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.
+(*** Type Well-Formedness judgement ***)
+
+inductive WFType : Env \to Typ \to Prop \def
+ | WFT_TFree : \forall X,G.(in_list ? X (fv_env G))
+ \to (WFType G (TFree X))
+ | WFT_Top : \forall G.(WFType G Top)
+ | WFT_Arrow : \forall G,T,U.(WFType G T) \to (WFType G U) \to
+ (WFType G (Arrow T U))
+ | WFT_Forall : \forall G,T,U.(WFType G T) \to
+ (\forall X:nat.
+ (\lnot (in_list ? X (fv_env G))) \to
+ (\lnot (in_list ? X (fv_type U))) \to
+ (WFType ((mk_bound true X T) :: G)
+ (subst_type_O U (TFree X)))) \to
+ (WFType G (Forall T U)).
+
+(*** Environment Well-Formedness judgement ***)
+
+inductive WFEnv : Env \to Prop \def
+ | WFE_Empty : (WFEnv Empty)
+ | WFE_cons : \forall B,X,T,G.(WFEnv G) \to
+ \lnot (in_list ? X (fv_env G)) \to
+ (WFType G T) \to (WFEnv ((mk_bound B X T) :: G)).
+
+(*** Subtyping judgement ***)
+inductive JSubtype : Env \to Typ \to Typ \to Prop \def
+ | SA_Top : \forall G:Env.\forall T:Typ.(WFEnv G) \to
+ (WFType G T) \to (JSubtype G T Top)
+ | SA_Refl_TVar : \forall G:Env.\forall X:nat.(WFEnv G) \to (var_in_env X G)
+ \to (JSubtype G (TFree X) (TFree X))
+ | SA_Trans_TVar : \forall G:Env.\forall X:nat.\forall T:Typ.
+ \forall U:Typ.
+ (var_bind_in_env (mk_bound true X U) G) \to
+ (JSubtype G U T) \to (JSubtype G (TFree X) T)
+ | SA_Arrow : \forall G:Env.\forall S1,S2,T1,T2:Typ.
+ (JSubtype G T1 S1) \to (JSubtype G S2 T2) \to
+ (JSubtype G (Arrow S1 S2) (Arrow T1 T2))
+ | SA_All : \forall G:Env.\forall S1,S2,T1,T2:Typ.
+ (JSubtype G T1 S1) \to
+ (\forall X:nat.\lnot (var_in_env X G) \to
+ (JSubtype ((mk_bound true X T1) :: G)
+ (subst_type_O S2 (TFree X)) (subst_type_O T2 (TFree X)))) \to
+ (JSubtype G (Forall S1 S2) (Forall T1 T2)).
+
+(*** Typing judgement ***)
+inductive JType : Env \to Term \to Typ \to Prop \def
+ | T_Var : \forall G:Env.\forall x:nat.\forall T:Typ.
+ (WFEnv G) \to (var_bind_in_env (mk_bound false x T) G) \to
+ (JType G (Free x) T)
+ | T_Abs : \forall G.\forall T1,T2:Typ.\forall t2:Term.
+ \forall x:nat.
+ (JType ((mk_bound false x T1)::G) (subst_term_O t2 (Free x)) T2) \to
+ (JType G (Abs T1 t2) (Arrow T1 T2))
+ | T_App : \forall G.\forall t1,t2:Term.\forall T2:Typ.
+ \forall T1:Typ.(JType G t1 (Arrow T1 T2)) \to (JType G t2 T1) \to
+ (JType G (App t1 t2) T2)
+ | T_TAbs : \forall G:Env.\forall T1,T2:Typ.\forall t2:Term.
+ \forall X:nat.
+ (JType ((mk_bound true X T1)::G)
+ (subst_term_tO t2 (TFree X)) (subst_type_O T2 (TFree X)))
+ \to (JType G (TAbs T1 t2) (Forall T1 T2))
+ | T_TApp : \forall G:Env.\forall t1:Term.\forall T2,T12:Typ.
+ \forall X:nat.\forall T11:Typ.
+ (JType G t1 (Forall T11 (subst_type_tfree_type T12 X (TVar O)))) \to
+ (JSubtype G T2 T11)
+ \to (JType G (TApp t1 T2) (subst_type_tfree_type T12 X T2))
+ | T_Sub : \forall G:Env.\forall t:Term.\forall T:Typ.
+ \forall S:Typ.(JType G t S) \to (JSubtype G S T) \to (JType G t T).
+
+(*** 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)).
+intros;elim T;simplify;reflexivity;
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]]
+(*** 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 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]
+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 *)
+
+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.
[apply a2|apply ex_intro [apply (a1 :: a3)|simplify;reflexivity]]]
qed.
-(*** Type Well-Formedness judgement ***)
-
-inductive WFType : Env \to Typ \to Prop \def
- | WFT_TFree : \forall X,G.(in_list ? X (fv_env G))
- \to (WFType G (TFree X))
- | WFT_Top : \forall G.(WFType G Top)
- | WFT_Arrow : \forall G,T,U.(WFType G T) \to (WFType G U) \to
- (WFType G (Arrow T U))
- | WFT_Forall : \forall G,T,U.(WFType G T) \to
- (\forall X:nat.
- (\lnot (in_list ? X (fv_env G))) \to
- (\lnot (in_list ? X (fv_type U))) \to
- (WFType ((mk_bound true X T) :: G)
- (subst_type_O U (TFree X)))) \to
- (WFType G (Forall T U)).
-
-(*** Environment Well-Formedness judgement ***)
-
-inductive WFEnv : Env \to Prop \def
- | WFE_Empty : (WFEnv Empty)
- | WFE_cons : \forall B,X,T,G.(WFEnv G) \to
- \lnot (in_list ? X (fv_env G)) \to
- (WFType G T) \to (WFEnv ((mk_bound B X T) :: G)).
-
-(*** Subtyping judgement ***)
-inductive JSubtype : Env \to Typ \to Typ \to Prop \def
- | SA_Top : \forall G:Env.\forall T:Typ.(WFEnv G) \to
- (WFType G T) \to (JSubtype G T Top)
- | SA_Refl_TVar : \forall G:Env.\forall X:nat.(WFEnv G) \to (var_in_env X G)
- \to (JSubtype G (TFree X) (TFree X))
- | SA_Trans_TVar : \forall G:Env.\forall X:nat.\forall T:Typ.
- \forall U:Typ.
- (var_bind_in_env (mk_bound true X U) G) \to
- (JSubtype G U T) \to (JSubtype G (TFree X) T)
- | SA_Arrow : \forall G:Env.\forall S1,S2,T1,T2:Typ.
- (JSubtype G T1 S1) \to (JSubtype G S2 T2) \to
- (JSubtype G (Arrow S1 S2) (Arrow T1 T2))
- | SA_All : \forall G:Env.\forall S1,S2,T1,T2:Typ.
- (JSubtype G T1 S1) \to
- (\forall X:nat.\lnot (var_in_env X G) \to
- (JSubtype ((mk_bound true X T1) :: G)
- (subst_type_O S2 (TFree X)) (subst_type_O T2 (TFree X)))) \to
- (JSubtype G (Forall S1 S2) (Forall T1 T2)).
-
-(*** Typing judgement ***)
-inductive JType : Env \to Term \to Typ \to Prop \def
- | T_Var : \forall G:Env.\forall x:nat.\forall T:Typ.
- (WFEnv G) \to (var_bind_in_env (mk_bound false x T) G) \to
- (JType G (Free x) T)
- | T_Abs : \forall G.\forall T1,T2:Typ.\forall t2:Term.
- \forall x:nat.
- (JType ((mk_bound false x T1)::G) (subst_term_O t2 (Free x)) T2) \to
- (JType G (Abs T1 t2) (Arrow T1 T2))
- | T_App : \forall G.\forall t1,t2:Term.\forall T2:Typ.
- \forall T1:Typ.(JType G t1 (Arrow T1 T2)) \to (JType G t2 T1) \to
- (JType G (App t1 t2) T2)
- | T_TAbs : \forall G:Env.\forall T1,T2:Typ.\forall t2:Term.
- \forall X:nat.
- (JType ((mk_bound true X T1)::G)
- (subst_term_tO t2 (TFree X)) (subst_type_O T2 (TFree X)))
- \to (JType G (TAbs T1 t2) (Forall T1 T2))
- | T_TApp : \forall G:Env.\forall t1:Term.\forall T2,T12:Typ.
- \forall X:nat.\forall T11:Typ.
- (JType G t1 (Forall T11 (subst_type_tfree_type T12 X (TVar O)))) \to
- (JSubtype G T2 T11)
- \to (JType G (TApp t1 T2) (subst_type_tfree_type T12 X T2))
- | T_Sub : \forall G:Env.\forall t:Term.\forall T:Typ.
- \forall S:Typ.(JType G t S) \to (JSubtype G S T) \to (JType G t T).
-
lemma WFT_env_incl : \forall G,T.(WFType G T) \to
\forall H.(incl ? (fv_env G) (fv_env H)) \to (WFType H T).
|simplify;apply (incl_nat_cons ? ? ? H6)]]]
qed.
-(*** definitions and theorems about swaps ***)
-
-definition swap : nat \to nat \to nat \to nat \def
- \lambda u,v,x.match (eqb x u) with
- [true \Rightarrow v
- |false \Rightarrow match (eqb x v) with
- [true \Rightarrow u
- |false \Rightarrow x]].
-
-lemma swap_left : \forall x,y.(swap x y x) = y.
-intros;unfold swap;rewrite > eqb_n_n;simplify;reflexivity;
+lemma fv_env_extends : \forall H,x,B,C,T,U,G.
+ (fv_env (H @ ((mk_bound B x T) :: G))) =
+ (fv_env (H @ ((mk_bound C x U) :: G))).
+intros;elim H
+ [simplify;reflexivity
+ |elim s;simplify;rewrite > H1;reflexivity]
qed.
-lemma 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 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 swap_other : \forall x,y,z.(z \neq x) \to (z \neq y) \to (swap x y z) = z.
-intros;unfold swap;elim (eq_eqb_case z x)
- [elim H2;lapply (H H3);elim Hletin
- |elim H2;rewrite > H4;simplify;elim (eq_eqb_case z y)
- [elim H5;lapply (H1 H6);elim Hletin
- |elim H5;rewrite > H7;simplify;reflexivity]]
-qed.
-
-lemma swap_inv : \forall u,v,x.(swap u v (swap u v x)) = x.
-intros;unfold in match (swap u v x);elim (eq_eqb_case x u)
- [elim H;rewrite > H2;simplify;rewrite > H1;apply swap_right
- |elim H;rewrite > H2;simplify;elim (eq_eqb_case x v)
- [elim H3;rewrite > H5;simplify;rewrite > H4;apply swap_left
- |elim H3;rewrite > H5;simplify;apply (swap_other ? ? ? H1 H4)]]
-qed.
-
-lemma swap_inj : \forall u,v,x,y.(swap u v x) = (swap u v y) \to x = y.
-intros;unfold swap in H;elim (eq_eqb_case x u)
- [elim H1;elim (eq_eqb_case y u)
- [elim H4;rewrite > H5;assumption
- |elim H4;rewrite > H3 in H;rewrite > H6 in H;simplify in H;
- elim (eq_eqb_case y v)
- [elim H7;rewrite > H9 in H;simplify in H;rewrite > H in H8;
- lapply (H5 H8);elim Hletin
- |elim H7;rewrite > H9 in H;simplify in H;elim H8;symmetry;assumption]]
- |elim H1;elim (eq_eqb_case y u)
- [elim H4;rewrite > H3 in H;rewrite > H6 in H;simplify in H;
- elim (eq_eqb_case x v)
- [elim H7;rewrite > H9 in H;simplify in H;rewrite < H in H8;
- elim H2;assumption
- |elim H7;rewrite > H9 in H;simplify in H;elim H8;assumption]
- |elim H4;rewrite > H3 in H;rewrite > H6 in H;simplify in H;
- elim (eq_eqb_case x v)
- [elim H7;rewrite > H9 in H;elim (eq_eqb_case y v)
- [elim H10;rewrite > H11;assumption
- |elim H10;rewrite > H12 in H;simplify in H;elim H5;symmetry;
- assumption]
- |elim H7;rewrite > H9 in H;elim (eq_eqb_case y v)
- [elim H10;rewrite > H12 in H;simplify in H;elim H2;assumption
- |elim H10;rewrite > H12 in H;simplify in H;assumption]]]]
-qed.
+(*** 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)
- |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)]]
+ |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
intros;rewrite > subst_O_nat;apply (fv_subst_type_nat ? ? ? ? H);
qed.
-let rec swap_Typ u v T on T \def
- match T with
- [(TVar n) \Rightarrow (TVar n)
- |(TFree X) \Rightarrow (TFree (swap u v X))
- |Top \Rightarrow Top
- |(Arrow T1 T2) \Rightarrow (Arrow (swap_Typ u v T1) (swap_Typ u v T2))
- |(Forall T1 T2) \Rightarrow (Forall (swap_Typ u v T1) (swap_Typ u v T2))].
-
lemma swap_Typ_inv : \forall u,v,T.(swap_Typ u v (swap_Typ u v T)) = T.
intros;elim T
- [simplify;reflexivity
+ [1,3:simplify;reflexivity
|simplify;rewrite > swap_inv;reflexivity
- |simplify;reflexivity
- |simplify;rewrite > H;rewrite > H1;reflexivity
- |simplify;rewrite > H;rewrite > H1;reflexivity]
+ |*:simplify;rewrite > H;rewrite > H1;reflexivity]
qed.
lemma swap_Typ_not_free : \forall u,v,T.\lnot (in_list ? u (fv_type T)) \to
\lnot (in_list ? v (fv_type T)) \to (swap_Typ u v T) = T.
intros 3;elim T 0
- [intros;simplify;reflexivity
+ [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;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
+ |*: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);
+ [1,3: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]]]
+ |*:split
+ [1,3: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.
(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]
+ |2,3:simplify;reflexivity
+ |*:simplify;rewrite > H;rewrite > H1;reflexivity]
qed.
lemma subst_type_O_swap : \forall u,v,T,X.
(in_list ? x (fv_type T))).
intros;split
[elim T 0
- [simplify;intros;elim (in_list_nil ? ? H)
+ [1,3:simplify;intros;elim (in_list_nil ? ? H)
|simplify;intros;cut (x = n)
[rewrite > Hcut;apply in_Base
|inversion H
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)]]
+ |*: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
- [simplify;intros;elim (in_list_nil ? ? H)
+ [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
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)]]]
+ |*: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.
-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
[assumption|apply nil_cons]
|intros;lapply (sym_eq ? ? ? H4);absurd (a1::l = [])
[assumption|apply nil_cons]]
- |simplify;simplify in H;assumption
- |simplify in H;simplify;assumption
- |simplify in H2;simplify;apply natinG_or_inH_to_natinGH;
+ |2,3:simplify;simplify in H;assumption
+ |*:simplify in H2;simplify;apply natinG_or_inH_to_natinGH;
lapply (nat_in_list_case ? ? ? H2);elim Hletin
- [left;apply (H1 ? H3)
- |right;apply (H ? H3)]
- |simplify in H2;simplify;apply natinG_or_inH_to_natinGH;
- lapply (nat_in_list_case ? ? ? H2);elim Hletin
- [left;apply (H1 ? H3)
- |right;apply (H ? H3)]]
+ [1,3:left;apply (H1 ? H3)
+ |*:right;apply (H ? H3)]]
qed.
lemma in_dom_swap : \forall u,v,x,G.
|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
(*** 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]]
+ [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 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;
+ [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))
- [simplify;unfold;unfold;constructor 2;unfold in H1;unfold in H1;assumption
- |simplify;unfold;unfold;constructor 2;unfold in H;unfold in H;assumption]]
+ [1,3: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.
[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)).
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;
|unfold;intro;apply H5;apply (fv_WFT ? ? ? Hletin3 H8)]]
qed.
-(*** alternative "constructor" for universal types' well-formedness ***)
+(*** alternate "constructor" for universal types' well-formedness ***)
lemma WFT_Forall2 : \forall G,X,T,T1,T2.
(WFEnv G) \to
apply fv_subst_type_O;assumption]
qed.
-(*** alternative "constructor" for subtyping between universal types ***)
+(*** 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
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 s;simplify in H4;
+ inversion H4
+ [intros;absurd (mk_bound b n t::l@(mk_bound B x T::G)=Empty)
+ [assumption
+ |apply nil_cons]
+ |intros;lapply (inj_tail ? ? ? ? ? H9);lapply (inj_head ? ? ? ? H9);
+ destruct Hletin1;apply WFE_cons
+ [apply H1
+ [rewrite > Hletin;assumption
+ |assumption]
+ |rewrite > Hcut1;generalize in match H7;rewrite < Hletin;
+ rewrite > (fv_env_extends ? x B C T U);intro;assumption
+ |rewrite < Hletin in H8;rewrite > Hcut2;
+ apply (WFT_env_incl ? ? H8);rewrite > (fv_env_extends ? x B C T U);
+ unfold;intros;assumption]]]
+qed.
+
+lemma 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.
+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.
+
projects / helm.git / commitdiff