+++ /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 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⓧ* [ term 46 n ] term 46 L )"
- non associative with precedence 46
- for @{ 'VoidStar $n $L }.
--- /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 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⓧ* [ term 46 n1 ] term 46 L1 ≋ ⓧ* [ term 46 n2 ] term 46 L2 )"
+ non associative with precedence 45
+ for @{ 'RVoidStar $n1 $L1 $n2 $L2 }.
(* FREE VARIABLES INCLUSION FOR RESTRICTED CLOSURES *************************)
-inductive fle (T2) (L2) (T1): predicate lenv ≝
-| fle_intro: ∀f1,f2,L1,n. ⓧ*[n]L1 ⊢ 𝐅*⦃T1⦄ ≡ f1 → L2 ⊢ 𝐅*⦃T2⦄ ≡ f2 →
- |L1| = |L2| → ⫱*[n]f1 ⊆ f2 → fle T2 L2 T1 (ⓧ*[n]L1)
+inductive fle (T1) (T2): relation lenv ≝
+| fle_intro: ∀f1,f2,L1,L2,n1,n2. ⓧ*[n1]L1 ⊢ 𝐅*⦃T1⦄ ≡ f1 → ⓧ*[n2]L2 ⊢ 𝐅*⦃T2⦄ ≡ f2 →
+ |L1| = |L2| → ⫱*[n1]f1 ⊆ ⫱*[n2]f2 → fle T1 T2 (ⓧ*[n1]L1) (ⓧ*[n2]L2)
.
interpretation "free variables inclusion (restricted closure)"
- 'SubSetEq L1 T1 L2 T2 = (fle T2 L2 T1 L1).
+ 'SubSetEq L1 T1 L2 T2 = (fle T1 T2 L1 L2).
(* Basic properties *********************************************************)
(* Basic inversion lemmas ***************************************************)
-fact fle_inv_voids_sn_aux: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊆ ⦃L2, T2⦄ →
- ∀K1,n. |K1| = |L2| → L1 = ⓧ*[n]K1 →
- ∃∃f1,f2. ⓧ*[n]K1 ⊢ 𝐅*⦃T1⦄ ≡ f1 & L2 ⊢ 𝐅*⦃T2⦄ ≡ f2 & ⫱*[n]f1 ⊆ f2.
-#L1 #L2 #T1 #T2 * -L1 #f1 #f2 #L1 #n #Hf1 #Hf2 #HL12 #Hf12 #Y #x #HY #H destruct
-elim (voids_inj_length … H) // -H -HL12 -HY #H1 #H2 destruct
+fact fle_inv_voids_aux: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊆ ⦃L2, T2⦄ →
+ ∀K1,K2,n1,n2. |K1| = |K2| → L1 = ⓧ*[n1]K1 → L2 = ⓧ*[n2]K2 →
+ ∃∃f1,f2. ⓧ*[n1]K1 ⊢ 𝐅*⦃T1⦄ ≡ f1 & ⓧ*[n2]K2 ⊢ 𝐅*⦃T2⦄ ≡ f2 & ⫱*[n1]f1 ⊆ ⫱*[n2]f2.
+#L1 #L2 #T1 #T2 * -L1 -L2
+#f1 #f2 #L1 #L2 #n1 #n2 #Hf1 #Hf2 #HL12 #Hf12 #Y1 #Y2 #x1 #x2 #HY12 #H1 #H2 destruct
+>H1 in Hf1; >H2 in Hf2; #Hf2 #Hf1
+@(ex3_2_intro … Hf1 Hf2) -Hf1 -Hf2
+
+elim (voids_inj_length … H1) // -H -HL12 -HY #H1 #H2 destruct
/2 width=5 by ex3_2_intro/
qed-.
(* *)
(**************************************************************************)
-include "basic_2/notation/functions/voidstar_2.ma".
+include "basic_2/notation/relations/rvoidstar_4.ma".
include "basic_2/syntax/lenv.ma".
-(* EXTENSION OF A LOCAL ENVIRONMENT WITH EXCLUSION BINDERS ******************)
+(* EQUIVALENCE FOR LOCAL ENVIRONMENTS UP TO EXCLUSION BINDERS ***************)
-rec definition voids (L:lenv) (n:nat) on n: lenv ≝ match n with
-[ O ⇒ L | S m ⇒ (voids L m).ⓧ ].
+inductive voids: bi_relation nat lenv ≝
+| voids_atom : voids 0 (⋆) 0 (⋆)
+| voids_pair_sn: ∀I1,I2,K1,K2,V1,n. voids n K1 n K2 →
+ voids 0 (K1.ⓑ{I1}V1) 0 (K2.ⓘ{I2})
+| voids_pair_dx: ∀I1,I2,K1,K2,V2,n. voids n K1 n K2 →
+ voids 0 (K1.ⓘ{I1}) 0 (K2.ⓑ{I2}V2)
+| voids_void_sn: ∀K1,K2,n1,n2. voids n1 K1 n2 K2 →
+ voids (⫯n1) (K1.ⓧ) n2 K2
+| voids_void_dx: ∀K1,K2,n1,n2. voids n1 K1 n2 K2 →
+ voids n1 K1 (⫯n2) (K2.ⓧ)
+.
-interpretation "extension with exclusion binders (local environment)"
- 'VoidStar n L = (voids L n).
+interpretation "equivalence up to exclusion binders (local environment)"
+ 'RVoidStar n1 L1 n2 L2 = (voids n1 L1 n2 L2).
(* Basic properties *********************************************************)
+lemma voids_refl: ∀L. ∃n. ⓧ*[n]L ≋ ⓧ*[n]L.
+#L elim L -L /2 width=2 by ex_intro, voids_atom/
+#L #I * #n #IH cases I -I /3 width=2 by ex_intro, voids_pair_dx/
+* /4 width=2 by ex_intro, voids_void_sn, voids_void_dx/
+qed-.
+
+lemma voids_sym: bi_symmetric … voids.
+#n1 #n2 #L1 #L2 #H elim H -L1 -L2 -n1 -n2
+/2 width=2 by voids_atom, voids_pair_sn, voids_pair_dx, voids_void_sn, voids_void_dx/
+qed-.
+
+(*
+
+
+
+
lemma voids_zero: ∀L. L = ⓧ*[0]L.
// qed.
elim (destruct_lbind_lbind_aux … H) -H (**) (* destruct lemma needed *)
/2 width=1 by/
qed-.
+*)
\ No newline at end of file
include "basic_2/syntax/lenv_length.ma".
include "basic_2/syntax/voids.ma".
-(* EXTENSION OF A LOCAL ENVIRONMENT WITH EXCLUSION BINDERS ******************)
+(* EQUIVALENCE FOR LOCAL ENVIRONMENTS UP TO EXCLUSION BINDERS ***************)
-(* Properties with length for local environments ****************************)
+(* Forward lemmas with length for local environments ************************)
-lemma length_void: ∀L,n. n+|L| = |ⓧ*[n]L|.
-#L #n elim n -n //
-#n #IH <voids_succ >length_bind <IH -IH //
-qed.
+lemma voids_fwd_length: ∀L1,L2,n1,n2. ⓧ*[n1]L1 ≋ ⓧ*[n2]L2 →
+ |L1| + n2 = |L2| + n1.
+#L1 #L2 #n1 #n2 #H elim H -L1 -L2 -n1 -n2 normalize //
+#I1 #I2 #K1 #K2 #V #n #_ #IH
(* Main forward properties with length for local environments ***************)
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