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Diffstat (limited to 'plugins/nsatz/Nsatz_domain.v')
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diff --git a/plugins/nsatz/Nsatz_domain.v b/plugins/nsatz/Nsatz_domain.v new file mode 100644 index 00000000..11f905f9 --- /dev/null +++ b/plugins/nsatz/Nsatz_domain.v @@ -0,0 +1,558 @@ +(************************************************************************) +(* 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 *) +(************************************************************************) + +(* + Tactic nsatz: proofs of polynomials equalities with variables in R. + Uses Hilbert Nullstellensatz and Buchberger algorithm. + Thanks to B.Gregoire for the verification of the certicate + and L.Thery for help on ring tactic, + and to B.Barras for modularization of the ocaml code. + Example: see test-suite/success/Nsatz.v + L.Pottier, june 2010 +*) + +Require Import List. +Require Import Setoid. +Require Import BinPos. +Require Import BinList. +Require Import Znumtheory. +Require Import Ring_polynom Ring_tac InitialRing. + +Declare ML Module "nsatz_plugin". + + +Class Zero (A : Type) := {zero : A}. +Notation "0" := zero. +Class One (A : Type) := {one : A}. +Notation "1" := one. +Class Addition (A : Type) := {addition : A -> A -> A}. +Notation "x + y" := (addition x y). +Class Multiplication (A : Type) := {multiplication : A -> A -> A}. +Notation "x * y" := (multiplication x y). +Class Subtraction (A : Type) := {subtraction : A -> A -> A}. +Notation "x - y" := (subtraction x y). +Class Opposite (A : Type) := {opposite : A -> A}. +Notation "- x" := (opposite x). + +Class Ring (R:Type) := { + ring0: R; ring1: R; + ring_plus: R->R->R; ring_mult: R->R->R; + ring_sub: R->R->R; ring_opp: R->R; + ring_ring: + ring_theory ring0 ring1 ring_plus ring_mult ring_sub + ring_opp (@eq R)}. + +Class Domain (R : Type) := { + domain_ring:> Ring R; + domain_axiom_product: + forall x y, ring_mult x y = ring0 -> x = ring0 \/ y = ring0; + domain_axiom_one_zero: ring1 <> ring0}. + +Ltac ring2 := simpl; ring. + +Section domain. + +Variable R: Type. +Variable Rd: Domain R. +Add Ring Rr: (@ring_ring R (@domain_ring R Rd)). + +Instance zero_ring : Zero R := {zero := ring0}. +Instance one_ring : One R := {one := ring1}. +Instance addition_ring : Addition R := {addition x y := ring_plus x y}. +Instance multiplication_ring : Multiplication R := {multiplication x y := ring_mult x y}. +Instance subtraction_ring : Subtraction R := {subtraction x y := ring_sub x y}. +Instance opposite_ring : Opposite R := {opposite x := ring_opp x}. + +Lemma psos_r1b: forall x y:R, x - y = 0 -> x = y. +intros x y H; replace x with ((x - y) + y); + [rewrite H | idtac]; ring2. +Qed. + +Lemma psos_r1: forall x y, x = y -> x - y = 0. +intros x y H; rewrite H; ring2. +Qed. + + +Lemma nsatzR_diff: forall x y:R, x<>y -> x - y<>0. +intros. +intro; apply H. +replace x with ((x - y) + y) by ring2. +rewrite H0; ring2. +Qed. + +(* code de Benjamin *) +Require Import ZArith. + +Definition PolZ := Pol Z. +Definition PEZ := PExpr Z. + +Definition P0Z : PolZ := @P0 Z 0%Z. + +Definition PolZadd : PolZ -> PolZ -> PolZ := + @Padd Z 0%Z Zplus Zeq_bool. + +Definition PolZmul : PolZ -> PolZ -> PolZ := + @Pmul Z 0%Z 1%Z Zplus Zmult Zeq_bool. + +Definition PolZeq := @Peq Z Zeq_bool. + +Definition norm := + @norm_aux Z 0%Z 1%Z Zplus Zmult Zminus Zopp Zeq_bool. + +Fixpoint mult_l (la : list PEZ) (lp: list PolZ) : PolZ := + match la, lp with + | a::la, p::lp => PolZadd (PolZmul (norm a) p) (mult_l la lp) + | _, _ => P0Z + end. + +Fixpoint compute_list (lla: list (list PEZ)) (lp:list PolZ) := + match lla with + | List.nil => lp + | la::lla => compute_list lla ((mult_l la lp)::lp) + end. + +Definition check (lpe:list PEZ) (qe:PEZ) (certif: list (list PEZ) * list PEZ) := + let (lla, lq) := certif in + let lp := List.map norm lpe in + PolZeq (norm qe) (mult_l lq (compute_list lla lp)). + + +(* Correction *) +Definition PhiR : list R -> PolZ -> R := + (Pphi 0 ring_plus ring_mult (gen_phiZ 0 1 ring_plus ring_mult ring_opp)). + +Definition pow (r : R) (n : nat) := pow_N 1 ring_mult r (Nnat.N_of_nat n). + +Definition PEevalR : list R -> PEZ -> R := + PEeval 0 ring_plus ring_mult ring_sub ring_opp + (gen_phiZ 0 1 ring_plus ring_mult ring_opp) + Nnat.nat_of_N pow. + +Lemma P0Z_correct : forall l, PhiR l P0Z = 0. +Proof. trivial. Qed. + +Lemma Rext: ring_eq_ext ring_plus ring_mult ring_opp eq. +apply mk_reqe. intros. rewrite H; rewrite H0; trivial. + intros. rewrite H; rewrite H0; trivial. +intros. rewrite H; trivial. Qed. + +Lemma Rset : Setoid_Theory R eq. +apply Eqsth. +Qed. + +Lemma PolZadd_correct : forall P' P l, + PhiR l (PolZadd P P') = ((PhiR l P) + (PhiR l P')). +Proof. + refine (Padd_ok Rset Rext (Rth_ARth Rset Rext (@ring_ring _ (@domain_ring _ Rd))) + (gen_phiZ_morph Rset Rext (@ring_ring _ (@domain_ring _ Rd)))). +Qed. + +Lemma PolZmul_correct : forall P P' l, + PhiR l (PolZmul P P') = ((PhiR l P) * (PhiR l P')). +Proof. + refine (Pmul_ok Rset Rext (Rth_ARth Rset Rext (@ring_ring _ (@domain_ring _ Rd))) + (gen_phiZ_morph Rset Rext (@ring_ring _ (@domain_ring _ Rd)))). +Qed. + +Lemma R_power_theory + : power_theory 1 ring_mult eq Nnat.nat_of_N pow. +apply mkpow_th. unfold pow. intros. rewrite Nnat.N_of_nat_of_N. trivial. Qed. + +Lemma norm_correct : + forall (l : list R) (pe : PEZ), PEevalR l pe = PhiR l (norm pe). +Proof. + intros;apply (norm_aux_spec Rset Rext (Rth_ARth Rset Rext (@ring_ring _ (@domain_ring _ Rd))) + (gen_phiZ_morph Rset Rext (@ring_ring _ (@domain_ring _ Rd))) R_power_theory) + with (lmp:= List.nil). + compute;trivial. +Qed. + +Lemma PolZeq_correct : forall P P' l, + PolZeq P P' = true -> + PhiR l P = PhiR l P'. +Proof. + intros;apply + (Peq_ok Rset Rext (gen_phiZ_morph Rset Rext (@ring_ring _ (@domain_ring _ Rd))));trivial. +Qed. + +Fixpoint Cond0 (A:Type) (Interp:A->R) (l:list A) : Prop := + match l with + | List.nil => True + | a::l => Interp a = 0 /\ Cond0 A Interp l + end. + +Lemma mult_l_correct : forall l la lp, + Cond0 PolZ (PhiR l) lp -> + PhiR l (mult_l la lp) = 0. +Proof. + induction la;simpl;intros;trivial. + destruct lp;trivial. + simpl in H;destruct H. + rewrite PolZadd_correct, PolZmul_correct, H, IHla;[ring2 | trivial]. +Qed. + +Lemma compute_list_correct : forall l lla lp, + Cond0 PolZ (PhiR l) lp -> + Cond0 PolZ (PhiR l) (compute_list lla lp). +Proof. + induction lla;simpl;intros;trivial. + apply IHlla;simpl;split;trivial. + apply mult_l_correct;trivial. +Qed. + +Lemma check_correct : + forall l lpe qe certif, + check lpe qe certif = true -> + Cond0 PEZ (PEevalR l) lpe -> + PEevalR l qe = 0. +Proof. + unfold check;intros l lpe qe (lla, lq) H2 H1. + apply PolZeq_correct with (l:=l) in H2. + rewrite norm_correct, H2. + apply mult_l_correct. + apply compute_list_correct. + clear H2 lq lla qe;induction lpe;simpl;trivial. + simpl in H1;destruct H1. + rewrite <- norm_correct;auto. +Qed. + +(* fin du code de Benjamin *) + +Lemma pow_not_zero: forall p n, pow p n = 0 -> p = 0. +induction n. unfold pow; simpl. intros. absurd (1 = 0). +simpl. apply domain_axiom_one_zero. + trivial. replace (pow p (S n)) with (p * (pow p n)). intros. +case (@domain_axiom_product _ _ _ _ H). trivial. trivial. +unfold pow; simpl. +clear IHn. induction n; try ring2. simpl. + rewrite pow_pos_Psucc. trivial. exact Rset. + intros. rewrite H; rewrite H0; trivial. + intros. ring2. intros. ring2. Qed. + +Lemma Rdomain_pow: forall c p r, ~c= 0 -> c * (pow p r)= 0 -> p = ring0. +intros. case (@domain_axiom_product _ _ _ _ H0). intros; absurd (c = 0); auto. +intros. apply pow_not_zero with r. trivial. Qed. + +Definition R2:= 1 + 1. + +Fixpoint IPR p {struct p}: R := + match p with + xH => 1 + | xO xH => 1 + 1 + | xO p1 => R2 + (IPR p1) + | xI xH => 1 + (1 + 1) + | xI p1 => 1 + (R2 * (IPR p1)) + end. + +Definition IZR1 z := + match z with Z0 => 0 + | Zpos p => IPR p + | Zneg p => -(IPR p) + end. + +Fixpoint interpret3 t fv {struct t}: R := + match t with + | (PEadd t1 t2) => + let v1 := interpret3 t1 fv in + let v2 := interpret3 t2 fv in (v1 + v2) + | (PEmul t1 t2) => + let v1 := interpret3 t1 fv in + let v2 := interpret3 t2 fv in (v1 * v2) + | (PEsub t1 t2) => + let v1 := interpret3 t1 fv in + let v2 := interpret3 t2 fv in (v1 - v2) + | (PEopp t1) => + let v1 := interpret3 t1 fv in (- v1) + | (PEpow t1 t2) => + let v1 := interpret3 t1 fv in pow v1 (Nnat.nat_of_N t2) + | (PEc t1) => (IZR1 t1) + | (PEX n) => List.nth (pred (nat_of_P n)) fv 0 + end. + + +End domain. + +Ltac equalities_to_goal := + lazymatch goal with + | H: (@eq _ ?x 0) |- _ => try revert H + | H: (@eq _ 0 ?x) |- _ => + try generalize (sym_equal H); clear H + | H: (@eq _ ?x ?y) |- _ => + try generalize (@psos_r1 _ _ _ _ H); clear H + end. + +Ltac nsatz_domain_begin tacsimpl:= + intros; + try apply (@psos_r1b _ _); + repeat equalities_to_goal; + tacsimpl. + +Ltac generalise_eq_hyps:= + repeat + (match goal with + |h : (?p = ?q)|- _ => revert h + end). + +Ltac lpol_goal t := + match t with + | ?a = ring0 -> ?b => + let r:= lpol_goal b in + constr:(a::r) + | ?a = ring0 => constr:(a::nil) + end. + +(* lp est incluse dans fv. La met en tete. *) + +Ltac parametres_en_tete fv lp := + match fv with + | (@nil _) => lp + | (@cons _ ?x ?fv1) => + let res := AddFvTail x lp in + parametres_en_tete fv1 res + end. + +Ltac append1 a l := + match l with + | (@nil _) => constr:(cons a l) + | (cons ?x ?l) => let l' := append1 a l in constr:(cons x l') + end. + +Ltac rev l := + match l with + |(@nil _) => l + | (cons ?x ?l) => let l' := rev l in append1 x l' + end. + +Ltac nsatz_call_n info nparam p rr lp kont := + let ll := constr:(PEc info :: PEc nparam :: PEpow p rr :: lp) in + nsatz_compute ll; + match goal with + | |- (?c::PEpow _ ?r::?lq0)::?lci0 = _ -> _ => + intros _; + set (lci:=lci0); + set (lq:=lq0); + kont c rr lq lci + end. + +Ltac nsatz_call radicalmax info nparam p lp kont := + let rec try_n n := + lazymatch n with + | 0%N => fail + | _ => +(* idtac "Trying power: " n;*) + (let r := eval compute in (Nminus radicalmax (Npred n)) in + nsatz_call_n info nparam p r lp kont) || + let n' := eval compute in (Npred n) in try_n n' + end in + try_n radicalmax. + + +Set Implicit Arguments. +Class Cclosed_seq T (l:list T) := {}. +Instance Iclosed_nil T : Cclosed_seq (T:=T) nil. +Instance Iclosed_cons T t l `{Cclosed_seq (T:=T) l} : Cclosed_seq (T:=T) (t::l). + +Class Cfind_at (R:Type) (b:R) (l:list R) (i:nat) := {}. +Instance Ifind0 (R:Type) (b:R) l: Cfind_at b (b::l) 0. +Instance IfindS (R:Type) (b2 b1:R) l i `{Cfind_at R b1 l i} : Cfind_at b1 (b2::l) (S i) | 1. +Definition Ifind0' := Ifind0. +Definition IfindS' := IfindS. + +Definition li_find_at (R:Type) (b:R) l i `{Cfind_at R b l i} {H:Cclosed_seq (T:=R) l} := (l,i). + +Class Creify (R:Type) (e:PExpr Z) (l:list R) (b:R) := {}. +Instance Ireify_zero (R:Type) (Rd:Domain R) l : Creify (PEc 0%Z) l ring0. +Instance Ireify_one (R:Type) (Rd:Domain R) l : Creify (PEc 1%Z) l ring1. +Instance Ireify_plus (R:Type) (Rd:Domain R) e1 l b1 e2 b2 `{Creify R e1 l b1} `{Creify R e2 l b2} + : Creify (PEadd e1 e2) l (ring_plus b1 b2). +Instance Ireify_mult (R:Type) (Rd:Domain R) e1 l b1 e2 b2 `{Creify R e1 l b1} `{Creify R e2 l b2} + : Creify (PEmul e1 e2) l (ring_mult b1 b2). +Instance Ireify_sub (R:Type) (Rd:Domain R) e1 l b1 e2 b2 `{Creify R e1 l b1} `{Creify R e2 l b2} + : Creify (PEsub e1 e2) l (ring_sub b1 b2). +Instance Ireify_opp (R:Type) (Rd:Domain R) e1 l b1 `{Creify R e1 l b1} + : Creify (PEopp e1) l (ring_opp b1). +Instance Ireify_var (R:Type) b l i `{Cfind_at R b l i} + : Creify (PEX _ (P_of_succ_nat i)) l b | 100. + + +Class Creifylist (R:Type) (le:list (PExpr Z)) (l:list R) (lb:list R) := {}. +Instance Creify_nil (R:Type) l : Creifylist nil l (@nil R). +Instance Creify_cons (R:Type) e1 l b1 le2 lb2 `{Creify R e1 l b1} `{Creifylist R le2 l lb2} + : Creifylist (e1::le2) l (b1::lb2). + +Definition li_reifyl (R:Type) le l lb `{Creifylist R le l lb} + {H:Cclosed_seq (T:=R) l} := (l,le). + +Unset Implicit Arguments. + +Ltac lterm_goal g := + match g with + ?b1 = ?b2 => constr:(b1::b2::nil) + | ?b1 = ?b2 -> ?g => let l := lterm_goal g in constr:(b1::b2::l) + end. + +Ltac reify_goal l le lb:= + match le with + nil => idtac + | ?e::?le1 => + match lb with + ?b::?lb1 => + let x := fresh "B" in + set (x:= b) at 1; + change x with (@interpret3 _ _ e l); + clear x; + reify_goal l le1 lb1 + end + end. + +Ltac get_lpol g := + match g with + (interpret3 _ _ ?p _) = _ => constr:(p::nil) + | (interpret3 _ _ ?p _) = _ -> ?g => + let l := get_lpol g in constr:(p::l) + end. + +Ltac nsatz_domain_generic radicalmax info lparam lvar tacsimpl Rd := + match goal with + |- ?g => let lb := lterm_goal g in + (*idtac "lb"; idtac lb;*) + match eval red in (li_reifyl (lb:=lb)) with + | (?fv, ?le) => + let fv := match lvar with + (@nil _) => fv + | _ => lvar + end in + (* idtac "variables:";idtac fv;*) + let nparam := eval compute in (Z_of_nat (List.length lparam)) in + let fv := parametres_en_tete fv lparam in + (*idtac "variables:"; idtac fv; + idtac "nparam:"; idtac nparam;*) + match eval red in (li_reifyl (l:=fv) (lb:=lb)) with + | (?fv, ?le) => + idtac "variables:";idtac fv; + reify_goal fv le lb; + match goal with + |- ?g => + let lp := get_lpol g in + let lpol := eval compute in (List.rev lp) in + (*idtac "polynomes:"; idtac lpol;*) + tacsimpl; intros; + + let SplitPolyList kont := + match lpol with + | ?p2::?lp2 => kont p2 lp2 + | _ => idtac "polynomial not in the ideal" + end in + tacsimpl; + SplitPolyList ltac:(fun p lp => + set (p21:=p) ; + set (lp21:=lp); + (*idtac "lp:"; idtac lp; *) + nsatz_call radicalmax info nparam p lp ltac:(fun c r lq lci => + set (q := PEmul c (PEpow p21 r)); + let Hg := fresh "Hg" in + assert (Hg:check lp21 q (lci,lq) = true); + [ (vm_compute;reflexivity) || idtac "invalid nsatz certificate" + | let Hg2 := fresh "Hg" in + assert (Hg2: interpret3 _ _ q fv = ring0); + [ tacsimpl; + apply (@check_correct _ Rd fv lp21 q (lci,lq) Hg); + tacsimpl; + repeat (split;[assumption|idtac]); exact I + | simpl in Hg2; tacsimpl; + apply Rdomain_pow with (interpret3 _ _ c fv) (Nnat.nat_of_N r); tacsimpl; + [ apply domain_axiom_one_zero || idtac "could not prove discrimination result" + | exact Hg2] + ] + ] +) +) +end end end end . + +Ltac nsatz_domainpv radicalmax info lparam lvar tacsimpl rd:= + nsatz_domain_begin tacsimpl; + nsatz_domain_generic radicalmax info lparam lvar tacsimpl rd. + +Ltac nsatz_domain:= + intros; + match goal with + |- (@eq ?r _ _ ) => + let a := constr:(@Ireify_zero _ _ (@nil r)) in + match a with + (@Ireify_zero _ ?rd _) => + nsatz_domainpv 6%N 1%Z (@nil r) (@nil r) ltac:(simpl) rd + end + end. + + + +(* Dans Z *) +Instance Zri : Ring Z := { + ring0 := 0%Z; + ring1 := 1%Z; + ring_plus := Zplus; + ring_mult := Zmult; + ring_sub := Zminus; + ring_opp := Zopp; + ring_ring := Zth}. + +Lemma Zaxiom_one_zero: 1%Z <> 0%Z. +discriminate. +Qed. + +Instance Zdi : Domain Z := { + domain_ring := Zri; + domain_axiom_product := Zmult_integral; + domain_axiom_one_zero := Zaxiom_one_zero}. + + +Ltac simplZ:= + simpl; +replace 0%Z with (@ring0 _ (@domain_ring _ Zdi));[idtac|reflexivity]; +replace 1%Z with (@ring1 _ (@domain_ring _ Zdi));[idtac|reflexivity]; +replace Zplus with (@ring_plus _ (@domain_ring _ Zdi));[idtac|reflexivity]; +replace Zmult with (@ring_mult _ (@domain_ring _ Zdi));[idtac|reflexivity]; +replace Zminus with (@ring_sub _ (@domain_ring _ Zdi));[idtac|reflexivity]; +replace Zopp with (@ring_opp _ (@domain_ring _ Zdi));[idtac|reflexivity]. + +Ltac nsatz_domainZ:= nsatz_domainpv 6%N 1%Z (@nil Z) (@nil Z) ltac:simplZ Zdi. + + +(* Dans R *) +Require Import Reals. +Require Import RealField. + +Instance Rri : Ring R := { + ring0 := 0%R; + ring1 := 1%R; + ring_plus := Rplus; + ring_mult := Rmult; + ring_sub := Rminus; + ring_opp := Ropp; + ring_ring := RTheory}. + +Lemma Raxiom_one_zero: 1%R <> 0%R. +discrR. +Qed. + +Instance Rdi : Domain R := { + domain_ring := Rri; + domain_axiom_product := Rmult_integral; + domain_axiom_one_zero := Raxiom_one_zero}. + + +Ltac simplR:= + simpl; +replace 0%R with (@ring0 _ (@domain_ring _ Rdi));[idtac|reflexivity]; +replace 1%R with (@ring1 _ (@domain_ring _ Rdi));[idtac|reflexivity]; +replace Rplus with (@ring_plus _ (@domain_ring _ Rdi));[idtac|reflexivity]; +replace Rmult with (@ring_mult _ (@domain_ring _ Rdi));[idtac|reflexivity]; +replace Rminus with (@ring_sub _ (@domain_ring _ Rdi));[idtac|reflexivity]; +replace Ropp with (@ring_opp _ (@domain_ring _ Rdi));[idtac|reflexivity]. + +Ltac nsatz_domainR:= nsatz_domainpv 6%N 1%Z (@List.nil R) (@List.nil R) ltac:simplR Rdi. |