(**************************************************************************)
set "baseuri" "cic:/matita/Fsub/part1a/".
-include "logic/equality.ma".
-include "nat/nat.ma".
-include "datatypes/bool.ma".
-include "nat/compare.ma".
include "Fsub/defn.ma".
-theorem JS_Refl : \forall T,G.(WFType G T) \to (WFEnv G) \to (JSubtype G T T).
-apply Typ_len_ind;intro;elim U
- [(* FIXME *) generalize in match H1;intro;inversion H1
- [intros;destruct H6
- |intros;destruct H5
- |intros;destruct H9
- |intros;destruct H9]
- |apply (SA_Refl_TVar ? ? H2);(*FIXME*)generalize in match H1;intro;
- inversion H1
- [intros;destruct H6;rewrite > Hcut;assumption
- |intros;destruct H5
- |intros;destruct H9
- |intros;destruct H9]
- |apply (SA_Top ? ? H2 H1)
- |cut ((WFType G t) \land (WFType G t1))
- [elim Hcut;apply SA_Arrow
- [apply H2
- [apply t_len_arrow1
- |assumption
- |assumption]
- |apply H2
- [apply t_len_arrow2
- |assumption
- |assumption]]
- |(*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
- |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;auto]]
-qed.
+(*** Lemma A.1 (Reflexivity) ***)
-lemma env_append_weaken : \forall G,H.(WFEnv (H @ G)) \to
- (incl ? G (H @ G)).
-intros 2;elim H
- [simplify;unfold;intros;assumption
- |simplify in H2;simplify;unfold;intros;apply in_Skip;apply H1
- [apply (WFE_consG_WFE_G ? ? H2)
- |assumption]]
+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]
+ |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).
intros 4;elim H
|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
- [assumption
- |assumption
- |lapply (incl_bound_fv ? ? H7);apply (WFT_env_incl ? ? ? ? Hletin1);
- apply (JS_to_WFT1 ? ? ? H1)]
- |unfold;intros;inversion H9
- [intros;lapply (inj_head ? ? ? ? H11);rewrite > Hletin1;apply in_Base
- |intros;lapply (inj_tail ? ? ? ? ? H13);rewrite < Hletin1 in H10;
- apply in_Skip;apply (H7 ? H10)]]]
-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 s;simplify;rewrite > H1;reflexivity]
+ [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)]]]
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));lapply (Hletin H4);
- elim Hletin1
- |intros;lapply (inj_tail ? ? ? ? ? H8);lapply (inj_head ? ? ? ? H8);
- destruct Hletin1;rewrite < Hletin in H6;rewrite < Hletin in H4;
- rewrite < Hcut1 in H6;apply (WFE_cons ? ? ? ? H4 H6 H2)]
- |intros;simplify;generalize in match H2;elim s;simplify in H4;
- inversion H4
- [intros;absurd (mk_bound b n t::l@(mk_bound B x T::G)=Empty)
- [assumption
- |apply nil_cons]
- |intros;lapply (inj_tail ? ? ? ? ? H9);lapply (inj_head ? ? ? ? H9);
- destruct Hletin1;apply WFE_cons
- [apply H1
- [rewrite > Hletin;assumption
- |assumption]
- |rewrite > Hcut1;generalize in match H7;rewrite < Hletin;
- rewrite > (fv_env_extends ? x B C T U);intro;assumption
- |rewrite < Hletin in H8;rewrite > Hcut2;
- apply (WFT_env_incl ? ? H8);rewrite > (fv_env_extends ? x B C T U);
- unfold;intros;assumption]]]
-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;(*FIXME*)generalize in match H1;intro;inversion H1
- [intros;lapply (inj_head ? ? ? ? H5);rewrite < H4 in Hletin;
- destruct Hletin;absurd (y = x) [symmetry;assumption|assumption]
- |intros;simplify;lapply (inj_tail ? ? ? ? ? H7);rewrite > Hletin;
- apply in_Skip;assumption]
- |(*FIXME*)generalize in match H2;intro;inversion H2
- [intros;simplify in H6;lapply (inj_head ? ? ? ? H6);rewrite > Hletin;
- simplify;apply in_Base
- |simplify;intros;lapply (inj_tail ? ? ? ? ? H8);rewrite > Hletin in H1;
- rewrite > H7 in H1;apply in_Skip;apply (H1 H5 H3)]]
-qed.
-
-lemma t_len_pred: \forall T,m.(S (t_len T)) \leq m \to (t_len T) \leq (pred m).
-intros 2;elim m
- [elim (not_le_Sn_O ? H)
- |simplify;apply (le_S_S_to_le ? ? H1)]
-qed.
-
-lemma pred_m_lt_m : \forall m,T.(t_len T) \leq m \to (pred m) < m.
-intros 2;elim m 0
+lemma decidable_eq_Typ : \forall S,T:Typ.(S = T) ∨ (S ≠ T).
+intro;elim S
[elim T
- [simplify in H;elim (not_le_Sn_n ? H)
- |simplify in H;elim (not_le_Sn_n ? H)
- |simplify in H;elim (not_le_Sn_n ? H)
- |simplify in H2;elim (not_le_Sn_O ? H2)
- |simplify in H2;elim (not_le_Sn_O ? H2)]
- |intros;simplify;unfold;constructor 1]
+ [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]]]
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 (in_list_nil ? ? H1);elim Hletin
- |inversion H6
- [intros;rewrite < H7 in H8;lapply (inj_head ? ? ? ? H8);
- rewrite > Hletin in H5;inversion H5
- [intros;rewrite < H9 in H10;lapply (inj_head ? ? ? ? H10);
- destruct Hletin1;symmetry;assumption
- |intros;lapply (inj_tail ? ? ? ? ? H12);rewrite < Hletin1 in H9;
- rewrite < H11 in H9;lapply (boundinenv_natinfv x e)
- [destruct Hletin;rewrite < Hcut1 in Hletin2;lapply (H3 Hletin2);
- elim Hletin3
- |apply ex_intro
- [apply B|apply ex_intro [apply T|assumption]]]]
- |intros;lapply (inj_tail ? ? ? ? ? H10);rewrite < H9 in H7;
- rewrite < Hletin in H7;(*FIXME*)generalize in match H5;intro;inversion H5
- [intros;rewrite < H12 in H13;lapply (inj_head ? ? ? ? H13);
- destruct Hletin1;rewrite < Hcut1 in H7;
- lapply (boundinenv_natinfv n e)
- [lapply (H3 Hletin2);elim Hletin3
- |apply ex_intro
- [apply B|apply ex_intro [apply U|assumption]]]
- |intros;apply (H2 ? H7);rewrite > H14;lapply (inj_tail ? ? ? ? ? H15);
- rewrite > Hletin1;assumption]]]
-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 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 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,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 s 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 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
- |intros;destruct H11]
+ |*: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)
- |assumption
- |assumption
- |assumption
- |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)]]]
- |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_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]]]]]
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