X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Flibrary%2FFsub%2Fpart1a.ma;h=8558725cc883569783da832862df8b0d1ed8e969;hb=34d2f477be65e3fd26bfb6d43a3dd0807274549b;hp=622b8f702acf7adfc519663ad710f69fbbba4aaa;hpb=16b9c591af0d9a188a916140e5fcd2b58805277f;p=helm.git diff --git a/helm/software/matita/library/Fsub/part1a.ma b/helm/software/matita/library/Fsub/part1a.ma index 622b8f702..8558725cc 100644 --- a/helm/software/matita/library/Fsub/part1a.ma +++ b/helm/software/matita/library/Fsub/part1a.ma @@ -13,368 +13,122 @@ (**************************************************************************) 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.