(* *)
(**************************************************************************)
-set "baseuri" "cic:/matita/Fsub/part1a/".
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 [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]]
+theorem JS_Refl : ∀T,G.(G ⊢ T) → G ⊢ ♦ → G ⊢ T ⊴ T.
+intros 3; elim H;try autobatch;
+apply SA_All; [ autobatch | intros;autobatch depth=4 size=10]
qed.
(*
* set inclusion.
*)
-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);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_list_head
- |destruct H13;apply in_list_cons;apply (H7 ? H10)]]]
+lemma JS_weakening : ∀G,T,U.G ⊢ T ⊴ U → ∀H.H ⊢ ♦ → G ⊆ H → H ⊢ T ⊴ U.
+intros 4; elim H;try autobatch depth=4 size=7;
+apply (SA_All ? ? ? ? ? (H2 ? H6 H7));
+intros; autobatch depth=6 width=4 size=13;
qed.
+inverter JS_indinv for JSubtype (%?%).
+
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]]
+ ∀l.G=l@ !X ⊴ U::G1 → l@ !X ⊴ P::G1 ⊢ M ⊴ N.
+intros 10.elim H2; destruct;
+ [letin x \def fv_env. letin y ≝incl. autobatch depth=4 size=8.
+ | autobatch depth=4 size=7;
+ | elim (decidable_eq_nat X n)
+ [apply (SA_Trans_TVar ? ? ? P); destruct;
+ [ autobatch
+ | lapply (WFE_bound_bound X t1 U ? ? H3); autobatch]
+ | apply (SA_Trans_TVar ? ? ? t1); autobatch]
+ | autobatch
+ | apply SA_All;
+ [ autobatch
+ | intros; apply (H6 ? ? (mk_bound true X1 t2::l1)); autobatch]]
qed.
-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
- [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]]]]]
+lemma JS_trans_prova: ∀T,G1.(G1 ⊢ T) →
+ ∀G,R,U.fv_env G1 ⊆ fv_env G → G ⊢ R ⊴ T → G ⊢ T ⊴ U → G ⊢ R ⊴ U.
+intros 3;elim H;clear H;
+ [elim H3 using JS_indinv;destruct;autobatch
+ |inversion H3; intros; destruct; assumption
+ |*:elim H6 using JS_indinv;destruct;
+ [1,3: autobatch
+ |*: inversion H7; intros; destruct;
+ [1,2: autobatch depth=4 width=4 size=9
+ | apply SA_Top
+ [ assumption
+ | apply WFT_Forall;intros;autobatch depth=4]
+ | apply SA_All
+ [ autobatch
+ | intros;apply (H4 X);simplify;
+ [4: apply (narrowing X (mk_bound true X t::G) ? ? ? ? ? H11 ? ? [])
+ [intros;apply H2;try unfold;intros;autobatch;
+ |*:autobatch]
+ |3:apply incl_cons;apply H5
+ |*:autobatch]]]]]
qed.
theorem JS_trans : ∀G,T,U,V.G ⊢ T ⊴ U → G ⊢ U ⊴ V → G ⊢ T ⊴ V.
-intros 5;apply (JS_trans_prova ? G);autobatch;
+intros 5; apply (JS_trans_prova ? G); autobatch depth=2.
qed.
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.
+ G2 @ !X ⊴ Q :: G1 ⊢ T ⊴ U → G1 ⊢ P ⊴ Q →
+ G2 @ !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.
+intros;apply (JS_trans ? ? ? ? ? H2);apply (JS_weakening ? ? ? H1);autobatch.
+qed.
\ No newline at end of file
projects / helm.git / blobdiff