}.
(* representation of Fsub typing environments *)
-definition Env \def (list bound).
+(*definition Env \def (list bound).
definition Empty \def (nil bound).
definition Cons \def \lambda G,X,T.((mk_bound false X T) :: G).
definition TCons \def \lambda G,X,T.((mk_bound true X T) :: G).
-definition env_append : Env \to Env \to Env \def \lambda G,H.(H @ G).
+definition env_append : Env \to Env \to Env \def \lambda G,H.(H @ G). *)
-(* notation "hvbox(\Forall S. break T)"
- non associative with precedence 90
-for @{ 'forall $S $T}.
-
-notation "hvbox(#x)"
- with precedence 60
- for @{'var $x}.
-
-notation "hvbox(##x)"
- with precedence 61
- for @{'tvar $x}.
-
-notation "hvbox(!x)"
- with precedence 60
- for @{'name $x}.
-
-notation "hvbox(!!x)"
- with precedence 61
- for @{'tname $x}.
-
-notation "hvbox(s break \mapsto t)"
- right associative with precedence 55
- for @{ 'arrow $s $t }.
-
-interpretation "universal type" 'forall S T = (cic:/matita/Fsub/defn/Typ.ind#xpointer(1/1/5) S T).
-
-interpretation "bound var" 'var x = (cic:/matita/Fsub/defn/Typ.ind#xpointer(1/1/1) x).
-
-interpretation "bound tvar" 'tvar x = (cic:/matita/Fsub/defn/Typ.ind#xpointer(1/1/3) x).
-
-interpretation "bound tname" 'tname x = (cic:/matita/Fsub/defn/Typ.ind#xpointer(1/1/2) x).
-
-interpretation "arrow type" 'arrow S T = (cic:/matita/Fsub/defn/Typ.ind#xpointer(1/1/4) S T). *)
-
(*** Various kinds of substitution, not all will be used probably ***)
(* substitutes i-th dangling index in type T with type U *)
| (Forall T1 T2) \Rightarrow (Forall (subst_type_nat T1 U i) (subst_type_nat T2 U (S i))) ].
(* substitutes 0-th dangling index in type T with type U *)
-let rec subst_type_O T U \def subst_type_nat T U O.
+(*let rec subst_type_O T U \def subst_type_nat T U O.*)
(* substitutes 0-th dangling index in term t with term u *)
-let rec subst_term_O t u \def
+(*let rec subst_term_O t u \def
let rec aux t0 i \def
match t0 with
[ (Var n) \Rightarrow match (eqb n i) with
| (Arrow T1 T2) \Rightarrow (Arrow (subst_type_tfree_type T1 X U)
(subst_type_tfree_type T2 X U))
| (Forall T1 T2) \Rightarrow (Forall (subst_type_tfree_type T1 X U)
- (subst_type_tfree_type T2 X U)) ].
+ (subst_type_tfree_type T2 X U)) ].*)
(*** height of T's syntactic tree ***)
(*** definitions about lists ***)
-(* var binding is in env judgement *)
-definition var_bind_in_env : bound \to Env \to Prop \def
- \lambda b,G.(in_list bound b G).
+(*(* var binding is in env judgement *)
+definition var_bind_in_env : bound \to (list bound) \to Prop \def
+ \lambda b,G.(in_list bound b G).*)
definition fv_env : (list bound) \to (list nat) \def
\lambda G.(map ? ? (\lambda b.match b with
[(mk_bound B X T) \Rightarrow X]) G).
-(* variable is in env judgement *)
-definition var_in_env : nat \to Env \to Prop \def
+(*(* variable is in env judgement *)
+definition var_in_env : nat \to (list bound) \to Prop \def
\lambda x,G.(in_list nat x (fv_env G)).
-definition var_type_in_env : nat \to Env \to Prop \def
- \lambda x,G.\exists T.(var_bind_in_env (mk_bound true x T) G).
+definition var_type_in_env : nat \to (list bound) \to Prop \def
+ \lambda x,G.\exists T.(var_bind_in_env (mk_bound true x T) G).*)
let rec fv_type T \def
match T with
(*** Type Well-Formedness judgement ***)
-inductive WFType : Env \to Typ \to Prop \def
+inductive WFType : (list bound) \to Typ \to Prop \def
| WFT_TFree : \forall X,G.(in_list ? X (fv_env G))
\to (WFType G (TFree X))
| WFT_Top : \forall G.(WFType G Top)
(\lnot (in_list ? X (fv_env G))) \to
(\lnot (in_list ? X (fv_type U))) \to
(WFType ((mk_bound true X T) :: G)
- (subst_type_O U (TFree X)))) \to
+ (subst_type_nat U (TFree X) O))) \to
(WFType G (Forall T U)).
(*** Environment Well-Formedness judgement ***)
-inductive WFEnv : Env \to Prop \def
- | WFE_Empty : (WFEnv Empty)
+inductive WFEnv : (list bound) \to Prop \def
+ | WFE_Empty : (WFEnv (nil ?))
| WFE_cons : \forall B,X,T,G.(WFEnv G) \to
\lnot (in_list ? X (fv_env G)) \to
(WFType G T) \to (WFEnv ((mk_bound B X T) :: G)).
(*** Subtyping judgement ***)
-inductive JSubtype : Env \to Typ \to Typ \to Prop \def
- | SA_Top : \forall G:Env.\forall T:Typ.(WFEnv G) \to
+inductive JSubtype : (list bound) \to Typ \to Typ \to Prop \def
+ | SA_Top : \forall G.\forall T:Typ.(WFEnv G) \to
(WFType G T) \to (JSubtype G T Top)
- | SA_Refl_TVar : \forall G:Env.\forall X:nat.(WFEnv G) \to (var_in_env X G)
+ | SA_Refl_TVar : \forall G.\forall X:nat.(WFEnv G)
+ \to (in_list ? X (fv_env G))
\to (JSubtype G (TFree X) (TFree X))
- | SA_Trans_TVar : \forall G:Env.\forall X:nat.\forall T:Typ.
+ | SA_Trans_TVar : \forall G.\forall X:nat.\forall T:Typ.
\forall U:Typ.
- (var_bind_in_env (mk_bound true X U) G) \to
+ (in_list ? (mk_bound true X U) G) \to
(JSubtype G U T) \to (JSubtype G (TFree X) T)
- | SA_Arrow : \forall G:Env.\forall S1,S2,T1,T2:Typ.
+ | SA_Arrow : \forall G.\forall S1,S2,T1,T2:Typ.
(JSubtype G T1 S1) \to (JSubtype G S2 T2) \to
(JSubtype G (Arrow S1 S2) (Arrow T1 T2))
- | SA_All : \forall G:Env.\forall S1,S2,T1,T2:Typ.
+ | SA_All : \forall G.\forall S1,S2,T1,T2:Typ.
(JSubtype G T1 S1) \to
- (\forall X:nat.\lnot (var_in_env X G) \to
+ (\forall X:nat.\lnot (in_list ? X (fv_env G)) \to
(JSubtype ((mk_bound true X T1) :: G)
- (subst_type_O S2 (TFree X)) (subst_type_O T2 (TFree X)))) \to
+ (subst_type_nat S2 (TFree X) O) (subst_type_nat T2 (TFree X) O))) \to
(JSubtype G (Forall S1 S2) (Forall T1 T2)).
-(*
-notation < "hvbox(e break ⊢ ta \nbsp 'V' \nbsp tb (= \above \alpha))"
- non associative with precedence 30 for @{ 'subjudg $e $ta $tb }.
-notation > "hvbox(e break ⊢ ta 'Fall' break tb)"
- non associative with precedence 30 for @{ 'subjudg $e $ta $tb }.
-notation "hvbox(e break ⊢ ta \lessdot break tb)"
+notation "hvbox(e ⊢ break ta ⊴ break tb)"
non associative with precedence 30 for @{ 'subjudg $e $ta $tb }.
interpretation "Fsub subtype judgement" 'subjudg e ta tb =
(cic:/matita/Fsub/defn/JSubtype.ind#xpointer(1/1) e ta tb).
-lemma xx : \forall e,ta,tb. e \vdash ta Fall tb.
-*)
-
-(*** Typing judgement ***)
-inductive JType : Env \to Term \to Typ \to Prop \def
- | T_Var : \forall G:Env.\forall x:nat.\forall T:Typ.
- (WFEnv G) \to (var_bind_in_env (mk_bound false x T) G) \to
- (JType G (Free x) T)
- | T_Abs : \forall G.\forall T1,T2:Typ.\forall t2:Term.
- \forall x:nat.
- (JType ((mk_bound false x T1)::G) (subst_term_O t2 (Free x)) T2) \to
- (JType G (Abs T1 t2) (Arrow T1 T2))
- | T_App : \forall G.\forall t1,t2:Term.\forall T2:Typ.
- \forall T1:Typ.(JType G t1 (Arrow T1 T2)) \to (JType G t2 T1) \to
- (JType G (App t1 t2) T2)
- | T_TAbs : \forall G:Env.\forall T1,T2:Typ.\forall t2:Term.
- \forall X:nat.
- (JType ((mk_bound true X T1)::G)
- (subst_term_tO t2 (TFree X)) (subst_type_O T2 (TFree X)))
- \to (JType G (TAbs T1 t2) (Forall T1 T2))
- | T_TApp : \forall G:Env.\forall t1:Term.\forall T2,T12:Typ.
- \forall X:nat.\forall T11:Typ.
- (JType G t1 (Forall T11 (subst_type_tfree_type T12 X (TVar O)))) \to
- (JSubtype G T2 T11)
- \to (JType G (TApp t1 T2) (subst_type_tfree_type T12 X T2))
- | T_Sub : \forall G:Env.\forall t:Term.\forall T:Typ.
- \forall S:Typ.(JType G t S) \to (JSubtype G S T) \to (JType G t T).
+notation > "hvbox(\Forall S.T)"
+ non associative with precedence 60 for @{ 'forall $S $T}.
+notation < "hvbox('All' \sub S. break T)"
+ non associative with precedence 60 for @{ 'forall $S $T}.
+interpretation "universal type" 'forall S T =
+ (cic:/matita/Fsub/defn/Typ.ind#xpointer(1/1/5) S T).
+
+notation "#x" with precedence 79 for @{'tvar $x}.
+interpretation "bound tvar" 'tvar x =
+ (cic:/matita/Fsub/defn/Typ.ind#xpointer(1/1/1) x).
+
+notation "!x" with precedence 79 for @{'tname $x}.
+interpretation "bound tname" 'tname x =
+ (cic:/matita/Fsub/defn/Typ.ind#xpointer(1/1/2) x).
+
+notation "⊤" with precedence 90 for @{'toptype}.
+interpretation "toptype" 'toptype =
+ (cic:/matita/Fsub/defn/Typ.ind#xpointer(1/1/3)).
+
+notation "hvbox(s break ⇛ t)"
+ right associative with precedence 55 for @{ 'arrow $s $t }.
+interpretation "arrow type" 'arrow S T =
+ (cic:/matita/Fsub/defn/Typ.ind#xpointer(1/1/4) S T).
+
+notation "hvbox(S [# n ↦ T])"
+ non associative with precedence 80 for @{ 'substvar $S $T $n }.
+interpretation "subst bound var" 'substvar S T n =
+ (cic:/matita/Fsub/defn/subst_type_nat.con S T n).
+
+notation "hvbox(|T|)"
+ non associative with precedence 30 for @{ 'tlen $T }.
+interpretation "type length" 'tlen T =
+ (cic:/matita/Fsub/defn/t_len.con T).
+
+notation > "hvbox(x ∈ l)"
+ non associative with precedence 30 for @{ 'inlist $x $l }.
+notation < "hvbox(x \nbsp ∈ \nbsp l)"
+ non associative with precedence 30 for @{ 'inlist $x $l }.
+interpretation "item in list" 'inlist x l =
+ (cic:/matita/Fsub/util/in_list.ind#xpointer(1/1) _ x l).
+
+notation "hvbox(!X ⊴ T)"
+ non associative with precedence 60 for @{ 'subtypebound $X $T }.
+interpretation "subtyping bound" 'subtypebound X T =
+ (cic:/matita/Fsub/defn/bound.ind#xpointer(1/1/1) true X T).
+
+(*notation < "hvbox(e break ⊢ ta \nbsp 'V' \nbsp tb (= \above \alpha))"
+ non associative with precedence 30 for @{ 'subjudg $e $ta $tb }.
+notation > "hvbox(e break ⊢ ta 'Fall' break tb)"
+ non associative with precedence 30 for @{ 'subjudg $e $ta $tb }.
+notation "hvbox(e break ⊢ ta \lessdot break tb)"
+ non associative with precedence 30 for @{ 'subjudg $e $ta $tb }.*)
(****** PROOFS ********)
-lemma subst_O_nat : \forall T,U.((subst_type_O T U) = (subst_type_nat T U O)).
+(*lemma subst_O_nat : \forall T,U.((subst_type_O T U) = T[#O↦U]).
intros;elim T;simplify;reflexivity;
-qed.
+qed.*)
(*** theorems about lists ***)
(* FIXME: these definitions shouldn't be part of the poplmark challenge
- use destruct instead, when hopefully it will get fixed... *)
-lemma inj_head : \forall h1,h2:bound.\forall t1,t2:Env.
- ((h1::t1) = (h2::t2)) \to (h1 = h2).
+lemma inj_head : \forall h1,h2:bound.\forall t1,t2:(list bound).
+ (h1::t1 = h2::t2) \to h1 = h2.
intros.
lapply (eq_f ? ? head ? ? H).simplify in Hletin.assumption.
qed.
lemma inj_head_nat : \forall h1,h2:nat.\forall t1,t2:(list nat).
- ((h1::t1) = (h2::t2)) \to (h1 = h2).
+ (h1::t1 = h2::t2) \to (h1 = h2).
intros.
lapply (eq_f ? ? head_nat ? ? H).simplify in Hletin.assumption.
qed.
[apply (H4 H6)
|apply (H2 H6)]
|simplify in H5;lapply (nat_in_list_case ? ? ? H5);elim Hletin
- [lapply (fresh_name ((fv_type t1) @ (fv_env e)));elim Hletin1;
+ [lapply (fresh_name ((fv_type t1) @ (fv_env l)));elim Hletin1;
cut ((\lnot (in_list ? a (fv_type t1))) \land
- (\lnot (in_list ? a (fv_env e))))
+ (\lnot (in_list ? a (fv_env l))))
[elim Hcut;lapply (H4 ? H9 H8)
[cut (x \neq a)
[simplify in Hletin2;
|intros;lapply (inj_tail ? ? ? ? ? H14);rewrite > Hletin3;
assumption]
|unfold;intro;apply H8;rewrite < H10;assumption]
- |rewrite > subst_O_nat;apply in_FV_subst;assumption]
+ |apply in_FV_subst;assumption]
|split
[unfold;intro;apply H7;apply natinG_or_inH_to_natinGH;right;
assumption
intros.simplify.
(* FIXME!!! BUG?!?! *)
cut ((max (t_len T1) (t_len T2)) = match (leb (t_len T1) (t_len T2)) with
- [ false \Rightarrow (t_len T2)
- | true \Rightarrow (t_len T1) ])
+ [ false ⇒ (t_len T2)
+ | true ⇒ (t_len T1) ])
[rewrite > Hcut;cut ((leb (t_len T1) (t_len T2)) = false \lor
(leb (t_len T1) (t_len T2)) = true)
[lapply (leb_to_Prop (t_len T1) (t_len T2));elim Hcut1
(*** lemmata relating subtyping and well-formedness ***)
-lemma JS_to_WFE : \forall G,T,U.(G \vdash T \lessdot U) \to (WFEnv G).
+lemma JS_to_WFE : \forall G,T,U.(G \vdash T ⊴ U) \to (WFEnv G).
intros;elim H;assumption.
qed.
rewrite < Hcut1 in H6;apply (WFE_cons ? ? ? ? H4 H6 H2)]
|intros;simplify;generalize in match H2;elim t;simplify in H4;
inversion H4
- [intros;absurd (mk_bound b n t1::l@(mk_bound B x T::G)=Empty)
+ [intros;absurd (mk_bound b n t1::l@(mk_bound B x T::G)=[])
[assumption
|apply nil_cons]
|intros;lapply (inj_tail ? ? ? ? ? H9);lapply (inj_head ? ? ? ? H9);
[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)
+ rewrite < H11 in H9;lapply (boundinenv_natinfv x l)
[destruct Hletin;rewrite < Hcut1 in Hletin2;lapply (H3 Hletin2);
elim Hletin3
|apply ex_intro
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 (boundinenv_natinfv n l)
[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.
\ No newline at end of file
+qed.
[lapply (H2 t ? ? Hcut H4)
[apply t_len_forall1
|apply (SA_All ? ? ? ? ? Hletin);intros;apply H2
- [rewrite > subst_O_nat;rewrite < eq_t_len_TFree_subst;
+ [rewrite < eq_t_len_TFree_subst;
apply t_len_forall2
|generalize in match H3;intro;inversion H3
[intros;destruct H9
|intros;destruct H12;subst;apply H9
[assumption
|unfold;intro;apply H5;
- elim (fresh_name ((fv_env e)@(fv_type t3)));
- cut ((\lnot (in_list ? a (fv_env e))) \land
+ elim (fresh_name ((fv_env l)@(fv_type t3)));
+ cut ((\lnot (in_list ? a (fv_env l))) \land
(\lnot (in_list ? a (fv_type t3))))
[elim Hcut1;lapply (H9 ? H13 H14);
lapply (fv_WFT ? X ? Hletin1)
elim (H14 H11)
|intros;lapply (inj_tail ? ? ? ? ? H18);
rewrite < Hletin3 in H15;assumption]
- |rewrite >subst_O_nat;apply varinT_varinT_subst;
+ |apply varinT_varinT_subst;
assumption]
|split;unfold;intro;apply H12;apply natinG_or_inH_to_natinGH
[right;assumption
|rewrite > H8 in H5;apply (H7 ? H5)]
|elim (decidable_eq_nat X n)
[apply (SA_Trans_TVar ? ? ? P)
- [rewrite < H10;elim l
+ [rewrite < H10;elim l1
[simplify;constructor 1
|simplify;constructor 2;assumption]
|apply H7
[lapply (H6 ? H7 H8 H9);lapply (JS_to_WFE ? ? ? Hletin);
apply (JS_weakening ? ? ? H3 ? Hletin1);unfold;intros;
- elim l;simplify;constructor 2;assumption
+ elim l1;simplify;constructor 2;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 l;simplify
+ |rewrite > H9;rewrite > H10;elim l1;simplify
[constructor 1
|constructor 2;assumption]]]]
|apply (SA_Trans_TVar ? ? ? t1)
|apply (H7 ? H8 H9 H10)]
|apply SA_All
[apply (H5 ? H8 H9 H10)
- |intros;apply (H7 ? ? (mk_bound true X1 t2::l) H8)
- [rewrite > H10;cut ((fv_env (l@(mk_bound true X P::G1))) =
- (fv_env (l@(mk_bound true X U::G1))))
- [unfold;intro;apply H11;unfold;rewrite > Hcut2;assumption
- |elim l
+ |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)
[elim Hletin;apply (H12 ? ? ? H8 H2)
|apply t_len_forall1]
|intros;(*destruct H12;*)subst;
- lapply (H6 (subst_type_O t5 (TFree X)))
+ lapply (H6 (subst_type_nat t5 (TFree X) O))
[elim Hletin;apply H13
[lapply (H6 t4)
- [elim Hletin1;apply (H16 e1 [] X t6);
+ [elim Hletin1;apply (H16 l1 [] X t6);
[simplify;apply H4;assumption
|assumption]
|apply t_len_forall1]
|apply (H10 ? H12)]
- |rewrite > subst_O_nat;rewrite < eq_t_len_TFree_subst;
+ |rewrite < eq_t_len_TFree_subst;
apply t_len_forall2]]]]]
qed.
projects / helm.git / commitdiff