X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Flibrary%2FFsub%2Fdefn.ma;h=0a95b31b5115ebb3082f1aba61f83250114d4b06;hb=ed936515481f5035fde443f4aee55b86e427cef4;hp=550f8271e236324182697029b591ac9f3d8a1d09;hpb=a180bddcd4a8f35de3d7292162ba05d0077723aa;p=helm.git diff --git a/helm/software/matita/library/Fsub/defn.ma b/helm/software/matita/library/Fsub/defn.ma index 550f8271e..0a95b31b5 100644 --- a/helm/software/matita/library/Fsub/defn.ma +++ b/helm/software/matita/library/Fsub/defn.ma @@ -21,7 +21,7 @@ include "list/list.ma". include "Fsub/util.ma". (*** representation of Fsub types ***) -inductive Typ : Type \def +inductive Typ : Set \def | TVar : nat \to Typ (* type var *) | TFree: nat \to Typ (* free type name *) | Top : Typ (* maximum type *) @@ -29,7 +29,7 @@ inductive Typ : Type \def | Forall : Typ \to Typ \to Typ. (* universal type *) (*** representation of Fsub terms ***) -inductive Term : Type \def +inductive Term : Set \def | Var : nat \to Term (* variable *) | Free : nat \to Term (* free name *) | Abs : Typ \to Term \to Term (* abstraction *) @@ -39,7 +39,7 @@ inductive Term : Type \def (* representation of bounds *) -record bound : Type \def { +record bound : Set \def { istype : bool; (* is subtyping bound? *) name : nat ; (* name *) btype : Typ (* type to which the name is bound *) @@ -257,23 +257,6 @@ inductive JType : Env \to Term \to Typ \to Prop \def | T_Sub : \forall G:Env.\forall t:Term.\forall T:Typ. \forall S:Typ.(JType G t S) \to (JSubtype G S T) \to (JType G t T). -(*** definitions about swaps ***) - -let rec swap_Typ u v T on T \def - match T with - [(TVar n) \Rightarrow (TVar n) - |(TFree X) \Rightarrow (TFree (swap u v X)) - |Top \Rightarrow Top - |(Arrow T1 T2) \Rightarrow (Arrow (swap_Typ u v T1) (swap_Typ u v T2)) - |(Forall T1 T2) \Rightarrow (Forall (swap_Typ u v T1) (swap_Typ u v T2))]. - -definition swap_bound : nat \to nat \to bound \to bound \def - \lambda u,v,b.match b with - [(mk_bound B X T) \Rightarrow (mk_bound B (swap u v X) (swap_Typ u v T))]. - -definition swap_Env : nat \to nat \to Env \to Env \def - \lambda u,v,G.(map ? ? (\lambda b.(swap_bound u v b)) G). - (****** PROOFS ********) lemma subst_O_nat : \forall T,U.((subst_type_O T U) = (subst_type_nat T U O)). @@ -304,12 +287,6 @@ qed. (* end of fixme *) -lemma var_notinbG_notinG : \forall G,x,b. - (\lnot (var_in_env x (b::G))) - \to \lnot (var_in_env x G). -intros 3.elim b.unfold.intro.elim H.unfold.simplify.constructor 2.exact H1. -qed. - lemma boundinenv_natinfv : \forall x,G. (\exists B,T.(in_list ? (mk_bound B x T) G)) \to (in_list ? x (fv_env G)). @@ -368,7 +345,24 @@ intros 2;elim G 0 [apply a3 |apply in_Skip;rewrite < H4;assumption]]]] qed. - + +theorem varinT_varinT_subst : \forall X,Y,T. + (in_list ? X (fv_type T)) \to \forall n. + (in_list ? X (fv_type (subst_type_nat T (TFree Y) n))). +intros 3;elim T + [simplify in H;elim (in_list_nil ? ? H) + |simplify in H;simplify;assumption + |simplify in H;elim (in_list_nil ? ? H) + |simplify in H2;simplify;elim (nat_in_list_case ? ? ? H2); + apply natinG_or_inH_to_natinGH; + [left;apply (H1 H3) + |right;apply (H H3)] + |simplify in H2;simplify;elim (nat_in_list_case ? ? ? H2); + apply natinG_or_inH_to_natinGH; + [left;apply (H1 H3); + |right;apply (H H3)]] +qed. + lemma incl_bound_fv : \forall l1,l2.(incl ? l1 l2) \to (incl ? (fv_env l1) (fv_env l2)). intros.unfold in H.unfold.intros.apply boundinenv_natinfv. @@ -379,14 +373,6 @@ lapply (natinfv_boundinenv ? ? H1).elim Hletin.elim H2.apply ex_intro |apply (H ? H3)]] qed. -(* lemma incl_cons : \forall x,l1,l2. - (incl bound l1 l2) \to (incl bound (x :: l1) (x :: l2)). -intros.unfold in H.unfold.intros.inversion H1 - [intros;lapply (inj_head ? ? ? ? H3);rewrite > Hletin;apply in_Base - |intros;apply in_Skip;apply H;lapply (inj_tail ? ? ? ? ? H5);rewrite > Hletin; - assumption] -qed. *) - lemma incl_nat_cons : \forall x,l1,l2. (incl nat l1 l2) \to (incl nat (x :: l1) (x :: l2)). intros.unfold in H.unfold.intros.inversion H1 @@ -395,83 +381,16 @@ intros.unfold in H.unfold.intros.inversion H1 assumption] qed. -lemma boundin_envappend_case : \forall G,H,b.(var_bind_in_env b (H @ G)) \to - (var_bind_in_env b G) \lor (var_bind_in_env b H). -intros 3.elim H - [simplify in H1;left;assumption - |unfold in H2;inversion H2 - [intros;simplify in H4;lapply (inj_head ? ? ? ? H4);rewrite > Hletin; - right;apply in_Base - |intros;simplify in H6;lapply (inj_tail ? ? ? ? ? H6);rewrite < Hletin in H3; - rewrite > H5 in H1;lapply (H1 H3);elim Hletin1 - [left;assumption|right;apply in_Skip;assumption]]] -qed. - -lemma varin_envappend_case: \forall G,H,x.(var_in_env x (H @ G)) \to - (var_in_env x G) \lor (var_in_env x H). -intros 3.elim H 0 - [simplify;intro;left;assumption - |intros 2;elim t;simplify in H2;inversion H2 - [intros;lapply (inj_head_nat ? ? ? ? H4);rewrite > Hletin;right; - simplify;constructor 1 - |intros;lapply (inj_tail ? ? ? ? ? H6); - lapply H1 - [rewrite < H5;elim Hletin1 - [left;assumption|right;simplify;constructor 2;assumption] - |unfold var_in_env;unfold fv_env;rewrite > Hletin;rewrite > H5; - assumption]]] -qed. - -lemma boundinG_or_boundinH_to_boundinGH : \forall G,H,b. - (var_bind_in_env b G) \lor (var_bind_in_env b H) \to - (var_bind_in_env b (H @ G)). -intros.elim H1 - [elim H - [simplify;assumption - |simplify;apply in_Skip;assumption] - |generalize in match H2;elim H2 - [simplify;apply in_Base - |lapply (H4 H3);simplify;apply in_Skip;assumption]] -qed. - - -lemma varinG_or_varinH_to_varinGH : \forall G,H,x. - (var_in_env x G) \lor (var_in_env x H) \to - (var_in_env x (H @ G)). -intros.elim H1 0 - [elim H - [simplify;assumption - |elim t;simplify;constructor 2;apply (H2 H3)] - |elim H 0 - [simplify;intro;lapply (in_list_nil nat x H2);elim Hletin - |intros 2;elim t;simplify in H3;inversion H3 - [intros;lapply (inj_head_nat ? ? ? ? H5);rewrite > Hletin;simplify; - constructor 1 - |intros;simplify;constructor 2;rewrite < H6;apply H2; - lapply (inj_tail ? ? ? ? ? H7);rewrite > H6;unfold;unfold fv_env; - rewrite > Hletin;assumption]]] -qed. - -lemma varbind_to_append : \forall G,b.(var_bind_in_env b G) \to - \exists G1,G2.(G = (G2 @ (b :: G1))). -intros.generalize in match H.elim H - [apply ex_intro [apply l|apply ex_intro [apply Empty|reflexivity]] - |lapply (H2 H1);elim Hletin;elim H4;rewrite > H5; - apply ex_intro - [apply a2|apply ex_intro [apply (a1 :: a3)|simplify;reflexivity]]] -qed. - - lemma WFT_env_incl : \forall G,T.(WFType G T) \to \forall H.(incl ? (fv_env G) (fv_env H)) \to (WFType H T). -intros 4.generalize in match H1.elim H - [apply WFT_TFree;unfold in H3;apply (H3 ? H2) +intros 3.elim H + [apply WFT_TFree;unfold in H3;apply (H3 ? H1) |apply WFT_Top - |apply WFT_Arrow [apply (H3 ? H6)|apply (H5 ? H6)] + |apply WFT_Arrow [apply (H2 ? H6)|apply (H4 ? H6)] |apply WFT_Forall - [apply (H3 ? H6) - |intros;apply H5 - [unfold;intro;unfold in H7;apply H7;unfold in H6;apply(H6 ? H9) + [apply (H2 ? H6) + |intros;apply H4 + [unfold;intro;apply H7;apply(H6 ? H9) |assumption |simplify;apply (incl_nat_cons ? ? ? H6)]]] qed. @@ -501,110 +420,6 @@ intros 10;elim H rewrite > H7 in H1;apply in_Skip;apply (H1 H5 H3)]] qed. - -(*** theorems about swaps ***) - -lemma fv_subst_type_nat : \forall x,T,y,n.(in_list ? x (fv_type T)) \to - (in_list ? x (fv_type (subst_type_nat T (TFree y) n))). -intros 3;elim T 0 - [intros;simplify in H;elim (in_list_nil ? ? H) - |2,3:simplify;intros;assumption - |*:intros;simplify in H2;elim (nat_in_list_case ? ? ? H2) - [1,3:simplify;apply natinG_or_inH_to_natinGH;left;apply (H1 ? H3) - |*:simplify;apply natinG_or_inH_to_natinGH;right;apply (H ? H3)]] -qed. - -lemma fv_subst_type_O : \forall x,T,y.(in_list ? x (fv_type T)) \to - (in_list ? x (fv_type (subst_type_O T (TFree y)))). -intros;rewrite > subst_O_nat;apply (fv_subst_type_nat ? ? ? ? H); -qed. - -lemma swap_Typ_inv : \forall u,v,T.(swap_Typ u v (swap_Typ u v T)) = T. -intros;elim T - [1,3:simplify;reflexivity - |simplify;rewrite > swap_inv;reflexivity - |*:simplify;rewrite > H;rewrite > H1;reflexivity] -qed. - -lemma swap_Typ_not_free : \forall u,v,T.\lnot (in_list ? u (fv_type T)) \to - \lnot (in_list ? v (fv_type T)) \to (swap_Typ u v T) = T. -intros 3;elim T 0 - [1,3:intros;simplify;reflexivity - |simplify;intros;cut (n \neq u \land n \neq v) - [elim Hcut;rewrite > (swap_other ? ? ? H2 H3);reflexivity - |split - [unfold;intro;apply H;rewrite > H2;apply in_Base - |unfold;intro;apply H1;rewrite > H2;apply in_Base]] - |*:simplify;intros;cut ((\lnot (in_list ? u (fv_type t)) \land - \lnot (in_list ? u (fv_type t1))) \land - (\lnot (in_list ? v (fv_type t)) \land - \lnot (in_list ? v (fv_type t1)))) - [1,3:elim Hcut;elim H4;elim H5;clear Hcut H4 H5;rewrite > (H H6 H8); - rewrite > (H1 H7 H9);reflexivity - |*:split - [1,3:split;unfold;intro;apply H2;apply natinG_or_inH_to_natinGH;autobatch - |*:split;unfold;intro;apply H3;apply natinG_or_inH_to_natinGH;autobatch]]] -qed. - -lemma subst_type_nat_swap : \forall u,v,T,X,m. - (swap_Typ u v (subst_type_nat T (TFree X) m)) = - (subst_type_nat (swap_Typ u v T) (TFree (swap u v X)) m). -intros 4;elim T - [simplify;elim (eqb_case n m);rewrite > H;simplify;reflexivity - |2,3:simplify;reflexivity - |*:simplify;rewrite > H;rewrite > H1;reflexivity] -qed. - -lemma subst_type_O_swap : \forall u,v,T,X. - (swap_Typ u v (subst_type_O T (TFree X))) = - (subst_type_O (swap_Typ u v T) (TFree (swap u v X))). -intros 4;rewrite > (subst_O_nat (swap_Typ u v T));rewrite > (subst_O_nat T); -apply subst_type_nat_swap; -qed. - -lemma in_fv_type_swap : \forall u,v,x,T.((in_list ? x (fv_type T)) \to - (in_list ? (swap u v x) (fv_type (swap_Typ u v T)))) \land - ((in_list ? (swap u v x) (fv_type (swap_Typ u v T))) \to - (in_list ? x (fv_type T))). -intros;split - [elim T 0 - [1,3:simplify;intros;elim (in_list_nil ? ? H) - |simplify;intros;cut (x = n) - [rewrite > Hcut;apply in_Base - |inversion H - [intros;lapply (inj_head_nat ? ? ? ? H2);rewrite > Hletin; - reflexivity - |intros;lapply (inj_tail ? ? ? ? ? H4);rewrite < Hletin in H1; - elim (in_list_nil ? ? H1)]] - |*:simplify;intros;elim (nat_in_list_case ? ? ? H2) - [1,3:apply natinG_or_inH_to_natinGH;left;apply (H1 H3) - |*:apply natinG_or_inH_to_natinGH;right;apply (H H3)]] - |elim T 0 - [1,3:simplify;intros;elim (in_list_nil ? ? H) - |simplify;intros;cut ((swap u v x) = (swap u v n)) - [lapply (swap_inj ? ? ? ? Hcut);rewrite > Hletin;apply in_Base - |inversion H - [intros;lapply (inj_head_nat ? ? ? ? H2);rewrite > Hletin; - reflexivity - |intros;lapply (inj_tail ? ? ? ? ? H4);rewrite < Hletin in H1; - elim (in_list_nil ? ? H1)]] - |*:simplify;intros;elim (nat_in_list_case ? ? ? H2) - [1,3:apply natinG_or_inH_to_natinGH;left;apply (H1 H3) - |*:apply natinG_or_inH_to_natinGH;right;apply (H H3)]]] -qed. - -lemma lookup_swap : \forall x,u,v,T,B,G.(in_list ? (mk_bound B x T) G) \to - (in_list ? (mk_bound B (swap u v x) (swap_Typ u v T)) (swap_Env u v G)). -intros 6;elim G 0 - [intros;elim (in_list_nil ? ? H) - |intro;elim t;simplify;inversion H1 - [intros;lapply (inj_head ? ? ? ? H3);rewrite < H2 in Hletin; - destruct Hletin;rewrite > Hcut;rewrite > Hcut1;rewrite > Hcut2; - apply in_Base - |intros;lapply (inj_tail ? ? ? ? ? H5);rewrite < Hletin in H2; - rewrite < H4 in H2;apply in_Skip;apply (H H2)]] -qed. - lemma in_FV_subst : \forall x,T,U,n.(in_list ? x (fv_type T)) \to (in_list ? x (fv_type (subst_type_nat T U n))). intros 3;elim T @@ -620,27 +435,6 @@ intros 3;elim T |*:right;apply (H ? H3)]] qed. -lemma in_dom_swap : \forall u,v,x,G. - ((in_list ? x (fv_env G)) \to - (in_list ? (swap u v x) (fv_env (swap_Env u v G)))) \land - ((in_list ? (swap u v x) (fv_env (swap_Env u v G))) \to - (in_list ? x (fv_env G))). -intros;split - [elim G 0 - [simplify;intro;elim (in_list_nil ? ? H) - |intro;elim t 0;simplify;intros;inversion H1 - [intros;lapply (inj_head_nat ? ? ? ? H3);rewrite > Hletin;apply in_Base - |intros;lapply (inj_tail ? ? ? ? ? H5);rewrite < Hletin in H2; - rewrite > H4 in H;apply in_Skip;apply (H H2)]] - |elim G 0 - [simplify;intro;elim (in_list_nil ? ? H) - |intro;elim t 0;simplify;intros;inversion H1 - [intros;lapply (inj_head_nat ? ? ? ? H3);rewrite < H2 in Hletin; - lapply (swap_inj ? ? ? ? Hletin);rewrite > Hletin1;apply in_Base - |intros;lapply (inj_tail ? ? ? ? ? H5);rewrite < Hletin in H2; - rewrite > H4 in H;apply in_Skip;apply (H H2)]]] -qed. - (*** lemma on fresh names ***) lemma fresh_name : \forall l:(list nat).\exists n.\lnot (in_list ? n l). @@ -696,7 +490,7 @@ cut (\forall l:(list nat).\exists n.\forall m. |elim (leb a t);autobatch]]]] qed. -(*** lemmas on well-formedness ***) +(*** lemmata on well-formedness ***) lemma fv_WFT : \forall T,x,G.(WFType G T) \to (in_list ? x (fv_type T)) \to (in_list ? x (fv_env G)). @@ -736,80 +530,6 @@ intros 4.elim H |apply (H2 H6)]] qed. -lemma WFE_consG_to_WFT : \forall G.\forall b,X,T. - (WFEnv ((mk_bound b X T)::G)) \to (WFType G T). -intros. -inversion H - [intro;reduce in H1;destruct H1 - |intros;lapply (inj_head ? ? ? ? H5);lapply (inj_tail ? ? ? ? ? H5); - destruct Hletin;rewrite > Hletin1;rewrite > Hcut2;assumption] -qed. - -lemma WFE_consG_WFE_G : \forall G.\forall b. - (WFEnv (b::G)) \to (WFEnv G). -intros. -inversion H - [intro;reduce in H1;destruct H1 - |intros;lapply (inj_tail ? ? ? ? ? H5);rewrite > Hletin;assumption] -qed. - -(* silly, but later useful *) - -lemma env_append_weaken : \forall G,H.(WFEnv (H @ G)) \to - (incl ? G (H @ G)). -intros 2;elim H - [simplify;unfold;intros;assumption - |simplify in H2;simplify;unfold;intros;apply in_Skip;apply H1 - [apply (WFE_consG_WFE_G ? ? H2) - |assumption]] -qed. - -lemma WFT_swap : \forall u,v,G,T.(WFType G T) \to - (WFType (swap_Env u v G) (swap_Typ u v T)). -intros.elim H - [simplify;apply WFT_TFree;lapply (natinfv_boundinenv ? ? H1);elim Hletin; - elim H2;apply boundinenv_natinfv;apply ex_intro - [apply a - |apply ex_intro - [apply (swap_Typ u v a1) - |apply lookup_swap;assumption]] - |simplify;apply WFT_Top - |simplify;apply WFT_Arrow - [assumption|assumption] - |simplify;apply WFT_Forall - [assumption - |intros;rewrite < (swap_inv u v); - cut (\lnot (in_list ? (swap u v X) (fv_type t1))) - [cut (\lnot (in_list ? (swap u v X) (fv_env e))) - [generalize in match (H4 ? Hcut1 Hcut);simplify; - rewrite > subst_type_O_swap;intro;assumption - |lapply (in_dom_swap u v (swap u v X) e);elim Hletin;unfold; - intros;lapply (H7 H9);rewrite > (swap_inv u v) in Hletin1; - apply (H5 Hletin1)] - |generalize in match (in_fv_type_swap u v (swap u v X) t1);intros; - elim H7;unfold;intro;lapply (H8 H10); - rewrite > (swap_inv u v) in Hletin;apply (H6 Hletin)]]] -qed. - -lemma WFE_swap : \forall u,v,G.(WFEnv G) \to (WFEnv (swap_Env u v G)). -intros 3.elim G 0 - [intro;simplify;assumption - |intros 2;elim t;simplify;constructor 2 - [apply H;apply (WFE_consG_WFE_G ? ? H1) - |unfold;intro;lapply (in_dom_swap u v n l);elim Hletin;lapply (H4 H2); - (* FIXME trick *)generalize in match H1;intro;inversion H1 - [intros;absurd ((mk_bound b n t1)::l = []) - [assumption|apply nil_cons] - |intros;lapply (inj_head ? ? ? ? H10);lapply (inj_tail ? ? ? ? ? H10); - destruct Hletin2;rewrite < Hcut1 in H8;rewrite < Hletin3 in H8; - apply (H8 Hletin1)] - |apply (WFT_swap u v l t1);inversion H1 - [intro;absurd ((mk_bound b n t1)::l = []) - [assumption|apply nil_cons] - |intros;lapply (inj_head ? ? ? ? H6);lapply (inj_tail ? ? ? ? ? H6); - destruct Hletin;rewrite > Hletin1;rewrite > Hcut2;assumption]]] -qed. - (*** some exotic inductions and related lemmas ***) lemma not_t_len_lt_SO : \forall T.\lnot (t_len T) < (S O). @@ -820,15 +540,6 @@ intros;elim T |*:simplify in H2;apply H;apply (trans_lt ? ? ? ? H2);unfold;constructor 1]] qed. -lemma t_len_gt_O : \forall T.(t_len T) > O. -intro;elim T - [1,2,3:simplify;unfold;unfold;constructor 1 - |*:simplify;lapply (max_case (t_len t) (t_len t1));rewrite > Hletin; - elim (leb (t_len t) (t_len t1)) - [1,3:simplify;unfold;unfold;constructor 2;unfold in H1;unfold in H1;assumption - |*:simplify;unfold;unfold;constructor 2;unfold in H;unfold in H;assumption]] -qed. - lemma Typ_len_ind : \forall P:Typ \to Prop. (\forall U.(\forall V.((t_len V) < (t_len U)) \to (P V)) \to (P U)) @@ -923,47 +634,7 @@ intro.elim T rewrite < Hletin1;reflexivity] qed. -lemma swap_env_not_free : \forall u,v,G.(WFEnv G) \to - \lnot (in_list ? u (fv_env G)) \to - \lnot (in_list ? v (fv_env G)) \to - (swap_Env u v G) = G. -intros 3.elim G 0 - [simplify;intros;reflexivity - |intros 2;elim t 0;simplify;intros;lapply (notin_cons ? ? ? ? H2); - lapply (notin_cons ? ? ? ? H3);elim Hletin;elim Hletin1; - lapply (swap_other ? ? ? H4 H6);lapply (WFE_consG_to_WFT ? ? ? ? H1); - cut (\lnot (in_list ? u (fv_type t1))) - [cut (\lnot (in_list ? v (fv_type t1))) - [lapply (swap_Typ_not_free ? ? ? Hcut Hcut1); - lapply (WFE_consG_WFE_G ? ? H1); - lapply (H Hletin5 H5 H7); - rewrite > Hletin2;rewrite > Hletin4;rewrite > Hletin6;reflexivity - |unfold;intro;apply H7; - apply (fv_WFT ? ? ? Hletin3 H8)] - |unfold;intro;apply H5;apply (fv_WFT ? ? ? Hletin3 H8)]] -qed. - -(*** alternate "constructor" for universal types' well-formedness ***) - -lemma WFT_Forall2 : \forall G,X,T,T1,T2. - (WFEnv G) \to - (WFType G T1) \to - \lnot (in_list ? X (fv_type T2)) \to - \lnot (in_list ? X (fv_env G)) \to - (WFType ((mk_bound true X T)::G) - (subst_type_O T2 (TFree X))) \to - (WFType G (Forall T1 T2)). -intros.apply WFT_Forall - [assumption - |intros;generalize in match (WFT_swap X X1 ? ? H4);simplify; - rewrite > swap_left; - rewrite > (swap_env_not_free X X1 G H H3 H5); - rewrite > subst_type_O_swap;rewrite > swap_left; - rewrite > (swap_Typ_not_free ? ? T2 H2 H6); - intro;apply (WFT_env_incl ? ? H7);unfold;simplify;intros;assumption] -qed. - -(*** lemmas relating subtyping and well-formedness ***) +(*** lemmata relating subtyping and well-formedness ***) lemma JS_to_WFE : \forall G,T,U.(JSubtype G T U) \to (WFEnv G). intros;elim H;assumption. @@ -980,22 +651,10 @@ intros;elim H |elim H3;assumption] |elim H2;elim H4;split;apply WFT_Arrow;assumption |elim H2;split - [lapply (fresh_name ((fv_env e) @ (fv_type t1))); - elim Hletin;cut ((\lnot (in_list ? a (fv_env e))) \land - (\lnot (in_list ? a (fv_type t1)))) - [elim Hcut;apply (WFT_Forall2 ? a t2 ? ? (JS_to_WFE ? ? ? H1) H6 H9 H8); - lapply (H4 ? H8);elim Hletin1;assumption - |split;unfold;intro;apply H7;apply natinG_or_inH_to_natinGH - [right;assumption - |left;assumption]] - |lapply (fresh_name ((fv_env e) @ (fv_type t3))); - elim Hletin;cut ((\lnot (in_list ? a (fv_env e))) \land - (\lnot (in_list ? a (fv_type t3)))) - [elim Hcut;apply (WFT_Forall2 ? a t2 ? ? (JS_to_WFE ? ? ? H1) H5 H9 H8); - lapply (H4 ? H8);elim Hletin1;assumption - |split;unfold;intro;apply H7;apply natinG_or_inH_to_natinGH - [right;assumption - |left;assumption]]]] + [apply (WFT_Forall ? ? ? H6);intros;elim (H4 X H7); + apply (WFT_env_incl ? ? H9);simplify;unfold;intros;assumption + |apply (WFT_Forall ? ? ? H5);intros;elim (H4 X H7); + apply (WFT_env_incl ? ? H10);simplify;unfold;intros;assumption]] qed. lemma JS_to_WFT1 : \forall G,T,U.(JSubtype G T U) \to (WFType G T). @@ -1006,113 +665,6 @@ lemma JS_to_WFT2 : \forall G,T,U.(JSubtype G T U) \to (WFType G U). intros;lapply (JS_to_WFT ? ? ? H);elim Hletin;assumption. qed. -(*** lemma relating subtyping and swaps ***) - -lemma JS_swap : \forall u,v,G,T,U.(JSubtype G T U) \to - (JSubtype (swap_Env u v G) (swap_Typ u v T) (swap_Typ u v U)). -intros 6.elim H - [simplify;apply SA_Top - [apply WFE_swap;assumption - |apply WFT_swap;assumption] - |simplify;apply SA_Refl_TVar - [apply WFE_swap;assumption - |unfold in H2;unfold;lapply (in_dom_swap u v n e);elim Hletin; - apply (H3 H2)] - |simplify;apply SA_Trans_TVar - [apply (swap_Typ u v t1) - |apply lookup_swap;assumption - |assumption] - |simplify;apply SA_Arrow;assumption - |simplify;apply SA_All - [assumption - |intros;lapply (H4 (swap u v X)) - [simplify in Hletin;rewrite > subst_type_O_swap in Hletin; - rewrite > subst_type_O_swap in Hletin;rewrite > swap_inv in Hletin; - assumption - |unfold;intro;apply H5;unfold; - lapply (in_dom_swap u v (swap u v X) e); - elim Hletin;rewrite > swap_inv in H7;apply H7;assumption]]] -qed. - -lemma fresh_WFT : \forall x,G,T.(WFType G T) \to \lnot (in_list ? x (fv_env G)) - \to \lnot (in_list ? x (fv_type T)). -intros;unfold;intro;apply H1;apply (fv_WFT ? ? ? H H2); -qed. - -lemma fresh_subst_type_O : \forall x,T1,B,G,T,y. - (WFType ((mk_bound B x T1)::G) (subst_type_O T (TFree x))) \to - \lnot (in_list ? y (fv_env G)) \to (x \neq y) \to - \lnot (in_list ? y (fv_type T)). -intros;unfold;intro; -cut (in_list ? y (fv_env ((mk_bound B x T1) :: G))) - [simplify in Hcut;inversion Hcut - [intros;apply H2;lapply (inj_head_nat ? ? ? ? H5);rewrite < H4 in Hletin; - assumption - |intros;apply H1;rewrite > H6;lapply (inj_tail ? ? ? ? ? H7); - rewrite > Hletin;assumption] - |apply (fv_WFT (subst_type_O T (TFree x)) ? ? H); - apply fv_subst_type_O;assumption] -qed. - -(*** alternate "constructor" for subtyping between universal types ***) - -lemma SA_All2 : \forall G,S1,S2,T1,T2,X.(JSubtype G T1 S1) \to - \lnot (in_list ? X (fv_env G)) \to - \lnot (in_list ? X (fv_type S2)) \to - \lnot (in_list ? X (fv_type T2)) \to - (JSubtype ((mk_bound true X T1) :: G) - (subst_type_O S2 (TFree X)) - (subst_type_O T2 (TFree X))) \to - (JSubtype G (Forall S1 S2) (Forall T1 T2)). -intros;apply (SA_All ? ? ? ? ? H);intros; -lapply (decidable_eq_nat X X1);elim Hletin - [rewrite < H6;assumption - |elim (JS_to_WFT ? ? ? H);elim (JS_to_WFT ? ? ? H4); - cut (\lnot (in_list ? X1 (fv_type S2))) - [cut (\lnot (in_list ? X1 (fv_type T2))) - [cut (((mk_bound true X1 T1)::G) = - (swap_Env X X1 ((mk_bound true X T1)::G))) - [rewrite > Hcut2; - cut (((subst_type_O S2 (TFree X1)) = - (swap_Typ X X1 (subst_type_O S2 (TFree X)))) \land - ((subst_type_O T2 (TFree X1)) = - (swap_Typ X X1 (subst_type_O T2 (TFree X))))) - [elim Hcut3;rewrite > H11;rewrite > H12;apply JS_swap; - assumption - |split - [rewrite > (subst_type_O_swap X X1 S2 X); - rewrite > (swap_Typ_not_free X X1 S2 H2 Hcut); - rewrite > swap_left;reflexivity - |rewrite > (subst_type_O_swap X X1 T2 X); - rewrite > (swap_Typ_not_free X X1 T2 H3 Hcut1); - rewrite > swap_left;reflexivity]] - |simplify;lapply (JS_to_WFE ? ? ? H); - rewrite > (swap_env_not_free X X1 G Hletin1 H1 H5); - cut ((\lnot (in_list ? X (fv_type T1))) \land - (\lnot (in_list ? X1 (fv_type T1)))) - [elim Hcut2;rewrite > (swap_Typ_not_free X X1 T1 H11 H12); - rewrite > swap_left;reflexivity - |split - [unfold;intro;apply H1;apply (fv_WFT T1 X G H7 H11) - |unfold;intro;apply H5;apply (fv_WFT T1 X1 G H7 H11)]]] - |unfold;intro;apply H5;lapply (fv_WFT ? X1 ? H10) - [inversion Hletin1 - [intros;simplify in H13;lapply (inj_head_nat ? ? ? ? H13); - rewrite < H12 in Hletin2;lapply (H6 Hletin2);elim Hletin3 - |intros;simplify in H15;lapply (inj_tail ? ? ? ? ? H15); - rewrite < Hletin2 in H12;rewrite < H14 in H12;lapply (H5 H12); - elim Hletin3] - |rewrite > subst_O_nat;apply in_FV_subst;assumption]] - |unfold;intro;apply H5;lapply (fv_WFT ? X1 ? H9) - [inversion Hletin1 - [intros;simplify in H13;lapply (inj_head_nat ? ? ? ? H13); - rewrite < H12 in Hletin2;lapply (H6 Hletin2);elim Hletin3 - |intros;simplify in H15;lapply (inj_tail ? ? ? ? ? H15); - rewrite < Hletin2 in H12;rewrite < H14 in H12;lapply (H5 H12); - elim Hletin3] - |rewrite > subst_O_nat;apply in_FV_subst;assumption]]] -qed. - lemma WFE_Typ_subst : \forall H,x,B,C,T,U,G. (WFEnv (H @ ((mk_bound B x T) :: G))) \to (WFType G U) \to (WFEnv (H @ ((mk_bound C x U) :: G))). @@ -1140,20 +692,6 @@ intros 7;elim H 0 unfold;intros;assumption]]] qed. -lemma t_len_pred: \forall T,m.(S (t_len T)) \leq m \to (t_len T) \leq (pred m). -intros 2;elim m - [elim (not_le_Sn_O ? H) - |simplify;apply (le_S_S_to_le ? ? H1)] -qed. - -lemma pred_m_lt_m : \forall m,T.(t_len T) \leq m \to (pred m) < m. -intros 2;elim m 0 - [elim T - [4,5:simplify in H2;elim (not_le_Sn_O ? H2) - |*:simplify in H;elim (not_le_Sn_n ? H)] - |intros;simplify;unfold;constructor 1] -qed. - lemma WFE_bound_bound : \forall B,x,T,U,G. (WFEnv G) \to (in_list ? (mk_bound B x T) G) \to (in_list ? (mk_bound B x U) G) \to T = U. @@ -1180,5 +718,4 @@ intros 6;elim H [apply B|apply ex_intro [apply U|assumption]]] |intros;apply (H2 ? H7);rewrite > H14;lapply (inj_tail ? ? ? ? ? H15); rewrite > Hletin1;assumption]]] -qed. - +qed. \ No newline at end of file