diff options
Diffstat (limited to 'plugins/micromega/ZMicromega.v')
| -rw-r--r-- | plugins/micromega/ZMicromega.v | 491 |
1 files changed, 265 insertions, 226 deletions
diff --git a/plugins/micromega/ZMicromega.v b/plugins/micromega/ZMicromega.v index d6245681..bdc4671d 100644 --- a/plugins/micromega/ZMicromega.v +++ b/plugins/micromega/ZMicromega.v @@ -1,6 +1,6 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2011 *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012 *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) @@ -8,7 +8,7 @@ (* *) (* Micromega: A reflexive tactic using the Positivstellensatz *) (* *) -(* Frédéric Besson (Irisa/Inria) 2006-2008 *) +(* Frédéric Besson (Irisa/Inria) 2006-2011 *) (* *) (************************************************************************) @@ -34,20 +34,20 @@ Require Import EnvRing. Open Scope Z_scope. -Lemma Zsor : SOR 0 1 Zplus Zmult Zminus Zopp (@eq Z) Zle Zlt. +Lemma Zsor : SOR 0 1 Z.add Z.mul Z.sub Z.opp (@eq Z) Z.le Z.lt. Proof. constructor ; intros ; subst ; try (intuition (auto with zarith)). apply Zsth. apply Zth. - destruct (Ztrichotomy n m) ; intuition (auto with zarith). - apply Zmult_lt_0_compat ; auto. + destruct (Z.lt_trichotomy n m) ; intuition. + apply Z.mul_pos_pos ; auto. Qed. Lemma ZSORaddon : - SORaddon 0 1 Zplus Zmult Zminus Zopp (@eq Z) Zle (* ring elements *) - 0%Z 1%Z Zplus Zmult Zminus Zopp (* coefficients *) - Zeq_bool Zle_bool - (fun x => x) (fun x => x) (pow_N 1 Zmult). + SORaddon 0 1 Z.add Z.mul Z.sub Z.opp (@eq Z) Z.le (* ring elements *) + 0%Z 1%Z Z.add Z.mul Z.sub Z.opp (* coefficients *) + Zeq_bool Z.leb + (fun x => x) (fun x => x) (pow_N 1 Z.mul). Proof. constructor. constructor ; intros ; try reflexivity. @@ -65,20 +65,20 @@ Fixpoint Zeval_expr (env : PolEnv Z) (e: PExpr Z) : Z := | PEX x => env x | PEadd e1 e2 => Zeval_expr env e1 + Zeval_expr env e2 | PEmul e1 e2 => Zeval_expr env e1 * Zeval_expr env e2 - | PEpow e1 n => Zpower (Zeval_expr env e1) (Z_of_N n) + | PEpow e1 n => Z.pow (Zeval_expr env e1) (Z.of_N n) | PEsub e1 e2 => (Zeval_expr env e1) - (Zeval_expr env e2) - | PEopp e => Zopp (Zeval_expr env e) + | PEopp e => Z.opp (Zeval_expr env e) end. -Definition eval_expr := eval_pexpr Zplus Zmult Zminus Zopp (fun x => x) (fun x => x) (pow_N 1 Zmult). +Definition eval_expr := eval_pexpr Z.add Z.mul Z.sub Z.opp (fun x => x) (fun x => x) (pow_N 1 Z.mul). -Lemma ZNpower : forall r n, r ^ Z_of_N n = pow_N 1 Zmult r n. +Lemma ZNpower : forall r n, r ^ Z.of_N n = pow_N 1 Z.mul r n. Proof. destruct n. reflexivity. simpl. - unfold Zpower_pos. - replace (pow_pos Zmult r p) with (1 * (pow_pos Zmult r p)) by ring. + unfold Z.pow_pos. + replace (pow_pos Z.mul r p) with (1 * (pow_pos Z.mul r p)) by ring. generalize 1. induction p; simpl ; intros ; repeat rewrite IHp ; ring. Qed. @@ -94,10 +94,10 @@ Definition Zeval_op2 (o : Op2) : Z -> Z -> Prop := match o with | OpEq => @eq Z | OpNEq => fun x y => ~ x = y -| OpLe => Zle -| OpGe => Zge -| OpLt => Zlt -| OpGt => Zgt +| OpLe => Z.le +| OpGe => Z.ge +| OpLt => Z.lt +| OpGt => Z.gt end. Definition Zeval_formula (env : PolEnv Z) (f : Formula Z):= @@ -105,23 +105,23 @@ Definition Zeval_formula (env : PolEnv Z) (f : Formula Z):= (Zeval_op2 op) (Zeval_expr env lhs) (Zeval_expr env rhs). Definition Zeval_formula' := - eval_formula Zplus Zmult Zminus Zopp (@eq Z) Zle Zlt (fun x => x) (fun x => x) (pow_N 1 Zmult). + eval_formula Z.add Z.mul Z.sub Z.opp (@eq Z) Z.le Z.lt (fun x => x) (fun x => x) (pow_N 1 Z.mul). Lemma Zeval_formula_compat : forall env f, Zeval_formula env f <-> Zeval_formula' env f. Proof. destruct f ; simpl. rewrite Zeval_expr_compat. rewrite Zeval_expr_compat. unfold eval_expr. - generalize (eval_pexpr Zplus Zmult Zminus Zopp (fun x : Z => x) - (fun x : N => x) (pow_N 1 Zmult) env Flhs). - generalize ((eval_pexpr Zplus Zmult Zminus Zopp (fun x : Z => x) - (fun x : N => x) (pow_N 1 Zmult) env Frhs)). + generalize (eval_pexpr Z.add Z.mul Z.sub Z.opp (fun x : Z => x) + (fun x : N => x) (pow_N 1 Z.mul) env Flhs). + generalize ((eval_pexpr Z.add Z.mul Z.sub Z.opp (fun x : Z => x) + (fun x : N => x) (pow_N 1 Z.mul) env Frhs)). destruct Fop ; simpl; intros ; intuition (auto with zarith). Qed. Definition eval_nformula := - eval_nformula 0 Zplus Zmult (@eq Z) Zle Zlt (fun x => x) . + eval_nformula 0 Z.add Z.mul (@eq Z) Z.le Z.lt (fun x => x) . Definition Zeval_op1 (o : Op1) : Z -> Prop := match o with @@ -140,7 +140,7 @@ Qed. Definition ZWitness := Psatz Z. -Definition ZWeakChecker := check_normalised_formulas 0 1 Zplus Zmult Zeq_bool Zle_bool. +Definition ZWeakChecker := check_normalised_formulas 0 1 Z.add Z.mul Zeq_bool Z.leb. Lemma ZWeakChecker_sound : forall (l : list (NFormula Z)) (cm : ZWitness), ZWeakChecker l cm = true -> @@ -154,13 +154,13 @@ Proof. exact H. Qed. -Definition psub := psub Z0 Zplus Zminus Zopp Zeq_bool. +Definition psub := psub Z0 Z.add Z.sub Z.opp Zeq_bool. -Definition padd := padd Z0 Zplus Zeq_bool. +Definition padd := padd Z0 Z.add Zeq_bool. -Definition norm := norm 0 1 Zplus Zmult Zminus Zopp Zeq_bool. +Definition norm := norm 0 1 Z.add Z.mul Z.sub Z.opp Zeq_bool. -Definition eval_pol := eval_pol 0 Zplus Zmult (fun x => x). +Definition eval_pol := eval_pol Z.add Z.mul (fun x => x). Lemma eval_pol_sub : forall env lhs rhs, eval_pol env (psub lhs rhs) = eval_pol env lhs - eval_pol env rhs. Proof. @@ -194,27 +194,27 @@ Definition xnormalise (t:Formula Z) : list (NFormula Z) := | OpLe => (psub lhs (padd rhs (Pc 1)),NonStrict) :: nil end. -Require Import Tauto. +Require Import Tauto BinNums. Definition normalise (t:Formula Z) : cnf (NFormula Z) := List.map (fun x => x::nil) (xnormalise t). -Lemma normalise_correct : forall env t, eval_cnf (eval_nformula env) (normalise t) <-> Zeval_formula env t. +Lemma normalise_correct : forall env t, eval_cnf eval_nformula env (normalise t) <-> Zeval_formula env t. Proof. Opaque padd. unfold normalise, xnormalise ; simpl; intros env t. rewrite Zeval_formula_compat. - unfold eval_cnf. + unfold eval_cnf, eval_clause. destruct t as [lhs o rhs]; case_eq o; simpl; repeat rewrite eval_pol_sub; repeat rewrite eval_pol_add; repeat rewrite <- eval_pol_norm ; simpl in *; unfold eval_expr; - generalize ( eval_pexpr Zplus Zmult Zminus Zopp (fun x : Z => x) - (fun x : BinNat.N => x) (pow_N 1 Zmult) env lhs); - generalize (eval_pexpr Zplus Zmult Zminus Zopp (fun x : Z => x) - (fun x : BinNat.N => x) (pow_N 1 Zmult) env rhs) ; intros z1 z2 ; intros ; subst; + generalize ( eval_pexpr Z.add Z.mul Z.sub Z.opp (fun x : Z => x) + (fun x : N => x) (pow_N 1 Z.mul) env lhs); + generalize (eval_pexpr Z.add Z.mul Z.sub Z.opp (fun x : Z => x) + (fun x : N => x) (pow_N 1 Z.mul) env rhs) ; intros z1 z2 ; intros ; subst; intuition (auto with zarith). Transparent padd. Qed. @@ -235,31 +235,34 @@ Definition xnegate (t:RingMicromega.Formula Z) : list (NFormula Z) := Definition negate (t:RingMicromega.Formula Z) : cnf (NFormula Z) := List.map (fun x => x::nil) (xnegate t). -Lemma negate_correct : forall env t, eval_cnf (eval_nformula env) (negate t) <-> ~ Zeval_formula env t. +Lemma negate_correct : forall env t, eval_cnf eval_nformula env (negate t) <-> ~ Zeval_formula env t. Proof. Proof. Opaque padd. intros env t. rewrite Zeval_formula_compat. unfold negate, xnegate ; simpl. - unfold eval_cnf. + unfold eval_cnf,eval_clause. destruct t as [lhs o rhs]; case_eq o; simpl; repeat rewrite eval_pol_sub; repeat rewrite eval_pol_add; repeat rewrite <- eval_pol_norm ; simpl in *; unfold eval_expr; - generalize ( eval_pexpr Zplus Zmult Zminus Zopp (fun x : Z => x) - (fun x : BinNat.N => x) (pow_N 1 Zmult) env lhs); - generalize (eval_pexpr Zplus Zmult Zminus Zopp (fun x : Z => x) - (fun x : BinNat.N => x) (pow_N 1 Zmult) env rhs) ; intros z1 z2 ; intros ; subst; + generalize ( eval_pexpr Z.add Z.mul Z.sub Z.opp (fun x : Z => x) + (fun x : N => x) (pow_N 1 Z.mul) env lhs); + generalize (eval_pexpr Z.add Z.mul Z.sub Z.opp (fun x : Z => x) + (fun x : N => x) (pow_N 1 Z.mul) env rhs) ; intros z1 z2 ; intros ; subst; intuition (auto with zarith). Transparent padd. Qed. +Definition Zunsat := check_inconsistent 0 Zeq_bool Z.leb. + +Definition Zdeduce := nformula_plus_nformula 0 Z.add Zeq_bool. Definition ZweakTautoChecker (w: list ZWitness) (f : BFormula (Formula Z)) : bool := - @tauto_checker (Formula Z) (NFormula Z) normalise negate ZWitness ZWeakChecker f w. + @tauto_checker (Formula Z) (NFormula Z) Zunsat Zdeduce normalise negate ZWitness ZWeakChecker f w. (* To get a complete checker, the proof format has to be enriched *) @@ -267,36 +270,47 @@ Require Import Zdiv. Open Scope Z_scope. Definition ceiling (a b:Z) : Z := - let (q,r) := Zdiv_eucl a b in + let (q,r) := Z.div_eucl a b in match r with | Z0 => q | _ => q + 1 end. -Lemma narrow_interval_lower_bound : forall a b x, a > 0 -> a * x >= b -> x >= ceiling b a. + +Require Import Znumtheory. + +Lemma Zdivide_ceiling : forall a b, (b | a) -> ceiling a b = Z.div a b. Proof. unfold ceiling. intros. - generalize (Z_div_mod b a H). - destruct (Zdiv_eucl b a). + apply Zdivide_mod in H. + case_eq (Z.div_eucl a b). intros. - destruct H1. - destruct H2. - subst. - destruct (Ztrichotomy z0 0) as [ HH1 | [HH2 | HH3]]; destruct z0 ; try auto with zarith ; try discriminate. - assert (HH :x >= z \/ x < z) by (destruct (Ztrichotomy x z) ; auto with zarith). - destruct HH ;auto. - generalize (Zmult_lt_compat_l _ _ _ H3 H1). - auto with zarith. - clear H2. - assert (HH :x >= z +1 \/ x <= z) by (destruct (Ztrichotomy x z) ; intuition (auto with zarith)). - destruct HH ;auto. - assert (0 < a) by auto with zarith. - generalize (Zmult_lt_0_le_compat_r _ _ _ H2 H1). - intros. - rewrite Zmult_comm in H4. - rewrite (Zmult_comm z) in H4. - auto with zarith. + change z with (fst (z,z0)). + rewrite <- H0. + change (fst (Z.div_eucl a b)) with (Z.div a b). + change z0 with (snd (z,z0)). + rewrite <- H0. + change (snd (Z.div_eucl a b)) with (Z.modulo a b). + rewrite H. + reflexivity. +Qed. + +Lemma narrow_interval_lower_bound a b x : + a > 0 -> a * x >= b -> x >= ceiling b a. +Proof. + rewrite !Z.ge_le_iff. + unfold ceiling. + intros Ha H. + generalize (Z_div_mod b a Ha). + destruct (Z.div_eucl b a) as (q,r). intros (->,(H1,H2)). + destruct r as [|r|r]. + - rewrite Z.add_0_r in H. + apply Z.mul_le_mono_pos_l in H; auto with zarith. + - assert (0 < Z.pos r) by easy. + rewrite Z.add_1_r, Z.le_succ_l. + apply Z.mul_lt_mono_pos_l with a; auto with zarith. + - now elim H1. Qed. (** NB: narrow_interval_upper_bound is Zdiv.Zdiv_le_lower_bound *) @@ -307,40 +321,13 @@ Inductive ZArithProof : Type := | DoneProof | RatProof : ZWitness -> ZArithProof -> ZArithProof | CutProof : ZWitness -> ZArithProof -> ZArithProof -| EnumProof : ZWitness -> ZWitness -> list ZArithProof -> ZArithProof. - -(* n/d <= x -> d*x - n >= 0 *) -(* -Definition makeLb (v:PExpr Z) (q:Q) : NFormula Z := - let (n,d) := q in (PEsub (PEmul (PEc (Zpos d)) v) (PEc n),NonStrict). - -(* x <= n/d -> d * x <= d *) -Definition makeUb (v:PExpr Z) (q:Q) : NFormula Z := - let (n,d) := q in - (PEsub (PEc n) (PEmul (PEc (Zpos d)) v), NonStrict). +| EnumProof : ZWitness -> ZWitness -> list ZArithProof -> ZArithProof +(*| SplitProof : PolC Z -> ZArithProof -> ZArithProof -> ZArithProof*). -Definition qceiling (q:Q) : Z := - let (n,d) := q in ceiling n (Zpos d). -Definition qfloor (q:Q) : Z := - let (n,d) := q in Zdiv n (Zpos d). -Definition makeLbCut (v:PExprC Z) (q:Q) : NFormula Z := - (PEsub v (PEc (qceiling q)), NonStrict). - -Definition neg_nformula (f : NFormula Z) := - let (e,o) := f in - (PEopp (PEadd e (PEc 1%Z)), o). +(* n/d <= x -> d*x - n >= 0 *) -Lemma neg_nformula_sound : forall env f, snd f = NonStrict ->( ~ (Zeval_nformula env (neg_nformula f)) <-> Zeval_nformula env f). -Proof. - unfold neg_nformula. - destruct f. - simpl. - intros ; subst ; simpl in *. - split; auto with zarith. -Qed. -*) (* In order to compute the 'cut', we need to express a polynomial P as a * Q + b. - b is the constant @@ -364,7 +351,7 @@ Proof. destruct x ; simpl ; intuition congruence. Qed. -Definition ZgcdM (x y : Z) := Zmax (Zgcd x y) 1. +Definition ZgcdM (x y : Z) := Z.max (Z.gcd x y) 1. Fixpoint Zgcd_pol (p : PolC Z) : (Z * Z) := @@ -382,7 +369,7 @@ Fixpoint Zgcd_pol (p : PolC Z) : (Z * Z) := Fixpoint Zdiv_pol (p:PolC Z) (x:Z) : PolC Z := match p with - | Pc c => Pc (Zdiv c x) + | Pc c => Pc (Z.div c x) | Pinj j p => Pinj j (Zdiv_pol p x) | PX p j q => PX (Zdiv_pol p x) j (Zdiv_pol q x) end. @@ -425,10 +412,10 @@ Proof. intros. simpl. unfold ZgcdM. - generalize (Zgcd_is_pos z1 z2). - generalize (Zmax_spec (Zgcd z1 z2) 1). - generalize (Zgcd_is_pos (Zmax (Zgcd z1 z2) 1) z). - generalize (Zmax_spec (Zgcd (Zmax (Zgcd z1 z2) 1) z) 1). + generalize (Z.gcd_nonneg z1 z2). + generalize (Zmax_spec (Z.gcd z1 z2) 1). + generalize (Z.gcd_nonneg (Z.max (Z.gcd z1 z2) 1) z). + generalize (Zmax_spec (Z.gcd (Z.max (Z.gcd z1 z2) 1) z) 1). auto with zarith. Qed. @@ -437,7 +424,7 @@ Proof. intros. induction H. constructor. - apply Zdivide_trans with (1:= H0) ; assumption. + apply Z.divide_trans with (1:= H0) ; assumption. constructor. auto. constructor ; auto. Qed. @@ -448,20 +435,20 @@ Proof. exists c. ring. Qed. -Lemma Zgcd_minus : forall a b c, (a | c - b ) -> (Zgcd a b | c). +Lemma Zgcd_minus : forall a b c, (a | c - b ) -> (Z.gcd a b | c). Proof. intros a b c (q,Hq). destruct (Zgcd_is_gcd a b) as [(a',Ha) (b',Hb) _]. - set (g:=Zgcd a b) in *; clearbody g. + set (g:=Z.gcd a b) in *; clearbody g. exists (q * a' + b'). - symmetry in Hq. rewrite <- Zeq_plus_swap in Hq. + symmetry in Hq. rewrite <- Z.add_move_r in Hq. rewrite <- Hq, Hb, Ha. ring. Qed. Lemma Zdivide_pol_sub : forall p a b, - 0 < Zgcd a b -> - Zdivide_pol a (PsubC Zminus p b) -> - Zdivide_pol (Zgcd a b) p. + 0 < Z.gcd a b -> + Zdivide_pol a (PsubC Z.sub p b) -> + Zdivide_pol (Z.gcd a b) p. Proof. induction p. simpl. @@ -481,7 +468,7 @@ Proof. Qed. Lemma Zdivide_pol_sub_0 : forall p a, - Zdivide_pol a (PsubC Zminus p 0) -> + Zdivide_pol a (PsubC Z.sub p 0) -> Zdivide_pol a p. Proof. induction p. @@ -500,7 +487,7 @@ Qed. Lemma Zgcd_pol_div : forall p g c, - Zgcd_pol p = (g, c) -> Zdivide_pol g (PsubC Zminus p c). + Zgcd_pol p = (g, c) -> Zdivide_pol g (PsubC Z.sub p c). Proof. induction p ; simpl. (* Pc *) @@ -515,12 +502,12 @@ Proof. case_eq (Zgcd_pol p1) ; case_eq (Zgcd_pol p3) ; intros. inv H1. unfold ZgcdM at 1. - destruct (Zmax_spec (Zgcd (ZgcdM z1 z2) z) 1) as [HH1 | HH1]; + destruct (Zmax_spec (Z.gcd (ZgcdM z1 z2) z) 1) as [HH1 | HH1]; destruct HH1 as [HH1 HH1'] ; rewrite HH1'. constructor. apply Zdivide_pol_Zdivide with (x:= ZgcdM z1 z2). unfold ZgcdM. - destruct (Zmax_spec (Zgcd z1 z2) 1) as [HH2 | HH2]. + destruct (Zmax_spec (Z.gcd z1 z2) 1) as [HH2 | HH2]. destruct HH2. rewrite H2. apply Zdivide_pol_sub ; auto. @@ -528,9 +515,9 @@ Proof. destruct HH2. rewrite H2. apply Zdivide_pol_one. unfold ZgcdM in HH1. unfold ZgcdM. - destruct (Zmax_spec (Zgcd z1 z2) 1) as [HH2 | HH2]. + destruct (Zmax_spec (Z.gcd z1 z2) 1) as [HH2 | HH2]. destruct HH2. rewrite H2 in *. - destruct (Zgcd_is_gcd (Zgcd z1 z2) z); auto. + destruct (Zgcd_is_gcd (Z.gcd z1 z2) z); auto. destruct HH2. rewrite H2. destruct (Zgcd_is_gcd 1 z); auto. apply Zdivide_pol_Zdivide with (x:= z). @@ -543,7 +530,7 @@ Qed. -Lemma Zgcd_pol_correct_lt : forall p env g c, Zgcd_pol p = (g,c) -> 0 < g -> eval_pol env p = g * (eval_pol env (Zdiv_pol (PsubC Zminus p c) g)) + c. +Lemma Zgcd_pol_correct_lt : forall p env g c, Zgcd_pol p = (g,c) -> 0 < g -> eval_pol env p = g * (eval_pol env (Zdiv_pol (PsubC Z.sub p c) g)) + c. Proof. intros. rewrite <- Zdiv_pol_correct ; auto. @@ -557,8 +544,8 @@ Qed. Definition makeCuttingPlane (p : PolC Z) : PolC Z * Z := let (g,c) := Zgcd_pol p in - if Zgt_bool g Z0 - then (Zdiv_pol (PsubC Zminus p c) g , Zopp (ceiling (Zopp c) g)) + if Z.gtb g Z0 + then (Zdiv_pol (PsubC Z.sub p c) g , Z.opp (ceiling (Z.opp c) g)) else (p,Z0). @@ -566,11 +553,13 @@ Definition genCuttingPlane (f : NFormula Z) : option (PolC Z * Z * Op1) := let (e,op) := f in match op with | Equal => let (g,c) := Zgcd_pol e in - if andb (Zgt_bool g Z0) (andb (Zgt_bool c Z0) (negb (Zeq_bool (Zgcd g c) g))) + if andb (Z.gtb g Z0) (andb (negb (Zeq_bool c Z0)) (negb (Zeq_bool (Z.gcd g c) g))) then None (* inconsistent *) - else Some (e, Z0,op) (* It could still be inconsistent -- but not a cut *) + else (* Could be optimised Zgcd_pol is recomputed *) + let (p,c) := makeCuttingPlane e in + Some (p,c,Equal) | NonEqual => Some (e,Z0,op) - | Strict => let (p,c) := makeCuttingPlane (PsubC Zminus e 1) in + | Strict => let (p,c) := makeCuttingPlane (PsubC Z.sub e 1) in Some (p,c,NonStrict) | NonStrict => let (p,c) := makeCuttingPlane e in Some (p,c,NonStrict) @@ -596,16 +585,16 @@ Proof. Qed. - - - Definition eval_Psatz : list (NFormula Z) -> ZWitness -> option (NFormula Z) := - eval_Psatz 0 1 Zplus Zmult Zeq_bool Zle_bool. - - -Definition check_inconsistent := check_inconsistent 0 Zeq_bool Zle_bool. + eval_Psatz 0 1 Z.add Z.mul Zeq_bool Z.leb. +Definition valid_cut_sign (op:Op1) := + match op with + | Equal => true + | NonStrict => true + | _ => false + end. Fixpoint ZChecker (l:list (NFormula Z)) (pf : ZArithProof) {struct pf} : bool := match pf with @@ -614,7 +603,7 @@ Fixpoint ZChecker (l:list (NFormula Z)) (pf : ZArithProof) {struct pf} : bool : match eval_Psatz l w with | None => false | Some f => - if check_inconsistent f then true + if Zunsat f then true else ZChecker (f::l) pf end | CutProof w pf => @@ -627,29 +616,24 @@ Fixpoint ZChecker (l:list (NFormula Z)) (pf : ZArithProof) {struct pf} : bool : end end | EnumProof w1 w2 pf => - match eval_Psatz l w1 , eval_Psatz l w2 with - | Some f1 , Some f2 => - match genCuttingPlane f1 , genCuttingPlane f2 with - |Some (e1,z1,op1) , Some (e2,z2,op2) => - match op1 , op2 with - | NonStrict , NonStrict => - if is_pol_Z0 (padd e1 e2) - then - (fix label (pfs:list ZArithProof) := - fun lb ub => - match pfs with - | nil => if Zgt_bool lb ub then true else false - | pf::rsr => andb (ZChecker ((psub e1 (Pc lb), Equal) :: l) pf) (label rsr (Zplus lb 1%Z) ub) - end) - pf (Zopp z1) z2 - else false - | _ , _ => false - end - | _ , _ => false - end - | _ , _ => false - end - end. + match eval_Psatz l w1 , eval_Psatz l w2 with + | Some f1 , Some f2 => + match genCuttingPlane f1 , genCuttingPlane f2 with + |Some (e1,z1,op1) , Some (e2,z2,op2) => + if (valid_cut_sign op1 && valid_cut_sign op2 && is_pol_Z0 (padd e1 e2)) + then + (fix label (pfs:list ZArithProof) := + fun lb ub => + match pfs with + | nil => if Z.gtb lb ub then true else false + | pf::rsr => andb (ZChecker ((psub e1 (Pc lb), Equal) :: l) pf) (label rsr (Z.add lb 1%Z) ub) + end) pf (Z.opp z1) z2 + else false + | _ , _ => true + end + | _ , _ => false + end +end. @@ -702,7 +686,7 @@ Proof. apply make_conj_in ; auto. Qed. -Lemma makeCuttingPlane_sound : forall env e e' c, +Lemma makeCuttingPlane_ns_sound : forall env e e' c, eval_nformula env (e, NonStrict) -> makeCuttingPlane e = (e',c) -> eval_nformula env (nformula_of_cutting_plane (e', c, NonStrict)). @@ -717,19 +701,18 @@ Proof. unfold makeCuttingPlane in H0. revert H0. case_eq (Zgcd_pol e) ; intros g c0. - generalize (Zgt_cases g 0) ; destruct (Zgt_bool g 0). + generalize (Zgt_cases g 0) ; destruct (Z.gtb g 0). intros. inv H2. - change (RingMicromega.eval_pol 0 Zplus Zmult (fun x : Z => x)) with eval_pol in *. + change (RingMicromega.eval_pol Z.add Z.mul (fun x : Z => x)) with eval_pol in *. apply Zgcd_pol_correct_lt with (env:=env) in H1. - generalize (narrow_interval_lower_bound g (- c0) (eval_pol env (Zdiv_pol (PsubC Zminus e c0) g)) H0). + generalize (narrow_interval_lower_bound g (- c0) (eval_pol env (Zdiv_pol (PsubC Z.sub e c0) g)) H0). auto with zarith. auto with zarith. (* g <= 0 *) intros. inv H2. auto with zarith. Qed. - Lemma cutting_plane_sound : forall env f p, eval_nformula env f -> genCuttingPlane f = Some p -> @@ -741,13 +724,50 @@ Proof. (* Equal *) destruct p as [[e' z] op]. case_eq (Zgcd_pol e) ; intros g c. - destruct (Zgt_bool g 0 && (Zgt_bool c 0 && negb (Zeq_bool (Zgcd g c) g))) ; [discriminate|]. - intros. inv H1. unfold nformula_of_cutting_plane. - unfold eval_nformula in *. - unfold RingMicromega.eval_nformula in *. - unfold eval_op1 in *. - rewrite (RingMicromega.eval_pol_add Zsor ZSORaddon). - simpl. rewrite H0. reflexivity. + case_eq (Z.gtb g 0 && (negb (Zeq_bool c 0) && negb (Zeq_bool (Z.gcd g c) g))) ; [discriminate|]. + case_eq (makeCuttingPlane e). + intros. + inv H3. + unfold makeCuttingPlane in H. + rewrite H1 in H. + revert H. + change (eval_pol env e = 0) in H2. + case_eq (Z.gtb g 0). + intros. + rewrite <- Zgt_is_gt_bool in H. + rewrite Zgcd_pol_correct_lt with (1:= H1) in H2; auto with zarith. + unfold nformula_of_cutting_plane. + change (eval_pol env (padd e' (Pc z)) = 0). + inv H3. + rewrite eval_pol_add. + set (x:=eval_pol env (Zdiv_pol (PsubC Z.sub e c) g)) in *; clearbody x. + simpl. + rewrite andb_false_iff in H0. + destruct H0. + rewrite Zgt_is_gt_bool in H ; congruence. + rewrite andb_false_iff in H0. + destruct H0. + rewrite negb_false_iff in H0. + apply Zeq_bool_eq in H0. + subst. simpl. + rewrite Z.add_0_r, Z.mul_eq_0 in H2. + intuition auto with zarith. + rewrite negb_false_iff in H0. + apply Zeq_bool_eq in H0. + assert (HH := Zgcd_is_gcd g c). + rewrite H0 in HH. + inv HH. + apply Zdivide_opp_r in H4. + rewrite Zdivide_ceiling ; auto. + apply Z.sub_move_0_r. + apply Z.div_unique_exact ; auto with zarith. + intros. + unfold nformula_of_cutting_plane. + inv H3. + change (eval_pol env (padd e' (Pc 0)) = 0). + rewrite eval_pol_add. + simpl. + auto with zarith. (* NonEqual *) intros. inv H0. @@ -759,10 +779,10 @@ Proof. simpl. auto with zarith. (* Strict *) destruct p as [[e' z] op]. - case_eq (makeCuttingPlane (PsubC Zminus e 1)). + case_eq (makeCuttingPlane (PsubC Z.sub e 1)). intros. inv H1. - apply makeCuttingPlane_sound with (env:=env) (2:= H). + apply makeCuttingPlane_ns_sound with (env:=env) (2:= H). simpl in *. rewrite (RingMicromega.PsubC_ok Zsor ZSORaddon). auto with zarith. @@ -771,7 +791,7 @@ Proof. case_eq (makeCuttingPlane e). intros. inv H1. - apply makeCuttingPlane_sound with (env:=env) (2:= H). + apply makeCuttingPlane_ns_sound with (env:=env) (2:= H). assumption. Qed. @@ -783,25 +803,26 @@ Proof. destruct f. destruct o. case_eq (Zgcd_pol p) ; intros g c. - case_eq (Zgt_bool g 0 && (Zgt_bool c 0 && negb (Zeq_bool (Zgcd g c) g))). + case_eq (Z.gtb g 0 && (negb (Zeq_bool c 0) && negb (Zeq_bool (Z.gcd g c) g))). intros. flatten_bool. rewrite negb_true_iff in H5. apply Zeq_bool_neq in H5. - contradict H5. rewrite <- Zgt_is_gt_bool in H3. - rewrite <- Zgt_is_gt_bool in H. - apply Zis_gcd_gcd; auto with zarith. - constructor; auto with zarith. + rewrite negb_true_iff in H. + apply Zeq_bool_neq in H. change (eval_pol env p = 0) in H2. rewrite Zgcd_pol_correct_lt with (1:= H0) in H2; auto with zarith. - set (x:=eval_pol env (Zdiv_pol (PsubC Zminus p c) g)) in *; clearbody x. + set (x:=eval_pol env (Zdiv_pol (PsubC Z.sub p c) g)) in *; clearbody x. + contradict H5. + apply Zis_gcd_gcd; auto with zarith. + constructor; auto with zarith. exists (-x). - rewrite <- Zopp_mult_distr_l, Zmult_comm; auto with zarith. + rewrite Z.mul_opp_l, Z.mul_comm; auto with zarith. (**) + destruct (makeCuttingPlane p); discriminate. discriminate. - discriminate. - destruct (makeCuttingPlane (PsubC Zminus p 1)) ; discriminate. + destruct (makeCuttingPlane (PsubC Z.sub p 1)) ; discriminate. destruct (makeCuttingPlane p) ; discriminate. Qed. @@ -816,11 +837,11 @@ Proof. simpl. intro l. case_eq (eval_Psatz l w) ; [| discriminate]. intros f Hf. - case_eq (check_inconsistent f). + case_eq (Zunsat f). intros. apply (checker_nf_sound Zsor ZSORaddon l w). unfold check_normalised_formulas. unfold eval_Psatz in Hf. rewrite Hf. - unfold check_inconsistent in H0. assumption. + unfold Zunsat in H0. assumption. intros. assert (make_impl (eval_nformula env) (f::l) False). apply H with (2:= H1). @@ -868,60 +889,59 @@ Proof. case_eq (eval_Psatz l w1) ; [ | discriminate]. case_eq (eval_Psatz l w2) ; [ | discriminate]. intros f1 Hf1 f2 Hf2. - case_eq (genCuttingPlane f2) ; [ | discriminate]. + case_eq (genCuttingPlane f2). destruct p as [ [p1 z1] op1]. - case_eq (genCuttingPlane f1) ; [ | discriminate]. + case_eq (genCuttingPlane f1). destruct p as [ [p2 z2] op2]. - case_eq op1 ; case_eq op2 ; try discriminate. - case_eq (is_pol_Z0 (padd p1 p2)) ; try discriminate. - intros. + case_eq (valid_cut_sign op1 && valid_cut_sign op2 && is_pol_Z0 (padd p1 p2)). + intros Hcond. + flatten_bool. + rename H1 into HZ0. + rename H2 into Hop1. + rename H3 into Hop2. + intros HCutL HCutR Hfix env. (* get the bounds of the enum *) rewrite <- make_conj_impl. intro. assert (-z1 <= eval_pol env p1 <= z2). split. apply eval_Psatz_sound with (env:=env) in Hf2 ; auto. - apply cutting_plane_sound with (1:= Hf2) in H4. - unfold nformula_of_cutting_plane in H4. - unfold eval_nformula in H4. - unfold RingMicromega.eval_nformula in H4. - change (RingMicromega.eval_pol 0 Zplus Zmult (fun x : Z => x)) with eval_pol in H4. - unfold eval_op1 in H4. - rewrite eval_pol_add in H4. simpl in H4. - auto with zarith. + apply cutting_plane_sound with (1:= Hf2) in HCutR. + unfold nformula_of_cutting_plane in HCutR. + unfold eval_nformula in HCutR. + unfold RingMicromega.eval_nformula in HCutR. + change (RingMicromega.eval_pol Z.add Z.mul (fun x : Z => x)) with eval_pol in HCutR. + unfold eval_op1 in HCutR. + destruct op1 ; simpl in Hop1 ; try discriminate; + rewrite eval_pol_add in HCutR; simpl in HCutR; auto with zarith. (**) - apply is_pol_Z0_eval_pol with (env := env) in H0. - rewrite eval_pol_add in H0. + apply is_pol_Z0_eval_pol with (env := env) in HZ0. + rewrite eval_pol_add in HZ0. replace (eval_pol env p1) with (- eval_pol env p2) by omega. apply eval_Psatz_sound with (env:=env) in Hf1 ; auto. - apply cutting_plane_sound with (1:= Hf1) in H3. - unfold nformula_of_cutting_plane in H3. - unfold eval_nformula in H3. - unfold RingMicromega.eval_nformula in H3. - change (RingMicromega.eval_pol 0 Zplus Zmult (fun x : Z => x)) with eval_pol in H3. - unfold eval_op1 in H3. - rewrite eval_pol_add in H3. simpl in H3. - omega. - revert H5. - set (FF := (fix label (pfs : list ZArithProof) (lb ub : Z) {struct pfs} : bool := - match pfs with - | nil => if Z_gt_dec lb ub then true else false - | pf :: rsr => - (ZChecker ((PsubC Zminus p1 lb, Equal) :: l) pf && - label rsr (lb + 1)%Z ub)%bool - end)). + apply cutting_plane_sound with (1:= Hf1) in HCutL. + unfold nformula_of_cutting_plane in HCutL. + unfold eval_nformula in HCutL. + unfold RingMicromega.eval_nformula in HCutL. + change (RingMicromega.eval_pol Z.add Z.mul (fun x : Z => x)) with eval_pol in HCutL. + unfold eval_op1 in HCutL. + rewrite eval_pol_add in HCutL. simpl in HCutL. + destruct op2 ; simpl in Hop2 ; try discriminate ; omega. + revert Hfix. + match goal with + | |- context[?F pf (-z1) z2 = true] => set (FF := F) + end. intros. assert (HH :forall x, -z1 <= x <= z2 -> exists pr, (In pr pf /\ - ZChecker ((PsubC Zminus p1 x,Equal) :: l) pr = true)%Z). - clear H. - clear H0 H1 H2 H3 H4 H7. - revert H5. + ZChecker ((PsubC Z.sub p1 x,Equal) :: l) pr = true)%Z). + clear HZ0 Hop1 Hop2 HCutL HCutR H0 H1. + revert Hfix. generalize (-z1). clear z1. intro z1. revert z1 z2. induction pf;simpl ;intros. generalize (Zgt_cases z1 z2). - destruct (Zgt_bool z1 z2). + destruct (Z.gtb z1 z2). intros. apply False_ind ; omega. discriminate. @@ -931,16 +951,22 @@ Proof. subst. exists a ; auto. assert (z1 + 1 <= x <= z2)%Z by omega. - destruct (IHpf _ _ H1 _ H3). + elim IHpf with (2:=H2) (3:= H4). destruct H4. - exists x0 ; split;auto. + intros. + exists x0 ; split;tauto. + intros until 1. + apply H ; auto. + unfold ltof in *. + simpl in *. + zify. omega. (*/asser *) - destruct (HH _ H7) as [pr [Hin Hcheker]]. - assert (make_impl (eval_nformula env) ((PsubC Zminus p1 (eval_pol env p1),Equal) :: l) False). + destruct (HH _ H1) as [pr [Hin Hcheker]]. + assert (make_impl (eval_nformula env) ((PsubC Z.sub p1 (eval_pol env p1),Equal) :: l) False). apply (H pr);auto. apply in_bdepth ; auto. - rewrite <- make_conj_impl in H8. - apply H8. + rewrite <- make_conj_impl in H2. + apply H2. rewrite make_conj_cons. split ;auto. unfold eval_nformula. @@ -948,10 +974,23 @@ Proof. simpl. rewrite (RingMicromega.PsubC_ok Zsor ZSORaddon). unfold eval_pol. ring. + discriminate. + (* No cutting plane *) + intros. + rewrite <- make_conj_impl. + intros. + apply eval_Psatz_sound with (2:= Hf1) in H3. + apply genCuttingPlaneNone with (2:= H3) ; auto. + (* No Cutting plane (bis) *) + intros. + rewrite <- make_conj_impl. + intros. + apply eval_Psatz_sound with (2:= Hf2) in H2. + apply genCuttingPlaneNone with (2:= H2) ; auto. Qed. Definition ZTautoChecker (f : BFormula (Formula Z)) (w: list ZArithProof): bool := - @tauto_checker (Formula Z) (NFormula Z) normalise negate ZArithProof ZChecker f w. + @tauto_checker (Formula Z) (NFormula Z) Zunsat Zdeduce normalise negate ZArithProof ZChecker f w. Lemma ZTautoChecker_sound : forall f w, ZTautoChecker f w = true -> forall env, eval_f (Zeval_formula env) f. Proof. @@ -959,6 +998,11 @@ Proof. unfold ZTautoChecker. apply (tauto_checker_sound Zeval_formula eval_nformula). apply Zeval_nformula_dec. + intros until env. + unfold eval_nformula. unfold RingMicromega.eval_nformula. + destruct t. + apply (check_inconsistent_sound Zsor ZSORaddon) ; auto. + unfold Zdeduce. apply (nformula_plus_nformula_correct Zsor ZSORaddon). intros env t. rewrite normalise_correct ; auto. intros env t. @@ -1009,12 +1053,7 @@ Definition eval := eval_formula. Definition prod_pos_nat := prod positive nat. -Definition n_of_Z (z:Z) : BinNat.N := - match z with - | Z0 => N0 - | Zpos p => Npos p - | Zneg p => N0 - end. +Notation n_of_Z := Z.to_N (only parsing). (* Local Variables: *) (* coding: utf-8 *) |