X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Flibrary%2FFsub%2Fdefn.ma;h=0a95b31b5115ebb3082f1aba61f83250114d4b06;hb=9f0b789ee03d53d9f00d82f57c64466788504d61;hp=ef94097fb18ab94cd3ec4112a22092f3aa9cd779;hpb=94b16d13221f5ea3618b453e5f86b787d04d664e;p=helm.git diff --git a/helm/software/matita/library/Fsub/defn.ma b/helm/software/matita/library/Fsub/defn.ma index ef94097fb..0a95b31b5 100644 --- a/helm/software/matita/library/Fsub/defn.ma +++ b/helm/software/matita/library/Fsub/defn.ma @@ -18,24 +18,7 @@ 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]. +include "Fsub/util.ma". (*** representation of Fsub types ***) inductive Typ : Set \def @@ -70,7 +53,7 @@ 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)" +(* notation "hvbox(\Forall S. break T)" non associative with precedence 90 for @{ 'forall $S $T}. @@ -94,15 +77,15 @@ 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 "universal type" 'forall S T = (cic:/matita/Fsub/defn/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 var" 'var x = (cic:/matita/Fsub/defn/Typ.ind#xpointer(1/1/1) x). -interpretation "bound tvar" 'tvar x = (cic:/matita/test/Typ.ind#xpointer(1/1/3) x). +interpretation "bound tvar" 'tvar x = (cic:/matita/Fsub/defn/Typ.ind#xpointer(1/1/3) x). -interpretation "bound tname" 'tname x = (cic:/matita/test/Typ.ind#xpointer(1/1/2) x). +interpretation "bound tname" 'tname x = (cic:/matita/Fsub/defn/Typ.ind#xpointer(1/1/2) x). -interpretation "arrow type" 'arrow S T = (cic:/matita/test/Typ.ind#xpointer(1/1/4) S T). +interpretation "arrow type" 'arrow S T = (cic:/matita/Fsub/defn/Typ.ind#xpointer(1/1/4) S T). *) (*** Various kinds of substitution, not all will be used probably ***) @@ -170,104 +153,6 @@ let rec t_len T \def |(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) @@ -278,43 +163,12 @@ definition head_nat \def [ 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)). +(*** definitions about lists ***) (* 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). @@ -326,9 +180,6 @@ definition var_in_env : nat \to Env \to Prop \def 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 [] @@ -337,178 +188,6 @@ let rec fv_type T \def |(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 @@ -578,222 +257,167 @@ 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). +(****** PROOFS ********) -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)]]] +lemma subst_O_nat : \forall T,U.((subst_type_O T U) = (subst_type_nat T U O)). +intros;elim T;simplify;reflexivity; 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; +(*** theorems about lists ***) + +(* FIXME: these definitions shouldn't be part of the poplmark challenge + - use destruct instead, when hopefully it will get fixed... *) + +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 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] +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 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)]] +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. -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)]] +(* end of fixme *) + +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 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); +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. -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] +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 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]]] +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 t;simplify in H1;inversion H1 + [intros;rewrite < H2;simplify;apply ex_intro + [apply b + |apply ex_intro + [apply t1 + |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 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] + +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 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; +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 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)]]] +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. - -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)]] + +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 3.elim H + [apply WFT_TFree;unfold in H3;apply (H3 ? H1) + |apply WFT_Top + |apply WFT_Arrow [apply (H2 ? H6)|apply (H4 ? H6)] + |apply WFT_Forall + [apply (H2 ? H6) + |intros;apply H4 + [unfold;intro;apply H7;apply(H6 ? H9) + |assumption + |simplify;apply (incl_nat_cons ? ? ? H6)]]] +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 t;simplify;rewrite > H1;reflexivity] +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 in_FV_subst : \forall x,T,U,n.(in_list ? x (fv_type T)) \to @@ -804,37 +428,11 @@ intros 3;elim T [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; + |2,3:simplify;simplify in H;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)]]] + [1,3:left;apply (H1 ? H3) + |*:right;apply (H ? H3)]] qed. (*** lemma on fresh names ***) @@ -853,7 +451,7 @@ cut (\forall l:(list nat).\exists n.\forall m. [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 + |elim H;lapply (decidable_eq_nat a t);elim Hletin [apply ex_intro [apply (S a) |intros;unfold;intro;inversion H4 @@ -864,23 +462,23 @@ cut (\forall l:(list nat).\exists n.\forall m. 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) + |cut ((leb a t) = true \lor (leb a t) = false) [elim Hcut [apply ex_intro - [apply (S s) + [apply (S t) |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; + |lapply (leb_to_Prop a t);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; + |intros;lapply (leb_to_Prop a t);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); @@ -889,10 +487,10 @@ cut (\forall l:(list nat).\exists n.\forall m. |intros;lapply (inj_tail ? ? ? ? ? H9); rewrite < Hletin3 in H6;rewrite < H8 in H6; apply (H1 ? H4 H6)]]] - |elim (leb a s);auto]]]] + |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)). @@ -932,105 +530,14 @@ 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. - -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 ***) -(* TODO : relocate the following 3 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]] + [1,2,3: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)) + [1,3: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 Typ_len_ind : \forall P:Typ \to Prop. @@ -1044,15 +551,10 @@ cut (\forall P:Typ \to Prop. [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]]] + [1,2,3:apply H;intros;simplify in H4;elim (not_t_len_lt_SO ? H4) + |*:apply H;intros;apply (H1 (t_len V)) + [1,3:rewrite > H5;assumption + |*:reflexivity]]] qed. lemma t_len_arrow1 : \forall T1,T2.(t_len T1) < (t_len (Arrow T1 T2)). @@ -1067,7 +569,7 @@ cut ((max (t_len T1) (t_len T2)) = match (leb (t_len T1) (t_len T2)) with [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 (leb (t_len T1) (t_len T2));autobatch] |elim T1;simplify;reflexivity] qed. @@ -1084,7 +586,7 @@ cut ((max (t_len T1) (t_len T2)) = match (leb (t_len T1) (t_len T2)) with 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 (leb (t_len T1) (t_len T2));autobatch] |elim T1;simplify;reflexivity] qed. @@ -1100,7 +602,7 @@ cut ((max (t_len T1) (t_len T2)) = match (leb (t_len T1) (t_len T2)) with [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 (leb (t_len T1) (t_len T2));autobatch] |elim T1;simplify;reflexivity] qed. @@ -1117,65 +619,22 @@ cut ((max (t_len T1) (t_len T2)) = match (leb (t_len T1) (t_len T2)) with 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 (leb (t_len T1) (t_len T2));autobatch] |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;elim (eqb n n1);simplify;reflexivity + |2,3: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 ***) +(*** 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. @@ -1192,22 +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). @@ -1218,109 +665,57 @@ 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]]] +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 t;simplify in H4; + inversion H4 + [intros;absurd (mk_bound b n t1::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 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. \ No newline at end of file