From: Wilmer Ricciotti Date: Thu, 30 Nov 2006 16:14:12 +0000 (+0000) Subject: library: added solution to POPLMark challenge part 1a (transitivity of Fsub subtyping) X-Git-Tag: make_still_working~6619 X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=commitdiff_plain;h=16b9c591af0d9a188a916140e5fcd2b58805277f;p=helm.git library: added solution to POPLMark challenge part 1a (transitivity of Fsub subtyping) --- diff --git a/helm/software/matita/library/Fsub/defn.ma b/helm/software/matita/library/Fsub/defn.ma new file mode 100644 index 000000000..bef16a566 --- /dev/null +++ b/helm/software/matita/library/Fsub/defn.ma @@ -0,0 +1,1356 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +set "baseuri" "cic:/matita/library/Fsub/defn". +include "logic/equality.ma". +include "nat/nat.ma". +include "datatypes/bool.ma". +include "nat/compare.ma". +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);auto; +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;auto + |rewrite > H in Hletin;simplify in Hletin;right;auto] +qed. + +let rec max m n \def + match (leb m n) with + [true \Rightarrow n + |false \Rightarrow m]. + +(*** representation of Fsub types ***) +inductive Typ : Set \def + | TVar : nat \to Typ (* type var *) + | TFree: nat \to Typ (* free type name *) + | Top : Typ (* maximum type *) + | Arrow : Typ \to Typ \to Typ (* functions *) + | Forall : Typ \to Typ \to Typ. (* universal type *) + +(*** representation of Fsub terms ***) +inductive Term : Set \def + | Var : nat \to Term (* variable *) + | Free : nat \to Term (* free name *) + | Abs : Typ \to Term \to Term (* abstraction *) + | App : Term \to Term \to Term (* function application *) + | TAbs : Typ \to Term \to Term (* type abstraction *) + | TApp : Term \to Typ \to Term. (* type application *) + +(* representation of bounds *) + +record bound : Set \def { + istype : bool; (* is subtyping bound? *) + name : nat ; (* name *) + btype : Typ (* type to which the name is bound *) + }. + +(* representation of Fsub typing environments *) +definition Env \def (list bound). +definition Empty \def (nil bound). +definition Cons \def \lambda G,X,T.((mk_bound false X T) :: G). +definition TCons \def \lambda G,X,T.((mk_bound true X T) :: G). + +definition env_append : Env \to Env \to Env \def \lambda G,H.(H @ G). + +notation "hvbox(\Forall S. break T)" + non associative with precedence 90 +for @{ 'forall $S $T}. + +notation "hvbox(#x)" + with precedence 60 + for @{'var $x}. + +notation "hvbox(##x)" + with precedence 61 + for @{'tvar $x}. + +notation "hvbox(!x)" + with precedence 60 + for @{'name $x}. + +notation "hvbox(!!x)" + with precedence 61 + for @{'tname $x}. + +notation "hvbox(s break \mapsto t)" + right associative with precedence 55 + for @{ 'arrow $s $t }. + +interpretation "universal type" 'forall S T = (cic:/matita/test/Typ.ind#xpointer(1/1/5) S T). + +interpretation "bound var" 'var x = (cic:/matita/test/Typ.ind#xpointer(1/1/1) x). + +interpretation "bound tvar" 'tvar x = (cic:/matita/test/Typ.ind#xpointer(1/1/3) x). + +interpretation "bound tname" 'tname x = (cic:/matita/test/Typ.ind#xpointer(1/1/2) x). + +interpretation "arrow type" 'arrow S T = (cic:/matita/test/Typ.ind#xpointer(1/1/4) S T). + +(*** Various kinds of substitution, not all will be used probably ***) + +(* substitutes i-th dangling index in type T with type U *) +let rec subst_type_nat T U i \def + match T with + [ (TVar n) \Rightarrow match (eqb n i) with + [ true \Rightarrow U + | false \Rightarrow T] + | (TFree X) \Rightarrow T + | Top \Rightarrow T + | (Arrow T1 T2) \Rightarrow (Arrow (subst_type_nat T1 U i) (subst_type_nat T2 U i)) + | (Forall T1 T2) \Rightarrow (Forall (subst_type_nat T1 U i) (subst_type_nat T2 U (S i))) ]. + +(* substitutes 0-th dangling index in type T with type U *) +let rec subst_type_O T U \def subst_type_nat T U O. + +(* substitutes 0-th dangling index in term t with term u *) +let rec subst_term_O t u \def + let rec aux t0 i \def + match t0 with + [ (Var n) \Rightarrow match (eqb n i) with + [ true \Rightarrow u + | false \Rightarrow t0] + | (Free X) \Rightarrow t0 + | (Abs T1 t1) \Rightarrow (Abs T1 (aux t1 (S i))) + | (App t1 t2) \Rightarrow (App (aux t1 i) (aux t2 i)) + | (TAbs T1 t1) \Rightarrow (TAbs T1 (aux t1 (S i))) + | (TApp t1 T1) \Rightarrow (TApp (aux t1 i) T1) ] + in aux t O. + +(* substitutes 0-th dangling index in term T, which shall be a TVar, + with type U *) +let rec subst_term_tO t T \def + let rec aux t0 i \def + match t0 with + [ (Var n) \Rightarrow t0 + | (Free X) \Rightarrow t0 + | (Abs T1 t1) \Rightarrow (Abs (subst_type_nat T1 T i) (aux t1 (S i))) + | (App t1 t2) \Rightarrow (App (aux t1 i) (aux t2 i)) + | (TAbs T1 t1) \Rightarrow (TAbs (subst_type_nat T1 T i) (aux t1 (S i))) + | (TApp t1 T1) \Rightarrow (TApp (aux t1 i) (subst_type_nat T1 T i)) ] + in aux t O. + +(* substitutes (TFree X) in type T with type U *) +let rec subst_type_tfree_type T X U on T \def + match T with + [ (TVar n) \Rightarrow T + | (TFree Y) \Rightarrow match (eqb X Y) with + [ true \Rightarrow U + | false \Rightarrow T ] + | Top \Rightarrow T + | (Arrow T1 T2) \Rightarrow (Arrow (subst_type_tfree_type T1 X U) + (subst_type_tfree_type T2 X U)) + | (Forall T1 T2) \Rightarrow (Forall (subst_type_tfree_type T1 X U) + (subst_type_tfree_type T2 X U)) ]. + +(*** height of T's syntactic tree ***) + +let rec t_len T \def + match T with + [(TVar n) \Rightarrow (S O) + |(TFree X) \Rightarrow (S O) + |Top \Rightarrow (S O) + |(Arrow T1 T2) \Rightarrow (S (max (t_len T1) (t_len T2))) + |(Forall T1 T2) \Rightarrow (S (max (t_len T1) (t_len T2)))]. + +(* +let rec fresh_name G n \def + match G with + [ nil \Rightarrow n + | (cons b H) \Rightarrow match (leb (fresh_name H n) (name b)) with + [ true \Rightarrow (S (name b)) + | false \Rightarrow (fresh_name H n) ]]. + +lemma freshname_Gn_geq_n : \forall G,n.((fresh_name G n) >= n). +intro;elim G + [simplify;unfold;constructor 1 + |simplify;cut ((leb (fresh_name l n) (name s)) = true \lor + (leb (fresh_name l n) (name s) = false)) + [elim Hcut + [lapply (leb_to_Prop (fresh_name l n) (name s));rewrite > H1 in Hletin; + simplify in Hletin;rewrite > H1;simplify;lapply (H n); + unfold in Hletin1;unfold; + apply (trans_le ? ? ? Hletin1); + apply (trans_le ? ? ? Hletin);constructor 2;constructor 1 + |rewrite > H1;simplify;apply H] + |elim (leb (fresh_name l n) (name s)) [left;reflexivity|right;reflexivity]]] +qed. + +lemma freshname_consGX_gt_X : \forall G,X,T,b,n. + (fresh_name (cons ? (mk_bound b X T) G) n) > X. +intros.unfold.unfold.simplify.cut ((leb (fresh_name G n) X) = true \lor + (leb (fresh_name G n) X) = false) + [elim Hcut + [rewrite > H;simplify;constructor 1 + |rewrite > H;simplify;lapply (leb_to_Prop (fresh_name G n) X); + rewrite > H in Hletin;simplify in Hletin; + lapply (not_le_to_lt ? ? Hletin);unfold in Hletin1;assumption] + |elim (leb (fresh_name G n) X) [left;reflexivity|right;reflexivity]] +qed. + +lemma freshname_case : \forall G,X,T,b,n. + (fresh_name ((mk_bound b X T) :: G) n) = (fresh_name G n) \lor + (fresh_name ((mk_bound b X T) :: G) n) = (S X). +intros.simplify.cut ((leb (fresh_name G n) X) = true \lor + (leb (fresh_name G n) X) = false) + [elim Hcut + [rewrite > H;simplify;right;reflexivity + |rewrite > H;simplify;left;reflexivity] + |elim (leb (fresh_name G n) X) + [left;reflexivity|right;reflexivity]] +qed. + +lemma freshname_monotone_n : \forall G,m,n.(m \leq n) \to + ((fresh_name G m) \leq (fresh_name G n)). +intros.elim G + [simplify;assumption + |simplify;cut ((leb (fresh_name l m) (name s)) = true \lor + (leb (fresh_name l m) (name s)) = false) + [cut ((leb (fresh_name l n) (name s)) = true \lor + (leb (fresh_name l n) (name s)) = false) + [elim Hcut + [rewrite > H2;simplify;elim Hcut1 + [rewrite > H3;simplify;constructor 1 + |rewrite > H3;simplify; + lapply (leb_to_Prop (fresh_name l n) (name s)); + rewrite > H3 in Hletin;simplify in Hletin; + lapply (not_le_to_lt ? ? Hletin);unfold in Hletin1;assumption] + |rewrite > H2;simplify;elim Hcut1 + [rewrite > H3;simplify; + lapply (leb_to_Prop (fresh_name l m) (name s)); + rewrite > H2 in Hletin;simplify in Hletin; + lapply (not_le_to_lt ? ? Hletin);unfold in Hletin1; + lapply (leb_to_Prop (fresh_name l n) (name s)); + rewrite > H3 in Hletin2; + simplify in Hletin2;lapply (trans_le ? ? ? Hletin1 H1); + lapply (trans_le ? ? ? Hletin3 Hletin2); + absurd ((S (name s)) \leq (name s)) + [assumption|apply not_le_Sn_n] + |rewrite > H3;simplify;assumption]] + |elim (leb (fresh_name l n) (name s)) + [left;reflexivity|right;reflexivity]] + |elim (leb (fresh_name l m) (name s)) [left;reflexivity|right;reflexivity]]] +qed. + +lemma freshname_monotone_G : \forall G,X,T,b,n. + (fresh_name G n) \leq (fresh_name ((mk_bound b X T) :: G) n). +intros.simplify.cut ((leb (fresh_name G n) X) = true \lor + (leb (fresh_name G n) X) = false) + [elim Hcut + [rewrite > H;simplify;lapply (leb_to_Prop (fresh_name G n) X); + rewrite > H in Hletin;simplify in Hletin;constructor 2;assumption + |rewrite > H;simplify;constructor 1] + |elim (leb (fresh_name G n) X) + [left;reflexivity|right;reflexivity]] +qed.*) + +lemma subst_O_nat : \forall T,U.((subst_type_O T U) = (subst_type_nat T U O)). +intros;elim T;simplify;reflexivity; +qed. + +(* FIXME: these definitions shouldn't be part of the poplmark challenge + - use destruct instead, when hopefully it will get fixed... *) + +definition head \def + \lambda G:(list bound).match G with + [ nil \Rightarrow (mk_bound false O Top) + | (cons b H) \Rightarrow b]. + +definition head_nat \def + \lambda G:(list nat).match G with + [ nil \Rightarrow O + | (cons n H) \Rightarrow n]. + +lemma inj_head : \forall h1,h2:bound.\forall t1,t2:Env. + ((h1::t1) = (h2::t2)) \to (h1 = h2). +intros.lapply (eq_f ? ? head ? ? H).simplify in Hletin.assumption. +qed. + +lemma inj_head_nat : \forall h1,h2:nat.\forall t1,t2:(list nat). + ((h1::t1) = (h2::t2)) \to (h1 = h2). +intros.lapply (eq_f ? ? head_nat ? ? H).simplify in Hletin.assumption. +qed. + +lemma inj_tail : \forall A.\forall h1,h2:A.\forall t1,t2:(list A). + ((h1::t1) = (h2::t2)) \to (t1 = t2). +intros.lapply (eq_f ? ? (tail ?) ? ? H).simplify in Hletin.assumption. +qed. + +(* end of fixme *) + +(*** definitions and theorems about lists ***) + +inductive in_list (A : Set) : A \to (list A) \to Prop \def + | in_Base : \forall x:A.\forall l:(list A). + (in_list A x (x :: l)) + | in_Skip : \forall x,y:A.\forall l:(list A). + (in_list A x l) \to (in_list A x (y :: l)). + +(* var binding is in env judgement *) +definition var_bind_in_env : bound \to Env \to Prop \def + \lambda b,G.(in_list bound b G). + +(* FIXME: use the map in library/list (when there will be one) *) +definition map : \forall A,B,f.((list A) \to (list B)) \def + \lambda A,B,f.let rec map (l : (list A)) : (list B) \def + match l in list return \lambda l0:(list A).(list B) with + [nil \Rightarrow (nil B) + |(cons (a:A) (t:(list A))) \Rightarrow + (cons B (f a) (map t))] in map. + +definition fv_env : (list bound) \to (list nat) \def + \lambda G.(map ? ? (\lambda b.match b with + [(mk_bound B X T) \Rightarrow X]) G). + +(* variable is in env judgement *) +definition var_in_env : nat \to Env \to Prop \def + \lambda x,G.(in_list nat x (fv_env G)). + +definition var_type_in_env : nat \to Env \to Prop \def + \lambda x,G.\exists T.(var_bind_in_env (mk_bound true x T) G). + +definition incl : \forall A.(list A) \to (list A) \to Prop \def + \lambda A,l,m.\forall x.(in_list A x l) \to (in_list A x m). + +let rec fv_type T \def + match T with + [(TVar n) \Rightarrow [] + |(TFree x) \Rightarrow [x] + |Top \Rightarrow [] + |(Arrow U V) \Rightarrow ((fv_type U) @ (fv_type V)) + |(Forall U V) \Rightarrow ((fv_type U) @ (fv_type V))]. + +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 in_list_nil : \forall A,x.\lnot (in_list A x []). +intros.unfold.intro.inversion H + [intros;lapply (sym_eq ? ? ? H2);absurd (a::l = []) + [assumption|apply nil_cons] + |intros;lapply (sym_eq ? ? ? H4);absurd (a1::l = []) + [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 boundinenv_natinfv : \forall x,G. + (\exists B,T.(in_list ? (mk_bound B x T) G)) \to + (in_list ? x (fv_env G)). +intros 2;elim G + [elim H;elim H1;lapply (in_list_nil ? ? H2);elim Hletin + |elim H1;elim H2;inversion H3 + [intros;rewrite < H4;simplify;apply in_Base + |intros;elim a3;simplify;apply in_Skip; + lapply (inj_tail ? ? ? ? ? H7);rewrite > Hletin in H;apply H; + apply ex_intro + [apply a + |apply ex_intro + [apply a1 + |rewrite > H6;assumption]]]] +qed. + +lemma nat_in_list_case : \forall G,H,n.(in_list nat n (H @ G)) \to + (in_list nat n G) \lor (in_list nat n H). +intros 3.elim H + [simplify in H1;left;assumption + |simplify in H2;inversion H2 + [intros;lapply (inj_head_nat ? ? ? ? H4);rewrite > Hletin; + right;apply in_Base + |intros;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 natinG_or_inH_to_natinGH : \forall G,H,n. + (in_list nat n G) \lor (in_list nat n H) \to + (in_list nat n (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 natinfv_boundinenv : \forall x,G.(in_list ? x (fv_env G)) \to + \exists B,T.(in_list ? (mk_bound B x T) G). +intros 2;elim G 0 + [simplify;intro;lapply (in_list_nil ? ? H);elim Hletin + |intros 3;elim s;simplify in H1;inversion H1 + [intros;rewrite < H2;simplify;apply ex_intro + [apply b + |apply ex_intro + [apply t + |lapply (inj_head_nat ? ? ? ? H3);rewrite > H2;rewrite < Hletin; + apply in_Base]] + |intros;lapply (inj_tail ? ? ? ? ? H5);rewrite < Hletin in H2; + rewrite < H4 in H2;lapply (H H2);elim Hletin1;elim H6;apply ex_intro + [apply a2 + |apply ex_intro + [apply a3 + |apply in_Skip;rewrite < H4;assumption]]]] +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. +lapply (natinfv_boundinenv ? ? H1).elim Hletin.elim H2.apply ex_intro + [apply a + |apply ex_intro + [apply a1 + |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 + [intros;lapply (inj_head_nat ? ? ? ? H3);rewrite > Hletin;apply in_Base + |intros;apply in_Skip;apply H;lapply (inj_tail ? ? ? ? ? H5);rewrite > Hletin; + 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 s;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 s;simplify;constructor 2;apply (H2 H3)] + |elim H 0 + [simplify;intro;lapply (in_list_nil nat x H2);elim Hletin + |intros 2;elim s;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. + +(*** Type Well-Formedness judgement ***) + +inductive WFType : Env \to Typ \to Prop \def + | WFT_TFree : \forall X,G.(in_list ? X (fv_env G)) + \to (WFType G (TFree X)) + | WFT_Top : \forall G.(WFType G Top) + | WFT_Arrow : \forall G,T,U.(WFType G T) \to (WFType G U) \to + (WFType G (Arrow T U)) + | WFT_Forall : \forall G,T,U.(WFType G T) \to + (\forall X:nat. + (\lnot (in_list ? X (fv_env G))) \to + (\lnot (in_list ? X (fv_type U))) \to + (WFType ((mk_bound true X T) :: G) + (subst_type_O U (TFree X)))) \to + (WFType G (Forall T U)). + +(*** Environment Well-Formedness judgement ***) + +inductive WFEnv : Env \to Prop \def + | WFE_Empty : (WFEnv Empty) + | WFE_cons : \forall B,X,T,G.(WFEnv G) \to + \lnot (in_list ? X (fv_env G)) \to + (WFType G T) \to (WFEnv ((mk_bound B X T) :: G)). + +(*** Subtyping judgement ***) +inductive JSubtype : Env \to Typ \to Typ \to Prop \def + | SA_Top : \forall G:Env.\forall T:Typ.(WFEnv G) \to + (WFType G T) \to (JSubtype G T Top) + | SA_Refl_TVar : \forall G:Env.\forall X:nat.(WFEnv G) \to (var_in_env X G) + \to (JSubtype G (TFree X) (TFree X)) + | SA_Trans_TVar : \forall G:Env.\forall X:nat.\forall T:Typ. + \forall U:Typ. + (var_bind_in_env (mk_bound true X U) G) \to + (JSubtype G U T) \to (JSubtype G (TFree X) T) + | SA_Arrow : \forall G:Env.\forall S1,S2,T1,T2:Typ. + (JSubtype G T1 S1) \to (JSubtype G S2 T2) \to + (JSubtype G (Arrow S1 S2) (Arrow T1 T2)) + | SA_All : \forall G:Env.\forall S1,S2,T1,T2:Typ. + (JSubtype G T1 S1) \to + (\forall X:nat.\lnot (var_in_env X G) \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)). + +(*** Typing judgement ***) +inductive JType : Env \to Term \to Typ \to Prop \def + | T_Var : \forall G:Env.\forall x:nat.\forall T:Typ. + (WFEnv G) \to (var_bind_in_env (mk_bound false x T) G) \to + (JType G (Free x) T) + | T_Abs : \forall G.\forall T1,T2:Typ.\forall t2:Term. + \forall x:nat. + (JType ((mk_bound false x T1)::G) (subst_term_O t2 (Free x)) T2) \to + (JType G (Abs T1 t2) (Arrow T1 T2)) + | T_App : \forall G.\forall t1,t2:Term.\forall T2:Typ. + \forall T1:Typ.(JType G t1 (Arrow T1 T2)) \to (JType G t2 T1) \to + (JType G (App t1 t2) T2) + | T_TAbs : \forall G:Env.\forall T1,T2:Typ.\forall t2:Term. + \forall X:nat. + (JType ((mk_bound true X T1)::G) + (subst_term_tO t2 (TFree X)) (subst_type_O T2 (TFree X))) + \to (JType G (TAbs T1 t2) (Forall T1 T2)) + | T_TApp : \forall G:Env.\forall t1:Term.\forall T2,T12:Typ. + \forall X:nat.\forall T11:Typ. + (JType G t1 (Forall T11 (subst_type_tfree_type T12 X (TVar O)))) \to + (JSubtype G T2 T11) + \to (JType G (TApp t1 T2) (subst_type_tfree_type T12 X T2)) + | 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). + + +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) + |apply WFT_Top + |apply WFT_Arrow [apply (H3 ? H6)|apply (H5 ? H6)] + |apply WFT_Forall + [apply (H3 ? H6) + |intros;apply H5 + [unfold;intro;unfold in H7;apply H7;unfold in H6;apply(H6 ? H9) + |assumption + |simplify;apply (incl_nat_cons ? ? ? H6)]]] +qed. + +(*** definitions and theorems about swaps ***) + +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 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 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) + |simplify;intros;assumption + |simplify;intros;assumption + |intros;simplify in H2;elim (nat_in_list_case ? ? ? H2) + [simplify;apply natinG_or_inH_to_natinGH;left;apply (H1 ? H3) + |simplify;apply natinG_or_inH_to_natinGH;right;apply (H ? H3)] + |intros;simplify in H2;elim (nat_in_list_case ? ? ? H2) + [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. + +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))]. + +lemma swap_Typ_inv : \forall u,v,T.(swap_Typ u v (swap_Typ u v T)) = T. +intros;elim T + [simplify;reflexivity + |simplify;rewrite > swap_inv;reflexivity + |simplify;reflexivity + |simplify;rewrite > H;rewrite > H1;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 + [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;reflexivity + |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)))) + [elim Hcut;elim H4;elim H5;clear Hcut H4 H5;rewrite > (H H6 H8); + rewrite > (H1 H7 H9);reflexivity + |split + [split;unfold;intro;apply H2;apply natinG_or_inH_to_natinGH;auto + |split;unfold;intro;apply H3;apply natinG_or_inH_to_natinGH;auto]] + |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)))) + [elim Hcut;elim H4;elim H5;clear Hcut H4 H5;rewrite > (H H6 H8); + rewrite > (H1 H7 H9);reflexivity + |split + [split;unfold;intro;apply H2;apply natinG_or_inH_to_natinGH;auto + |split;unfold;intro;apply H3;apply natinG_or_inH_to_natinGH;auto]]] +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 + |simplify;reflexivity + |simplify;reflexivity + |simplify;rewrite > H;rewrite > H1;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 + [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;intro;elim (in_list_nil ? ? H) + |simplify;intros;elim (nat_in_list_case ? ? ? H2) + [apply natinG_or_inH_to_natinGH;left;apply (H1 H3) + |apply natinG_or_inH_to_natinGH;right;apply (H H3)] + |simplify;intros;elim (nat_in_list_case ? ? ? H2) + [apply natinG_or_inH_to_natinGH;left;apply (H1 H3) + |apply natinG_or_inH_to_natinGH;right;apply (H H3)]] + |elim T 0 + [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;intro;elim (in_list_nil ? ? H) + |simplify;intros;elim (nat_in_list_case ? ? ? H2) + [apply natinG_or_inH_to_natinGH;left;apply (H1 H3) + |apply natinG_or_inH_to_natinGH;right;apply (H H3)] + |simplify;intros;elim (nat_in_list_case ? ? ? H2) + [apply natinG_or_inH_to_natinGH;left;apply (H1 H3) + |apply natinG_or_inH_to_natinGH;right;apply (H H3)]]] +qed. + +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). + +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 s;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 + [simplify in H;inversion H + [intros;lapply (sym_eq ? ? ? H2);absurd (a::l = []) + [assumption|apply nil_cons] + |intros;lapply (sym_eq ? ? ? H4);absurd (a1::l = []) + [assumption|apply nil_cons]] + |simplify;simplify in H;assumption + |simplify in H;simplify;assumption + |simplify in H2;simplify;apply natinG_or_inH_to_natinGH; + lapply (nat_in_list_case ? ? ? H2);elim Hletin + [left;apply (H1 ? H3) + |right;apply (H ? H3)] + |simplify in H2;simplify;apply natinG_or_inH_to_natinGH; + lapply (nat_in_list_case ? ? ? H2);elim Hletin + [left;apply (H1 ? H3) + |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 s 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 s 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). +cut (\forall l:(list nat).\exists n.\forall m. + (n \leq m) \to \lnot (in_list ? m l)) + [intros;lapply (Hcut l);elim Hletin;apply ex_intro + [apply a + |apply H;constructor 1] + |intros;elim l + [apply ex_intro + [apply O + |intros;unfold;intro;inversion H1 + [intros;lapply (sym_eq ? ? ? H3);absurd (a::l1 = []) + [assumption|apply nil_cons] + |intros;lapply (sym_eq ? ? ? H5);absurd (a1::l1 = []) + [assumption|apply nil_cons]]] + |elim H;lapply (decidable_eq_nat a s);elim Hletin + [apply ex_intro + [apply (S a) + |intros;unfold;intro;inversion H4 + [intros;lapply (inj_head_nat ? ? ? ? H6);rewrite < Hletin1 in H5; + rewrite < H2 in H5;rewrite > H5 in H3; + apply (not_le_Sn_n ? H3) + |intros;lapply (inj_tail ? ? ? ? ? H8);rewrite < Hletin1 in H5; + rewrite < H7 in H5; + apply (H1 m ? H5);lapply (le_S ? ? H3); + apply (le_S_S_to_le ? ? Hletin2)]] + |cut ((leb a s) = true \lor (leb a s) = false) + [elim Hcut + [apply ex_intro + [apply (S s) + |intros;unfold;intro;inversion H5 + [intros;lapply (inj_head_nat ? ? ? ? H7);rewrite > H6 in H4; + rewrite < Hletin1 in H4;apply (not_le_Sn_n ? H4) + |intros;lapply (inj_tail ? ? ? ? ? H9); + rewrite < Hletin1 in H6;lapply (H1 a1) + [apply (Hletin2 H6) + |lapply (leb_to_Prop a s);rewrite > H3 in Hletin2; + simplify in Hletin2;rewrite < H8; + apply (trans_le ? ? ? Hletin2); + apply (trans_le ? ? ? ? H4);constructor 2;constructor 1]]] + |apply ex_intro + [apply a + |intros;lapply (leb_to_Prop a s);rewrite > H3 in Hletin1; + simplify in Hletin1;lapply (not_le_to_lt ? ? Hletin1); + unfold in Hletin2;unfold;intro;inversion H5 + [intros;lapply (inj_head_nat ? ? ? ? H7); + rewrite < Hletin3 in H6;rewrite > H6 in H4; + apply (Hletin1 H4) + |intros;lapply (inj_tail ? ? ? ? ? H9); + rewrite < Hletin3 in H6;rewrite < H8 in H6; + apply (H1 ? H4 H6)]]] + |elim (leb a s);auto]]]] +qed. + +(*** lemmas 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)). +intros 4.elim H + [simplify in H2;inversion H2 + [intros;lapply (inj_head_nat ? ? ? ? H4);rewrite < Hletin;assumption + |intros;lapply (inj_tail ? ? ? ? ? H6);rewrite < Hletin in H3; + inversion H3 + [intros;lapply (sym_eq ? ? ? H8);absurd (a2 :: l2 = []) + [assumption|apply nil_cons] + |intros;lapply (sym_eq ? ? ? H10); + absurd (a3 :: l2 = []) [assumption|apply nil_cons]]] + |simplify in H1;lapply (in_list_nil ? x H1);elim Hletin + |simplify in H5;lapply (nat_in_list_case ? ? ? H5);elim Hletin + [apply (H4 H6) + |apply (H2 H6)] + |simplify in H5;lapply (nat_in_list_case ? ? ? H5);elim Hletin + [lapply (fresh_name ((fv_type t1) @ (fv_env e)));elim Hletin1; + cut ((\lnot (in_list ? a (fv_type t1))) \land + (\lnot (in_list ? a (fv_env e)))) + [elim Hcut;lapply (H4 ? H9 H8) + [cut (x \neq a) + [simplify in Hletin2; + (* FIXME trick *);generalize in match Hletin2;intro; + inversion Hletin2 + [intros;lapply (inj_head_nat ? ? ? ? H12); + rewrite < Hletin3 in H11;lapply (Hcut1 H11);elim Hletin4 + |intros;lapply (inj_tail ? ? ? ? ? H14);rewrite > Hletin3; + assumption] + |unfold;intro;apply H8;rewrite < H10;assumption] + |rewrite > subst_O_nat;apply in_FV_subst;assumption] + |split + [unfold;intro;apply H7;apply natinG_or_inH_to_natinGH;right; + assumption + |unfold;intro;apply H7;apply natinG_or_inH_to_natinGH;left; + assumption]] + |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. + +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 s;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 t)::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 t);inversion H1 + [intro;absurd ((mk_bound b n t)::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 ***) +(* FIXME : use nat_elim1 instead *) + +lemma nat_ind_strong: \forall P:nat \to Prop. + (P O) \to + (\forall m.(\forall n.(n < m) \to (P n)) \to (P m)) \to + \forall n.(P n). +cut (\forall P:nat \to Prop.\forall n. + (P O) \to (\forall m.(P m) \to (P (S m))) \to (P n)) + [intros;cut ((\lambda l.\forall m.(m < (S l)) \to (P m)) n) + [simplify in Hcut1;apply Hcut1;unfold;constructor 1 + |cut (\forall m:nat.(m < (S O)) \to (P m)) + [cut (\forall n. + ((\lambda l.\forall m.(m < (S l)) \to (P m)) n) \to + ((\lambda l.\forall m.(m < (S l)) \to (P m)) (S n))) + [apply (Hcut ? ? Hcut1 Hcut2) + |simplify;intros 2;lapply (H1 ? H2);intros;unfold in H3; + lapply (le_S_S_to_le ? ? H3); + lapply (le_to_or_lt_eq ? ? Hletin1); + elim Hletin2 + [apply (H2 ? H4) + |rewrite > H4;assumption]] + |intros;unfold in H2;lapply (le_S_S_to_le ? ? H2); + generalize in match Hletin;elim m + [assumption + |absurd ((S n1) \leq O) + [assumption|apply not_le_Sn_O]]]] + |intros 2;elim n + [assumption + |apply (H2 ? (H H1 H2))]] +qed. + +(* TODO : relocate the following 2 lemmas *) + +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. + +lemma not_t_len_lt_SO : \forall T.\lnot (t_len T) < (S O). +intros;elim T + [simplify;unfold;intro;unfold in H;elim (not_le_Sn_n ? H) + |simplify;unfold;intro;unfold in H;elim (not_le_Sn_n ? H) + |simplify;unfold;intro;unfold in H;elim (not_le_Sn_n ? H) + |simplify;unfold;rewrite > max_case;elim (leb (t_len t) (t_len t1)) + [simplify in H2;apply H1;apply (trans_lt ? ? ? ? H2);unfold;constructor 1 + |simplify in H2;apply H;apply (trans_lt ? ? ? ? H2);unfold;constructor 1] + |simplify;unfold;rewrite > max_case;elim (leb (t_len t) (t_len t1)) + [simplify in H2;apply H1;apply (trans_lt ? ? ? ? H2);unfold;constructor 1 + |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 + [simplify;unfold;unfold;constructor 1 + |simplify;unfold;unfold;constructor 1 + |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)) + [simplify;unfold;unfold;constructor 2;unfold in H1;unfold in H1;assumption + |simplify;unfold;unfold;constructor 2;unfold in H;unfold in H;assumption] + |simplify;lapply (max_case (t_len t) (t_len t1));rewrite > Hletin; + elim (leb (t_len t) (t_len t1)) + [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)) + \to \forall T.(P T). +cut (\forall P:Typ \to Prop. + (\forall U.(\forall V.((t_len V) < (t_len U)) \to (P V)) + \to (P U)) + \to \forall T,n.(n = (t_len T)) \to (P T)) + [intros;apply (Hcut ? H ? (t_len T));reflexivity + |intros 4;generalize in match T;apply (nat_elim1 n);intros; + generalize in match H2;elim t + [apply H;intros;simplify in H4;elim (not_t_len_lt_SO ? H4) + |apply H;intros;simplify in H4;elim (not_t_len_lt_SO ? H4) + |apply H;intros;simplify in H4;elim (not_t_len_lt_SO ? H4) + |apply H;intros;apply (H1 (t_len V)) + [rewrite > H5;assumption + |reflexivity] + |apply H;intros;apply (H1 (t_len V)) + [rewrite > H5;assumption + |reflexivity]]] +qed. + +lemma t_len_arrow1 : \forall T1,T2.(t_len T1) < (t_len (Arrow T1 T2)). +intros.simplify. +(* FIXME!!! BUG?!?! *) +cut ((max (t_len T1) (t_len T2)) = match (leb (t_len T1) (t_len T2)) with + [ false \Rightarrow (t_len T2) + | true \Rightarrow (t_len T1) ]) + [rewrite > Hcut;cut ((leb (t_len T1) (t_len T2)) = false \lor + (leb (t_len T1) (t_len T2)) = true) + [lapply (leb_to_Prop (t_len T1) (t_len T2));elim Hcut1 + [rewrite > H;rewrite > H in Hletin;simplify;constructor 1 + |rewrite > H;rewrite > H in Hletin;simplify;simplify in Hletin; + unfold;apply le_S_S;assumption] + |elim (leb (t_len T1) (t_len T2));auto] + |elim T1;simplify;reflexivity] +qed. + +lemma t_len_arrow2 : \forall T1,T2.(t_len T2) < (t_len (Arrow T1 T2)). +intros.simplify. +(* FIXME!!! BUG?!?! *) +cut ((max (t_len T1) (t_len T2)) = match (leb (t_len T1) (t_len T2)) with + [ false \Rightarrow (t_len T2) + | true \Rightarrow (t_len T1) ]) + [rewrite > Hcut;cut ((leb (t_len T1) (t_len T2)) = false \lor + (leb (t_len T1) (t_len T2)) = true) + [lapply (leb_to_Prop (t_len T1) (t_len T2));elim Hcut1 + [rewrite > H;rewrite > H in Hletin;simplify;simplify in Hletin; + lapply (not_le_to_lt ? ? Hletin);unfold in Hletin1;unfold; + constructor 2;assumption + |rewrite > H;simplify;unfold;constructor 1] + |elim (leb (t_len T1) (t_len T2));auto] + |elim T1;simplify;reflexivity] +qed. + +lemma t_len_forall1 : \forall T1,T2.(t_len T1) < (t_len (Forall T1 T2)). +intros.simplify. +(* FIXME!!! BUG?!?! *) +cut ((max (t_len T1) (t_len T2)) = match (leb (t_len T1) (t_len T2)) with + [ false \Rightarrow (t_len T2) + | true \Rightarrow (t_len T1) ]) + [rewrite > Hcut;cut ((leb (t_len T1) (t_len T2)) = false \lor + (leb (t_len T1) (t_len T2)) = true) + [lapply (leb_to_Prop (t_len T1) (t_len T2));elim Hcut1 + [rewrite > H;rewrite > H in Hletin;simplify;constructor 1 + |rewrite > H;rewrite > H in Hletin;simplify;simplify in Hletin; + unfold;apply le_S_S;assumption] + |elim (leb (t_len T1) (t_len T2));auto] + |elim T1;simplify;reflexivity] +qed. + +lemma t_len_forall2 : \forall T1,T2.(t_len T2) < (t_len (Forall T1 T2)). +intros.simplify. +(* FIXME!!! BUG?!?! *) +cut ((max (t_len T1) (t_len T2)) = match (leb (t_len T1) (t_len T2)) with + [ false \Rightarrow (t_len T2) + | true \Rightarrow (t_len T1) ]) + [rewrite > Hcut;cut ((leb (t_len T1) (t_len T2)) = false \lor + (leb (t_len T1) (t_len T2)) = true) + [lapply (leb_to_Prop (t_len T1) (t_len T2));elim Hcut1 + [rewrite > H;rewrite > H in Hletin;simplify;simplify in Hletin; + lapply (not_le_to_lt ? ? Hletin);unfold in Hletin1;unfold; + constructor 2;assumption + |rewrite > H;simplify;unfold;constructor 1] + |elim (leb (t_len T1) (t_len T2));auto] + |elim T1;simplify;reflexivity] +qed. + +lemma eq_t_len_TFree_subst : \forall T,n,X.(t_len T) = + (t_len (subst_type_nat T (TFree X) n)). +intro.elim T + [simplify;elim (eqb n n1) + [simplify;reflexivity + |simplify;reflexivity] + |simplify;reflexivity + |simplify;reflexivity + |simplify;lapply (H n X);lapply (H1 n X);rewrite < Hletin;rewrite < Hletin1; + reflexivity + |simplify;lapply (H n X);lapply (H1 (S n) X);rewrite < Hletin; + 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 s 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 t))) + [cut (\lnot (in_list ? v (fv_type t))) + [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. + +(*** alternative "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 ***) + +lemma JS_to_WFE : \forall G,T,U.(JSubtype G T U) \to (WFEnv G). +intros;elim H;assumption. +qed. + +lemma JS_to_WFT : \forall G,T,U.(JSubtype G T U) \to ((WFType G T) \land + (WFType G U)). +intros;elim H + [split [assumption|apply WFT_Top] + |split;apply WFT_TFree;assumption + |split + [apply WFT_TFree;apply boundinenv_natinfv;apply ex_intro + [apply true | apply ex_intro [apply t1 |assumption]] + |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]]]] +qed. + +lemma JS_to_WFT1 : \forall G,T,U.(JSubtype G T U) \to (WFType G T). +intros;lapply (JS_to_WFT ? ? ? H);elim Hletin;assumption. +qed. + +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. + +(*** alternative "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. \ No newline at end of file diff --git a/helm/software/matita/library/Fsub/part1a.ma b/helm/software/matita/library/Fsub/part1a.ma new file mode 100644 index 000000000..622b8f702 --- /dev/null +++ b/helm/software/matita/library/Fsub/part1a.ma @@ -0,0 +1,380 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +set "baseuri" "cic:/matita/Fsub/part1a/". +include "library/logic/equality.ma". +include "library/nat/nat.ma". +include "library/datatypes/bool.ma". +include "library/nat/compare.ma". +include "Fsub/defn.ma". + +theorem JS_Refl : \forall T,G.(WFType G T) \to (WFEnv G) \to (JSubtype G T T). +apply Typ_len_ind;intro;elim U + [(* FIXME *) generalize in match H1;intro;inversion H1 + [intros;destruct H6 + |intros;destruct H5 + |intros;destruct H9 + |intros;destruct H9] + |apply (SA_Refl_TVar ? ? H2);(*FIXME*)generalize in match H1;intro; + inversion H1 + [intros;destruct H6;rewrite > Hcut;assumption + |intros;destruct H5 + |intros;destruct H9 + |intros;destruct H9] + |apply (SA_Top ? ? H2 H1) + |cut ((WFType G t) \land (WFType G t1)) + [elim Hcut;apply SA_Arrow + [apply H2 + [apply t_len_arrow1 + |assumption + |assumption] + |apply H2 + [apply t_len_arrow2 + |assumption + |assumption]] + |(*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 + |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;auto]] +qed. + +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 JS_weakening : \forall G,T,U.(JSubtype G T U) \to + \forall H.(WFEnv H) \to (incl ? G H) \to (JSubtype H T U). +intros 4;elim H + [apply (SA_Top ? ? H4);lapply (incl_bound_fv ? ? H5); + apply (WFT_env_incl ? ? H2 ? Hletin) + |apply (SA_Refl_TVar ? ? H4);lapply (incl_bound_fv ? ? H5); + apply (Hletin ? H2) + |lapply (H3 ? H5 H6);lapply (H6 ? H1); + apply (SA_Trans_TVar ? ? ? ? Hletin1 Hletin) + |lapply (H2 ? H6 H7);lapply (H4 ? H6 H7); + apply (SA_Arrow ? ? ? ? ? Hletin Hletin1) + |lapply (H2 ? H6 H7);apply (SA_All ? ? ? ? ? Hletin);intros;apply H4 + [unfold;intro;apply H8;lapply (incl_bound_fv ? ? H7);apply (Hletin1 ? H9) + |apply WFE_cons + [assumption + |assumption + |lapply (incl_bound_fv ? ? H7);apply (WFT_env_incl ? ? ? ? Hletin1); + apply (JS_to_WFT1 ? ? ? H1)] + |unfold;intros;inversion H9 + [intros;lapply (inj_head ? ? ? ? H11);rewrite > Hletin1;apply in_Base + |intros;lapply (inj_tail ? ? ? ? ? H13);rewrite < Hletin1 in H10; + apply in_Skip;apply (H7 ? H10)]]] +qed. + +lemma fv_env_extends : \forall H,x,B,C,T,U,G. + (fv_env (H @ ((mk_bound B x T) :: G))) = + (fv_env (H @ ((mk_bound C x U) :: G))). +intros;elim H + [simplify;reflexivity + |elim s;simplify;rewrite > H1;reflexivity] +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))). +intros 7;elim H 0 + [simplify;intros;(*FIXME*)generalize in match H1;intro;inversion H1 + [intros;lapply (nil_cons ? G (mk_bound B x T));lapply (Hletin H4); + elim Hletin1 + |intros;lapply (inj_tail ? ? ? ? ? H8);lapply (inj_head ? ? ? ? H8); + destruct Hletin1;rewrite < Hletin in H6;rewrite < Hletin in H4; + rewrite < Hcut1 in H6;apply (WFE_cons ? ? ? ? H4 H6 H2)] + |intros;simplify;generalize in match H2;elim s;simplify in H4; + inversion H4 + [intros;absurd (mk_bound b n t::l@(mk_bound B x T::G)=Empty) + [assumption + |apply nil_cons] + |intros;lapply (inj_tail ? ? ? ? ? H9);lapply (inj_head ? ? ? ? H9); + destruct Hletin1;apply WFE_cons + [apply H1 + [rewrite > Hletin;assumption + |assumption] + |rewrite > Hcut1;generalize in match H7;rewrite < Hletin; + rewrite > (fv_env_extends ? x B C T U);intro;assumption + |rewrite < Hletin in H8;rewrite > Hcut2; + apply (WFT_env_incl ? ? H8);rewrite > (fv_env_extends ? x B C T U); + unfold;intros;assumption]]] +qed. + +lemma lookup_env_extends : \forall G,H,B,C,D,T,U,V,x,y. + (in_list ? (mk_bound D y V) (H @ ((mk_bound C x U) :: G))) \to + (y \neq x) \to + (in_list ? (mk_bound D y V) (H @ ((mk_bound B x T) :: G))). +intros 10;elim H + [simplify in H1;(*FIXME*)generalize in match H1;intro;inversion H1 + [intros;lapply (inj_head ? ? ? ? H5);rewrite < H4 in Hletin; + destruct Hletin;absurd (y = x) [symmetry;assumption|assumption] + |intros;simplify;lapply (inj_tail ? ? ? ? ? H7);rewrite > Hletin; + apply in_Skip;assumption] + |(*FIXME*)generalize in match H2;intro;inversion H2 + [intros;simplify in H6;lapply (inj_head ? ? ? ? H6);rewrite > Hletin; + simplify;apply in_Base + |simplify;intros;lapply (inj_tail ? ? ? ? ? H8);rewrite > Hletin in H1; + rewrite > H7 in H1;apply in_Skip;apply (H1 H5 H3)]] +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 + [simplify in H;elim (not_le_Sn_n ? H) + |simplify in H;elim (not_le_Sn_n ? H) + |simplify in H;elim (not_le_Sn_n ? H) + |simplify in H2;elim (not_le_Sn_O ? H2) + |simplify in H2;elim (not_le_Sn_O ? H2)] + |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. +intros 6;elim H + [lapply (in_list_nil ? ? H1);elim Hletin + |inversion H6 + [intros;rewrite < H7 in H8;lapply (inj_head ? ? ? ? H8); + rewrite > Hletin in H5;inversion H5 + [intros;rewrite < H9 in H10;lapply (inj_head ? ? ? ? H10); + destruct Hletin1;symmetry;assumption + |intros;lapply (inj_tail ? ? ? ? ? H12);rewrite < Hletin1 in H9; + rewrite < H11 in H9;lapply (boundinenv_natinfv x e) + [destruct Hletin;rewrite < Hcut1 in Hletin2;lapply (H3 Hletin2); + elim Hletin3 + |apply ex_intro + [apply B|apply ex_intro [apply T|assumption]]]] + |intros;lapply (inj_tail ? ? ? ? ? H10);rewrite < H9 in H7; + rewrite < Hletin in H7;(*FIXME*)generalize in match H5;intro;inversion H5 + [intros;rewrite < H12 in H13;lapply (inj_head ? ? ? ? H13); + destruct Hletin1;rewrite < Hcut1 in H7; + lapply (boundinenv_natinfv n e) + [lapply (H3 Hletin2);elim Hletin3 + |apply ex_intro + [apply B|apply ex_intro [apply U|assumption]]] + |intros;apply (H2 ? H7);rewrite > H14;lapply (inj_tail ? ? ? ? ? H15); + rewrite > Hletin1;assumption]]] +qed. + +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 + (JSubtype G T U)) \land + (\forall G,H,X,P,Q,M,N. + (t_len Q) \leq n \to + (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 + (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 s 0;simplify;intros;apply incl_nat_cons;assumption]]] + |intros 4;(*generalize in match H1;*)elim H1 + [inversion H5 + [intros;rewrite < H8;apply (SA_Top ? ? H2 H3) + |intros;destruct H9 + |intros;destruct H10 + |intros;destruct H11 + |intros;destruct H11] + |assumption + |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 + [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) + |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] + |inversion H7 + [intros;apply (SA_Top ? ? H8);rewrite < H10;apply WFT_Forall + [apply (JS_to_WFT2 ? ? ? H2) + |intros;lapply (H4 ? H13);lapply (JS_to_WFT1 ? ? ? Hletin); + apply (WFT_env_incl ? ? Hletin1);simplify;unfold;intros; + assumption] + |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) + |assumption + |assumption + |assumption + |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)]]] + |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)]]]] +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; +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; +qed. \ No newline at end of file