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OperationalSemantics: equivalence of big and small
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Adam Chlipala
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1 changed files with 223 additions and 0 deletions
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@ -225,3 +225,226 @@ Proof.
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f_equal.
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ring.
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Qed.
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(** * Small-step semantics *)
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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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(* Here's a small-step factorial execution. *)
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Theorem factorial_2_small : exists v, step^* ($0 $+ ("input", 2), factorial) (v, Skip)
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/\ v $? "output" = Some 2.
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Proof.
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eexists; propositional.
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econstructor.
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econstructor.
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econstructor.
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econstructor.
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apply StepSeq2.
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econstructor.
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econstructor.
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simplify.
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equality.
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econstructor.
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econstructor.
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econstructor.
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econstructor.
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econstructor.
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econstructor.
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apply StepSeq2.
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econstructor.
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econstructor.
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econstructor.
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econstructor.
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apply StepSeq2.
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econstructor.
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econstructor.
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simplify.
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equality.
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econstructor.
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econstructor.
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econstructor.
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econstructor.
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econstructor.
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econstructor.
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apply StepSeq2.
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econstructor.
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econstructor.
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econstructor.
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econstructor.
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apply StepSeq2.
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econstructor.
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apply StepWhileFalse.
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simplify.
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equality.
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econstructor.
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simplify.
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equality.
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Qed.
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Ltac step1 :=
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apply TrcRefl || eapply TrcFront
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|| apply StepAssign || apply StepSeq2 || eapply StepSeq1
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|| (apply StepIfTrue; [ simplify; equality ])
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|| (apply StepIfFalse; [ simplify; equality ])
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|| (eapply StepWhileTrue; [ simplify; equality ])
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|| (apply StepWhileFalse; [ simplify; equality ]).
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Ltac stepper := simplify; try equality; repeat step1.
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Theorem factorial_2_small_snazzy : exists v, step^* ($0 $+ ("input", 2), factorial) (v, Skip)
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/\ v $? "output" = Some 2.
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Proof.
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eexists; propositional.
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stepper.
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stepper.
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Qed.
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(* It turns out that these two semantics styles are equivalent. Let's prove
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* it. *)
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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.
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constructor.
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cases y.
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econstructor.
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econstructor.
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eassumption.
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apply IHtrc.
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equality.
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equality.
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Qed.
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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; simplify.
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constructor.
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econstructor.
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constructor.
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constructor.
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eapply trc_trans.
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apply step_star_Seq.
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eassumption.
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econstructor.
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apply StepSeq2.
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assumption.
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econstructor.
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constructor.
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assumption.
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assumption.
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econstructor.
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apply StepIfFalse.
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assumption.
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assumption.
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econstructor.
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constructor.
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assumption.
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eapply trc_trans.
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apply step_star_Seq.
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eassumption.
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econstructor.
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apply StepSeq2.
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assumption.
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econstructor.
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apply StepWhileFalse.
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assumption.
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constructor.
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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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invert H.
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constructor.
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invert H0.
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econstructor.
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apply IHstep.
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eassumption.
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assumption.
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econstructor.
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constructor.
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assumption.
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constructor.
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assumption.
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assumption.
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apply EvalIfFalse.
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assumption.
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assumption.
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invert H0.
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econstructor.
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assumption.
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eassumption.
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assumption.
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invert H0.
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apply EvalWhileFalse.
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assumption.
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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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trivial.
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cases y.
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eapply small_big''.
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eassumption.
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eapply IHtrc.
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equality.
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equality.
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assumption.
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Qed.
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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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simplify.
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eapply small_big'.
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eassumption.
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constructor.
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Qed.
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