(*** definitions about lists ***)
+definition filter_types : list bound → list bound ≝
+ λG.(filter ? G (λB.match B with [mk_bound B X T ⇒ B])).
+
definition fv_env : list bound → list nat ≝
- λG.(map ? ? (λb.match b with [mk_bound B X T ⇒ X]) G).
+ λG.(map ? ? (λb.match b with [mk_bound B X T ⇒ X]) (filter_types G)).
let rec fv_type T ≝
match T with
(subst_type_nat S2 (TFree X) O) (subst_type_nat T2 (TFree X) O)) →
JSubtype G (Forall S1 S2) (Forall T1 T2).
-notation "hvbox(e ⊢ break ta ⊴ break tb)"
+notation "mstyle color #007f00 (hvbox(e ⊢ break ta ⊴ break tb))"
non associative with precedence 30 for @{ 'subjudg $e $ta $tb }.
interpretation "Fsub subtype judgement" 'subjudg e ta tb = (JSubtype e ta tb).
-notation > "hvbox(\Forall S.T)"
+notation "mstyle color #007f00 (hvbox(e ⊢ ♦))"
+ non associative with precedence 30 for @{ 'wfejudg $e }.
+interpretation "Fsub WF env judgement" 'wfejudg e = (WFEnv e).
+
+notation "mstyle color #007f00 (hvbox(e ⊢ break t))"
+ non associative with precedence 30 for @{ 'wftjudg $e $t }.
+interpretation "Fsub WF type judgement" 'wftjudg e t = (WFType e t).
+
+notation > "\Forall S.T"
non associative with precedence 60 for @{ 'forall $S $T}.
-notation < "hvbox('All' \sub S. break T)"
+notation < "hvbox(⊓ \sub S. break T)"
non associative with precedence 60 for @{ 'forall $S $T}.
interpretation "universal type" 'forall S T = (Forall S T).
right associative with precedence 55 for @{ 'arrow $s $t }.
interpretation "arrow type" 'arrow S T = (Arrow S T).
-notation "hvbox(S [# n ↦ T])"
+notation "hvbox(S [#n ↦ T])"
non associative with precedence 80 for @{ 'substvar $S $T $n }.
interpretation "subst bound var" 'substvar S T n = (subst_type_nat S T n).
(*** theorems about lists ***)
-lemma boundinenv_natinfv : ∀x,G.(∃B,T.in_list ? (mk_bound B x T) G) →
- in_list ? x (fv_env G).
+lemma boundinenv_natinfv : ∀x,G.(∃T.!x ⊴ T ∈ G) → x ∈ (fv_env G).
intros 2;elim G;decompose
- [elim (not_in_list_nil ? ? H)
- |elim (in_list_cons_case ? ? ? ? H1)
- [rewrite < H2;simplify;apply in_list_head
- |simplify;apply in_list_cons;apply H;autobatch]]
+ [elim (not_in_list_nil ? ? H1)
+ |elim (in_list_cons_case ? ? ? ? H2)
+ [rewrite < H1;simplify;apply in_list_head
+ |elim a;apply (bool_elim ? b);intro;simplify;try apply in_list_cons;
+ apply H;autobatch]]
qed.
-lemma natinfv_boundinenv : ∀x,G.in_list ? x (fv_env G) →
- ∃B,T.in_list ? (mk_bound B x T) G.
+lemma natinfv_boundinenv : ∀x,G.x ∈ (fv_env G) → ∃T.!x ⊴ T ∈ G.
intros 2;elim G 0
[simplify;intro;lapply (not_in_list_nil ? ? H);elim Hletin
|intros 3;
- elim a;simplify in H1;elim (in_list_cons_case ? ? ? ? H1)
+ elim a;simplify in H1;elim b in H1;simplify in H1
+ [elim (in_list_cons_case ? ? ? ? H1)
[rewrite < H2;autobatch
- |elim (H H2);elim H3;apply ex_intro[apply a1];autobatch]]
+ |elim (H H2);autobatch]
+ |elim (H H1);autobatch]]
qed.
-lemma incl_bound_fv : ∀l1,l2.incl ? l1 l2 → incl ? (fv_env l1) (fv_env l2).
+lemma incl_bound_fv : ∀l1,l2.l1 ⊆ l2 → (fv_env l1) ⊆ (fv_env l2).
intros;unfold in H;unfold;intros;apply boundinenv_natinfv;
lapply (natinfv_boundinenv ? ? H1);decompose;autobatch depth=4;
qed.
-lemma incl_cons : ∀x,l1,l2.incl ? l1 l2 → incl nat (x :: l1) (x :: l2).
-intros.unfold in H.unfold.intros.elim (in_list_cons_case ? ? ? ? H1)
- [applyS in_list_head|autobatch]
-qed.
-
-lemma WFT_env_incl : ∀G,T.WFType G T →
- ∀H.incl ? (fv_env G) (fv_env H) → WFType H T.
+lemma WFT_env_incl : ∀G,T.(G ⊢ T) → ∀H.fv_env G ⊆ fv_env H → (H ⊢ T).
intros 3.elim H
[apply WFT_TFree;unfold in H3;apply (H3 ? H1)
|apply WFT_Top
[apply (H2 ? H6)
|intros;apply (H4 ? ? H8)
[unfold;intro;autobatch
- |simplify;apply (incl_cons ? ? ? H6)]]]
+ |simplify;apply (incl_cons ???? H6)]]]
qed.
-lemma fv_env_extends : ∀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
- [reflexivity|elim a;simplify;rewrite > H1;reflexivity]
+lemma fv_env_extends : ∀H,x,T,U,G,B.
+ fv_env (H @ mk_bound B x T :: G) =
+ fv_env (H @ mk_bound B x U :: G).
+intros 5;elim H;elim B
+ [1,2:reflexivity
+ |*:elim a;elim b;simplify;lapply (H1 true);lapply (H1 false);
+ try rewrite > Hletin;try rewrite > Hletin1;reflexivity]
qed.
lemma lookup_env_extends : ∀G,H,B,C,D,T,U,V,x,y.
- in_list ? (mk_bound D y V) (H @ ((mk_bound C x U) :: G)) → y ≠ x →
- in_list ? (mk_bound D y V) (H @ ((mk_bound B x T) :: G)).
+ mk_bound D y V ∈ H @ mk_bound C x U :: G → y ≠ x →
+ mk_bound D y V ∈ H @ mk_bound B x T :: G.
intros 10;elim H
[simplify in H1;elim (in_list_cons_case ? ? ? ? H1)
[destruct H3;elim H2;reflexivity
|apply (in_list_cons ? ? ? ? (H1 H4 H3))]]
qed.
-lemma in_FV_subst : ∀x,T,U,n.in_list ? x (fv_type T) →
- in_list ? x (fv_type (subst_type_nat T U n)).
+lemma in_FV_subst : ∀x,T,U,n.x ∈ fv_type T → x ∈ fv_type (subst_type_nat T U n).
intros 3;elim T
[simplify in H;elim (not_in_list_nil ? ? H)
|2,3:simplify;simplify in H;assumption
(*** lemma on fresh names ***)
-lemma fresh_name : ∀l:list nat.∃n.¬in_list ? n l.
+lemma fresh_name : ∀l:list nat.∃n.n ∉ l.
cut (∀l:list nat.∃n.∀m.n ≤ m → ¬ in_list ? m l);intros
[lapply (Hcut l);elim Hletin;apply ex_intro;autobatch
|elim l
(*** lemmata on well-formedness ***)
-lemma fv_WFT : ∀T,x,G.WFType G T → in_list ? x (fv_type T) →
- in_list ? x (fv_env G).
+lemma fv_WFT : ∀T,x,G.(G ⊢ T) → x ∈ fv_type T → x ∈ fv_env G.
intros 4.elim H
[simplify in H2;elim (in_list_cons_case ? ? ? ? H2)
[applyS H1|elim (not_in_list_nil ? ? H3)]
(*** lemmata relating subtyping and well-formedness ***)
-lemma JS_to_WFE : ∀G,T,U.G ⊢ T ⊴ U → WFEnv G.
+lemma JS_to_WFE : ∀G,T,U.G ⊢ T ⊴ U → G ⊢ ♦.
intros;elim H;assumption.
qed.
-lemma JS_to_WFT : ∀G,T,U.G ⊢ T ⊴ U → WFType G T ∧ WFType G U.
+lemma JS_to_WFT : ∀G,T,U.G ⊢ T ⊴ U → (G ⊢ T) ∧ (G ⊢ U).
intros;elim H
[1,2:autobatch
|split
apply (WFT_env_incl ? ? H10);simplify;unfold;intros;assumption]]
qed.
-lemma JS_to_WFT1 : ∀G,T,U.G ⊢ T ⊴ U → WFType G T.
+lemma JS_to_WFT1 : ∀G,T,U.G ⊢ T ⊴ U → G ⊢ T.
intros;elim (JS_to_WFT ? ? ? H);assumption;
qed.
-lemma JS_to_WFT2 : ∀G,T,U.G ⊢ T ⊴ U → WFType G U.
+lemma JS_to_WFT2 : ∀G,T,U.G ⊢ T ⊴ U → G ⊢ U.
intros;elim (JS_to_WFT ? ? ? H);assumption;
qed.
-lemma WFE_Typ_subst : ∀H,x,B,C,T,U,G.
- WFEnv (H @ ((mk_bound B x T) :: G)) → WFType G U →
- WFEnv (H @ ((mk_bound C x U) :: G)).
-intros 7;elim H 0
+lemma WFE_Typ_subst : ∀H,x,B,T,U,G.
+ H @ mk_bound B x T :: G ⊢ ♦ → (G ⊢ U) →
+ H @ mk_bound B x U :: G ⊢ ♦.
+intros 6;elim H 0
[simplify;intros;inversion H1;intros
[elim (nil_cons ? G (mk_bound B x T) H3)
|destruct H7;autobatch]
[destruct H5
|destruct H9;apply WFE_cons
[apply (H1 H5 H3)
- |rewrite < (fv_env_extends ? x B C T U); assumption
+ |rewrite < (fv_env_extends ? x T U); assumption
|apply (WFT_env_incl ? ? H8);
- rewrite < (fv_env_extends ? x B C T U);unfold;intros;
+ rewrite < (fv_env_extends ? x T U);unfold;intros;
assumption]]]
qed.
-lemma WFE_bound_bound : ∀B,x,T,U,G.WFEnv G → in_list ? (mk_bound B x T) G →
- in_list ? (mk_bound B x U) G → T = U.
-intros 6;elim H
+lemma WFE_bound_bound : ∀x,T,U,G.G ⊢ ♦ → !x ⊴ T ∈ G → !x ⊴ U ∈ G → T = U.
+intros 5;elim H
[lapply (not_in_list_nil ? ? H1);elim Hletin
|elim (in_list_cons_case ? ? ? ? H6)
[destruct H7;destruct;elim (in_list_cons_case ? ? ? ? H5)
|apply (H2 H8 H7)]]]
qed.
-lemma WFT_to_incl: ∀G,T,U.(∀X.¬in_list ? X (fv_env G) → ¬in_list ? X (fv_type U) →
- WFType (mk_bound true X T::G) (subst_type_nat U (TFree X) O))
- → incl ? (fv_type U) (fv_env G).
+lemma WFT_to_incl: ∀G,T,U.(∀X.X ∉ fv_env G → X ∉ fv_type U →
+ (mk_bound true X T::G ⊢ (subst_type_nat U (TFree X) O))) →
+ fv_type U ⊆ fv_env G.
intros;elim (fresh_name (fv_type U@fv_env G));lapply(H a)
[unfold;intros;lapply (fv_WFT ? x ? Hletin)
[simplify in Hletin1;inversion Hletin1;intros
qed.
lemma incl_fv_env: ∀X,G,G1,U,P.
- incl ? (fv_env (G1@(mk_bound true X U::G)))
- (fv_env (G1@(mk_bound true X P::G))).
+ fv_env (G1@ !X ⊴ U::G) ⊆ fv_env (G1@ !X ⊴ P::G).
intros.rewrite < fv_env_extends.apply incl_A_A.
qed.
lemma fv_append : ∀G,H.fv_env (G @ H) = fv_env G @ fv_env H.
-intro;elim G;simplify;autobatch paramodulation;
+intro;elim G;simplify;
+[reflexivity
+|elim a;simplify;elim b;simplify;lapply (H H1);rewrite > Hletin;reflexivity]
qed.
\ No newline at end of file
projects / helm.git / blobdiff