mirror of
https://github.com/achlipala/frap.git
synced 2024-11-10 00:07:51 +00:00
Add AbstractInterpret; fix 8.4 compatibility
This commit is contained in:
parent
3657865469
commit
c9cedde15f
4 changed files with 562 additions and 4 deletions
557
AbstractInterpret.v
Normal file
557
AbstractInterpret.v
Normal file
|
@ -0,0 +1,557 @@
|
|||
(** Formal Reasoning About Programs <http://adam.chlipala.net/frap/>
|
||||
* Chapter 7: Abstract Interpretation and Dataflow Analysis
|
||||
* Author: Adam Chlipala
|
||||
* License: https://creativecommons.org/licenses/by-nc-nd/4.0/ *)
|
||||
|
||||
Require Import Frap Imp.
|
||||
Export Imp.
|
||||
|
||||
Set Implicit Arguments.
|
||||
|
||||
|
||||
(* Reduced version of code from AbstractInterpretation.v *)
|
||||
|
||||
Record absint := {
|
||||
Domain :> Set;
|
||||
Top : Domain;
|
||||
Constant : nat -> Domain;
|
||||
Add : Domain -> Domain -> Domain;
|
||||
Subtract : Domain -> Domain -> Domain;
|
||||
Multiply : Domain -> Domain -> Domain;
|
||||
Join : Domain -> Domain -> Domain;
|
||||
Represents : nat -> Domain -> Prop
|
||||
}.
|
||||
|
||||
Record absint_sound (a : absint) : Prop := {
|
||||
TopSound : forall n, a.(Represents) n a.(Top);
|
||||
|
||||
ConstSound : forall n, a.(Represents) n (a.(Constant) n);
|
||||
|
||||
AddSound : forall n na m ma, a.(Represents) n na
|
||||
-> a.(Represents) m ma
|
||||
-> a.(Represents) (n + m) (a.(Add) na ma);
|
||||
SubtractSound: forall n na m ma, a.(Represents) n na
|
||||
-> a.(Represents) m ma
|
||||
-> a.(Represents) (n - m) (a.(Subtract) na ma);
|
||||
MultiplySound : forall n na m ma, a.(Represents) n na
|
||||
-> a.(Represents) m ma
|
||||
-> a.(Represents) (n * m) (a.(Multiply) na ma);
|
||||
|
||||
AddMonotone : forall na na' ma ma', (forall n, a.(Represents) n na -> a.(Represents) n na')
|
||||
-> (forall n, a.(Represents) n ma -> a.(Represents) n ma')
|
||||
-> (forall n, a.(Represents) n (a.(Add) na ma)
|
||||
-> a.(Represents) n (a.(Add) na' ma'));
|
||||
SubtractMonotone : forall na na' ma ma', (forall n, a.(Represents) n na -> a.(Represents) n na')
|
||||
-> (forall n, a.(Represents) n ma -> a.(Represents) n ma')
|
||||
-> (forall n, a.(Represents) n (a.(Subtract) na ma)
|
||||
-> a.(Represents) n (a.(Subtract) na' ma'));
|
||||
MultiplyMonotone : forall na na' ma ma', (forall n, a.(Represents) n na -> a.(Represents) n na')
|
||||
-> (forall n, a.(Represents) n ma -> a.(Represents) n ma')
|
||||
-> (forall n, a.(Represents) n (a.(Multiply) na ma)
|
||||
-> a.(Represents) n (a.(Multiply) na' ma'));
|
||||
|
||||
JoinSoundLeft : forall x y n, a.(Represents) n x
|
||||
-> a.(Represents) n (a.(Join) x y);
|
||||
JoinSoundRight : forall x y n, a.(Represents) n y
|
||||
-> a.(Represents) n (a.(Join) x y)
|
||||
}.
|
||||
|
||||
Hint Resolve TopSound ConstSound AddSound SubtractSound MultiplySound
|
||||
AddMonotone SubtractMonotone MultiplyMonotone
|
||||
JoinSoundLeft JoinSoundRight.
|
||||
|
||||
|
||||
|
||||
Definition astate (a : absint) := fmap var a.
|
||||
|
||||
Fixpoint absint_interp (e : arith) a (s : astate a) : a :=
|
||||
match e with
|
||||
| Const n => a.(Constant) n
|
||||
| Var x => match s $? x with
|
||||
| None => a.(Top)
|
||||
| Some xa => xa
|
||||
end
|
||||
| Plus e1 e2 => a.(Add) (absint_interp e1 s) (absint_interp e2 s)
|
||||
| Minus e1 e2 => a.(Subtract) (absint_interp e1 s) (absint_interp e2 s)
|
||||
| Times e1 e2 => a.(Multiply) (absint_interp e1 s) (absint_interp e2 s)
|
||||
end.
|
||||
|
||||
Definition merge_astate a : astate a -> astate a -> astate a :=
|
||||
merge (fun x y =>
|
||||
match x with
|
||||
| None => None
|
||||
| Some x' =>
|
||||
match y with
|
||||
| None => None
|
||||
| Some y' => Some (a.(Join) x' y')
|
||||
end
|
||||
end).
|
||||
|
||||
Definition subsumed a (s1 s2 : astate a) :=
|
||||
forall x, match s1 $? x with
|
||||
| None => s2 $? x = None
|
||||
| Some xa1 =>
|
||||
forall xa2, s2 $? x = Some xa2
|
||||
-> forall n, a.(Represents) n xa1
|
||||
-> a.(Represents) n xa2
|
||||
end.
|
||||
|
||||
Theorem subsumed_refl : forall a (s : astate a),
|
||||
subsumed s s.
|
||||
Proof.
|
||||
unfold subsumed; simplify.
|
||||
cases (s $? x); equality.
|
||||
Qed.
|
||||
|
||||
Hint Resolve subsumed_refl.
|
||||
|
||||
Lemma subsumed_use : forall a (s s' : astate a) x n t0 t,
|
||||
s $? x = Some t0
|
||||
-> subsumed s s'
|
||||
-> s' $? x = Some t
|
||||
-> Represents a n t0
|
||||
-> Represents a n t.
|
||||
Proof.
|
||||
unfold subsumed; simplify.
|
||||
specialize (H0 x).
|
||||
rewrite H in H0.
|
||||
eauto.
|
||||
Qed.
|
||||
|
||||
Lemma subsumed_use_empty : forall a (s s' : astate a) x n t0 t,
|
||||
s $? x = None
|
||||
-> subsumed s s'
|
||||
-> s' $? x = Some t
|
||||
-> Represents a n t0
|
||||
-> Represents a n t.
|
||||
Proof.
|
||||
unfold subsumed; simplify.
|
||||
specialize (H0 x).
|
||||
rewrite H in H0.
|
||||
equality.
|
||||
Qed.
|
||||
|
||||
Hint Resolve subsumed_use subsumed_use_empty.
|
||||
|
||||
Lemma subsumed_trans : forall a (s1 s2 s3 : astate a),
|
||||
subsumed s1 s2
|
||||
-> subsumed s2 s3
|
||||
-> subsumed s1 s3.
|
||||
Proof.
|
||||
unfold subsumed; simplify.
|
||||
specialize (H x); specialize (H0 x).
|
||||
cases (s1 $? x); simplify.
|
||||
cases (s2 $? x); eauto.
|
||||
cases (s2 $? x); eauto.
|
||||
equality.
|
||||
Qed.
|
||||
|
||||
Lemma subsumed_merge_left : forall a, absint_sound a
|
||||
-> forall s1 s2 : astate a,
|
||||
subsumed s1 (merge_astate s1 s2).
|
||||
Proof.
|
||||
unfold subsumed, merge_astate; simplify.
|
||||
cases (s1 $? x); trivial.
|
||||
cases (s2 $? x); simplify; try equality.
|
||||
invert H0; eauto.
|
||||
Qed.
|
||||
|
||||
Hint Resolve subsumed_merge_left.
|
||||
|
||||
Lemma subsumed_add : forall a, absint_sound a
|
||||
-> forall (s1 s2 : astate a) x v1 v2,
|
||||
subsumed s1 s2
|
||||
-> (forall n, a.(Represents) n v1 -> a.(Represents) n v2)
|
||||
-> subsumed (s1 $+ (x, v1)) (s2 $+ (x, v2)).
|
||||
Proof.
|
||||
unfold subsumed; simplify.
|
||||
cases (x ==v x0); subst; simplify; eauto.
|
||||
invert H2; eauto.
|
||||
specialize (H0 x0); eauto.
|
||||
Qed.
|
||||
|
||||
Hint Resolve subsumed_add.
|
||||
|
||||
|
||||
(** * Flow-sensitive analysis *)
|
||||
|
||||
Definition compatible a (s : astate a) (v : valuation) : Prop :=
|
||||
forall x xa, s $? x = Some xa
|
||||
-> exists n, v $? x = Some n
|
||||
/\ a.(Represents) n xa.
|
||||
|
||||
Lemma compatible_add : forall a (s : astate a) v x na n,
|
||||
compatible s v
|
||||
-> a.(Represents) n na
|
||||
-> compatible (s $+ (x, na)) (v $+ (x, n)).
|
||||
Proof.
|
||||
unfold compatible; simplify.
|
||||
cases (x ==v x0); simplify; eauto.
|
||||
invert H1; eauto.
|
||||
Qed.
|
||||
|
||||
Hint Resolve compatible_add.
|
||||
|
||||
(* A similar result follows about soundness of expression interpretation. *)
|
||||
Theorem absint_interp_ok : forall a, absint_sound a
|
||||
-> forall (s : astate a) v e,
|
||||
compatible s v
|
||||
-> a.(Represents) (interp e v) (absint_interp e s).
|
||||
Proof.
|
||||
induct e; simplify; eauto.
|
||||
cases (s $? x); auto.
|
||||
unfold compatible in H0.
|
||||
apply H0 in Heq.
|
||||
invert Heq.
|
||||
propositional.
|
||||
rewrite H2.
|
||||
assumption.
|
||||
Qed.
|
||||
|
||||
Hint Resolve absint_interp_ok.
|
||||
|
||||
Definition astates (a : absint) := fmap cmd (astate a).
|
||||
|
||||
Fixpoint absint_step a (s : astate a) (c : cmd) (wrap : cmd -> cmd) : option (astates a) :=
|
||||
match c with
|
||||
| Skip => None
|
||||
| Assign x e => Some ($0 $+ (wrap Skip, s $+ (x, absint_interp e s)))
|
||||
| Sequence c1 c2 =>
|
||||
match absint_step s c1 (fun c => wrap (Sequence c c2)) with
|
||||
| None => Some ($0 $+ (wrap c2, s))
|
||||
| v => v
|
||||
end
|
||||
| If _ then_ else_ => Some ($0 $+ (wrap then_, s) $+ (wrap else_, s))
|
||||
| While e body => Some ($0 $+ (wrap Skip, s) $+ (wrap (Sequence body (While e body)), s))
|
||||
end.
|
||||
|
||||
Lemma command_equal : forall c1 c2 : cmd, sumbool (c1 = c2) (c1 <> c2).
|
||||
Proof.
|
||||
repeat decide equality.
|
||||
Qed.
|
||||
|
||||
Theorem absint_step_ok : forall a, absint_sound a
|
||||
-> forall (s : astate a) v, compatible s v
|
||||
-> forall c v' c', step (v, c) (v', c')
|
||||
-> forall wrap, exists ss s', absint_step s c wrap = Some ss
|
||||
/\ ss $? wrap c' = Some s'
|
||||
/\ compatible s' v'.
|
||||
Proof.
|
||||
induct 2; simplify.
|
||||
|
||||
do 2 eexists; propositional.
|
||||
simplify; equality.
|
||||
eauto.
|
||||
|
||||
eapply IHstep in H0; auto.
|
||||
invert H0.
|
||||
invert H2.
|
||||
propositional.
|
||||
rewrite H2.
|
||||
eauto.
|
||||
|
||||
do 2 eexists; propositional.
|
||||
simplify; equality.
|
||||
assumption.
|
||||
|
||||
do 2 eexists; propositional.
|
||||
cases (command_equal (wrap c') (wrap else_)).
|
||||
simplify; equality.
|
||||
simplify; equality.
|
||||
assumption.
|
||||
|
||||
do 2 eexists; propositional.
|
||||
simplify; equality.
|
||||
assumption.
|
||||
|
||||
do 2 eexists; propositional.
|
||||
simplify; equality.
|
||||
assumption.
|
||||
|
||||
do 2 eexists; propositional.
|
||||
cases (command_equal (wrap Skip) (wrap (body;; while e loop body done))).
|
||||
simplify; equality.
|
||||
simplify; equality.
|
||||
assumption.
|
||||
Qed.
|
||||
|
||||
Inductive abs_step a : astate a * cmd -> astate a * cmd -> Prop :=
|
||||
| AbsStep : forall s c ss s' c',
|
||||
absint_step s c (fun x => x) = Some ss
|
||||
-> ss $? c' = Some s'
|
||||
-> abs_step (s, c) (s', c').
|
||||
|
||||
Hint Constructors abs_step.
|
||||
|
||||
Definition absint_trsys a (c : cmd) := {|
|
||||
Initial := {($0, c)};
|
||||
Step := abs_step (a := a)
|
||||
|}.
|
||||
|
||||
Inductive Rabsint a : valuation * cmd -> astate a * cmd -> Prop :=
|
||||
| RAbsint : forall v s c,
|
||||
compatible s v
|
||||
-> Rabsint (v, c) (s, c).
|
||||
|
||||
Hint Constructors abs_step Rabsint.
|
||||
|
||||
Theorem absint_simulates : forall a v c,
|
||||
absint_sound a
|
||||
-> simulates (Rabsint (a := a)) (trsys_of v c) (absint_trsys a c).
|
||||
Proof.
|
||||
simplify.
|
||||
constructor; simplify.
|
||||
|
||||
exists ($0, c); propositional.
|
||||
subst.
|
||||
constructor.
|
||||
unfold compatible.
|
||||
simplify.
|
||||
equality.
|
||||
|
||||
invert H0.
|
||||
cases st1'.
|
||||
eapply absint_step_ok in H1; eauto.
|
||||
invert H1.
|
||||
invert H0.
|
||||
propositional.
|
||||
eauto.
|
||||
Qed.
|
||||
|
||||
Definition merge_astates a : astates a -> astates a -> astates a :=
|
||||
merge (fun x y =>
|
||||
match x with
|
||||
| None => y
|
||||
| Some x' =>
|
||||
match y with
|
||||
| None => Some x'
|
||||
| Some y' => Some (merge_astate x' y')
|
||||
end
|
||||
end).
|
||||
|
||||
Inductive oneStepClosure a : astates a -> astates a -> Prop :=
|
||||
| OscNil :
|
||||
oneStepClosure $0 $0
|
||||
| OscCons : forall ss c s ss' ss'',
|
||||
oneStepClosure ss ss'
|
||||
-> match absint_step s c (fun x => x) with
|
||||
| None => ss'
|
||||
| Some ss'' => merge_astates ss'' ss'
|
||||
end = ss''
|
||||
-> oneStepClosure (ss $+ (c, s)) ss''.
|
||||
|
||||
Definition subsumeds a (ss1 ss2 : astates a) :=
|
||||
forall c s1, ss1 $? c = Some s1
|
||||
-> exists s2, ss2 $? c = Some s2
|
||||
/\ subsumed s1 s2.
|
||||
|
||||
Theorem subsumeds_refl : forall a (ss : astates a),
|
||||
subsumeds ss ss.
|
||||
Proof.
|
||||
unfold subsumeds; simplify; eauto.
|
||||
Qed.
|
||||
|
||||
Hint Resolve subsumeds_refl.
|
||||
|
||||
Lemma subsumeds_add : forall a (ss1 ss2 : astates a) c s1 s2,
|
||||
subsumeds ss1 ss2
|
||||
-> subsumed s1 s2
|
||||
-> subsumeds (ss1 $+ (c, s1)) (ss2 $+ (c, s2)).
|
||||
Proof.
|
||||
unfold subsumeds; simplify.
|
||||
cases (command_equal c c0); subst; simplify; eauto.
|
||||
invert H1; eauto.
|
||||
Qed.
|
||||
|
||||
Hint Resolve subsumeds_add.
|
||||
|
||||
Lemma subsumeds_empty : forall a (ss : astates a),
|
||||
subsumeds $0 ss.
|
||||
Proof.
|
||||
unfold subsumeds; simplify.
|
||||
equality.
|
||||
Qed.
|
||||
|
||||
Lemma subsumeds_add_left : forall a (ss1 ss2 : astates a) c s,
|
||||
ss2 $? c = Some s
|
||||
-> subsumeds ss1 ss2
|
||||
-> subsumeds (ss1 $+ (c, s)) ss2.
|
||||
Proof.
|
||||
unfold subsumeds; simplify.
|
||||
cases (command_equal c c0); subst; simplify; eauto.
|
||||
invert H1; eauto.
|
||||
Qed.
|
||||
|
||||
Inductive interpret a : astates a -> astates a -> astates a -> Prop :=
|
||||
| InterpretDone : forall ss1 any ss2,
|
||||
oneStepClosure ss1 ss2
|
||||
-> subsumeds ss2 ss1
|
||||
-> interpret ss1 any ss1
|
||||
| InterpretStep : forall ss worklist ss' ss'',
|
||||
oneStepClosure worklist ss'
|
||||
-> interpret (merge_astates ss ss') ss' ss''
|
||||
-> interpret ss worklist ss''.
|
||||
|
||||
Lemma oneStepClosure_sound : forall a, absint_sound a
|
||||
-> forall ss ss' : astates a, oneStepClosure ss ss'
|
||||
-> forall c s s' c', ss $? c = Some s
|
||||
-> abs_step (s, c) (s', c')
|
||||
-> exists s'', ss' $? c' = Some s''
|
||||
/\ subsumed s' s''.
|
||||
Proof.
|
||||
induct 2; simplify.
|
||||
|
||||
equality.
|
||||
|
||||
cases (command_equal c c0); subst; simplify.
|
||||
|
||||
invert H2.
|
||||
invert H3.
|
||||
rewrite H5.
|
||||
unfold merge_astates; simplify.
|
||||
rewrite H7.
|
||||
cases (ss' $? c').
|
||||
eexists; propositional.
|
||||
unfold subsumed; simplify.
|
||||
unfold merge_astate; simplify.
|
||||
cases (s' $? x); try equality.
|
||||
cases (a0 $? x); simplify; try equality.
|
||||
invert H1; eauto.
|
||||
eauto.
|
||||
|
||||
apply IHoneStepClosure in H3; auto.
|
||||
invert H3; propositional.
|
||||
cases (absint_step s c (fun x => x)); eauto.
|
||||
unfold merge_astates; simplify.
|
||||
rewrite H3.
|
||||
cases (a0 $? c'); eauto.
|
||||
eexists; propositional.
|
||||
unfold subsumed; simplify.
|
||||
unfold merge_astate; simplify.
|
||||
specialize (H4 x0).
|
||||
cases (s' $? x0).
|
||||
cases (a1 $? x0); try equality.
|
||||
cases (x $? x0); try equality.
|
||||
invert 1.
|
||||
eauto.
|
||||
|
||||
rewrite H4.
|
||||
cases (a1 $? x0); equality.
|
||||
Qed.
|
||||
|
||||
Lemma absint_step_monotone_None : forall a (s : astate a) c wrap,
|
||||
absint_step s c wrap = None
|
||||
-> forall s' : astate a, absint_step s' c wrap = None.
|
||||
Proof.
|
||||
induct c; simplify; try equality.
|
||||
cases (absint_step s c1 (fun c => wrap (c;; c2))); equality.
|
||||
Qed.
|
||||
|
||||
Lemma absint_interp_monotone : forall a, absint_sound a
|
||||
-> forall (s : astate a) e s' n,
|
||||
a.(Represents) n (absint_interp e s)
|
||||
-> subsumed s s'
|
||||
-> a.(Represents) n (absint_interp e s').
|
||||
Proof.
|
||||
induct e; simplify; eauto.
|
||||
|
||||
cases (s' $? x); eauto.
|
||||
cases (s $? x); eauto.
|
||||
Qed.
|
||||
|
||||
Hint Resolve absint_interp_monotone.
|
||||
|
||||
Lemma absint_step_monotone : forall a, absint_sound a
|
||||
-> forall (s : astate a) c wrap ss,
|
||||
absint_step s c wrap = Some ss
|
||||
-> forall s', subsumed s s'
|
||||
-> exists ss', absint_step s' c wrap = Some ss'
|
||||
/\ subsumeds ss ss'.
|
||||
Proof.
|
||||
induct c; simplify.
|
||||
|
||||
equality.
|
||||
|
||||
invert H0.
|
||||
eexists; propositional.
|
||||
eauto.
|
||||
apply subsumeds_add; eauto.
|
||||
|
||||
cases (absint_step s c1 (fun c => wrap (c;; c2))).
|
||||
|
||||
invert H0.
|
||||
eapply IHc1 in Heq; eauto.
|
||||
invert Heq; propositional.
|
||||
rewrite H2; eauto.
|
||||
|
||||
invert H0.
|
||||
eapply absint_step_monotone_None in Heq; eauto.
|
||||
rewrite Heq; eauto.
|
||||
|
||||
invert H0; eauto.
|
||||
|
||||
invert H0; eauto.
|
||||
Qed.
|
||||
|
||||
Lemma abs_step_monotone : forall a, absint_sound a
|
||||
-> forall (s : astate a) c s' c',
|
||||
abs_step (s, c) (s', c')
|
||||
-> forall s1, subsumed s s1
|
||||
-> exists s1', abs_step (s1, c) (s1', c')
|
||||
/\ subsumed s' s1'.
|
||||
Proof.
|
||||
invert 2; simplify.
|
||||
eapply absint_step_monotone in H4; eauto.
|
||||
invert H4; propositional.
|
||||
apply H3 in H6.
|
||||
invert H6; propositional; eauto.
|
||||
Qed.
|
||||
|
||||
Lemma interpret_sound' : forall c a, absint_sound a
|
||||
-> forall ss worklist ss' : astates a, interpret ss worklist ss'
|
||||
-> ss $? c = Some $0
|
||||
-> invariantFor (absint_trsys a c) (fun p => exists s, ss' $? snd p = Some s
|
||||
/\ subsumed (fst p) s).
|
||||
Proof.
|
||||
induct 2; simplify; subst.
|
||||
|
||||
apply invariant_induction; simplify; propositional; subst; simplify; eauto.
|
||||
|
||||
invert H3; propositional.
|
||||
cases s.
|
||||
cases s'.
|
||||
simplify.
|
||||
eapply abs_step_monotone in H4; eauto.
|
||||
invert H4; propositional.
|
||||
eapply oneStepClosure_sound in H4; eauto.
|
||||
invert H4; propositional.
|
||||
eapply H1 in H4.
|
||||
invert H4; propositional.
|
||||
eauto using subsumed_trans.
|
||||
|
||||
apply IHinterpret.
|
||||
unfold merge_astates; simplify.
|
||||
rewrite H2.
|
||||
cases (ss' $? c); trivial.
|
||||
unfold merge_astate; simplify; equality.
|
||||
Qed.
|
||||
|
||||
Theorem interpret_sound : forall c a (ss : astates a),
|
||||
absint_sound a
|
||||
-> interpret ($0 $+ (c, $0)) ($0 $+ (c, $0)) ss
|
||||
-> invariantFor (absint_trsys a c) (fun p => exists s, ss $? snd p = Some s
|
||||
/\ subsumed (fst p) s).
|
||||
Proof.
|
||||
simplify.
|
||||
eapply interpret_sound'; eauto.
|
||||
simplify; equality.
|
||||
Qed.
|
||||
|
||||
Ltac interpret_simpl := unfold merge_astates, merge_astate;
|
||||
simplify; repeat simplify_map.
|
||||
Ltac oneStepClosure := apply OscNil
|
||||
|| (eapply OscCons; [ oneStepClosure
|
||||
| interpret_simpl; reflexivity ]).
|
||||
Ltac interpret1 := eapply InterpretStep; [ oneStepClosure | interpret_simpl ].
|
||||
Ltac interpret_done := eapply InterpretDone; [ oneStepClosure
|
||||
| repeat (apply subsumeds_add_left || apply subsumeds_empty); (simplify; equality) ].
|
|
@ -1885,12 +1885,12 @@ Proof.
|
|||
| [ _ : impossible x = _ |- _ ] => fail 1
|
||||
| _ => cases (impossible x); simplify
|
||||
end
|
||||
end; propositional; constructor; simplify;
|
||||
end; propositional; try constructor; simplify;
|
||||
repeat match goal with
|
||||
| [ H : Some _ = Some _ |- _] => invert H
|
||||
| [ _ : context[match ?X with _ => _ end] |- _ ] => cases X
|
||||
| [ |- context[match ?X with _ => _ end] ] => cases X
|
||||
end; eauto; try equality; linear_arithmetic).
|
||||
end; eauto; try equality; try linear_arithmetic).
|
||||
Qed.
|
||||
|
||||
(* As before, two helpful lemmas to feed the book library's automation about
|
||||
|
@ -2083,7 +2083,7 @@ Proof.
|
|||
| [ _ : impossible x = _ |- _ ] => fail 1
|
||||
| _ => cases (impossible x); simplify
|
||||
end
|
||||
end; propositional; constructor; simplify;
|
||||
end; propositional; try constructor; simplify;
|
||||
repeat match goal with
|
||||
| [ H : Some _ = Some _ |- _] => invert H
|
||||
| [ _ : context[match ?X with _ => _ end] |- _ ] => cases X
|
||||
|
|
2
Makefile
2
Makefile
|
@ -13,7 +13,7 @@ coq: Makefile.coq
|
|||
$(MAKE) -f Makefile.coq
|
||||
|
||||
lib: Makefile.coq
|
||||
$(MAKE) -f Makefile.coq Frap.vo
|
||||
$(MAKE) -f Makefile.coq Frap.vo AbstractInterpret.vo
|
||||
|
||||
Makefile.coq: Makefile _CoqProject *.v
|
||||
coq_makefile -f _CoqProject -o Makefile.coq
|
||||
|
|
|
@ -6,6 +6,7 @@ Relations.v
|
|||
Invariant.v
|
||||
ModelCheck.v
|
||||
Imp.v
|
||||
AbstractInterpret.v
|
||||
Frap.v
|
||||
BasicSyntax_template.v
|
||||
BasicSyntax.v
|
||||
|
|
Loading…
Reference in a new issue