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 *)
+inductive Typ : Set ≝
+ | TVar : nat → Typ (* type var *)
+ | TFree: nat → Typ (* free type name *)
| Top : Typ (* maximum type *)
- | Arrow : Typ \to Typ \to Typ (* functions *)
- | Forall : Typ \to Typ \to Typ. (* universal type *)
+ | Arrow : Typ → Typ → Typ (* functions *)
+ | Forall : Typ → Typ → Typ. (* universal type *)
(* representation of bounds *)
-record bound : Set \def {
+record bound : Set ≝ {
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
+let rec subst_type_nat T U i ≝
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))) ].
+ [ TVar n ⇒ match eqb n i with
+ [ true ⇒ U
+ | false ⇒ T]
+ | TFree X ⇒ T
+ | Top ⇒ T
+ | Arrow T1 T2 ⇒ Arrow (subst_type_nat T1 U i) (subst_type_nat T2 U i)
+ | Forall T1 T2 ⇒ 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).
+definition fv_env : list bound → list nat ≝
+ λG.(map ? ? (λb.match b with [mk_bound B X T ⇒ X]) G).
-let rec fv_type T \def
+let rec fv_type T ≝
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))].
+ [TVar n ⇒ []
+ |TFree x ⇒ [x]
+ |Top ⇒ []
+ |Arrow U V ⇒ fv_type U @ fv_type V
+ |Forall U V ⇒ 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
+inductive WFType : list bound → Typ → Prop ≝
+ | WFT_TFree : ∀X,G.in_list ? X (fv_env G) → WFType G (TFree X)
+ | WFT_Top : ∀G.WFType G Top
+ | WFT_Arrow : ∀G,T,U.WFType G T → WFType G U → WFType G (Arrow T U)
+ | WFT_Forall : ∀G,T,U.WFType G T →
+ (∀X:nat.
+ (¬ (in_list ? X (fv_env G))) →
+ (¬ (in_list ? X (fv_type U))) →
+ (WFType ((mk_bound true X T) :: G)
+ (subst_type_nat U (TFree X) O))) →
(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)).
+inductive WFEnv : list bound → Prop ≝
+ | WFE_Empty : WFEnv (nil ?)
+ | WFE_cons : ∀B,X,T,G.WFEnv G → ¬ (in_list ? X (fv_env G)) →
+ WFType G T → 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)).
+inductive JSubtype : list bound → Typ → Typ → Prop ≝
+ | SA_Top : ∀G,T.WFEnv G → WFType G T → JSubtype G T Top
+ | SA_Refl_TVar : ∀G,X.WFEnv G → in_list ? X (fv_env G)
+ → JSubtype G (TFree X) (TFree X)
+ | SA_Trans_TVar : ∀G,X,T,U.in_list ? (mk_bound true X U) G →
+ JSubtype G U T → JSubtype G (TFree X) T
+ | SA_Arrow : ∀G,S1,S2,T1,T2. JSubtype G T1 S1 → JSubtype G S2 T2 →
+ JSubtype G (Arrow S1 S2) (Arrow T1 T2)
+ | SA_All : ∀G,S1,S2,T1,T2. JSubtype G T1 S1 →
+ (∀X.¬ (in_list ? X (fv_env G)) →
+ JSubtype ((mk_bound true X T1) :: G)
+ (subst_type_nat S2 (TFree X) O) (subst_type_nat T2 (TFree X) O)) →
+ 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/defn/JSubtype.ind#xpointer(1/1) e ta tb).
+interpretation "Fsub subtype judgement" 'subjudg e ta tb = (JSubtype 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/defn/Typ.ind#xpointer(1/1/5) S T).
+interpretation "universal type" 'forall S T = (Forall S T).
notation "#x" with precedence 79 for @{'tvar $x}.
-interpretation "bound tvar" 'tvar x =
- (cic:/matita/Fsub/defn/Typ.ind#xpointer(1/1/1) x).
+interpretation "bound tvar" 'tvar x = (TVar x).
notation "!x" with precedence 79 for @{'tname $x}.
-interpretation "bound tname" 'tname x =
- (cic:/matita/Fsub/defn/Typ.ind#xpointer(1/1/2) x).
+interpretation "bound tname" 'tname x = (TFree x).
notation "⊤" with precedence 90 for @{'toptype}.
-interpretation "toptype" 'toptype =
- (cic:/matita/Fsub/defn/Typ.ind#xpointer(1/1/3)).
+interpretation "toptype" 'toptype = Top.
notation "hvbox(s break ⇛ t)"
right associative with precedence 55 for @{ 'arrow $s $t }.
-interpretation "arrow type" 'arrow S T =
- (cic:/matita/Fsub/defn/Typ.ind#xpointer(1/1/4) S T).
+interpretation "arrow type" 'arrow S T = (Arrow 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/defn/subst_type_nat.con S T n).
+interpretation "subst bound var" 'substvar S T n = (subst_type_nat 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/defn/bound.ind#xpointer(1/1/1) true X T).
+interpretation "subtyping bound" 'subtypebound X T = (mk_bound 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]]
+lemma boundinenv_natinfv : ∀x,G.(∃B,T.in_list ? (mk_bound B x T) G) →
+ in_list ? x (fv_env G).
+intros 2;elim G;decompose
+ [elim (not_in_list_nil ? ? H)
+ |elim (in_list_cons_case ? ? ? ? H1)
+ [rewrite < H2;simplify;apply in_list_head
+ |simplify;apply in_list_cons;apply H;autobatch]]
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).
+lemma natinfv_boundinenv : ∀x,G.in_list ? x (fv_env G) →
+ ∃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]]
+ [rewrite < H2;autobatch
+ |elim (H H2);elim H3;apply ex_intro[apply a1];autobatch]]
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)]]
+lemma incl_bound_fv : ∀l1,l2.incl ? l1 l2 → incl ? (fv_env l1) (fv_env l2).
+intros;unfold in H;unfold;intros;apply boundinenv_natinfv;
+lapply (natinfv_boundinenv ? ? H1);decompose;autobatch depth=4;
qed.
-lemma incl_cons : \forall x,l1,l2.
- (incl ? l1 l2) \to (incl nat (x :: l1) (x :: l2)).
+lemma incl_cons : ∀x,l1,l2.incl ? l1 l2 → 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)]
+ [applyS in_list_head|autobatch]
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).
+lemma WFT_env_incl : ∀G,T.WFType G T →
+ ∀H.incl ? (fv_env G) (fv_env H) → 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_Arrow;autobatch
|apply WFT_Forall
[apply (H2 ? H6)
|intros;apply (H4 ? ? H8)
- [unfold;intro;apply H7;apply(H6 ? H9)
+ [unfold;intro;autobatch
|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))).
+lemma fv_env_extends : ∀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]
+ [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))).
+lemma lookup_env_extends : ∀G,H,B,C,D,T,U,V,x,y.
+ in_list ? (mk_bound D y V) (H @ ((mk_bound C x U) :: G)) → y ≠ x →
+ 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
+ [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))).
+lemma in_FV_subst : ∀x,T,U,n.in_list ? x (fv_type T) →
+ 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)]]
+ |*:simplify in H2;simplify;elim (in_list_append_to_or_in_list ? ? ? ? H2);
+ autobatch]
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.
+lemma fresh_name : ∀l:list nat.∃n.¬in_list ? n l.
+cut (∀l:list nat.∃n.∀m.n ≤ m → ¬ in_list ? m l);intros
+ [lapply (Hcut l);elim Hletin;apply ex_intro;autobatch
+ |elim l
+ [apply ex_intro[apply O];intros;unfold;intro;elim (not_in_list_nil ? ? H1)
+ |elim H;apply ex_intro[apply (S (max a1 a))];
+ intros;unfold;intro;
elim (in_list_cons_case ? ? ? ? H3)
- [rewrite > H4 in H2.autobatch
+ [rewrite > H4 in H2;autobatch
|elim H4
- [apply (H1 m ? H4).apply (trans_le ? (max a1 a));autobatch
+ [apply (H1 m ? H4);autobatch
|assumption]]]]
qed.
(*** lemmata on well-formedness ***)
-lemma fv_WFT : \forall T,x,G.(WFType G T) → x ∈ fv_type T → x ∈ fv_env G.
+lemma fv_WFT : ∀T,x,G.WFType G T → in_list ? x (fv_type T) →
+ 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)]
+ [applyS H1|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 (fresh_name (fv_type t1 @ fv_env l));
+ cut (¬ in_list ? a (fv_type t1) ∧ ¬ in_list ? 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]]]]
+ |autobatch]
+ |split;intro;apply H7;autobatch]]]
qed.
(*** lemmata relating subtyping and well-formedness ***)
-lemma JS_to_WFE : \forall G,T,U.(G \vdash T ⊴ U) \to (WFEnv G).
+lemma JS_to_WFE : ∀G,T,U.G ⊢ T ⊴ U → 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)).
+lemma JS_to_WFT : ∀G,T,U.G ⊢ T ⊴ U → WFType G T ∧ WFType G U.
intros;elim H
- [split [assumption|apply WFT_Top]
- |split;apply WFT_TFree;assumption
+ [1,2:autobatch
|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
+ [apply WFT_TFree;(* FIXME! qui bastava autobatch, ma si e` rotto *) apply boundinenv_natinfv;autobatch
+ |elim H3;assumption]
+ |decompose;autobatch size=7
|elim H2;split
[apply (WFT_Forall ? ? ? H6);intros;elim (H4 X H7);
apply (WFT_env_incl ? ? H9);simplify;unfold;intros;assumption
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.
+lemma JS_to_WFT1 : ∀G,T,U.G ⊢ T ⊴ U → WFType G T.
+intros;elim (JS_to_WFT ? ? ? H);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.
+lemma JS_to_WFT2 : ∀G,T,U.G ⊢ T ⊴ U → WFType G U.
+intros;elim (JS_to_WFT ? ? ? H);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))).
+lemma WFE_Typ_subst : ∀H,x,B,C,T,U,G.
+ WFEnv (H @ ((mk_bound B x T) :: G)) → WFType G U →
+ 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)]
+ [simplify;intros;inversion H1;intros
+ [elim (nil_cons ? G (mk_bound B x T) H3)
+ |destruct H7;autobatch]
|intros;simplify;generalize in match H2;elim a;simplify in H4;
inversion H4;intros
[destruct H5
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.
+lemma WFE_bound_bound : ∀B,x,T,U,G.WFEnv G → in_list ? (mk_bound B x T) G →
+ in_list ? (mk_bound B x U) G → 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 H7;elim H3;apply boundinenv_natinfv;autobatch]
|elim (in_list_cons_case ? ? ? ? H5)
- [destruct H8;elim H3;apply boundinenv_natinfv;apply (ex_intro ? ? B);
- apply (ex_intro ? ? U);assumption
+ [destruct H8;elim H3;apply boundinenv_natinfv;autobatch
|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)))
+lemma WFT_to_incl: ∀G,T,U.(∀X.¬in_list ? X (fv_env G) → ¬in_list ? 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)
+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
intros.rewrite < fv_env_extends.apply incl_A_A.
qed.
-lemma JSubtype_Top: ∀G,P.G ⊢ ⊤ ⊴ P → P = ⊤.
-intros.inversion H;intros
- [assumption|reflexivity
- |destruct H5|*:destruct H6]
-qed.
-
-(*
-(* elim vs inversion *)
-lemma JS_trans_TFree: ∀G,T,X.G ⊢ T ⊴ (TFree X) →
- ∀U.G ⊢ (TFree X) ⊴ U → G ⊢ T ⊴ U.
-intros 4.cut (∀Y.TFree Y = TFree X → ∀U.G ⊢ (TFree Y) ⊴ U → G ⊢ T ⊴ U)
- [apply Hcut;reflexivity
- |elim H;intros
- [rewrite > H3 in H4;rewrite > (JSubtype_Top ? ? H4);apply SA_Top;assumption
- |rewrite < H3;assumption
- |apply (SA_Trans_TVar ? ? ? ? H1);apply (H3 Y);assumption
- |*:destruct H5]]
-qed.
-*)
-
-lemma fv_append : ∀G,H.fv_env (G @ H) = (fv_env G @ fv_env H).
+lemma fv_append : ∀G,H.fv_env (G @ H) = fv_env G @ fv_env H.
intro;elim G;simplify;autobatch paramodulation;
-qed.
+qed.
\ No newline at end of file
include "Fsub/defn.ma".
+axiom daemon : False.
+
(*** Lemma A.1 (Reflexivity) ***)
theorem JS_Refl : ∀T,G.WFType G T → WFEnv G → G ⊢ T ⊴ T.
-intros 3.elim H
- [apply SA_Refl_TVar [apply H2|assumption]
- |apply SA_Top [assumption|apply WFT_Top]
- |apply (SA_Arrow ? ? ? ? ? (H2 H5) (H4 H5))
- |apply (SA_All ? ? ? ? ? (H2 H5));intros;apply (H4 ? H6)
- [intro;apply H6;apply (fv_WFT ? ? ? (WFT_Forall ? ? ? H1 H3));
- simplify;autobatch
- |autobatch]]
+intros 3; elim H;
+ [1,2,3: autobatch
+ | apply SA_All; [ autobatch | intros;autobatch depth=4 size=10]]
qed.
(*
*)
lemma JS_weakening : ∀G,T,U.G ⊢ T ⊴ U → ∀H.WFEnv H → incl ? G H → H ⊢ T ⊴ U.
-intros 4;elim H
- [apply (SA_Top ? ? H4);apply (WFT_env_incl ? ? H2 ? (incl_bound_fv ? ? H5))
- |apply (SA_Refl_TVar ? ? H4);apply (incl_bound_fv ? ? H5 ? H2)
- |apply (SA_Trans_TVar ? ? ? ? ? (H3 ? H5 H6));apply H6;assumption
- |apply (SA_Arrow ? ? ? ? ? (H2 ? H6 H7) (H4 ? H6 H7))
- |apply (SA_All ? ? ? ? ? (H2 ? H6 H7));intros;apply H4
- [unfold;intro;apply H8;apply (incl_bound_fv ? ? H7 ? H9)
- |apply (WFE_cons ? ? ? ? H6 H8);autobatch
- |unfold;intros;inversion H9;intros
- [destruct H11;apply in_list_head
- |destruct H13;apply in_list_cons;apply (H7 ? H10)]]]
+intros 4; elim H;
+ [1,2,3,4: autobatch depth=4 size=7
+ | apply (SA_All ? ? ? ? ? (H2 ? H6 H7));
+ intros; apply H4;autobatch depth=4 size=7]
qed.
+inverter JS_indinv for JSubtype (%?%).
+
theorem narrowing:∀X,G,G1,U,P,M,N.
G1 ⊢ P ⊴ U → (∀G2,T.G2@G1 ⊢ U ⊴ T → G2@G1 ⊢ P ⊴ T) → G ⊢ M ⊴ N →
∀l.G=l@(mk_bound true X U::G1) → l@(mk_bound true X P::G1) ⊢ M ⊴ N.
-intros 10.elim H2
- [letin x \def fv_env. letin y ≝incl.
- (* autobatch depth=4 size=8 by SA_Top, WFE_Typ_subst, H3, JS_to_WFT1, H, H4, WFT_env_incl, incl_fv_env]*)
- apply SA_Top
- [autobatch by WFE_Typ_subst, H3, JS_to_WFT1, H.
- (*
- rewrite > H5 in H3;
- apply (WFE_Typ_subst ? ? ? ? ? ? ? H3 (JS_to_WFT1 ? ? ? H)) *)
- |autobatch by H4, WFT_env_incl, incl_fv_env]
- (* rewrite > H5 in H4;apply (WFT_env_incl ? ? H4);apply incl_fv_env] *)
- |autobatch depth=4 by SA_Refl_TVar, WFE_Typ_subst, H3, JS_to_WFT1, H, H4.
- (*
- apply SA_Refl_TVar;
- [autobatch by WFE_Typ_subst, H3, JS_to_WFT1, H.
- (*
- rewrite > H5 in H3;apply (WFE_Typ_subst ? ? ? ? ? ? ? H3);
- apply (JS_to_WFT1 ? ? ? H) *)
- |autobatch by H4. (* rewrite > H5 in H4;rewrite < fv_env_extends;apply H4*)] *)
- |elim (decidable_eq_nat X n)
- [apply (SA_Trans_TVar ? ? ? P)
- [rewrite < H7;elim l1;simplify
- [constructor 1|constructor 2;assumption]
- |applyS H1.
- lapply (WFE_bound_bound true n t1 U ? ? H3);
- [autobatch. (* apply (JS_to_WFE ? ? ? H4) *)
- |autobatch. (* rewrite < Hletin;rewrite < append_cons;apply (H5 ? H6) *)
- |destruct.elim l1;autobatch.
- ]]
- |(* autobatch depth=4 size=7 by SA_Trans_TVar, lookup_env_extends, H3, sym_neq, H5, H6, H7. *)
- apply (SA_Trans_TVar ? ? ? t1);
- [autobatch by lookup_env_extends, H3, sym_neq, H7.
- (* rewrite > H6 in H3; apply (lookup_env_extends ? ? ? ? ? ? ? ? ? ? H3);
- unfold;intro;apply H7;symmetry;assumption *)
- |apply (H5 ? H6)]]
- |autobatch; (* apply (SA_Arrow ? ? ? ? ? (H4 ? H7) (H6 ? H7)) *)
- |apply (SA_All ? ? ? ? ? (H4 ? H7));intros;autobatch]
- (*
- apply (H6 ? ? (mk_bound true X1 t2::l1))
- [rewrite > H7;rewrite > fv_env_extends;apply H8
- |simplify;rewrite < H7;reflexivity]] *)
+intros 10.elim H2; destruct;
+ [letin x \def fv_env. letin y ≝incl. autobatch depth=4 size=8.
+ | autobatch depth=4 size=7;
+ | elim (decidable_eq_nat X n)
+ [apply (SA_Trans_TVar ? ? ? P); destruct;
+ [ autobatch
+ | lapply (WFE_bound_bound true X t1 U ? ? H3); autobatch]
+ | apply (SA_Trans_TVar ? ? ? t1); autobatch]
+ | autobatch
+ | apply SA_All;
+ [ autobatch
+ | intros; apply (H6 ? ? (mk_bound true X1 t2::l1)); autobatch]]
qed.
lemma JS_trans_prova: ∀T,G1.WFType G1 T →
∀G,R,U.incl ? (fv_env G1) (fv_env G) → G ⊢ R ⊴ T → G ⊢ T ⊴ U → G ⊢ R ⊴ U.
-intros 3;elim H;clear H; try autobatch;
- [
- rewrite > (JSubtype_Top ? ? H3);autobatch
- |generalize in match H7;generalize in match H4;generalize in match H2;
- generalize in match H5;clear H7 H4 H2 H5;
- generalize in match (refl_eq ? (Arrow t t1));
- elim H6 in ⊢ (? ? ? %→%); clear H6; intros; destruct;
- [apply (SA_Trans_TVar ? ? ? ? H);apply (H4 ? ? H8 H9);autobatch
- |inversion H11;intros; destruct; autobatch depth=4 width=4 size=9;
- ]
- |generalize in match H7;generalize in match H4;generalize in match H2;
- generalize in match H5;clear H7 H4 H2 H5;
- generalize in match (refl_eq ? (Forall t t1));elim H6 in ⊢ (? ? ? %→%);destruct;
- [apply (SA_Trans_TVar ? ? ? ? H);apply (H4 ? H7 H8 H9 H10);reflexivity
- |inversion H11;intros;destruct;
- [apply SA_Top
- [autobatch
- |apply WFT_Forall
- [autobatch
- |intros;lapply (H4 ? H13);autobatch]]
- |apply SA_All
- [autobatch paramodulation
- |intros;apply (H10 X)
- [intro;apply H15;apply H8;assumption
- |intro;apply H15;apply H8;apply (WFT_to_incl ? ? ? H3);
- assumption
- |simplify;autobatch
- |apply (narrowing X (mk_bound true X t::l1)
- ? ? ? ? ? H7 ? ? [])
- [intros;apply H9
- [unfold;intros;lapply (H8 ? H17);rewrite > fv_append;
- autobatch
- |apply (JS_weakening ? ? ? H7)
- [autobatch
- |unfold;intros;autobatch]
- |assumption]
- |*:autobatch]
- |autobatch]]]]]
+intros 3;elim H;clear H;
+ [elim H3 using JS_indinv;destruct;autobatch
+ |inversion H3; intros; destruct; assumption
+ |*: elim H6 using JS_indinv;destruct;
+ [1,3: autobatch
+ |*: inversion H7; intros; destruct;
+ [1,2: autobatch depth=4 width=4 size=9
+ | apply SA_Top
+ [ assumption
+ | apply WFT_Forall;intros;autobatch depth=4]
+ | apply SA_All
+ [ autobatch
+ | intros;apply (H4 X);simplify;
+ [4: apply (narrowing X (mk_bound true X t::G) ? ? ? ? ? H11 ? ? [])
+ [intros;apply H2;try unfold;intros;autobatch;
+ |*:autobatch]
+ |*:autobatch]]]]]
qed.
theorem JS_trans : ∀G,T,U,V.G ⊢ T ⊴ U → G ⊢ U ⊴ V → G ⊢ T ⊴ V.
-intros 5;apply (JS_trans_prova ? G);autobatch;
+intros 5; apply (JS_trans_prova ? G); autobatch depth=2.
qed.
theorem JS_narrow : ∀G1,G2,X,P,Q,T,U.
(G2 @ (mk_bound true X Q :: G1)) ⊢ T ⊴ U → G1 ⊢ P ⊴ Q →
(G2 @ (mk_bound true X P :: G1)) ⊢ T ⊴ U.
intros;apply (narrowing ? ? ? ? ? ? ? H1 ? H) [|autobatch]
-intros;apply (JS_trans ? ? ? ? ? H2);apply (JS_weakening ? ? ? H1);
- [autobatch|unfold;intros;autobatch]
+intros;apply (JS_trans ? ? ? ? ? H2);apply (JS_weakening ? ? ? H1);autobatch.
qed.
projects / helm.git / commitdiff