X-Git-Url: http://matita.cs.unibo.it/gitweb/?p=helm.git;a=blobdiff_plain;f=matita%2Fcontribs%2FPOPLmark%2FFsub%2Fdefn2.ma;fp=matita%2Fcontribs%2FPOPLmark%2FFsub%2Fdefn2.ma;h=eaeb47491d3a03af95bb40365f41d844fb4788ab;hp=0000000000000000000000000000000000000000;hb=f61af501fb4608cc4fb062a0864c774e677f0d76;hpb=58ae1809c352e71e7b5530dc41e2bfc834e1aef1 diff --git a/matita/contribs/POPLmark/Fsub/defn2.ma b/matita/contribs/POPLmark/Fsub/defn2.ma new file mode 100644 index 000000000..eaeb47491 --- /dev/null +++ b/matita/contribs/POPLmark/Fsub/defn2.ma @@ -0,0 +1,359 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "Fsub/util.ma". + +(*** representation of Fsub types ***) +inductive Typ : Set \def + | TVar : nat \to Typ (* type var *) + | TFree: nat \to Typ (* free type name *) + | Top : Typ (* maximum type *) + | Arrow : Typ \to Typ \to Typ (* functions *) + | Forall : Typ \to Typ \to Typ. (* universal type *) + +(* representation of bounds *) + +record bound : Set \def { + istype : bool; (* is subtyping bound? *) + name : nat ; (* name *) + btype : Typ (* type to which the name is bound *) + }. + +(*** Various kinds of substitution, not all will be used probably ***) + +(* substitutes i-th dangling index in type T with type U *) +let rec subst_type_nat T U i \def + match T with + [ (TVar n) \Rightarrow match (eqb n i) with + [ true \Rightarrow U + | false \Rightarrow T] + | (TFree X) \Rightarrow T + | Top \Rightarrow T + | (Arrow T1 T2) \Rightarrow (Arrow (subst_type_nat T1 U i) (subst_type_nat T2 U i)) + | (Forall T1 T2) \Rightarrow (Forall (subst_type_nat T1 U i) (subst_type_nat T2 U (S i))) ]. + +(*** definitions about lists ***) + +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). + +let rec fv_type T \def + match T with + [(TVar n) \Rightarrow [] + |(TFree x) \Rightarrow [x] + |Top \Rightarrow [] + |(Arrow U V) \Rightarrow ((fv_type U) @ (fv_type V)) + |(Forall U V) \Rightarrow ((fv_type U) @ (fv_type V))]. + +(*** Type Well-Formedness judgement ***) + +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) + | WFT_Arrow : \forall G,T,U.(WFType G T) \to (WFType G U) \to + (WFType G (Arrow T U)) + | WFT_Forall : \forall G,T,U.(WFType G T) \to + (\forall X:nat. + (\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_nat U (TFree X) O))) \to + (WFType G (Forall T U)). + +(*** Environment Well-Formedness judgement ***) + +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 : (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.\forall X:nat.(WFEnv G) + \to (in_list ? X (fv_env G)) + \to (JSubtype G (TFree X) (TFree X)) + | SA_Trans_TVar : \forall G.\forall X:nat.\forall T:Typ. + \forall U:Typ. + (in_list ? (mk_bound true X U) G) \to + (JSubtype G U T) \to (JSubtype G (TFree X) T) + | 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.\forall S1,S2,T1,T2:Typ. + (JSubtype G T1 S1) \to + (\forall X:nat.\lnot (in_list ? X (fv_env G)) \to + (JSubtype ((mk_bound true X T1) :: G) + (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 ⊴ break tb)" + non associative with precedence 30 for @{ 'subjudg $e $ta $tb }. +interpretation "Fsub subtype judgement" 'subjudg e ta tb = + (cic:/matita/Fsub/defn2/JSubtype.ind#xpointer(1/1) e ta tb). + +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/defn2/Typ.ind#xpointer(1/1/5) S T). + +notation "#x" with precedence 79 for @{'tvar $x}. +interpretation "bound tvar" 'tvar x = + (cic:/matita/Fsub/defn2/Typ.ind#xpointer(1/1/1) x). + +notation "!x" with precedence 79 for @{'tname $x}. +interpretation "bound tname" 'tname x = + (cic:/matita/Fsub/defn2/Typ.ind#xpointer(1/1/2) x). + +notation "⊤" with precedence 90 for @{'toptype}. +interpretation "toptype" 'toptype = + (cic:/matita/Fsub/defn2/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/defn2/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/defn2/subst_type_nat.con S T n). + +notation "hvbox(!X ⊴ T)" + non associative with precedence 60 for @{ 'subtypebound $X $T }. +interpretation "subtyping bound" 'subtypebound X T = + (cic:/matita/Fsub/defn2/bound.ind#xpointer(1/1/1) true X T). + +(****** PROOFS ********) + +(*** theorems about lists ***) + +lemma boundinenv_natinfv : \forall x,G. + (\exists B,T.(in_list ? (mk_bound B x T) G)) \to + (in_list ? x (fv_env G)). +intros 2;elim G + [elim H;elim H1;lapply (not_in_list_nil ? ? H2);elim Hletin + |elim H1;elim H2;elim (in_list_cons_case ? ? ? ? H3) + [rewrite < H4;simplify;apply in_list_head + |simplify;apply in_list_cons;apply H;apply (ex_intro ? ? a); + apply (ex_intro ? ? a1);assumption]] +qed. + +lemma natinfv_boundinenv : \forall x,G.(in_list ? x (fv_env G)) \to + \exists B,T.(in_list ? (mk_bound B x T) G). +intros 2;elim G 0 + [simplify;intro;lapply (not_in_list_nil ? ? H);elim Hletin + |intros 3;elim t;simplify in H1;elim (in_list_cons_case ? ? ? ? H1) + [rewrite < H2;apply (ex_intro ? ? b);apply (ex_intro ? ? t1);apply in_list_head + |elim (H H2);elim H3;apply (ex_intro ? ? a); + apply (ex_intro ? ? a1);apply in_list_cons;assumption]] +qed. + +lemma incl_bound_fv : \forall l1,l2.(incl ? l1 l2) \to + (incl ? (fv_env l1) (fv_env l2)). +intros.unfold in H.unfold.intros.apply boundinenv_natinfv. +lapply (natinfv_boundinenv ? ? H1).elim Hletin.elim H2.apply ex_intro + [apply a + |apply ex_intro + [apply a1 + |apply (H ? H3)]] +qed. + +lemma incl_cons : \forall x,l1,l2. + (incl ? l1 l2) \to (incl nat (x :: l1) (x :: l2)). +intros.unfold in H.unfold.intros.elim (in_list_cons_case ? ? ? ? H1) + [rewrite > H2;apply in_list_head|apply in_list_cons;apply (H ? H2)] +qed. + +lemma WFT_env_incl : \forall G,T.(WFType G T) \to + \forall H.(incl ? (fv_env G) (fv_env H)) \to (WFType H T). +intros 3.elim H + [apply WFT_TFree;unfold in H3;apply (H3 ? H1) + |apply WFT_Top + |apply WFT_Arrow [apply (H2 ? H6)|apply (H4 ? H6)] + |apply WFT_Forall + [apply (H2 ? H6) + |intros;apply (H4 ? ? H8) + [unfold;intro;apply H7;apply(H6 ? H9) + |simplify;apply (incl_cons ? ? ? H6)]]] +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 t;simplify;rewrite > H1;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;elim (in_list_cons_case ? ? ? ? H1) + [destruct H3;elim (H2);reflexivity + |simplify;apply (in_list_cons ? ? ? ? H3);] + |simplify in H2;simplify;elim (in_list_cons_case ? ? ? ? H2) + [rewrite > H4;apply in_list_head + |apply (in_list_cons ? ? ? ? (H1 H4 H3))]] +qed. + +lemma in_FV_subst : \forall x,T,U,n.(in_list ? x (fv_type T)) \to + (in_list ? 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 + |*:simplify in H2;simplify;elim (in_list_append_to_or_in_list ? ? ? ? H2) + [1,3:apply in_list_to_in_list_append_l;apply (H ? H3) + |*:apply in_list_to_in_list_append_r;apply (H1 ? H3)]] +qed. + +(*** lemma on fresh names ***) + +lemma fresh_name : \forall l:(list nat).\exists n.\lnot (in_list ? n l). +cut (\forall l:(list nat).\exists n.\forall m. + (n \leq m) \to \lnot (in_list ? m l)) + [intros;lapply (Hcut l);elim Hletin;apply ex_intro + [apply a + |apply H;constructor 1] + |intros;elim l + [apply (ex_intro ? ? O);intros;unfold;intro;elim (not_in_list_nil ? ? H1) + |elim H; + apply (ex_intro ? ? (S (max a t))). + intros.unfold. intro. + elim (in_list_cons_case ? ? ? ? H3) + [rewrite > H4 in H2.autobatch + |elim H4 + [apply (H1 m ? H4).apply (trans_le ? (max a t));autobatch + |assumption]]]] +qed. + +(*** lemmata on well-formedness ***) + +lemma fv_WFT : \forall T,x,G.(WFType G T) \to (in_list ? x (fv_type T)) \to + (in_list ? x (fv_env G)). +intros 4.elim H + [simplify in H2;elim (in_list_cons_case ? ? ? ? H2) + [rewrite > H3;assumption|elim (not_in_list_nil ? ? H3)] + |simplify in H1;elim (not_in_list_nil ? x H1) + |simplify in H5;elim (in_list_append_to_or_in_list ? ? ? ? H5);autobatch + |simplify in H5;elim (in_list_append_to_or_in_list ? ? ? ? H5) + [apply (H2 H6) + |elim (fresh_name ((fv_type t1) @ (fv_env l))); + cut (¬ (a ∈ (fv_type t1)) ∧ ¬ (a ∈ (fv_env l))) + [elim Hcut;lapply (H4 ? H9 H8) + [cut (x ≠ a) + [simplify in Hletin;elim (in_list_cons_case ? ? ? ? Hletin) + [elim (Hcut1 H10) + |assumption] + |intro;apply H8;applyS H6] + |apply in_FV_subst;assumption] + |split + [intro;apply H7;apply in_list_to_in_list_append_l;assumption + |intro;apply H7;apply in_list_to_in_list_append_r;assumption]]]] +qed. + +(*** lemmata relating subtyping and well-formedness ***) + +lemma JS_to_WFE : \forall G,T,U.(G \vdash T ⊴ U) \to (WFEnv G). +intros;elim H;assumption. +qed. + +lemma JS_to_WFT : \forall G,T,U.(JSubtype G T U) \to ((WFType G T) \land + (WFType G U)). +intros;elim H + [split [assumption|apply WFT_Top] + |split;apply WFT_TFree;assumption + |split + [apply WFT_TFree;apply boundinenv_natinfv;apply ex_intro + [apply true | apply ex_intro [apply t1 |assumption]] + |elim H3;assumption] + |elim H2;elim H4;split;apply WFT_Arrow;assumption + |elim H2;split + [apply (WFT_Forall ? ? ? H6);intros;elim (H4 X H7); + apply (WFT_env_incl ? ? H9);simplify;unfold;intros;assumption + |apply (WFT_Forall ? ? ? H5);intros;elim (H4 X H7); + apply (WFT_env_incl ? ? H10);simplify;unfold;intros;assumption]] +qed. + +lemma JS_to_WFT1 : \forall G,T,U.(JSubtype G T U) \to (WFType G T). +intros;lapply (JS_to_WFT ? ? ? H);elim Hletin;assumption. +qed. + +lemma JS_to_WFT2 : \forall G,T,U.(JSubtype G T U) \to (WFType G U). +intros;lapply (JS_to_WFT ? ? ? H);elim Hletin;assumption. +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));elim (Hletin H4) + |destruct H8;apply (WFE_cons ? ? ? ? H4 H6 H2)] + |intros;simplify;generalize in match H2;elim t;simplify in H4; + inversion H4;intros + [destruct H5 + |destruct H9;apply WFE_cons + [apply (H1 H5 H3) + |rewrite < (fv_env_extends ? x B C T U); assumption + |apply (WFT_env_incl ? ? H8); + rewrite < (fv_env_extends ? x B C T U);unfold;intros; + assumption]]] +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 (not_in_list_nil ? ? H1);elim Hletin + |elim (in_list_cons_case ? ? ? ? H6) + [destruct H7;destruct;elim (in_list_cons_case ? ? ? ? H5) + [destruct H7;reflexivity + |elim H7;elim H3;apply boundinenv_natinfv;apply (ex_intro ? ? B); + apply (ex_intro ? ? T);assumption] + |elim (in_list_cons_case ? ? ? ? H5) + [destruct H8;elim H3;apply boundinenv_natinfv;apply (ex_intro ? ? B); + apply (ex_intro ? ? U);assumption + |apply (H2 H8 H7)]]] +qed. + +lemma WFT_to_incl: ∀G,T,U. + (∀X.(¬(X ∈ fv_env G)) → (¬(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). +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 + [destruct H4;elim H1;autobatch + |destruct H6;assumption] + |apply in_FV_subst;assumption] + |*:intro;apply H1;autobatch] +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))). +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; +qed.