--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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 *)
+(* *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/Fsub/defn".
+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))) ].
+
+(*** height of T's syntactic tree ***)
+
+let rec t_len T \def
+ match T with
+ [(TVar n) \Rightarrow (S O)
+ |(TFree X) \Rightarrow (S O)
+ |Top \Rightarrow (S O)
+ |(Arrow T1 T2) \Rightarrow (S (max (t_len T1) (t_len T2)))
+ |(Forall T1 T2) \Rightarrow (S (max (t_len T1) (t_len T2)))].
+
+(*** 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/defn/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/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 ⊴ 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).
+
+(****** 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 (in_list_nil ? ? H2);elim Hletin
+ |elim H1;elim H2;elim (in_cons_case ? ? ? ? H3)
+ [rewrite < H4;simplify;apply in_Base
+ |simplify;apply in_Skip;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 (in_list_nil ? ? H);elim Hletin
+ |intros 3;elim t;simplify in H1;elim (in_cons_case ? ? ? ? H1)
+ [rewrite < H2;apply (ex_intro ? ? b);apply (ex_intro ? ? t1);apply in_Base
+ |elim (H H2);elim H3;apply (ex_intro ? ? a);
+ apply (ex_intro ? ? a1);apply in_Skip;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_cons_case ? ? ? ? H1)
+ [rewrite > H2;apply in_Base|apply in_Skip;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_cons_case ? ? ? ? H1)
+ [destruct H3;elim (H2);reflexivity
+ |simplify;apply (in_Skip ? ? ? ? H3);]
+ |simplify in H2;simplify;elim (in_cons_case ? ? ? ? H2)
+ [rewrite > H4;apply in_Base
+ |apply (in_Skip ? ? ? ? (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 (in_list_nil ? ? H)
+ |2,3:simplify;simplify in H;assumption
+ |*:simplify in H2;simplify;elim (append_to_or_in_list ? ? ? ? H2)
+ [1,3:apply in_list_append1;apply (H ? H3)
+ |*:apply in_list_append2;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 (in_list_nil ? ? H1)
+ |elim H;
+ apply (ex_intro ? ? (S (max a t))).
+ intros.unfold. intro.
+ elim (in_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_cons_case ? ? ? ? H2)
+ [rewrite > H3;assumption|elim (in_list_nil ? ? H3)]
+ |simplify in H1;elim (in_list_nil ? x H1)
+ |simplify in H5;elim (append_to_or_in_list ? ? ? ? H5);autobatch
+ |simplify in H5;elim (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_cons_case ? ? ? ? Hletin)
+ [elim (Hcut1 H10)
+ |assumption]
+ |intro;apply H8;applyS H6]
+ |apply in_FV_subst;assumption]
+ |split
+ [intro;apply H7;apply in_list_append1;assumption
+ |intro;apply H7;apply in_list_append2;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 (in_list_nil ? ? H1);elim Hletin
+ |elim (in_cons_case ? ? ? ? H6)
+ [destruct H7;destruct;elim (in_cons_case ? ? ? ? H5)
+ [destruct H7;reflexivity
+ |elim H7;elim H3;apply boundinenv_natinfv;apply (ex_intro ? ? B);
+ apply (ex_intro ? ? T);assumption]
+ |elim (in_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 JSubtype_Top: ∀G,P.G ⊢ ⊤ ⊴ P → P = ⊤.
+intros.inversion H;intros
+ [assumption|reflexivity
+ |destruct H5|*:destruct H6]
+qed.
+
+(* elim vs inversion *)
+lemma JS_trans_TFree: ∀G,T,X.G ⊢ T ⊴ (TFree X) →
+ ∀U.G ⊢ (TFree X) ⊴ U → G ⊢ T ⊴ U.
+intros 4.cut (∀Y.TFree Y = TFree X → ∀U.G ⊢ (TFree Y) ⊴ U → G ⊢ T ⊴ U)
+ [apply Hcut;reflexivity
+ |elim H;intros
+ [rewrite > H3 in H4;rewrite > (JSubtype_Top ? ? H4);apply SA_Top;assumption
+ |rewrite < H3;assumption
+ |apply (SA_Trans_TVar ? ? ? ? H1);apply (H3 Y);assumption
+ |*:destruct H5]]
+qed.
+
+lemma fv_append : ∀G,H.fv_env (G @ H) = (fv_env G @ fv_env H).
+intro;elim G;simplify;autobatch paramodulation;
+qed.
\ No newline at end of file
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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 *)
+(* *)
+(**************************************************************************)
+
+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]]
+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 : ∀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_Base
+ |destruct H13;apply in_Skip;apply (H7 ? H10)]]]
+qed.
+
+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 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]]]]]
+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;
+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.
+intros;apply (narrowing ? ? ? ? ? ? ? H1 ? H) [|autobatch]
+intros;apply (JS_trans ? ? ? ? ? H2);apply (JS_weakening ? ? ? H1);
+ [autobatch|unfold;intros;autobatch]
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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 *)
+(* *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/Fsub/part1a_inversion/".
+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]]
+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 : ∀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_Base
+ |destruct H13;apply in_Skip;apply (H7 ? H10)]]]
+qed.
+
+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 JSubtype_Arrow_inv:
+ ∀G:list bound.∀T1,T2,T3:Typ.
+ ∀P:list bound → Typ → Prop.
+ (∀n,t1.
+ (mk_bound true n t1) ∈ G → G ⊢ t1 ⊴ (Arrow T2 T3) → P G t1 → P G (TFree n)) →
+ (∀t,t1. G ⊢ T2 ⊴ t → G ⊢ t1 ⊴ T3 → P G (Arrow t t1)) →
+ G ⊢ T1 ⊴ (Arrow T2 T3) → P G T1.
+ intros;
+ generalize in match (refl_eq ? (Arrow T2 T3));
+ change in ⊢ (% → ?) with (Arrow T2 T3 = Arrow T2 T3);
+ generalize in match (refl_eq ? G);
+ change in ⊢ (% → ?) with (G = G);
+ elim H2 in ⊢ (? ? ? % → ? ? ? % → %);
+ [1,2: destruct H6
+ |5: destruct H8
+ | lapply (H5 H6 H7); destruct; clear H5;
+ apply H;
+ assumption
+ | destruct;
+ clear H4 H6;
+ apply H1;
+ assumption
+ ]
+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
+ |apply (JSubtype_Arrow_inv ? ? ? ? ? ? ? H6); intros;
+ [ autobatch
+ | inversion H7;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]]]]]
+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;
+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.
+intros;apply (narrowing ? ? ? ? ? ? ? H1 ? H) [|autobatch]
+intros;apply (JS_trans ? ? ? ? ? H2);apply (JS_weakening ? ? ? H1);
+ [autobatch|unfold;intros;autobatch]
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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 *)
+(* *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/Fsub/util".
+include "logic/equality.ma".
+include "nat/compare.ma".
+include "list/list.ma".
+
+(*** useful definitions and lemmas not really related to Fsub ***)
+
+definition max \def
+\lambda m,n.
+ match (leb m n) with
+ [true \Rightarrow n
+ |false \Rightarrow m].
+
+lemma le_n_max_m_n: \forall m,n:nat. n \le max m n.
+intros.
+unfold max.
+apply (leb_elim m n)
+ [simplify.intros.apply le_n
+ |simplify.intros.autobatch
+ ]
+qed.
+
+lemma le_m_max_m_n: \forall m,n:nat. m \le max m n.
+intros.
+unfold max.
+apply (leb_elim m n)
+ [simplify.intro.assumption
+ |simplify.intros.autobatch
+ ]
+qed.
+
+inductive in_list (A:Type): A → (list A) → Prop ≝
+| in_Base : ∀ x,l.(in_list A x (x::l))
+| in_Skip : ∀ x,y,l.(in_list A x l) → (in_list A x (y::l)).
+
+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).
+
+definition incl : \forall A.(list A) \to (list A) \to Prop \def
+ \lambda A,l,m.\forall x.(in_list A x l) \to (in_list A x m).
+
+(* FIXME: use the map in library/list (when there will be one) *)
+definition map : \forall A,B,f.((list A) \to (list B)) \def
+ \lambda A,B,f.let rec map (l : (list A)) : (list B) \def
+ match l in list return \lambda l0:(list A).(list B) with
+ [nil \Rightarrow (nil B)
+ |(cons (a:A) (t:(list A))) \Rightarrow
+ (cons B (f a) (map t))] in map.
+
+lemma in_list_nil : \forall A,x.\lnot (in_list A x []).
+intros.unfold.intro.inversion H
+ [intros;lapply (sym_eq ? ? ? H2);destruct Hletin
+ |intros;destruct H4]
+qed.
+
+lemma in_cons_case : ∀A.∀x,h:A.∀t:list A.x ∈ h::t → x = h ∨ (x ∈ t).
+intros;inversion H;intros
+ [destruct H2;left;symmetry;reflexivity
+ |destruct H4;right;applyS H1]
+qed.
+
+lemma append_nil:\forall A:Type.\forall l:list A. l@[]=l.
+intros.
+elim l;intros;autobatch.
+qed.
+
+lemma append_cons:\forall A.\forall a:A.\forall l,l1.
+l@(a::l1)=(l@[a])@l1.
+intros.
+rewrite > associative_append.
+reflexivity.
+qed.
+
+lemma in_list_append1: \forall A.\forall x:A.
+\forall l1,l2. x \in l1 \to x \in l1@l2.
+intros.
+elim H
+ [simplify.apply in_Base
+ |simplify.apply in_Skip.assumption
+ ]
+qed.
+
+lemma in_list_append2: \forall A.\forall x:A.
+\forall l1,l2. x \in l2 \to x \in l1@l2.
+intros 3.
+elim l1
+ [simplify.assumption
+ |simplify.apply in_Skip.apply H.assumption
+ ]
+qed.
+
+theorem append_to_or_in_list: \forall A:Type.\forall x:A.
+\forall l,l1. x \in l@l1 \to (x \in l) \lor (x \in l1).
+intros 3.
+elim l
+ [right.apply H
+ |simplify in H1.inversion H1;intros; destruct;
+ [left.apply in_Base
+ | elim (H l2)
+ [left.apply in_Skip. assumption
+ |right.assumption
+ |assumption
+ ]
+ ]
+ ]
+qed.
+
+lemma max_case : \forall m,n.(max m n) = match (leb m n) with
+ [ true \Rightarrow n
+ | false \Rightarrow m ].
+intros;elim m;simplify;reflexivity;
+qed.
+
+lemma incl_A_A: ∀T,A.incl T A A.
+intros.unfold incl.intros.assumption.
+qed.
\ No newline at end of file
--- /dev/null
+include ../Makefile.defs
+
+DIR=$(shell basename $$PWD)
+
+$(DIR) all:
+ $(BIN)matitac
+$(DIR).opt opt all.opt:
+ $(BIN)matitac.opt
+clean:
+ $(BIN)matitaclean
+clean.opt:
+ $(BIN)matitaclean.opt
+depend:
+ $(BIN)matitadep
+depend.opt:
+ $(BIN)matitadep.opt
--- /dev/null
+Fsub/part1a.ma Fsub/defn.ma
+Fsub/util.ma list/list.ma logic/equality.ma nat/compare.ma
+Fsub/defn.ma Fsub/util.ma
+Fsub/part1a_inversion.ma Fsub/defn.ma
+list/list.ma
+logic/equality.ma
+nat/compare.ma
--- /dev/null
+baseuri=cic:/matita
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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 *)
-(* *)
-(**************************************************************************)
-
-set "baseuri" "cic:/matita/Fsub/defn".
-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))) ].
-
-(*** height of T's syntactic tree ***)
-
-let rec t_len T \def
- match T with
- [(TVar n) \Rightarrow (S O)
- |(TFree X) \Rightarrow (S O)
- |Top \Rightarrow (S O)
- |(Arrow T1 T2) \Rightarrow (S (max (t_len T1) (t_len T2)))
- |(Forall T1 T2) \Rightarrow (S (max (t_len T1) (t_len T2)))].
-
-(*** 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/defn/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/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 ⊴ 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).
-
-(****** 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 (in_list_nil ? ? H2);elim Hletin
- |elim H1;elim H2;elim (in_cons_case ? ? ? ? H3)
- [rewrite < H4;simplify;apply in_Base
- |simplify;apply in_Skip;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 (in_list_nil ? ? H);elim Hletin
- |intros 3;elim t;simplify in H1;elim (in_cons_case ? ? ? ? H1)
- [rewrite < H2;apply (ex_intro ? ? b);apply (ex_intro ? ? t1);apply in_Base
- |elim (H H2);elim H3;apply (ex_intro ? ? a);
- apply (ex_intro ? ? a1);apply in_Skip;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_cons_case ? ? ? ? H1)
- [rewrite > H2;apply in_Base|apply in_Skip;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_cons_case ? ? ? ? H1)
- [destruct H3;elim (H2);reflexivity
- |simplify;apply (in_Skip ? ? ? ? H3);]
- |simplify in H2;simplify;elim (in_cons_case ? ? ? ? H2)
- [rewrite > H4;apply in_Base
- |apply (in_Skip ? ? ? ? (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 (in_list_nil ? ? H)
- |2,3:simplify;simplify in H;assumption
- |*:simplify in H2;simplify;elim (append_to_or_in_list ? ? ? ? H2)
- [1,3:apply in_list_append1;apply (H ? H3)
- |*:apply in_list_append2;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 (in_list_nil ? ? H1)
- |elim H;
- apply (ex_intro ? ? (S (max a t))).
- intros.unfold. intro.
- elim (in_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_cons_case ? ? ? ? H2)
- [rewrite > H3;assumption|elim (in_list_nil ? ? H3)]
- |simplify in H1;elim (in_list_nil ? x H1)
- |simplify in H5;elim (append_to_or_in_list ? ? ? ? H5);autobatch
- |simplify in H5;elim (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_cons_case ? ? ? ? Hletin)
- [elim (Hcut1 H10)
- |assumption]
- |intro;apply H8;applyS H6]
- |apply in_FV_subst;assumption]
- |split
- [intro;apply H7;apply in_list_append1;assumption
- |intro;apply H7;apply in_list_append2;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 (in_list_nil ? ? H1);elim Hletin
- |elim (in_cons_case ? ? ? ? H6)
- [destruct H7;destruct;elim (in_cons_case ? ? ? ? H5)
- [destruct H7;reflexivity
- |elim H7;elim H3;apply boundinenv_natinfv;apply (ex_intro ? ? B);
- apply (ex_intro ? ? T);assumption]
- |elim (in_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 JSubtype_Top: ∀G,P.G ⊢ ⊤ ⊴ P → P = ⊤.
-intros.inversion H;intros
- [assumption|reflexivity
- |destruct H5|*:destruct H6]
-qed.
-
-(* elim vs inversion *)
-lemma JS_trans_TFree: ∀G,T,X.G ⊢ T ⊴ (TFree X) →
- ∀U.G ⊢ (TFree X) ⊴ U → G ⊢ T ⊴ U.
-intros 4.cut (∀Y.TFree Y = TFree X → ∀U.G ⊢ (TFree Y) ⊴ U → G ⊢ T ⊴ U)
- [apply Hcut;reflexivity
- |elim H;intros
- [rewrite > H3 in H4;rewrite > (JSubtype_Top ? ? H4);apply SA_Top;assumption
- |rewrite < H3;assumption
- |apply (SA_Trans_TVar ? ? ? ? H1);apply (H3 Y);assumption
- |*:destruct H5]]
-qed.
-
-lemma fv_append : ∀G,H.fv_env (G @ H) = (fv_env G @ fv_env H).
-intro;elim G;simplify;autobatch paramodulation;
-qed.
\ No newline at end of file
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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 *)
-(* *)
-(**************************************************************************)
-
-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]]
-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 : ∀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_Base
- |destruct H13;apply in_Skip;apply (H7 ? H10)]]]
-qed.
-
-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 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]]]]]
-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;
-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.
-intros;apply (narrowing ? ? ? ? ? ? ? H1 ? H) [|autobatch]
-intros;apply (JS_trans ? ? ? ? ? H2);apply (JS_weakening ? ? ? H1);
- [autobatch|unfold;intros;autobatch]
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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 *)
-(* *)
-(**************************************************************************)
-
-set "baseuri" "cic:/matita/Fsub/part1a_inversion/".
-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]]
-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 : ∀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_Base
- |destruct H13;apply in_Skip;apply (H7 ? H10)]]]
-qed.
-
-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 JSubtype_Arrow_inv:
- ∀G:list bound.∀T1,T2,T3:Typ.
- ∀P:list bound → Typ → Prop.
- (∀n,t1.
- (mk_bound true n t1) ∈ G → G ⊢ t1 ⊴ (Arrow T2 T3) → P G t1 → P G (TFree n)) →
- (∀t,t1. G ⊢ T2 ⊴ t → G ⊢ t1 ⊴ T3 → P G (Arrow t t1)) →
- G ⊢ T1 ⊴ (Arrow T2 T3) → P G T1.
- intros;
- generalize in match (refl_eq ? (Arrow T2 T3));
- change in ⊢ (% → ?) with (Arrow T2 T3 = Arrow T2 T3);
- generalize in match (refl_eq ? G);
- change in ⊢ (% → ?) with (G = G);
- elim H2 in ⊢ (? ? ? % → ? ? ? % → %);
- [1,2: destruct H6
- |5: destruct H8
- | lapply (H5 H6 H7); destruct; clear H5;
- apply H;
- assumption
- | destruct;
- clear H4 H6;
- apply H1;
- assumption
- ]
-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
- |apply (JSubtype_Arrow_inv ? ? ? ? ? ? ? H6); intros;
- [ autobatch
- | inversion H7;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]]]]]
-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;
-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.
-intros;apply (narrowing ? ? ? ? ? ? ? H1 ? H) [|autobatch]
-intros;apply (JS_trans ? ? ? ? ? H2);apply (JS_weakening ? ? ? H1);
- [autobatch|unfold;intros;autobatch]
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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 *)
-(* *)
-(**************************************************************************)
-
-set "baseuri" "cic:/matita/Fsub/util".
-include "logic/equality.ma".
-include "nat/compare.ma".
-include "list/list.ma".
-
-(*** useful definitions and lemmas not really related to Fsub ***)
-
-definition max \def
-\lambda m,n.
- match (leb m n) with
- [true \Rightarrow n
- |false \Rightarrow m].
-
-lemma le_n_max_m_n: \forall m,n:nat. n \le max m n.
-intros.
-unfold max.
-apply (leb_elim m n)
- [simplify.intros.apply le_n
- |simplify.intros.autobatch
- ]
-qed.
-
-lemma le_m_max_m_n: \forall m,n:nat. m \le max m n.
-intros.
-unfold max.
-apply (leb_elim m n)
- [simplify.intro.assumption
- |simplify.intros.autobatch
- ]
-qed.
-
-inductive in_list (A:Type): A → (list A) → Prop ≝
-| in_Base : ∀ x,l.(in_list A x (x::l))
-| in_Skip : ∀ x,y,l.(in_list A x l) → (in_list A x (y::l)).
-
-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).
-
-definition incl : \forall A.(list A) \to (list A) \to Prop \def
- \lambda A,l,m.\forall x.(in_list A x l) \to (in_list A x m).
-
-(* FIXME: use the map in library/list (when there will be one) *)
-definition map : \forall A,B,f.((list A) \to (list B)) \def
- \lambda A,B,f.let rec map (l : (list A)) : (list B) \def
- match l in list return \lambda l0:(list A).(list B) with
- [nil \Rightarrow (nil B)
- |(cons (a:A) (t:(list A))) \Rightarrow
- (cons B (f a) (map t))] in map.
-
-lemma in_list_nil : \forall A,x.\lnot (in_list A x []).
-intros.unfold.intro.inversion H
- [intros;lapply (sym_eq ? ? ? H2);destruct Hletin
- |intros;destruct H4]
-qed.
-
-lemma in_cons_case : ∀A.∀x,h:A.∀t:list A.x ∈ h::t → x = h ∨ (x ∈ t).
-intros;inversion H;intros
- [destruct H2;left;symmetry;reflexivity
- |destruct H4;right;applyS H1]
-qed.
-
-lemma append_nil:\forall A:Type.\forall l:list A. l@[]=l.
-intros.
-elim l;intros;autobatch.
-qed.
-
-lemma append_cons:\forall A.\forall a:A.\forall l,l1.
-l@(a::l1)=(l@[a])@l1.
-intros.
-rewrite > associative_append.
-reflexivity.
-qed.
-
-lemma in_list_append1: \forall A.\forall x:A.
-\forall l1,l2. x \in l1 \to x \in l1@l2.
-intros.
-elim H
- [simplify.apply in_Base
- |simplify.apply in_Skip.assumption
- ]
-qed.
-
-lemma in_list_append2: \forall A.\forall x:A.
-\forall l1,l2. x \in l2 \to x \in l1@l2.
-intros 3.
-elim l1
- [simplify.assumption
- |simplify.apply in_Skip.apply H.assumption
- ]
-qed.
-
-theorem append_to_or_in_list: \forall A:Type.\forall x:A.
-\forall l,l1. x \in l@l1 \to (x \in l) \lor (x \in l1).
-intros 3.
-elim l
- [right.apply H
- |simplify in H1.inversion H1;intros; destruct;
- [left.apply in_Base
- | elim (H l2)
- [left.apply in_Skip. assumption
- |right.assumption
- |assumption
- ]
- ]
- ]
-qed.
-
-lemma max_case : \forall m,n.(max m n) = match (leb m n) with
- [ true \Rightarrow n
- | false \Rightarrow m ].
-intros;elim m;simplify;reflexivity;
-qed.
-
-lemma incl_A_A: ∀T,A.incl T A A.
-intros.unfold incl.intros.assumption.
-qed.
\ No newline at end of file
Q/Qaxioms.ma Z/compare.ma Z/times.ma nat/iteration2.ma
Q/q.ma Z/compare.ma Z/plus.ma
technicalities/setoids.ma datatypes/constructors.ma logic/coimplication.ma logic/connectives2.ma
-Fsub/part1a.ma Fsub/defn.ma
-Fsub/util.ma list/list.ma logic/equality.ma nat/compare.ma
-Fsub/defn.ma Fsub/util.ma
-Fsub/part1a_inversion.ma Fsub/defn.ma
decidable_kit/streicher.ma logic/connectives.ma logic/equality.ma
decidable_kit/decidable.ma datatypes/bool.ma decidable_kit/streicher.ma logic/connectives.ma nat/compare.ma
decidable_kit/fgraph.ma decidable_kit/fintype.ma
projects / helm.git / commitdiff