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Adapt to PR tify
This is the companion of https://github.com/fajb/coq/tree/tify - Update the reflexive proof verifier for lia,lra etc. - Define instances for the [rify] tactic (Require Import Rify)
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‎test-suite/micromega/milp.v‎

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Require Import Rbase Rify isZ Lra Zfloor.
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Local Open Scope R_scope.
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Lemma up0 : up 0%R = 1%Z.
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Proof. intros. rify. lra. Qed.
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Lemma up_succ r : up (r + 1)%R = (up r + 1)%Z.
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Proof. intros. rify. lra. Qed.
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Lemma up_IZR z : up (IZR z) = (z + 1)%Z.
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Proof.
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rify. lra.
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Qed.
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Lemma up_shiftz r z : up (r + IZR z)%R = (up r + z)%Z.
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Proof.
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rify. lra.
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Qed.
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Lemma up_succ2 r n : Zfloor (r + IZR n) = (Zfloor r + n)%Z.
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Proof.
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rify. lra.
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Qed.
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Goal forall x1 x2 x3 x5 x6
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(H: -x2 > 0)
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(H0: x3 >= 0)
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(H1: -x1 + x5 + x6 = 0)
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(H2: x1 + -x5 + x6 = 0)
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(H4: -x1 > 0)
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(H6: x2 + -x3 = 0)
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(H7: 3 + 3*x1 = 0),
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False.
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Proof.
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intros.
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lra.
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Qed.
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Lemma spurious_isZ : forall
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(x l X : R)
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(delta : R)
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(iZ1 : isZ x)
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(H13 : - (X - l) <= 0)
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(iZ2 : isZ X)
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(H4 : l > 0)
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(iZ2 : isZ X)
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(H9 : delta / 2 <> 0)
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(iZ2 : isZ X)
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(H11 : 0 < delta / 2)
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(iZ1 : isZ x)
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(H10 : (X <= 0)),
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False.
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Proof.
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intros.
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lra.
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Qed.
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Goal forall (k x:R) (P:Prop)
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(Hyp : 0 <= k < 1)
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(H2 : k < x < 1 /\ P)
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(H3 : k < x < 1),
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0 < x.
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Proof.
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intros.
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lra.
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Qed.
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Ltac cnf := match goal with
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| |- Tauto.cnf_checker _ ?F _ = true =>
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let f:= fresh "f" in
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set(f:=F); compute in f; unfold f;clear f
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end.
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Ltac collect :=
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match goal with
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| |- context [ RMicromega.xcollect_isZ ?A ?B ?F] =>
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let f := fresh "f" in
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set (f:= RMicromega.xcollect_isZ A B F) ; compute in f ; unfold f; clear f
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end.
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Goal forall (x y:R)
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(H : x <= y)
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(Hlt : x < y),
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y - x = 0 -> False.
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Proof.
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intros.
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lra.
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Qed.
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Lemma two_x_1 : forall (x:R)
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(IZ : isZ x)
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(H : 2 * x + 1 = 0),
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False.
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Proof.
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intros.
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lra.
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Qed.
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Goal forall x1 x2 x3 x4,
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isZ x1 ->
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isZ x2 ->
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isZ x3 ->
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isZ x4 ->
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- x1 + x2 - x3 + x4 >= 0 ->
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x1 - x2 + x3 - x4 + 1 > 0 ->
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- x1 + x2 - x3 + x4 > 0 ->
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False.
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Proof.
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intros. lra.
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Qed.
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Goal forall z1 z2 z3, Zfloor (IZR z1 - IZR z2 + IZR z3)%R = (z1 - z2 + z3)%Z.
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Proof.
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intros.
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rify.
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(** Now, some work over isZ *)
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lra.
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Qed.
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Goal forall (z : Z) x, IZR z <= x -> (z <= Zfloor x)%Z.
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Proof.
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rify.
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lra.
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Qed.
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Goal forall (z : Z) x, IZR z <= x < IZR z + 1 -> Zfloor x = z.
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Proof.
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intros.
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rify.
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lra.
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Qed.
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Goal R1 + (R1 + R1) = 3.
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Proof.
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intros.
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rify.
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reflexivity.
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Qed.
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Goal forall x,
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x * 3 = 1 ->
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False.
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Proof.
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intros.
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Fail lra.
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Abort.
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Goal forall x,
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isZ x ->
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x * 3 = 1 ->
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False.
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Proof.
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intros.
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lra.
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Qed.
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Require Import Rfunctions R_sqrt.
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Goal forall e x
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(He : 0 < e)
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(ef := Rmin (e / (3 * (Rabs (x) + 1))) (sqrt (e / 3)) : R)
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(H : sqrt (e / 3) > 0)
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(H0 : 3 * (Rabs ( x) + 1) >= 1),
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0 < ef.
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Proof.
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intros.
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rify. nra.
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Qed.
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Goal forall x v,
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v * (3 * x) = 1 ->
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x >= 1 ->
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v = 1 / (3 * x).
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Proof.
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intros.
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rify. nra.
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Qed.
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‎test-suite/output/MExtraction.v‎

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@@ -57,11 +57,12 @@ Extract Constant Rinv => "fun x -> 1 / x".
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extraction is only performed as a test in the test suite. *)
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Recursive Extraction
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Tauto.mapX Tauto.foldA Tauto.collect_annot Tauto.ids_of_formula Tauto.map_bformula
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map_eFormula eFormula
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Tauto.abst_form
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ZMicromega.cnfZ ZMicromega.Zeval_const QMicromega.cnfQ
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ZMicromega.cnfZ ZMicromega.Zeval_const QMicromega.cnfQ RMicromega.cnfR
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List.map simpl_cone (*map_cone indexes*)
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denorm QArith_base.Qpower vm_add
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normZ normQ normQ Z.to_N N.of_nat ZTautoChecker ZWeakChecker QTautoChecker RTautoChecker find.
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normZ normQ (*normQ*) Z.to_N N.of_nat ZTautoChecker ZWeakChecker QTautoChecker RTautoChecker find.
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(* Local Variables: *)
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(* coding: utf-8 *)

‎test-suite/success/tify.v‎

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From Stdlib Require Import Reals.
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From Stdlib Require Import ZArith.
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From Stdlib Require Import ZifyClasses Tify.
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(* Inject nat -> Z *)
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Instance Inj_nat_Z : InjTyp nat Z.
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Proof.
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apply (mkinj _ _ Z.of_nat (fun x => (0 <= x)%Z)).
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apply Nat2Z.is_nonneg.
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Defined.
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Add Tify InjTyp Inj_nat_Z.
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Instance Op_nat_add_Z : BinOp Nat.add.
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Proof.
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apply mkbop with (TBOp := Z.add).
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simpl. apply Nat2Z.inj_add.
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Defined.
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Add Tify BinOp Op_nat_add_Z.
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Instance Op_eq_nat_eq_Z : BinRel (@eq nat).
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Proof.
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apply mkbrel with (TR := @eq Z).
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simpl. split. congruence.
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apply Nat2Z.inj.
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Defined.
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Add Tify BinRel Op_eq_nat_eq_Z.
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(** Inject nat -> R *)
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Definition isZ (r:R) := IZR (Zfloor r) = r.
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Instance Inj_nat_R : InjTyp nat R.
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Proof.
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apply (mkinj _ _ INR (fun r => isZ r /\ 0 <= r)).
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- intros. split. unfold isZ.
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rewrite INR_IZR_INZ.
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rewrite ZfloorZ. reflexivity.
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apply pos_INR.
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Defined.
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Add Tify InjTyp Inj_nat_R.
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Instance Op_add_R : BinOp Nat.add.
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Proof.
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apply mkbop with (TBOp := Rplus).
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simpl. intros. apply plus_INR.
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Defined.
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Add Tify BinOp Op_add_R.
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Instance Op_eq_nat_eq_R : BinRel (@eq nat).
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Proof.
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apply mkbrel with (TR := @eq R).
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simpl. split; intro. congruence.
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apply INR_eq; auto.
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Defined.
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Add Tify BinRel Op_eq_nat_eq_R.
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(* tify selects the right injection *)
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Goal forall (x y:nat), (x + y = y + x)%nat.
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Proof.
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intros.
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tify R.
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apply Rplus_comm.
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Qed.
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Goal forall (x y:nat), (x + y = y + x)%nat.
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Proof.
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intros.
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tify Z.
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apply Zplus_comm.
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Qed.
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(* The R instances do not break lia. *)
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From Stdlib Require Import Lia.
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Goal forall (x y:nat), (x + y = y + x)%nat.
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Proof.
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intros.
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lia.
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Qed.

‎theories/All/All.v‎

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From Stdlib Require Export micromega.Lra.
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From Stdlib Require Export micromega.OrderedRing.
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From Stdlib Require Export micromega.Psatz.
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From Stdlib Require Export micromega.MMicromega.
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From Stdlib Require Export micromega.QMicromega.
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From Stdlib Require Export micromega.RMicromega.
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From Stdlib Require Export micromega.Refl.
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From Stdlib Require Export micromega.ZCoeff.
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From Stdlib Require Export micromega.ZMicromega.
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From Stdlib Require Export micromega.Zify.
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From Stdlib Require Export micromega.Tify.
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From Stdlib Require Export micromega.ZifyBool.
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From Stdlib Require Export micromega.ZifyClasses.
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From Stdlib Require Export micromega.TifyClasses.
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From Stdlib Require Export micromega.ZifyComparison.
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From Stdlib Require Export micromega.ZifyInst.
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From Stdlib Require Export micromega.ZifyN.
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From Stdlib Require Export micromega.ZifySint63.
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From Stdlib Require Export micromega.ZifyUint63.
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From Stdlib Require Export micromega.Ztac.
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From Stdlib Require Export micromega.isZ.
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From Stdlib Require Export micromega.Rify.
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From Stdlib Require Export funind.FunInd.
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From Stdlib Require Export funind.Recdef.
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From Stdlib Require Export extraction.ExtrHaskellBasic.

‎theories/Reals/RIneq.v‎

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Lemma succ_IZR : forall n:Z, IZR (Z.succ n) = IZR n + 1.
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Proof. now intros n; unfold Z.succ; apply plus_IZR. Qed.
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Lemma pred_IZR : forall n:Z, IZR (Z.pred n) = IZR n - 1.
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Proof. now intros n; unfold Z.pred; apply plus_IZR. Qed.
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Lemma opp_IZR : forall n:Z, IZR (- n) = - IZR n.
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Proof.
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intros [| p | p]; unfold IZR; simpl.

‎theories/Strings/PString.v‎

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#[local] Open Scope list_scope.
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#[local] Arguments to_Z _/ : simpl nomatch.
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#[local] Instance Op_max_length : ZifyClasses.CstOp max_length :=
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#[local] Instance Op_max_length : TifyClasses.CstOp max_length :=
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{ TCst := 16777211%Z ; TCstInj := eq_refl }.
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Add Zify CstOp Op_max_length.
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Add Tify CstOp Op_max_length.
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#[local] Ltac case_if :=
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lazymatch goal with

‎theories/micromega/Lia.v‎

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(************************************************************************)
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(* * The Rocq Prover / The Rocq Development Team *)
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(* v * Copyright INRIA, CNRS and contributors *)

‎theories/micromega/Lra.v‎

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@@ -29,7 +29,7 @@ Ltac rchange :=
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let __varmap := fresh "__varmap" in
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let __ff := fresh "__ff" in
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intros __wit __varmap __ff ;
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change (Tauto.eval_bf (Reval_formula (@find R 0%R __varmap)) __ff) ;
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change (Tauto.eval_bf (eReval_formula (@find R 0%R __varmap)) __ff) ;
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apply (RTautoChecker_sound __ff __wit).
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Ltac rchecker_no_abstract := rchange ; vm_compute ; reflexivity.

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