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
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}.
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).
+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 ***)
|(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)
[ 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).
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
| 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 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 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]]]
+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_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_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 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_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 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 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.
-
-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 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
[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;
+ |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)]]
-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 ***)
[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
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);
|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)).
|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.
[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)).
[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.
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.
[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.
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.
|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.
-
-(*** 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_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
projects / helm.git / blobdiff