198 lines
3.6 KiB
Coq
198 lines
3.6 KiB
Coq
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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 ~> 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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Lemma open_rec_lc_core : forall τ i σ1 j σ2,
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i <> j ->
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{i ~> σ1} τ = {j ~> σ2} ({i ~> σ1} τ) ->
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({j ~> σ2} τ) = τ.
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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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(* id *)
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- reflexivity.
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(* free var *)
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- reflexivity.
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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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(* univ *)
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- simpl in *.
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inversion H.
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f_equal.
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apply IHτ with (i:=S i) (j:=S j) (σ1:=σ1).
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auto.
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apply H1.
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(* spec *)
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- simpl in *.
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inversion H.
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f_equal.
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* apply IHτ1 with (i:=i) (j:=j) (σ1:=σ1).
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auto.
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apply H1.
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* apply IHτ2 with (i:=i) (j:=j) (σ1:=σ1).
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auto.
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apply H2.
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(* func *)
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- simpl in *.
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inversion H.
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f_equal.
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* apply IHτ1 with (i:=i) (j:=j) (σ1:=σ1).
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auto.
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apply H1.
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* apply IHτ2 with (i:=i) (j:=j) (σ1:=σ1).
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auto.
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apply H2.
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(* morph *)
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- simpl in *.
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inversion H.
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f_equal.
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* apply IHτ1 with (i:=i) (j:=j) (σ1:=σ1).
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auto.
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apply H1.
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* apply IHτ2 with (i:=i) (j:=j) (σ1:=σ1).
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auto.
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apply H2.
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(* ladder *)
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- simpl in *.
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inversion H.
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f_equal.
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* apply IHτ1 with (i:=i) (j:=j) (σ1:=σ1).
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auto.
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apply H1.
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* apply IHτ2 with (i:=i) (j:=j) (σ1:=σ1).
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auto.
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apply H2.
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Qed.
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Lemma type_open_rec_lc : forall k σ τ,
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type_lc τ ->
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({ k ~> 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.
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(* id *)
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- auto.
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(* free var *)
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- auto.
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(* univ *)
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- simpl.
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intro k.
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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 open_rec_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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- simpl. intro. rewrite IHtype_lc1. rewrite IHtype_lc2. reflexivity.
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- simpl. intro. rewrite IHtype_lc1. rewrite IHtype_lc2. reflexivity.
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- simpl. intro. rewrite IHtype_lc1. rewrite IHtype_lc2. reflexivity.
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- simpl. intro. rewrite IHtype_lc1. rewrite IHtype_lc2. reflexivity.
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Qed.
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Lemma type_subst_open_rec : forall τ σ1 σ2 x k,
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type_lc σ2 ->
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[x ~> σ2] ({k ~> σ1} τ)
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=
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{k ~> [x ~> σ2] σ1} ([x ~> 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_rec_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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