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Replace omega with lia
This commit is contained in:
Adam Chlipala
parent
d41373e0cb
commit
5a28d4fe6a
5 changed files with 40 additions and 41 deletions
1
.gitignore
vendored
1
.gitignore
vendored
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@ -22,3 +22,4 @@ Deep.ml*
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Deeper.ml*
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DeeperWithFail.ml*
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*.dir-locals.el
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*.cache
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@ -1538,7 +1538,7 @@ Proof.
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repeat erewrite interp_agree in * by eassumption; eauto 10.
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assert (Hnu : noUnderscoreArith e = true) by assumption.
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eapply flatten_Assign with (tempCount := 0) (dst := x) in Hnu; eauto.
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eapply flatten_Assign with (tempCount := 0) (dst := x) in Hnu; try eassumption.
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first_order.
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do 3 eexists.
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split.
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@ -1659,8 +1659,8 @@ Proof.
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simplify; propositional; eauto.
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invert H1; simplify; subst.
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assert (noUnderscore c2 = true) by eauto.
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eapply flatten_ok' in H5; eauto.
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assert (noUnderscore c2 = true) by eauto 4.
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eapply flatten_ok' in H5; eauto 4.
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first_order.
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cases vc2; simplify; subst.
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pose proof (plug_total x0 (flattenContext C)).
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@ -1,4 +1,4 @@
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Require Import Eqdep String Arith Omega Program Sets Relations Map Var Invariant Bool ModelCheck.
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Require Import Eqdep String Arith Lia Program Sets Relations Map Var Invariant Bool ModelCheck.
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Export Ascii String Arith Sets Relations Map Var Invariant Bool ModelCheck.
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Require Import List.
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Export List ListNotations.
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@ -177,8 +177,8 @@ Ltac doSubtract :=
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Ltac simpl_maps :=
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repeat match goal with
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| [ |- context[add ?m ?k1 ?v $? ?k2] ] =>
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(rewrite (@lookup_add_ne _ _ m k1 k2 v) by (congruence || omega))
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|| (rewrite (@lookup_add_eq _ _ m k1 k2 v) by (congruence || omega))
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(rewrite (@lookup_add_ne _ _ m k1 k2 v) by (congruence || lia))
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|| (rewrite (@lookup_add_eq _ _ m k1 k2 v) by (congruence || lia))
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end.
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Ltac simplify := repeat (unifyTails; pose proof I);
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@ -203,7 +203,7 @@ Ltac linear_arithmetic := intros;
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| [ _ : context[min ?a ?b] |- _ ] =>
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let Heq := fresh "Heq" in destruct (Min.min_spec a b) as [[? Heq] | [? Heq]];
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rewrite Heq in *; clear Heq
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end; omega.
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end; lia.
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Ltac equality := intuition congruence.
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@ -333,8 +333,8 @@ Inductive ordering (n m : nat) :=
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| Gt (_ : n > m).
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Local Hint Constructors ordering.
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Local Hint Extern 1 (_ < _) => omega.
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Local Hint Extern 1 (_ > _) => omega.
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Local Hint Extern 1 (_ < _) => lia.
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Local Hint Extern 1 (_ > _) => lia.
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Theorem totally_ordered : forall n m, ordering n m.
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Proof.
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15
HoareLogic.v
15
HoareLogic.v
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@ -170,6 +170,7 @@ Qed.
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(* BEGIN syntax macros that won't be explained *)
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Coercion Const : nat >-> exp.
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Coercion Var : string >-> exp.
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Declare Scope cmd_scope.
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Notation "*[ e ]" := (Read e) : cmd_scope.
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Infix "+" := Plus : cmd_scope.
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Infix "-" := Minus : cmd_scope.
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@ -189,6 +190,7 @@ Notation "{{ I }} 'while' b 'loop' body 'done'" := (While_ I b body) (at level 7
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Notation "'assert' {{ I }}" := (Assert I) (at level 75).
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Delimit Scope cmd_scope with cmd.
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Declare Scope reset_scope.
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Infix "+" := plus : reset_scope.
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Infix "-" := Init.Nat.sub : reset_scope.
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Infix "*" := mult : reset_scope.
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@ -351,8 +353,6 @@ Proof.
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apply HtAssign.
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simplify.
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t.
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ring [H0].
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(* This variant of [ring] suggests a hypothesis to use in the proof. *)
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simplify.
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t.
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Qed.
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@ -368,7 +368,6 @@ Theorem fact_ok_snazzy : forall n,
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{{_&v ~> v $! "acc" = fact n}}.
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Proof.
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ht.
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ring [H0].
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Qed.
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(** ** Selection sort *)
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@ -384,10 +383,10 @@ Proof.
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ht.
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Qed.
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Hint Resolve leq_f.
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Hint Extern 1 (@eq nat _ _) => linear_arithmetic.
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Hint Extern 1 (_ < _) => linear_arithmetic.
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Hint Extern 1 (_ <= _) => linear_arithmetic.
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Hint Resolve leq_f : core.
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Hint Extern 1 (@eq nat _ _) => linear_arithmetic : core.
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Hint Extern 1 (_ < _) => linear_arithmetic : core.
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Hint Extern 1 (_ <= _) => linear_arithmetic : core.
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(* We also register [linear_arithmetic] as a step to try during proof search. *)
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(* These invariants are fairly hairy, but probably the best way to understand
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@ -471,7 +470,7 @@ Inductive step : heap * valuation * cmd -> heap * valuation * cmd -> Prop :=
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a h v
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-> step (h, v, Assert a) (h, v, Skip).
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Hint Constructors step.
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Hint Constructors step : core.
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Definition trsys_of (st : heap * valuation * cmd) := {|
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Initial := {st};
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@ -205,7 +205,7 @@ Module References.
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-> heapty ht h.
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Hint Constructors value plug step0 step hasty heapty.
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Hint Constructors value plug step0 step hasty heapty : core.
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(* Perhaps surprisingly, this language admits well-typed, nonterminating
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@ -222,7 +222,7 @@ Module References.
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repeat (econstructor; simplify).
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Qed.
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Hint Resolve lookup_add_eq.
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Hint Resolve lookup_add_eq : core.
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Ltac loopy := propositional; subst; simplify;
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repeat match goal with
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@ -293,7 +293,7 @@ Module References.
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Ltac t := simplify; propositional; repeat (t0; simplify); try equality; eauto 7.
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Hint Extern 2 (exists _ : _ * _, _) => eexists (_ $+ (_, _), _).
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Hint Extern 2 (exists _ : _ * _, _) => eexists (_ $+ (_, _), _) : core.
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(* Progress is quite straightforward. *)
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Lemma progress : forall ht h, heapty ht h
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@ -323,7 +323,7 @@ Module References.
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cases (x ==v x'); simplify; eauto.
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Qed.
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Hint Resolve weakening_override.
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Hint Resolve weakening_override : core.
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Lemma weakening : forall H G e t,
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hasty H G e t
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@ -333,7 +333,7 @@ Module References.
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induct 1; t.
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Qed.
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Hint Resolve weakening.
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Hint Resolve weakening : core.
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Lemma hasty_change : forall H G e t,
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hasty H G e t
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@ -343,7 +343,7 @@ Module References.
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t.
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Qed.
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Hint Resolve hasty_change.
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Hint Resolve hasty_change : core.
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Lemma substitution : forall H G x t' e t e',
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hasty H (G $+ (x, t')) e t
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@ -353,7 +353,7 @@ Module References.
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induct 1; t.
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Qed.
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Hint Resolve substitution.
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Hint Resolve substitution : core.
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(* A new property: expanding the heap typing preserves typing. *)
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Lemma heap_weakening : forall H G e t,
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@ -364,7 +364,7 @@ Module References.
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induct 1; t.
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Qed.
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Hint Resolve heap_weakening.
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Hint Resolve heap_weakening : core.
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(* A property about extending heap typings *)
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Lemma heap_override : forall H h k t t0 l,
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@ -378,7 +378,7 @@ Module References.
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apply H2 in H0; t.
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Qed.
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Hint Resolve heap_override.
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Hint Resolve heap_override : core.
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(* A higher-level property, stated via [heapty] *)
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Lemma heapty_extend : forall H h l t v,
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@ -394,12 +394,11 @@ Module References.
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invert H0; eauto 6.
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apply H3 in H0; t.
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rewrite lookup_add_ne by linear_arithmetic.
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apply H4.
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linear_arithmetic.
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Qed.
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Hint Resolve heapty_extend.
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Hint Resolve heapty_extend : core.
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(* The old cases of preservation proceed as before, and we need to fiddle with
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* the heap in the new cases. Note a crucial change to the theorem statement:
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@ -445,7 +444,7 @@ Module References.
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assumption.
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Qed.
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Hint Resolve preservation0.
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Hint Resolve preservation0 : core.
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(* This lemma gets more complicated, too, to accommodate heap typings. *)
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Lemma generalize_plug : forall H e1 C e1',
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@ -482,7 +481,7 @@ Module References.
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eauto.
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Qed.
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Hint Resolve progress preservation.
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Hint Resolve progress preservation : core.
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(* We'll need this fact for the base case of invariant induction. *)
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Lemma heapty_empty : heapty $0 $0.
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@ -490,7 +489,7 @@ Module References.
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exists 0; t.
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Qed.
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Hint Resolve heapty_empty.
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Hint Resolve heapty_empty : core.
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(* Now there isn't much more to the proof of type safety. The crucial overall
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* insight is a strengthened invariant that quantifies existentially over a
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@ -504,7 +503,7 @@ Module References.
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apply invariant_weaken with (invariant1 := fun he' => exists H,
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hasty H $0 (snd he') t
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/\ heapty H (fst he')); eauto.
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apply invariant_induction; simplify; eauto.
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apply invariant_induction; simplify.
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propositional.
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subst; simplify.
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eauto.
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@ -593,7 +592,7 @@ Module GarbageCollection.
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-> step (h, e) (h', e).
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Hint Constructors step.
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Hint Constructors step : core.
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Definition trsys_of (e : exp) := {|
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Initial := {($0, e)};
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@ -632,10 +631,10 @@ Module GarbageCollection.
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assumption.
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Qed.
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Hint Resolve reachableLocFromExp_trans.
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Hint Extern 1 (_ \in _) => simplify; solve [ sets ].
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Hint Extern 1 (_ \subseteq _) => simplify; solve [ sets ].
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Hint Constructors reachableLoc reachableLocFromExp.
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Hint Resolve reachableLocFromExp_trans : core.
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Hint Extern 1 (_ \in _) => simplify; solve [ sets ] : core.
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Hint Extern 1 (_ \subseteq _) => simplify; solve [ sets ] : core.
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Hint Constructors reachableLoc reachableLocFromExp : core.
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(* Typing is preserved by moving to a heap typing that only needs to preserve
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* the mappings for *reachable* locations. *)
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@ -685,9 +684,9 @@ Module GarbageCollection.
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(* The case for the original [step] rule proceeds exactly the same way as
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* before. *)
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eapply generalize_plug in H; eauto.
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eapply generalize_plug in H; eauto 3.
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invert H; propositional.
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eapply preservation0 in H6; eauto.
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eapply preservation0 in H6; eauto 3.
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invert H6; propositional.
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eauto.
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@ -731,7 +730,7 @@ Module GarbageCollection.
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equality.
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Qed.
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Hint Resolve progress preservation.
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Hint Resolve progress preservation : core.
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(* The safety proof itself is anticlimactic, looking the same as before. *)
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Theorem safety : forall e t, hasty $0 $0 e t
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@ -743,7 +742,7 @@ Module GarbageCollection.
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apply invariant_weaken with (invariant1 := fun he' => exists H,
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hasty H $0 (snd he') t
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/\ heapty H (fst he')); eauto.
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apply invariant_induction; simplify; eauto.
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apply invariant_induction; simplify.
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propositional.
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subst; simplify.
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eauto.
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