(**************************************************************************)
set "baseuri" "cic:/matita/Fsub/part1a/".
-include "library/logic/equality.ma".
-include "library/nat/nat.ma".
-include "library/datatypes/bool.ma".
-include "library/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]]
+(*** 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]]
qed.
-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]]
-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
- [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]
+ [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 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]]]
+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 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
- [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]
-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 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
- (JSubtype G T U)) \land
- (\forall G,H,X,P,Q,M,N.
- (t_len Q) \leq n \to
- (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
- (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)]
+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
- [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
- [inversion H5
- [intros;rewrite < H8;apply (SA_Top ? ? H2 H3)
- |intros;destruct H9
- |intros;destruct H10
- |intros;destruct H11
- |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;elim (H (pred m))
- [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)
- |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]
- |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;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)]]]]
+ [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]]]]]
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;
+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 (t_len Q));apply (H3 ? ? ? ? ? ? ? ? H H1);
-constructor 1;
-qed.
\ No newline at end of file
+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.
projects / helm.git / blobdiff