183 lines
3.4 KiB
Coq
183 lines
3.4 KiB
Coq
Require Import terms_debruijn.
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Require Import Atom.
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Require Import Metatheory.
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Require Import FSetNotin.
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(*
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* Substitution has no effect if the variable is not free
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*)
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Lemma type_subst_fresh : forall (x : atom) (τ:type_DeBruijn) (σ:type_DeBruijn),
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x `notin` (type_fv τ) ->
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([ x ~tt~> σ ] τ) = τ
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.
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Proof.
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intros.
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induction τ.
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- reflexivity.
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- unfold type_fv in H.
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apply AtomSetNotin.elim_notin_singleton in H.
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simpl.
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case_eq (x == a).
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congruence.
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reflexivity.
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- reflexivity.
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- simpl. rewrite IHτ.
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reflexivity.
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apply H.
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- simpl; rewrite IHτ1, IHτ2.
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reflexivity.
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simpl type_fv in H; fsetdec.
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simpl type_fv in H; fsetdec.
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- simpl. rewrite IHτ1, IHτ2.
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reflexivity.
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simpl type_fv in H; fsetdec.
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simpl type_fv in H; fsetdec.
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- simpl. rewrite IHτ1, IHτ2.
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reflexivity.
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simpl type_fv in H; fsetdec.
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simpl type_fv in H; fsetdec.
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- simpl. rewrite IHτ1, IHτ2.
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reflexivity.
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simpl type_fv in H; fsetdec.
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simpl type_fv in H; fsetdec.
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Qed.
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(*
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* Substitution has no effect if the variable is not free
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*)
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Lemma expr_subst_fresh : forall (x : atom) (t:expr_DeBruijn) (e:expr_DeBruijn),
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x `notin` (expr_fv e) ->
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([ x ~ee~> t ] e) = e
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.
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Proof.
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intros.
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induction e.
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- unfold expr_fv in H.
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apply AtomSetNotin.elim_notin_singleton in H.
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simpl.
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case_eq (x == a).
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congruence.
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reflexivity.
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- reflexivity.
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- simpl. rewrite IHe.
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all: auto.
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- simpl; rewrite IHe.
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auto. auto.
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- simpl; rewrite IHe.
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auto. auto.
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- simpl; rewrite IHe.
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auto. auto.
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- simpl. rewrite IHe1, IHe2.
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reflexivity.
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simpl expr_fv in H; fsetdec.
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simpl expr_fv in H; fsetdec.
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- simpl. rewrite IHe1, IHe2.
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reflexivity.
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simpl expr_fv in H; fsetdec.
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simpl expr_fv in H; fsetdec.
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- simpl; rewrite IHe.
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auto. auto.
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- simpl; rewrite IHe.
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auto. auto.
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Qed.
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Lemma type_open_lc_core : forall (τ:type_DeBruijn) i (σ1:type_DeBruijn) j (σ2:type_DeBruijn),
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i <> j ->
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{i ~tt~> σ1} τ = {j ~tt~> σ2} ({i ~tt~> σ1} τ) ->
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({j ~tt~> σ2} τ) = τ
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.
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Proof with eauto*.
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induction τ;
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intros i σ1 j σ2 Neq H;
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simpl in *;
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inversion H;
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f_equal; eauto.
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(* bound var *)
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- simpl in *.
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destruct (j === n).
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destruct (i === n).
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3:reflexivity.
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rewrite e,e0 in Neq.
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contradiction Neq.
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reflexivity.
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rewrite H,e.
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simpl.
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destruct (n===n).
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reflexivity.
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contradict n1.
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reflexivity.
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Qed.
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(*
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* opening is idempotent on locally closed types
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*)
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Lemma type_open_lc : forall k σ τ,
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type_lc τ ->
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({ k ~tt~> σ } τ) = τ
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.
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Proof.
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intros.
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generalize dependent k.
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induction H; eauto; simpl in *; intro k.
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(* univ *)
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f_equal.
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unfold type_open in *.
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pick fresh x for L.
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apply type_open_lc_core with
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(i := 0) (σ1 := (ty_fvar x))
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(j := S k) (σ2 := σ).
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trivial.
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apply eq_sym, H0, Fr.
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(* rest *)
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all: rewrite IHtype_lc1; rewrite IHtype_lc2; reflexivity.
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Qed.
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(*
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* type substitution distributes over opening
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*)
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Lemma type_subst_open : forall τ σ1 σ2 x k,
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type_lc σ2 ->
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[x ~tt~> σ2] ({k ~tt~> σ1} τ)
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=
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{k ~tt~> [x ~tt~> σ2] σ1} ([x ~tt~> σ2] τ).
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Proof.
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induction τ;
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intros; simpl; f_equal; auto.
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(* free var *)
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- destruct (x == a).
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subst.
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apply eq_sym, type_open_lc.
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assumption.
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trivial.
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(* bound var *)
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- destruct (k === n).
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reflexivity.
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trivial.
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Qed.
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