2EPmult(A_27a),V11p),V12p0)
& ap(ap(V4f3,V11p),V12p0) = V6v ) )
| ? [V13p] :
( mem(V13p,ty_2EringNorm_2Epolynom(A_27a))
& V0x = ap(c_2EringNorm_2EPopp(A_27a),V13p)
& ap(V5f4,V13p) = V6v ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(ax_thm_2EringNorm_2Epolynom__normalize__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ( ! [V0r] :
( mem(V0r,ty_2Ering_2Ering(A_27a))
=> ! [V1i] :
( mem(V1i,ty_2Equote_2Eindex)
=> ap(ap(c_2EringNorm_2Epolynom__normalize(A_27a),V0r),ap(c_2EringNorm_2EPvar(A_27a),V1i)) = ap(ap(c_2Ecanonical_2ECons__varlist(A_27a),ap(ap(c_2Elist_2ECONS(ty_2Equote_2Eindex),V1i),c_2Elist_2ENIL(ty_2Equote_2Eindex))),c_2Ecanonical_2ENil__monom(A_27a)) ) )
& ! [V2r] :
( mem(V2r,ty_2Ering_2Ering(A_27a))
=> ! [V3c] :
( mem(V3c,A_27a)
=> ap(ap(c_2EringNorm_2Epolynom__normalize(A_27a),V2r),ap(c_2EringNorm_2EPconst(A_27a),V3c)) = ap(ap(ap(c_2Ecanonical_2ECons__monom(A_27a),V3c),c_2Elist_2ENIL(ty_2Equote_2Eindex)),c_2Ecanonical_2ENil__monom(A_27a)) ) )
& ! [V4r] :
( mem(V4r,ty_2Ering_2Ering(A_27a))
=> ! [V5pl] :
( mem(V5pl,ty_2EringNorm_2Epolynom(A_27a))
=> ! [V6pr] :
( mem(V6pr,ty_2EringNorm_2Epolynom(A_27a))
=> ap(ap(c_2EringNorm_2Epolynom__normalize(A_27a),V4r),ap(ap(c_2EringNorm_2EPplus(A_27a),V5pl),V6pr)) = ap(ap(ap(c_2EringNorm_2Er__canonical__sum__merge(A_27a),V4r),ap(ap(c_2EringNorm_2Epolynom__normalize(A_27a),V4r),V5pl)),ap(ap(c_2EringNorm_2Epolynom__normalize(A_27a),V4r),V6pr)) ) ) )
& ! [V7r] :
( mem(V7r,ty_2Ering_2Ering(A_27a))
=> ! [V8pl] :
( mem(V8pl,ty_2EringNorm_2Epolynom(A_27a))
=> ! [V9pr] :
( mem(V9pr,ty_2EringNorm_2Epolynom(A_27a))
=> ap(ap(c_2EringNorm_2Epolynom__normalize(A_27a),V7r),ap(ap(c_2EringNorm_2EPmult(A_27a),V8pl),V9pr)) = ap(ap(ap(c_2EringNorm_2Er__canonical__sum__prod(A_27a),V7r),ap(ap(c_2EringNorm_2Epolynom__normalize(A_27a),V7r),V8pl)),ap(ap(c_2EringNorm_2Epolynom__normalize(A_27a),V7r),V9pr)) ) ) )
& ! [V10r] :
( mem(V10r,ty_2Ering_2Ering(A_27a))
=> ! [V11p] :
( mem(V11p,ty_2EringNorm_2Epolynom(A_27a))
=> ap(ap(c_2EringNorm_2Epolynom__normalize(A_27a),V10r),ap(c_2EringNorm_2EPopp(A_27a),V11p)) = ap(ap(ap(ap(c_2EringNorm_2Er__canonical__sum__scalar3(A_27a),V10r),ap(ap(c_2Ering_2Ering__RN(A_27a),V10r),ap(c_2Ering_2Ering__R1(A_27a),V10r))),c_2Elist_2ENIL(ty_2Equote_2Eindex)),ap(ap(c_2EringNorm_2Epolynom__normalize(A_27a),V10r),V11p)) ) ) ) ) ).
fof(ax_thm_2EringNorm_2Epolynom__simplify__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0r] :
( mem(V0r,ty_2Ering_2Ering(A_27a))
=> ! [V1x] :
( mem(V1x,ty_2EringNorm_2Epolynom(A_27a))
=> ap(ap(c_2EringNorm_2Epolynom__simplify(A_27a),V0r),V1x) = ap(ap(c_2EringNorm_2Er__canonical__sum__simplify(A_27a),V0r),ap(ap(c_2EringNorm_2Epolynom__normalize(A_27a),V0r),V1x)) ) ) ) ).
fof(ax_thm_2EringNorm_2Einterp__p__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ( ! [V0r] :
( mem(V0r,ty_2Ering_2Ering(A_27a))
=> ! [V1vm] :
( mem(V1vm,ty_2Equote_2Evarmap(A_27a))
=> ! [V2c] :
( mem(V2c,A_27a)
=> ap(ap(ap(c_2EringNorm_2Einterp__p(A_27a),V0r),V1vm),ap(c_2EringNorm_2EPconst(A_27a),V2c)) = V2c ) ) )
& ! [V3r] :
( mem(V3r,ty_2Ering_2Ering(A_27a))
=> ! [V4vm] :
( mem(V4vm,ty_2Equote_2Evarmap(A_27a))
=> ! [V5i] :
( mem(V5i,ty_2Equote_2Eindex)
=> ap(ap(ap(c_2EringNorm_2Einterp__p(A_27a),V3r),V4vm),ap(c_2EringNorm_2EPvar(A_27a),V5i)) = ap(ap(c_2Equote_2Evarmap__find(A_27a),V5i),V4vm) ) ) )
& ! [V6r] :
( mem(V6r,ty_2Ering_2Ering(A_27a))
=> ! [V7vm] :
( mem(V7vm,ty_2Equote_2Evarmap(A_27a))
=> ! [V8p1] :
( mem(V8p1,ty_2EringNorm_2Epolynom(A_27a))
=> ! [V9p2] :
( mem(V9p2,ty_2EringNorm_2Epolynom(A_27a))
=> ap(ap(ap(c_2EringNorm_2Einterp__p(A_27a),V6r),V7vm),ap(ap(c_2EringNorm_2EPplus(A_27a),V8p1),V9p2)) = ap(ap(ap(c_2Ering_2Ering__RP(A_27a),V6r),ap(ap(ap(c_2EringNorm_2Einterp__p(A_27a),V6r),V7vm),V8p1)),ap(ap(ap(c_2EringNorm_2Einterp__p(A_27a),V6r),V7vm),V9p2)) ) ) ) )
& ! [V10r] :
( mem(V10r,ty_2Ering_2Ering(A_27a))
=> ! [V11vm] :
( mem(V11vm,ty_2Equote_2Evarmap(A_27a))
=> ! [V12p1] :
( mem(V12p1,ty_2EringNorm_2Epolynom(A_27a))
=> ! [V13p2] :
( mem(V13p2,ty_2EringNorm_2Epolynom(A_27a))
=> ap(ap(ap(c_2EringNorm_2Einterp__p(A_27a),V10r),V11vm),ap(ap(c_2EringNorm_2EPmult(A_27a),V12p1),V13p2)) = ap(ap(ap(c_2Ering_2Ering__RM(A_27a),V10r),ap(ap(ap(c_2EringNorm_2Einterp__p(A_27a),V10r),V11vm),V12p1)),ap(ap(ap(c_2EringNorm_2Einterp__p(A_27a),V10r),V11vm),V13p2)) ) ) ) )
& ! [V14r] :
( mem(V14r,ty_2Ering_2Ering(A_27a))
=> ! [V15vm] :
( mem(V15vm,ty_2Equote_2Evarmap(A_27a))
=> ! [V16p1] :
( mem(V16p1,ty_2EringNorm_2Epolynom(A_27a))
=> ap(ap(ap(c_2EringNorm_2Einterp__p(A_27a),V14r),V15vm),ap(c_2EringNorm_2EPopp(A_27a),V16p1)) = ap(ap(c_2Ering_2Ering__RN(A_27a),V14r),ap(ap(ap(c_2EringNorm_2Einterp__p(A_27a),V14r),V15vm),V16p1)) ) ) ) ) ) ).
fof(conj_thm_2EringNorm_2Epolynom__normalize__ok,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0r] :
( mem(V0r,ty_2Ering_2Ering(A_27a))
=> ( p(ap(c_2Ering_2Eis__ring(A_27a),V0r))
=> ! [V1vm] :
( mem(V1vm,ty_2Equote_2Evarmap(A_27a))
=> ! [V2p] :
( mem(V2p,ty_2EringNorm_2Epolynom(A_27a))
=> ap(ap(ap(c_2EringNorm_2Er__interp__cs(A_27a),V0r),V1vm),ap(ap(c_2EringNorm_2Epolynom__normalize(A_27a),V0r),V2p)) = ap(ap(ap(c_2EringNorm_2Einterp__p(A_27a),V0r),V1vm),V2p) ) ) ) ) ) ).
fof(conj_thm_2EringNorm_2Epolynom__simplify__ok,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0r] :
( mem(V0r,ty_2Ering_2Ering(A_27a))
=> ( p(ap(c_2Ering_2Eis__ring(A_27a),V0r))
=> ! [V1vm] :
( mem(V1vm,ty_2Equote_2Evarmap(A_27a))
=> ! [V2p] :
( mem(V2p,ty_2EringNorm_2Epolynom(A_27a))
=> ap(ap(ap(c_2EringNorm_2Er__interp__cs(A_27a),V0r),V1vm),ap(ap(c_2EringNorm_2Epolynom__simplify(A_27a),V0r),V2p)) = ap(ap(ap(c_2EringNorm_2Einterp__p(A_27a),V0r),V1vm),V2p) ) ) ) ) ) ).
%------------------------------------------------------------------------------