153 lines
3.4 KiB
Coq
153 lines
3.4 KiB
Coq
Require Import MyTactics.
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Require Import LambdaCalculusSyntax.
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(* This technical lemma states that the renaming [lift 1] is injective. *)
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Lemma lift_inj_Var:
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forall t x,
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lift 1 t = Var (S x) <-> t = Var x.
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Proof.
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split; intros.
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{ eauto using lift_inj. }
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{ subst. eauto. }
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Qed.
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(* -------------------------------------------------------------------------- *)
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(* The predicate [fv k t] means that the free variables of the term [t] are
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contained in the semi-open interval [0..k). *)
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Definition fv k t :=
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t.[upn k (ren (+1))] = t.
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(* The predicate [fv] is characterized by the following lemmas. *)
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Lemma fv_Var_eq:
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forall k x,
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fv k (Var x) <-> x < k.
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Proof.
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unfold fv. asimpl. induction k; intros.
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(* Base case. *)
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{ asimpl. split; intros; false.
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{ unfold ids, Ids_term in *. injections. omega. }
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{ omega. }
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}
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(* Step. *)
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{ destruct x; asimpl.
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{ split; intros. { omega. } { reflexivity. } }
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rewrite lift_inj_Var. rewrite IHk. omega. }
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Qed.
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Lemma fv_Lam_eq:
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forall k t,
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fv k (Lam t) <-> fv (S k) t.
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Proof.
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unfold fv. intros. asimpl. split; intros.
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{ injections. eauto. }
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{ unpack. congruence. }
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Qed.
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Lemma fv_App_eq:
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forall k t1 t2,
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fv k (App t1 t2) <-> fv k t1 /\ fv k t2.
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Proof.
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unfold fv. intros. asimpl. split; intros.
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{ injections. eauto. }
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{ unpack. congruence. }
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Qed.
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Lemma fv_Let_eq:
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forall k t1 t2,
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fv k (Let t1 t2) <-> fv k t1 /\ fv (S k) t2.
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Proof.
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unfold fv. intros. asimpl. split; intros.
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{ injections. eauto. }
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{ unpack. congruence. }
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Qed.
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Hint Rewrite fv_Var_eq fv_Lam_eq fv_App_eq fv_Let_eq : fv.
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(* The tactic [fv] uses the above lemmas as rewriting rules. *)
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Ltac fv :=
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autorewrite with fv in *.
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(* -------------------------------------------------------------------------- *)
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(* The predicate [closed t] means that the term [t] is closed, that is, [t]
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has no free variables. *)
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Definition closed :=
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fv 0.
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(* [closed t] is equivalent to [lift 1 t = t]. *)
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Lemma closed_eq:
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forall t,
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closed t <-> lift 1 t = t.
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Proof.
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unfold closed, fv. asimpl. tauto.
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Qed.
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(* The following lemmas allow decomposing a closedness hypothesis.
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Because [closed] is not an inductive notion, there is no lemma
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for [Lam] and for the right-hand side of [Let]. *)
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Lemma closed_Var:
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forall x,
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~ closed (Var x).
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Proof.
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unfold closed; intros; fv. omega.
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Qed.
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Lemma closed_AppL:
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forall t1 t2,
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closed (App t1 t2) ->
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closed t1.
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Proof.
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unfold closed; intros; fv. tauto.
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Qed.
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Lemma closed_AppR:
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forall t1 t2,
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closed (App t1 t2) ->
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closed t2.
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Proof.
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unfold closed; intros; fv. tauto.
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Qed.
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Lemma closed_LetL:
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forall t1 t2,
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closed (Let t1 t2) ->
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closed t1.
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Proof.
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unfold closed; intros; fv. tauto.
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Qed.
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Hint Resolve closed_Var closed_AppL closed_AppR closed_LetL : closed.
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(* -------------------------------------------------------------------------- *)
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(* If the free variables of the term [t] are below [k], then [t] is unaffected
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by a substitution of the form [upn k sigma]. *)
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Lemma fv_unaffected:
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forall t k sigma,
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fv k t ->
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t.[upn k sigma] = t.
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Proof.
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induction t; intros; fv; unpack; asimpl;
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try solve [ eauto using upn_k_sigma_x with typeclass_instances
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| f_equal; eauto ].
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Qed.
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(* If the term [t] is closed, then [t] is unaffected by any substitution. *)
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Lemma closed_unaffected:
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forall t sigma,
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closed t ->
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t.[sigma] = t.
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Proof.
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intros. eapply fv_unaffected with (k := 0). eauto.
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Qed.
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