|apply H2
[apply t_len_arrow2
|*:assumption]]
- |(*FIXME*)generalize in match H3;intro;inversion H3
+ |(* no shortcut? *)
+ (*FIXME*)generalize in match H3;intro;inversion H3
[intros;destruct H8
|intros;destruct H7
|intros;destruct H11;rewrite > Hcut;rewrite > Hcut1;split;assumption
|intros;destruct H11]]
- |elim (fresh_name ((fv_type t1) @ (fv_env G)));
- cut ((\lnot (in_list ? a (fv_type t1))) \land
- (\lnot (in_list ? a (fv_env G))))
- [elim Hcut;cut (WFType G t)
- [apply (SA_All2 ? ? ? ? ? a ? H7 H6 H6)
- [apply H2
- [apply t_len_forall1
- |*:assumption]
- |apply H2
- [rewrite > subst_O_nat;rewrite < eq_t_len_TFree_subst;
- apply t_len_forall2
- |(*FIXME*)generalize in match H3;intro;inversion H3
- [intros;destruct H11
- |intros;destruct H10
- |intros;destruct H14
- |intros;destruct H14;rewrite < Hcut2 in H11;
- rewrite < Hcut3 in H11;rewrite < H13;rewrite < H13 in H11;
- apply (H11 ? H7 H6)]
- |apply WFE_cons;assumption]]
- |(*FIXME*)generalize in match H3;intro;inversion H3
- [intros;destruct H11
- |intros;destruct H10
- |intros;destruct H14
- |intros;destruct H14;rewrite > Hcut1;assumption]]
- |split;unfold;intro;apply H5;apply natinG_or_inH_to_natinGH;autobatch]]
+ |cut (WFType G t)
+ [lapply (H2 t ? ? Hcut H4)
+ [apply t_len_forall1
+ |apply (SA_All ? ? ? ? ? Hletin);intros;apply H2
+ [rewrite > subst_O_nat;rewrite < eq_t_len_TFree_subst;
+ apply t_len_forall2
+ |generalize in match H3;intro;inversion H3
+ [intros;destruct H9
+ |intros;destruct H8
+ |intros;destruct H12
+ |intros;destruct H12;subst;apply H9
+ [assumption
+ |unfold;intro;apply H5;
+ elim (fresh_name ((fv_env e)@(fv_type t3)));
+ cut ((\lnot (in_list ? a (fv_env e))) \land
+ (\lnot (in_list ? a (fv_type t3))))
+ [elim Hcut1;lapply (H9 ? H13 H14);
+ lapply (fv_WFT ? X ? Hletin1)
+ [simplify in Hletin2;inversion Hletin2
+ [intros;lapply (inj_head_nat ? ? ? ? H16);subst;
+ elim (H14 H11)
+ |intros;lapply (inj_tail ? ? ? ? ? H18);
+ rewrite < Hletin3 in H15;assumption]
+ |rewrite >subst_O_nat;apply varinT_varinT_subst;
+ assumption]
+ |split;unfold;intro;apply H12;apply natinG_or_inH_to_natinGH
+ [right;assumption
+ |left;assumption]]]]
+ |apply WFE_cons;assumption]]
+ |(*FIXME*)generalize in match H3;intro;inversion H3
+ [intros;destruct H8
+ |intros;destruct H7
+ |intros;destruct H11
+ |intros;destruct H11;subst;assumption]]]
qed.
(*
[unfold;intro;apply H8;lapply (incl_bound_fv ? ? H7);apply (Hletin1 ? H9)
|apply WFE_cons
[1,2:assumption
- |lapply (incl_bound_fv ? ? H7);apply (WFT_env_incl ? ? ? ? Hletin1);
- apply (JS_to_WFT1 ? ? ? H1)]
+ |apply (JS_to_WFT1 ? ? ? Hletin)]
|unfold;intros;inversion H9
[intros;lapply (inj_head ? ? ? ? H11);rewrite > Hletin1;apply in_Base
|intros;lapply (inj_tail ? ? ? ? ? H13);rewrite < Hletin1 in H10;
(* Lemma A.3 (Transitivity and Narrowing) *)
-lemma JS_trans_narrow : \forall n.
- (\forall G,T,Q,U.
- (t_len Q) \leq n \to (JSubtype G T Q) \to (JSubtype G Q U) \to
+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,Q,M,N.
- (t_len Q) \leq n \to
+ (\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)).
-intro;apply (nat_elim1 n);intros 2;
-cut (\forall G,T,Q.(JSubtype G T Q) \to
- \forall U.(t_len Q \leq m) \to (JSubtype G Q U) \to (JSubtype G T U))
- [cut (\forall G,M,N.(JSubtype G M N) \to
- \forall G1,X,Q,G2,P.
- (G = G2 @ ((mk_bound true X Q) :: G1)) \to (t_len Q) \leq m \to
- (JSubtype G1 P Q) \to
+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))
- [split
- [intros;apply (Hcut ? ? ? H2 ? H1 H3)
- |intros;apply (Hcut1 ? ? ? H3 ? ? ? ? ? ? H2 H4);reflexivity]
- |intros 9;cut (incl ? (fv_env (G2 @ ((mk_bound true X Q)::G1)))
- (fv_env (G2 @ ((mk_bound true X P)::G1))))
- [intros;
-(* [rewrite > H6 in H2;lapply (JS_to_WFT1 ? ? ? H8);
- apply (WFE_Typ_subst ? ? ? ? ? ? ? H2 Hletin) *)
- generalize in match Hcut1;generalize in match H2;
- generalize in match H1;generalize in match H4;
- generalize in match G1;generalize in match G2;elim H1
- [apply SA_Top
- [rewrite > H9 in H5;lapply (JS_to_WFT1 ? ? ? H7);
- apply (WFE_Typ_subst ? ? ? ? ? ? ? H5 Hletin)
- |rewrite > H9 in H6;apply (WFT_env_incl ? ? H6);elim l
- [simplify;unfold;intros;assumption
- |simplify;apply (incl_nat_cons ? ? ? H11)]]
- |apply SA_Refl_TVar
- [rewrite > H9 in H5;lapply (JS_to_WFT1 ? ? ? H7);
- apply (WFE_Typ_subst ? ? ? ? ? ? ? H5 Hletin)
- |apply H10;rewrite < H9;assumption]
- |elim (decidable_eq_nat X n1)
- [apply (SA_Trans_TVar ? ? ? P)
- [rewrite < H12;elim l
- [simplify;apply in_Base
- |simplify;apply in_Skip;assumption]
- |lapply (JS_to_WFE ? ? ? H9);rewrite > H10 in Hletin;
- rewrite > H10 in H5;
- lapply (WFE_bound_bound ? ? ? Q ? Hletin H5)
- [lapply (H7 ? ? H8 H6 H10 H11);rewrite > Hletin1 in Hletin2;
- apply (Hcut ? ? ? ? ? H3 Hletin2);
- lapply (JS_to_WFE ? ? ? Hletin2);
- apply (JS_weakening ? ? ? H8 ? Hletin3);unfold;intros;
- elim l;simplify;apply in_Skip;assumption
- |rewrite > H12;elim l
- [simplify;apply in_Base
- |simplify;apply in_Skip;assumption]]]
- |rewrite > H10 in H5;apply (SA_Trans_TVar ? ? ? t1)
- [apply (lookup_env_extends ? ? ? ? ? ? ? ? ? ? H5);unfold;
- intro;apply H12;symmetry;assumption
- |apply (H7 ? ? H8 H6 H10 H11)]]
- |apply SA_Arrow
- [apply (H6 ? ? H9 H5 H11 H12)
- |apply (H8 ? ? H9 H7 H11 H12)]
- |apply SA_All
- [apply (H6 ? ? H9 H5 H11 H12)
- |intros;apply (H8 ? ? (mk_bound true X1 t2::l) l1)
- [unfold;intro;apply H13;rewrite > H11 in H14;apply (H12 ? H14)
- |assumption
- |apply H7;rewrite > H11;unfold;intro;apply H13;apply (H12 ? H14)
- |simplify;rewrite < H11;reflexivity
- |simplify;apply incl_nat_cons;assumption]]]
- |elim G2 0
- [simplify;unfold;intros;assumption
- |intro;elim t 0;simplify;intros;apply incl_nat_cons;assumption]]]
- |intros 4;(*generalize in match H1;*)elim H1
+ [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 l
+ [simplify;constructor 1
+ |simplify;constructor 2;assumption]
+ |apply H7
+ [lapply (H6 ? H7 H8 H9);lapply (JS_to_WFE ? ? ? Hletin);
+ apply (JS_weakening ? ? ? H3 ? Hletin1);unfold;intros;
+ elim l;simplify;constructor 2;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 l;simplify
+ [constructor 1
+ |constructor 2;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::l) H8)
+ [rewrite > H10;cut ((fv_env (l@(mk_bound true X P::G1))) =
+ (fv_env (l@(mk_bound true X U::G1))))
+ [unfold;intro;apply H11;unfold;rewrite > Hcut2;assumption
+ |elim l
+ [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)
+ |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;elim (H (pred m))
- [apply SA_Arrow
+ |intros;destruct H13;apply SA_Arrow
[rewrite > H12 in H2;rewrite < Hcut in H8;
- apply (H15 ? ? ? ? ? H8 H2);lapply (t_len_arrow1 t2 t3);
- unfold in Hletin;lapply (trans_le ? ? ? Hletin H6);
- apply (t_len_pred ? ? Hletin1)
+ lapply (H6 t2)
+ [elim Hletin;apply (H15 ? ? ? H8 H2)
+ |apply (t_len_arrow1 t2 t3)]
|rewrite > H12 in H4;rewrite < Hcut1 in H10;
- apply (H15 ? ? ? ? ? H4 H10);lapply (t_len_arrow2 t2 t3);
- unfold in Hletin;lapply (trans_le ? ? ? Hletin H6);
- apply (t_len_pred ? ? Hletin1)]
- |apply (pred_m_lt_m ? ? H6)]
- |intros;destruct H13]
+ 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;destruct H11
|intros;destruct H12
|intros;destruct H13
- |intros;destruct H13;elim (H (pred m))
- [elim (fresh_name ((fv_env e1) @ (fv_type t1) @ (fv_type t7)));
- cut ((\lnot (in_list ? a (fv_env e1))) \land
- (\lnot (in_list ? a (fv_type t1))) \land
- (\lnot (in_list ? a (fv_type t7))))
- [elim Hcut2;elim H18;clear Hcut2 H18;apply (SA_All2 ? ? ? ? ? a)
- [rewrite < Hcut in H8;rewrite > H12 in H2;
- apply (H15 ? ? ? ? ? H8 H2);lapply (t_len_forall1 t2 t3);
- unfold in Hletin;lapply (trans_le ? ? ? Hletin H6);
- apply (t_len_pred ? ? Hletin1)
- |5:lapply (H10 ? H20);rewrite > H12 in H5;
- lapply (H5 ? H20 (subst_type_O t5 (TFree a)))
- [apply (H15 ? ? ? ? ? ? Hletin)
- [rewrite < Hcut1;rewrite > subst_O_nat;
- rewrite < eq_t_len_TFree_subst;
- lapply (t_len_forall2 t2 t3);unfold in Hletin2;
- lapply (trans_le ? ? ? Hletin2 H6);
- apply (t_len_pred ? ? Hletin3)
- |rewrite < Hcut in H8;
- apply (H16 e1 (nil ?) a t6 t2 ? ? ? Hletin1 H8);
- lapply (t_len_forall1 t2 t3);unfold in Hletin2;
- lapply (trans_le ? ? ? Hletin2 H6);
- apply (t_len_pred ? ? Hletin3)]
- |rewrite > subst_O_nat;rewrite < eq_t_len_TFree_subst;
- lapply (t_len_forall2 t2 t3);unfold in Hletin1;
- lapply (trans_le ? ? ? Hletin1 H6);
- apply (trans_le ? ? ? ? Hletin2);constructor 2;
- constructor 1
- |rewrite > Hcut1;rewrite > H12 in H4;
- lapply (H4 ? H20);rewrite < Hcut1;apply JS_Refl
- [apply (JS_to_WFT2 ? ? ? Hletin1)
- |apply (JS_to_WFE ? ? ? Hletin1)]]
- |*:assumption]
- |split
- [split
- [unfold;intro;apply H17;
- apply (natinG_or_inH_to_natinGH ? (fv_env e1));right;
- assumption
- |unfold;intro;apply H17;
- apply (natinG_or_inH_to_natinGH
- ((fv_type t1) @ (fv_type t7)));left;
- apply natinG_or_inH_to_natinGH;right;assumption]
- |unfold;intro;apply H17;
- apply (natinG_or_inH_to_natinGH
- ((fv_type t1) @ (fv_type t7)));left;
- apply natinG_or_inH_to_natinGH;left;assumption]]
- |apply (pred_m_lt_m ? ? H6)]]]]
+ |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_O t5 (TFree X)))
+ [elim Hletin;apply H13
+ [lapply (H6 t4)
+ [elim Hletin1;apply (H16 e1 [] X t6);
+ [simplify;apply H4;assumption
+ |assumption]
+ |apply t_len_forall1]
+ |apply (H10 ? H12)]
+ |rewrite > subst_O_nat;rewrite < eq_t_len_TFree_subst;
+ apply t_len_forall2]]]]]
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 (t_len U));apply (H2 ? ? ? ? ? H H1);constructor 1;
+intros;elim JS_trans_narrow;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 (t_len Q));apply (H3 ? ? ? ? ? ? ? ? H H1);
-constructor 1;
+intros;elim JS_trans_narrow;autobatch;
qed.
projects / helm.git / blobdiff