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Diffstat (limited to 'contrib7/ring/Setoid_ring_normalize.v')
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diff --git a/contrib7/ring/Setoid_ring_normalize.v b/contrib7/ring/Setoid_ring_normalize.v deleted file mode 100644 index b6b79dae..00000000 --- a/contrib7/ring/Setoid_ring_normalize.v +++ /dev/null @@ -1,1141 +0,0 @@ -(************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) -(* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(************************************************************************) - -(* $Id: Setoid_ring_normalize.v,v 1.1.2.1 2004/07/16 19:30:19 herbelin Exp $ *) - -Require Setoid_ring_theory. -Require Quote. - -Set Implicit Arguments. - -Lemma index_eq_prop: (n,m:index)(Is_true (index_eq n m)) -> n=m. -Proof. - Induction n; Induction m; Simpl; Try (Reflexivity Orelse Contradiction). - Intros; Rewrite (H i0); Trivial. - Intros; Rewrite (H i0); Trivial. -Save. - -Section setoid. - -Variable A : Type. -Variable Aequiv : A -> A -> Prop. -Variable Aplus : A -> A -> A. -Variable Amult : A -> A -> A. -Variable Aone : A. -Variable Azero : A. -Variable Aopp : A -> A. -Variable Aeq : A -> A -> bool. - -Variable S : (Setoid_Theory A Aequiv). - -Add Setoid A Aequiv S. - -Variable plus_morph : (a,a0,a1,a2:A) - (Aequiv a a0)->(Aequiv a1 a2)->(Aequiv (Aplus a a1) (Aplus a0 a2)). -Variable mult_morph : (a,a0,a1,a2:A) - (Aequiv a a0)->(Aequiv a1 a2)->(Aequiv (Amult a a1) (Amult a0 a2)). -Variable opp_morph : (a,a0:A) - (Aequiv a a0)->(Aequiv (Aopp a) (Aopp a0)). - -Add Morphism Aplus : Aplus_ext. -Exact plus_morph. -Save. - -Add Morphism Amult : Amult_ext. -Exact mult_morph. -Save. - -Add Morphism Aopp : Aopp_ext. -Exact opp_morph. -Save. - -Local equiv_refl := (Seq_refl A Aequiv S). -Local equiv_sym := (Seq_sym A Aequiv S). -Local equiv_trans := (Seq_trans A Aequiv S). - -Hints Resolve equiv_refl equiv_trans. -Hints Immediate equiv_sym. - -Section semi_setoid_rings. - -(* Section definitions. *) - - -(******************************************) -(* Normal abtract Polynomials *) -(******************************************) -(* DEFINITIONS : -- A varlist is a sorted product of one or more variables : x, x*y*z -- A monom is a constant, a varlist or the product of a constant by a varlist - variables. 2*x*y, x*y*z, 3 are monoms : 2*3, x*3*y, 4*x*3 are NOT. -- A canonical sum is either a monom or an ordered sum of monoms - (the order on monoms is defined later) -- A normal polynomial it either a constant or a canonical sum or a constant - plus a canonical sum -*) - -(* varlist is isomorphic to (list var), but we built a special inductive - for efficiency *) -Inductive varlist : Type := -| Nil_var : varlist -| Cons_var : index -> varlist -> varlist -. - -Inductive canonical_sum : Type := -| Nil_monom : canonical_sum -| Cons_monom : A -> varlist -> canonical_sum -> canonical_sum -| Cons_varlist : varlist -> canonical_sum -> canonical_sum -. - -(* Order on monoms *) - -(* That's the lexicographic order on varlist, extended by : - - A constant is less than every monom - - The relation between two varlist is preserved by multiplication by a - constant. - - Examples : - 3 < x < y - x*y < x*y*y*z - 2*x*y < x*y*y*z - x*y < 54*x*y*y*z - 4*x*y < 59*x*y*y*z -*) - -Fixpoint varlist_eq [x,y:varlist] : bool := - Cases x y of - | Nil_var Nil_var => true - | (Cons_var i xrest) (Cons_var j yrest) => - (andb (index_eq i j) (varlist_eq xrest yrest)) - | _ _ => false - end. - -Fixpoint varlist_lt [x,y:varlist] : bool := - Cases x y of - | Nil_var (Cons_var _ _) => true - | (Cons_var i xrest) (Cons_var j yrest) => - if (index_lt i j) then true - else (andb (index_eq i j) (varlist_lt xrest yrest)) - | _ _ => false - end. - -(* merges two variables lists *) -Fixpoint varlist_merge [l1:varlist] : varlist -> varlist := - Cases l1 of - | (Cons_var v1 t1) => - Fix vm_aux {vm_aux [l2:varlist] : varlist := - Cases l2 of - | (Cons_var v2 t2) => - if (index_lt v1 v2) - then (Cons_var v1 (varlist_merge t1 l2)) - else (Cons_var v2 (vm_aux t2)) - | Nil_var => l1 - end} - | Nil_var => [l2]l2 - end. - -(* returns the sum of two canonical sums *) -Fixpoint canonical_sum_merge [s1:canonical_sum] - : canonical_sum -> canonical_sum := -Cases s1 of -| (Cons_monom c1 l1 t1) => - Fix csm_aux{csm_aux[s2:canonical_sum] : canonical_sum := - Cases s2 of - | (Cons_monom c2 l2 t2) => - if (varlist_eq l1 l2) - then (Cons_monom (Aplus c1 c2) l1 - (canonical_sum_merge t1 t2)) - else if (varlist_lt l1 l2) - then (Cons_monom c1 l1 (canonical_sum_merge t1 s2)) - else (Cons_monom c2 l2 (csm_aux t2)) - | (Cons_varlist l2 t2) => - if (varlist_eq l1 l2) - then (Cons_monom (Aplus c1 Aone) l1 - (canonical_sum_merge t1 t2)) - else if (varlist_lt l1 l2) - then (Cons_monom c1 l1 (canonical_sum_merge t1 s2)) - else (Cons_varlist l2 (csm_aux t2)) - | Nil_monom => s1 - end} -| (Cons_varlist l1 t1) => - Fix csm_aux2{csm_aux2[s2:canonical_sum] : canonical_sum := - Cases s2 of - | (Cons_monom c2 l2 t2) => - if (varlist_eq l1 l2) - then (Cons_monom (Aplus Aone c2) l1 - (canonical_sum_merge t1 t2)) - else if (varlist_lt l1 l2) - then (Cons_varlist l1 (canonical_sum_merge t1 s2)) - else (Cons_monom c2 l2 (csm_aux2 t2)) - | (Cons_varlist l2 t2) => - if (varlist_eq l1 l2) - then (Cons_monom (Aplus Aone Aone) l1 - (canonical_sum_merge t1 t2)) - else if (varlist_lt l1 l2) - then (Cons_varlist l1 (canonical_sum_merge t1 s2)) - else (Cons_varlist l2 (csm_aux2 t2)) - | Nil_monom => s1 - end} -| Nil_monom => [s2]s2 -end. - -(* Insertion of a monom in a canonical sum *) -Fixpoint monom_insert [c1:A; l1:varlist; s2 : canonical_sum] - : canonical_sum := - Cases s2 of - | (Cons_monom c2 l2 t2) => - if (varlist_eq l1 l2) - then (Cons_monom (Aplus c1 c2) l1 t2) - else if (varlist_lt l1 l2) - then (Cons_monom c1 l1 s2) - else (Cons_monom c2 l2 (monom_insert c1 l1 t2)) - | (Cons_varlist l2 t2) => - if (varlist_eq l1 l2) - then (Cons_monom (Aplus c1 Aone) l1 t2) - else if (varlist_lt l1 l2) - then (Cons_monom c1 l1 s2) - else (Cons_varlist l2 (monom_insert c1 l1 t2)) - | Nil_monom => (Cons_monom c1 l1 Nil_monom) - end. - -Fixpoint varlist_insert [l1:varlist; s2:canonical_sum] - : canonical_sum := - Cases s2 of - | (Cons_monom c2 l2 t2) => - if (varlist_eq l1 l2) - then (Cons_monom (Aplus Aone c2) l1 t2) - else if (varlist_lt l1 l2) - then (Cons_varlist l1 s2) - else (Cons_monom c2 l2 (varlist_insert l1 t2)) - | (Cons_varlist l2 t2) => - if (varlist_eq l1 l2) - then (Cons_monom (Aplus Aone Aone) l1 t2) - else if (varlist_lt l1 l2) - then (Cons_varlist l1 s2) - else (Cons_varlist l2 (varlist_insert l1 t2)) - | Nil_monom => (Cons_varlist l1 Nil_monom) - end. - -(* Computes c0*s *) -Fixpoint canonical_sum_scalar [c0:A; s:canonical_sum] : canonical_sum := - Cases s of - | (Cons_monom c l t) => - (Cons_monom (Amult c0 c) l (canonical_sum_scalar c0 t)) - | (Cons_varlist l t) => - (Cons_monom c0 l (canonical_sum_scalar c0 t)) - | Nil_monom => Nil_monom - end. - -(* Computes l0*s *) -Fixpoint canonical_sum_scalar2 [l0:varlist; s:canonical_sum] - : canonical_sum := - Cases s of - | (Cons_monom c l t) => - (monom_insert c (varlist_merge l0 l) (canonical_sum_scalar2 l0 t)) - | (Cons_varlist l t) => - (varlist_insert (varlist_merge l0 l) (canonical_sum_scalar2 l0 t)) - | Nil_monom => Nil_monom - end. - -(* Computes c0*l0*s *) -Fixpoint canonical_sum_scalar3 [c0:A;l0:varlist; s:canonical_sum] - : canonical_sum := - Cases s of - | (Cons_monom c l t) => - (monom_insert (Amult c0 c) (varlist_merge l0 l) - (canonical_sum_scalar3 c0 l0 t)) - | (Cons_varlist l t) => - (monom_insert c0 (varlist_merge l0 l) - (canonical_sum_scalar3 c0 l0 t)) - | Nil_monom => Nil_monom - end. - -(* returns the product of two canonical sums *) -Fixpoint canonical_sum_prod [s1:canonical_sum] - : canonical_sum -> canonical_sum := - [s2]Cases s1 of - | (Cons_monom c1 l1 t1) => - (canonical_sum_merge (canonical_sum_scalar3 c1 l1 s2) - (canonical_sum_prod t1 s2)) - | (Cons_varlist l1 t1) => - (canonical_sum_merge (canonical_sum_scalar2 l1 s2) - (canonical_sum_prod t1 s2)) - | Nil_monom => Nil_monom - end. - -(* The type to represent concrete semi-setoid-ring polynomials *) - -Inductive Type setspolynomial := - SetSPvar : index -> setspolynomial -| SetSPconst : A -> setspolynomial -| SetSPplus : setspolynomial -> setspolynomial -> setspolynomial -| SetSPmult : setspolynomial -> setspolynomial -> setspolynomial. - -Fixpoint setspolynomial_normalize [p:setspolynomial] : canonical_sum := - Cases p of - | (SetSPplus l r) => (canonical_sum_merge (setspolynomial_normalize l) (setspolynomial_normalize r)) - | (SetSPmult l r) => (canonical_sum_prod (setspolynomial_normalize l) (setspolynomial_normalize r)) - | (SetSPconst c) => (Cons_monom c Nil_var Nil_monom) - | (SetSPvar i) => (Cons_varlist (Cons_var i Nil_var) Nil_monom) - end. - -Fixpoint canonical_sum_simplify [ s:canonical_sum] : canonical_sum := - Cases s of - | (Cons_monom c l t) => - if (Aeq c Azero) - then (canonical_sum_simplify t) - else if (Aeq c Aone) - then (Cons_varlist l (canonical_sum_simplify t)) - else (Cons_monom c l (canonical_sum_simplify t)) - | (Cons_varlist l t) => (Cons_varlist l (canonical_sum_simplify t)) - | Nil_monom => Nil_monom - end. - -Definition setspolynomial_simplify := - [x:setspolynomial] (canonical_sum_simplify (setspolynomial_normalize x)). - -Variable vm : (varmap A). - -Definition interp_var [i:index] := (varmap_find Azero i vm). - -Definition ivl_aux := Fix ivl_aux {ivl_aux[x:index; t:varlist] : A := - Cases t of - | Nil_var => (interp_var x) - | (Cons_var x' t') => (Amult (interp_var x) (ivl_aux x' t')) - end}. - -Definition interp_vl := [l:varlist] - Cases l of - | Nil_var => Aone - | (Cons_var x t) => (ivl_aux x t) - end. - -Definition interp_m := [c:A][l:varlist] - Cases l of - | Nil_var => c - | (Cons_var x t) => - (Amult c (ivl_aux x t)) - end. - -Definition ics_aux := Fix ics_aux{ics_aux[a:A; s:canonical_sum] : A := - Cases s of - | Nil_monom => a - | (Cons_varlist l t) => (Aplus a (ics_aux (interp_vl l) t)) - | (Cons_monom c l t) => (Aplus a (ics_aux (interp_m c l) t)) - end}. - -Definition interp_setcs : canonical_sum -> A := - [s]Cases s of - | Nil_monom => Azero - | (Cons_varlist l t) => - (ics_aux (interp_vl l) t) - | (Cons_monom c l t) => - (ics_aux (interp_m c l) t) - end. - -Fixpoint interp_setsp [p:setspolynomial] : A := - Cases p of - | (SetSPconst c) => c - | (SetSPvar i) => (interp_var i) - | (SetSPplus p1 p2) => (Aplus (interp_setsp p1) (interp_setsp p2)) - | (SetSPmult p1 p2) => (Amult (interp_setsp p1) (interp_setsp p2)) - end. - -(* End interpretation. *) - -Unset Implicit Arguments. - -(* Section properties. *) - -Variable T : (Semi_Setoid_Ring_Theory Aequiv Aplus Amult Aone Azero Aeq). - -Hint SSR_plus_sym_T := Resolve (SSR_plus_sym T). -Hint SSR_plus_assoc_T := Resolve (SSR_plus_assoc T). -Hint SSR_plus_assoc2_T := Resolve (SSR_plus_assoc2 S T). -Hint SSR_mult_sym_T := Resolve (SSR_mult_sym T). -Hint SSR_mult_assoc_T := Resolve (SSR_mult_assoc T). -Hint SSR_mult_assoc2_T := Resolve (SSR_mult_assoc2 S T). -Hint SSR_plus_zero_left_T := Resolve (SSR_plus_zero_left T). -Hint SSR_plus_zero_left2_T := Resolve (SSR_plus_zero_left2 S T). -Hint SSR_mult_one_left_T := Resolve (SSR_mult_one_left T). -Hint SSR_mult_one_left2_T := Resolve (SSR_mult_one_left2 S T). -Hint SSR_mult_zero_left_T := Resolve (SSR_mult_zero_left T). -Hint SSR_mult_zero_left2_T := Resolve (SSR_mult_zero_left2 S T). -Hint SSR_distr_left_T := Resolve (SSR_distr_left T). -Hint SSR_distr_left2_T := Resolve (SSR_distr_left2 S T). -Hint SSR_plus_reg_left_T := Resolve (SSR_plus_reg_left T). -Hint SSR_plus_permute_T := Resolve (SSR_plus_permute S plus_morph T). -Hint SSR_mult_permute_T := Resolve (SSR_mult_permute S mult_morph T). -Hint SSR_distr_right_T := Resolve (SSR_distr_right S plus_morph T). -Hint SSR_distr_right2_T := Resolve (SSR_distr_right2 S plus_morph T). -Hint SSR_mult_zero_right_T := Resolve (SSR_mult_zero_right S T). -Hint SSR_mult_zero_right2_T := Resolve (SSR_mult_zero_right2 S T). -Hint SSR_plus_zero_right_T := Resolve (SSR_plus_zero_right S T). -Hint SSR_plus_zero_right2_T := Resolve (SSR_plus_zero_right2 S T). -Hint SSR_mult_one_right_T := Resolve (SSR_mult_one_right S T). -Hint SSR_mult_one_right2_T := Resolve (SSR_mult_one_right2 S T). -Hint SSR_plus_reg_right_T := Resolve (SSR_plus_reg_right S T). -Hints Resolve refl_equal sym_equal trans_equal. -(*Hints Resolve refl_eqT sym_eqT trans_eqT.*) -Hints Immediate T. - -Lemma varlist_eq_prop : (x,y:varlist) - (Is_true (varlist_eq x y))->x==y. -Proof. - Induction x; Induction y; Contradiction Orelse Try Reflexivity. - Simpl; Intros. - Generalize (andb_prop2 ? ? H1); Intros; Elim H2; Intros. - Rewrite (index_eq_prop H3); Rewrite (H v0 H4); Reflexivity. -Save. - -Remark ivl_aux_ok : (v:varlist)(i:index) - (Aequiv (ivl_aux i v) (Amult (interp_var i) (interp_vl v))). -Proof. - Induction v; Simpl; Intros. - Trivial. - Rewrite (H i); Trivial. -Save. - -Lemma varlist_merge_ok : (x,y:varlist) - (Aequiv (interp_vl (varlist_merge x y)) (Amult (interp_vl x) (interp_vl y))). -Proof. - Induction x. - Simpl; Trivial. - Induction y. - Simpl; Trivial. - Simpl; Intros. - Elim (index_lt i i0); Simpl; Intros. - - Rewrite (ivl_aux_ok v i). - Rewrite (ivl_aux_ok v0 i0). - Rewrite (ivl_aux_ok (varlist_merge v (Cons_var i0 v0)) i). - Rewrite (H (Cons_var i0 v0)). - Simpl. - Rewrite (ivl_aux_ok v0 i0). - EAuto. - - Rewrite (ivl_aux_ok v i). - Rewrite (ivl_aux_ok v0 i0). - Rewrite (ivl_aux_ok - (Fix vm_aux - {vm_aux [l2:varlist] : varlist := - Cases (l2) of - Nil_var => (Cons_var i v) - | (Cons_var v2 t2) => - (if (index_lt i v2) - then (Cons_var i (varlist_merge v l2)) - else (Cons_var v2 (vm_aux t2))) - end} v0) i0). - Rewrite H0. - Rewrite (ivl_aux_ok v i). - EAuto. -Save. - -Remark ics_aux_ok : (x:A)(s:canonical_sum) - (Aequiv (ics_aux x s) (Aplus x (interp_setcs s))). -Proof. - Induction s; Simpl; Intros;Trivial. -Save. - -Remark interp_m_ok : (x:A)(l:varlist) - (Aequiv (interp_m x l) (Amult x (interp_vl l))). -Proof. - NewDestruct l;Trivial. -Save. - -Hint ivl_aux_ok_ := Resolve ivl_aux_ok. -Hint ics_aux_ok_ := Resolve ics_aux_ok. -Hint interp_m_ok_ := Resolve interp_m_ok. - -(* Hints Resolve ivl_aux_ok ics_aux_ok interp_m_ok. *) - -Lemma canonical_sum_merge_ok : (x,y:canonical_sum) - (Aequiv (interp_setcs (canonical_sum_merge x y)) - (Aplus (interp_setcs x) (interp_setcs y))). -Proof. -Induction x; Simpl. -Trivial. - -Induction y; Simpl; Intros. -EAuto. - -Generalize (varlist_eq_prop v v0). -Elim (varlist_eq v v0). -Intros; Rewrite (H1 I). -Simpl. -Rewrite (ics_aux_ok (interp_m a v0) c). -Rewrite (ics_aux_ok (interp_m a0 v0) c0). -Rewrite (ics_aux_ok (interp_m (Aplus a a0) v0) - (canonical_sum_merge c c0)). -Rewrite (H c0). -Rewrite (interp_m_ok (Aplus a a0) v0). -Rewrite (interp_m_ok a v0). -Rewrite (interp_m_ok a0 v0). -Setoid_replace (Amult (Aplus a a0) (interp_vl v0)) - with (Aplus (Amult a (interp_vl v0)) (Amult a0 (interp_vl v0))). -Setoid_replace (Aplus - (Aplus (Amult a (interp_vl v0)) - (Amult a0 (interp_vl v0))) - (Aplus (interp_setcs c) (interp_setcs c0))) - with (Aplus (Amult a (interp_vl v0)) - (Aplus (Amult a0 (interp_vl v0)) - (Aplus (interp_setcs c) (interp_setcs c0)))). -Setoid_replace (Aplus (Aplus (Amult a (interp_vl v0)) (interp_setcs c)) - (Aplus (Amult a0 (interp_vl v0)) (interp_setcs c0))) - with (Aplus (Amult a (interp_vl v0)) - (Aplus (interp_setcs c) - (Aplus (Amult a0 (interp_vl v0)) (interp_setcs c0)))). -Auto. - -Elim (varlist_lt v v0); Simpl. -Intro. -Rewrite (ics_aux_ok (interp_m a v) - (canonical_sum_merge c (Cons_monom a0 v0 c0))). -Rewrite (ics_aux_ok (interp_m a v) c). -Rewrite (ics_aux_ok (interp_m a0 v0) c0). -Rewrite (H (Cons_monom a0 v0 c0)); Simpl. -Rewrite (ics_aux_ok (interp_m a0 v0) c0); Auto. - -Intro. -Rewrite (ics_aux_ok (interp_m a0 v0) - (Fix csm_aux - {csm_aux [s2:canonical_sum] : canonical_sum := - Cases (s2) of - Nil_monom => (Cons_monom a v c) - | (Cons_monom c2 l2 t2) => - (if (varlist_eq v l2) - then - (Cons_monom (Aplus a c2) v - (canonical_sum_merge c t2)) - else - (if (varlist_lt v l2) - then - (Cons_monom a v - (canonical_sum_merge c s2)) - else (Cons_monom c2 l2 (csm_aux t2)))) - | (Cons_varlist l2 t2) => - (if (varlist_eq v l2) - then - (Cons_monom (Aplus a Aone) v - (canonical_sum_merge c t2)) - else - (if (varlist_lt v l2) - then - (Cons_monom a v - (canonical_sum_merge c s2)) - else (Cons_varlist l2 (csm_aux t2)))) - end} c0)). -Rewrite H0. -Rewrite (ics_aux_ok (interp_m a v) c); -Rewrite (ics_aux_ok (interp_m a0 v0) c0); Simpl; Auto. - -Generalize (varlist_eq_prop v v0). -Elim (varlist_eq v v0). -Intros; Rewrite (H1 I). -Simpl. -Rewrite (ics_aux_ok (interp_m (Aplus a Aone) v0) - (canonical_sum_merge c c0)); -Rewrite (ics_aux_ok (interp_m a v0) c); -Rewrite (ics_aux_ok (interp_vl v0) c0). -Rewrite (H c0). -Rewrite (interp_m_ok (Aplus a Aone) v0). -Rewrite (interp_m_ok a v0). -Setoid_replace (Amult (Aplus a Aone) (interp_vl v0)) - with (Aplus (Amult a (interp_vl v0)) (Amult Aone (interp_vl v0))). -Setoid_replace (Aplus - (Aplus (Amult a (interp_vl v0)) - (Amult Aone (interp_vl v0))) - (Aplus (interp_setcs c) (interp_setcs c0))) - with (Aplus (Amult a (interp_vl v0)) - (Aplus (Amult Aone (interp_vl v0)) - (Aplus (interp_setcs c) (interp_setcs c0)))). -Setoid_replace (Aplus (Aplus (Amult a (interp_vl v0)) (interp_setcs c)) - (Aplus (interp_vl v0) (interp_setcs c0))) - with (Aplus (Amult a (interp_vl v0)) - (Aplus (interp_setcs c) (Aplus (interp_vl v0) (interp_setcs c0)))). -Setoid_replace (Amult Aone (interp_vl v0)) with (interp_vl v0). -Auto. - -Elim (varlist_lt v v0); Simpl. -Intro. -Rewrite (ics_aux_ok (interp_m a v) - (canonical_sum_merge c (Cons_varlist v0 c0))); -Rewrite (ics_aux_ok (interp_m a v) c); -Rewrite (ics_aux_ok (interp_vl v0) c0). -Rewrite (H (Cons_varlist v0 c0)); Simpl. -Rewrite (ics_aux_ok (interp_vl v0) c0). -Auto. - -Intro. -Rewrite (ics_aux_ok (interp_vl v0) - (Fix csm_aux - {csm_aux [s2:canonical_sum] : canonical_sum := - Cases (s2) of - Nil_monom => (Cons_monom a v c) - | (Cons_monom c2 l2 t2) => - (if (varlist_eq v l2) - then - (Cons_monom (Aplus a c2) v - (canonical_sum_merge c t2)) - else - (if (varlist_lt v l2) - then - (Cons_monom a v - (canonical_sum_merge c s2)) - else (Cons_monom c2 l2 (csm_aux t2)))) - | (Cons_varlist l2 t2) => - (if (varlist_eq v l2) - then - (Cons_monom (Aplus a Aone) v - (canonical_sum_merge c t2)) - else - (if (varlist_lt v l2) - then - (Cons_monom a v - (canonical_sum_merge c s2)) - else (Cons_varlist l2 (csm_aux t2)))) - end} c0)); Rewrite H0. -Rewrite (ics_aux_ok (interp_m a v) c); -Rewrite (ics_aux_ok (interp_vl v0) c0); Simpl. -Auto. - -Induction y; Simpl; Intros. -Trivial. - -Generalize (varlist_eq_prop v v0). -Elim (varlist_eq v v0). -Intros; Rewrite (H1 I). -Simpl. -Rewrite (ics_aux_ok (interp_m (Aplus Aone a) v0) - (canonical_sum_merge c c0)); -Rewrite (ics_aux_ok (interp_vl v0) c); -Rewrite (ics_aux_ok (interp_m a v0) c0); Rewrite ( -H c0). -Rewrite (interp_m_ok (Aplus Aone a) v0); -Rewrite (interp_m_ok a v0). -Setoid_replace (Amult (Aplus Aone a) (interp_vl v0)) - with (Aplus (Amult Aone (interp_vl v0)) (Amult a (interp_vl v0))); -Setoid_replace (Aplus - (Aplus (Amult Aone (interp_vl v0)) - (Amult a (interp_vl v0))) - (Aplus (interp_setcs c) (interp_setcs c0))) - with (Aplus (Amult Aone (interp_vl v0)) - (Aplus (Amult a (interp_vl v0)) - (Aplus (interp_setcs c) (interp_setcs c0)))); -Setoid_replace (Aplus (Aplus (interp_vl v0) (interp_setcs c)) - (Aplus (Amult a (interp_vl v0)) (interp_setcs c0))) - with (Aplus (interp_vl v0) - (Aplus (interp_setcs c) - (Aplus (Amult a (interp_vl v0)) (interp_setcs c0)))). -Auto. - -Elim (varlist_lt v v0); Simpl; Intros. -Rewrite (ics_aux_ok (interp_vl v) - (canonical_sum_merge c (Cons_monom a v0 c0))); -Rewrite (ics_aux_ok (interp_vl v) c); -Rewrite (ics_aux_ok (interp_m a v0) c0). -Rewrite (H (Cons_monom a v0 c0)); Simpl. -Rewrite (ics_aux_ok (interp_m a v0) c0); Auto. - -Rewrite (ics_aux_ok (interp_m a v0) - (Fix csm_aux2 - {csm_aux2 [s2:canonical_sum] : canonical_sum := - Cases (s2) of - Nil_monom => (Cons_varlist v c) - | (Cons_monom c2 l2 t2) => - (if (varlist_eq v l2) - then - (Cons_monom (Aplus Aone c2) v - (canonical_sum_merge c t2)) - else - (if (varlist_lt v l2) - then - (Cons_varlist v - (canonical_sum_merge c s2)) - else (Cons_monom c2 l2 (csm_aux2 t2)))) - | (Cons_varlist l2 t2) => - (if (varlist_eq v l2) - then - (Cons_monom (Aplus Aone Aone) v - (canonical_sum_merge c t2)) - else - (if (varlist_lt v l2) - then - (Cons_varlist v - (canonical_sum_merge c s2)) - else (Cons_varlist l2 (csm_aux2 t2)))) - end} c0)); Rewrite H0. -Rewrite (ics_aux_ok (interp_vl v) c); -Rewrite (ics_aux_ok (interp_m a v0) c0); Simpl; Auto. - -Generalize (varlist_eq_prop v v0). -Elim (varlist_eq v v0); Intros. -Rewrite (H1 I); Simpl. -Rewrite (ics_aux_ok (interp_m (Aplus Aone Aone) v0) - (canonical_sum_merge c c0)); -Rewrite (ics_aux_ok (interp_vl v0) c); -Rewrite (ics_aux_ok (interp_vl v0) c0); Rewrite ( -H c0). -Rewrite (interp_m_ok (Aplus Aone Aone) v0). -Setoid_replace (Amult (Aplus Aone Aone) (interp_vl v0)) - with (Aplus (Amult Aone (interp_vl v0)) (Amult Aone (interp_vl v0))); -Setoid_replace (Aplus - (Aplus (Amult Aone (interp_vl v0)) - (Amult Aone (interp_vl v0))) - (Aplus (interp_setcs c) (interp_setcs c0))) - with (Aplus (Amult Aone (interp_vl v0)) - (Aplus (Amult Aone (interp_vl v0)) - (Aplus (interp_setcs c) (interp_setcs c0)))); -Setoid_replace (Aplus (Aplus (interp_vl v0) (interp_setcs c)) - (Aplus (interp_vl v0) (interp_setcs c0))) - with (Aplus (interp_vl v0) - (Aplus (interp_setcs c) (Aplus (interp_vl v0) (interp_setcs c0)))). -Setoid_replace (Amult Aone (interp_vl v0)) with (interp_vl v0); Auto. - -Elim (varlist_lt v v0); Simpl. -Rewrite (ics_aux_ok (interp_vl v) - (canonical_sum_merge c (Cons_varlist v0 c0))); -Rewrite (ics_aux_ok (interp_vl v) c); -Rewrite (ics_aux_ok (interp_vl v0) c0); -Rewrite (H (Cons_varlist v0 c0)); Simpl. -Rewrite (ics_aux_ok (interp_vl v0) c0); Auto. - -Rewrite (ics_aux_ok (interp_vl v0) - (Fix csm_aux2 - {csm_aux2 [s2:canonical_sum] : canonical_sum := - Cases (s2) of - Nil_monom => (Cons_varlist v c) - | (Cons_monom c2 l2 t2) => - (if (varlist_eq v l2) - then - (Cons_monom (Aplus Aone c2) v - (canonical_sum_merge c t2)) - else - (if (varlist_lt v l2) - then - (Cons_varlist v - (canonical_sum_merge c s2)) - else (Cons_monom c2 l2 (csm_aux2 t2)))) - | (Cons_varlist l2 t2) => - (if (varlist_eq v l2) - then - (Cons_monom (Aplus Aone Aone) v - (canonical_sum_merge c t2)) - else - (if (varlist_lt v l2) - then - (Cons_varlist v - (canonical_sum_merge c s2)) - else (Cons_varlist l2 (csm_aux2 t2)))) - end} c0)); Rewrite H0. -Rewrite (ics_aux_ok (interp_vl v) c); -Rewrite (ics_aux_ok (interp_vl v0) c0); Simpl; Auto. -Save. - -Lemma monom_insert_ok: (a:A)(l:varlist)(s:canonical_sum) - (Aequiv (interp_setcs (monom_insert a l s)) - (Aplus (Amult a (interp_vl l)) (interp_setcs s))). -Proof. -Induction s; Intros. -Simpl; Rewrite (interp_m_ok a l); Trivial. - -Simpl; Generalize (varlist_eq_prop l v); Elim (varlist_eq l v). -Intro Hr; Rewrite (Hr I); Simpl. -Rewrite (ics_aux_ok (interp_m (Aplus a a0) v) c); -Rewrite (ics_aux_ok (interp_m a0 v) c). -Rewrite (interp_m_ok (Aplus a a0) v); -Rewrite (interp_m_ok a0 v). -Setoid_replace (Amult (Aplus a a0) (interp_vl v)) - with (Aplus (Amult a (interp_vl v)) (Amult a0 (interp_vl v))). -Auto. - -Elim (varlist_lt l v); Simpl; Intros. -Rewrite (ics_aux_ok (interp_m a0 v) c). -Rewrite (interp_m_ok a0 v); Rewrite (interp_m_ok a l). -Auto. - -Rewrite (ics_aux_ok (interp_m a0 v) (monom_insert a l c)); -Rewrite (ics_aux_ok (interp_m a0 v) c); Rewrite H. -Auto. - -Simpl. -Generalize (varlist_eq_prop l v); Elim (varlist_eq l v). -Intro Hr; Rewrite (Hr I); Simpl. -Rewrite (ics_aux_ok (interp_m (Aplus a Aone) v) c); -Rewrite (ics_aux_ok (interp_vl v) c). -Rewrite (interp_m_ok (Aplus a Aone) v). -Setoid_replace (Amult (Aplus a Aone) (interp_vl v)) - with (Aplus (Amult a (interp_vl v)) (Amult Aone (interp_vl v))). -Setoid_replace (Amult Aone (interp_vl v)) with (interp_vl v). -Auto. - -Elim (varlist_lt l v); Simpl; Intros; Auto. -Rewrite (ics_aux_ok (interp_vl v) (monom_insert a l c)); -Rewrite H. -Rewrite (ics_aux_ok (interp_vl v) c); Auto. -Save. - -Lemma varlist_insert_ok : - (l:varlist)(s:canonical_sum) - (Aequiv (interp_setcs (varlist_insert l s)) - (Aplus (interp_vl l) (interp_setcs s))). -Proof. -Induction s; Simpl; Intros. -Trivial. - -Generalize (varlist_eq_prop l v); Elim (varlist_eq l v). -Intro Hr; Rewrite (Hr I); Simpl. -Rewrite (ics_aux_ok (interp_m (Aplus Aone a) v) c); -Rewrite (ics_aux_ok (interp_m a v) c). -Rewrite (interp_m_ok (Aplus Aone a) v); -Rewrite (interp_m_ok a v). -Setoid_replace (Amult (Aplus Aone a) (interp_vl v)) - with (Aplus (Amult Aone (interp_vl v)) (Amult a (interp_vl v))). -Setoid_replace (Amult Aone (interp_vl v)) with (interp_vl v); Auto. - -Elim (varlist_lt l v); Simpl; Intros; Auto. -Rewrite (ics_aux_ok (interp_m a v) (varlist_insert l c)); -Rewrite (ics_aux_ok (interp_m a v) c). -Rewrite (interp_m_ok a v). -Rewrite H; Auto. - -Generalize (varlist_eq_prop l v); Elim (varlist_eq l v). -Intro Hr; Rewrite (Hr I); Simpl. -Rewrite (ics_aux_ok (interp_m (Aplus Aone Aone) v) c); -Rewrite (ics_aux_ok (interp_vl v) c). -Rewrite (interp_m_ok (Aplus Aone Aone) v). -Setoid_replace (Amult (Aplus Aone Aone) (interp_vl v)) - with (Aplus (Amult Aone (interp_vl v)) (Amult Aone (interp_vl v))). -Setoid_replace (Amult Aone (interp_vl v)) with (interp_vl v); Auto. - -Elim (varlist_lt l v); Simpl; Intros; Auto. -Rewrite (ics_aux_ok (interp_vl v) (varlist_insert l c)). -Rewrite H. -Rewrite (ics_aux_ok (interp_vl v) c); Auto. -Save. - -Lemma canonical_sum_scalar_ok : (a:A)(s:canonical_sum) - (Aequiv (interp_setcs (canonical_sum_scalar a s)) (Amult a (interp_setcs s))). -Proof. -Induction s; Simpl; Intros. -Trivial. - -Rewrite (ics_aux_ok (interp_m (Amult a a0) v) - (canonical_sum_scalar a c)); -Rewrite (ics_aux_ok (interp_m a0 v) c). -Rewrite (interp_m_ok (Amult a a0) v); -Rewrite (interp_m_ok a0 v). -Rewrite H. -Setoid_replace (Amult a (Aplus (Amult a0 (interp_vl v)) (interp_setcs c))) - with (Aplus (Amult a (Amult a0 (interp_vl v))) (Amult a (interp_setcs c))). -Auto. - -Rewrite (ics_aux_ok (interp_m a v) (canonical_sum_scalar a c)); -Rewrite (ics_aux_ok (interp_vl v) c); Rewrite H. -Rewrite (interp_m_ok a v). -Auto. -Save. - -Lemma canonical_sum_scalar2_ok : (l:varlist; s:canonical_sum) - (Aequiv (interp_setcs (canonical_sum_scalar2 l s)) (Amult (interp_vl l) (interp_setcs s))). -Proof. -Induction s; Simpl; Intros; Auto. -Rewrite (monom_insert_ok a (varlist_merge l v) - (canonical_sum_scalar2 l c)). -Rewrite (ics_aux_ok (interp_m a v) c). -Rewrite (interp_m_ok a v). -Rewrite H. -Rewrite (varlist_merge_ok l v). -Setoid_replace (Amult (interp_vl l) - (Aplus (Amult a (interp_vl v)) (interp_setcs c))) - with (Aplus (Amult (interp_vl l) (Amult a (interp_vl v))) - (Amult (interp_vl l) (interp_setcs c))). -Auto. - -Rewrite (varlist_insert_ok (varlist_merge l v) - (canonical_sum_scalar2 l c)). -Rewrite (ics_aux_ok (interp_vl v) c). -Rewrite H. -Rewrite (varlist_merge_ok l v). -Auto. -Save. - -Lemma canonical_sum_scalar3_ok : (c:A; l:varlist; s:canonical_sum) - (Aequiv (interp_setcs (canonical_sum_scalar3 c l s)) (Amult c (Amult (interp_vl l) (interp_setcs s)))). -Proof. -Induction s; Simpl; Intros. -Rewrite (SSR_mult_zero_right S T (interp_vl l)). -Auto. - -Rewrite (monom_insert_ok (Amult c a) (varlist_merge l v) - (canonical_sum_scalar3 c l c0)). -Rewrite (ics_aux_ok (interp_m a v) c0). -Rewrite (interp_m_ok a v). -Rewrite H. -Rewrite (varlist_merge_ok l v). -Setoid_replace (Amult (interp_vl l) - (Aplus (Amult a (interp_vl v)) (interp_setcs c0))) - with (Aplus (Amult (interp_vl l) (Amult a (interp_vl v))) - (Amult (interp_vl l) (interp_setcs c0))). -Setoid_replace (Amult c - (Aplus (Amult (interp_vl l) (Amult a (interp_vl v))) - (Amult (interp_vl l) (interp_setcs c0)))) - with (Aplus (Amult c (Amult (interp_vl l) (Amult a (interp_vl v)))) - (Amult c (Amult (interp_vl l) (interp_setcs c0)))). -Setoid_replace (Amult (Amult c a) (Amult (interp_vl l) (interp_vl v))) - with (Amult c (Amult a (Amult (interp_vl l) (interp_vl v)))). -Auto. - -Rewrite (monom_insert_ok c (varlist_merge l v) - (canonical_sum_scalar3 c l c0)). -Rewrite (ics_aux_ok (interp_vl v) c0). -Rewrite H. -Rewrite (varlist_merge_ok l v). -Setoid_replace (Aplus (Amult c (Amult (interp_vl l) (interp_vl v))) - (Amult c (Amult (interp_vl l) (interp_setcs c0)))) - with (Amult c - (Aplus (Amult (interp_vl l) (interp_vl v)) - (Amult (interp_vl l) (interp_setcs c0)))). -Auto. -Save. - -Lemma canonical_sum_prod_ok : (x,y:canonical_sum) - (Aequiv (interp_setcs (canonical_sum_prod x y)) (Amult (interp_setcs x) (interp_setcs y))). -Proof. -Induction x; Simpl; Intros. -Trivial. - -Rewrite (canonical_sum_merge_ok (canonical_sum_scalar3 a v y) - (canonical_sum_prod c y)). -Rewrite (canonical_sum_scalar3_ok a v y). -Rewrite (ics_aux_ok (interp_m a v) c). -Rewrite (interp_m_ok a v). -Rewrite (H y). -Setoid_replace (Amult a (Amult (interp_vl v) (interp_setcs y))) - with (Amult (Amult a (interp_vl v)) (interp_setcs y)). -Setoid_replace (Amult (Aplus (Amult a (interp_vl v)) (interp_setcs c)) - (interp_setcs y)) - with (Aplus (Amult (Amult a (interp_vl v)) (interp_setcs y)) - (Amult (interp_setcs c) (interp_setcs y))). -Trivial. - -Rewrite (canonical_sum_merge_ok (canonical_sum_scalar2 v y) - (canonical_sum_prod c y)). -Rewrite (canonical_sum_scalar2_ok v y). -Rewrite (ics_aux_ok (interp_vl v) c). -Rewrite (H y). -Trivial. -Save. - -Theorem setspolynomial_normalize_ok : (p:setspolynomial) - (Aequiv (interp_setcs (setspolynomial_normalize p)) (interp_setsp p)). -Proof. -Induction p; Simpl; Intros; Trivial. -Rewrite (canonical_sum_merge_ok (setspolynomial_normalize s) - (setspolynomial_normalize s0)). -Rewrite H; Rewrite H0; Trivial. - -Rewrite (canonical_sum_prod_ok (setspolynomial_normalize s) - (setspolynomial_normalize s0)). -Rewrite H; Rewrite H0; Trivial. -Save. - -Lemma canonical_sum_simplify_ok : (s:canonical_sum) - (Aequiv (interp_setcs (canonical_sum_simplify s)) (interp_setcs s)). -Proof. -Induction s; Simpl; Intros. -Trivial. - -Generalize (SSR_eq_prop T 9!a 10!Azero). -Elim (Aeq a Azero). -Simpl. -Intros. -Rewrite (ics_aux_ok (interp_m a v) c). -Rewrite (interp_m_ok a v). -Rewrite (H0 I). -Setoid_replace (Amult Azero (interp_vl v)) with Azero. -Rewrite H. -Trivial. - -Intros; Simpl. -Generalize (SSR_eq_prop T 9!a 10!Aone). -Elim (Aeq a Aone). -Intros. -Rewrite (ics_aux_ok (interp_m a v) c). -Rewrite (interp_m_ok a v). -Rewrite (H1 I). -Simpl. -Rewrite (ics_aux_ok (interp_vl v) (canonical_sum_simplify c)). -Rewrite H. -Auto. - -Simpl. -Intros. -Rewrite (ics_aux_ok (interp_m a v) (canonical_sum_simplify c)). -Rewrite (ics_aux_ok (interp_m a v) c). -Rewrite H; Trivial. - -Rewrite (ics_aux_ok (interp_vl v) (canonical_sum_simplify c)). -Rewrite H. -Auto. -Save. - -Theorem setspolynomial_simplify_ok : (p:setspolynomial) - (Aequiv (interp_setcs (setspolynomial_simplify p)) (interp_setsp p)). -Proof. -Intro. -Unfold setspolynomial_simplify. -Rewrite (canonical_sum_simplify_ok (setspolynomial_normalize p)). -Exact (setspolynomial_normalize_ok p). -Save. - -End semi_setoid_rings. - -Implicits Cons_varlist. -Implicits Cons_monom. -Implicits SetSPconst. -Implicits SetSPplus. -Implicits SetSPmult. - - - -Section setoid_rings. - -Set Implicit Arguments. - -Variable vm : (varmap A). -Variable T : (Setoid_Ring_Theory Aequiv Aplus Amult Aone Azero Aopp Aeq). - -Hint STh_plus_sym_T := Resolve (STh_plus_sym T). -Hint STh_plus_assoc_T := Resolve (STh_plus_assoc T). -Hint STh_plus_assoc2_T := Resolve (STh_plus_assoc2 S T). -Hint STh_mult_sym_T := Resolve (STh_mult_sym T). -Hint STh_mult_assoc_T := Resolve (STh_mult_assoc T). -Hint STh_mult_assoc2_T := Resolve (STh_mult_assoc2 S T). -Hint STh_plus_zero_left_T := Resolve (STh_plus_zero_left T). -Hint STh_plus_zero_left2_T := Resolve (STh_plus_zero_left2 S T). -Hint STh_mult_one_left_T := Resolve (STh_mult_one_left T). -Hint STh_mult_one_left2_T := Resolve (STh_mult_one_left2 S T). -Hint STh_mult_zero_left_T := Resolve (STh_mult_zero_left S plus_morph mult_morph T). -Hint STh_mult_zero_left2_T := Resolve (STh_mult_zero_left2 S plus_morph mult_morph T). -Hint STh_distr_left_T := Resolve (STh_distr_left T). -Hint STh_distr_left2_T := Resolve (STh_distr_left2 S T). -Hint STh_plus_reg_left_T := Resolve (STh_plus_reg_left S plus_morph T). -Hint STh_plus_permute_T := Resolve (STh_plus_permute S plus_morph T). -Hint STh_mult_permute_T := Resolve (STh_mult_permute S mult_morph T). -Hint STh_distr_right_T := Resolve (STh_distr_right S plus_morph T). -Hint STh_distr_right2_T := Resolve (STh_distr_right2 S plus_morph T). -Hint STh_mult_zero_right_T := Resolve (STh_mult_zero_right S plus_morph mult_morph T). -Hint STh_mult_zero_right2_T := Resolve (STh_mult_zero_right2 S plus_morph mult_morph T). -Hint STh_plus_zero_right_T := Resolve (STh_plus_zero_right S T). -Hint STh_plus_zero_right2_T := Resolve (STh_plus_zero_right2 S T). -Hint STh_mult_one_right_T := Resolve (STh_mult_one_right S T). -Hint STh_mult_one_right2_T := Resolve (STh_mult_one_right2 S T). -Hint STh_plus_reg_right_T := Resolve (STh_plus_reg_right S plus_morph T). -Hints Resolve refl_equal sym_equal trans_equal. -(*Hints Resolve refl_eqT sym_eqT trans_eqT.*) -Hints Immediate T. - - -(*** Definitions *) - -Inductive Type setpolynomial := - SetPvar : index -> setpolynomial -| SetPconst : A -> setpolynomial -| SetPplus : setpolynomial -> setpolynomial -> setpolynomial -| SetPmult : setpolynomial -> setpolynomial -> setpolynomial -| SetPopp : setpolynomial -> setpolynomial. - -Fixpoint setpolynomial_normalize [x:setpolynomial] : canonical_sum := - Cases x of - | (SetPplus l r) => (canonical_sum_merge - (setpolynomial_normalize l) - (setpolynomial_normalize r)) - | (SetPmult l r) => (canonical_sum_prod - (setpolynomial_normalize l) - (setpolynomial_normalize r)) - | (SetPconst c) => (Cons_monom c Nil_var Nil_monom) - | (SetPvar i) => (Cons_varlist (Cons_var i Nil_var) Nil_monom) - | (SetPopp p) => (canonical_sum_scalar3 - (Aopp Aone) Nil_var - (setpolynomial_normalize p)) - end. - -Definition setpolynomial_simplify := - [x:setpolynomial](canonical_sum_simplify (setpolynomial_normalize x)). - -Fixpoint setspolynomial_of [x:setpolynomial] : setspolynomial := - Cases x of - | (SetPplus l r) => (SetSPplus (setspolynomial_of l) (setspolynomial_of r)) - | (SetPmult l r) => (SetSPmult (setspolynomial_of l) (setspolynomial_of r)) - | (SetPconst c) => (SetSPconst c) - | (SetPvar i) => (SetSPvar i) - | (SetPopp p) => (SetSPmult (SetSPconst (Aopp Aone)) (setspolynomial_of p)) - end. - -(*** Interpretation *) - -Fixpoint interp_setp [p:setpolynomial] : A := - Cases p of - | (SetPconst c) => c - | (SetPvar i) => (varmap_find Azero i vm) - | (SetPplus p1 p2) => (Aplus (interp_setp p1) (interp_setp p2)) - | (SetPmult p1 p2) => (Amult (interp_setp p1) (interp_setp p2)) - | (SetPopp p1) => (Aopp (interp_setp p1)) - end. - -(*** Properties *) - -Unset Implicit Arguments. - -Lemma setspolynomial_of_ok : (p:setpolynomial) - (Aequiv (interp_setp p) (interp_setsp vm (setspolynomial_of p))). -Induction p; Trivial; Simpl; Intros. -Rewrite H; Rewrite H0; Trivial. -Rewrite H; Rewrite H0; Trivial. -Rewrite H. -Rewrite (STh_opp_mult_left2 S plus_morph mult_morph T Aone - (interp_setsp vm (setspolynomial_of s))). -Rewrite (STh_mult_one_left T - (interp_setsp vm (setspolynomial_of s))). -Trivial. -Save. - -Theorem setpolynomial_normalize_ok : (p:setpolynomial) - (setpolynomial_normalize p) - ==(setspolynomial_normalize (setspolynomial_of p)). -Induction p; Trivial; Simpl; Intros. -Rewrite H; Rewrite H0; Reflexivity. -Rewrite H; Rewrite H0; Reflexivity. -Rewrite H; Simpl. -Elim (canonical_sum_scalar3 (Aopp Aone) Nil_var - (setspolynomial_normalize (setspolynomial_of s))); - [ Reflexivity - | Simpl; Intros; Rewrite H0; Reflexivity - | Simpl; Intros; Rewrite H0; Reflexivity ]. -Save. - -Theorem setpolynomial_simplify_ok : (p:setpolynomial) - (Aequiv (interp_setcs vm (setpolynomial_simplify p)) (interp_setp p)). -Intro. -Unfold setpolynomial_simplify. -Rewrite (setspolynomial_of_ok p). -Rewrite setpolynomial_normalize_ok. -Rewrite (canonical_sum_simplify_ok vm - (Semi_Setoid_Ring_Theory_of A Aequiv S Aplus Amult Aone Azero Aopp - Aeq plus_morph mult_morph T) - (setspolynomial_normalize (setspolynomial_of p))). -Rewrite (setspolynomial_normalize_ok vm - (Semi_Setoid_Ring_Theory_of A Aequiv S Aplus Amult Aone Azero Aopp - Aeq plus_morph mult_morph T) (setspolynomial_of p)). -Trivial. -Save. - -End setoid_rings. - -End setoid. |