From: Wilmer Ricciotti Date: Thu, 21 Jun 2007 16:33:15 +0000 (+0000) Subject: PoplMark challenge part 1a: new, shorter version w/o equivariance proofs. X-Git-Tag: 0.4.95@7852~393 X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=commitdiff_plain;h=d93aae3c356d9f74876374443a8ee79545486cf4;p=helm.git PoplMark challenge part 1a: new, shorter version w/o equivariance proofs. --- diff --git a/matita/library/Fsub/defn.ma b/matita/library/Fsub/defn.ma index 550f8271e..0a95b31b5 100644 --- a/matita/library/Fsub/defn.ma +++ b/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 diff --git a/matita/library/Fsub/part1a.ma b/matita/library/Fsub/part1a.ma index 55d97c266..e1026f3d7 100644 --- a/matita/library/Fsub/part1a.ma +++ b/matita/library/Fsub/part1a.ma @@ -42,36 +42,46 @@ apply Typ_len_ind;intro;elim U |apply H2 [apply t_len_arrow2 |*:assumption]] - |(*FIXME*)generalize in match H3;intro;inversion H3 + |(* no shortcut? *) + (*FIXME*)generalize in match H3;intro;inversion H3 [intros;destruct H8 |intros;destruct H7 |intros;destruct H11;rewrite > Hcut;rewrite > Hcut1;split;assumption |intros;destruct H11]] - |elim (fresh_name ((fv_type t1) @ (fv_env G))); - cut ((\lnot (in_list ? a (fv_type t1))) \land - (\lnot (in_list ? a (fv_env G)))) - [elim Hcut;cut (WFType G t) - [apply (SA_All2 ? ? ? ? ? a ? H7 H6 H6) - [apply H2 - [apply t_len_forall1 - |*:assumption] - |apply H2 - [rewrite > subst_O_nat;rewrite < eq_t_len_TFree_subst; - apply t_len_forall2 - |(*FIXME*)generalize in match H3;intro;inversion H3 - [intros;destruct H11 - |intros;destruct H10 - |intros;destruct H14 - |intros;destruct H14;rewrite < Hcut2 in H11; - rewrite < Hcut3 in H11;rewrite < H13;rewrite < H13 in H11; - apply (H11 ? H7 H6)] - |apply WFE_cons;assumption]] - |(*FIXME*)generalize in match H3;intro;inversion H3 - [intros;destruct H11 - |intros;destruct H10 - |intros;destruct H14 - |intros;destruct H14;rewrite > Hcut1;assumption]] - |split;unfold;intro;apply H5;apply natinG_or_inH_to_natinGH;autobatch]] + |cut (WFType G t) + [lapply (H2 t ? ? Hcut H4) + [apply t_len_forall1 + |apply (SA_All ? ? ? ? ? Hletin);intros;apply H2 + [rewrite > subst_O_nat;rewrite < eq_t_len_TFree_subst; + apply t_len_forall2 + |generalize in match H3;intro;inversion H3 + [intros;destruct H9 + |intros;destruct H8 + |intros;destruct H12 + |intros;destruct H12;subst;apply H9 + [assumption + |unfold;intro;apply H5; + elim (fresh_name ((fv_env e)@(fv_type t3))); + cut ((\lnot (in_list ? a (fv_env e))) \land + (\lnot (in_list ? a (fv_type t3)))) + [elim Hcut1;lapply (H9 ? H13 H14); + lapply (fv_WFT ? X ? Hletin1) + [simplify in Hletin2;inversion Hletin2 + [intros;lapply (inj_head_nat ? ? ? ? H16);subst; + elim (H14 H11) + |intros;lapply (inj_tail ? ? ? ? ? H18); + rewrite < Hletin3 in H15;assumption] + |rewrite >subst_O_nat;apply varinT_varinT_subst; + assumption] + |split;unfold;intro;apply H12;apply natinG_or_inH_to_natinGH + [right;assumption + |left;assumption]]]] + |apply WFE_cons;assumption]] + |(*FIXME*)generalize in match H3;intro;inversion H3 + [intros;destruct H8 + |intros;destruct H7 + |intros;destruct H11 + |intros;destruct H11;subst;assumption]]] qed. (* @@ -95,8 +105,7 @@ intros 4;elim H [unfold;intro;apply H8;lapply (incl_bound_fv ? ? H7);apply (Hletin1 ? H9) |apply WFE_cons [1,2:assumption - |lapply (incl_bound_fv ? ? H7);apply (WFT_env_incl ? ? ? ? Hletin1); - apply (JS_to_WFT1 ? ? ? H1)] + |apply (JS_to_WFT1 ? ? ? Hletin)] |unfold;intros;inversion H9 [intros;lapply (inj_head ? ? ? ? H11);rewrite > Hletin1;apply in_Base |intros;lapply (inj_tail ? ? ? ? ? H13);rewrite < Hletin1 in H10; @@ -105,104 +114,103 @@ qed. (* Lemma A.3 (Transitivity and Narrowing) *) -lemma JS_trans_narrow : \forall n. - (\forall G,T,Q,U. - (t_len Q) \leq n \to (JSubtype G T Q) \to (JSubtype G Q U) \to +lemma JS_trans_narrow : \forall Q. + (\forall G,T,U. + (JSubtype G T Q) \to (JSubtype G Q U) \to (JSubtype G T U)) \land - (\forall G,H,X,P,Q,M,N. - (t_len Q) \leq n \to + (\forall G,H,X,P,M,N. (JSubtype (H @ ((mk_bound true X Q) :: G)) M N) \to (JSubtype G P Q) \to (JSubtype (H @ ((mk_bound true X P) :: G)) M N)). -intro;apply (nat_elim1 n);intros 2; -cut (\forall G,T,Q.(JSubtype G T Q) \to - \forall U.(t_len Q \leq m) \to (JSubtype G Q U) \to (JSubtype G T U)) - [cut (\forall G,M,N.(JSubtype G M N) \to - \forall G1,X,Q,G2,P. - (G = G2 @ ((mk_bound true X Q) :: G1)) \to (t_len Q) \leq m \to - (JSubtype G1 P Q) \to +apply Typ_len_ind;intros 2; +cut (\forall G,T,P. + (JSubtype G T U) \to + (JSubtype G U P) \to + (JSubtype G T P)) + [split + [assumption + |cut (\forall G,M,N.(JSubtype G M N) \to + \forall G1,X,G2,P. + (G = G2 @ ((mk_bound true X U) :: G1)) \to + (JSubtype G1 P U) \to (JSubtype (G2 @ ((mk_bound true X P) :: G1)) M N)) - [split - [intros;apply (Hcut ? ? ? H2 ? H1 H3) - |intros;apply (Hcut1 ? ? ? H3 ? ? ? ? ? ? H2 H4);reflexivity] - |intros 9;cut (incl ? (fv_env (G2 @ ((mk_bound true X Q)::G1))) - (fv_env (G2 @ ((mk_bound true X P)::G1)))) - [intros; -(* [rewrite > H6 in H2;lapply (JS_to_WFT1 ? ? ? H8); - apply (WFE_Typ_subst ? ? ? ? ? ? ? H2 Hletin) *) - generalize in match Hcut1;generalize in match H2; - generalize in match H1;generalize in match H4; - generalize in match G1;generalize in match G2;elim H1 - [apply SA_Top - [rewrite > H9 in H5;lapply (JS_to_WFT1 ? ? ? H7); - apply (WFE_Typ_subst ? ? ? ? ? ? ? H5 Hletin) - |rewrite > H9 in H6;apply (WFT_env_incl ? ? H6);elim l - [simplify;unfold;intros;assumption - |simplify;apply (incl_nat_cons ? ? ? H11)]] - |apply SA_Refl_TVar - [rewrite > H9 in H5;lapply (JS_to_WFT1 ? ? ? H7); - apply (WFE_Typ_subst ? ? ? ? ? ? ? H5 Hletin) - |apply H10;rewrite < H9;assumption] - |elim (decidable_eq_nat X n1) - [apply (SA_Trans_TVar ? ? ? P) - [rewrite < H12;elim l - [simplify;apply in_Base - |simplify;apply in_Skip;assumption] - |lapply (JS_to_WFE ? ? ? H9);rewrite > H10 in Hletin; - rewrite > H10 in H5; - lapply (WFE_bound_bound ? ? ? Q ? Hletin H5) - [lapply (H7 ? ? H8 H6 H10 H11);rewrite > Hletin1 in Hletin2; - apply (Hcut ? ? ? ? ? H3 Hletin2); - lapply (JS_to_WFE ? ? ? Hletin2); - apply (JS_weakening ? ? ? H8 ? Hletin3);unfold;intros; - elim l;simplify;apply in_Skip;assumption - |rewrite > H12;elim l - [simplify;apply in_Base - |simplify;apply in_Skip;assumption]]] - |rewrite > H10 in H5;apply (SA_Trans_TVar ? ? ? t1) - [apply (lookup_env_extends ? ? ? ? ? ? ? ? ? ? H5);unfold; - intro;apply H12;symmetry;assumption - |apply (H7 ? ? H8 H6 H10 H11)]] - |apply SA_Arrow - [apply (H6 ? ? H9 H5 H11 H12) - |apply (H8 ? ? H9 H7 H11 H12)] - |apply SA_All - [apply (H6 ? ? H9 H5 H11 H12) - |intros;apply (H8 ? ? (mk_bound true X1 t2::l) l1) - [unfold;intro;apply H13;rewrite > H11 in H14;apply (H12 ? H14) - |assumption - |apply H7;rewrite > H11;unfold;intro;apply H13;apply (H12 ? H14) - |simplify;rewrite < H11;reflexivity - |simplify;apply incl_nat_cons;assumption]]] - |elim G2 0 - [simplify;unfold;intros;assumption - |intro;elim t 0;simplify;intros;apply incl_nat_cons;assumption]]] - |intros 4;(*generalize in match H1;*)elim H1 + [intros;apply (Hcut1 ? ? ? H2 ? ? ? ? ? H3);reflexivity + |intros;cut (incl ? (fv_env (G2 @ ((mk_bound true X U)::G1))) + (fv_env (G2 @ ((mk_bound true X P)::G1)))) + [intros;generalize in match H2;generalize in match Hcut1; + generalize in match Hcut;generalize in match G2;elim H1 + [apply SA_Top + [rewrite > H8 in H4;lapply (JS_to_WFT1 ? ? ? H3); + apply (WFE_Typ_subst ? ? ? ? ? ? ? H4 Hletin) + |rewrite > H8 in H5;apply (WFT_env_incl ? ? H5 ? H7)] + |apply SA_Refl_TVar + [rewrite > H8 in H4;apply (WFE_Typ_subst ? ? ? ? ? ? ? H4); + apply (JS_to_WFT1 ? ? ? H3) + |rewrite > H8 in H5;apply (H7 ? H5)] + |elim (decidable_eq_nat X n) + [apply (SA_Trans_TVar ? ? ? P) + [rewrite < H10;elim l + [simplify;constructor 1 + |simplify;constructor 2;assumption] + |apply H7 + [lapply (H6 ? H7 H8 H9);lapply (JS_to_WFE ? ? ? Hletin); + apply (JS_weakening ? ? ? H3 ? Hletin1);unfold;intros; + elim l;simplify;constructor 2;assumption + |lapply (WFE_bound_bound true n t1 U ? ? H4) + [apply (JS_to_WFE ? ? ? H5) + |rewrite < Hletin;apply (H6 ? H7 H8 H9) + |rewrite > H9;rewrite > H10;elim l;simplify + [constructor 1 + |constructor 2;assumption]]]] + |apply (SA_Trans_TVar ? ? ? t1) + [rewrite > H9 in H4; + apply (lookup_env_extends ? ? ? ? ? ? ? ? ? ? H4); + unfold;intro;apply H10;symmetry;assumption + |apply (H6 ? H7 H8 H9)]] + |apply SA_Arrow + [apply (H5 ? H8 H9 H10) + |apply (H7 ? H8 H9 H10)] + |apply SA_All + [apply (H5 ? H8 H9 H10) + |intros;apply (H7 ? ? (mk_bound true X1 t2::l) H8) + [rewrite > H10;cut ((fv_env (l@(mk_bound true X P::G1))) = + (fv_env (l@(mk_bound true X U::G1)))) + [unfold;intro;apply H11;unfold;rewrite > Hcut2;assumption + |elim l + [simplify;reflexivity + |elim t4;simplify;rewrite > H12;reflexivity]] + |simplify;apply (incl_nat_cons ? ? ? H9) + |simplify;rewrite < H10;reflexivity]]] + |cut ((fv_env (G2@(mk_bound true X U::G1))) = + (fv_env (G2@(mk_bound true X P::G1)))) + [rewrite > Hcut1;unfold;intros;assumption + |elim G2 + [simplify;reflexivity + |elim t;simplify;rewrite > H4;reflexivity]]]]] + |intros 4;generalize in match H;elim H1 [inversion H5 [intros;rewrite < H8;apply (SA_Top ? ? H2 H3) |intros;destruct H9 |intros;destruct H10 |*:intros;destruct H11] |assumption - |apply (SA_Trans_TVar ? ? ? ? H2);apply (H4 ? H5 H6) + |apply (SA_Trans_TVar ? ? ? ? H2);apply (H4 H5 H6) |inversion H7 [intros;apply (SA_Top ? ? H8);rewrite < H10;apply WFT_Arrow [apply (JS_to_WFT2 ? ? ? H2) |apply (JS_to_WFT1 ? ? ? H4)] |intros;destruct H11 |intros;destruct H12 - |intros;destruct H13;elim (H (pred m)) - [apply SA_Arrow + |intros;destruct H13;apply SA_Arrow [rewrite > H12 in H2;rewrite < Hcut in H8; - apply (H15 ? ? ? ? ? H8 H2);lapply (t_len_arrow1 t2 t3); - unfold in Hletin;lapply (trans_le ? ? ? Hletin H6); - apply (t_len_pred ? ? Hletin1) + lapply (H6 t2) + [elim Hletin;apply (H15 ? ? ? H8 H2) + |apply (t_len_arrow1 t2 t3)] |rewrite > H12 in H4;rewrite < Hcut1 in H10; - apply (H15 ? ? ? ? ? H4 H10);lapply (t_len_arrow2 t2 t3); - unfold in Hletin;lapply (trans_le ? ? ? Hletin H6); - apply (t_len_pred ? ? Hletin1)] - |apply (pred_m_lt_m ? ? H6)] - |intros;destruct H13] + lapply (H6 t3) + [elim Hletin;apply (H15 ? ? ? H4 H10) + |apply (t_len_arrow2 t2 t3)]] + |intros;destruct H13] |inversion H7 [intros;apply (SA_Top ? ? H8);rewrite < H10;apply WFT_Forall [apply (JS_to_WFT2 ? ? ? H2) @@ -212,65 +220,32 @@ cut (\forall G,T,Q.(JSubtype G T Q) \to |intros;destruct H11 |intros;destruct H12 |intros;destruct H13 - |intros;destruct H13;elim (H (pred m)) - [elim (fresh_name ((fv_env e1) @ (fv_type t1) @ (fv_type t7))); - cut ((\lnot (in_list ? a (fv_env e1))) \land - (\lnot (in_list ? a (fv_type t1))) \land - (\lnot (in_list ? a (fv_type t7)))) - [elim Hcut2;elim H18;clear Hcut2 H18;apply (SA_All2 ? ? ? ? ? a) - [rewrite < Hcut in H8;rewrite > H12 in H2; - apply (H15 ? ? ? ? ? H8 H2);lapply (t_len_forall1 t2 t3); - unfold in Hletin;lapply (trans_le ? ? ? Hletin H6); - apply (t_len_pred ? ? Hletin1) - |5:lapply (H10 ? H20);rewrite > H12 in H5; - lapply (H5 ? H20 (subst_type_O t5 (TFree a))) - [apply (H15 ? ? ? ? ? ? Hletin) - [rewrite < Hcut1;rewrite > subst_O_nat; - rewrite < eq_t_len_TFree_subst; - lapply (t_len_forall2 t2 t3);unfold in Hletin2; - lapply (trans_le ? ? ? Hletin2 H6); - apply (t_len_pred ? ? Hletin3) - |rewrite < Hcut in H8; - apply (H16 e1 (nil ?) a t6 t2 ? ? ? Hletin1 H8); - lapply (t_len_forall1 t2 t3);unfold in Hletin2; - lapply (trans_le ? ? ? Hletin2 H6); - apply (t_len_pred ? ? Hletin3)] - |rewrite > subst_O_nat;rewrite < eq_t_len_TFree_subst; - lapply (t_len_forall2 t2 t3);unfold in Hletin1; - lapply (trans_le ? ? ? Hletin1 H6); - apply (trans_le ? ? ? ? Hletin2);constructor 2; - constructor 1 - |rewrite > Hcut1;rewrite > H12 in H4; - lapply (H4 ? H20);rewrite < Hcut1;apply JS_Refl - [apply (JS_to_WFT2 ? ? ? Hletin1) - |apply (JS_to_WFE ? ? ? Hletin1)]] - |*:assumption] - |split - [split - [unfold;intro;apply H17; - apply (natinG_or_inH_to_natinGH ? (fv_env e1));right; - assumption - |unfold;intro;apply H17; - apply (natinG_or_inH_to_natinGH - ((fv_type t1) @ (fv_type t7)));left; - apply natinG_or_inH_to_natinGH;right;assumption] - |unfold;intro;apply H17; - apply (natinG_or_inH_to_natinGH - ((fv_type t1) @ (fv_type t7)));left; - apply natinG_or_inH_to_natinGH;left;assumption]] - |apply (pred_m_lt_m ? ? H6)]]]] + |intros;destruct H13;subst;apply SA_All + [lapply (H6 t4) + [elim Hletin;apply (H12 ? ? ? H8 H2) + |apply t_len_forall1] + |intros;(*destruct H12;*)subst; + lapply (H6 (subst_type_O t5 (TFree X))) + [elim Hletin;apply H13 + [lapply (H6 t4) + [elim Hletin1;apply (H16 e1 [] X t6); + [simplify;apply H4;assumption + |assumption] + |apply t_len_forall1] + |apply (H10 ? H12)] + |rewrite > subst_O_nat;rewrite < eq_t_len_TFree_subst; + apply t_len_forall2]]]]] qed. theorem JS_trans : \forall G,T,U,V.(JSubtype G T U) \to (JSubtype G U V) \to (JSubtype G T V). -intros;elim (JS_trans_narrow (t_len U));apply (H2 ? ? ? ? ? H H1);constructor 1; +intros;elim JS_trans_narrow;autobatch; qed. theorem JS_narrow : \forall G1,G2,X,P,Q,T,U. (JSubtype (G2 @ (mk_bound true X Q :: G1)) T U) \to (JSubtype G1 P Q) \to (JSubtype (G2 @ (mk_bound true X P :: G1)) T U). -intros;elim (JS_trans_narrow (t_len Q));apply (H3 ? ? ? ? ? ? ? ? H H1); -constructor 1; +intros;elim JS_trans_narrow;autobatch; qed. diff --git a/matita/library/Fsub/util.ma b/matita/library/Fsub/util.ma index 2e50ed5c0..fa59484f5 100644 --- a/matita/library/Fsub/util.ma +++ b/matita/library/Fsub/util.ma @@ -19,17 +19,6 @@ include "list/list.ma". (*** useful definitions and lemmas not really related to Fsub ***) -lemma eqb_case : \forall x,y.(eqb x y) = true \lor (eqb x y) = false. -intros;elim (eqb x y);autobatch; -qed. - -lemma eq_eqb_case : \forall x,y.((x = y) \land (eqb x y) = true) \lor - ((x \neq y) \land (eqb x y) = false). -intros;lapply (eqb_to_Prop x y);elim (eqb_case x y) - [rewrite > H in Hletin;simplify in Hletin;left;autobatch - |rewrite > H in Hletin;simplify in Hletin;right;autobatch] -qed. - let rec max m n \def match (leb m n) with [true \Rightarrow n @@ -52,13 +41,6 @@ definition map : \forall A,B,f.((list A) \to (list B)) \def |(cons (a:A) (t:(list A))) \Rightarrow (cons B (f a) (map t))] in map. -definition swap : nat \to nat \to nat \to nat \def - \lambda u,v,x.match (eqb x u) with - [true \Rightarrow v - |false \Rightarrow match (eqb x v) with - [true \Rightarrow u - |false \Rightarrow x]]. - lemma in_list_nil : \forall A,x.\lnot (in_list A x []). intros.unfold.intro.inversion H [intros;lapply (sym_eq ? ? ? H2);absurd (a::l = []) @@ -67,67 +49,8 @@ intros.unfold.intro.inversion H [assumption|apply nil_cons]] qed. -lemma notin_cons : \forall A,x,y,l.\lnot (in_list A x (y::l)) \to - (y \neq x) \land \lnot (in_list A x l). -intros.split - [unfold;intro;apply H;rewrite > H1;constructor 1 - |unfold;intro;apply H;constructor 2;assumption] -qed. - -lemma swap_left : \forall x,y.(swap x y x) = y. -intros;unfold swap;rewrite > eqb_n_n;simplify;reflexivity; -qed. - -lemma swap_right : \forall x,y.(swap x y y) = x. -intros;unfold swap;elim (eq_eqb_case y x) - [elim H;rewrite > H2;simplify;rewrite > H1;reflexivity - |elim H;rewrite > H2;simplify;rewrite > eqb_n_n;simplify;reflexivity] -qed. - -lemma swap_other : \forall x,y,z.(z \neq x) \to (z \neq y) \to (swap x y z) = z. -intros;unfold swap;elim (eq_eqb_case z x) - [elim H2;lapply (H H3);elim Hletin - |elim H2;rewrite > H4;simplify;elim (eq_eqb_case z y) - [elim H5;lapply (H1 H6);elim Hletin - |elim H5;rewrite > H7;simplify;reflexivity]] -qed. - -lemma swap_inv : \forall u,v,x.(swap u v (swap u v x)) = x. -intros;unfold in match (swap u v x);elim (eq_eqb_case x u) - [elim H;rewrite > H2;simplify;rewrite > H1;apply swap_right - |elim H;rewrite > H2;simplify;elim (eq_eqb_case x v) - [elim H3;rewrite > H5;simplify;rewrite > H4;apply swap_left - |elim H3;rewrite > H5;simplify;apply (swap_other ? ? ? H1 H4)]] -qed. - -lemma swap_inj : \forall u,v,x,y.(swap u v x) = (swap u v y) \to x = y. -intros;unfold swap in H;elim (eq_eqb_case x u) - [elim H1;elim (eq_eqb_case y u) - [elim H4;rewrite > H5;assumption - |elim H4;rewrite > H3 in H;rewrite > H6 in H;simplify in H; - elim (eq_eqb_case y v) - [elim H7;rewrite > H9 in H;simplify in H;rewrite > H in H8; - lapply (H5 H8);elim Hletin - |elim H7;rewrite > H9 in H;simplify in H;elim H8;symmetry;assumption]] - |elim H1;elim (eq_eqb_case y u) - [elim H4;rewrite > H3 in H;rewrite > H6 in H;simplify in H; - elim (eq_eqb_case x v) - [elim H7;rewrite > H9 in H;simplify in H;rewrite < H in H8; - elim H2;assumption - |elim H7;rewrite > H9 in H;simplify in H;elim H8;assumption] - |elim H4;rewrite > H3 in H;rewrite > H6 in H;simplify in H; - elim (eq_eqb_case x v) - [elim H7;rewrite > H9 in H;elim (eq_eqb_case y v) - [elim H10;rewrite > H11;assumption - |elim H10;rewrite > H12 in H;simplify in H;elim H5;symmetry; - assumption] - |elim H7;rewrite > H9 in H;elim (eq_eqb_case y v) - [elim H10;rewrite > H12 in H;simplify in H;elim H2;assumption - |elim H10;rewrite > H12 in H;simplify in H;assumption]]]] -qed. - lemma max_case : \forall m,n.(max m n) = match (leb m n) with [ false \Rightarrow n | true \Rightarrow m ]. intros;elim m;simplify;reflexivity; -qed. \ No newline at end of file +qed. \ No newline at end of file