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diff --git a/contrib/setoid_ring/Field_theory.v b/contrib/setoid_ring/Field_theory.v new file mode 100644 index 00000000..f810859c --- /dev/null +++ b/contrib/setoid_ring/Field_theory.v @@ -0,0 +1,1460 @@ +(************************************************************************) +(* 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 *) +(************************************************************************) + +Require Ring. +Import Ring_polynom Ring_tac Ring_theory InitialRing Setoid List. +Require Import ZArith_base. +Set Implicit Arguments. + +Section MakeFieldPol. + +(* Field elements *) + Variable R:Type. + Variable (rO rI : R) (radd rmul rsub: R->R->R) (ropp : R->R). + Variable (rdiv : R -> R -> R) (rinv : R -> R). + Variable req : R -> R -> Prop. + + Notation "0" := rO. Notation "1" := rI. + Notation "x + y" := (radd x y). Notation "x * y " := (rmul x y). + Notation "x - y " := (rsub x y). Notation "x / y" := (rdiv x y). + Notation "- x" := (ropp x). Notation "/ x" := (rinv x). + Notation "x == y" := (req x y) (at level 70, no associativity). + + (* Equality properties *) + Variable Rsth : Setoid_Theory R req. + Variable Reqe : ring_eq_ext radd rmul ropp req. + Variable SRinv_ext : forall p q, p == q -> / p == / q. + + (* Field properties *) + Record almost_field_theory : Prop := mk_afield { + AF_AR : almost_ring_theory rO rI radd rmul rsub ropp req; + AF_1_neq_0 : ~ 1 == 0; + AFdiv_def : forall p q, p / q == p * / q; + AFinv_l : forall p, ~ p == 0 -> / p * p == 1 + }. + +Section AlmostField. + + Variable AFth : almost_field_theory. + Let ARth := AFth.(AF_AR). + Let rI_neq_rO := AFth.(AF_1_neq_0). + Let rdiv_def := AFth.(AFdiv_def). + Let rinv_l := AFth.(AFinv_l). + + (* Coefficients *) + Variable C: Type. + Variable (cO cI: C) (cadd cmul csub : C->C->C) (copp : C->C). + Variable ceqb : C->C->bool. + Variable phi : C -> R. + + Variable CRmorph : ring_morph rO rI radd rmul rsub ropp req + cO cI cadd cmul csub copp ceqb phi. + +Lemma ceqb_rect : forall c1 c2 (A:Type) (x y:A) (P:A->Type), + (phi c1 == phi c2 -> P x) -> P y -> P (if ceqb c1 c2 then x else y). +Proof. +intros. +generalize (fun h => X (morph_eq CRmorph c1 c2 h)). +case (ceqb c1 c2); auto. +Qed. + + + (* C notations *) + Notation "x +! y" := (cadd x y) (at level 50). + Notation "x *! y " := (cmul x y) (at level 40). + Notation "x -! y " := (csub x y) (at level 50). + Notation "-! x" := (copp x) (at level 35). + Notation " x ?=! y" := (ceqb x y) (at level 70, no associativity). + Notation "[ x ]" := (phi x) (at level 0). + + + (* Usefull tactics *) + Add Setoid R req Rsth as R_set1. + Add Morphism radd : radd_ext. exact (Radd_ext Reqe). Qed. + Add Morphism rmul : rmul_ext. exact (Rmul_ext Reqe). Qed. + Add Morphism ropp : ropp_ext. exact (Ropp_ext Reqe). Qed. + Add Morphism rsub : rsub_ext. exact (ARsub_ext Rsth Reqe ARth). Qed. + Add Morphism rinv : rinv_ext. exact SRinv_ext. Qed. + +Let eq_trans := Setoid.Seq_trans _ _ Rsth. +Let eq_sym := Setoid.Seq_sym _ _ Rsth. +Let eq_refl := Setoid.Seq_refl _ _ Rsth. + +Hint Resolve eq_refl rdiv_def rinv_l rI_neq_rO CRmorph.(morph1) . +Hint Resolve (Rmul_ext Reqe) (Rmul_ext Reqe) (Radd_ext Reqe) + (ARsub_ext Rsth Reqe ARth) (Ropp_ext Reqe) SRinv_ext. +Hint Resolve (ARadd_0_l ARth) (ARadd_comm ARth) (ARadd_assoc ARth) + (ARmul_1_l ARth) (ARmul_0_l ARth) + (ARmul_comm ARth) (ARmul_assoc ARth) (ARdistr_l ARth) + (ARopp_mul_l ARth) (ARopp_add ARth) + (ARsub_def ARth) . + +Notation NPEeval := (PEeval rO radd rmul rsub ropp phi). +Notation Nnorm := (norm cO cI cadd cmul csub copp ceqb). +Notation NPphi_dev := (Pphi_dev rO rI radd rmul cO cI ceqb phi). + +(* add abstract semi-ring to help with some proofs *) +Add Ring Rring : (ARth_SRth ARth). + + +(* additional ring properties *) + +Lemma rsub_0_l : forall r, 0 - r == - r. +intros; rewrite (ARsub_def ARth) in |- *; ring. +Qed. + +Lemma rsub_0_r : forall r, r - 0 == r. +intros; rewrite (ARsub_def ARth) in |- *. +rewrite (ARopp_zero Rsth Reqe ARth) in |- *; ring. +Qed. + +(*************************************************************************** + + Properties of division + + ***************************************************************************) + +Theorem rdiv_simpl: forall p q, ~ q == 0 -> q * (p / q) == p. +intros p q H. +rewrite rdiv_def in |- *. +transitivity (/ q * q * p); [ ring | idtac ]. +rewrite rinv_l in |- *; auto. +Qed. +Hint Resolve rdiv_simpl . + +Theorem SRdiv_ext: + forall p1 p2, p1 == p2 -> forall q1 q2, q1 == q2 -> p1 / q1 == p2 / q2. +intros p1 p2 H q1 q2 H0. +transitivity (p1 * / q1); auto. +transitivity (p2 * / q2); auto. +Qed. +Hint Resolve SRdiv_ext . + + Add Morphism rdiv : rdiv_ext. exact SRdiv_ext. Qed. + +Lemma rmul_reg_l : forall p q1 q2, + ~ p == 0 -> p * q1 == p * q2 -> q1 == q2. +intros. +rewrite <- (@rdiv_simpl q1 p) in |- *; trivial. +rewrite <- (@rdiv_simpl q2 p) in |- *; trivial. +repeat rewrite rdiv_def in |- *. +repeat rewrite (ARmul_assoc ARth) in |- *. +auto. +Qed. + +Theorem field_is_integral_domain : forall r1 r2, + ~ r1 == 0 -> ~ r2 == 0 -> ~ r1 * r2 == 0. +Proof. +red in |- *; intros. +apply H0. +transitivity (1 * r2); auto. +transitivity (/ r1 * r1 * r2); auto. +rewrite <- (ARmul_assoc ARth) in |- *. +rewrite H1 in |- *. +apply ARmul_0_r with (1 := Rsth) (2 := ARth). +Qed. + +Theorem ropp_neq_0 : forall r, + ~ -(1) == 0 -> ~ r == 0 -> ~ -r == 0. +intros. +setoid_replace (- r) with (- (1) * r). + apply field_is_integral_domain; trivial. + rewrite <- (ARopp_mul_l ARth) in |- *. + rewrite (ARmul_1_l ARth) in |- *. + reflexivity. +Qed. + +Theorem rdiv_r_r : forall r, ~ r == 0 -> r / r == 1. +intros. +rewrite (AFdiv_def AFth) in |- *. +rewrite (ARmul_comm ARth) in |- *. +apply (AFinv_l AFth). +trivial. +Qed. + +Theorem rdiv1: forall r, r == r / 1. +intros r; transitivity (1 * (r / 1)); auto. +Qed. + +Theorem rdiv2: + forall r1 r2 r3 r4, + ~ r2 == 0 -> + ~ r4 == 0 -> + r1 / r2 + r3 / r4 == (r1 * r4 + r3 * r2) / (r2 * r4). +Proof. +intros r1 r2 r3 r4 H H0. +assert (~ r2 * r4 == 0) by complete (apply field_is_integral_domain; trivial). +apply rmul_reg_l with (r2 * r4); trivial. +rewrite rdiv_simpl in |- *; trivial. +rewrite (ARdistr_r Rsth Reqe ARth) in |- *. +apply (Radd_ext Reqe). + transitivity (r2 * (r1 / r2) * r4); [ ring | auto ]. + transitivity (r2 * (r4 * (r3 / r4))); auto. + transitivity (r2 * r3); auto. +Qed. + + +Theorem rdiv2b: + forall r1 r2 r3 r4 r5, + ~ (r2*r5) == 0 -> + ~ (r4*r5) == 0 -> + r1 / (r2*r5) + r3 / (r4*r5) == (r1 * r4 + r3 * r2) / (r2 * (r4 * r5)). +Proof. +intros r1 r2 r3 r4 r5 H H0. +assert (HH1: ~ r2 == 0) by (intros HH; case H; rewrite HH; ring). +assert (HH2: ~ r5 == 0) by (intros HH; case H; rewrite HH; ring). +assert (HH3: ~ r4 == 0) by (intros HH; case H0; rewrite HH; ring). +assert (HH4: ~ r2 * (r4 * r5) == 0) + by complete (repeat apply field_is_integral_domain; trivial). +apply rmul_reg_l with (r2 * (r4 * r5)); trivial. +rewrite rdiv_simpl in |- *; trivial. +rewrite (ARdistr_r Rsth Reqe ARth) in |- *. +apply (Radd_ext Reqe). + transitivity ((r2 * r5) * (r1 / (r2 * r5)) * r4); [ ring | auto ]. + transitivity ((r4 * r5) * (r3 / (r4 * r5)) * r2); [ ring | auto ]. +Qed. + +Theorem rdiv5: forall r1 r2, - (r1 / r2) == - r1 / r2. +intros r1 r2. +transitivity (- (r1 * / r2)); auto. +transitivity (- r1 * / r2); auto. +Qed. +Hint Resolve rdiv5 . + +Theorem rdiv3: + forall r1 r2 r3 r4, + ~ r2 == 0 -> + ~ r4 == 0 -> + r1 / r2 - r3 / r4 == (r1 * r4 - r3 * r2) / (r2 * r4). +intros r1 r2 r3 r4 H H0. +assert (~ r2 * r4 == 0) by (apply field_is_integral_domain; trivial). +transitivity (r1 / r2 + - (r3 / r4)); auto. +transitivity (r1 / r2 + - r3 / r4); auto. +transitivity ((r1 * r4 + - r3 * r2) / (r2 * r4)); auto. +apply rdiv2; auto. +apply SRdiv_ext; auto. +transitivity (r1 * r4 + - (r3 * r2)); symmetry; auto. +Qed. + + +Theorem rdiv3b: + forall r1 r2 r3 r4 r5, + ~ (r2 * r5) == 0 -> + ~ (r4 * r5) == 0 -> + r1 / (r2*r5) - r3 / (r4*r5) == (r1 * r4 - r3 * r2) / (r2 * (r4 * r5)). +Proof. +intros r1 r2 r3 r4 r5 H H0. +transitivity (r1 / (r2 * r5) + - (r3 / (r4 * r5))); auto. +transitivity (r1 / (r2 * r5) + - r3 / (r4 * r5)); auto. +transitivity ((r1 * r4 + - r3 * r2) / (r2 * (r4 * r5))). +apply rdiv2b; auto; try ring. +apply (SRdiv_ext); auto. +transitivity (r1 * r4 + - (r3 * r2)); symmetry; auto. +Qed. + +Theorem rdiv6: + forall r1 r2, + ~ r1 == 0 -> ~ r2 == 0 -> / (r1 / r2) == r2 / r1. +intros r1 r2 H H0. +assert (~ r1 / r2 == 0) as Hk. + intros H1; case H. + transitivity (r2 * (r1 / r2)); auto. + rewrite H1 in |- *; ring. + apply rmul_reg_l with (r1 / r2); auto. + transitivity (/ (r1 / r2) * (r1 / r2)); auto. + transitivity 1; auto. + repeat rewrite rdiv_def in |- *. + transitivity (/ r1 * r1 * (/ r2 * r2)); [ idtac | ring ]. + repeat rewrite rinv_l in |- *; auto. +Qed. +Hint Resolve rdiv6 . + + Theorem rdiv4: + forall r1 r2 r3 r4, + ~ r2 == 0 -> + ~ r4 == 0 -> + (r1 / r2) * (r3 / r4) == (r1 * r3) / (r2 * r4). +Proof. +intros r1 r2 r3 r4 H H0. +assert (~ r2 * r4 == 0) by complete (apply field_is_integral_domain; trivial). +apply rmul_reg_l with (r2 * r4); trivial. +rewrite rdiv_simpl in |- *; trivial. +transitivity (r2 * (r1 / r2) * (r4 * (r3 / r4))); [ ring | idtac ]. +repeat rewrite rdiv_simpl in |- *; trivial. +Qed. + + Theorem rdiv7: + forall r1 r2 r3 r4, + ~ r2 == 0 -> + ~ r3 == 0 -> + ~ r4 == 0 -> + (r1 / r2) / (r3 / r4) == (r1 * r4) / (r2 * r3). +Proof. +intros. +rewrite (rdiv_def (r1 / r2)) in |- *. +rewrite rdiv6 in |- *; trivial. +apply rdiv4; trivial. +Qed. + +Theorem rdiv8: forall r1 r2, ~ r2 == 0 -> r1 == 0 -> r1 / r2 == 0. +intros r1 r2 H H0. +transitivity (r1 * / r2); auto. +transitivity (0 * / r2); auto. +Qed. + + +Theorem cross_product_eq : forall r1 r2 r3 r4, + ~ r2 == 0 -> ~ r4 == 0 -> r1 * r4 == r3 * r2 -> r1 / r2 == r3 / r4. +intros. +transitivity (r1 / r2 * (r4 / r4)). + rewrite rdiv_r_r in |- *; trivial. + symmetry in |- *. + apply (ARmul_1_r Rsth ARth). + rewrite rdiv4 in |- *; trivial. + rewrite H1 in |- *. + rewrite (ARmul_comm ARth r2 r4) in |- *. + rewrite <- rdiv4 in |- *; trivial. + rewrite rdiv_r_r in |- *. + trivial. + apply (ARmul_1_r Rsth ARth). +Qed. + +(*************************************************************************** + + Some equality test + + ***************************************************************************) + +Fixpoint positive_eq (p1 p2 : positive) {struct p1} : bool := + match p1, p2 with + xH, xH => true + | xO p3, xO p4 => positive_eq p3 p4 + | xI p3, xI p4 => positive_eq p3 p4 + | _, _ => false + end. + +Theorem positive_eq_correct: + forall p1 p2, if positive_eq p1 p2 then p1 = p2 else p1 <> p2. +intros p1; elim p1; + (try (intros p2; case p2; simpl; auto; intros; discriminate)). +intros p3 rec p2; case p2; simpl; auto; (try (intros; discriminate)); intros p4. +generalize (rec p4); case (positive_eq p3 p4); auto. +intros H1; apply f_equal with ( f := xI ); auto. +intros H1 H2; case H1; injection H2; auto. +intros p3 rec p2; case p2; simpl; auto; (try (intros; discriminate)); intros p4. +generalize (rec p4); case (positive_eq p3 p4); auto. +intros H1; apply f_equal with ( f := xO ); auto. +intros H1 H2; case H1; injection H2; auto. +Qed. + +(* equality test *) +Fixpoint PExpr_eq (e1 e2 : PExpr C) {struct e1} : bool := + match e1, e2 with + PEc c1, PEc c2 => ceqb c1 c2 + | PEX p1, PEX p2 => positive_eq p1 p2 + | PEadd e3 e5, PEadd e4 e6 => if PExpr_eq e3 e4 then PExpr_eq e5 e6 else false + | PEsub e3 e5, PEsub e4 e6 => if PExpr_eq e3 e4 then PExpr_eq e5 e6 else false + | PEmul e3 e5, PEmul e4 e6 => if PExpr_eq e3 e4 then PExpr_eq e5 e6 else false + | PEopp e3, PEopp e4 => PExpr_eq e3 e4 + | _, _ => false + end. + +Theorem PExpr_eq_semi_correct: + forall l e1 e2, PExpr_eq e1 e2 = true -> NPEeval l e1 == NPEeval l e2. +intros l e1; elim e1. +intros c1; intros e2; elim e2; simpl; (try (intros; discriminate)). +intros c2; apply (morph_eq CRmorph). +intros p1; intros e2; elim e2; simpl; (try (intros; discriminate)). +intros p2; generalize (positive_eq_correct p1 p2); case (positive_eq p1 p2); + (try (intros; discriminate)); intros H; rewrite H; auto. +intros e3 rec1 e5 rec2 e2; case e2; simpl; (try (intros; discriminate)). +intros e4 e6; generalize (rec1 e4); case (PExpr_eq e3 e4); + (try (intros; discriminate)); generalize (rec2 e6); case (PExpr_eq e5 e6); + (try (intros; discriminate)); auto. +intros e3 rec1 e5 rec2 e2; case e2; simpl; (try (intros; discriminate)). +intros e4 e6; generalize (rec1 e4); case (PExpr_eq e3 e4); + (try (intros; discriminate)); generalize (rec2 e6); case (PExpr_eq e5 e6); + (try (intros; discriminate)); auto. +intros e3 rec1 e5 rec2 e2; case e2; simpl; (try (intros; discriminate)). +intros e4 e6; generalize (rec1 e4); case (PExpr_eq e3 e4); + (try (intros; discriminate)); generalize (rec2 e6); case (PExpr_eq e5 e6); + (try (intros; discriminate)); auto. +intros e3 rec e2; (case e2; simpl; (try (intros; discriminate))). +intros e4; generalize (rec e4); case (PExpr_eq e3 e4); + (try (intros; discriminate)); auto. +Qed. + +(* add *) +Definition NPEadd e1 e2 := + match e1, e2 with + PEc c1, PEc c2 => PEc (cadd c1 c2) + | PEc c, _ => if ceqb c cO then e2 else PEadd e1 e2 + | _, PEc c => if ceqb c cO then e1 else PEadd e1 e2 + | _, _ => PEadd e1 e2 + end. + +Theorem NPEadd_correct: + forall l e1 e2, NPEeval l (NPEadd e1 e2) == NPEeval l (PEadd e1 e2). +Proof. +intros l e1 e2. +destruct e1; destruct e2; simpl in |- *; try reflexivity; try apply ceqb_rect; + try (intro eq_c; rewrite eq_c in |- *); simpl in |- *; + try rewrite (morph0 CRmorph) in |- *; try ring. +apply (morph_add CRmorph). +Qed. + +(* mul *) +Definition NPEmul x y := + match x, y with + PEc c1, PEc c2 => PEc (cmul c1 c2) + | PEc c, _ => + if ceqb c cI then y else if ceqb c cO then PEc cO else PEmul x y + | _, PEc c => + if ceqb c cI then x else if ceqb c cO then PEc cO else PEmul x y + | _, _ => PEmul x y + end. + +Theorem NPEmul_correct : forall l e1 e2, + NPEeval l (NPEmul e1 e2) == NPEeval l (PEmul e1 e2). +intros l e1 e2. +destruct e1; destruct e2; simpl in |- *; try reflexivity; + repeat apply ceqb_rect; + try (intro eq_c; rewrite eq_c in |- *); simpl in |- *; + try rewrite (morph0 CRmorph) in |- *; + try rewrite (morph1 CRmorph) in |- *; + try ring. +apply (morph_mul CRmorph). +Qed. + +(* sub *) +Definition NPEsub e1 e2 := + match e1, e2 with + PEc c1, PEc c2 => PEc (csub c1 c2) + | PEc c, _ => if ceqb c cO then PEopp e2 else PEsub e1 e2 + | _, PEc c => if ceqb c cO then e1 else PEsub e1 e2 + | _, _ => PEsub e1 e2 + end. + +Theorem NPEsub_correct: + forall l e1 e2, NPEeval l (NPEsub e1 e2) == NPEeval l (PEsub e1 e2). +intros l e1 e2. +destruct e1; destruct e2; simpl in |- *; try reflexivity; try apply ceqb_rect; + try (intro eq_c; rewrite eq_c in |- *); simpl in |- *; + try rewrite (morph0 CRmorph) in |- *; try reflexivity; + try (symmetry; apply rsub_0_l); try (symmetry; apply rsub_0_r). +apply (morph_sub CRmorph). +Qed. + +(* opp *) +Definition NPEopp e1 := + match e1 with PEc c1 => PEc (copp c1) | _ => PEopp e1 end. + +Theorem NPEopp_correct: + forall l e1, NPEeval l (NPEopp e1) == NPEeval l (PEopp e1). +intros l e1; case e1; simpl; auto. +intros; apply (morph_opp CRmorph). +Qed. + +(* simplification *) +Fixpoint PExpr_simp (e : PExpr C) : PExpr C := + match e with + PEadd e1 e2 => NPEadd (PExpr_simp e1) (PExpr_simp e2) + | PEmul e1 e2 => NPEmul (PExpr_simp e1) (PExpr_simp e2) + | PEsub e1 e2 => NPEsub (PExpr_simp e1) (PExpr_simp e2) + | PEopp e1 => NPEopp (PExpr_simp e1) + | _ => e + end. + +Theorem PExpr_simp_correct: + forall l e, NPEeval l (PExpr_simp e) == NPEeval l e. +intros l e; elim e; simpl; auto. +intros e1 He1 e2 He2. +transitivity (NPEeval l (PEadd (PExpr_simp e1) (PExpr_simp e2))); auto. +apply NPEadd_correct. +simpl; auto. +intros e1 He1 e2 He2. +transitivity (NPEeval l (PEsub (PExpr_simp e1) (PExpr_simp e2))); auto. +apply NPEsub_correct. +simpl; auto. +intros e1 He1 e2 He2. +transitivity (NPEeval l (PEmul (PExpr_simp e1) (PExpr_simp e2))); auto. +apply NPEmul_correct. +simpl; auto. +intros e1 He1. +transitivity (NPEeval l (PEopp (PExpr_simp e1))); auto. +apply NPEopp_correct. +simpl; auto. +Qed. + + +(**************************************************************************** + + Datastructure + + ***************************************************************************) + +(* The input: syntax of a field expression *) + +Inductive FExpr : Type := + FEc: C -> FExpr + | FEX: positive -> FExpr + | FEadd: FExpr -> FExpr -> FExpr + | FEsub: FExpr -> FExpr -> FExpr + | FEmul: FExpr -> FExpr -> FExpr + | FEopp: FExpr -> FExpr + | FEinv: FExpr -> FExpr + | FEdiv: FExpr -> FExpr -> FExpr . + +Fixpoint FEeval (l : list R) (pe : FExpr) {struct pe} : R := + match pe with + | FEc c => phi c + | FEX x => BinList.nth 0 x l + | FEadd x y => FEeval l x + FEeval l y + | FEsub x y => FEeval l x - FEeval l y + | FEmul x y => FEeval l x * FEeval l y + | FEopp x => - FEeval l x + | FEinv x => / FEeval l x + | FEdiv x y => FEeval l x / FEeval l y + end. + +(* The result of the normalisation *) + +Record linear : Type := mk_linear { + num : PExpr C; + denum : PExpr C; + condition : list (PExpr C) }. + +(*************************************************************************** + + Semantics and properties of side condition + + ***************************************************************************) + +Fixpoint PCond (l : list R) (le : list (PExpr C)) {struct le} : Prop := + match le with + | nil => True + | e1 :: nil => ~ req (PEeval rO radd rmul rsub ropp phi l e1) rO + | e1 :: l1 => ~ req (PEeval rO radd rmul rsub ropp phi l e1) rO /\ PCond l l1 + end. + +Theorem PCond_cons_inv_l : + forall l a l1, PCond l (a::l1) -> ~ NPEeval l a == 0. +intros l a l1 H. +destruct l1; simpl in H |- *; trivial. +destruct H; trivial. +Qed. + +Theorem PCond_cons_inv_r : forall l a l1, PCond l (a :: l1) -> PCond l l1. +intros l a l1 H. +destruct l1; simpl in H |- *; trivial. +destruct H; trivial. +Qed. + +Theorem PCond_app_inv_l: forall l l1 l2, PCond l (l1 ++ l2) -> PCond l l1. +intros l l1 l2; elim l1; simpl app in |- *. + simpl in |- *; auto. + destruct l0; simpl in *. + destruct l2; firstorder. + firstorder. +Qed. + +Theorem PCond_app_inv_r: forall l l1 l2, PCond l (l1 ++ l2) -> PCond l l2. +intros l l1 l2; elim l1; simpl app; auto. +intros a l0 H H0; apply H; apply PCond_cons_inv_r with ( 1 := H0 ). +Qed. + +(* An unsatisfiable condition: issued when a division by zero is detected *) +Definition absurd_PCond := cons (PEc cO) nil. + +Lemma absurd_PCond_bottom : forall l, ~ PCond l absurd_PCond. +unfold absurd_PCond in |- *; simpl in |- *. +red in |- *; intros. +apply H. +apply (morph0 CRmorph). +Qed. + +(*************************************************************************** + + Normalisation + + ***************************************************************************) + + +Fixpoint isIn (e1 e2: PExpr C) {struct e2}: option (PExpr C) := + match e2 with + | PEmul e3 e4 => + match isIn e1 e3 with + Some e5 => Some (NPEmul e5 e4) + | None => match isIn e1 e4 with + | Some e5 => Some (NPEmul e3 e5) + | None => None + end + end + | _ => + if PExpr_eq e1 e2 then Some (PEc cI) else None + end. + +Theorem isIn_correct: forall l e1 e2, + match isIn e1 e2 with + (Some e3) => NPEeval l e2 == NPEeval l (NPEmul e1 e3) + | _ => True + end. +Proof. +intros l e1 e2; elim e2; simpl; auto. + intros c; + generalize (PExpr_eq_semi_correct l e1 (PEc c)); + case (PExpr_eq e1 (PEc c)); simpl; auto; intros H. + rewrite NPEmul_correct; simpl; auto. + rewrite H; auto; simpl. + rewrite (morph1 CRmorph); rewrite (ARmul_1_r Rsth ARth); auto. + intros p; + generalize (PExpr_eq_semi_correct l e1 (PEX C p)); + case (PExpr_eq e1 (PEX C p)); simpl; auto; intros H. + rewrite NPEmul_correct; simpl; auto. + rewrite H; auto; simpl. + rewrite (morph1 CRmorph); rewrite (ARmul_1_r Rsth ARth); auto. + intros p Hrec p1 Hrec1. + generalize (PExpr_eq_semi_correct l e1 (PEadd p p1)); + case (PExpr_eq e1 (PEadd p p1)); simpl; auto; intros H. + rewrite NPEmul_correct; simpl; auto. + rewrite H; auto; simpl. + rewrite (morph1 CRmorph); rewrite (ARmul_1_r Rsth ARth); auto. + intros p Hrec p1 Hrec1. + generalize (PExpr_eq_semi_correct l e1 (PEsub p p1)); + case (PExpr_eq e1 (PEsub p p1)); simpl; auto; intros H. + rewrite NPEmul_correct; simpl; auto. + rewrite H; auto; simpl. + rewrite (morph1 CRmorph); rewrite (ARmul_1_r Rsth ARth); auto. + intros p; case (isIn e1 p). + intros p2 Hrec p1 Hrec1. + rewrite Hrec; auto; simpl. + repeat (rewrite NPEmul_correct; simpl; auto). + intros _ p1; case (isIn e1 p1); auto. + intros p2 H; rewrite H. + repeat (rewrite NPEmul_correct; simpl; auto). + ring. + intros p; + generalize (PExpr_eq_semi_correct l e1 (PEopp p)); + case (PExpr_eq e1 (PEopp p)); simpl; auto; intros H. + rewrite NPEmul_correct; simpl; auto. + rewrite H; auto; simpl. + rewrite (morph1 CRmorph); rewrite (ARmul_1_r Rsth ARth); auto. +Qed. + +Record rsplit : Type := mk_rsplit { + rsplit_left : PExpr C; + rsplit_common : PExpr C; + rsplit_right : PExpr C}. + +(* Stupid name clash *) +Let left := rsplit_left. +Let right := rsplit_right. +Let common := rsplit_common. + +Fixpoint split (e1 e2: PExpr C) {struct e1}: rsplit := + match e1 with + | PEmul e3 e4 => + let r1 := split e3 e2 in + let r2 := split e4 (right r1) in + mk_rsplit (NPEmul (left r1) (left r2)) + (NPEmul (common r1) (common r2)) + (right r2) + | _ => + match isIn e1 e2 with + Some e3 => mk_rsplit (PEc cI) e1 e3 + | None => mk_rsplit e1 (PEc cI) e2 + end + end. + +Theorem split_correct: forall l e1 e2, + NPEeval l e1 == NPEeval l (NPEmul (left (split e1 e2)) + (common (split e1 e2))) +/\ + NPEeval l e2 == NPEeval l (NPEmul (right (split e1 e2)) + (common (split e1 e2))). +Proof. +intros l e1; elim e1; simpl; auto. + intros c e2; generalize (isIn_correct l (PEc c) e2); + case (isIn (PEc c) e2); auto; intros p; + [intros Hp1; rewrite Hp1 | idtac]; + simpl left; simpl common; simpl right; auto; + repeat rewrite NPEmul_correct; simpl; split; + try rewrite (morph1 CRmorph); ring. + intros p e2; generalize (isIn_correct l (PEX C p) e2); + case (isIn (PEX C p) e2); auto; intros p1; + [intros Hp1; rewrite Hp1 | idtac]; + simpl left; simpl common; simpl right; auto; + repeat rewrite NPEmul_correct; simpl; split; + try rewrite (morph1 CRmorph); ring. + intros p1 _ p2 _ e2; generalize (isIn_correct l (PEadd p1 p2) e2); + case (isIn (PEadd p1 p2) e2); auto; intros p; + [intros Hp1; rewrite Hp1 | idtac]; + simpl left; simpl common; simpl right; auto; + repeat rewrite NPEmul_correct; simpl; split; + try rewrite (morph1 CRmorph); ring. + intros p1 _ p2 _ e2; generalize (isIn_correct l (PEsub p1 p2) e2); + case (isIn (PEsub p1 p2) e2); auto; intros p; + [intros Hp1; rewrite Hp1 | idtac]; + simpl left; simpl common; simpl right; auto; + repeat rewrite NPEmul_correct; simpl; split; + try rewrite (morph1 CRmorph); ring. + intros p1 Hp1 p2 Hp2 e2. + repeat rewrite NPEmul_correct; simpl; split. + case (Hp1 e2); case (Hp2 (right (split p1 e2))). + intros tmp1 _ tmp2 _; rewrite tmp1; rewrite tmp2. + repeat rewrite NPEmul_correct; simpl. + ring. + case (Hp1 e2); case (Hp2 (right (split p1 e2))). + intros _ tmp1 _ tmp2; rewrite tmp2; + repeat rewrite NPEmul_correct; simpl. + rewrite tmp1. + repeat rewrite NPEmul_correct; simpl. + ring. + intros p _ e2; generalize (isIn_correct l (PEopp p) e2); + case (isIn (PEopp p) e2); auto; intros p1; + [intros Hp1; rewrite Hp1 | idtac]; + simpl left; simpl common; simpl right; auto; + repeat rewrite NPEmul_correct; simpl; split; + try rewrite (morph1 CRmorph); ring. +Qed. + + +Theorem split_correct_l: forall l e1 e2, + NPEeval l e1 == NPEeval l (NPEmul (left (split e1 e2)) + (common (split e1 e2))). +Proof. +intros l e1 e2; case (split_correct l e1 e2); auto. +Qed. + +Theorem split_correct_r: forall l e1 e2, + NPEeval l e2 == NPEeval l (NPEmul (right (split e1 e2)) + (common (split e1 e2))). +Proof. +intros l e1 e2; case (split_correct l e1 e2); auto. +Qed. + +Fixpoint Fnorm (e : FExpr) : linear := + match e with + | FEc c => mk_linear (PEc c) (PEc cI) nil + | FEX x => mk_linear (PEX C x) (PEc cI) nil + | FEadd e1 e2 => + let x := Fnorm e1 in + let y := Fnorm e2 in + let s := split (denum x) (denum y) in + mk_linear + (NPEadd (NPEmul (num x) (right s)) (NPEmul (num y) (left s))) + (NPEmul (left s) (NPEmul (right s) (common s))) + (condition x ++ condition y) + + | FEsub e1 e2 => + let x := Fnorm e1 in + let y := Fnorm e2 in + let s := split (denum x) (denum y) in + mk_linear + (NPEsub (NPEmul (num x) (right s)) (NPEmul (num y) (left s))) + (NPEmul (left s) (NPEmul (right s) (common s))) + (condition x ++ condition y) + | FEmul e1 e2 => + let x := Fnorm e1 in + let y := Fnorm e2 in + mk_linear (NPEmul (num x) (num y)) + (NPEmul (denum x) (denum y)) + (condition x ++ condition y) + | FEopp e1 => + let x := Fnorm e1 in + mk_linear (NPEopp (num x)) (denum x) (condition x) + | FEinv e1 => + let x := Fnorm e1 in + mk_linear (denum x) (num x) (num x :: condition x) + | FEdiv e1 e2 => + let x := Fnorm e1 in + let y := Fnorm e2 in + mk_linear (NPEmul (num x) (denum y)) + (NPEmul (denum x) (num y)) + (num y :: condition x ++ condition y) + end. + + +(* Example *) +(* +Eval compute + in (Fnorm + (FEdiv + (FEc cI) + (FEadd (FEinv (FEX xH%positive)) (FEinv (FEX (xO xH)%positive))))). +*) + +Theorem Pcond_Fnorm: + forall l e, + PCond l (condition (Fnorm e)) -> ~ NPEeval l (denum (Fnorm e)) == 0. +intros l e; elim e. + simpl in |- *; intros _ _; rewrite (morph1 CRmorph) in |- *; exact rI_neq_rO. + simpl in |- *; intros _ _; rewrite (morph1 CRmorph) in |- *; exact rI_neq_rO. + intros e1 Hrec1 e2 Hrec2 Hcond. + simpl condition in Hcond. + simpl denum in |- *. + rewrite NPEmul_correct in |- *. + simpl in |- *. + apply field_is_integral_domain. + intros HH; case Hrec1; auto. + apply PCond_app_inv_l with (1 := Hcond). + rewrite (split_correct_l l (denum (Fnorm e1)) (denum (Fnorm e2))). + rewrite NPEmul_correct; simpl; rewrite HH; ring. + intros HH; case Hrec2; auto. + apply PCond_app_inv_r with (1 := Hcond). + rewrite (split_correct_r l (denum (Fnorm e1)) (denum (Fnorm e2))); auto. + intros e1 Hrec1 e2 Hrec2 Hcond. + simpl condition in Hcond. + simpl denum in |- *. + rewrite NPEmul_correct in |- *. + simpl in |- *. + apply field_is_integral_domain. + intros HH; case Hrec1; auto. + apply PCond_app_inv_l with (1 := Hcond). + rewrite (split_correct_l l (denum (Fnorm e1)) (denum (Fnorm e2))). + rewrite NPEmul_correct; simpl; rewrite HH; ring. + intros HH; case Hrec2; auto. + apply PCond_app_inv_r with (1 := Hcond). + rewrite (split_correct_r l (denum (Fnorm e1)) (denum (Fnorm e2))); auto. + intros e1 Hrec1 e2 Hrec2 Hcond. + simpl condition in Hcond. + simpl denum in |- *. + rewrite NPEmul_correct in |- *. + simpl in |- *. + apply field_is_integral_domain. + apply Hrec1. + apply PCond_app_inv_l with (1 := Hcond). + apply Hrec2. + apply PCond_app_inv_r with (1 := Hcond). + intros e1 Hrec1 Hcond. + simpl condition in Hcond. + simpl denum in |- *. + auto. + intros e1 Hrec1 Hcond. + simpl condition in Hcond. + simpl denum in |- *. + apply PCond_cons_inv_l with (1:=Hcond). + intros e1 Hrec1 e2 Hrec2 Hcond. + simpl condition in Hcond. + simpl denum in |- *. + rewrite NPEmul_correct in |- *. + simpl in |- *. + apply field_is_integral_domain. + apply Hrec1. + specialize PCond_cons_inv_r with (1:=Hcond); intro Hcond1. + apply PCond_app_inv_l with (1 := Hcond1). + apply PCond_cons_inv_l with (1:=Hcond). +Qed. +Hint Resolve Pcond_Fnorm. + + +(*************************************************************************** + + Main theorem + + ***************************************************************************) + +Theorem Fnorm_FEeval_PEeval: + forall l fe, + PCond l (condition (Fnorm fe)) -> + FEeval l fe == NPEeval l (num (Fnorm fe)) / NPEeval l (denum (Fnorm fe)). +Proof. +intros l fe; elim fe; simpl. +intros c H; rewrite CRmorph.(morph1); apply rdiv1. +intros p H; rewrite CRmorph.(morph1); apply rdiv1. +intros e1 He1 e2 He2 HH. +assert (HH1: PCond l (condition (Fnorm e1))). +apply PCond_app_inv_l with ( 1 := HH ). +assert (HH2: PCond l (condition (Fnorm e2))). +apply PCond_app_inv_r with ( 1 := HH ). +rewrite (He1 HH1); rewrite (He2 HH2). +rewrite NPEadd_correct; simpl. +repeat rewrite NPEmul_correct; simpl. +generalize (split_correct_l l (denum (Fnorm e1)) (denum (Fnorm e2))) + (split_correct_r l (denum (Fnorm e1)) (denum (Fnorm e2))). +repeat rewrite NPEmul_correct; simpl. +intros U1 U2; rewrite U1; rewrite U2. +apply rdiv2b; auto. + rewrite <- U1; auto. + rewrite <- U2; auto. + +intros e1 He1 e2 He2 HH. +assert (HH1: PCond l (condition (Fnorm e1))). +apply PCond_app_inv_l with ( 1 := HH ). +assert (HH2: PCond l (condition (Fnorm e2))). +apply PCond_app_inv_r with ( 1 := HH ). +rewrite (He1 HH1); rewrite (He2 HH2). +rewrite NPEsub_correct; simpl. +repeat rewrite NPEmul_correct; simpl. +generalize (split_correct_l l (denum (Fnorm e1)) (denum (Fnorm e2))) + (split_correct_r l (denum (Fnorm e1)) (denum (Fnorm e2))). +repeat rewrite NPEmul_correct; simpl. +intros U1 U2; rewrite U1; rewrite U2. +apply rdiv3b; auto. + rewrite <- U1; auto. + rewrite <- U2; auto. + +intros e1 He1 e2 He2 HH. +assert (HH1: PCond l (condition (Fnorm e1))). +apply PCond_app_inv_l with ( 1 := HH ). +assert (HH2: PCond l (condition (Fnorm e2))). +apply PCond_app_inv_r with ( 1 := HH ). +rewrite (He1 HH1); rewrite (He2 HH2). +repeat rewrite NPEmul_correct; simpl. +apply rdiv4; auto. + +intros e1 He1 HH. +rewrite NPEopp_correct; simpl; rewrite (He1 HH); apply rdiv5; auto. + +intros e1 He1 HH. +assert (HH1: PCond l (condition (Fnorm e1))). +apply PCond_cons_inv_r with ( 1 := HH ). +rewrite (He1 HH1); apply rdiv6; auto. +apply PCond_cons_inv_l with ( 1 := HH ). + +intros e1 He1 e2 He2 HH. +assert (HH1: PCond l (condition (Fnorm e1))). +apply PCond_app_inv_l with (condition (Fnorm e2)). +apply PCond_cons_inv_r with ( 1 := HH ). +assert (HH2: PCond l (condition (Fnorm e2))). +apply PCond_app_inv_r with (condition (Fnorm e1)). +apply PCond_cons_inv_r with ( 1 := HH ). +rewrite (He1 HH1); rewrite (He2 HH2). +repeat rewrite NPEmul_correct;simpl. +apply rdiv7; auto. +apply PCond_cons_inv_l with ( 1 := HH ). +Qed. + +Theorem Fnorm_crossproduct: + forall l fe1 fe2, + let nfe1 := Fnorm fe1 in + let nfe2 := Fnorm fe2 in + NPEeval l (PEmul (num nfe1) (denum nfe2)) == + NPEeval l (PEmul (num nfe2) (denum nfe1)) -> + PCond l (condition nfe1 ++ condition nfe2) -> + FEeval l fe1 == FEeval l fe2. +intros l fe1 fe2 nfe1 nfe2 Hcrossprod Hcond; subst nfe1 nfe2. +rewrite Fnorm_FEeval_PEeval in |- *. + apply PCond_app_inv_l with (1 := Hcond). + rewrite Fnorm_FEeval_PEeval in |- *. + apply PCond_app_inv_r with (1 := Hcond). + apply cross_product_eq; trivial. + apply Pcond_Fnorm. + apply PCond_app_inv_l with (1 := Hcond). + apply Pcond_Fnorm. + apply PCond_app_inv_r with (1 := Hcond). +Qed. + +(* Correctness lemmas of reflexive tactics *) + +Theorem Fnorm_correct: + forall l fe, + Peq ceqb (Nnorm (num (Fnorm fe))) (Pc cO) = true -> + PCond l (condition (Fnorm fe)) -> FEeval l fe == 0. +intros l fe H H1; + apply eq_trans with (1 := Fnorm_FEeval_PEeval l fe H1). +apply rdiv8; auto. +transitivity (NPEeval l (PEc cO)); auto. +apply (ring_correct Rsth Reqe ARth CRmorph); auto. +simpl; apply (morph0 CRmorph); auto. +Qed. + +(* simplify a field expression into a fraction *) +(* TODO: simplify when den is constant... *) +Definition display_linear l num den := + NPphi_dev l num / NPphi_dev l den. + +Theorem Pphi_dev_div_ok: + forall l fe nfe, + Fnorm fe = nfe -> + PCond l (condition nfe) -> + FEeval l fe == display_linear l (Nnorm (num nfe)) (Nnorm (denum nfe)). +Proof. + intros l fe nfe eq_nfe H; subst nfe. + apply eq_trans with (1 := Fnorm_FEeval_PEeval _ _ H). + unfold display_linear; apply SRdiv_ext; + apply (Pphi_dev_ok Rsth Reqe ARth CRmorph); reflexivity. +Qed. + +(* solving a field equation *) +Theorem Field_correct : + forall l fe1 fe2, + forall nfe1, Fnorm fe1 = nfe1 -> + forall nfe2, Fnorm fe2 = nfe2 -> + Peq ceqb (Nnorm (PEmul (num nfe1) (denum nfe2))) + (Nnorm (PEmul (num nfe2) (denum nfe1))) = true -> + PCond l (condition nfe1 ++ condition nfe2) -> + FEeval l fe1 == FEeval l fe2. +Proof. +intros l fe1 fe2 nfe1 eq1 nfe2 eq2 Hnorm Hcond; subst nfe1 nfe2. +apply Fnorm_crossproduct; trivial. +apply (ring_correct Rsth Reqe ARth CRmorph); trivial. +Qed. + +(* simplify a field equation : generate the crossproduct and simplify + polynomials *) +Theorem Field_simplify_eq_old_correct : + forall l fe1 fe2 nfe1 nfe2, + Fnorm fe1 = nfe1 -> + Fnorm fe2 = nfe2 -> + NPphi_dev l (Nnorm (PEmul (num nfe1) (denum nfe2))) == + NPphi_dev l (Nnorm (PEmul (num nfe2) (denum nfe1))) -> + PCond l (condition nfe1 ++ condition nfe2) -> + FEeval l fe1 == FEeval l fe2. +Proof. +intros l fe1 fe2 nfe1 nfe2 eq1 eq2 Hcrossprod Hcond; subst nfe1 nfe2. +apply Fnorm_crossproduct; trivial. +rewrite (Pphi_dev_gen_ok Rsth Reqe ARth CRmorph) in |- *. +rewrite (Pphi_dev_gen_ok Rsth Reqe ARth CRmorph) in |- *. +trivial. +Qed. + +Theorem Field_simplify_eq_correct : + forall l fe1 fe2, + forall nfe1, Fnorm fe1 = nfe1 -> + forall nfe2, Fnorm fe2 = nfe2 -> + forall den, split (denum nfe1) (denum nfe2) = den -> + NPphi_dev l (Nnorm (PEmul (num nfe1) (right den))) == + NPphi_dev l (Nnorm (PEmul (num nfe2) (left den))) -> + PCond l (condition nfe1 ++ condition nfe2) -> + FEeval l fe1 == FEeval l fe2. +Proof. +intros l fe1 fe2 nfe1 eq1 nfe2 eq2 den eq3 Hcrossprod Hcond; + subst nfe1 nfe2 den. +apply Fnorm_crossproduct; trivial. +simpl in |- *. +elim (split_correct l (denum (Fnorm fe1)) (denum (Fnorm fe2))); intros. +rewrite H in |- *. +rewrite H0 in |- *. +clear H H0. +rewrite NPEmul_correct in |- *. +rewrite NPEmul_correct in |- *. +simpl in |- *. +repeat rewrite (ARmul_assoc ARth) in |- *. +rewrite <- (Pphi_dev_gen_ok Rsth Reqe ARth CRmorph) in Hcrossprod. +rewrite <- (Pphi_dev_gen_ok Rsth Reqe ARth CRmorph) in Hcrossprod. +simpl in Hcrossprod. +rewrite Hcrossprod in |- *. +reflexivity. +Qed. + +Section Fcons_impl. + +Variable Fcons : PExpr C -> list (PExpr C) -> list (PExpr C). + +Hypothesis PCond_fcons_inv : forall l a l1, + PCond l (Fcons a l1) -> ~ NPEeval l a == 0 /\ PCond l l1. + +Fixpoint Fapp (l m:list (PExpr C)) {struct l} : list (PExpr C) := + match l with + | nil => m + | cons a l1 => Fcons a (Fapp l1 m) + end. + +Lemma fcons_correct : forall l l1, + PCond l (Fapp l1 nil) -> PCond l l1. +induction l1; simpl in |- *; intros. + trivial. + elim PCond_fcons_inv with (1 := H); intros. + destruct l1; auto. +Qed. + +End Fcons_impl. + +Section Fcons_simpl. + +(* Some general simpifications of the condition: eliminate duplicates, + split multiplications *) + +Fixpoint Fcons (e:PExpr C) (l:list (PExpr C)) {struct l} : list (PExpr C) := + match l with + nil => cons e nil + | cons a l1 => if PExpr_eq e a then l else cons a (Fcons e l1) + end. + +Theorem PFcons_fcons_inv: + forall l a l1, PCond l (Fcons a l1) -> ~ NPEeval l a == 0 /\ PCond l l1. +intros l a l1; elim l1; simpl Fcons; auto. +simpl; auto. +intros a0 l0. +generalize (PExpr_eq_semi_correct l a a0); case (PExpr_eq a a0). +intros H H0 H1; split; auto. +rewrite H; auto. +generalize (PCond_cons_inv_l _ _ _ H1); simpl; auto. +intros H H0 H1; + assert (Hp: ~ NPEeval l a0 == 0 /\ (~ NPEeval l a == 0 /\ PCond l l0)). +split. +generalize (PCond_cons_inv_l _ _ _ H1); simpl; auto. +apply H0. +generalize (PCond_cons_inv_r _ _ _ H1); simpl; auto. +generalize Hp; case l0; simpl; intuition. +Qed. + +(* equality of normal forms rather than syntactic equality *) +Fixpoint Fcons0 (e:PExpr C) (l:list (PExpr C)) {struct l} : list (PExpr C) := + match l with + nil => cons e nil + | cons a l1 => + if Peq ceqb (Nnorm e) (Nnorm a) then l else cons a (Fcons0 e l1) + end. + +Theorem PFcons0_fcons_inv: + forall l a l1, PCond l (Fcons0 a l1) -> ~ NPEeval l a == 0 /\ PCond l l1. +intros l a l1; elim l1; simpl Fcons0; auto. +simpl; auto. +intros a0 l0. +generalize (ring_correct Rsth Reqe ARth CRmorph l a a0); + case (Peq ceqb (Nnorm a) (Nnorm a0)). +intros H H0 H1; split; auto. +rewrite H; auto. +generalize (PCond_cons_inv_l _ _ _ H1); simpl; auto. +intros H H0 H1; + assert (Hp: ~ NPEeval l a0 == 0 /\ (~ NPEeval l a == 0 /\ PCond l l0)). +split. +generalize (PCond_cons_inv_l _ _ _ H1); simpl; auto. +apply H0. +generalize (PCond_cons_inv_r _ _ _ H1); simpl; auto. +generalize Hp; case l0; simpl; intuition. +Qed. + +Fixpoint Fcons00 (e:PExpr C) (l:list (PExpr C)) {struct e} : list (PExpr C) := + match e with + PEmul e1 e2 => Fcons00 e1 (Fcons00 e2 l) + | _ => Fcons0 e l + end. + +Theorem PFcons00_fcons_inv: + forall l a l1, PCond l (Fcons00 a l1) -> ~ NPEeval l a == 0 /\ PCond l l1. +intros l a; elim a; try (intros; apply PFcons0_fcons_inv; auto; fail). + intros p H p0 H0 l1 H1. + simpl in H1. + case (H _ H1); intros H2 H3. + case (H0 _ H3); intros H4 H5; split; auto. + simpl in |- *. + apply field_is_integral_domain; trivial. +Qed. + + +Definition Pcond_simpl_gen := + fcons_correct _ PFcons00_fcons_inv. + + +(* Specific case when the equality test of coefs is complete w.r.t. the + field equality: non-zero coefs can be eliminated, and opposite can + be simplified (if -1 <> 0) *) + +Hypothesis ceqb_complete : forall c1 c2, phi c1 == phi c2 -> ceqb c1 c2 = true. + +Lemma ceqb_rect_complete : forall c1 c2 (A:Type) (x y:A) (P:A->Type), + (phi c1 == phi c2 -> P x) -> + (~ phi c1 == phi c2 -> P y) -> + P (if ceqb c1 c2 then x else y). +Proof. +intros. +generalize (fun h => X (morph_eq CRmorph c1 c2 h)). +generalize (@ceqb_complete c1 c2). +case (c1 ?=! c2); auto; intros. +apply X0. +red in |- *; intro. +absurd (false = true); auto; discriminate. +Qed. + +Fixpoint Fcons1 (e:PExpr C) (l:list (PExpr C)) {struct e} : list (PExpr C) := + match e with + PEmul e1 e2 => Fcons1 e1 (Fcons1 e2 l) + | PEopp e => if ceqb (copp cI) cO then absurd_PCond else Fcons1 e l + | PEc c => if ceqb c cO then absurd_PCond else l + | _ => Fcons0 e l + end. + +Theorem PFcons1_fcons_inv: + forall l a l1, PCond l (Fcons1 a l1) -> ~ NPEeval l a == 0 /\ PCond l l1. +intros l a; elim a; try (intros; apply PFcons0_fcons_inv; auto; fail). + simpl in |- *; intros c l1. + apply ceqb_rect_complete; intros. + elim (@absurd_PCond_bottom l H0). + split; trivial. + rewrite <- (morph0 CRmorph) in |- *; trivial. + intros p H p0 H0 l1 H1. + simpl in H1. + case (H _ H1); intros H2 H3. + case (H0 _ H3); intros H4 H5; split; auto. + simpl in |- *. + apply field_is_integral_domain; trivial. + simpl in |- *; intros p H l1. + apply ceqb_rect_complete; intros. + elim (@absurd_PCond_bottom l H1). + destruct (H _ H1). + split; trivial. + apply ropp_neq_0; trivial. + rewrite (morph_opp CRmorph) in H0. + rewrite (morph1 CRmorph) in H0. + rewrite (morph0 CRmorph) in H0. + trivial. +Qed. + +Definition Fcons2 e l := Fcons1 (PExpr_simp e) l. + +Theorem PFcons2_fcons_inv: + forall l a l1, PCond l (Fcons2 a l1) -> ~ NPEeval l a == 0 /\ PCond l l1. +unfold Fcons2 in |- *; intros l a l1 H; split; + case (PFcons1_fcons_inv l (PExpr_simp a) l1); auto. +intros H1 H2 H3; case H1. +transitivity (NPEeval l a); trivial. +apply PExpr_simp_correct. +Qed. + +Definition Pcond_simpl_complete := + fcons_correct _ PFcons2_fcons_inv. + +End Fcons_simpl. + +Let Mpc := MPcond_map cO cI cadd cmul csub copp ceqb. +Let Mp := MPcond_dev rO rI radd rmul req cO cI ceqb phi. +Let Subst := PNSubstL cO cI cadd cmul ceqb. + +(* simplification + rewriting *) +Theorem Field_subst_correct : +forall l ul fe m n, + PCond l (Fapp Fcons00 (condition (Fnorm fe)) nil) -> + Mp (Mpc ul) l -> + Peq ceqb (Subst (Nnorm (num (Fnorm fe))) (Mpc ul) m n) (Pc cO) = true -> + FEeval l fe == 0. +intros l ul fe m n H H1 H2. +assert (H3 := (Pcond_simpl_gen _ _ H)). +apply eq_trans with (1 := Fnorm_FEeval_PEeval l fe + (Pcond_simpl_gen _ _ H)). +apply rdiv8; auto. +rewrite (PNSubstL_dev_ok Rsth Reqe ARth CRmorph m n + _ (num (Fnorm fe)) l H1). +rewrite <-(Ring_polynom.Pphi_Pphi_dev Rsth Reqe ARth CRmorph). +rewrite (fun x => Peq_ok Rsth Reqe CRmorph x (Pc cO)); auto. +simpl; apply (morph0 CRmorph); auto. +Qed. + + +End AlmostField. + +Section FieldAndSemiField. + + Record field_theory : Prop := mk_field { + F_R : ring_theory rO rI radd rmul rsub ropp req; + F_1_neq_0 : ~ 1 == 0; + Fdiv_def : forall p q, p / q == p * / q; + Finv_l : forall p, ~ p == 0 -> / p * p == 1 + }. + + Definition F2AF f := + mk_afield + (Rth_ARth Rsth Reqe f.(F_R)) f.(F_1_neq_0) f.(Fdiv_def) f.(Finv_l). + + Record semi_field_theory : Prop := mk_sfield { + SF_SR : semi_ring_theory rO rI radd rmul req; + SF_1_neq_0 : ~ 1 == 0; + SFdiv_def : forall p q, p / q == p * / q; + SFinv_l : forall p, ~ p == 0 -> / p * p == 1 + }. + +End FieldAndSemiField. + +End MakeFieldPol. + + Definition SF2AF R (rO rI:R) radd rmul rdiv rinv req Rsth + (sf:semi_field_theory rO rI radd rmul rdiv rinv req) := + mk_afield _ _ + (SRth_ARth Rsth sf.(SF_SR)) + sf.(SF_1_neq_0) + sf.(SFdiv_def) + sf.(SFinv_l). + + +Section Complete. + Variable R : Type. + Variable (rO rI : R) (radd rmul rsub: R->R->R) (ropp : R -> R). + Variable (rdiv : R -> R -> R) (rinv : R -> R). + Variable req : R -> R -> Prop. + Notation "0" := rO. Notation "1" := rI. + Notation "x + y" := (radd x y). Notation "x * y " := (rmul x y). + Notation "x - y " := (rsub x y). Notation "- x" := (ropp x). + Notation "x / y " := (rdiv x y). Notation "/ x" := (rinv x). + Notation "x == y" := (req x y) (at level 70, no associativity). + Variable Rsth : Setoid_Theory R req. + Add Setoid R req Rsth as R_setoid3. + Variable Reqe : ring_eq_ext radd rmul ropp req. + Add Morphism radd : radd_ext3. exact (Radd_ext Reqe). Qed. + Add Morphism rmul : rmul_ext3. exact (Rmul_ext Reqe). Qed. + Add Morphism ropp : ropp_ext3. exact (Ropp_ext Reqe). Qed. + +Section AlmostField. + + Variable AFth : almost_field_theory rO rI radd rmul rsub ropp rdiv rinv req. + Let ARth := AFth.(AF_AR). + Let rI_neq_rO := AFth.(AF_1_neq_0). + Let rdiv_def := AFth.(AFdiv_def). + Let rinv_l := AFth.(AFinv_l). + +Hypothesis S_inj : forall x y, 1+x==1+y -> x==y. + +Hypothesis gen_phiPOS_not_0 : forall p, ~ gen_phiPOS1 rI radd rmul p == 0. + +Lemma add_inj_r : forall p x y, + gen_phiPOS1 rI radd rmul p + x == gen_phiPOS1 rI radd rmul p + y -> x==y. +intros p x y. +elim p using Pind; simpl in |- *; intros. + apply S_inj; trivial. + apply H. + apply S_inj. + repeat rewrite (ARadd_assoc ARth) in |- *. + rewrite <- (ARgen_phiPOS_Psucc Rsth Reqe ARth) in |- *; trivial. +Qed. + +Lemma gen_phiPOS_inj : forall x y, + gen_phiPOS rI radd rmul x == gen_phiPOS rI radd rmul y -> + x = y. +intros x y. +repeat rewrite <- (same_gen Rsth Reqe ARth) in |- *. +ElimPcompare x y; intro. + intros. + apply Pcompare_Eq_eq; trivial. + intro. + elim gen_phiPOS_not_0 with (y - x)%positive. + apply add_inj_r with x. + symmetry in |- *. + rewrite (ARadd_0_r Rsth ARth) in |- *. + rewrite <- (ARgen_phiPOS_add Rsth Reqe ARth) in |- *. + rewrite Pplus_minus in |- *; trivial. + change Eq with (CompOpp Eq) in |- *. + rewrite <- Pcompare_antisym in |- *; trivial. + rewrite H in |- *; trivial. + intro. + elim gen_phiPOS_not_0 with (x - y)%positive. + apply add_inj_r with y. + rewrite (ARadd_0_r Rsth ARth) in |- *. + rewrite <- (ARgen_phiPOS_add Rsth Reqe ARth) in |- *. + rewrite Pplus_minus in |- *; trivial. +Qed. + + +Lemma gen_phiN_inj : forall x y, + gen_phiN rO rI radd rmul x == gen_phiN rO rI radd rmul y -> + x = y. +destruct x; destruct y; simpl in |- *; intros; trivial. + elim gen_phiPOS_not_0 with p. + symmetry in |- *. + rewrite (same_gen Rsth Reqe ARth) in |- *; trivial. + elim gen_phiPOS_not_0 with p. + rewrite (same_gen Rsth Reqe ARth) in |- *; trivial. + rewrite gen_phiPOS_inj with (1 := H) in |- *; trivial. +Qed. + +Lemma gen_phiN_complete : forall x y, + gen_phiN rO rI radd rmul x == gen_phiN rO rI radd rmul y -> + Neq_bool x y = true. +intros. + replace y with x. + unfold Neq_bool in |- *. + rewrite Ncompare_refl in |- *; trivial. + apply gen_phiN_inj; trivial. +Qed. + +End AlmostField. + +Section Field. + + Variable Fth : field_theory rO rI radd rmul rsub ropp rdiv rinv req. + Let Rth := Fth.(F_R). + Let rI_neq_rO := Fth.(F_1_neq_0). + Let rdiv_def := Fth.(Fdiv_def). + Let rinv_l := Fth.(Finv_l). + Let AFth := F2AF Rsth Reqe Fth. + Let ARth := Rth_ARth Rsth Reqe Rth. + +Lemma ring_S_inj : forall x y, 1+x==1+y -> x==y. +intros. +transitivity (x + (1 + - (1))). + rewrite (Ropp_def Rth) in |- *. + symmetry in |- *. + apply (ARadd_0_r Rsth ARth). + transitivity (y + (1 + - (1))). + repeat rewrite <- (ARplus_assoc ARth) in |- *. + repeat rewrite (ARadd_assoc ARth) in |- *. + apply (Radd_ext Reqe). + repeat rewrite <- (ARadd_comm ARth 1) in |- *. + trivial. + reflexivity. + rewrite (Ropp_def Rth) in |- *. + apply (ARadd_0_r Rsth ARth). +Qed. + + + Hypothesis gen_phiPOS_not_0 : forall p, ~ gen_phiPOS1 rI radd rmul p == 0. + +Let gen_phiPOS_inject := + gen_phiPOS_inj AFth ring_S_inj gen_phiPOS_not_0. + +Lemma gen_phiPOS_discr_sgn : forall x y, + ~ gen_phiPOS rI radd rmul x == - gen_phiPOS rI radd rmul y. +red in |- *; intros. +apply gen_phiPOS_not_0 with (y + x)%positive. +rewrite (ARgen_phiPOS_add Rsth Reqe ARth) in |- *. +transitivity (gen_phiPOS1 1 radd rmul y + - gen_phiPOS1 1 radd rmul y). + apply (Radd_ext Reqe); trivial. + reflexivity. + rewrite (same_gen Rsth Reqe ARth) in |- *. + rewrite (same_gen Rsth Reqe ARth) in |- *. + trivial. + apply (Ropp_def Rth). +Qed. + +Lemma gen_phiZ_inj : forall x y, + gen_phiZ rO rI radd rmul ropp x == gen_phiZ rO rI radd rmul ropp y -> + x = y. +destruct x; destruct y; simpl in |- *; intros. + trivial. + elim gen_phiPOS_not_0 with p. + rewrite (same_gen Rsth Reqe ARth) in |- *. + symmetry in |- *; trivial. + elim gen_phiPOS_not_0 with p. + rewrite (same_gen Rsth Reqe ARth) in |- *. + rewrite <- (Ropp_opp Rsth Reqe Rth (gen_phiPOS 1 radd rmul p)) in |- *. + rewrite <- H in |- *. + apply (ARopp_zero Rsth Reqe ARth). + elim gen_phiPOS_not_0 with p. + rewrite (same_gen Rsth Reqe ARth) in |- *. + trivial. + rewrite gen_phiPOS_inject with (1 := H) in |- *; trivial. + elim gen_phiPOS_discr_sgn with (1 := H). + elim gen_phiPOS_not_0 with p. + rewrite (same_gen Rsth Reqe ARth) in |- *. + rewrite <- (Ropp_opp Rsth Reqe Rth (gen_phiPOS 1 radd rmul p)) in |- *. + rewrite H in |- *. + apply (ARopp_zero Rsth Reqe ARth). + elim gen_phiPOS_discr_sgn with p0 p. + symmetry in |- *; trivial. + replace p0 with p; trivial. + apply gen_phiPOS_inject. + rewrite <- (Ropp_opp Rsth Reqe Rth (gen_phiPOS 1 radd rmul p)) in |- *. + rewrite <- (Ropp_opp Rsth Reqe Rth (gen_phiPOS 1 radd rmul p0)) in |- *. + rewrite H in |- *; trivial. + reflexivity. +Qed. + +Lemma gen_phiZ_complete : forall x y, + gen_phiZ rO rI radd rmul ropp x == gen_phiZ rO rI radd rmul ropp y -> + Zeq_bool x y = true. +intros. + replace y with x. + unfold Zeq_bool in |- *. + rewrite Zcompare_refl in |- *; trivial. + apply gen_phiZ_inj; trivial. +Qed. + +End Field. + +End Complete. + |