[elim H;elim H1;lapply (in_list_nil ? ? H2);elim Hletin
|elim H1;elim H2;elim (in_cons_case ? ? ? ? H3)
[rewrite < H4;simplify;apply in_Base
- |elim H4;elim t;simplify;apply in_Skip2;apply H;apply (ex_intro ? ? a);
+ |simplify;apply in_Skip;apply H;apply (ex_intro ? ? a);
apply (ex_intro ? ? a1);assumption]]
qed.
-lemma nat_in_list_case : \forall G,H,n.(in_list nat n (H @ G)) \to
- (in_list nat n G) \lor (in_list nat n H).
-intros 3.elim H
- [simplify in H1;left;assumption
- |simplify in H2;elim (in_cons_case ? ? ? ? H2)
- [right;rewrite > H3;apply in_Base
- |elim H3;elim (H1 H5) [left;assumption|right;apply in_Skip2;assumption]]]
-qed.
-
-lemma natinG_or_inH_to_natinGH : \forall G,H,n.
- (in_list nat n G) \lor (in_list nat n H) \to
- (in_list nat n (H @ G)).
-intros.elim H1
- [elim H
- [simplify;assumption
- |simplify;apply in_Skip2;assumption]
- |generalize in match H2;elim H2
- [simplify;apply in_Base
- |lapply (H4 H3);simplify;apply in_Skip;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 (in_list_nil ? ? H);elim Hletin
|intros 3;elim t;simplify in H1;elim (in_cons_case ? ? ? ? H1)
[rewrite < H2;apply (ex_intro ? ? b);apply (ex_intro ? ? t1);apply in_Base
- |elim H2;elim (H H4);elim H5;apply (ex_intro ? ? a);
- apply (ex_intro ? ? a1);apply in_Skip
- [assumption
- |intro;destruct H7;elim (H3 Hcut1)]]]
-qed.
-
-theorem varinT_varinT_subst : \forall X,Y,T.
- (in_list ? X (fv_type T)) \to \forall n.
- (in_list ? X (fv_type (subst_type_nat T (TFree Y) n))).
-intros 3;elim T
- [simplify in H;elim (in_list_nil ? ? H)
- |simplify in H;simplify;assumption
- |simplify in H;elim (in_list_nil ? ? H)
- |simplify in H2;simplify;elim (nat_in_list_case ? ? ? H2);
- apply natinG_or_inH_to_natinGH;
- [left;apply (H1 H3)
- |right;apply (H H3)]
- |simplify in H2;simplify;elim (nat_in_list_case ? ? ? H2);
- apply natinG_or_inH_to_natinGH;
- [left;apply (H1 H3);
- |right;apply (H H3)]]
+ |elim (H H2);elim H3;apply (ex_intro ? ? a);
+ apply (ex_intro ? ? a1);apply in_Skip;assumption]]
qed.
lemma incl_bound_fv : \forall l1,l2.(incl ? l1 l2) \to
|apply (H ? H3)]]
qed.
-lemma incl_nat_cons : \forall x,l1,l2.
- (incl nat l1 l2) \to (incl nat (x :: l1) (x :: l2)).
+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_cons_case ? ? ? ? H1)
- [rewrite > H2;apply in_Base|elim H2;apply in_Skip2;apply (H ? H4)]
+ [rewrite > H2;apply in_Base|apply in_Skip;apply (H ? H2)]
qed.
lemma WFT_env_incl : \forall G,T.(WFType G T) \to
[apply (H2 ? H6)
|intros;apply (H4 ? ? H8)
[unfold;intro;apply H7;apply(H6 ? H9)
- |simplify;apply (incl_nat_cons ? ? ? H6)]]]
+ |simplify;apply (incl_cons ? ? ? H6)]]]
qed.
lemma fv_env_extends : \forall H,x,B,C,T,U,G.
intros 10;elim H
[simplify in H1;elim (in_cons_case ? ? ? ? H1)
[destruct H3;elim (H2 Hcut1)
- |simplify;elim H3;apply (in_Skip ? ? ? ? H5);intro;destruct H6;
- apply (H2 Hcut1)]
+ |simplify;apply (in_Skip ? ? ? ? H3);]
|simplify in H2;simplify;elim (in_cons_case ? ? ? ? H2)
[rewrite > H4;apply in_Base
- |elim H4;apply (in_Skip ? ? ? ? (H1 H6 H3) H5)]]
+ |apply (in_Skip ? ? ? ? (H1 H4 H3))]]
qed.
lemma in_FV_subst : \forall x,T,U,n.(in_list ? x (fv_type T)) \to
intros 3;elim T
[simplify in H;elim (in_list_nil ? ? H)
|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
- [1,3:left;apply (H1 ? H3)
- |*:right;apply (H ? H3)]]
+ |*:simplify in H2;simplify;elim (append_to_or_in_list ? ? ? ? H2)
+ [1,3:apply in_list_append1;apply (H ? H3)
+ |*:apply in_list_append2;apply (H1 ? H3)]]
qed.
(*** lemma on fresh names ***)
intros.unfold. intro.
elim (in_cons_case ? ? ? ? H3)
[rewrite > H4 in H2.autobatch
- |elim H4.apply (H1 m ? H6).
- apply (trans_le ? (max a t));autobatch]]]
+ |elim H4
+ [apply (H1 m ? H4).apply (trans_le ? (max a t));autobatch
+ |assumption]]]]
qed.
(*** lemmata on well-formedness ***)
(in_list ? x (fv_env G)).
intros 4.elim H
[simplify in H2;elim (in_cons_case ? ? ? ? H2)
- [rewrite > H3;assumption|elim H3;elim (in_list_nil ? ? H5)]
+ [rewrite > H3;assumption|elim (in_list_nil ? ? H3)]
|simplify in H1;elim (in_list_nil ? x H1)
- |simplify in H5;elim (nat_in_list_case ? ? ? H5);autobatch
- |simplify in H5;elim (nat_in_list_case ? ? ? H5)
- [elim (fresh_name ((fv_type t1) @ (fv_env l)));
- cut ((¬ (in_list ? a (fv_type t1))) ∧
- (¬ (in_list ? a (fv_env l))))
+ |simplify in H5;elim (append_to_or_in_list ? ? ? ? H5);autobatch
+ |simplify in H5;elim (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_cons_case ? ? ? ? Hletin)
- [elim (Hcut1 H10)|elim H10;assumption]
- |intro;apply H8;rewrite < H10;assumption]
+ [elim (Hcut1 H10)
+ |assumption]
+ |intro;apply H8;applyS H6]
|apply in_FV_subst;assumption]
|split
- [intro;apply H7;apply natinG_or_inH_to_natinGH;right;assumption
- |intro;apply H7;apply natinG_or_inH_to_natinGH;left;assumption]]
- |apply (H2 H6)]]
-qed.
-
-(*** some exotic inductions and related lemmas ***)
-
-lemma O_lt_t_len: \forall T.O < (t_len T).
-intros;elim T
- [1,2,3:simplify;apply le_n
- |*:simplify;apply lt_O_S]
-qed.
-
-(*
-lemma not_t_len_lt_SO : \forall T.\lnot (t_len T) < (S O).
-intros;elim T
- [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 Typ_len_ind : \forall P:Typ \to Prop.
- (\forall U.(\forall V.((t_len V) < (t_len U)) \to (P V))
- \to (P U))
- \to \forall T.(P T).
-cut (\forall P:Typ \to Prop.
- (\forall U.(\forall V.((t_len V) < (t_len U)) \to (P V))
- \to (P U))
- \to \forall T,n.(n = (t_len T)) \to (P T))
- [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
- [1,2,3:apply H;intros;simplify in H4;elim (lt_to_not_le ? ? H4 (O_lt_t_len ?))
- |*: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)).
-intros.simplify.
-apply le_S_S.apply le_m_max_m_n.
-qed.
-
-lemma t_len_arrow2 : \forall T1,T2.(t_len T2) < (t_len (Arrow T1 T2)).
-intros.simplify.
-apply le_S_S.apply le_n_max_m_n.
-qed.
-
-lemma t_len_forall1 : \forall T1,T2.(t_len T1) < (t_len (Forall T1 T2)).
-intros.simplify.
-apply le_S_S.apply le_m_max_m_n.
-qed.
-
-lemma t_len_forall2 : \forall T1,T2.(t_len T2) < (t_len (Forall T1 T2)).
-intros.simplify.
-apply le_S_S.apply le_n_max_m_n.
-qed.
-
-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
- |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;
- rewrite < Hletin1;reflexivity]
+ [intro;apply H7;apply in_list_append1;assumption
+ |intro;apply H7;apply in_list_append2;assumption]]]]
qed.
(*** lemmata relating subtyping and well-formedness ***)
[destruct H7;assumption
|elim H7;elim H3;apply boundinenv_natinfv;apply (ex_intro ? ? b);
apply (ex_intro ? ? T);assumption]
- |elim H7;elim (in_cons_case ? ? ? ? H5)
- [destruct H10;elim H3;apply boundinenv_natinfv;apply (ex_intro ? ? B);
+ |elim (in_cons_case ? ? ? ? H5)
+ [destruct H8;elim H3;apply boundinenv_natinfv;apply (ex_intro ? ? B);
apply (ex_intro ? ? U);rewrite < Hcut1;assumption
- |elim H10;apply (H2 H12 H9)]]]
+ |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;rewrite > Hcut;rewrite < H3.autobatch
+ |destruct H6;rewrite > Hcut1;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 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).
+intro;elim G;simplify;autobatch paramodulation;
+qed.
\ No newline at end of file
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;rewrite > Hcut;apply in_Base
+ |destruct H13;rewrite < Hcut1 in H10;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
- |assumption]
- |apply t_len_forall1]
- |apply (H10 ? H12)]
- |rewrite < eq_t_len_TFree_subst;
- apply t_len_forall2]]]]]
+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
+ [apply (JS_trans_TFree ? ? ? H3 ? H4)
+ |rewrite > (JSubtype_Top ? ? H3);apply SA_Top;autobatch
+ |cut (∃T.T = Arrow t t1) as Htriv
+ [elim Htriv;clear Htriv;rewrite < H in H6;rewrite < H in H7;
+ generalize in match H7;generalize in match H4;generalize in match H2;
+ generalize in match H5;generalize in match H;clear H7 H4 H2 H5 H;elim H6
+ [1,2:destruct H4
+ |apply (SA_Trans_TVar ? ? ? ? H);apply (H4 H5 H7 H8 H9);assumption
+ |inversion H11;intros
+ [apply SA_Top;rewrite < H14;autobatch
+ |destruct H15
+ |destruct H16
+ |destruct H17;apply SA_Arrow;rewrite < H16;destruct H7
+ [apply H9
+ [autobatch
+ |rewrite < Hcut2;rewrite > Hcut;rewrite > H16;assumption
+ |rewrite < Hcut2;assumption]
+ |apply H10
+ [autobatch
+ |rewrite < Hcut3;rewrite < Hcut1;rewrite < H16;assumption
+ |rewrite > H16;rewrite < Hcut3;rewrite > Hcut1;assumption]]
+ |destruct H17]
+ |destruct H7]
+ |apply (ex_intro ? ? (Arrow t t1));reflexivity]
+ |cut (∃T.T = Forall t t1) as Htriv
+ [elim Htriv;clear Htriv;rewrite < H in H7;rewrite < H in H6.
+ generalize in match H7;generalize in match H4;generalize in match H2;
+ generalize in match H5;generalize in match H;clear H7 H4 H2 H5 H;
+ elim H6
+ [1,2:destruct H4
+ |apply (SA_Trans_TVar ? ? ? ? H);apply (H4 H5 H7 H8 H9 H10)
+ |destruct H7
+ |inversion H11;intros
+ [apply SA_Top
+ [assumption
+ |rewrite < H14;apply WFT_Forall
+ [autobatch
+ |intros;lapply (H4 ? H17);autobatch]]
+ |destruct H15
+ |destruct H16
+ |destruct H17
+ |destruct H17;apply SA_All;destruct H7
+ [rewrite < H16;apply H9;
+ [autobatch
+ |rewrite < Hcut2;rewrite > Hcut;rewrite > H16;assumption
+ |rewrite < Hcut2. assumption]
+ |intros;apply (H10 X)
+ [intro;apply H19;rewrite < H16;apply H8;assumption
+ |intro;apply H19;rewrite < H16;apply H8;
+ apply (WFT_to_incl ? ? ? H3);assumption
+ |simplify;apply incl_cons;rewrite < H16;assumption
+ |apply (narrowing X ((mk_bound true X t6)::l2)
+ ? ? ? ? ? H12 ? ? [])
+ [intros;apply H9
+ [unfold;intros;lapply (H8 ? H21);rewrite < H16;
+ rewrite > fv_append;autobatch
+ |rewrite < Hcut2;rewrite > Hcut;
+ apply (JS_weakening ? ? ? H12)
+ [autobatch
+ |unfold;intros;autobatch]
+ |rewrite < Hcut2;rewrite > Hcut;assumption]
+ |rewrite < Hcut;rewrite < Hcut3;rewrite < H16;apply H4;
+ rewrite > H16;assumption
+ |reflexivity]
+ |rewrite < Hcut3;rewrite > Hcut1;apply H14;assumption]]]]
+ |apply (ex_intro ? ? (Forall t t1));reflexivity]]
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.
inductive in_list (A:Type): A → (list A) → Prop ≝
| in_Base : ∀ x,l.(in_list A x (x::l))
-| in_Skip : ∀ x,y,l.(in_list A x l) → (x ≠ y) → (in_list A x (y::l)).
+| in_Skip : ∀ x,y,l.(in_list A x l) → (in_list A x (y::l)).
notation > "hvbox(x ∈ l)"
non associative with precedence 30 for @{ 'inlist $x $l }.
interpretation "item in list" 'inlist x l =
(cic:/matita/Fsub/util/in_list.ind#xpointer(1/1) _ x l).
-lemma in_Skip2 : ∀x,y,l.(in_list nat x l) → (in_list nat x (y::l)).
-intros;elim (decidable_eq_nat x y)
- [rewrite > H1;apply in_Base
- |apply (in_Skip ? ? ? ? H H1)]
-qed.
-
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).
lemma in_list_nil : \forall A,x.\lnot (in_list A x []).
intros.unfold.intro.inversion H
[intros;lapply (sym_eq ? ? ? H2);destruct Hletin
- |intros;destruct H5]
+ |intros;destruct H4]
qed.
-lemma in_cons_case : â\88\80A.â\88\80x,h:A.â\88\80t:list A.x â\88\88 h::t â\86\92 x = h â\88¨ (x â\89 h â\88§ (x â\88\88 t)).
+lemma in_cons_case : â\88\80A.â\88\80x,h:A.â\88\80t:list A.x â\88\88 h::t â\86\92 x = h â\88¨ (x â\88\88 t).
intros;inversion H;intros
[destruct H2;left;symmetry;assumption
- |destruct H5;right;split [rewrite > Hcut|rewrite > Hcut1] assumption ]
+ |destruct H4;right;applyS H1]
+qed.
+
+lemma append_nil:\forall A:Type.\forall l:list A. l@[]=l.
+intros.
+elim l;intros;autobatch.
+qed.
+
+lemma append_cons:\forall A.\forall a:A.\forall l,l1.
+l@(a::l1)=(l@[a])@l1.
+intros.
+rewrite > associative_append.
+reflexivity.
+qed.
+
+lemma in_list_append1: \forall A.\forall x:A.
+\forall l1,l2. x \in l1 \to x \in l1@l2.
+intros.
+elim H
+ [simplify.apply in_Base
+ |simplify.apply in_Skip.assumption
+ ]
+qed.
+
+lemma in_list_append2: \forall A.\forall x:A.
+\forall l1,l2. x \in l2 \to x \in l1@l2.
+intros 3.
+elim l1
+ [simplify.assumption
+ |simplify.apply in_Skip.apply H.assumption
+ ]
+qed.
+
+theorem append_to_or_in_list: \forall A:Type.\forall x:A.
+\forall l,l1. x \in l@l1 \to (x \in l) \lor (x \in l1).
+intros 3.
+elim l
+ [right.apply H
+ |simplify in H1.inversion H1;intros
+ [destruct H3.left.rewrite < Hcut.
+ apply in_Base
+ |destruct H5.
+ elim (H l2)
+ [left.apply in_Skip.
+ rewrite < H4.assumption
+ |right.rewrite < H4.assumption
+ |rewrite > Hcut1.rewrite > H4.assumption
+ ]
+ ]
+ ]
qed.
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 incl_A_A: ∀T,A.incl T A A.
+intros.unfold incl.intros.assumption.
qed.
\ No newline at end of file
projects / helm.git / commitdiff