include "Fsub/defn.ma".
(*** 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 l n H2 H1);
- |apply (SA_Top l Top H1 (WFT_Top l));
- |apply (SA_Arrow l t t1 t t1 (H2 H5) (H4 H5))
- |apply (SA_All ? ? ? ? ? (H2 H5));intros;apply (H4 X H6)
- [intro;apply H6;apply (fv_WFT (Forall t t1) X l)
- [apply (WFT_Forall ? ? ? H1 H3)
- |simplify;autobatch]
+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]]
qed.
-(*
+(*
* A slightly more general variant to lemma A.2.2, where weakening isn't
* defined as concatenation of any two disjoint environments, but as
* set inclusion.
*)
-
-lemma JS_weakening : \forall G,T,U.(JSubtype G T U) \to
- \forall H.(WFEnv H) \to (incl ? G H) \to (JSubtype H T U).
+
+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);lapply (incl_bound_fv ? ? H5);
- apply (WFT_env_incl ? ? H2 ? Hletin)
- |apply (SA_Refl_TVar ? ? H4);lapply (incl_bound_fv ? ? H5);
- apply (Hletin ? H2)
- |lapply (H3 ? H5 H6);lapply (H6 ? H1);
- apply (SA_Trans_TVar ? ? ? ? Hletin1 Hletin)
- |lapply (H2 ? H6 H7);lapply (H4 ? H6 H7);
- apply (SA_Arrow ? ? ? ? ? Hletin Hletin1)
- |lapply (H2 ? H6 H7);apply (SA_All ? ? ? ? ? Hletin);intros;apply H4
- [unfold;intro;apply H8;lapply (incl_bound_fv ? ? H7);apply (Hletin1 ? H9)
- |apply WFE_cons
- [1,2:assumption
- |apply (JS_to_WFT1 ? ? ? Hletin)]
- |unfold;intros;elim (in_cons_case ? ? ? ? H9)
- [rewrite > H10;apply in_Base
- |elim H10;apply (in_Skip ? ? ? ? ? H11);apply (H7 ? H12)]]]
+ [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_Base
+ |destruct H13;apply in_Skip;apply (H7 ? H10)]]]
qed.
-lemma decidable_eq_Typ : \forall S,T:Typ.(S = T) ∨ (S ≠ T).
-intro;elim S
- [elim T
- [elim (decidable_eq_nat n n1)
- [rewrite > H;left;reflexivity
- |right;intro;destruct H1;apply (H Hcut)]
- |2,3:right;intro;destruct H
- |*:right;intro;destruct H2]
- |elim T
- [2:elim (decidable_eq_nat n n1)
- [rewrite > H;left;reflexivity
- |right;intro;destruct H1;apply (H Hcut)]
- |1,3:right;intro;destruct H
- |*:right;intro;destruct H2]
- |elim T
- [3:left;reflexivity
- |1,2:right;intro;destruct H
- |*:right;intro;destruct H2]
- |elim T
- [1,2,3:right;intro;destruct H2
- |elim (H t2)
- [rewrite > H4;elim (H1 t3)
- [rewrite > H5;left;reflexivity
- |right;intro;apply H5;destruct H6;assumption]
- |right;intro;apply H4;destruct H5;assumption]
- |right;intro;destruct H4]
- |elim T
- [1,2,3:right;intro;destruct H2
- |right;intro;destruct H4
- |elim (H t2)
- [rewrite > H4;elim (H1 t3)
- [rewrite > H5;left;reflexivity
- |right;intro;apply H5;destruct H6;assumption]
- |right;intro;apply H4;destruct H5;assumption]]]
+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
+ [apply SA_Top
+ [rewrite > H5 in H3;
+ apply (WFE_Typ_subst ? ? ? ? ? ? ? H3 (JS_to_WFT1 ? ? ? H))
+ |rewrite > H5 in H4;apply (WFT_env_incl ? ? H4);apply incl_fv_env]
+ |apply SA_Refl_TVar
+ [rewrite > H5 in H3;apply (WFE_Typ_subst ? ? ? ? ? ? ? H3);
+ apply (JS_to_WFT1 ? ? ? H)
+ |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]
+ |rewrite > append_cons;apply H1;
+ lapply (WFE_bound_bound true n t1 U ? ? H3)
+ [apply (JS_to_WFE ? ? ? H4)
+ |rewrite < Hletin;rewrite < append_cons;apply (H5 ? H6)
+ |rewrite < H7;rewrite > H6;elim l1;simplify
+ [constructor 1|constructor 2;assumption]]]
+ |apply (SA_Trans_TVar ? ? ? t1)
+ [rewrite > H6 in H3;apply (lookup_env_extends ? ? ? ? ? ? ? ? ? ? H3);
+ unfold;intro;apply H7;symmetry;assumption
+ |apply (H5 ? H6)]]
+ |apply (SA_Arrow ? ? ? ? ? (H4 ? H7) (H6 ? H7))
+ |apply (SA_All ? ? ? ? ? (H4 ? H7));intros;
+ apply (H6 ? ? (mk_bound true X1 t2::l1))
+ [rewrite > H7;rewrite > fv_env_extends;apply H8
+ |simplify;rewrite < H7;reflexivity]]
qed.
-lemma decidable_eq_bound: ∀b1,b2:bound.(b1 = b2) ∨ (b1 ≠ b2).
-intros;elim b1;elim b2;elim (decidable_eq_nat n n1)
- [rewrite < H;elim (decidable_eq_Typ t t1)
- [rewrite < H1;elim (bool_to_decidable_eq b b3)
- [rewrite > H2;left;reflexivity
- |right;intro;destruct H3;apply (H2 Hcut)]
- |right;intro;destruct H2;apply (H1 Hcut1)]
- |right;intro;destruct H1;apply (H Hcut1)]
-qed.
-
-(* Lemma A.3 (Transitivity and Narrowing) *)
-
-lemma JS_trans_narrow : \forall Q.
- (\forall G,T,U.
- (JSubtype G T Q) \to (JSubtype G Q U) \to
- (JSubtype G T U)) \land
- (\forall G,H,X,P,M,N.
- (JSubtype (H @ ((mk_bound true X Q) :: G)) M N) \to
- (JSubtype G P Q) \to
- (JSubtype (H @ ((mk_bound true X P) :: G)) M N)).
-apply Typ_len_ind;intros 2;
-cut (\forall G,T,P.
- (JSubtype G T U) \to
- (JSubtype G U P) \to
- (JSubtype G T P))
- [split
- [assumption
- |cut (\forall G,M,N.(JSubtype G M N) \to
- \forall G1,X,G2,P.
- (G = G2 @ ((mk_bound true X U) :: G1)) \to
- (JSubtype G1 P U) \to
- (JSubtype (G2 @ ((mk_bound true X P) :: G1)) M N))
- [intros;apply (Hcut1 ? ? ? H2 ? ? ? ? ? H3);reflexivity
- |intros;cut (incl ? (fv_env (G2 @ ((mk_bound true X U)::G1)))
- (fv_env (G2 @ ((mk_bound true X P)::G1))))
- [intros;generalize in match H2;generalize in match Hcut1;
- generalize in match Hcut;generalize in match G2;elim H1
- [apply SA_Top
- [rewrite > H8 in H4;lapply (JS_to_WFT1 ? ? ? H3);
- apply (WFE_Typ_subst ? ? ? ? ? ? ? H4 Hletin)
- |rewrite > H8 in H5;apply (WFT_env_incl ? ? H5 ? H7)]
- |apply SA_Refl_TVar
- [rewrite > H8 in H4;apply (WFE_Typ_subst ? ? ? ? ? ? ? H4);
- apply (JS_to_WFT1 ? ? ? H3)
- |rewrite > H8 in H5;apply (H7 ? H5)]
- |elim (decidable_eq_nat X n)
- [apply (SA_Trans_TVar ? ? ? P)
- [rewrite < H10;elim l1
- [simplify;constructor 1
- |simplify;elim (decidable_eq_bound (mk_bound true X P) t2)
- [rewrite < H12;apply in_Base
- |apply (in_Skip ? ? ? ? ? H12);assumption]]
- |apply H7
- [lapply (H6 ? H7 H8 H9);lapply (JS_to_WFE ? ? ? Hletin);
- apply (JS_weakening ? ? ? H3 ? Hletin1);unfold;intros;
- elim l1
- [simplify;
- elim (decidable_eq_bound x (mk_bound true X P))
- [rewrite < H12;apply in_Base
- |apply (in_Skip ? ? ? ? ? H12);assumption]
- |simplify;elim (decidable_eq_bound x t2)
- [rewrite < H13;apply in_Base
- |apply (in_Skip ? ? ? ? ? H13);assumption]]
- |lapply (WFE_bound_bound true n t1 U ? ? H4)
- [apply (JS_to_WFE ? ? ? H5)
- |rewrite < Hletin;apply (H6 ? H7 H8 H9)
- |rewrite > H9;rewrite > H10;elim l1;simplify
- [constructor 1
- |elim (decidable_eq_bound (mk_bound true n U) t2)
- [rewrite > H12;apply in_Base
- |apply (in_Skip ? ? ? ? ? H12);assumption]]]]]
- |apply (SA_Trans_TVar ? ? ? t1)
- [rewrite > H9 in H4;
- apply (lookup_env_extends ? ? ? ? ? ? ? ? ? ? H4);
- unfold;intro;apply H10;symmetry;assumption
- |apply (H6 ? H7 H8 H9)]]
- |apply SA_Arrow
- [apply (H5 ? H8 H9 H10)
- |apply (H7 ? H8 H9 H10)]
- |apply SA_All
- [apply (H5 ? H8 H9 H10)
- |intros;apply (H7 ? ? (mk_bound true X1 t2::l1) H8)
- [rewrite > H10;cut ((fv_env (l1@(mk_bound true X P::G1))) =
- (fv_env (l1@(mk_bound true X U::G1))))
- [unfold;intro;apply H11;rewrite > Hcut2;assumption
- |elim l1
- [simplify;reflexivity
- |elim t4;simplify;rewrite > H12;reflexivity]]
- |simplify;apply (incl_nat_cons ? ? ? H9)
- |simplify;rewrite < H10;reflexivity]]]
- |cut ((fv_env (G2@(mk_bound true X U::G1))) =
- (fv_env (G2@(mk_bound true X P::G1))))
- [rewrite > Hcut1;unfold;intros;assumption
- |elim G2
- [simplify;reflexivity
- |elim t;simplify;rewrite > H4;reflexivity]]]]]
- |intros 4;generalize in match H;elim H1
- [inversion H5
- [intros;rewrite < H8;apply (SA_Top ? ? H2 H3)
- |intros;destruct H9
- |intros;destruct H10
- |*:intros;destruct H11]
- |assumption
- |apply (SA_Trans_TVar ? ? ? ? H2);apply (H4 H5 H6)
- |inversion H7
- [intros;apply (SA_Top ? ? H8);rewrite < H10;apply WFT_Arrow
- [apply (JS_to_WFT2 ? ? ? H2)
- |apply (JS_to_WFT1 ? ? ? H4)]
- |intros;destruct H11
- |intros;destruct H12
- |intros;destruct H13;apply SA_Arrow
- [rewrite > H12 in H2;rewrite < Hcut in H8;
- lapply (H6 t2)
- [elim Hletin;apply (H15 ? ? ? H8 H2)
- |apply (t_len_arrow1 t2 t3)]
- |rewrite > H12 in H4;rewrite < Hcut1 in H10;
- lapply (H6 t3)
- [elim Hletin;apply (H15 ? ? ? H4 H10)
- |apply (t_len_arrow2 t2 t3)]]
- |intros;destruct H13]
- |inversion H7
- [intros;apply (SA_Top ? ? H8);rewrite < H10;apply WFT_Forall
- [apply (JS_to_WFT2 ? ? ? H2)
- |intros;lapply (H4 ? H13);lapply (JS_to_WFT1 ? ? ? Hletin);
- apply (WFT_env_incl ? ? Hletin1);simplify;unfold;intros;
- assumption]
- |intros;destruct H11
- |intros;destruct H12
- |intros;destruct H13
- |intros;destruct H13;subst;apply SA_All
- [lapply (H6 t4)
- [elim Hletin;apply (H12 ? ? ? H8 H2)
- |apply t_len_forall1]
- |intros;(*destruct H12;*)subst;
- lapply (H6 (subst_type_nat t5 (TFree X) O))
- [elim Hletin;apply H13
- [lapply (H6 t4)
- [elim Hletin1;apply (H16 l1 [] X t6);
- [simplify;apply H4;assumption
+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]
- |apply t_len_forall1]
- |apply (H10 ? H12)]
- |rewrite < eq_t_len_TFree_subst;
- apply t_len_forall2]]]]]
+ |*:autobatch]
+ |autobatch]]]]]
qed.
-theorem JS_trans : \forall G,T,U,V.(JSubtype G T U) \to
- (JSubtype G U V) \to
- (JSubtype G T V).
-intros;elim JS_trans_narrow;autobatch;
+theorem JS_trans : ∀G,T,U,V.G ⊢ T ⊴ U → G ⊢ U ⊴ V → G ⊢ T ⊴ V.
+intros 5;apply (JS_trans_prova ? G);autobatch;
qed.
-theorem JS_narrow : \forall G1,G2,X,P,Q,T,U.
- (JSubtype (G2 @ (mk_bound true X Q :: G1)) T U) \to
- (JSubtype G1 P Q) \to
- (JSubtype (G2 @ (mk_bound true X P :: G1)) T U).
-intros;elim JS_trans_narrow;autobatch;
+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]
qed.
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