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Add Imp, recapping OperationalSemantics object language and semantics
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Adam Chlipala
parent
96327eb9aa
commit
e06af75c78
3 changed files with 225 additions and 6 deletions
218
Imp.v
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218
Imp.v
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Require Import Frap.
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Set Implicit Arguments.
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Inductive arith : Set :=
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| Const (n : nat)
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| Var (x : var)
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| Plus (e1 e2 : arith)
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| Minus (e1 e2 : arith)
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| Times (e1 e2 : arith).
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Inductive cmd :=
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| Skip
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| Assign (x : var) (e : arith)
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| Sequence (c1 c2 : cmd)
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| If (e : arith) (then_ else_ : cmd)
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| While (e : arith) (body : cmd).
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Coercion Const : nat >-> arith.
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Coercion Var : var >-> arith.
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Infix "+" := Plus : arith_scope.
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Infix "-" := Minus : arith_scope.
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Infix "*" := Times : arith_scope.
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Delimit Scope arith_scope with arith.
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Notation "x <- e" := (Assign x e%arith) (at level 75).
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Infix ";;" := Sequence (at level 76). (* This one changed slightly, to avoid parsing clashes. *)
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Notation "'when' e 'then' then_ 'else' else_ 'done'" := (If e%arith then_ else_) (at level 75, e at level 0).
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Notation "'while' e 'loop' body 'done'" := (While e%arith body) (at level 75).
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Definition valuation := fmap var nat.
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Fixpoint interp (e : arith) (v : valuation) : nat :=
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match e with
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| Const n => n
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| Var x =>
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match v $? x with
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| None => 0
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| Some n => n
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end
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| Plus e1 e2 => interp e1 v + interp e2 v
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| Minus e1 e2 => interp e1 v - interp e2 v
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| Times e1 e2 => interp e1 v * interp e2 v
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end.
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Inductive eval : valuation -> cmd -> valuation -> Prop :=
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| EvalSkip : forall v,
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eval v Skip v
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| EvalAssign : forall v x e,
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eval v (Assign x e) (v $+ (x, interp e v))
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| EvalSeq : forall v c1 v1 c2 v2,
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eval v c1 v1
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-> eval v1 c2 v2
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-> eval v (Sequence c1 c2) v2
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| EvalIfTrue : forall v e then_ else_ v',
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interp e v <> 0
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-> eval v then_ v'
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-> eval v (If e then_ else_) v'
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| EvalIfFalse : forall v e then_ else_ v',
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interp e v = 0
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-> eval v else_ v'
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-> eval v (If e then_ else_) v'
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| EvalWhileTrue : forall v e body v' v'',
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interp e v <> 0
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-> eval v body v'
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-> eval v' (While e body) v''
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-> eval v (While e body) v''
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| EvalWhileFalse : forall v e body,
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interp e v = 0
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-> eval v (While e body) v.
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Inductive step : valuation * cmd -> valuation * cmd -> Prop :=
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| StepAssign : forall v x e,
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step (v, Assign x e) (v $+ (x, interp e v), Skip)
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| StepSeq1 : forall v c1 c2 v' c1',
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step (v, c1) (v', c1')
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-> step (v, Sequence c1 c2) (v', Sequence c1' c2)
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| StepSeq2 : forall v c2,
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step (v, Sequence Skip c2) (v, c2)
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| StepIfTrue : forall v e then_ else_,
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interp e v <> 0
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-> step (v, If e then_ else_) (v, then_)
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| StepIfFalse : forall v e then_ else_,
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interp e v = 0
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-> step (v, If e then_ else_) (v, else_)
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| StepWhileTrue : forall v e body,
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interp e v <> 0
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-> step (v, While e body) (v, Sequence body (While e body))
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| StepWhileFalse : forall v e body,
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interp e v = 0
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-> step (v, While e body) (v, Skip).
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Hint Constructors trc step eval.
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Lemma step_star_Seq : forall v c1 c2 v' c1',
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step^* (v, c1) (v', c1')
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-> step^* (v, Sequence c1 c2) (v', Sequence c1' c2).
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Proof.
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induct 1; eauto.
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cases y; eauto.
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Qed.
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Hint Resolve step_star_Seq.
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Theorem big_small : forall v c v', eval v c v'
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-> step^* (v, c) (v', Skip).
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Proof.
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induct 1; eauto 6 using trc_trans.
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Qed.
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Lemma small_big'' : forall v c v' c', step (v, c) (v', c')
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-> forall v'', eval v' c' v''
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-> eval v c v''.
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Proof.
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induct 1; simplify;
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repeat match goal with
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| [ H : eval _ _ _ |- _ ] => invert1 H
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end; eauto.
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Qed.
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Hint Resolve small_big''.
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Lemma small_big' : forall v c v' c', step^* (v, c) (v', c')
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-> forall v'', eval v' c' v''
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-> eval v c v''.
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Proof.
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induct 1; eauto.
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cases y; eauto.
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Qed.
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Hint Resolve small_big'.
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Theorem small_big : forall v c v', step^* (v, c) (v', Skip)
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-> eval v c v'.
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Proof.
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eauto.
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Qed.
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Definition trsys_of (v : valuation) (c : cmd) : trsys (valuation * cmd) := {|
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Initial := {(v, c)};
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Step := step
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|}.
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Inductive context :=
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| Hole
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| CSeq (C : context) (c : cmd).
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Inductive plug : context -> cmd -> cmd -> Prop :=
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| PlugHole : forall c, plug Hole c c
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| PlugSeq : forall c C c' c2,
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plug C c c'
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-> plug (CSeq C c2) c (Sequence c' c2).
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Inductive step0 : valuation * cmd -> valuation * cmd -> Prop :=
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| Step0Assign : forall v x e,
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step0 (v, Assign x e) (v $+ (x, interp e v), Skip)
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| Step0Seq : forall v c2,
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step0 (v, Sequence Skip c2) (v, c2)
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| Step0IfTrue : forall v e then_ else_,
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interp e v <> 0
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-> step0 (v, If e then_ else_) (v, then_)
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| Step0IfFalse : forall v e then_ else_,
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interp e v = 0
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-> step0 (v, If e then_ else_) (v, else_)
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| Step0WhileTrue : forall v e body,
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interp e v <> 0
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-> step0 (v, While e body) (v, Sequence body (While e body))
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| Step0WhileFalse : forall v e body,
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interp e v = 0
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-> step0 (v, While e body) (v, Skip).
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Inductive cstep : valuation * cmd -> valuation * cmd -> Prop :=
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| CStep : forall C v c v' c' c1 c2,
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plug C c c1
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-> step0 (v, c) (v', c')
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-> plug C c' c2
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-> cstep (v, c1) (v', c2).
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Hint Constructors plug step0 cstep.
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Theorem step_cstep : forall v c v' c',
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step (v, c) (v', c')
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-> cstep (v, c) (v', c').
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Proof.
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induct 1; repeat match goal with
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| [ H : cstep _ _ |- _ ] => invert H
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end; eauto.
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Qed.
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Hint Resolve step_cstep.
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Lemma step0_step : forall v c v' c',
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step0 (v, c) (v', c')
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-> step (v, c) (v', c').
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Proof.
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invert 1; eauto.
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Qed.
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Hint Resolve step0_step.
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Lemma cstep_step' : forall C c0 c,
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plug C c0 c
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-> forall v' c'0 v c', step0 (v, c0) (v', c'0)
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-> plug C c'0 c'
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-> step (v, c) (v', c').
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Proof.
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induct 1; simplify; repeat match goal with
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| [ H : plug _ _ _ |- _ ] => invert1 H
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end; eauto.
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Qed.
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Hint Resolve cstep_step'.
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Theorem cstep_step : forall v c v' c',
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cstep (v, c) (v', c')
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-> step (v, c) (v', c').
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Proof.
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invert 1; eauto.
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Qed.
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@ -827,7 +827,7 @@ Lemma step0_step : forall v c v' c',
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step0 (v, c) (v', c')
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-> step (v, c) (v', c').
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Proof.
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induct 1; constructor; assumption.
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invert 1; constructor; assumption.
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Qed.
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Lemma cstep_step' : forall C c0 c,
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@ -853,7 +853,7 @@ Theorem cstep_step : forall v c v' c',
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cstep (v, c) (v', c')
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-> step (v, c) (v', c').
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Proof.
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induct 1.
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invert 1.
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eapply cstep_step'.
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eassumption.
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eassumption.
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@ -880,7 +880,7 @@ Lemma step0_step_snazzy : forall v c v' c',
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step0 (v, c) (v', c')
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-> step (v, c) (v', c').
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Proof.
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induct 1; eauto.
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invert 1; eauto.
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Qed.
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Hint Resolve step0_step_snazzy.
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@ -902,7 +902,7 @@ Theorem cstep_step_snazzy : forall v c v' c',
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cstep (v, c) (v', c')
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-> step (v, c) (v', c').
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Proof.
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induct 1; eauto.
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invert 1; eauto.
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Qed.
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@ -1166,7 +1166,7 @@ Module Concurrent.
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step0 (v, c) (v', c')
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-> step (v, c) (v', c').
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Proof.
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induct 1; eauto.
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invert 1; eauto.
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Qed.
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Hint Resolve step0_step.
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@ -1188,6 +1188,6 @@ Module Concurrent.
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cstep (v, c) (v', c')
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-> step (v, c) (v', c').
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Proof.
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induct 1; eauto.
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invert 1; eauto.
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Qed.
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End Concurrent.
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@ -5,6 +5,7 @@ Sets.v
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Relations.v
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Invariant.v
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ModelCheck.v
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Imp.v
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Frap.v
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BasicSyntax_template.v
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BasicSyntax.v
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