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LambdaESLab.v
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LambdaESLab.v
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(********************************************************************************
* Formalization of ES calculi *
* *
* Flavio L. C. de Moura & Daniel L. Ventura & Washington R. Segundo, 2014 *
* Flavio L. C. de Moura & Lucas de Moura Amaral, 2017 *
********************************************************************************)
Set Implicit Arguments.
Require Import Metatheory LambdaES_Defs LambdaES_Infra Lambda_Ex.
Require Import Rewriting Equation_C.
Require Export Morphisms.
(** Grammar of labelled pre-terms. Labelled terms extend the ordinary
terms with a new constructor for marked explicit substitutions. *)
Inductive lab_term : pterm -> Prop :=
| lab_term_var : forall x,
lab_term (pterm_fvar x)
| lab_term_app : forall t1 t2,
lab_term t1 ->
lab_term t2 ->
lab_term (pterm_app t1 t2)
| lab_term_abs : forall L t1,
(forall x, x \notin L -> lab_term (t1 ^ x)) ->
lab_term (pterm_abs t1)
| lab_term_sub : forall L t1 t2,
(forall x, x \notin L -> lab_term (t1 ^ x)) ->
lab_term t2 -> lab_term (t1[t2])
| lab_term_sub' : forall L t1 t2,
(forall x, x \notin L -> lab_term (t1 ^ x)) ->
(term t2) -> (SN lex t2) ->
lab_term (t1 [[ t2 ]]).
(** Every term is also a labelled term because the grammar with labels
extends the term grammar. *)
Lemma term_is_lab_term: forall t, term t -> lab_term t.
Proof.
intros t Ht. apply term_size_induction.
- intro x. apply lab_term_var.
- intros t1 Hbody Hind.
apply lab_term_abs with (fv t1).
intros x Hfv. apply Hind.
+ assumption.
+ trivial.
+ apply body_to_term; assumption.
- intros t1 t2 Ht1 Hlab1 Ht2 Hlab2.
apply lab_term_app; assumption.
- intros t1 t3 Ht3 Hlab3 Hb1 Hind.
apply lab_term_sub with (fv t1).
+ intros x Hfvt1. apply Hind.
* assumption.
* trivial.
* apply body_to_term; assumption.
+ assumption.
- assumption.
Qed.
Definition lab_body (t : pterm) :=
exists L, forall x, x \notin L -> lab_term (t ^ x).
Lemma body_is_lab_body : forall t, body t -> lab_body t.
Proof.
intros. case H; clear H; intros L H; exists L.
intros. apply term_is_lab_term. apply H; trivial.
Qed.
(** Alternative definition for local closure *)
(** Local closure for labelled terms. *)
Fixpoint lc_at' (k:nat) (t:pterm) : Prop :=
match t with
| pterm_bvar i => i < k
| pterm_fvar x => True
| pterm_app t1 t2 => lc_at' k t1 /\ lc_at' k t2
| pterm_abs t1 => lc_at' (S k) t1
| pterm_sub t1 t2 => (lc_at' (S k) t1) /\ lc_at' k t2
| pterm_sub' t1 t2 => (lc_at k t2) /\ (SN lex t2) /\ (lc_at' (S k) t1)
end.
Definition term'' t := lc_at' 0 t.
Definition body'' t := lc_at' 1 t.
(** Labelled equations extends eqc. Note that the equations defined form labelled substitutions are not used alone. *)
Inductive lab_eqc : pterm -> pterm -> Prop :=
| lab_eqc_def : forall t u v, lab_term u -> lab_term v -> lab_eqc (t[u][v]) ((& t)[v][u])
| lab_eqc_rx1 : forall t u v,
lab_term u -> term v -> lab_eqc (t[u][[v]]) ((& t)[[v]][u])
| lab_eqc_rx2 : forall t u v,
term u -> lab_term v -> lab_eqc (t[[u]][v]) ((& t)[v][[u]])
| lab_eqc_rx3 : forall t u v,
term u -> term v -> lab_eqc (t[[u]][[v]]) ((& t)[[v]][[u]]).
(** Correctness of the above definition. *)
(* forall u,0 notin fi(u) <-> lab_term u *)
Lemma eqc_is_lab_eqc: forall t u, eqc t u -> lab_eqc t u.
Proof.
introv Heqc. induction Heqc.
apply lab_eqc_def.
- apply term_is_lab_term; assumption.
- apply term_is_lab_term; assumption.
Qed.
Lemma eqc_ctx_is_lab_eqc_ctx: forall t u, t =c u -> ES_contextual_closure lab_eqc t u.
Proof.
introv Heqc. induction Heqc.
- apply ES_redex. apply eqc_is_lab_eqc; assumption.
- apply ES_app_left; assumption.
- apply ES_app_right; assumption.
- apply ES_abs_in with L; assumption.
- apply ES_subst_left with L; assumption.
- apply ES_subst_right; assumption.
Qed.
Lemma eqc_ctx_trans_is_lab_eqc_ctx_trans: forall t u, t =c+ u -> trans_closure (ES_contextual_closure lab_eqc) t u.
Proof.
introv Heqc. induction Heqc.
- apply one_step_reduction. apply eqc_ctx_is_lab_eqc_ctx; assumption.
- apply transitive_reduction with u.
+ apply eqc_ctx_is_lab_eqc_ctx; assumption.
+ assumption.
Qed.
Lemma lab_eqc_sym : forall t u, lab_eqc t u -> lab_eqc u t.
Proof.
introv Heqc. destruct Heqc.
- replace ((t [u]) [v]) with (((& (& t)) [u]) [v]).
apply lab_eqc_def; assumption. rewrite bswap_idemp; trivial.
- replace ((t [u]) [[v]]) with (((& (& t)) [u]) [[v]]).
apply lab_eqc_rx2; assumption. rewrite bswap_idemp; trivial.
- replace ((t [[u]]) [v]) with (((& (& t)) [[u]]) [v]).
apply lab_eqc_rx1; assumption. rewrite bswap_idemp; trivial.
- replace ((t [[u]]) [[v]]) with (((& (& t)) [[u]]) [[v]]).
apply lab_eqc_rx3; assumption. rewrite bswap_idemp; trivial.
Qed.
(** lab_eqC: Reflexive-transitive of the contextual closure of lab_eqc *)
Definition lab_eqC (t: pterm) (u : pterm) := star_closure (ESlab_contextual_closure lab_eqc) t u .
Notation "t =~e u" := (lab_eqC t u) (at level 66).
(** =~e is an equivalence relation *)
Lemma lab_eqC_rf : forall t, t =~e t.
Proof.
intro t. apply reflexive_reduction.
Qed.
(*
Lemma lab_eqc_preserves_SN_lex: forall t t', SN lex t -> lab_eqc t t' -> SN lex t'.
Proof.
introv HSN Heqc. induction Heqc.
induction H1. assumption.
inversion H0.
inversion H2.
unfold SN.
exists x.
apply SN_intro.
intros t' H4.
apply H3.
assert (lab_eqc ((t [u]) [[v]]) (((& t) [[v]]) [u])).
apply lab_eqc_rx1; assumption.
Admitted.
*)
Lemma lab_ctx_eqc_sym : forall t u, (ESlab_contextual_closure lab_eqc t u) -> ESlab_contextual_closure lab_eqc u t.
Proof.
introv H. induction H.
- apply ESlab_redex. apply lab_eqc_sym. assumption.
- apply ESlab_app_left; trivial.
- apply ESlab_app_right; trivial.
- apply ESlab_abs_in with L; trivial.
- apply ESlab_subst_left with L; trivial.
- apply ESlab_subst_right; trivial.
- apply ESlab_lsubst_left with L; assumption.
- apply ESlab_lsubst_right; trivial.
Qed.
Lemma lab_eqC_sym : forall t u, t =~e u -> u =~e t.
Proof.
introv H. unfold lab_eqC in *. induction H.
- apply lab_eqC_rf.
- induction H.
+ apply star_trans_reduction.
apply one_step_reduction.
apply lab_ctx_eqc_sym; assumption.
+ apply lab_ctx_eqc_sym in H.
apply star_transitive_closure' with u; assumption.
Qed.
Lemma lab_eqC_trans : forall t u v, t =~e u -> u =~e v -> t =~e v.
Proof.
introv Htu Huv.
apply star_closure_composition with u; trivial.
Qed.
Instance lab_eqC_eq : Equivalence lab_eqC.
Proof.
split; intros_all.
apply lab_eqC_rf.
apply lab_eqC_sym; trivial.
apply lab_eqC_trans with y; trivial.
Qed.
Lemma eqc_ctx_is_lab_eqC: forall t u, t =e u -> t =~e u.
Proof.
introv Heqc. induction Heqc.
- apply lab_eqC_rf.
- destruct H.
+ apply eqc_ctx_is_lab_eqc_ctx in H.
unfold lab_eqC. apply star_trans_reduction.
apply one_step_reduction. apply ES_cc_is_ESlab_cc in H; assumption.
+ unfold lab_eqC. apply star_trans_reduction.
apply transitive_reduction with u.
* apply eqc_ctx_is_lab_eqc_ctx in H.
apply ES_cc_is_ESlab_cc in H; assumption.
* apply eqc_ctx_trans_is_lab_eqc_ctx_trans in H0.
apply ES_cc_trans_is_ESlab_cc_trans in H0; assumption.
Qed.
(*
Lemma lab_eqc_trans : forall t u v, lab_eqc t u -> lab_eqc u v -> lab_eqc t v.
Proof.
intros t u v Htu Huv.
destruct Huv.
assumption.
inversion Htu.
apply lab_eqc_rx1; assumption.
rewrite bswap_idemp; trivial.
apply lab_eqc_rf.
inversion Htu.
apply lab_eqc_rx2; assumption.
rewrite bswap_idemp; trivial.
apply lab_eqc_rf.
inversion Htu.
apply lab_eqc_rx3; assumption.
rewrite bswap_idemp; trivial.
apply lab_eqc_rf.
Qed.
*)
(** The extended reduction system. This system is used to propagate
terminating labelled substitutions. *)
Inductive lab_sys_x : pterm -> pterm -> Prop :=
| lab_reg_rule_var : forall t, lab_sys_x (pterm_bvar 0 [[t]]) t
| lab_reg_rule_gc : forall t u, lab_term t -> lab_sys_x (t[[u]]) t
| lab_reg_rule_app : forall t1 t2 u,
lab_sys_x ((pterm_app t1 t2)[[u]]) (pterm_app (t1[[u]]) (t2[[u]]))
| lab_reg_rule_lamb : forall t u,
lab_sys_x ((pterm_abs t)[[u]]) (pterm_abs ((& t)[[u]]))
| lab_reg_rule_comp : forall t u v, has_free_index 0 u ->
lab_sys_x (t[u][[v]]) (((& t)[[v]])[u[[v]]]).
Notation "t ->_lab_x u" := (lab_sys_x t u) (at level 59, left associativity).
(** Internal reductions: lab_x plus lex inside labelled substitutions *)
Inductive lab_x_i: pterm -> pterm -> Prop :=
| xi_from_bx_in_les: forall t u u', u -->lex u' ->
lab_x_i (t [[ u ]]) (t [[ u' ]])
| xi_from_x: forall t t', t ->_lab_x t' -> lab_x_i t t'.
(** Reduction on labelled terms: lex plus lab_x *)
Inductive lab_sys_lx: pterm -> pterm -> Prop :=
| lab_B_lx : forall t u, t ->_B u -> lab_sys_lx t u
| lab_sys_x_lx : forall t u, t ->_x u -> lab_sys_lx t u
| lab_sys_x_lab_lx : forall t u, t ->_lab_x u -> lab_sys_lx t u.
Lemma Bx_is_lab_lx: forall t u, sys_Bx t u -> lab_sys_lx t u.
Proof.
introv HBx. destruct HBx.
- apply lab_B_lx; assumption.
- apply lab_sys_x_lx; assumption.
Qed.
Lemma ESLab_Bx_is_lab_lx: forall t u, ESlab_contextual_closure sys_Bx t u -> ESlab_contextual_closure lab_sys_lx t u.
Proof.
introv HES. induction HES.
- apply ESlab_redex. apply Bx_is_lab_lx in H; assumption.
- apply ESlab_app_left; assumption.
- apply ESlab_app_right; assumption.
- apply ESlab_abs_in with L; assumption.
- apply ESlab_subst_left with L; assumption.
- apply ESlab_subst_right; assumption.
- apply ESlab_lsubst_left with L; assumption.
- apply ESlab_lsubst_right; assumption.
Qed.
(* ********************************************************************** *)
(** pterm lists properties *)
(* Fixpoint cr_lc_at_list (n : nat) (l : list pterm) {struct l} : Prop := *)
(* match l with *)
(* | nil => True *)
(* | t::lu => lc_at' n t /\ (cr_lc_at_list (S n) lu) *)
(* end. *)
(* Lemma lc_at_mult_sub : forall n t lu, lc_at' n (t//[lu]) <-> (lc_at' (n + length lu) t /\ cr_lc_at_list n lu). *)
(* Proof. *)
(* intros. generalize n; clear n. induction lu; simpl. *)
(* split. intro. assert (Q : n + 0 = n); try omega. rewrite Q. split; trivial. *)
(* intro. assert (Q : n + 0 = n); try omega. rewrite Q in H. apply H. *)
(* intro n. replace (n + S (length lu)) with ((S n) + length lu). split. *)
(* intro H. destruct H. split. *)
(* apply IHlu; trivial. split; trivial. apply IHlu; trivial. *)
(* intro H. destruct H. destruct H0. split; trivial. apply IHlu. split; trivial. *)
(* omega. *)
(* Qed. *)
(** Reduction modulo lab_eqC *)
Definition red_ctx_mod_lab_eqC (R: pterm -> pterm -> Prop) (t: pterm) (u : pterm) :=
exists t' u', (t =~e t')/\(ESlab_contextual_closure R t' u')/\(u' =~e u).
(** External reduction context (reductions everywhere except inside labelled substitutions) modulo lab_eqC *)
Definition ext_lab_ee_ctx_red (R: pterm -> pterm -> Prop) (t: pterm) (u : pterm) :=
exists t' u', (t =~e t')/\(ext_lab_contextual_closure R t' u')/\(u' =~e u).
(** Reduction lab_sys_x modulo lab_eqC *)
Definition lab_ex := red_ctx_mod_lab_eqC lab_sys_x.
Notation "t -->[ex] u" := (lab_ex t u) (at level 59, left associativity).
Definition lab_lex := red_ctx_mod_lab_eqC lab_sys_lx.
Notation "t -->[lex] u" := (lab_lex t u) (at level 59, left associativity).
Definition lab_x_i_eq := red_ctx_mod_lab_eqC lab_x_i.
Notation "t -->[lx_i] u" := (lab_x_i_eq t u) (at level 59, left associativity).
Definition lab_x_e_eq := ext_lab_ee_ctx_red sys_Bx.
Notation "t -->[lx_e] u" := (lab_x_e_eq t u) (at level 59, left associativity).
Definition red_lab_wregular (R : pterm -> pterm -> Prop) :=
forall t t', R t t' -> lab_term t -> lab_term t'.
Instance rw_eqC_sub' : Proper (eqC ==> eqC ==> eqC) pterm_sub'.
Proof.
intros_all. apply star_closure_composition with (y [[x0]]).
- inversion H; subst.
+ reflexivity.
+ apply star_trans_reduction. inversion H1; subst.
apply one_step_reduction. pick_fresh z.
(* apply ES_lsubst_left with (fv x \u fv y \u fv x0 \u fv y0). *)
(* introv Hfv. apply eqc_ctx_open; assumption. *)
(* assert (x =c+ y). *)
(* { apply transitive_reduction with u; assumption. } *)
(* pick_fresh z. *)
(* apply eqc_trans_sub_left with (fv x \u fv y \u fv x0 \u fv y0 \u fv u). *)
(* introv Hfv. apply eqc_trans_open; assumption. *)
(* - induction H0. *)
(* + reflexivity. *)
(* + apply star_trans_reduction. induction H0. *)
(* * apply one_step_reduction. *)
(* apply ES_subst_right; assumption. *)
(* * apply transitive_reduction with (y[u]). *)
(* ** apply ES_subst_right; assumption. *)
(* ** assumption. *)
Admitted. (* Fabrício *)
(*
Definition red_lab_regular' (R : pterm -> pterm -> Prop) :=
forall t t', R t t' -> (lab_term t <-> lab_term t').
*)
(** Unlabelled reduction is in the corresponding labelled reduction. *)
Lemma sys_Bx_is_lab_sys_lx: forall t t', t -->lex t' -> t -->[lex] t'.
Proof.
introv Hlex. induction Hlex.
destruct H as [ x' [Hee [Hes Hee' ] ] ].
unfold lab_lex. unfold red_ctx_mod_lab_eqC.
exists x x'. split.
- apply eqc_ctx_is_lab_eqC; assumption.
- split.
+ apply ES_cc_is_ESlab_cc in Hes.
apply ESLab_Bx_is_lab_lx in Hes; assumption.
+ apply eqc_ctx_is_lab_eqC; assumption.
Qed.
Lemma open_lab_term : forall t x k, lab_term t -> ({k ~> pterm_fvar x}t) = t.
Proof.
intros t x k H. generalize k; clear k.
induction H; intros; simpl; trivial.
rewrite IHlab_term1; rewrite IHlab_term2; trivial.
case var_fresh with (L := L \u (fv t1) \u fv ({Datatypes.S k ~> pterm_fvar x}t1)). intros z Hz.
apply notin_union in Hz. destruct Hz. apply notin_union in H1. destruct H1.
assert (Q : {Datatypes.S k ~> pterm_fvar x}t1 ^ z = t1 ^ z). apply H0; trivial.
unfold open in Q. rewrite subst_com in Q; try omega; trivial.
apply open_var_inj in Q; trivial. rewrite Q; trivial.
rewrite IHlab_term.
case var_fresh with (L := L \u (fv t1) \u fv ({Datatypes.S k ~> pterm_fvar x}t1)). intros z Hz.
apply notin_union in Hz. destruct Hz. apply notin_union in H2. destruct H2.
assert (Q : {Datatypes.S k ~> pterm_fvar x}t1 ^ z = t1 ^ z). apply H0; trivial.
unfold open in Q. rewrite subst_com in Q; try omega; trivial.
apply open_var_inj in Q; trivial. rewrite Q; trivial.
rewrite <- open_rec_term with (t := t2); trivial.
case var_fresh with (L := L \u (fv t1) \u fv ({Datatypes.S k ~> pterm_fvar x}t1)). intros z Hz.
apply notin_union in Hz. destruct Hz. apply notin_union in H3. destruct H3.
assert (Q : {Datatypes.S k ~> pterm_fvar x}t1 ^ z = t1 ^ z).
apply H0; trivial.
unfold open in Q. rewrite subst_com in Q; try omega; trivial.
apply open_var_inj in Q; trivial.
rewrite Q; trivial.
Qed.
Lemma SN_open: forall t x k, SN lex t -> SN lex ({k ~> (pterm_fvar x)} t).
Proof.
Admitted.
Lemma SN_open_rename: forall t x y k, SN lex ({k ~> (pterm_fvar x)} t) -> SN lex ({k ~> (pterm_fvar y)} t).
Proof.
Admitted.
Lemma lc_at'_open_rec_rename: forall t x y m n, lc_at' m (open_rec n (pterm_fvar x) t) -> lc_at' m (open_rec n (pterm_fvar y) t).
Proof.
induction t. simpl. introv H. case_nat. constructor. assumption.
simpl. intros; trivial. simpl. introv H. destruct H.
apply (IHt1 x y) in H. apply (IHt2 x y) in H0.
split; assumption. simpl.
introv H. apply IHt with x. assumption. simpl.
introv H. destruct H. split. apply IHt1 with x; assumption. apply IHt2 with x; assumption.
introv H. destruct H. split. Admitted.
(* apply lc_at_open_rec_rename with x; auto. *)
(* destruct H0. split. apply SN_open_rename with (x := x); auto. *)
(* apply IHt1 with x; assumption. *)
(* Qed. *)
Corollary lab_term_open_rename: forall t x y, lab_term (t^x) -> lab_term (t^y).
Proof.
introv H. Admitted.
(* apply lab_term_eq_term'' in H. apply lab_term_eq_term''. *)
(* apply lc_at'_open_rec_rename with x. assumption. *)
(* Qed. *)
Lemma lab_sys_x_i_e: forall t t0 t1 t', (t =~e t0) -> lab_sys_lx t0 t1 -> (t1 =~e t') -> (t -->[lx_i] t' \/ t -->[lx_e] t').
Proof.
introv Hee Hred Hee'. induction Hred.
- right. exists t0 u. split*. split*.
+ apply ext_lab_redex. apply B_lx; assumption.
- right. exists t0 u. split*. split*.
apply ext_lab_redex. apply sys_x_lx; assumption.
- inversion H.
+ subst. right. (* rewrite Hee. *)
(* exists (pterm_bvar 0 [u]) u. split*. split. *)
(* + *)
(* + *)
(* + *)
(* + *)
(* * apply ext_lab_redex. apply sys_x_lx. apply reg_rule_var. *)
(* * assumption. *)
(* + right. inversion Hee. *)
(* * Admitted. *)
Admitted. (* Fabrício *)
(* rewrite Hee. *)
(* + *)
(* + *)
(* + *)
(* + *)
(* - *)
(* constructor 1. constructor 1. auto. auto. *)
(* constructor 2. exists t u. split*. split. constructor 1. constructor 2. auto. auto. *)
(* constructor 1. exists t u. split*. split. constructor 1. auto. constructor 2. auto. auto. *)
(* Qed. *)
Lemma lab_open_close_var : forall (x : var) (t : pterm), lab_term t -> t = close t x ^ x.
Proof.
introv W. unfold close, open. generalize 0.
induction W; intros k; simpls; f_equal*.
case_var*. simpl. case_nat*.
let L := gather_vars in match goal with |- _ = ?t =>
destruct (var_fresh (L \u fv t)) as [y Fr] end.
apply* (@open_var_inj y).
apply notin_union in Fr. destruct Fr.
apply notin_union in H1. destruct H1. auto.
apply notin_union in Fr. destruct Fr. auto.
unfolds open. rewrite* close_var_rec_open. VSD.fsetdec.
(*Focus 3. fail.*)
let L := gather_vars in match goal with |- _ = ?t =>
destruct (var_fresh (L \u fv t)) as [y Fr] end.
apply* (@open_var_inj y).
auto. auto.
(*rewrite* close_var_rec_open. VSD.fsetdec.*)
unfolds open. rewrite* close_var_rec_open. VSD.fsetdec.
let L := gather_vars in match goal with |- _ = ?t =>
destruct (var_fresh (L \u fv t)) as [y Fr] end.
apply* (@open_var_inj y).
auto. auto.
unfolds open. rewrite* close_var_rec_open. VSD.fsetdec.
let L := gather_vars in match goal with |- _ = ?t =>
destruct (var_fresh (L \u fv t)) as [y Fr] end.
apply* (@open_var_inj y).
auto. auto.
unfolds open. Admitted.
(* rewrite <- open_close_var_gen with (x := x) (k := k); auto. *)
(* Qed. *)
(* Check if needed.
Lemma close_var_spec : forall t x, term t ->
exists u, t = u ^ x /\ body u /\ x \notin (fv u).
Proof.
intros.
exists (close t x).
rewrite <- open_close_var; auto.
split*. split*.
apply close_var_body; auto.
apply close_fresh; auto.
Qed.
Lemma close_var_lab_body : forall x t, lab_term t -> lab_body (close t x).
Proof.
introv W. exists {{x}}. intros y Fr.
unfold open, close. generalize 0. gen y.
induction W; intros y Fr k; simpls.
case_var; simpl. case_nat*.
auto*.
constructor 1.
constructor 1.
apply* lab_term_app.
apply IHW1; auto.
apply IHW2; auto.
apply_fresh* lab_term_abs.
unfolds open. rewrite* close_var_rec_open. VSD.fsetdec.
apply_fresh* lab_term_sub. unfolds open. rewrite* close_var_rec_open. VSD.fsetdec.
apply_fresh* lab_term_sub'. unfolds open. rewrite* close_var_rec_open. VSD.fsetdec.
Admitted.
*)
(* Lemma ee_presv_ie: forall t t' u u', t =EE u -> u' =EE t' -> ((u -->[lx_i] u' \/ u -->[lx_e] u') -> (t -->[lx_i] t' \/ t -->[lx_e] t')). *)
(* Proof. *)
(* intros. *)
(* destruct H1. destruct H1. destruct H1. destruct H1. destruct H2. *)
(* left. *)
(* exists x x0. *)
(* split*. *)
(* apply star_closure_composition with u; auto. *)
(* split*. *)
(* apply star_closure_composition with u'; auto. *)
(* destruct H1. destruct H1. destruct H1. destruct H2. *)
(* right. *)
(* exists x x0. *)
(* split*. *)
(* apply star_ctx_eqcc_sym in H. *)
(* apply star_ctx_eqcc_sym in H. *)
(* apply star_closure_composition with u; auto. *)
(* split*. *)
(* apply star_closure_composition with u'; auto. *)
(* Qed. *)
(* Fabrício *)
Lemma lab_ex_impl_i_e: forall t t', t -->[lex] t' -> (t -->[lx_i] t' \/ t -->[lx_e] t').
Proof.
introv Hred. destruct Hred as [t0 [t1 [Hee [Hes Hee'] ] ] ]. gen t t'.
induction Hes. (* Induction on the contextual closure of lab_sys_lx. *)
- (* lab_sys_lx is applied at the root of the term. *)
induction H. (* Application of Bx \cup lab_x at the root of the term. *)
+ (* Application of the rule B. *)
introv Hee Hee'. right. unfold lab_x_e_eq. exists t u. split.
* assumption.
* split.
apply ext_lab_redex. apply B_lx; assumption.
assumption.
+ (* Application of rules x. *)
introv Hee Hee'. right. exists t u. split.
* assumption.
* split.
apply ext_lab_redex. apply sys_x_lx; assumption.
assumption.
+ (* Application of rules lab_x. *)
introv Hee Hee'. left. exists t u. split.
* assumption.
* split.
apply ESlab_redex. apply xi_from_x; assumption.
assumption.
- admit.
(* - (* lab_sys_lx is applied to the left of an application. *) *)
(* introv Hee Hee'. *)
(* (* apply IHHes. *) *)
(* apply lab_sys_x_i_e with (t0:=t) (t1:=t'). auto. *)
(* apply EE_lab_term with t0; auto*. *)
(* - *)
(* - *)
(* - *)
(* - *)
(* - *)
(* - *)
(* - *)
(* app_left *)
-(* apply EE_presv_ie with (u := (pterm_app t u)) (u' := (pterm_app t' u)); auto. *)
(* assert (t-->[lx_i]t' \/ t-->[lx_e]t'). *)
(* apply IHlab_contextual_closure; auto. constructor 1; auto. *)
(* pose proof (EE_lab_term H0 H3); auto. inversion H4; subst; auto. *)
(* constructor 1; auto. *)
(* destruct H4. *)
(* left. apply EE_ext_clos_app_left. *)
(* pose proof (EE_lab_term H0 H3); auto. auto. *)
(* right. apply EE_ext_clos_app_left. *)
(* pose proof (EE_lab_term H0 H3); auto. auto. *)
(* (* app_right *) *)
(* apply EE_presv_ie with (u := (pterm_app t u)) (u' := (pterm_app t u')); auto. *)
(* assert (u-->[lx_i]u' \/ u-->[lx_e]u'). *)
(* apply IHlab_contextual_closure; auto. constructor 1; auto. *)
(* pose proof (EE_lab_term H0 H3); auto. inversion H4; subst; auto. *)
(* constructor 1; auto. *)
(* destruct H4. *)
(* left. apply EE_ext_clos_app_right. *)
(* pose proof (EE_lab_term H0 H3); auto. auto. *)
(* right. apply EE_ext_clos_app_right. *)
(* pose proof (EE_lab_term H0 H3); auto. auto. *)
(* (* abs *) *)
(* apply EE_presv_ie with (u := pterm_abs t) (u' := pterm_abs t'); auto. *)
(* pose proof (EE_lab_term H1 H3); auto. inversion H4; subst; auto. *)
(* pick_fresh z. *)
(* assert (t^z-->[lx_i]t'^z \/ t^z-->[lx_e]t'^z). *)
(* apply H0 with z; auto. constructor 1; auto. *)
(* constructor 1; auto. *)
(* apply notin_union in Fr; destruct Fr. *)
(* apply notin_union in H7; destruct H7. *)
(* apply notin_union in H7; destruct H7. *)
(* apply notin_union in H7; destruct H7. *)
(* apply notin_union in H7; destruct H7. *)
(* destruct H5. *)
(* left. apply EE_ext_clos_abs with L. exact red_rename_lab_xi. *)
(* exact red_lab_regular'_lab_xi. *)
(* apply EE_lab_term with t0; auto. *)
(* intros. *)
(* pose proof red_rename_lab_xi_eq. apply H14 with z; auto. *)
(* right. apply EE_ext_clos_abs with L. exact red_rename_sys_Bx. exact red_lab_regular'_sys_Bx. *)
(* apply EE_lab_term with t0; auto. *)
(* intros. pose proof red_rename_lab_xe_eq. *)
(* unfold red_rename in H14. *)
(* apply H14 with z; auto. *)
(* (* outer sub *) *)
(* apply EE_presv_ie with (u := t[u]) (u' := t'[u]); auto. *)
(* pose proof (EE_lab_term H3 H4); auto. inversion H5; subst; auto. *)
(* pick_fresh z. *)
(* assert (t^z-->[lx_i]t'^z \/ t^z-->[lx_e]t'^z). *)
(* apply H1 with z; auto. constructor 1; auto. constructor 1; auto. *)
(* apply notin_union in Fr; destruct Fr. *)
(* apply notin_union in H7; destruct H7. *)
(* apply notin_union in H7; destruct H7. *)
(* apply notin_union in H7; destruct H7. *)
(* apply notin_union in H7; destruct H7. *)
(* apply notin_union in H7; destruct H7. *)
(* destruct H6. *)
(* left. apply EE_ext_clos_outer_sub with L; auto. *)
(* exact red_rename_lab_xi. exact red_lab_regular'_lab_xi. *)
(* intros. *)
(* pose proof red_rename_lab_xi_eq. apply H17 with z; auto. *)
(* right. apply EE_ext_clos_outer_sub with L; auto. *)
(* exact red_rename_sys_Bx. exact red_lab_regular'_sys_Bx. *)
(* intros. pose proof red_rename_lab_xe_eq. apply H17 with z; auto. *)
(* (* inner sub *) *)
(* apply EE_presv_ie with (u := t[u]) (u' := t[u']); auto. *)
(* assert (u' =EE u'). constructor 1; auto. *)
(* assert (u =EE u). constructor 1; auto. *)
(* apply EE_lab_term in H3. inversion H3; subst. *)
(* pose proof (IHlab_contextual_closure u' H4 u H9 H5). *)
(* destruct H6. *)
(* left. apply EE_ext_clos_inner_sub; auto. *)
(* right. apply EE_ext_clos_inner_sub; auto. *)
(* auto. *)
(* (* outer lsub *) *)
(* apply EE_presv_ie with (u := t[[u]]) (u' := t'[[u]]); auto. *)
(* pose proof (EE_lab_term H3 H4); auto. inversion H5; subst; auto. *)
(* pick_fresh z. *)
(* assert (t^z-->[lx_i]t'^z \/ t^z-->[lx_e]t'^z). *)
(* apply H1 with z; auto. constructor 1; auto. constructor 1; auto. *)
(* apply notin_union in Fr; destruct Fr. *)
(* apply notin_union in H7; destruct H7. *)
(* apply notin_union in H7; destruct H7. *)
(* apply notin_union in H7; destruct H7. *)
(* apply notin_union in H7; destruct H7. *)
(* apply notin_union in H7; destruct H7. *)
(* destruct H6. *)
(* left. apply EE_ext_clos_outer_lsub with L. *)
(* apply EE_lab_term with t0; auto. *)
(* exact red_rename_lab_xi. exact red_lab_regular'_lab_xi. *)
(* intros. *)
(* pose proof red_rename_lab_xi_eq. apply H18 with z; auto. *)
(* right. apply EE_ext_clos_outer_lsub with L. *)
(* apply EE_lab_term with t0; auto. *)
(* exact red_rename_sys_Bx. exact red_lab_regular'_sys_Bx. *)
(* intros. pose proof red_rename_lab_xe_eq. apply H18 with z; auto. *)
(* (* inner lsub *) *)
(* left. exists (t [[u]]) (t [[u']]). split. auto. *)
(* split*. *)
(* apply EE_lab_term with t0 (t [[u]]) in H5. *)
(* inversion H3; subst. *)
(* apply lab_sys_lx_term_is_sys_Bx with u u' in H0; auto. *)
(* (*inversion H5; subst.*) *)
(* constructor 1. constructor 1. auto. auto. *)
(* constructor 1. constructor 1. auto. *)
(* apply lab_sys_lx_term_is_sys_Bx with u u' in H0; auto. *)
(* auto. *)
Admitted.
(* Qed. *)
Lemma lab_ie_impl_ex: forall t t', (t -->[lx_i] t' \/ t -->[lx_e] t') -> t -->[lex] t'.
Proof. (* Admitted. *)
Admitted.
(* intros. destruct H; destruct H; destruct H; destruct H; destruct H0; generalize dependent t; generalize dependent t'; induction H0; intros. *)
(* (*[> ------------------ Interna <]*) *)
(* (*[> Base <]*) *)
(* exists t s. *)
(* split*. split*. *)
(* inversion H; subst. *)
(* inversion H3; subst. constructor 8; auto. exists L; auto. *)
(* inversion H4; subst. *)
(* constructor 1; auto. constructor 2; subst. auto. *)
(* constructor 1; auto. constructor 3; auto. *)
(* (* app_left *) *)
(* apply EE_presv_lab_lex with (u := (pterm_app t u)) (u' := (pterm_app t' u)); auto. *)
(* apply EE_clos_app_left. *)
(* pose proof (EE_lab_term H0 H3); auto. *)
(* apply IHext_lab_contextual_closure; auto. constructor 1; auto. *)
(* pose proof (EE_lab_term H0 H3); auto. inversion H4; subst; auto. *)
(* constructor 1; auto. *)
(* (* app_right *) *)
(* apply EE_presv_lab_lex with (u := (pterm_app t u)) (u' := (pterm_app t u')); auto. *)
(* apply EE_clos_app_right. pose proof (EE_lab_term H0 H3); auto. *)
(* apply IHext_lab_contextual_closure; auto. constructor 1; auto. *)
(* pose proof (EE_lab_term H0 H3); auto. inversion H4; subst; auto. *)
(* constructor 1; auto. *)
(* (* abs *) *)
(* apply EE_presv_lab_lex with (u := pterm_abs t) (u' := pterm_abs t'); auto. *)
(* pose proof (EE_lab_term H1 H3); auto. inversion H4; subst; auto. *)
(* pick_fresh z. *)
(* assert (t^z-->[lex]t'^z). *)
(* apply H0 with z; auto. constructor 1; auto. constructor 1; auto. *)
(* apply notin_union in Fr; destruct Fr. *)
(* apply notin_union in H7; destruct H7. *)
(* apply notin_union in H7; destruct H7. *)
(* apply notin_union in H7; destruct H7. *)
(* apply notin_union in H7; destruct H7. *)
(* apply EE_clos_abs with L. *)
(* apply EE_lab_term with t0; auto. *)
(* intros. pose proof red_rename_lab_lex. *)
(* intros. apply red_rename_lab_lex with z; auto. *)
(* (* outer sub *) *)
(* apply EE_presv_lab_lex with (u := t[u]) (u' := t'[u]); auto. *)
(* pose proof (EE_lab_term H3 H4); auto. inversion H5; subst; auto. *)
(* pick_fresh z. *)
(* assert (t^z-->[lex]t'^z). *)
(* apply H1 with z; auto. constructor 1; auto. constructor 1; auto. *)
(* apply notin_union in Fr; destruct Fr. *)
(* apply notin_union in H7; destruct H7. *)
(* apply notin_union in H7; destruct H7. *)
(* apply notin_union in H7; destruct H7. *)
(* apply notin_union in H7; destruct H7. *)
(* apply notin_union in H7; destruct H7. *)
(* apply EE_clos_outer_sub with L; auto. *)
(* intros. apply red_rename_lab_lex with z; auto. *)
(* (* inner sub *) *)
(* apply EE_presv_lab_lex with (u := t[u]) (u' := t[u']); auto. *)
(* assert (u' =EE u'). constructor 1; auto. *)
(* assert (u =EE u). constructor 1; auto. *)
(* apply EE_lab_term in H3. inversion H3; subst. *)
(* pose proof (IHext_lab_contextual_closure u' H4 u H9 H5). *)
(* apply EE_clos_inner_sub; auto. *)
(* auto. *)
(* (* outer lsub *) *)
(* apply EE_presv_lab_lex with (u := t[[u]]) (u' := t'[[u]]); auto. *)
(* pose proof (EE_lab_term H3 H4); auto. inversion H5; subst; auto. *)
(* pick_fresh z. *)
(* assert (t^z-->[lex]t'^z). *)
(* apply H1 with z; auto. constructor 1; auto. constructor 1; auto. *)
(* apply notin_union in Fr; destruct Fr. *)
(* apply notin_union in H7; destruct H7. *)
(* apply notin_union in H7; destruct H7. *)
(* apply notin_union in H7; destruct H7. *)
(* apply notin_union in H7; destruct H7. *)
(* apply notin_union in H7; destruct H7. *)
(* apply EE_clos_outer_lsub with L; auto. *)
(* intros. apply red_rename_lab_lex with z; auto. *)
(* (*[> ------------------------------------------------------- Externa <]*) *)
(* (*[> Base <]*) *)
(* exists (t) (s). split*. split*. *)
(* inversion H; subst. *)
(* constructor 1. constructor 1; auto. *)
(* constructor 1. constructor 2; auto. *)
(* (* app_left *) *)
(* apply EE_presv_lab_lex with (u := (pterm_app t u)) (u' := (pterm_app t' u)); auto. *)
(* apply EE_clos_app_left. *)
(* pose proof (EE_lab_term H0 H3); auto. *)
(* apply IHext_lab_contextual_closure; auto. constructor 1; auto. *)
(* pose proof (EE_lab_term H0 H3); auto. inversion H4; subst; auto. *)
(* constructor 1; auto. *)
(* (* app_right *) *)
(* apply EE_presv_lab_lex with (u := (pterm_app t u)) (u' := (pterm_app t u')); auto. *)
(* apply EE_clos_app_right. *)
(* pose proof (EE_lab_term H0 H3); auto. *)
(* apply IHext_lab_contextual_closure; auto. constructor 1; auto. *)
(* pose proof (EE_lab_term H0 H3); auto. inversion H4; subst; auto. *)
(* constructor 1; auto. *)
(* (* abs *) *)
(* apply EE_presv_lab_lex with (u := pterm_abs t) (u' := pterm_abs t'); auto. *)
(* pose proof (EE_lab_term H1 H3); auto. inversion H4; subst; auto. *)
(* pick_fresh z. *)
(* assert (t^z-->[lex]t'^z). *)
(* apply H0 with z; auto. constructor 1; auto. constructor 1; auto. *)
(* apply notin_union in Fr; destruct Fr. *)
(* apply notin_union in H7; destruct H7. *)
(* apply notin_union in H7; destruct H7. *)
(* apply notin_union in H7; destruct H7. *)
(* apply notin_union in H7; destruct H7. *)
(* apply EE_clos_abs with L. *)
(* apply EE_lab_term with t0; auto. *)
(* intros. pose proof red_rename_lab_lex. *)
(* intros. apply red_rename_lab_lex with z; auto. *)
(* (* outer sub *) *)
(* apply EE_presv_lab_lex with (u := t[u]) (u' := t'[u]); auto. *)
(* pose proof (EE_lab_term H3 H4); auto. inversion H5; subst; auto. *)
(* pick_fresh z. *)
(* assert (t^z-->[lex]t'^z). *)
(* apply H1 with z; auto. constructor 1; auto. constructor 1; auto. *)
(* apply notin_union in Fr; destruct Fr. *)
(* apply notin_union in H7; destruct H7. *)
(* apply notin_union in H7; destruct H7. *)
(* apply notin_union in H7; destruct H7. *)
(* apply notin_union in H7; destruct H7. *)
(* apply notin_union in H7; destruct H7. *)
(* apply EE_clos_outer_sub with L; auto. *)
(* intros. apply red_rename_lab_lex with z; auto. *)
(* (* inner sub *) *)
(* apply EE_presv_lab_lex with (u := t[u]) (u' := t[u']); auto. *)
(* assert (u' =EE u'). constructor 1; auto. *)
(* assert (u =EE u). constructor 1; auto. *)
(* apply EE_lab_term in H3. inversion H3; subst. *)
(* pose proof (IHext_lab_contextual_closure u' H4 u H9 H5). *)
(* apply EE_clos_inner_sub; auto. *)
(* auto. *)
(* (* outer lsub *) *)
(* apply EE_presv_lab_lex with (u := t[[u]]) (u' := t'[[u]]); auto. *)
(* pose proof (EE_lab_term H3 H4); auto. inversion H5; subst; auto. *)
(* pick_fresh z. *)
(* assert (t^z-->[lex]t'^z). *)
(* apply H1 with z; auto. constructor 1; auto. constructor 1; auto. *)
(* apply notin_union in Fr; destruct Fr. *)
(* apply notin_union in H7; destruct H7. *)
(* apply notin_union in H7; destruct H7. *)
(* apply notin_union in H7; destruct H7. *)
(* apply notin_union in H7; destruct H7. *)
(* apply notin_union in H7; destruct H7. *)
(* apply EE_clos_outer_lsub with L; auto. *)
(* intros. apply red_rename_lab_lex with z; auto. *)
(* Qed. *)
Theorem lab_ex_eq_i_e: forall t t', lab_term t -> (t -->[lex] t' <-> (t -->[lx_i] t' \/ t -->[lx_e] t')).
Proof.
split.
intros; apply lab_ex_impl_i_e; auto.
intros; apply lab_ie_impl_ex; auto.
Qed.
(** Parei aqui (Flávio) *)
(* Lemma swap_lab_term : forall x y S t, x \in S -> y \in S -> *)
(* lab_term S t -> lab_term S ([(x,y)]t). *)
(* Proof. *)
(* intros x y S t Hx Hy H. induction H; simpl. *)
(* case (x0 == x); case (x0 == y); intros; apply lab_term_var. *)
(* apply lab_term_app; [apply IHlab_term1; trivial | apply IHlab_term2; trivial]. *)
(* apply lab_term_abs with (L := L \u {{x}} \u {{y}}). intros z Hz. *)
(* apply notin_union in Hz. destruct Hz. apply notin_union in H1. destruct H1. *)
(* apply notin_singleton in H2. apply notin_singleton in H3. *)
(* rewrite open_swap; trivial. apply H0; trivial; apply in_union; left; trivial. *)
(* apply lab_term_sub with (L := L \u {{x}} \u {{y}}); *)
(* try apply IHlab_term; trivial; intros z Hz. *)
(* apply notin_union in Hz. destruct Hz. apply notin_union in H2. destruct H2. *)
(* apply notin_singleton in H3. apply notin_singleton in H4. *)
(* rewrite open_swap; trivial. apply H0; trivial; apply in_union; left; trivial. *)
(* apply lab_term_sub' with (L := L \u {{x}} \u {{y}}); trivial. intros z Hz. *)
(* apply notin_union in Hz. destruct Hz. apply notin_union in H4. destruct H4. *)
(* apply notin_singleton in H5. apply notin_singleton in H6. *)
(* rewrite open_swap; trivial. apply H0; trivial; apply in_union; left; trivial. *)
(* apply swap_term; trivial. apply lex_SN_swap; trivial. *)
(* apply swap_fvar; trivial. *)
(* Qed. *)
(* Lemma swap_lab_term' : forall x y S t, x \in S -> y \in S -> *)
(* x \notin fv t -> y \notin fv t -> *)
(* lab_term S (t ^ x) -> lab_term S (t ^ y). *)
(* Proof. *)
(* intros. replace (t^y) with ([(x,y)](t^x)). apply swap_lab_term; trivial. *)
(* unfold open. rewrite open_swap_comm. rewrite swap_var_l. *)
(* rewrite swap_fresh; trivial. *)
(* Qed. *)
(* Lemma swap_lab_body : forall x y S t, x \in S -> y \in S -> *)
(* lab_body S t -> lab_body S ([(x,y)]t). *)
(* Proof. *)
(* intros. unfold lab_body in *|-*. case H1; clear H1; intros L H1. *)
(* exists (L \u {{x}} \u {{y}}). intros z Hz. *)
(* apply notin_union in Hz. destruct Hz. apply notin_union in H2. destruct H2. *)
(* apply notin_singleton in H3. apply notin_singleton in H4. rewrite open_swap; trivial. *)
(* apply swap_lab_term. apply in_union. left; trivial. apply in_union. left; trivial. *)
(* apply H1; trivial. *)
(* Qed. *)
(* Lemma bswap_rec_open : forall x k t, bswap_rec (S k) (t ^ x) = (bswap_rec (S k) t) ^ x. *)
(* Proof. *)
(* intros. *)
(* case var_fresh with (L := fv t \u {{x}}). intros y Fry. *)
(* case var_fresh with (L := fv t \u {{x}} \u {{y}}). intros z Frz. *)
(* apply notin_union in Fry. destruct Fry. apply notin_singleton in H0. *)
(* apply notin_union in Frz. destruct Frz. apply notin_union in H1. *)
(* destruct H1. apply notin_singleton in H2. apply notin_singleton in H3. *)
(* rewrite <- bswap_eq_openclose with (x := y) (y := z). unfold open. *)
(* rewrite subst_com with (i := S k) (j := 0); try omega; trivial. *)
(* rewrite subst_com with (i := S (S k)) (j := 0) ; try omega; trivial. *)
(* rewrite <- openclose_com; try omega; trivial. *)
(* rewrite <- openclose_com; try omega; trivial. *)
(* rewrite bswap_eq_openclose with (x := y) (y := z); trivial. *)
(* apply notin_union. split; trivial. apply notin_singleton; trivial. *)
(* intro; apply H0; rewrite H4; trivial. intro; apply H3; rewrite H4; trivial. *)
(* unfold open. rewrite fv_open_; trivial. apply notin_union. *)
(* split. unfold open. rewrite fv_open_; trivial. apply notin_singleton; trivial. *)
(* Qed. *)
(* (** Open a labeled locally closed term is the identity *) *)
(* Lemma open_rec_lab_term : forall S t u, lab_term S t -> forall k, t = {k ~> u}t. *)
(* Proof. *)
(* induction 1; intros; simpl; fequals*; unfolds open ; *)
(* case var_fresh with (L := L); intros x Fr; *)
(* try apply* (@open_rec_term_core t1 0 (pterm_fvar x)); try omega; *)
(* try apply H0; trivial. apply open_rec_term; trivial. *)
(* Qed. *)
(* Lemma red_regular_lab_x : forall t t', lab_sys_x t t' -> (exists S, lab_term S t /\ lab_term S t'). *)
(* Proof. *)
(* intros t t' H. induction H; exists S. *)
(* inversion H. split; trivial. apply term_to_lab_term; trivial. *)
(* split; trivial. *)
(* split. inversion H0. inversion H. apply lab_term_sub' with (L := L \u x); trivial. *)
(* intros z Fr; apply notin_union in Fr; destruct Fr. *)
(* unfold open. simpl. apply lab_term_app. apply H7; trivial. apply H3; trivial. *)
(* inversion H0. inversion H. apply lab_term_app. *)
(* apply lab_term_sub' with (L := x); trivial. apply lab_term_sub' with (L := L); trivial. *)
(* split; trivial. inversion H. clear H H0 H1 t1 t2. *)
(* apply lab_term_abs with (L := L \u (fv t)); intros x Hx. *)
(* apply notin_union in Hx. destruct Hx. *)
(* assert (Q : lab_term (S \u {{x}}) (pterm_abs t ^ x)). apply H2; trivial. clear H2 H L. *)
(* unfold open in *|-*. simpl. rewrite <- open_rec_term with (t := u); trivial. *)
(* inversion Q. clear Q H t1. *)
(* apply lab_term_sub' with (L := L \u (fv t)); trivial. intros y Hy. unfold open in *|-*. *)
(* apply notin_union in Hy. destruct Hy. *)
(* assert (Q : lab_term (S \u {{x}} \u {{y}}) ({0 ~> pterm_fvar y}({1 ~> pterm_fvar x}t))). *)
(* apply H1; trivial. clear H1 H L. *)
(* rewrite <- open_bswap; trivial. rewrite subst_com in Q; try omega; trivial. *)
(* rewrite swap_eq_open; trivial. apply swap_lab_term; trivial. *)
(* apply in_union. left. apply in_union. right. apply in_singleton; trivial. *)
(* apply in_union. right. apply in_singleton; trivial. *)
(* apply subset_trans with (F := S); trivial. *)
(* apply subset_union_weak_l. *)
(* split; trivial. inversion H. clear H H1 H2 t1 t2. *)
(* apply lab_term_sub with (L := L \u (fv t)). intros x Hx. *)
(* apply notin_union in Hx. destruct Hx. *)
(* assert (Q : lab_term (S \u {{x}}) ((t [u]) ^ x)). apply H3; trivial. clear H3 H L. *)
(* unfold open in *|-*. simpl in *|-*. rewrite <- open_rec_term with (t := v); trivial. *)
(* inversion Q. clear Q H H2 t1 t2. *)
(* apply lab_term_sub' with (L := L \u (fv t)); trivial. intros y Hy. unfold open in *|-*. *)
(* apply notin_union in Hy. destruct Hy. *)
(* assert (Q : lab_term (S \u {{x}} \u {{y}}) ({0 ~> pterm_fvar y}({1 ~> pterm_fvar x}t))). *)
(* apply H3; trivial. clear H3 H L. *)
(* rewrite <- open_bswap; trivial. rewrite subst_com in Q; try omega; trivial. *)
(* rewrite swap_eq_open; trivial. apply swap_lab_term; trivial. *)
(* apply in_union. left. apply in_union. right. apply in_singleton; trivial. *)
(* apply in_union. right. apply in_singleton; trivial. *)
(* apply subset_trans with (F := S); trivial. *)
(* apply subset_union_weak_l. *)
(* apply lab_term_sub' with (L := L); trivial; intros x Hx. *)
(* assert (Q : lab_term (S \u {{x}}) ((t [u]) ^ x)). apply H3; trivial. clear H3. *)
(* unfold open in Q. simpl in Q. inversion Q; trivial. *)
(* Qed. *)