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diff --git a/plugins/ring/Ring_normalize.v b/plugins/ring/Ring_normalize.v new file mode 100644 index 00000000..7aeee218 --- /dev/null +++ b/plugins/ring/Ring_normalize.v @@ -0,0 +1,902 @@ +(************************************************************************) +(* 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 *) +(************************************************************************) + +(* $Id$ *) + +Require Import LegacyRing_theory. +Require Import Quote. + +Set Implicit Arguments. +Unset Boxed Definitions. + +Lemma index_eq_prop : forall n m:index, Is_true (index_eq n m) -> n = m. +Proof. + intros. + apply index_eq_prop. + generalize H. + case (index_eq n m); simpl in |- *; trivial; intros. + contradiction. +Qed. + +Section semi_rings. + +Variable A : Type. +Variable Aplus : A -> A -> A. +Variable Amult : A -> A -> A. +Variable Aone : A. +Variable Azero : A. +Variable Aeq : A -> A -> bool. + +(* 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) {struct y} : bool := + match x, y with + | 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) {struct y} : bool := + match x, y with + | 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 := + match l1 with + | Cons_var v1 t1 => + (fix vm_aux (l2:varlist) : varlist := + match l2 with + | 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 => fun l2 => l2 + end. + +(* returns the sum of two canonical sums *) +Fixpoint canonical_sum_merge (s1:canonical_sum) : + canonical_sum -> canonical_sum := + match s1 with + | Cons_monom c1 l1 t1 => + (fix csm_aux (s2:canonical_sum) : canonical_sum := + match s2 with + | 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 (s2:canonical_sum) : canonical_sum := + match s2 with + | 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 => fun s2 => s2 + end. + +(* Insertion of a monom in a canonical sum *) +Fixpoint monom_insert (c1:A) (l1:varlist) (s2:canonical_sum) {struct s2} : + canonical_sum := + match s2 with + | 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) {struct s2} : + canonical_sum := + match s2 with + | 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) {struct s} : + canonical_sum := + match s with + | 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) {struct s} : + canonical_sum := + match s with + | 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) {struct s} : canonical_sum := + match s with + | 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 s2:canonical_sum) {struct s1} : + canonical_sum := + match s1 with + | 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-ring polynomials *) +Inductive spolynomial : Type := + | SPvar : index -> spolynomial + | SPconst : A -> spolynomial + | SPplus : spolynomial -> spolynomial -> spolynomial + | SPmult : spolynomial -> spolynomial -> spolynomial. + +Fixpoint spolynomial_normalize (p:spolynomial) : canonical_sum := + match p with + | SPvar i => Cons_varlist (Cons_var i Nil_var) Nil_monom + | SPconst c => Cons_monom c Nil_var Nil_monom + | SPplus l r => + canonical_sum_merge (spolynomial_normalize l) (spolynomial_normalize r) + | SPmult l r => + canonical_sum_prod (spolynomial_normalize l) (spolynomial_normalize r) + end. + +(* Deletion of useless 0 and 1 in canonical sums *) +Fixpoint canonical_sum_simplify (s:canonical_sum) : canonical_sum := + match s with + | 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 spolynomial_simplify (x:spolynomial) := + canonical_sum_simplify (spolynomial_normalize x). + +(* End definitions. *) + +(* Section interpretation. *) + +(*** Here a variable map is defined and the interpetation of a spolynom + acording to a certain variables map. Once again the choosen definition + is generic and could be changed ****) + +Variable vm : varmap A. + +(* Interpretation of list of variables + * [x1; ... ; xn ] is interpreted as (find v x1)* ... *(find v xn) + * The unbound variables are mapped to 0. Normally this case sould + * never occur. Since we want only to prove correctness theorems, which form + * is : for any varmap and any spolynom ... this is a safe and pain-saving + * choice *) +Definition interp_var (i:index) := varmap_find Azero i vm. + +(* Local *) Definition ivl_aux := + (fix ivl_aux (x:index) (t:varlist) {struct t} : A := + match t with + | Nil_var => interp_var x + | Cons_var x' t' => Amult (interp_var x) (ivl_aux x' t') + end). + +Definition interp_vl (l:varlist) := + match l with + | Nil_var => Aone + | Cons_var x t => ivl_aux x t + end. + +(* Local *) Definition interp_m (c:A) (l:varlist) := + match l with + | Nil_var => c + | Cons_var x t => Amult c (ivl_aux x t) + end. + +(* Local *) Definition ics_aux := + (fix ics_aux (a:A) (s:canonical_sum) {struct s} : A := + match s with + | 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). + +(* Interpretation of a canonical sum *) +Definition interp_cs (s:canonical_sum) : A := + match s with + | 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_sp (p:spolynomial) : A := + match p with + | SPconst c => c + | SPvar i => interp_var i + | SPplus p1 p2 => Aplus (interp_sp p1) (interp_sp p2) + | SPmult p1 p2 => Amult (interp_sp p1) (interp_sp p2) + end. + + +(* End interpretation. *) + +Unset Implicit Arguments. + +(* Section properties. *) + +Variable T : Semi_Ring_Theory Aplus Amult Aone Azero Aeq. + +Hint Resolve (SR_plus_comm T). +Hint Resolve (SR_plus_assoc T). +Hint Resolve (SR_plus_assoc2 T). +Hint Resolve (SR_mult_comm T). +Hint Resolve (SR_mult_assoc T). +Hint Resolve (SR_mult_assoc2 T). +Hint Resolve (SR_plus_zero_left T). +Hint Resolve (SR_plus_zero_left2 T). +Hint Resolve (SR_mult_one_left T). +Hint Resolve (SR_mult_one_left2 T). +Hint Resolve (SR_mult_zero_left T). +Hint Resolve (SR_mult_zero_left2 T). +Hint Resolve (SR_distr_left T). +Hint Resolve (SR_distr_left2 T). +(*Hint Resolve (SR_plus_reg_left T).*) +Hint Resolve (SR_plus_permute T). +Hint Resolve (SR_mult_permute T). +Hint Resolve (SR_distr_right T). +Hint Resolve (SR_distr_right2 T). +Hint Resolve (SR_mult_zero_right T). +Hint Resolve (SR_mult_zero_right2 T). +Hint Resolve (SR_plus_zero_right T). +Hint Resolve (SR_plus_zero_right2 T). +Hint Resolve (SR_mult_one_right T). +Hint Resolve (SR_mult_one_right2 T). +(*Hint Resolve (SR_plus_reg_right T).*) +Hint Resolve refl_equal sym_equal trans_equal. +(* Hints Resolve refl_eqT sym_eqT trans_eqT. *) +Hint Immediate T. + +Lemma varlist_eq_prop : forall x y:varlist, Is_true (varlist_eq x y) -> x = y. +Proof. + simple induction x; simple induction y; contradiction || (try reflexivity). + simpl in |- *; intros. + generalize (andb_prop2 _ _ H1); intros; elim H2; intros. + rewrite (index_eq_prop _ _ H3); rewrite (H v0 H4); reflexivity. +Qed. + +Remark ivl_aux_ok : + forall (v:varlist) (i:index), + ivl_aux i v = Amult (interp_var i) (interp_vl v). +Proof. + simple induction v; simpl in |- *; intros. + trivial. + rewrite H; trivial. +Qed. + +Lemma varlist_merge_ok : + forall x y:varlist, + interp_vl (varlist_merge x y) = Amult (interp_vl x) (interp_vl y). +Proof. + simple induction x. + simpl in |- *; trivial. + simple induction y. + simpl in |- *; trivial. + simpl in |- *; intros. + elim (index_lt i i0); simpl in |- *; intros. + + repeat rewrite ivl_aux_ok. + rewrite H. simpl in |- *. + rewrite ivl_aux_ok. + eauto. + + repeat rewrite ivl_aux_ok. + rewrite H0. + rewrite ivl_aux_ok. + eauto. +Qed. + +Remark ics_aux_ok : + forall (x:A) (s:canonical_sum), ics_aux x s = Aplus x (interp_cs s). +Proof. + simple induction s; simpl in |- *; intros. + trivial. + reflexivity. + reflexivity. +Qed. + +Remark interp_m_ok : + forall (x:A) (l:varlist), interp_m x l = Amult x (interp_vl l). +Proof. + destruct l as [| i v]. + simpl in |- *; trivial. + reflexivity. +Qed. + +Lemma canonical_sum_merge_ok : + forall x y:canonical_sum, + interp_cs (canonical_sum_merge x y) = Aplus (interp_cs x) (interp_cs y). + +simple induction x; simpl in |- *. +trivial. + +simple induction y; simpl in |- *; intros. +(* monom and nil *) +eauto. + +(* monom and monom *) +generalize (varlist_eq_prop v v0). +elim (varlist_eq v v0). +intros; rewrite (H1 I). +simpl in |- *; repeat rewrite ics_aux_ok; rewrite H. +repeat rewrite interp_m_ok. +rewrite (SR_distr_left T). +repeat rewrite <- (SR_plus_assoc T). +apply f_equal with (f := Aplus (Amult a (interp_vl v0))). +trivial. + +elim (varlist_lt v v0); simpl in |- *. +repeat rewrite ics_aux_ok. +rewrite H; simpl in |- *; rewrite ics_aux_ok; eauto. + +rewrite ics_aux_ok; rewrite H0; repeat rewrite ics_aux_ok; simpl in |- *; + eauto. + +(* monom and varlist *) +generalize (varlist_eq_prop v v0). +elim (varlist_eq v v0). +intros; rewrite (H1 I). +simpl in |- *; repeat rewrite ics_aux_ok; rewrite H. +repeat rewrite interp_m_ok. +rewrite (SR_distr_left T). +repeat rewrite <- (SR_plus_assoc T). +apply f_equal with (f := Aplus (Amult a (interp_vl v0))). +rewrite (SR_mult_one_left T). +trivial. + +elim (varlist_lt v v0); simpl in |- *. +repeat rewrite ics_aux_ok. +rewrite H; simpl in |- *; rewrite ics_aux_ok; eauto. +rewrite ics_aux_ok; rewrite H0; repeat rewrite ics_aux_ok; simpl in |- *; + eauto. + +simple induction y; simpl in |- *; intros. +(* varlist and nil *) +trivial. + +(* varlist and monom *) +generalize (varlist_eq_prop v v0). +elim (varlist_eq v v0). +intros; rewrite (H1 I). +simpl in |- *; repeat rewrite ics_aux_ok; rewrite H. +repeat rewrite interp_m_ok. +rewrite (SR_distr_left T). +repeat rewrite <- (SR_plus_assoc T). +rewrite (SR_mult_one_left T). +apply f_equal with (f := Aplus (interp_vl v0)). +trivial. + +elim (varlist_lt v v0); simpl in |- *. +repeat rewrite ics_aux_ok. +rewrite H; simpl in |- *; rewrite ics_aux_ok; eauto. +rewrite ics_aux_ok; rewrite H0; repeat rewrite ics_aux_ok; simpl in |- *; + eauto. + +(* varlist and varlist *) +generalize (varlist_eq_prop v v0). +elim (varlist_eq v v0). +intros; rewrite (H1 I). +simpl in |- *; repeat rewrite ics_aux_ok; rewrite H. +repeat rewrite interp_m_ok. +rewrite (SR_distr_left T). +repeat rewrite <- (SR_plus_assoc T). +rewrite (SR_mult_one_left T). +apply f_equal with (f := Aplus (interp_vl v0)). +trivial. + +elim (varlist_lt v v0); simpl in |- *. +repeat rewrite ics_aux_ok. +rewrite H; simpl in |- *; rewrite ics_aux_ok; eauto. +rewrite ics_aux_ok; rewrite H0; repeat rewrite ics_aux_ok; simpl in |- *; + eauto. +Qed. + +Lemma monom_insert_ok : + forall (a:A) (l:varlist) (s:canonical_sum), + interp_cs (monom_insert a l s) = + Aplus (Amult a (interp_vl l)) (interp_cs s). +intros; generalize s; simple induction s0. + +simpl in |- *; rewrite interp_m_ok; trivial. + +simpl in |- *; intros. +generalize (varlist_eq_prop l v); elim (varlist_eq l v). +intro Hr; rewrite (Hr I); simpl in |- *; rewrite interp_m_ok; + repeat rewrite ics_aux_ok; rewrite interp_m_ok; rewrite (SR_distr_left T); + eauto. +elim (varlist_lt l v); simpl in |- *; + [ repeat rewrite interp_m_ok; rewrite ics_aux_ok; eauto + | repeat rewrite interp_m_ok; rewrite ics_aux_ok; rewrite H; + rewrite ics_aux_ok; eauto ]. + +simpl in |- *; intros. +generalize (varlist_eq_prop l v); elim (varlist_eq l v). +intro Hr; rewrite (Hr I); simpl in |- *; rewrite interp_m_ok; + repeat rewrite ics_aux_ok; rewrite (SR_distr_left T); + rewrite (SR_mult_one_left T); eauto. +elim (varlist_lt l v); simpl in |- *; + [ repeat rewrite interp_m_ok; rewrite ics_aux_ok; eauto + | repeat rewrite interp_m_ok; rewrite ics_aux_ok; rewrite H; + rewrite ics_aux_ok; eauto ]. +Qed. + +Lemma varlist_insert_ok : + forall (l:varlist) (s:canonical_sum), + interp_cs (varlist_insert l s) = Aplus (interp_vl l) (interp_cs s). +intros; generalize s; simple induction s0. + +simpl in |- *; trivial. + +simpl in |- *; intros. +generalize (varlist_eq_prop l v); elim (varlist_eq l v). +intro Hr; rewrite (Hr I); simpl in |- *; rewrite interp_m_ok; + repeat rewrite ics_aux_ok; rewrite interp_m_ok; rewrite (SR_distr_left T); + rewrite (SR_mult_one_left T); eauto. +elim (varlist_lt l v); simpl in |- *; + [ repeat rewrite interp_m_ok; rewrite ics_aux_ok; eauto + | repeat rewrite interp_m_ok; rewrite ics_aux_ok; rewrite H; + rewrite ics_aux_ok; eauto ]. + +simpl in |- *; intros. +generalize (varlist_eq_prop l v); elim (varlist_eq l v). +intro Hr; rewrite (Hr I); simpl in |- *; rewrite interp_m_ok; + repeat rewrite ics_aux_ok; rewrite (SR_distr_left T); + rewrite (SR_mult_one_left T); eauto. +elim (varlist_lt l v); simpl in |- *; + [ repeat rewrite interp_m_ok; rewrite ics_aux_ok; eauto + | repeat rewrite interp_m_ok; rewrite ics_aux_ok; rewrite H; + rewrite ics_aux_ok; eauto ]. +Qed. + +Lemma canonical_sum_scalar_ok : + forall (a:A) (s:canonical_sum), + interp_cs (canonical_sum_scalar a s) = Amult a (interp_cs s). +simple induction s. +simpl in |- *; eauto. + +simpl in |- *; intros. +repeat rewrite ics_aux_ok. +repeat rewrite interp_m_ok. +rewrite H. +rewrite (SR_distr_right T). +repeat rewrite <- (SR_mult_assoc T). +reflexivity. + +simpl in |- *; intros. +repeat rewrite ics_aux_ok. +repeat rewrite interp_m_ok. +rewrite H. +rewrite (SR_distr_right T). +repeat rewrite <- (SR_mult_assoc T). +reflexivity. +Qed. + +Lemma canonical_sum_scalar2_ok : + forall (l:varlist) (s:canonical_sum), + interp_cs (canonical_sum_scalar2 l s) = Amult (interp_vl l) (interp_cs s). +simple induction s. +simpl in |- *; trivial. + +simpl in |- *; intros. +rewrite monom_insert_ok. +repeat rewrite ics_aux_ok. +repeat rewrite interp_m_ok. +rewrite H. +rewrite varlist_merge_ok. +repeat rewrite (SR_distr_right T). +repeat rewrite <- (SR_mult_assoc T). +repeat rewrite <- (SR_plus_assoc T). +rewrite (SR_mult_permute T a (interp_vl l) (interp_vl v)). +reflexivity. + +simpl in |- *; intros. +rewrite varlist_insert_ok. +repeat rewrite ics_aux_ok. +repeat rewrite interp_m_ok. +rewrite H. +rewrite varlist_merge_ok. +repeat rewrite (SR_distr_right T). +repeat rewrite <- (SR_mult_assoc T). +repeat rewrite <- (SR_plus_assoc T). +reflexivity. +Qed. + +Lemma canonical_sum_scalar3_ok : + forall (c:A) (l:varlist) (s:canonical_sum), + interp_cs (canonical_sum_scalar3 c l s) = + Amult c (Amult (interp_vl l) (interp_cs s)). +simple induction s. +simpl in |- *; repeat rewrite (SR_mult_zero_right T); reflexivity. + +simpl in |- *; intros. +rewrite monom_insert_ok. +repeat rewrite ics_aux_ok. +repeat rewrite interp_m_ok. +rewrite H. +rewrite varlist_merge_ok. +repeat rewrite (SR_distr_right T). +repeat rewrite <- (SR_mult_assoc T). +repeat rewrite <- (SR_plus_assoc T). +rewrite (SR_mult_permute T a (interp_vl l) (interp_vl v)). +reflexivity. + +simpl in |- *; intros. +rewrite monom_insert_ok. +repeat rewrite ics_aux_ok. +repeat rewrite interp_m_ok. +rewrite H. +rewrite varlist_merge_ok. +repeat rewrite (SR_distr_right T). +repeat rewrite <- (SR_mult_assoc T). +repeat rewrite <- (SR_plus_assoc T). +rewrite (SR_mult_permute T c (interp_vl l) (interp_vl v)). +reflexivity. +Qed. + +Lemma canonical_sum_prod_ok : + forall x y:canonical_sum, + interp_cs (canonical_sum_prod x y) = Amult (interp_cs x) (interp_cs y). +simple induction x; simpl in |- *; intros. +trivial. + +rewrite canonical_sum_merge_ok. +rewrite canonical_sum_scalar3_ok. +rewrite ics_aux_ok. +rewrite interp_m_ok. +rewrite H. +rewrite (SR_mult_assoc T a (interp_vl v) (interp_cs y)). +symmetry in |- *. +eauto. + +rewrite canonical_sum_merge_ok. +rewrite canonical_sum_scalar2_ok. +rewrite ics_aux_ok. +rewrite H. +trivial. +Qed. + +Theorem spolynomial_normalize_ok : + forall p:spolynomial, interp_cs (spolynomial_normalize p) = interp_sp p. +simple induction p; simpl in |- *; intros. + +reflexivity. +reflexivity. + +rewrite canonical_sum_merge_ok. +rewrite H; rewrite H0. +reflexivity. + +rewrite canonical_sum_prod_ok. +rewrite H; rewrite H0. +reflexivity. +Qed. + +Lemma canonical_sum_simplify_ok : + forall s:canonical_sum, interp_cs (canonical_sum_simplify s) = interp_cs s. +simple induction s. + +reflexivity. + +(* cons_monom *) +simpl in |- *; intros. +generalize (SR_eq_prop T a Azero). +elim (Aeq a Azero). +intro Heq; rewrite (Heq I). +rewrite H. +rewrite ics_aux_ok. +rewrite interp_m_ok. +rewrite (SR_mult_zero_left T). +trivial. + +intros; simpl in |- *. +generalize (SR_eq_prop T a Aone). +elim (Aeq a Aone). +intro Heq; rewrite (Heq I). +simpl in |- *. +repeat rewrite ics_aux_ok. +rewrite interp_m_ok. +rewrite H. +rewrite (SR_mult_one_left T). +reflexivity. + +simpl in |- *. +repeat rewrite ics_aux_ok. +rewrite interp_m_ok. +rewrite H. +reflexivity. + +(* cons_varlist *) +simpl in |- *; intros. +repeat rewrite ics_aux_ok. +rewrite H. +reflexivity. + +Qed. + +Theorem spolynomial_simplify_ok : + forall p:spolynomial, interp_cs (spolynomial_simplify p) = interp_sp p. +intro. +unfold spolynomial_simplify in |- *. +rewrite canonical_sum_simplify_ok. +apply spolynomial_normalize_ok. +Qed. + +(* End properties. *) +End semi_rings. + +Implicit Arguments Cons_varlist. +Implicit Arguments Cons_monom. +Implicit Arguments SPconst. +Implicit Arguments SPplus. +Implicit Arguments SPmult. + +Section rings. + +(* Here the coercion between Ring and Semi-Ring will be useful *) + +Set Implicit Arguments. + +Variable A : Type. +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 vm : varmap A. +Variable T : Ring_Theory Aplus Amult Aone Azero Aopp Aeq. + +Hint Resolve (Th_plus_comm T). +Hint Resolve (Th_plus_assoc T). +Hint Resolve (Th_plus_assoc2 T). +Hint Resolve (Th_mult_comm T). +Hint Resolve (Th_mult_assoc T). +Hint Resolve (Th_mult_assoc2 T). +Hint Resolve (Th_plus_zero_left T). +Hint Resolve (Th_plus_zero_left2 T). +Hint Resolve (Th_mult_one_left T). +Hint Resolve (Th_mult_one_left2 T). +Hint Resolve (Th_mult_zero_left T). +Hint Resolve (Th_mult_zero_left2 T). +Hint Resolve (Th_distr_left T). +Hint Resolve (Th_distr_left2 T). +(*Hint Resolve (Th_plus_reg_left T).*) +Hint Resolve (Th_plus_permute T). +Hint Resolve (Th_mult_permute T). +Hint Resolve (Th_distr_right T). +Hint Resolve (Th_distr_right2 T). +Hint Resolve (Th_mult_zero_right T). +Hint Resolve (Th_mult_zero_right2 T). +Hint Resolve (Th_plus_zero_right T). +Hint Resolve (Th_plus_zero_right2 T). +Hint Resolve (Th_mult_one_right T). +Hint Resolve (Th_mult_one_right2 T). +(*Hint Resolve (Th_plus_reg_right T).*) +Hint Resolve refl_equal sym_equal trans_equal. +(*Hints Resolve refl_eqT sym_eqT trans_eqT.*) +Hint Immediate T. + +(*** Definitions *) + +Inductive polynomial : Type := + | Pvar : index -> polynomial + | Pconst : A -> polynomial + | Pplus : polynomial -> polynomial -> polynomial + | Pmult : polynomial -> polynomial -> polynomial + | Popp : polynomial -> polynomial. + +Fixpoint polynomial_normalize (x:polynomial) : canonical_sum A := + match x with + | Pplus l r => + canonical_sum_merge Aplus Aone (polynomial_normalize l) + (polynomial_normalize r) + | Pmult l r => + canonical_sum_prod Aplus Amult Aone (polynomial_normalize l) + (polynomial_normalize r) + | Pconst c => Cons_monom c Nil_var (Nil_monom A) + | Pvar i => Cons_varlist (Cons_var i Nil_var) (Nil_monom A) + | Popp p => + canonical_sum_scalar3 Aplus Amult Aone (Aopp Aone) Nil_var + (polynomial_normalize p) + end. + +Definition polynomial_simplify (x:polynomial) := + canonical_sum_simplify Aone Azero Aeq (polynomial_normalize x). + +Fixpoint spolynomial_of (x:polynomial) : spolynomial A := + match x with + | Pplus l r => SPplus (spolynomial_of l) (spolynomial_of r) + | Pmult l r => SPmult (spolynomial_of l) (spolynomial_of r) + | Pconst c => SPconst c + | Pvar i => SPvar A i + | Popp p => SPmult (SPconst (Aopp Aone)) (spolynomial_of p) + end. + +(*** Interpretation *) + +Fixpoint interp_p (p:polynomial) : A := + match p with + | Pconst c => c + | Pvar i => varmap_find Azero i vm + | Pplus p1 p2 => Aplus (interp_p p1) (interp_p p2) + | Pmult p1 p2 => Amult (interp_p p1) (interp_p p2) + | Popp p1 => Aopp (interp_p p1) + end. + +(*** Properties *) + +Unset Implicit Arguments. + +Lemma spolynomial_of_ok : + forall p:polynomial, + interp_p p = interp_sp Aplus Amult Azero vm (spolynomial_of p). +simple induction p; reflexivity || (simpl in |- *; intros). +rewrite H; rewrite H0; reflexivity. +rewrite H; rewrite H0; reflexivity. +rewrite H. +rewrite (Th_opp_mult_left2 T). +rewrite (Th_mult_one_left T). +reflexivity. +Qed. + +Theorem polynomial_normalize_ok : + forall p:polynomial, + polynomial_normalize p = + spolynomial_normalize Aplus Amult Aone (spolynomial_of p). +simple induction p; reflexivity || (simpl in |- *; intros). +rewrite H; rewrite H0; reflexivity. +rewrite H; rewrite H0; reflexivity. +rewrite H; simpl in |- *. +elim + (canonical_sum_scalar3 Aplus Amult Aone (Aopp Aone) Nil_var + (spolynomial_normalize Aplus Amult Aone (spolynomial_of p0))); + [ reflexivity + | simpl in |- *; intros; rewrite H0; reflexivity + | simpl in |- *; intros; rewrite H0; reflexivity ]. +Qed. + +Theorem polynomial_simplify_ok : + forall p:polynomial, + interp_cs Aplus Amult Aone Azero vm (polynomial_simplify p) = interp_p p. +intro. +unfold polynomial_simplify in |- *. +rewrite spolynomial_of_ok. +rewrite polynomial_normalize_ok. +rewrite (canonical_sum_simplify_ok A Aplus Amult Aone Azero Aeq vm T). +rewrite (spolynomial_normalize_ok A Aplus Amult Aone Azero Aeq vm T). +reflexivity. +Qed. + +End rings. + +Infix "+" := Pplus : ring_scope. +Infix "*" := Pmult : ring_scope. +Notation "- x" := (Popp x) : ring_scope. +Notation "[ x ]" := (Pvar x) (at level 0) : ring_scope. + +Delimit Scope ring_scope with ring. |