mueck
1 year ago
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6a3451d399
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From stdpp Require Import gmap base relations tactics.
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From iris Require Import prelude.
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From semantics.ts.stlc Require Import lang notation.
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(** Exercise 2 (LN Exercise 1): Deterministic Operational Semantics *)
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Lemma val_no_step e e':
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step e e' → is_val e → False.
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Proof.
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by destruct 1.
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Qed.
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(* Note how the above lemma is another way to phrase the statement "values
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* cannot step" that doesn't need inversion. It can also be used to prove the
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* phrasing we had to use inversion for in the lecture: *)
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Lemma val_no_step' (v : val) (e : expr) :
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step v e -> False.
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Proof.
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intros H. eapply val_no_step; first eassumption.
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apply is_val_val.
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Qed.
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(* You might find the following tactic useful,
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which derives a contradiction when you have a [step e1 e2] assumption and
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[e1] is a value.
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Example:
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H1: step e e'
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H2: is_val e
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=====================
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???
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Then [val_no_step.] will solve the goal by applying the [val_no_step] lemma.
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(We neither expect you to understand how exactly the tactic does this nor to
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be able to write such a tactic yourself. Where useful, we will always
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provide you with custom tactics and explain in a comment what they do.) *)
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Ltac val_no_step :=
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match goal with
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| [H: step ?e1 ?e2 |- _] =>
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solve [exfalso; eapply (val_no_step _ _ H); done]
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end.
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Lemma step_det e e' e'':
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step e e' → step e e'' → e' = e''.
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Proof.
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(* TODO: exercise *)
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Admitted.
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(** Exercise 3 (LN Exercise 2): Call-by-name Semantics *)
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Inductive cbn_step : expr → expr → Prop :=
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| CBNStepBeta x e e' :
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cbn_step (App (Lam x e) e') (subst' x e' e)
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(* TODO: add more constructors *)
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.
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(* We make the eauto tactic aware of the constructors of cbn_step *)
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#[global] Hint Constructors cbn_step : core.
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Lemma different_results :
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∃ (e: expr) (e1 e2: expr), rtc cbn_step e e1 ∧ rtc step e e2 ∧ is_val e1 ∧ is_val e2 ∧ e1 ≠ e2.
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Proof.
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(* TODO: exercise *)
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Admitted.
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Lemma val_no_cbn_step e e':
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cbn_step e e' → is_val e → False.
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Proof.
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(* TODO: exercise *)
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Admitted.
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(* Same tactic as [val_no_step] but for cbn_step.*)
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Ltac val_no_cbn_step :=
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match goal with
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| [H: cbn_step ?e1 ?e2 |- _] =>
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solve [exfalso; eapply (val_no_cbn_step _ _ H); done]
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end.
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Lemma cbn_step_det e e' e'':
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cbn_step e e' → cbn_step e e'' → e' = e''.
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Proof.
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(* TODO: exercise *)
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Admitted.
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(** Exercise 4 (LN Exercise 3): Reflexive Transitive Closure *)
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Section rtc.
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Context {X : Type}.
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Inductive rtc (R : X → X → Prop) : X → X → Prop :=
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| rtc_base x : rtc R x x
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| rtc_step x y z : R x y → rtc R y z → rtc R x z.
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Goal rtc step ()
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Lemma rtc_reflexive R : Reflexive (rtc R).
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Proof.
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unfold Reflexive.
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(* TODO: exercise *)
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Admitted.
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Lemma rtc_transitive R : Transitive (rtc R).
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Proof.
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unfold Transitive.
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(* TODO: exercise *)
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Admitted.
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Lemma rtc_subrel (R: X → X → Prop) (x y : X): R x y → rtc R x y.
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Proof.
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(* TODO: exercise *)
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Admitted.
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Section typeclass.
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(* We can use Coq's typeclass mechanism to enable the use of the [transitivity] and [reflexivity] tactics on our goals.
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Typeclasses enable easy extensions of existing mechanisms -- in this case, by telling Coq to use the knowledge about our definition of [rtc].
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*)
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(* [Transitive] is a typeclass. With [Instance] we provide an instance of it. *)
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Global Instance rtc_transitive_inst R : Transitive (rtc R).
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Proof.
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apply rtc_transitive.
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Qed.
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Global Instance rtc_reflexive_inst R : Reflexive (rtc R).
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Proof.
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apply rtc_reflexive.
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Qed.
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End typeclass.
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End rtc.
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(* Let's put this to the test! *)
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Goal rtc step (LitInt 42) (LitInt 42).
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Proof.
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(* this uses the [rtc_reflexive_inst] instance we registered. *)
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reflexivity.
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Qed.
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Goal rtc step (LitInt 32 + (LitInt 5 + LitInt 5)%E)%E (LitInt 42).
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Proof.
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(* this uses the [rtc_transitive_inst] instance we registered. *)
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etransitivity.
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+ eapply rtc_step; eauto. reflexivity.
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+ eapply rtc_step; eauto. reflexivity.
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Qed.
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Section stdpp.
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(* In fact, [rtc] is so common that it is already provided by the [stdpp] library! *)
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Import stdpp.relations.
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Print rtc.
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(* The typeclass instances are also already registered. *)
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Goal rtc step (LitInt 42) (LitInt 42).
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Proof. reflexivity. Qed.
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End stdpp.
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(* Start by proving these lemmas. Understand why they are useful. *)
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Lemma plus_right e1 e2 e2':
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rtc step e2 e2' → rtc step (Plus e1 e2) (Plus e1 e2').
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Proof.
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(* TODO: exercise *)
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Admitted.
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Lemma plus_left e1 e1' n:
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rtc step e1 e1' → rtc step (Plus e1 (LitInt n)) (Plus e1' (LitInt n)).
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Proof.
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(* TODO: exercise *)
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Admitted.
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(* The exercise: *)
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Lemma plus_to_consts e1 e2 n m:
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rtc step e1 (LitInt n) → rtc step e2 (LitInt m) → rtc step (e1 + e2)%E (LitInt (n + m)%Z).
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Proof.
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(* TODO: exercise *)
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Admitted.
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(* Now that you have an understanding of how rtc works, we can make eauto aware of it. *)
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#[local] Hint Constructors rtc : core.
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(* See its power here: *)
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Lemma plus_right_eauto e1 e2 e2':
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rtc step e2 e2' → rtc step (Plus e1 e2) (Plus e1 e2').
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Proof.
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induction 1; eauto.
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Abort.
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(** Exercise 5 (LN Exercise 4): Big-step vs small-step semantics *)
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Lemma big_step_steps e v :
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big_step e v → rtc step e (of_val v).
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Proof.
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(* TODO: exercise *)
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Admitted.
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Lemma steps_big_step e (v: val):
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rtc step e v → big_step e v.
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Proof.
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(* Note how there is a coercion (automatic conversion) hidden in the
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* statement of this lemma: *)
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Set Printing Coercions.
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(* It is sometimes very useful to temporarily print coercions if rewrites or
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* destructs do not behave as expected. *)
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Unset Printing Coercions.
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(* TODO: exercise *)
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Admitted.
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(** Exercise 6 (LN Exercise 5): left-to-right evaluation *)
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Inductive ltr_step : expr → expr → Prop := .
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#[global] Hint Constructors ltr_step : core.
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Lemma different_steps_ltr_step :
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∃ (e: expr) (e1 e2: expr), ltr_step e e1 ∧ step e e2 ∧ e1 ≠ e2.
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Proof.
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(* TODO: exercise *)
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Admitted.
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Lemma big_step_ltr_steps e v :
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big_step e v → rtc ltr_step e (of_val v).
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Proof.
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(* TODO: exercise *)
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Admitted.
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Lemma ltr_steps_big_step e (v: val):
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rtc ltr_step e v → big_step e v.
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Proof.
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(* TODO: exercise *)
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Admitted.
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@ -0,0 +1,321 @@
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From stdpp Require Export binders strings.
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From stdpp Require Import options.
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From semantics.lib Require Import maps.
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(** * Simply Typed Lambda Calculus *)
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(** ** Expressions and values. *)
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(** [Z] is Coq's version of the integers.
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All the standard operations, like [+], are defined on it.
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The type [binder] is defined as [x ::= BNamed (s: string) | BAnon]
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where BAnon can be used if we don't want to use the variable in
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the function.
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*)
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Inductive expr :=
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(* Base lambda calculus *)
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| Var (x : string)
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| Lam (x : binder) (e : expr)
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| App (e1 e2 : expr)
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(* Base types and their operations *)
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| LitInt (n: Z)
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| Plus (e1 e2 : expr).
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Inductive val :=
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| LitIntV (n: Z)
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| LamV (x : binder) (e : expr).
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(* Injections into expr *)
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Definition of_val (v : val) : expr :=
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match v with
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| LitIntV n => LitInt n
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| LamV x e => Lam x e
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end.
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(* try to make an expr into a val *)
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Definition to_val (e : expr) : option val :=
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match e with
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| LitInt n => Some (LitIntV n)
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| Lam x e => Some (LamV x e)
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| _ => None
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end.
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Lemma to_of_val v : to_val (of_val v) = Some v.
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Proof.
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destruct v; simpl; reflexivity.
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Qed.
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Lemma of_to_val e v : to_val e = Some v → of_val v = e.
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Proof.
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destruct e; simpl; try congruence.
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all: injection 1 as <-; simpl; reflexivity.
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Qed.
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(* Inj is a type class for injective functions.
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It is defined as:
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[Inj R S f := ∀ x y, S (f x) (f y) → R x y]
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*)
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#[export] Instance of_val_inj : Inj (=) (=) of_val.
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Proof. by intros ?? Hv; apply (inj Some); rewrite <-!to_of_val, Hv. Qed.
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(* A predicate which holds true whenever an
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expression is a value. *)
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Definition is_val (e : expr) : Prop :=
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match e with
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| LitInt n => True
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| Lam x e => True
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| _ => False
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end.
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Lemma is_val_spec e : is_val e ↔ ∃ v, to_val e = Some v.
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Proof.
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destruct e; simpl.
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(* naive_solver is an automation tactic like intuition, firstorder, auto, ...
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It is provided by the stdpp library. *)
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all: naive_solver.
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Qed.
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Lemma is_val_of_val v : is_val (of_val v).
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Proof.
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apply is_val_spec. rewrite to_of_val. eauto.
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Qed.
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(* A small tactic that simplifies handling values. *)
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Ltac simplify_val :=
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repeat match goal with
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| H: to_val (of_val ?v) = ?o |- _ => rewrite to_of_val in H
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| H: is_val ?e |- _ => destruct (proj1 (is_val_spec e) H) as (? & ?); clear H
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end.
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(* values are values *)
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Lemma is_val_val (v: val): is_val (of_val v).
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Proof.
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destruct v; simpl; done.
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Qed.
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(* we tell eauto to use the lemma is_val_val *)
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#[global]
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Hint Immediate is_val_val : core.
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(** ** Operational Semantics *)
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(** *** Substitution *)
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Fixpoint subst (x : string) (es : expr) (e : expr) : expr :=
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match e with
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| LitInt n => LitInt n
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(* The function [decide] can be used to decide propositions.
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[decide P] is of type {P} + {¬ P}.
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It can only be applied to propositions for which, by type class inference,
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it can be determined that the proposition is decidable. *)
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| Var y => if decide (x = y) then es else Var y
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| Lam y e =>
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Lam y $ if decide (BNamed x = y) then e else subst x es e
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| App e1 e2 => App (subst x es e1) (subst x es e2)
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| Plus e1 e2 => Plus (subst x es e1) (subst x es e2)
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end.
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(* We lift substitution to binders. *)
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Definition subst' (mx : binder) (es : expr) : expr → expr :=
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match mx with BNamed x => subst x es | BAnon => id end.
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(** *** Small-Step Semantics *)
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(* We use right-to-left evaluation order,
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which means in a binary term (e.g., e1 + e2),
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the left side can only be reduced once the right
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side is fully evaluated (i.e., is a value). *)
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Inductive step : expr → expr → Prop :=
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| StepBeta x e e' :
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is_val e' →
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step (App (Lam x e) e') (subst' x e' e)
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| StepAppL e1 e1' e2 :
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is_val e2 →
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step e1 e1' →
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step (App e1 e2) (App e1' e2)
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| StepAppR e1 e2 e2' :
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step e2 e2' →
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step (App e1 e2) (App e1 e2')
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| StepPlusRed (n1 n2 n3: Z) :
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(n1 + n2)%Z = n3 →
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step (Plus (LitInt n1) (LitInt n2)) (LitInt n3)
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| StepPlusL e1 e1' e2 :
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is_val e2 →
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step e1 e1' →
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step (Plus e1 e2) (Plus e1' e2)
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| StepPlusR e1 e2 e2' :
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step e2 e2' →
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step (Plus e1 e2) (Plus e1 e2').
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(* We make the tactic eauto aware of the constructors of [step].
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Then it can automatically solve goals where we want to prove a step. *)
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#[global] Hint Constructors step : core.
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(* A term is reducible, if it can take a step. *)
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Definition reducible (e : expr) :=
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∃ e', step e e'.
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(** *** Big-Step Semantics *)
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Inductive big_step : expr → val → Prop :=
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| bs_lit (n : Z) :
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big_step (LitInt n) (LitIntV n)
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| bs_lam (x : binder) (e : expr) :
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big_step (Lam x e) (LamV x e)
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| bs_add e1 e2 (z1 z2 : Z) :
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big_step e1 (LitIntV z1) →
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big_step e2 (LitIntV z2) →
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big_step (Plus e1 e2) (LitIntV (z1 + z2))%Z
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| bs_app e1 e2 x e v2 v :
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big_step e1 (@LamV x e) →
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big_step e2 v2 →
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big_step (subst' x (of_val v2) e) v →
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big_step (App e1 e2) v
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.
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#[export] Hint Constructors big_step : core.
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Lemma big_step_vals (v: val): big_step (of_val v) v.
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Proof.
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induction v; econstructor.
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Qed.
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Lemma big_step_inv_vals (v w: val): big_step (of_val v) w → v = w.
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Proof.
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destruct v; inversion 1; eauto.
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Qed.
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(** *** Contextual Semantics *)
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(** Base reduction *)
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Inductive base_step : expr → expr → Prop :=
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| BetaS x e1 e2 e' :
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is_val e2 →
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e' = subst' x e2 e1 →
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base_step (App (Lam x e1) e2) e'
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| PlusS e1 e2 (n1 n2 n3 : Z):
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e1 = (LitInt n1) →
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e2 = (LitInt n2) →
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(n1 + n2)%Z = n3 →
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base_step (Plus e1 e2) (LitInt n3).
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Inductive ectx :=
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| HoleCtx
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| AppLCtx (K: ectx) (v2 : val)
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| AppRCtx (e1 : expr) (K: ectx)
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| PlusLCtx (K: ectx) (v2 : val)
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| PlusRCtx (e1 : expr) (K: ectx).
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Fixpoint fill (K : ectx) (e : expr) : expr :=
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match K with
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| HoleCtx => e
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| AppLCtx K v2 => App (fill K e) (of_val v2)
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| AppRCtx e1 K => App e1 (fill K e)
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| PlusLCtx K v2 => Plus (fill K e) (of_val v2)
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| PlusRCtx e1 K => Plus e1 (fill K e)
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end.
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(* filling a context with another context *)
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Fixpoint comp_ectx (Ko Ki: ectx) :=
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match Ko with
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| HoleCtx => Ki
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| AppLCtx K v2 => AppLCtx (comp_ectx K Ki) v2
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| AppRCtx e1 K => AppRCtx e1 (comp_ectx K Ki)
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| PlusLCtx K v2 => PlusLCtx (comp_ectx K Ki) v2
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| PlusRCtx e1 K => PlusRCtx e1 (comp_ectx K Ki)
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end.
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Inductive contextual_step (e1 : expr) (e2 : expr) : Prop :=
|
||||
Ectx_step K e1' e2' :
|
||||
e1 = fill K e1' →
|
||||
e2 = fill K e2' →
|
||||
base_step e1' e2' →
|
||||
contextual_step e1 e2.
|
||||
|
||||
Definition contextual_reducible (e : expr) :=
|
||||
∃ e', contextual_step e e'.
|
||||
|
||||
#[export] Hint Constructors base_step : core.
|
||||
#[export] Hint Constructors contextual_step : core.
|
||||
(* Lemmas about the contextual semantics *)
|
||||
Definition empty_ectx := HoleCtx.
|
||||
|
||||
Lemma fill_empty e : fill empty_ectx e = e.
|
||||
Proof. done. Qed.
|
||||
|
||||
Lemma base_contextual_step e1 e2 :
|
||||
base_step e1 e2 → contextual_step e1 e2.
|
||||
Proof. apply Ectx_step with empty_ectx; by rewrite ?fill_empty. Qed.
|
||||
|
||||
Lemma fill_comp (K1 K2 : ectx) e : fill K1 (fill K2 e) = fill (comp_ectx K1 K2) e.
|
||||
Proof. induction K1; simpl; congruence. Qed.
|
||||
|
||||
Lemma fill_contextual_step K e1 e2 :
|
||||
contextual_step e1 e2 → contextual_step (fill K e1) (fill K e2).
|
||||
Proof.
|
||||
destruct 1 as [K' e1' e2' -> ->].
|
||||
rewrite !fill_comp. by econstructor.
|
||||
Qed.
|
||||
|
||||
|
||||
(** Open and closed expressions *)
|
||||
Fixpoint is_closed (X : list string) (e : expr) : bool :=
|
||||
match e with
|
||||
| Var x => bool_decide (x ∈ X)
|
||||
| Lam x e => is_closed (x :b: X) e
|
||||
| LitInt _ => true
|
||||
| App e1 e2
|
||||
| Plus e1 e2 => is_closed X e1 && is_closed X e2
|
||||
end.
|
||||
|
||||
Notation closed X e := (Is_true (is_closed X e)).
|
||||
#[export] Instance closed_proof_irrel X e : ProofIrrel (closed X e).
|
||||
Proof. unfold closed. apply _. Qed.
|
||||
#[export] Instance closed_dec X e : Decision (closed X e).
|
||||
Proof. unfold closed. apply _. Defined.
|
||||
|
||||
Lemma closed_weaken X Y e : closed X e → X ⊆ Y → closed Y e.
|
||||
Proof. revert X Y; induction e; naive_solver (eauto; set_solver). Qed.
|
||||
|
||||
Lemma closed_weaken_nil X e : closed [] e → closed X e.
|
||||
Proof. intros. by apply closed_weaken with [], list_subseteq_nil. Qed.
|
||||
|
||||
Lemma closed_subst X Y e x es :
|
||||
closed Y es → closed (x :: X) e → closed (X ++ Y) (subst x es e).
|
||||
Proof.
|
||||
induction e as [y|y e IH|e1 e2|n|e1 e2]in X |-*; simpl; intros Hc1 Hc2; eauto.
|
||||
- eapply bool_decide_unpack, elem_of_cons in Hc2 as [->|Hc2].
|
||||
+ destruct decide; try congruence. eapply closed_weaken; eauto with set_solver.
|
||||
+ destruct decide.
|
||||
* eapply closed_weaken; eauto with set_solver.
|
||||
* simpl. eapply bool_decide_pack. set_solver.
|
||||
- destruct y as [|y]; simpl in *; eauto.
|
||||
destruct decide as [Heq|].
|
||||
+ injection Heq as ->. eapply closed_weaken; eauto. set_solver.
|
||||
+ rewrite app_comm_cons. eapply IH; eauto.
|
||||
eapply closed_weaken; eauto. set_solver.
|
||||
- eapply andb_True. eapply andb_True in Hc2 as [H1 H2].
|
||||
split; eauto.
|
||||
- eapply andb_True. eapply andb_True in Hc2 as [H1 H2].
|
||||
split; eauto.
|
||||
Qed.
|
||||
|
||||
Lemma closed_subst_nil X e x es :
|
||||
closed [] es → closed (x :: X) e → closed X (subst x es e).
|
||||
Proof.
|
||||
intros Hc1 Hc2. eapply closed_subst in Hc1; eauto.
|
||||
revert Hc1. rewrite right_id; [done|apply _].
|
||||
Qed.
|
||||
|
||||
Lemma closed_do_subst' X e x es :
|
||||
closed [] es → closed (x :b: X) e → closed X (subst' x es e).
|
||||
Proof. destruct x; eauto using closed_subst_nil. Qed.
|
||||
|
||||
Lemma subst_closed X e x es : closed X e → x ∉ X → subst x es e = e.
|
||||
Proof.
|
||||
induction e in X |-*; simpl; rewrite ?bool_decide_spec, ?andb_True; intros ??;
|
||||
repeat case_decide; simplify_eq; simpl; f_equal; intuition eauto with set_solver.
|
||||
Qed.
|
||||
|
||||
Lemma subst_closed_nil e x es : closed [] e → subst x es e = e.
|
||||
Proof. intros. apply subst_closed with []; set_solver. Qed.
|
||||
@ -0,0 +1,49 @@
|
||||
From semantics.ts.stlc Require Export lang.
|
||||
Set Default Proof Using "Type".
|
||||
|
||||
|
||||
(* We declare two notation scopes, one for values and one for expressions.
|
||||
Afterwards, we add notations to them, which then do not interfere with
|
||||
any existing Coq notations. *)
|
||||
Declare Scope expr_scope.
|
||||
Declare Scope val_scope.
|
||||
Delimit Scope expr_scope with E.
|
||||
Delimit Scope val_scope with V.
|
||||
Bind Scope expr_scope with expr.
|
||||
Bind Scope val_scope with val.
|
||||
|
||||
(* Values can be embedded into expressions with of_val
|
||||
and integers into expressions with LitInt. The following
|
||||
coercions allow us to omit the embeddings. With them
|
||||
(e1 v2 n3) desugars to (e3 (of_val v2) (LitInt n1)). *)
|
||||
Coercion of_val : val >-> expr.
|
||||
Coercion LitInt : Z >-> expr.
|
||||
Coercion LitIntV : Z >-> val.
|
||||
Coercion App : expr >-> Funclass.
|
||||
Coercion Var : string >-> expr.
|
||||
|
||||
Notation "e1 + e2" := (Plus e1%E e2%E) : expr_scope.
|
||||
|
||||
(* The breaking point '/ ' makes sure that the body of the λ: is indented
|
||||
by two spaces in case the whole λ: does not fit on a single line. *)
|
||||
Notation "λ: x , e" := (Lam x%binder e%E)
|
||||
(at level 200, x at level 1, e at level 200,
|
||||
format "'[' 'λ:' x , '/ ' e ']'") : expr_scope.
|
||||
Notation "λ: x y .. z , e" := (Lam x%binder (Lam y%binder .. (Lam z%binder e%E) ..))
|
||||
(at level 200, x, y, z at level 1, e at level 200,
|
||||
format "'[' 'λ:' x y .. z , '/ ' e ']'") : expr_scope.
|
||||
|
||||
Notation "λ: x , e" := (LamV x%binder e%E)
|
||||
(at level 200, x at level 1, e at level 200,
|
||||
format "'[' 'λ:' x , '/ ' e ']'") : val_scope.
|
||||
Notation "λ: x y .. z , e" := (LamV x%binder (Lam y%binder .. (Lam z%binder e%E) .. ))
|
||||
(at level 200, x, y, z at level 1, e at level 200,
|
||||
format "'[' 'λ:' x y .. z , '/ ' e ']'") : val_scope.
|
||||
|
||||
Notation "'let:' x := e1 'in' e2" := (Lam x%binder e2%E e1%E)
|
||||
(only parsing, at level 200, x at level 1, e1, e2 at level 200
|
||||
(*, format "'[' 'let:' x := '[' e1 ']' 'in' '/' e2 ']'" *)
|
||||
) : expr_scope.
|
||||
Notation "e1 ;; e2" := (Lam BAnon e2%E e1%E)
|
||||
(at level 100, e2 at level 200,
|
||||
format "'[' '[hv' '[' e1 ']' ;; ']' '/' e2 ']'") : expr_scope.
|
||||
Loading…
Reference in new issue