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(************************************************************************)
(* * The Coq Proof Assistant / The Coq Development Team *)
(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *)
(* <O___,, * (see CREDITS file for the list of authors) *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
(*
Tactic nsatz: proofs of polynomials equalities in an integral domain
(commutative ring without zero divisor).
Examples: see test-suite/success/Nsatz.v
Reification is done using type classes, defined in Ncring_tac.v
*)
Require Import List.
Require Import Setoid.
Require Import BinPos.
Require Import BinList.
Require Import Znumtheory.
Require Export Morphisms Setoid Bool.
Require Export Algebra_syntax.
Require Export Ncring.
Require Export Ncring_initial.
Require Export Ncring_tac.
Require Export Integral_domain.
Require Import DiscrR.
Require Import ZArith.
Declare ML Module "nsatz_plugin".
Section nsatz1.
Context {R:Type}`{Rid:Integral_domain R}.
Lemma psos_r1b: forall x y:R, x - y == 0 -> x == y.
intros x y H; setoid_replace x with ((x - y) + y); simpl;
[setoid_rewrite H | idtac]; simpl. cring. cring.
Qed.
Lemma psos_r1: forall x y, x == y -> x - y == 0.
intros x y H; simpl; setoid_rewrite H; simpl; cring.
Qed.
Lemma nsatzR_diff: forall x y:R, not (x == y) -> not (x - y == 0).
intros.
intro; apply H.
simpl; setoid_replace x with ((x - y) + y). simpl.
setoid_rewrite H0.
simpl; cring.
simpl. simpl; cring.
Qed.
(* adpatation du code de Benjamin aux setoides *)
Export Ring_polynom.
Export InitialRing.
Definition PolZ := Pol Z.
Definition PEZ := PExpr Z.
Definition P0Z : PolZ := P0 (C:=Z) 0%Z.
Definition PolZadd : PolZ -> PolZ -> PolZ :=
@Padd Z 0%Z Z.add Zeq_bool.
Definition PolZmul : PolZ -> PolZ -> PolZ :=
@Pmul Z 0%Z 1%Z Z.add Z.mul Zeq_bool.
Definition PolZeq := @Peq Z Zeq_bool.
Definition norm :=
@norm_aux Z 0%Z 1%Z Z.add Z.mul Z.sub Z.opp 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 ring0 add mul
(InitialRing.gen_phiZ ring0 ring1 add mul opp)).
Definition PEevalR : list R -> PEZ -> R :=
PEeval ring0 ring1 add mul sub opp
(gen_phiZ ring0 ring1 add mul opp)
N.to_nat pow.
Lemma P0Z_correct : forall l, PhiR l P0Z = 0.
Proof. trivial. Qed.
Lemma Rext: ring_eq_ext add mul opp _==_.
Proof.
constructor; solve_proper.
Qed.
Lemma Rset : Setoid_Theory R _==_.
apply ring_setoid.
Qed.
Definition Rtheory:ring_theory ring0 ring1 add mul sub opp _==_.
apply mk_rt.
apply ring_add_0_l.
apply ring_add_comm.
apply ring_add_assoc.
apply ring_mul_1_l.
apply cring_mul_comm.
apply ring_mul_assoc.
apply ring_distr_l.
apply ring_sub_def.
apply ring_opp_def.
Defined.
Lemma PolZadd_correct : forall P' P l,
PhiR l (PolZadd P P') == ((PhiR l P) + (PhiR l P')).
Proof.
unfold PolZadd, PhiR. intros. simpl.
refine (Padd_ok Rset Rext (Rth_ARth Rset Rext Rtheory)
(gen_phiZ_morph Rset Rext Rtheory) _ _ _).
Qed.
Lemma PolZmul_correct : forall P P' l,
PhiR l (PolZmul P P') == ((PhiR l P) * (PhiR l P')).
Proof.
unfold PolZmul, PhiR. intros.
refine (Pmul_ok Rset Rext (Rth_ARth Rset Rext Rtheory)
(gen_phiZ_morph Rset Rext Rtheory) _ _ _).
Qed.
Lemma R_power_theory
: Ring_theory.power_theory ring1 mul _==_ N.to_nat pow.
apply Ring_theory.mkpow_th. unfold pow. intros. rewrite Nnat.N2Nat.id.
reflexivity. 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 Rtheory)
(gen_phiZ_morph Rset Rext Rtheory) R_power_theory).
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 Rtheory));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. cring.
destruct lp;trivial. simpl. cring.
simpl in H;destruct H.
rewrite PolZadd_correct.
simpl. rewrite PolZmul_correct. simpl. rewrite H.
rewrite IHla. cring. 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 *)
Definition R2:= 1 + 1.
Fixpoint IPR p {struct p}: R :=
match p with
xH => ring1
| 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 (N.to_nat t2)
| (PEc t1) => (IZR1 t1)
| PEO => 0
| PEI => 1
| (PEX _ n) => List.nth (pred (Pos.to_nat n)) fv 0
end.
End nsatz1.
Ltac equality_to_goal H x y:=
(* eliminate trivial hypotheses, but it takes time!:
let h := fresh "nH" in
(assert (h:equality x y);
[solve [cring] | clear H; clear h])
|| *)
.
Ltac equalities_to_goal :=
lazymatch goal with
| H: (_ ?x ?y) |- _ => equality_to_goal H x y
| H: (_ _ ?x ?y) |- _ => equality_to_goal H x y
| H: (_ _ _ ?x ?y) |- _ => equality_to_goal H x y
| H: (_ _ _ _ ?x ?y) |- _ => equality_to_goal H x y
(* extension possible :-) *)
| H: (?x == ?y) |- _ => equality_to_goal H x y
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 :=
(* idtac "Trying power: " rr;*)
let ll := constr:(PEc info :: PEc nparam :: PEpow p rr :: lp) in
(* idtac "calcul...";*)
nsatz_compute ll;
(* idtac "done";*)
match goal with
| |- (?c::PEpow _ ?r::?lq0)::?lci0 = _ -> _ =>
intros _;
let lci := fresh "lci" in
set (lci:=lci0);
let lq := fresh "lq" in
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
| _ =>
(let r := eval compute in (N.sub radicalmax (N.pred n)) in
nsatz_call_n info nparam p r lp kont) ||
let n' := eval compute in (N.pred n) in try_n n'
end in
try_n radicalmax.
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 => (* idtac "b="; idtac b;*)
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_generic radicalmax info lparam lvar :=
let nparam := eval compute in (Z.of_nat (List.length lparam)) in
match goal with
|- ?g => let lb := lterm_goal g in
match (match lvar with
|(@nil _) =>
match lparam with
|(@nil _) =>
let r := eval red in (list_reifyl (lterm:=lb)) in r
|_ =>
match eval red in (list_reifyl (lterm:=lb)) with
|(?fv, ?le) =>
let fv := parametres_en_tete fv lparam in
(* we reify a second time, with the good order
for variables *)
let r := eval red in
(list_reifyl (lterm:=lb) (lvar:=fv)) in r
end
end
|_ =>
let fv := parametres_en_tete lvar lparam in
let r := eval red in (list_reifyl (lterm:=lb) (lvar:=fv)) in r
end) with
|(?fv, ?le) =>
reify_goal fv le lb ;
match goal with
|- ?g =>
let lp := get_lpol g in
let lpol := eval compute in (List.rev lp) in
intros;
let SplitPolyList kont :=
match lpol with
| ?p2::?lp2 => kont p2 lp2
| _ => idtac "polynomial not in the ideal"
end in
SplitPolyList ltac:(fun p lp =>
let p21 := fresh "p21" in
let lp21 := fresh "lp21" in
set (p21:=p) ;
set (lp21:=lp);
(* idtac "nparam:"; idtac nparam; idtac "p:"; idtac p; idtac "lp:"; idtac lp; *)
nsatz_call radicalmax info nparam p lp ltac:(fun c r lq lci =>
let q := fresh "q" in
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) == 0);
[ (*simpl*) idtac;
generalize (@check_correct _ _ _ _ _ _ _ _ _ _ _ fv lp21 q (lci,lq) Hg);
let cc := fresh "H" in
(*simpl*) idtac; intro cc; apply cc; clear cc;
(*simpl*) idtac;
repeat (split;[assumption|idtac]); exact I
| (*simpl in Hg2;*) (*simpl*) idtac;
apply Rintegral_domain_pow with (interpret3 c fv) (N.to_nat r);
(*simpl*) idtac;
try apply integral_domain_one_zero;
try apply integral_domain_minus_one_zero;
try trivial;
try exact integral_domain_one_zero;
try exact integral_domain_minus_one_zero
|| (solve [simpl; unfold R2, equality, eq_notation, addition, add_notation,
one, one_notation, multiplication, mul_notation, zero, zero_notation;
discrR || omega])
|| ((*simpl*) idtac) || idtac "could not prove discrimination result"
]
]
)
)
end end end .
Ltac nsatz_default:=
intros;
try apply (@psos_r1b _ _ _ _ _ _ _ _ _ _ _);
match goal with |- (@equality ?r _ _ _) =>
repeat equalities_to_goal;
nsatz_generic 6%N 1%Z (@nil r) (@nil r)
end.
Tactic Notation "nsatz" := nsatz_default.
Tactic Notation "nsatz" "with"
"radicalmax" ":=" constr(radicalmax)
"strategy" ":=" constr(info)
"parameters" ":=" constr(lparam)
"variables" ":=" constr(lvar):=
intros;
try apply (@psos_r1b _ _ _ _ _ _ _ _ _ _ _);
match goal with |- (@equality ?r _ _ _) =>
repeat equalities_to_goal;
nsatz_generic radicalmax info lparam lvar
end.
(* Real numbers *)
Require Import Reals.
Require Import RealField.
Lemma Rsth : Setoid_Theory R (@eq R).
constructor;red;intros;subst;trivial.
Qed.
Instance Rops: (@Ring_ops R 0%R 1%R Rplus Rmult Rminus Ropp (@eq R)).
Defined.
Instance Rri : (Ring (Ro:=Rops)).
constructor;
try (try apply Rsth;
try (unfold respectful, Proper; unfold equality; unfold eq_notation in *;
intros; try rewrite H; try rewrite H0; reflexivity)).
exact Rplus_0_l. exact Rplus_comm. symmetry. apply Rplus_assoc.
exact Rmult_1_l. exact Rmult_1_r. symmetry. apply Rmult_assoc.
exact Rmult_plus_distr_r. intros; apply Rmult_plus_distr_l.
exact Rplus_opp_r.
Defined.
Class can_compute_Z (z : Z) := dummy_can_compute_Z : True.
Hint Extern 0 (can_compute_Z ?v) =>
match isZcst v with true => exact I end : typeclass_instances.
Instance reify_IZR z lvar {_ : can_compute_Z z} : reify (PEc z) lvar (IZR z).
Defined.
Lemma R_one_zero: 1%R <> 0%R.
discrR.
Qed.
Instance Rcri: (Cring (Rr:=Rri)).
red. exact Rmult_comm. Defined.
Instance Rdi : (Integral_domain (Rcr:=Rcri)).
constructor.
exact Rmult_integral. exact R_one_zero. Defined.
(* Rational numbers *)
Require Import QArith.
Instance Qops: (@Ring_ops Q 0%Q 1%Q Qplus Qmult Qminus Qopp Qeq).
Defined.
Instance Qri : (Ring (Ro:=Qops)).
constructor.
try apply Q_Setoid.
apply Qplus_comp.
apply Qmult_comp.
apply Qminus_comp.
apply Qopp_comp.
exact Qplus_0_l. exact Qplus_comm. apply Qplus_assoc.
exact Qmult_1_l. exact Qmult_1_r. apply Qmult_assoc.
apply Qmult_plus_distr_l. intros. apply Qmult_plus_distr_r.
reflexivity. exact Qplus_opp_r.
Defined.
Lemma Q_one_zero: not (Qeq 1%Q 0%Q).
unfold Qeq. simpl. auto with *. Qed.
Instance Qcri: (Cring (Rr:=Qri)).
red. exact Qmult_comm. Defined.
Instance Qdi : (Integral_domain (Rcr:=Qcri)).
constructor.
exact Qmult_integral. exact Q_one_zero. Defined.
(* Integers *)
Lemma Z_one_zero: 1%Z <> 0%Z.
omega.
Qed.
Instance Zcri: (Cring (Rr:=Zr)).
red. exact Z.mul_comm. Defined.
Instance Zdi : (Integral_domain (Rcr:=Zcri)).
constructor.
exact Zmult_integral. exact Z_one_zero. Defined.
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