(**************************************************************************)
set "baseuri" "cic:/matita/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].
+include "Fsub/util.ma".
(*** representation of Fsub types ***)
inductive Typ : Set \def
| 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 {
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/Fsub/defn/Typ.ind#xpointer(1/1/5) S T).
-
-interpretation "bound var" 'var x = (cic:/matita/Fsub/defn/Typ.ind#xpointer(1/1/1) x).
-
-interpretation "bound tvar" 'tvar x = (cic:/matita/Fsub/defn/Typ.ind#xpointer(1/1/3) 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/Fsub/defn/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 *)
| (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
|(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.
+(*** definitions about lists ***)
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 []
|(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
+inductive WFType : (list bound) \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)
(\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
+ (subst_type_nat U (TFree X) O))) \to
(WFType G (Forall T U)).
(*** Environment Well-Formedness judgement ***)
-inductive WFEnv : Env \to Prop \def
- | WFE_Empty : (WFEnv Empty)
+inductive WFEnv : (list bound) \to Prop \def
+ | WFE_Empty : (WFEnv (nil ?))
| 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
+inductive JSubtype : (list bound) \to Typ \to Typ \to Prop \def
+ | SA_Top : \forall G.\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)
+ | SA_Refl_TVar : \forall G.\forall X:nat.(WFEnv G)
+ \to (in_list ? X (fv_env G))
\to (JSubtype G (TFree X) (TFree X))
- | SA_Trans_TVar : \forall G:Env.\forall X:nat.\forall T:Typ.
+ | SA_Trans_TVar : \forall G.\forall X:nat.\forall T:Typ.
\forall U:Typ.
- (var_bind_in_env (mk_bound true X U) G) \to
+ (in_list ? (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.
+ | SA_Arrow : \forall G.\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.
+ | SA_All : \forall G.\forall S1,S2,T1,T2:Typ.
(JSubtype G T1 S1) \to
- (\forall X:nat.\lnot (var_in_env X G) \to
+ (\forall X:nat.\lnot (in_list ? X (fv_env G)) \to
(JSubtype ((mk_bound true X T1) :: G)
- (subst_type_O S2 (TFree X)) (subst_type_O T2 (TFree X)))) \to
+ (subst_type_nat S2 (TFree X) O) (subst_type_nat T2 (TFree X) O))) \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 ***)
+notation "hvbox(e ⊢ break ta ⊴ break tb)"
+ non associative with precedence 30 for @{ 'subjudg $e $ta $tb }.
+interpretation "Fsub subtype judgement" 'subjudg e ta tb =
+ (cic:/matita/Fsub/defn/JSubtype.ind#xpointer(1/1) e ta tb).
+
+notation > "hvbox(\Forall S.T)"
+ non associative with precedence 60 for @{ 'forall $S $T}.
+notation < "hvbox('All' \sub S. break T)"
+ non associative with precedence 60 for @{ 'forall $S $T}.
+interpretation "universal type" 'forall S T =
+ (cic:/matita/Fsub/defn/Typ.ind#xpointer(1/1/5) S T).
+
+notation "#x" with precedence 79 for @{'tvar $x}.
+interpretation "bound tvar" 'tvar x =
+ (cic:/matita/Fsub/defn/Typ.ind#xpointer(1/1/1) x).
-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.
+notation "!x" with precedence 79 for @{'tname $x}.
+interpretation "bound tname" 'tname x =
+ (cic:/matita/Fsub/defn/Typ.ind#xpointer(1/1/2) x).
+
+notation "⊤" with precedence 90 for @{'toptype}.
+interpretation "toptype" 'toptype =
+ (cic:/matita/Fsub/defn/Typ.ind#xpointer(1/1/3)).
+
+notation "hvbox(s break ⇛ t)"
+ right associative with precedence 55 for @{ 'arrow $s $t }.
+interpretation "arrow type" 'arrow S T =
+ (cic:/matita/Fsub/defn/Typ.ind#xpointer(1/1/4) S T).
+
+notation "hvbox(S [# n ↦ T])"
+ non associative with precedence 80 for @{ 'substvar $S $T $n }.
+interpretation "subst bound var" 'substvar S T n =
+ (cic:/matita/Fsub/defn/subst_type_nat.con S T n).
-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.
+notation "hvbox(|T|)"
+ non associative with precedence 30 for @{ 'tlen $T }.
+interpretation "type length" 'tlen T =
+ (cic:/matita/Fsub/defn/t_len.con T).
-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.
+notation "hvbox(!X ⊴ T)"
+ non associative with precedence 60 for @{ 'subtypebound $X $T }.
+interpretation "subtyping bound" 'subtypebound X T =
+ (cic:/matita/Fsub/defn/bound.ind#xpointer(1/1/1) true X T).
-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.
+(****** PROOFS ********)
-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.
+(*** theorems about lists ***)
-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)]]
+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;elim (in_cons_case ? ? ? ? H3)
+ [rewrite < H4;simplify;apply in_Base
+ |simplify;apply in_Skip;apply H;apply (ex_intro ? ? a);
+ apply (ex_intro ? ? a1);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 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;elim (in_cons_case ? ? ? ? H1)
+ [rewrite < H2;apply (ex_intro ? ? b);apply (ex_intro ? ? t1);apply in_Base
+ |elim (H H2);elim H3;apply (ex_intro ? ? a);
+ apply (ex_intro ? ? a1);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 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 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]
+lemma incl_cons : \forall x,l1,l2.
+ (incl ? l1 l2) \to (incl nat (x :: l1) (x :: l2)).
+intros.unfold in H.unfold.intros.elim (in_cons_case ? ? ? ? H1)
+ [rewrite > H2;apply in_Base|apply in_Skip;apply (H ? H2)]
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 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 ? ? H8)
+ [unfold;intro;apply H7;apply(H6 ? H9)
+ |simplify;apply (incl_cons ? ? ? H6)]]]
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 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.
-
-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 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;elim (in_cons_case ? ? ? ? H1)
+ [destruct H3;elim (H2);reflexivity
+ |simplify;apply (in_Skip ? ? ? ? H3);]
+ |simplify in H2;simplify;elim (in_cons_case ? ? ? ? H2)
+ [rewrite > H4;apply in_Base
+ |apply (in_Skip ? ? ? ? (H1 H4 H3))]]
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)]]]
+ [simplify in H;elim (in_list_nil ? ? H)
+ |2,3:simplify;simplify in H;assumption
+ |*:simplify in H2;simplify;elim (append_to_or_in_list ? ? ? ? H2)
+ [1,3:apply in_list_append1;apply (H ? H3)
+ |*:apply in_list_append2;apply (H1 ? H3)]]
qed.
(*** lemma on fresh names ***)
[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]]]]
+ [apply (ex_intro ? ? O);intros;unfold;intro;elim (in_list_nil ? ? H1)
+ |elim H;
+ apply (ex_intro ? ? (S (max a t))).
+ intros.unfold. intro.
+ elim (in_cons_case ? ? ? ? H3)
+ [rewrite > H4 in H2.autobatch
+ |elim H4
+ [apply (H1 m ? H4).apply (trans_le ? (max a t));autobatch
+ |assumption]]]]
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)).
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))))
+ [simplify in H2;elim (in_cons_case ? ? ? ? H2)
+ [rewrite > H3;assumption|elim (in_list_nil ? ? H3)]
+ |simplify in H1;elim (in_list_nil ? x H1)
+ |simplify in H5;elim (append_to_or_in_list ? ? ? ? H5);autobatch
+ |simplify in H5;elim (append_to_or_in_list ? ? ? ? H5)
+ [apply (H2 H6)
+ |elim (fresh_name ((fv_type t1) @ (fv_env l)));
+ cut (¬ (a ∈ (fv_type t1)) ∧ ¬ (a ∈ (fv_env l)))
[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]
+ [cut (x ≠ a)
+ [simplify in Hletin;elim (in_cons_case ? ? ? ? Hletin)
+ [elim (Hcut1 H10)
+ |assumption]
+ |intro;apply H8;applyS H6]
+ |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 ***)
-
-(* 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]]
-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]]]
+ [intro;apply H7;apply in_list_append1;assumption
+ |intro;apply H7;apply in_list_append2;assumption]]]]
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 ***)
+(*** lemmata relating subtyping and 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).
+lemma JS_to_WFE : \forall G,T,U.(G \vdash T ⊴ U) \to (WFEnv G).
intros;elim H;assumption.
qed.
|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).
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]
+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));elim (Hletin H4)
+ |destruct H8;apply (WFE_cons ? ? ? ? H4 H6 H2)]
+ |intros;simplify;generalize in match H2;elim t;simplify in H4;
+ inversion H4;intros
+ [destruct H5
+ |destruct H9;apply WFE_cons
+ [apply (H1 H5 H3)
+ |rewrite < (fv_env_extends ? x B C T U); assumption
+ |apply (WFT_env_incl ? ? H8);
+ rewrite < (fv_env_extends ? x B C T U);unfold;intros;
+ 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.
+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
+ |elim (in_cons_case ? ? ? ? H6)
+ [destruct H7;destruct;elim (in_cons_case ? ? ? ? H5)
+ [destruct H7;reflexivity
+ |elim H7;elim H3;apply boundinenv_natinfv;apply (ex_intro ? ? B);
+ apply (ex_intro ? ? T);assumption]
+ |elim (in_cons_case ? ? ? ? H5)
+ [destruct H8;elim H3;apply boundinenv_natinfv;apply (ex_intro ? ? B);
+ apply (ex_intro ? ? U);assumption
+ |apply (H2 H8 H7)]]]
+qed.
+
+lemma WFT_to_incl: ∀G,T,U.
+ (∀X.(¬(X ∈ fv_env G)) → (¬(X ∈ fv_type U)) →
+ (WFType (mk_bound true X T::G) (subst_type_nat U (TFree X) O)))
+ → incl ? (fv_type U) (fv_env G).
+intros.elim (fresh_name ((fv_type U)@(fv_env G))).lapply(H a)
+ [unfold;intros;lapply (fv_WFT ? x ? Hletin)
+ [simplify in Hletin1;inversion Hletin1;intros
+ [destruct H4;elim H1;autobatch
+ |destruct H6;assumption]
+ |apply in_FV_subst;assumption]
+ |*:intro;apply H1;autobatch]
+qed.
+
+lemma incl_fv_env: ∀X,G,G1,U,P.
+ incl ? (fv_env (G1@(mk_bound true X U::G)))
+ (fv_env (G1@(mk_bound true X P::G))).
+intros.rewrite < fv_env_extends.apply incl_A_A.
+qed.
+
+lemma JSubtype_Top: ∀G,P.G ⊢ ⊤ ⊴ P → P = ⊤.
+intros.inversion H;intros
+ [assumption|reflexivity
+ |destruct H5|*:destruct H6]
+qed.
+
+(* elim vs inversion *)
+lemma JS_trans_TFree: ∀G,T,X.G ⊢ T ⊴ (TFree X) →
+ ∀U.G ⊢ (TFree X) ⊴ U → G ⊢ T ⊴ U.
+intros 4.cut (∀Y.TFree Y = TFree X → ∀U.G ⊢ (TFree Y) ⊴ U → G ⊢ T ⊴ U)
+ [apply Hcut;reflexivity
+ |elim H;intros
+ [rewrite > H3 in H4;rewrite > (JSubtype_Top ? ? H4);apply SA_Top;assumption
+ |rewrite < H3;assumption
+ |apply (SA_Trans_TVar ? ? ? ? H1);apply (H3 Y);assumption
+ |*:destruct H5]]
+qed.
+
+lemma fv_append : ∀G,H.fv_env (G @ H) = (fv_env G @ fv_env H).
+intro;elim G;simplify;autobatch paramodulation;
+qed.
\ No newline at end of file
projects / helm.git / blobdiff