--- /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 ? ? a);
+ apply (ex_intro ? ? a1);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 t;simplify in H1;elim (in_list_cons_case ? ? ? ? H1)
+ [rewrite < H2;apply (ex_intro ? ? b);apply (ex_intro ? ? t1);apply in_list_head
+ |elim (H H2);elim H3;apply (ex_intro ? ? a);
+ apply (ex_intro ? ? a1);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 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;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 a t))).
+ intros.unfold. intro.
+ elim (in_list_cons_case ? ? ? ? H3)
+ [rewrite > H4 in H2.autobatch
+ |elim H4
+ [apply (H1 m ? H4).apply (trans_le ? (max a t));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 t;simplify in H4;
+ inversion H4;intros
+ [destruct H5
+ |destruct H9;apply WFE_cons
+ [apply (H1 H5 H3)
+ |rewrite < (fv_env_extends ? x B C T U); assumption
+ |apply (WFT_env_incl ? ? H8);
+ rewrite < (fv_env_extends ? x B C T U);unfold;intros;
+ assumption]]]
+qed.
+
+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