+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "Fsub/util.ma".
-
-(*** representation of Fsub types ***)
-inductive Typ : Set \def
- | TVar : nat \to Typ (* type var *)
- | TFree: nat \to Typ (* free type name *)
- | Top : Typ (* maximum type *)
- | Arrow : Typ \to Typ \to Typ (* functions *)
- | Forall : Typ \to Typ \to Typ. (* universal type *)
-
-(* representation of bounds *)
-
-record bound : Set \def {
- istype : bool; (* is subtyping bound? *)
- name : nat ; (* name *)
- btype : Typ (* type to which the name is bound *)
- }.
-
-(*** Various kinds of substitution, not all will be used probably ***)
-
-(* substitutes i-th dangling index in type T with type U *)
-let rec subst_type_nat T U i \def
- match T with
- [ (TVar n) \Rightarrow match (eqb n i) with
- [ true \Rightarrow U
- | false \Rightarrow T]
- | (TFree X) \Rightarrow T
- | Top \Rightarrow T
- | (Arrow T1 T2) \Rightarrow (Arrow (subst_type_nat T1 U i) (subst_type_nat T2 U i))
- | (Forall T1 T2) \Rightarrow (Forall (subst_type_nat T1 U i) (subst_type_nat T2 U (S i))) ].
-
-(*** definitions about lists ***)
-
-definition fv_env : (list bound) \to (list nat) \def
- \lambda G.(map ? ? (\lambda b.match b with
- [(mk_bound B X T) \Rightarrow X]) G).
-
-let rec fv_type T \def
- match T with
- [(TVar n) \Rightarrow []
- |(TFree x) \Rightarrow [x]
- |Top \Rightarrow []
- |(Arrow U V) \Rightarrow ((fv_type U) @ (fv_type V))
- |(Forall U V) \Rightarrow ((fv_type U) @ (fv_type V))].
-
-(*** Type Well-Formedness judgement ***)
-
-inductive WFType : (list bound) \to Typ \to Prop \def
- | WFT_TFree : \forall X,G.(in_list ? X (fv_env G))
- \to (WFType G (TFree X))
- | WFT_Top : \forall G.(WFType G Top)
- | 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_nat U (TFree X) O))) \to
- (WFType G (Forall T U)).
-
-(*** Environment Well-Formedness judgement ***)
-
-inductive WFEnv : (list bound) \to Prop \def
- | WFE_Empty : (WFEnv (nil ?))
- | WFE_cons : \forall B,X,T,G.(WFEnv G) \to
- \lnot (in_list ? X (fv_env G)) \to
- (WFType G T) \to (WFEnv ((mk_bound B X T) :: G)).
-
-(*** Subtyping judgement ***)
-inductive JSubtype : (list bound) \to Typ \to Typ \to Prop \def
- | SA_Top : \forall G.\forall T:Typ.(WFEnv G) \to
- (WFType G T) \to (JSubtype G T Top)
- | SA_Refl_TVar : \forall G.\forall X:nat.(WFEnv G)
- \to (in_list ? X (fv_env G))
- \to (JSubtype G (TFree X) (TFree X))
- | SA_Trans_TVar : \forall G.\forall X:nat.\forall T:Typ.
- \forall U:Typ.
- (in_list ? (mk_bound true X U) G) \to
- (JSubtype G U T) \to (JSubtype G (TFree X) T)
- | SA_Arrow : \forall G.\forall S1,S2,T1,T2:Typ.
- (JSubtype G T1 S1) \to (JSubtype G S2 T2) \to
- (JSubtype G (Arrow S1 S2) (Arrow T1 T2))
- | SA_All : \forall G.\forall S1,S2,T1,T2:Typ.
- (JSubtype G T1 S1) \to
- (\forall X:nat.\lnot (in_list ? X (fv_env G)) \to
- (JSubtype ((mk_bound true X T1) :: G)
- (subst_type_nat S2 (TFree X) O) (subst_type_nat T2 (TFree X) O))) \to
- (JSubtype G (Forall S1 S2) (Forall T1 T2)).
-
-notation "hvbox(e ⊢ break ta ⊴ break tb)"
- non associative with precedence 30 for @{ 'subjudg $e $ta $tb }.
-interpretation "Fsub subtype judgement" 'subjudg e ta tb =
- (cic:/matita/Fsub/defn2/JSubtype.ind#xpointer(1/1) e ta tb).
-
-notation > "hvbox(\Forall S.T)"
- non associative with precedence 60 for @{ 'forall $S $T}.
-notation < "hvbox('All' \sub S. break T)"
- non associative with precedence 60 for @{ 'forall $S $T}.
-interpretation "universal type" 'forall S T =
- (cic:/matita/Fsub/defn2/Typ.ind#xpointer(1/1/5) S T).
-
-notation "#x" with precedence 79 for @{'tvar $x}.
-interpretation "bound tvar" 'tvar x =
- (cic:/matita/Fsub/defn2/Typ.ind#xpointer(1/1/1) x).
-
-notation "!x" with precedence 79 for @{'tname $x}.
-interpretation "bound tname" 'tname x =
- (cic:/matita/Fsub/defn2/Typ.ind#xpointer(1/1/2) x).
-
-notation "⊤" with precedence 90 for @{'toptype}.
-interpretation "toptype" 'toptype =
- (cic:/matita/Fsub/defn2/Typ.ind#xpointer(1/1/3)).
-
-notation "hvbox(s break ⇛ t)"
- right associative with precedence 55 for @{ 'arrow $s $t }.
-interpretation "arrow type" 'arrow S T =
- (cic:/matita/Fsub/defn2/Typ.ind#xpointer(1/1/4) S T).
-
-notation "hvbox(S [# n ↦ T])"
- non associative with precedence 80 for @{ 'substvar $S $T $n }.
-interpretation "subst bound var" 'substvar S T n =
- (cic:/matita/Fsub/defn2/subst_type_nat.con S T n).
-
-notation "hvbox(!X ⊴ T)"
- non associative with precedence 60 for @{ 'subtypebound $X $T }.
-interpretation "subtyping bound" 'subtypebound X T =
- (cic:/matita/Fsub/defn2/bound.ind#xpointer(1/1/1) true X T).
-
-(****** PROOFS ********)
-
-(*** theorems about lists ***)
-
-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 (not_in_list_nil ? ? H2);elim Hletin
- |elim H1;elim H2;elim (in_list_cons_case ? ? ? ? H3)
- [rewrite < H4;simplify;apply in_list_head
- |simplify;apply in_list_cons;apply H;apply (ex_intro ? ? a1);
- apply (ex_intro ? ? a2);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 (not_in_list_nil ? ? H);elim Hletin
- |intros 3;
- elim a;simplify in H1;elim (in_list_cons_case ? ? ? ? H1)
- [rewrite < H2;apply (ex_intro ? ? b);apply (ex_intro ? ? t);apply in_list_head
- |elim (H H2);elim H3;apply (ex_intro ? ? a1);
- apply (ex_intro ? ? a2);apply in_list_cons;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 ? l1 l2) \to (incl nat (x :: l1) (x :: l2)).
-intros.unfold in H.unfold.intros.elim (in_list_cons_case ? ? ? ? H1)
- [rewrite > H2;apply in_list_head|apply in_list_cons;apply (H ? H2)]
-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 3.elim H
- [apply WFT_TFree;unfold in H3;apply (H3 ? H1)
- |apply WFT_Top
- |apply WFT_Arrow [apply (H2 ? H6)|apply (H4 ? H6)]
- |apply WFT_Forall
- [apply (H2 ? H6)
- |intros;apply (H4 ? ? H8)
- [unfold;intro;apply H7;apply(H6 ? H9)
- |simplify;apply (incl_cons ? ? ? H6)]]]
-qed.
-
-lemma 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 a;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;elim (in_list_cons_case ? ? ? ? H1)
- [destruct H3;elim (H2);reflexivity
- |simplify;apply (in_list_cons ? ? ? ? H3);]
- |simplify in H2;simplify;elim (in_list_cons_case ? ? ? ? H2)
- [rewrite > H4;apply in_list_head
- |apply (in_list_cons ? ? ? ? (H1 H4 H3))]]
-qed.
-
-lemma in_FV_subst : \forall x,T,U,n.(in_list ? x (fv_type T)) \to
- (in_list ? x (fv_type (subst_type_nat T U n))).
-intros 3;elim T
- [simplify in H;elim (not_in_list_nil ? ? H)
- |2,3:simplify;simplify in H;assumption
- |*:simplify in H2;simplify;elim (in_list_append_to_or_in_list ? ? ? ? H2)
- [1,3:apply in_list_to_in_list_append_l;apply (H ? H3)
- |*:apply in_list_to_in_list_append_r;apply (H1 ? H3)]]
-qed.
-
-(*** lemma on fresh names ***)
-
-lemma fresh_name : \forall l:(list nat).\exists n.\lnot (in_list ? n l).
-cut (\forall l:(list nat).\exists n.\forall m.
- (n \leq m) \to \lnot (in_list ? m l))
- [intros;lapply (Hcut l);elim Hletin;apply ex_intro
- [apply a
- |apply H;constructor 1]
- |intros;elim l
- [apply (ex_intro ? ? O);intros;unfold;intro;elim (not_in_list_nil ? ? H1)
- |elim H;
- apply (ex_intro ? ? (S (max a1 a))).
- intros.unfold. intro.
- elim (in_list_cons_case ? ? ? ? H3)
- [rewrite > H4 in H2.autobatch
- |elim H4
- [apply (H1 m ? H4).apply (trans_le ? (max a1 a));autobatch
- |assumption]]]]
-qed.
-
-(*** lemmata on well-formedness ***)
-
-lemma fv_WFT : \forall T,x,G.(WFType G T) \to (in_list ? x (fv_type T)) \to
- (in_list ? x (fv_env G)).
-intros 4.elim H
- [simplify in H2;elim (in_list_cons_case ? ? ? ? H2)
- [rewrite > H3;assumption|elim (not_in_list_nil ? ? H3)]
- |simplify in H1;elim (not_in_list_nil ? x H1)
- |simplify in H5;elim (in_list_append_to_or_in_list ? ? ? ? H5);autobatch
- |simplify in H5;elim (in_list_append_to_or_in_list ? ? ? ? H5)
- [apply (H2 H6)
- |elim (fresh_name ((fv_type t1) @ (fv_env l)));
- cut (¬ (a ∈ (fv_type t1)) ∧ ¬ (a ∈ (fv_env l)))
- [elim Hcut;lapply (H4 ? H9 H8)
- [cut (x ≠ a)
- [simplify in Hletin;elim (in_list_cons_case ? ? ? ? Hletin)
- [elim (Hcut1 H10)
- |assumption]
- |intro;apply H8;applyS H6]
- |apply in_FV_subst;assumption]
- |split
- [intro;apply H7;apply in_list_to_in_list_append_l;assumption
- |intro;apply H7;apply in_list_to_in_list_append_r;assumption]]]]
-qed.
-
-(*** lemmata relating subtyping and well-formedness ***)
-
-lemma JS_to_WFE : \forall G,T,U.(G \vdash T ⊴ U) \to (WFEnv G).
-intros;elim H;assumption.
-qed.
-
-lemma JS_to_WFT : \forall G,T,U.(JSubtype G T U) \to ((WFType G T) \land
- (WFType G U)).
-intros;elim H
- [split [assumption|apply WFT_Top]
- |split;apply WFT_TFree;assumption
- |split
- [apply WFT_TFree;apply boundinenv_natinfv;apply ex_intro
- [apply true | apply ex_intro [apply t1 |assumption]]
- |elim H3;assumption]
- |elim H2;elim H4;split;apply WFT_Arrow;assumption
- |elim H2;split
- [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 JS_to_WFT2 : \forall G,T,U.(JSubtype G T U) \to (WFType G U).
-intros;lapply (JS_to_WFT ? ? ? H);elim Hletin;assumption.
-qed.
-
-lemma WFE_Typ_subst : \forall H,x,B,C,T,U,G.
- (WFEnv (H @ ((mk_bound B x T) :: G))) \to (WFType G U) \to
- (WFEnv (H @ ((mk_bound C x U) :: G))).
-intros 7;elim H 0
- [simplify;intros;(*FIXME*)generalize in match H1;intro;inversion H1;intros
- [lapply (nil_cons ? G (mk_bound B x T));elim (Hletin H4)
- |destruct H8;apply (WFE_cons ? ? ? ? H4 H6 H2)]
- |intros;simplify;generalize in match H2;elim a;simplify in H4;
- inversion H4;intros
- [destruct H5
- |destruct H9;apply WFE_cons
- [apply (H1 H5 H3)
- |rewrite < (fv_env_extends ? x B C T U); assumption
- |apply (WFT_env_incl ? ? H8);
- rewrite < (fv_env_extends ? x B C T U);unfold;intros;
- assumption]]]
-qed.
-
-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 (not_in_list_nil ? ? H1);elim Hletin
- |elim (in_list_cons_case ? ? ? ? H6)
- [destruct H7;destruct;elim (in_list_cons_case ? ? ? ? H5)
- [destruct H7;reflexivity
- |elim H7;elim H3;apply boundinenv_natinfv;apply (ex_intro ? ? B);
- apply (ex_intro ? ? T);assumption]
- |elim (in_list_cons_case ? ? ? ? H5)
- [destruct H8;elim H3;apply boundinenv_natinfv;apply (ex_intro ? ? B);
- apply (ex_intro ? ? U);assumption
- |apply (H2 H8 H7)]]]
-qed.
-
-lemma WFT_to_incl: ∀G,T,U.
- (∀X.(¬(X ∈ fv_env G)) → (¬(X ∈ fv_type U)) →
- (WFType (mk_bound true X T::G) (subst_type_nat U (TFree X) O)))
- → incl ? (fv_type U) (fv_env G).
-intros.elim (fresh_name ((fv_type U)@(fv_env G))).lapply(H a)
- [unfold;intros;lapply (fv_WFT ? x ? Hletin)
- [simplify in Hletin1;inversion Hletin1;intros
- [destruct H4;elim H1;autobatch
- |destruct H6;assumption]
- |apply in_FV_subst;assumption]
- |*:intro;apply H1;autobatch]
-qed.
-
-lemma incl_fv_env: ∀X,G,G1,U,P.
- incl ? (fv_env (G1@(mk_bound true X U::G)))
- (fv_env (G1@(mk_bound true X P::G))).
-intros.rewrite < fv_env_extends.apply incl_A_A.
-qed.
-
-lemma fv_append : ∀G,H.fv_env (G @ H) = (fv_env G @ fv_env H).
-intro;elim G;simplify;autobatch paramodulation;
-qed.
projects / helm.git / blobdiff