include "Fsub/util.ma".
(*** representation of Fsub types ***)
-inductive Typ : Type \def
+inductive Typ : Set \def
| TVar : nat \to Typ (* type var *)
| TFree: nat \to Typ (* free type name *)
| Top : Typ (* maximum type *)
| Forall : Typ \to Typ \to Typ. (* universal type *)
(*** representation of Fsub terms ***)
-inductive Term : Type \def
+inductive Term : Set \def
| Var : nat \to Term (* variable *)
| Free : nat \to Term (* free name *)
| Abs : Typ \to Term \to Term (* abstraction *)
(* representation of bounds *)
-record bound : Type \def {
+record bound : Set \def {
istype : bool; (* is subtyping bound? *)
name : nat ; (* name *)
btype : Typ (* type to which the name is bound *)
| T_Sub : \forall G:Env.\forall t:Term.\forall T:Typ.
\forall S:Typ.(JType G t S) \to (JSubtype G S T) \to (JType G t T).
-(*** definitions about swaps ***)
-
-let rec swap_Typ u v T on T \def
- match T with
- [(TVar n) \Rightarrow (TVar n)
- |(TFree X) \Rightarrow (TFree (swap u v X))
- |Top \Rightarrow Top
- |(Arrow T1 T2) \Rightarrow (Arrow (swap_Typ u v T1) (swap_Typ u v T2))
- |(Forall T1 T2) \Rightarrow (Forall (swap_Typ u v T1) (swap_Typ u v T2))].
-
-definition swap_bound : nat \to nat \to bound \to bound \def
- \lambda u,v,b.match b with
- [(mk_bound B X T) \Rightarrow (mk_bound B (swap u v X) (swap_Typ u v T))].
-
-definition swap_Env : nat \to nat \to Env \to Env \def
- \lambda u,v,G.(map ? ? (\lambda b.(swap_bound u v b)) G).
-
(****** PROOFS ********)
lemma subst_O_nat : \forall T,U.((subst_type_O T U) = (subst_type_nat T U O)).
(* end of fixme *)
-lemma var_notinbG_notinG : \forall G,x,b.
- (\lnot (var_in_env x (b::G)))
- \to \lnot (var_in_env x G).
-intros 3.elim b.unfold.intro.elim H.unfold.simplify.constructor 2.exact H1.
-qed.
-
lemma boundinenv_natinfv : \forall x,G.
(\exists B,T.(in_list ? (mk_bound B x T) G)) \to
(in_list ? x (fv_env G)).
[apply a3
|apply in_Skip;rewrite < H4;assumption]]]]
qed.
-
+
+theorem varinT_varinT_subst : \forall X,Y,T.
+ (in_list ? X (fv_type T)) \to \forall n.
+ (in_list ? X (fv_type (subst_type_nat T (TFree Y) n))).
+intros 3;elim T
+ [simplify in H;elim (in_list_nil ? ? H)
+ |simplify in H;simplify;assumption
+ |simplify in H;elim (in_list_nil ? ? H)
+ |simplify in H2;simplify;elim (nat_in_list_case ? ? ? H2);
+ apply natinG_or_inH_to_natinGH;
+ [left;apply (H1 H3)
+ |right;apply (H H3)]
+ |simplify in H2;simplify;elim (nat_in_list_case ? ? ? H2);
+ apply natinG_or_inH_to_natinGH;
+ [left;apply (H1 H3);
+ |right;apply (H H3)]]
+qed.
+
lemma incl_bound_fv : \forall l1,l2.(incl ? l1 l2) \to
(incl ? (fv_env l1) (fv_env l2)).
intros.unfold in H.unfold.intros.apply boundinenv_natinfv.
|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
assumption]
qed.
-lemma boundin_envappend_case : \forall G,H,b.(var_bind_in_env b (H @ G)) \to
- (var_bind_in_env b G) \lor (var_bind_in_env b H).
-intros 3.elim H
- [simplify in H1;left;assumption
- |unfold in H2;inversion H2
- [intros;simplify in H4;lapply (inj_head ? ? ? ? H4);rewrite > Hletin;
- right;apply in_Base
- |intros;simplify in H6;lapply (inj_tail ? ? ? ? ? H6);rewrite < Hletin in H3;
- rewrite > H5 in H1;lapply (H1 H3);elim Hletin1
- [left;assumption|right;apply in_Skip;assumption]]]
-qed.
-
-lemma varin_envappend_case: \forall G,H,x.(var_in_env x (H @ G)) \to
- (var_in_env x G) \lor (var_in_env x H).
-intros 3.elim H 0
- [simplify;intro;left;assumption
- |intros 2;elim t;simplify in H2;inversion H2
- [intros;lapply (inj_head_nat ? ? ? ? H4);rewrite > Hletin;right;
- simplify;constructor 1
- |intros;lapply (inj_tail ? ? ? ? ? H6);
- lapply H1
- [rewrite < H5;elim Hletin1
- [left;assumption|right;simplify;constructor 2;assumption]
- |unfold var_in_env;unfold fv_env;rewrite > Hletin;rewrite > H5;
- assumption]]]
-qed.
-
-lemma boundinG_or_boundinH_to_boundinGH : \forall G,H,b.
- (var_bind_in_env b G) \lor (var_bind_in_env b H) \to
- (var_bind_in_env b (H @ G)).
-intros.elim H1
- [elim H
- [simplify;assumption
- |simplify;apply in_Skip;assumption]
- |generalize in match H2;elim H2
- [simplify;apply in_Base
- |lapply (H4 H3);simplify;apply in_Skip;assumption]]
-qed.
-
-
-lemma varinG_or_varinH_to_varinGH : \forall G,H,x.
- (var_in_env x G) \lor (var_in_env x H) \to
- (var_in_env x (H @ G)).
-intros.elim H1 0
- [elim H
- [simplify;assumption
- |elim t;simplify;constructor 2;apply (H2 H3)]
- |elim H 0
- [simplify;intro;lapply (in_list_nil nat x H2);elim Hletin
- |intros 2;elim t;simplify in H3;inversion H3
- [intros;lapply (inj_head_nat ? ? ? ? H5);rewrite > Hletin;simplify;
- constructor 1
- |intros;simplify;constructor 2;rewrite < H6;apply H2;
- lapply (inj_tail ? ? ? ? ? H7);rewrite > H6;unfold;unfold fv_env;
- rewrite > Hletin;assumption]]]
-qed.
-
-lemma varbind_to_append : \forall G,b.(var_bind_in_env b G) \to
- \exists G1,G2.(G = (G2 @ (b :: G1))).
-intros.generalize in match H.elim H
- [apply ex_intro [apply l|apply ex_intro [apply Empty|reflexivity]]
- |lapply (H2 H1);elim Hletin;elim H4;rewrite > H5;
- apply ex_intro
- [apply a2|apply ex_intro [apply (a1 :: a3)|simplify;reflexivity]]]
-qed.
-
-
lemma WFT_env_incl : \forall G,T.(WFType G T) \to
\forall H.(incl ? (fv_env G) (fv_env H)) \to (WFType H T).
-intros 4.generalize in match H1.elim H
- [apply WFT_TFree;unfold in H3;apply (H3 ? H2)
+intros 3.elim H
+ [apply WFT_TFree;unfold in H3;apply (H3 ? H1)
|apply WFT_Top
- |apply WFT_Arrow [apply (H3 ? H6)|apply (H5 ? H6)]
+ |apply WFT_Arrow [apply (H2 ? H6)|apply (H4 ? H6)]
|apply WFT_Forall
- [apply (H3 ? H6)
- |intros;apply H5
- [unfold;intro;unfold in H7;apply H7;unfold in H6;apply(H6 ? H9)
+ [apply (H2 ? H6)
+ |intros;apply H4
+ [unfold;intro;apply H7;apply(H6 ? H9)
|assumption
|simplify;apply (incl_nat_cons ? ? ? H6)]]]
qed.
rewrite > H7 in H1;apply in_Skip;apply (H1 H5 H3)]]
qed.
-
-(*** theorems about swaps ***)
-
-lemma fv_subst_type_nat : \forall x,T,y,n.(in_list ? x (fv_type T)) \to
- (in_list ? x (fv_type (subst_type_nat T (TFree y) n))).
-intros 3;elim T 0
- [intros;simplify in H;elim (in_list_nil ? ? H)
- |2,3:simplify;intros;assumption
- |*:intros;simplify in H2;elim (nat_in_list_case ? ? ? H2)
- [1,3:simplify;apply natinG_or_inH_to_natinGH;left;apply (H1 ? H3)
- |*:simplify;apply natinG_or_inH_to_natinGH;right;apply (H ? H3)]]
-qed.
-
-lemma fv_subst_type_O : \forall x,T,y.(in_list ? x (fv_type T)) \to
- (in_list ? x (fv_type (subst_type_O T (TFree y)))).
-intros;rewrite > subst_O_nat;apply (fv_subst_type_nat ? ? ? ? H);
-qed.
-
-lemma swap_Typ_inv : \forall u,v,T.(swap_Typ u v (swap_Typ u v T)) = T.
-intros;elim T
- [1,3:simplify;reflexivity
- |simplify;rewrite > swap_inv;reflexivity
- |*:simplify;rewrite > H;rewrite > H1;reflexivity]
-qed.
-
-lemma swap_Typ_not_free : \forall u,v,T.\lnot (in_list ? u (fv_type T)) \to
- \lnot (in_list ? v (fv_type T)) \to (swap_Typ u v T) = T.
-intros 3;elim T 0
- [1,3:intros;simplify;reflexivity
- |simplify;intros;cut (n \neq u \land n \neq v)
- [elim Hcut;rewrite > (swap_other ? ? ? H2 H3);reflexivity
- |split
- [unfold;intro;apply H;rewrite > H2;apply in_Base
- |unfold;intro;apply H1;rewrite > H2;apply in_Base]]
- |*:simplify;intros;cut ((\lnot (in_list ? u (fv_type t)) \land
- \lnot (in_list ? u (fv_type t1))) \land
- (\lnot (in_list ? v (fv_type t)) \land
- \lnot (in_list ? v (fv_type t1))))
- [1,3:elim Hcut;elim H4;elim H5;clear Hcut H4 H5;rewrite > (H H6 H8);
- rewrite > (H1 H7 H9);reflexivity
- |*:split
- [1,3:split;unfold;intro;apply H2;apply natinG_or_inH_to_natinGH;autobatch
- |*:split;unfold;intro;apply H3;apply natinG_or_inH_to_natinGH;autobatch]]]
-qed.
-
-lemma subst_type_nat_swap : \forall u,v,T,X,m.
- (swap_Typ u v (subst_type_nat T (TFree X) m)) =
- (subst_type_nat (swap_Typ u v T) (TFree (swap u v X)) m).
-intros 4;elim T
- [simplify;elim (eqb_case n m);rewrite > H;simplify;reflexivity
- |2,3:simplify;reflexivity
- |*:simplify;rewrite > H;rewrite > H1;reflexivity]
-qed.
-
-lemma subst_type_O_swap : \forall u,v,T,X.
- (swap_Typ u v (subst_type_O T (TFree X))) =
- (subst_type_O (swap_Typ u v T) (TFree (swap u v X))).
-intros 4;rewrite > (subst_O_nat (swap_Typ u v T));rewrite > (subst_O_nat T);
-apply subst_type_nat_swap;
-qed.
-
-lemma in_fv_type_swap : \forall u,v,x,T.((in_list ? x (fv_type T)) \to
- (in_list ? (swap u v x) (fv_type (swap_Typ u v T)))) \land
- ((in_list ? (swap u v x) (fv_type (swap_Typ u v T))) \to
- (in_list ? x (fv_type T))).
-intros;split
- [elim T 0
- [1,3:simplify;intros;elim (in_list_nil ? ? H)
- |simplify;intros;cut (x = n)
- [rewrite > Hcut;apply in_Base
- |inversion H
- [intros;lapply (inj_head_nat ? ? ? ? H2);rewrite > Hletin;
- reflexivity
- |intros;lapply (inj_tail ? ? ? ? ? H4);rewrite < Hletin in H1;
- elim (in_list_nil ? ? H1)]]
- |*:simplify;intros;elim (nat_in_list_case ? ? ? H2)
- [1,3:apply natinG_or_inH_to_natinGH;left;apply (H1 H3)
- |*:apply natinG_or_inH_to_natinGH;right;apply (H H3)]]
- |elim T 0
- [1,3:simplify;intros;elim (in_list_nil ? ? H)
- |simplify;intros;cut ((swap u v x) = (swap u v n))
- [lapply (swap_inj ? ? ? ? Hcut);rewrite > Hletin;apply in_Base
- |inversion H
- [intros;lapply (inj_head_nat ? ? ? ? H2);rewrite > Hletin;
- reflexivity
- |intros;lapply (inj_tail ? ? ? ? ? H4);rewrite < Hletin in H1;
- elim (in_list_nil ? ? H1)]]
- |*:simplify;intros;elim (nat_in_list_case ? ? ? H2)
- [1,3:apply natinG_or_inH_to_natinGH;left;apply (H1 H3)
- |*:apply natinG_or_inH_to_natinGH;right;apply (H H3)]]]
-qed.
-
-lemma lookup_swap : \forall x,u,v,T,B,G.(in_list ? (mk_bound B x T) G) \to
- (in_list ? (mk_bound B (swap u v x) (swap_Typ u v T)) (swap_Env u v G)).
-intros 6;elim G 0
- [intros;elim (in_list_nil ? ? H)
- |intro;elim t;simplify;inversion H1
- [intros;lapply (inj_head ? ? ? ? H3);rewrite < H2 in Hletin;
- destruct Hletin;rewrite > Hcut;rewrite > Hcut1;rewrite > Hcut2;
- apply in_Base
- |intros;lapply (inj_tail ? ? ? ? ? H5);rewrite < Hletin in H2;
- rewrite < H4 in H2;apply in_Skip;apply (H H2)]]
-qed.
-
lemma in_FV_subst : \forall x,T,U,n.(in_list ? x (fv_type T)) \to
(in_list ? x (fv_type (subst_type_nat T U n))).
intros 3;elim T
|*:right;apply (H ? H3)]]
qed.
-lemma in_dom_swap : \forall u,v,x,G.
- ((in_list ? x (fv_env G)) \to
- (in_list ? (swap u v x) (fv_env (swap_Env u v G)))) \land
- ((in_list ? (swap u v x) (fv_env (swap_Env u v G))) \to
- (in_list ? x (fv_env G))).
-intros;split
- [elim G 0
- [simplify;intro;elim (in_list_nil ? ? H)
- |intro;elim t 0;simplify;intros;inversion H1
- [intros;lapply (inj_head_nat ? ? ? ? H3);rewrite > Hletin;apply in_Base
- |intros;lapply (inj_tail ? ? ? ? ? H5);rewrite < Hletin in H2;
- rewrite > H4 in H;apply in_Skip;apply (H H2)]]
- |elim G 0
- [simplify;intro;elim (in_list_nil ? ? H)
- |intro;elim t 0;simplify;intros;inversion H1
- [intros;lapply (inj_head_nat ? ? ? ? H3);rewrite < H2 in Hletin;
- lapply (swap_inj ? ? ? ? Hletin);rewrite > Hletin1;apply in_Base
- |intros;lapply (inj_tail ? ? ? ? ? H5);rewrite < Hletin in H2;
- rewrite > H4 in H;apply in_Skip;apply (H H2)]]]
-qed.
-
(*** lemma on fresh names ***)
lemma fresh_name : \forall l:(list nat).\exists n.\lnot (in_list ? n l).
|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)).
|apply (H2 H6)]]
qed.
-lemma WFE_consG_to_WFT : \forall G.\forall b,X,T.
- (WFEnv ((mk_bound b X T)::G)) \to (WFType G T).
-intros.
-inversion H
- [intro;reduce in H1;destruct H1
- |intros;lapply (inj_head ? ? ? ? H5);lapply (inj_tail ? ? ? ? ? H5);
- destruct Hletin;rewrite > Hletin1;rewrite > Hcut2;assumption]
-qed.
-
-lemma WFE_consG_WFE_G : \forall G.\forall b.
- (WFEnv (b::G)) \to (WFEnv G).
-intros.
-inversion H
- [intro;reduce in H1;destruct H1
- |intros;lapply (inj_tail ? ? ? ? ? H5);rewrite > Hletin;assumption]
-qed.
-
-(* silly, but later useful *)
-
-lemma env_append_weaken : \forall G,H.(WFEnv (H @ G)) \to
- (incl ? G (H @ G)).
-intros 2;elim H
- [simplify;unfold;intros;assumption
- |simplify in H2;simplify;unfold;intros;apply in_Skip;apply H1
- [apply (WFE_consG_WFE_G ? ? H2)
- |assumption]]
-qed.
-
-lemma WFT_swap : \forall u,v,G,T.(WFType G T) \to
- (WFType (swap_Env u v G) (swap_Typ u v T)).
-intros.elim H
- [simplify;apply WFT_TFree;lapply (natinfv_boundinenv ? ? H1);elim Hletin;
- elim H2;apply boundinenv_natinfv;apply ex_intro
- [apply a
- |apply ex_intro
- [apply (swap_Typ u v a1)
- |apply lookup_swap;assumption]]
- |simplify;apply WFT_Top
- |simplify;apply WFT_Arrow
- [assumption|assumption]
- |simplify;apply WFT_Forall
- [assumption
- |intros;rewrite < (swap_inv u v);
- cut (\lnot (in_list ? (swap u v X) (fv_type t1)))
- [cut (\lnot (in_list ? (swap u v X) (fv_env e)))
- [generalize in match (H4 ? Hcut1 Hcut);simplify;
- rewrite > subst_type_O_swap;intro;assumption
- |lapply (in_dom_swap u v (swap u v X) e);elim Hletin;unfold;
- intros;lapply (H7 H9);rewrite > (swap_inv u v) in Hletin1;
- apply (H5 Hletin1)]
- |generalize in match (in_fv_type_swap u v (swap u v X) t1);intros;
- elim H7;unfold;intro;lapply (H8 H10);
- rewrite > (swap_inv u v) in Hletin;apply (H6 Hletin)]]]
-qed.
-
-lemma WFE_swap : \forall u,v,G.(WFEnv G) \to (WFEnv (swap_Env u v G)).
-intros 3.elim G 0
- [intro;simplify;assumption
- |intros 2;elim t;simplify;constructor 2
- [apply H;apply (WFE_consG_WFE_G ? ? H1)
- |unfold;intro;lapply (in_dom_swap u v n l);elim Hletin;lapply (H4 H2);
- (* FIXME trick *)generalize in match H1;intro;inversion H1
- [intros;absurd ((mk_bound b n t1)::l = [])
- [assumption|apply nil_cons]
- |intros;lapply (inj_head ? ? ? ? H10);lapply (inj_tail ? ? ? ? ? H10);
- destruct Hletin2;rewrite < Hcut1 in H8;rewrite < Hletin3 in H8;
- apply (H8 Hletin1)]
- |apply (WFT_swap u v l t1);inversion H1
- [intro;absurd ((mk_bound b n t1)::l = [])
- [assumption|apply nil_cons]
- |intros;lapply (inj_head ? ? ? ? H6);lapply (inj_tail ? ? ? ? ? H6);
- destruct Hletin;rewrite > Hletin1;rewrite > Hcut2;assumption]]]
-qed.
-
(*** some exotic inductions and related lemmas ***)
lemma not_t_len_lt_SO : \forall T.\lnot (t_len T) < (S O).
|*:simplify in H2;apply H;apply (trans_lt ? ? ? ? H2);unfold;constructor 1]]
qed.
-lemma t_len_gt_O : \forall T.(t_len T) > O.
-intro;elim T
- [1,2,3:simplify;unfold;unfold;constructor 1
- |*:simplify;lapply (max_case (t_len t) (t_len t1));rewrite > Hletin;
- elim (leb (t_len t) (t_len t1))
- [1,3:simplify;unfold;unfold;constructor 2;unfold in H1;unfold in H1;assumption
- |*:simplify;unfold;unfold;constructor 2;unfold in H;unfold in H;assumption]]
-qed.
-
lemma Typ_len_ind : \forall P:Typ \to Prop.
(\forall U.(\forall V.((t_len V) < (t_len U)) \to (P V))
\to (P U))
rewrite < Hletin1;reflexivity]
qed.
-lemma swap_env_not_free : \forall u,v,G.(WFEnv G) \to
- \lnot (in_list ? u (fv_env G)) \to
- \lnot (in_list ? v (fv_env G)) \to
- (swap_Env u v G) = G.
-intros 3.elim G 0
- [simplify;intros;reflexivity
- |intros 2;elim t 0;simplify;intros;lapply (notin_cons ? ? ? ? H2);
- lapply (notin_cons ? ? ? ? H3);elim Hletin;elim Hletin1;
- lapply (swap_other ? ? ? H4 H6);lapply (WFE_consG_to_WFT ? ? ? ? H1);
- cut (\lnot (in_list ? u (fv_type t1)))
- [cut (\lnot (in_list ? v (fv_type t1)))
- [lapply (swap_Typ_not_free ? ? ? Hcut Hcut1);
- lapply (WFE_consG_WFE_G ? ? H1);
- lapply (H Hletin5 H5 H7);
- rewrite > Hletin2;rewrite > Hletin4;rewrite > Hletin6;reflexivity
- |unfold;intro;apply H7;
- apply (fv_WFT ? ? ? Hletin3 H8)]
- |unfold;intro;apply H5;apply (fv_WFT ? ? ? Hletin3 H8)]]
-qed.
-
-(*** alternate "constructor" for universal types' well-formedness ***)
-
-lemma WFT_Forall2 : \forall G,X,T,T1,T2.
- (WFEnv G) \to
- (WFType G T1) \to
- \lnot (in_list ? X (fv_type T2)) \to
- \lnot (in_list ? X (fv_env G)) \to
- (WFType ((mk_bound true X T)::G)
- (subst_type_O T2 (TFree X))) \to
- (WFType G (Forall T1 T2)).
-intros.apply WFT_Forall
- [assumption
- |intros;generalize in match (WFT_swap X X1 ? ? H4);simplify;
- rewrite > swap_left;
- rewrite > (swap_env_not_free X X1 G H H3 H5);
- rewrite > subst_type_O_swap;rewrite > swap_left;
- rewrite > (swap_Typ_not_free ? ? T2 H2 H6);
- intro;apply (WFT_env_incl ? ? H7);unfold;simplify;intros;assumption]
-qed.
-
-(*** lemmas relating subtyping and well-formedness ***)
+(*** lemmata relating subtyping and well-formedness ***)
lemma JS_to_WFE : \forall G,T,U.(JSubtype G T U) \to (WFEnv G).
intros;elim H;assumption.
|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]
-qed.
-
-(*** alternate "constructor" for subtyping between universal types ***)
-
-lemma SA_All2 : \forall G,S1,S2,T1,T2,X.(JSubtype G T1 S1) \to
- \lnot (in_list ? X (fv_env G)) \to
- \lnot (in_list ? X (fv_type S2)) \to
- \lnot (in_list ? X (fv_type T2)) \to
- (JSubtype ((mk_bound true X T1) :: G)
- (subst_type_O S2 (TFree X))
- (subst_type_O T2 (TFree X))) \to
- (JSubtype G (Forall S1 S2) (Forall T1 T2)).
-intros;apply (SA_All ? ? ? ? ? H);intros;
-lapply (decidable_eq_nat X X1);elim Hletin
- [rewrite < H6;assumption
- |elim (JS_to_WFT ? ? ? H);elim (JS_to_WFT ? ? ? H4);
- cut (\lnot (in_list ? X1 (fv_type S2)))
- [cut (\lnot (in_list ? X1 (fv_type T2)))
- [cut (((mk_bound true X1 T1)::G) =
- (swap_Env X X1 ((mk_bound true X T1)::G)))
- [rewrite > Hcut2;
- cut (((subst_type_O S2 (TFree X1)) =
- (swap_Typ X X1 (subst_type_O S2 (TFree X)))) \land
- ((subst_type_O T2 (TFree X1)) =
- (swap_Typ X X1 (subst_type_O T2 (TFree X)))))
- [elim Hcut3;rewrite > H11;rewrite > H12;apply JS_swap;
- assumption
- |split
- [rewrite > (subst_type_O_swap X X1 S2 X);
- rewrite > (swap_Typ_not_free X X1 S2 H2 Hcut);
- rewrite > swap_left;reflexivity
- |rewrite > (subst_type_O_swap X X1 T2 X);
- rewrite > (swap_Typ_not_free X X1 T2 H3 Hcut1);
- rewrite > swap_left;reflexivity]]
- |simplify;lapply (JS_to_WFE ? ? ? H);
- rewrite > (swap_env_not_free X X1 G Hletin1 H1 H5);
- cut ((\lnot (in_list ? X (fv_type T1))) \land
- (\lnot (in_list ? X1 (fv_type T1))))
- [elim Hcut2;rewrite > (swap_Typ_not_free X X1 T1 H11 H12);
- rewrite > swap_left;reflexivity
- |split
- [unfold;intro;apply H1;apply (fv_WFT T1 X G H7 H11)
- |unfold;intro;apply H5;apply (fv_WFT T1 X1 G H7 H11)]]]
- |unfold;intro;apply H5;lapply (fv_WFT ? X1 ? H10)
- [inversion Hletin1
- [intros;simplify in H13;lapply (inj_head_nat ? ? ? ? H13);
- rewrite < H12 in Hletin2;lapply (H6 Hletin2);elim Hletin3
- |intros;simplify in H15;lapply (inj_tail ? ? ? ? ? H15);
- rewrite < Hletin2 in H12;rewrite < H14 in H12;lapply (H5 H12);
- elim Hletin3]
- |rewrite > subst_O_nat;apply in_FV_subst;assumption]]
- |unfold;intro;apply H5;lapply (fv_WFT ? X1 ? H9)
- [inversion Hletin1
- [intros;simplify in H13;lapply (inj_head_nat ? ? ? ? H13);
- rewrite < H12 in Hletin2;lapply (H6 Hletin2);elim Hletin3
- |intros;simplify in H15;lapply (inj_tail ? ? ? ? ? H15);
- rewrite < Hletin2 in H12;rewrite < H14 in H12;lapply (H5 H12);
- elim Hletin3]
- |rewrite > subst_O_nat;apply in_FV_subst;assumption]]]
-qed.
-
lemma WFE_Typ_subst : \forall H,x,B,C,T,U,G.
(WFEnv (H @ ((mk_bound B x T) :: G))) \to (WFType G U) \to
(WFEnv (H @ ((mk_bound C x U) :: G))).
unfold;intros;assumption]]]
qed.
-lemma t_len_pred: \forall T,m.(S (t_len T)) \leq m \to (t_len T) \leq (pred m).
-intros 2;elim m
- [elim (not_le_Sn_O ? H)
- |simplify;apply (le_S_S_to_le ? ? H1)]
-qed.
-
-lemma pred_m_lt_m : \forall m,T.(t_len T) \leq m \to (pred m) < m.
-intros 2;elim m 0
- [elim T
- [4,5:simplify in H2;elim (not_le_Sn_O ? H2)
- |*:simplify in H;elim (not_le_Sn_n ? H)]
- |intros;simplify;unfold;constructor 1]
-qed.
-
lemma WFE_bound_bound : \forall B,x,T,U,G. (WFEnv G) \to
(in_list ? (mk_bound B x T) G) \to
(in_list ? (mk_bound B x U) G) \to T = U.
[apply B|apply ex_intro [apply U|assumption]]]
|intros;apply (H2 ? H7);rewrite > H14;lapply (inj_tail ? ? ? ? ? H15);
rewrite > Hletin1;assumption]]]
-qed.
-
+qed.
\ No newline at end of file
projects / helm.git / blobdiff