mirror of
https://github.com/achlipala/frap.git
synced 2024-11-10 00:07:51 +00:00
450 lines
10 KiB
Coq
450 lines
10 KiB
Coq
(** Formal Reasoning About Programs <http://adam.chlipala.net/frap/>
|
|
* Chapter 6: Operational Semantics
|
|
* Author: Adam Chlipala
|
|
* License: https://creativecommons.org/licenses/by-nc-nd/4.0/ *)
|
|
|
|
Require Import Frap.
|
|
|
|
Set Implicit Arguments.
|
|
|
|
|
|
(* OK, enough with defining transition relations manually. Let's return to our
|
|
* syntax of imperative programs from Chapter 3. *)
|
|
|
|
Inductive arith : Set :=
|
|
| Const (n : nat)
|
|
| Var (x : var)
|
|
| Plus (e1 e2 : arith)
|
|
| Minus (e1 e2 : arith)
|
|
| Times (e1 e2 : arith).
|
|
|
|
Inductive cmd :=
|
|
| Skip
|
|
| Assign (x : var) (e : arith)
|
|
| Sequence (c1 c2 : cmd)
|
|
| If (e : arith) (then_ else_ : cmd)
|
|
| While (e : arith) (body : cmd).
|
|
(* Important differences: we added [If] and switched [Repeat] to general
|
|
* [While]. *)
|
|
|
|
(* Here are some notations for the language, which again we won't really
|
|
* explain. *)
|
|
Coercion Const : nat >-> arith.
|
|
Coercion Var : var >-> arith.
|
|
Infix "+" := Plus : arith_scope.
|
|
Infix "-" := Minus : arith_scope.
|
|
Infix "*" := Times : arith_scope.
|
|
Delimit Scope arith_scope with arith.
|
|
Notation "x <- e" := (Assign x e%arith) (at level 75).
|
|
Infix ";" := Sequence (at level 76).
|
|
Notation "'when' e 'do' then_ 'else' else_ 'done'" := (If e%arith then_ else_) (at level 75, e at level 0).
|
|
Notation "'while' e 'do' body 'done'" := (While e%arith body) (at level 75).
|
|
|
|
(* Here's an adaptation of our factorial example from Chapter 3. *)
|
|
Example factorial :=
|
|
"output" <- 1;
|
|
while "input" do
|
|
"output" <- "output" * "input";
|
|
"input" <- "input" - 1
|
|
done.
|
|
|
|
(* Recall our use of a recursive function to interpret expressions. *)
|
|
Definition valuation := fmap var nat.
|
|
Fixpoint interp (e : arith) (v : valuation) : nat :=
|
|
match e with
|
|
| Const n => n
|
|
| Var x =>
|
|
match v $? x with
|
|
| None => 0
|
|
| Some n => n
|
|
end
|
|
| Plus e1 e2 => interp e1 v + interp e2 v
|
|
| Minus e1 e2 => interp e1 v - interp e2 v
|
|
| Times e1 e2 => interp e1 v * interp e2 v
|
|
end.
|
|
|
|
(* Our old trick of interpreters won't work for this new language, because of
|
|
* the general "while" loops. No such interpreter could terminate for all
|
|
* programs. Instead, we will use inductive predicates to explain program
|
|
* meanings. First, let's apply the most intuitive method, called
|
|
* *big-step operational semantics*. *)
|
|
Inductive eval : valuation -> cmd -> valuation -> Prop :=
|
|
| EvalSkip : forall v,
|
|
eval v Skip v
|
|
| EvalAssign : forall v x e,
|
|
eval v (Assign x e) (v $+ (x, interp e v))
|
|
| EvalSeq : forall v c1 v1 c2 v2,
|
|
eval v c1 v1
|
|
-> eval v1 c2 v2
|
|
-> eval v (Sequence c1 c2) v2
|
|
| EvalIfTrue : forall v e then_ else_ v',
|
|
interp e v <> 0
|
|
-> eval v then_ v'
|
|
-> eval v (If e then_ else_) v'
|
|
| EvalIfFalse : forall v e then_ else_ v',
|
|
interp e v = 0
|
|
-> eval v else_ v'
|
|
-> eval v (If e then_ else_) v'
|
|
| EvalWhileTrue : forall v e body v' v'',
|
|
interp e v <> 0
|
|
-> eval v body v'
|
|
-> eval v' (While e body) v''
|
|
-> eval v (While e body) v''
|
|
| EvalWhileFalse : forall v e body,
|
|
interp e v = 0
|
|
-> eval v (While e body) v.
|
|
|
|
(* Let's run the factorial program on a few inputs. *)
|
|
Theorem factorial_2 : exists v, eval ($0 $+ ("input", 2)) factorial v
|
|
/\ v $? "output" = Some 2.
|
|
Proof.
|
|
eexists; propositional.
|
|
|
|
econstructor.
|
|
econstructor.
|
|
econstructor.
|
|
simplify.
|
|
equality.
|
|
econstructor.
|
|
econstructor.
|
|
econstructor.
|
|
econstructor.
|
|
simplify.
|
|
equality.
|
|
econstructor.
|
|
econstructor.
|
|
econstructor.
|
|
apply EvalWhileFalse.
|
|
(* Note that, for this step, we had to specify which rule to use, since
|
|
* otherwise [econstructor] incorrectly guesses [EvalWhileTrue]. *)
|
|
simplify.
|
|
equality.
|
|
|
|
simplify.
|
|
equality.
|
|
Qed.
|
|
|
|
(* That was rather repetitive. It's easy to automate. *)
|
|
|
|
Ltac eval1 :=
|
|
apply EvalSkip || apply EvalAssign || eapply EvalSeq
|
|
|| (apply EvalIfTrue; [ simplify; equality | ])
|
|
|| (apply EvalIfFalse; [ simplify; equality | ])
|
|
|| (eapply EvalWhileTrue; [ simplify; equality | | ])
|
|
|| (apply EvalWhileFalse; [ simplify; equality ]).
|
|
Ltac evaluate := simplify; try equality; repeat eval1.
|
|
|
|
Theorem factorial_2_snazzy : exists v, eval ($0 $+ ("input", 2)) factorial v
|
|
/\ v $? "output" = Some 2.
|
|
Proof.
|
|
eexists; propositional.
|
|
evaluate.
|
|
evaluate.
|
|
Qed.
|
|
|
|
Theorem factorial_3 : exists v, eval ($0 $+ ("input", 3)) factorial v
|
|
/\ v $? "output" = Some 6.
|
|
Proof.
|
|
eexists; propositional.
|
|
evaluate.
|
|
evaluate.
|
|
Qed.
|
|
|
|
(* Instead of chugging through these relatively slow individual executions,
|
|
* let's prove once and for all that [factorial] is correct. *)
|
|
Fixpoint fact (n : nat) : nat :=
|
|
match n with
|
|
| O => 1
|
|
| S n' => n * fact n'
|
|
end.
|
|
|
|
Example factorial_loop :=
|
|
while "input" do
|
|
"output" <- "output" * "input";
|
|
"input" <- "input" - 1
|
|
done.
|
|
|
|
Lemma factorial_loop_correct : forall n v out, v $? "input" = Some n
|
|
-> v $? "output" = Some out
|
|
-> exists v', eval v factorial_loop v'
|
|
/\ v' $? "output" = Some (fact n * out).
|
|
Proof.
|
|
induct n; simplify.
|
|
|
|
exists v; propositional.
|
|
apply EvalWhileFalse.
|
|
simplify.
|
|
rewrite H.
|
|
equality.
|
|
rewrite H0.
|
|
f_equal.
|
|
ring.
|
|
|
|
assert (exists v', eval (v $+ ("output", out * S n) $+ ("input", n)) factorial_loop v'
|
|
/\ v' $? "output" = Some (fact n * (out * S n))).
|
|
apply IHn.
|
|
simplify; equality.
|
|
simplify; equality.
|
|
first_order.
|
|
eexists; propositional.
|
|
econstructor.
|
|
simplify.
|
|
rewrite H.
|
|
equality.
|
|
econstructor.
|
|
econstructor.
|
|
econstructor.
|
|
simplify.
|
|
rewrite H, H0.
|
|
replace (S n - 1) with n by linear_arithmetic.
|
|
(* [replace e1 with e2 by tac]: replace occurrences of [e1] with [e2], proving
|
|
* [e2 = e1] with tactic [tac]. *)
|
|
apply H1.
|
|
rewrite H2.
|
|
f_equal.
|
|
ring.
|
|
Qed.
|
|
|
|
Theorem factorial_correct : forall n v, v $? "input" = Some n
|
|
-> exists v', eval v factorial v'
|
|
/\ v' $? "output" = Some (fact n).
|
|
Proof.
|
|
simplify.
|
|
assert (exists v', eval (v $+ ("output", 1)) factorial_loop v'
|
|
/\ v' $? "output" = Some (fact n * 1)).
|
|
apply factorial_loop_correct.
|
|
simplify; equality.
|
|
simplify; equality.
|
|
first_order.
|
|
eexists; propositional.
|
|
econstructor.
|
|
econstructor.
|
|
simplify.
|
|
apply H0.
|
|
rewrite H1.
|
|
f_equal.
|
|
ring.
|
|
Qed.
|
|
|
|
|
|
(** * Small-step semantics *)
|
|
|
|
Inductive step : valuation * cmd -> valuation * cmd -> Prop :=
|
|
| StepAssign : forall v x e,
|
|
step (v, Assign x e) (v $+ (x, interp e v), Skip)
|
|
| StepSeq1 : forall v c1 c2 v' c1',
|
|
step (v, c1) (v', c1')
|
|
-> step (v, Sequence c1 c2) (v', Sequence c1' c2)
|
|
| StepSeq2 : forall v c2,
|
|
step (v, Sequence Skip c2) (v, c2)
|
|
| StepIfTrue : forall v e then_ else_,
|
|
interp e v <> 0
|
|
-> step (v, If e then_ else_) (v, then_)
|
|
| StepIfFalse : forall v e then_ else_,
|
|
interp e v = 0
|
|
-> step (v, If e then_ else_) (v, else_)
|
|
| StepWhileTrue : forall v e body,
|
|
interp e v <> 0
|
|
-> step (v, While e body) (v, Sequence body (While e body))
|
|
| StepWhileFalse : forall v e body,
|
|
interp e v = 0
|
|
-> step (v, While e body) (v, Skip).
|
|
|
|
(* Here's a small-step factorial execution. *)
|
|
Theorem factorial_2_small : exists v, step^* ($0 $+ ("input", 2), factorial) (v, Skip)
|
|
/\ v $? "output" = Some 2.
|
|
Proof.
|
|
eexists; propositional.
|
|
|
|
econstructor.
|
|
econstructor.
|
|
econstructor.
|
|
econstructor.
|
|
apply StepSeq2.
|
|
econstructor.
|
|
econstructor.
|
|
simplify.
|
|
equality.
|
|
econstructor.
|
|
econstructor.
|
|
econstructor.
|
|
econstructor.
|
|
econstructor.
|
|
econstructor.
|
|
apply StepSeq2.
|
|
econstructor.
|
|
econstructor.
|
|
econstructor.
|
|
econstructor.
|
|
apply StepSeq2.
|
|
econstructor.
|
|
econstructor.
|
|
simplify.
|
|
equality.
|
|
econstructor.
|
|
econstructor.
|
|
econstructor.
|
|
econstructor.
|
|
econstructor.
|
|
econstructor.
|
|
apply StepSeq2.
|
|
econstructor.
|
|
econstructor.
|
|
econstructor.
|
|
econstructor.
|
|
apply StepSeq2.
|
|
econstructor.
|
|
apply StepWhileFalse.
|
|
simplify.
|
|
equality.
|
|
econstructor.
|
|
|
|
simplify.
|
|
equality.
|
|
Qed.
|
|
|
|
Ltac step1 :=
|
|
apply TrcRefl || eapply TrcFront
|
|
|| apply StepAssign || apply StepSeq2 || eapply StepSeq1
|
|
|| (apply StepIfTrue; [ simplify; equality ])
|
|
|| (apply StepIfFalse; [ simplify; equality ])
|
|
|| (eapply StepWhileTrue; [ simplify; equality ])
|
|
|| (apply StepWhileFalse; [ simplify; equality ]).
|
|
Ltac stepper := simplify; try equality; repeat step1.
|
|
|
|
Theorem factorial_2_small_snazzy : exists v, step^* ($0 $+ ("input", 2), factorial) (v, Skip)
|
|
/\ v $? "output" = Some 2.
|
|
Proof.
|
|
eexists; propositional.
|
|
|
|
stepper.
|
|
stepper.
|
|
Qed.
|
|
|
|
(* It turns out that these two semantics styles are equivalent. Let's prove
|
|
* it. *)
|
|
|
|
Lemma step_star_Seq : forall v c1 c2 v' c1',
|
|
step^* (v, c1) (v', c1')
|
|
-> step^* (v, Sequence c1 c2) (v', Sequence c1' c2).
|
|
Proof.
|
|
induct 1.
|
|
|
|
constructor.
|
|
|
|
cases y.
|
|
econstructor.
|
|
econstructor.
|
|
eassumption.
|
|
apply IHtrc.
|
|
equality.
|
|
equality.
|
|
Qed.
|
|
|
|
Theorem big_small : forall v c v', eval v c v'
|
|
-> step^* (v, c) (v', Skip).
|
|
Proof.
|
|
induct 1; simplify.
|
|
|
|
constructor.
|
|
|
|
econstructor.
|
|
constructor.
|
|
constructor.
|
|
|
|
eapply trc_trans.
|
|
apply step_star_Seq.
|
|
eassumption.
|
|
econstructor.
|
|
apply StepSeq2.
|
|
assumption.
|
|
|
|
econstructor.
|
|
constructor.
|
|
assumption.
|
|
assumption.
|
|
|
|
econstructor.
|
|
apply StepIfFalse.
|
|
assumption.
|
|
assumption.
|
|
|
|
econstructor.
|
|
constructor.
|
|
assumption.
|
|
eapply trc_trans.
|
|
apply step_star_Seq.
|
|
eassumption.
|
|
econstructor.
|
|
apply StepSeq2.
|
|
assumption.
|
|
|
|
econstructor.
|
|
apply StepWhileFalse.
|
|
assumption.
|
|
constructor.
|
|
Qed.
|
|
|
|
Lemma small_big'' : forall v c v' c', step (v, c) (v', c')
|
|
-> forall v'', eval v' c' v''
|
|
-> eval v c v''.
|
|
Proof.
|
|
induct 1; simplify.
|
|
|
|
invert H.
|
|
constructor.
|
|
|
|
invert H0.
|
|
econstructor.
|
|
apply IHstep.
|
|
eassumption.
|
|
assumption.
|
|
|
|
econstructor.
|
|
constructor.
|
|
assumption.
|
|
|
|
constructor.
|
|
assumption.
|
|
assumption.
|
|
|
|
apply EvalIfFalse.
|
|
assumption.
|
|
assumption.
|
|
|
|
invert H0.
|
|
econstructor.
|
|
assumption.
|
|
eassumption.
|
|
assumption.
|
|
|
|
invert H0.
|
|
apply EvalWhileFalse.
|
|
assumption.
|
|
Qed.
|
|
|
|
Lemma small_big' : forall v c v' c', step^* (v, c) (v', c')
|
|
-> forall v'', eval v' c' v''
|
|
-> eval v c v''.
|
|
Proof.
|
|
induct 1; simplify.
|
|
|
|
trivial.
|
|
|
|
cases y.
|
|
eapply small_big''.
|
|
eassumption.
|
|
eapply IHtrc.
|
|
equality.
|
|
equality.
|
|
assumption.
|
|
Qed.
|
|
|
|
Theorem small_big : forall v c v', step^* (v, c) (v', Skip)
|
|
-> eval v c v'.
|
|
Proof.
|
|
simplify.
|
|
eapply small_big'.
|
|
eassumption.
|
|
constructor.
|
|
Qed.
|