work in progress with voids and lveq (was: the most recent voids)

[helm.git] / matita / matita / contribs / lambdadelta / basic_2 / syntax / lveq.ma

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+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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 "basic_2/notation/relations/voidstareq_4.ma".
+include "basic_2/syntax/lenv.ma".
+
+(* EQUIVALENCE FOR LOCAL ENVIRONMENTS UP TO EXCLUSION BINDERS ***************)
+
+inductive lveq: bi_relation nat lenv ≝
+| lveq_atom   : lveq 0 (⋆) 0 (⋆)
+| lveq_pair_sn: ∀I1,I2,K1,K2,V1,n. lveq n K1 n K2 →
+                lveq 0 (K1.ⓑ{I1}V1) 0 (K2.ⓘ{I2})
+| lveq_pair_dx: ∀I1,I2,K1,K2,V2,n. lveq n K1 n K2 →
+                lveq 0 (K1.ⓘ{I1}) 0 (K2.ⓑ{I2}V2)
+| lveq_void_sn: ∀K1,K2,n1,n2. lveq n1 K1 n2 K2 →
+                lveq (⫯n1) (K1.ⓧ) n2 K2
+| lveq_void_dx: ∀K1,K2,n1,n2. lveq n1 K1 n2 K2 →
+                lveq n1 K1 (⫯n2) (K2.ⓧ)
+.
+
+interpretation "equivalence up to exclusion binders (local environment)"
+   'VoidStarEq L1 n1 n2 L2 = (lveq n1 L1 n2 L2).
+
+(* Basic properties *********************************************************)
+
+lemma lveq_refl: ∀L. ∃n. L ≋ⓧ*[n, n] L.
+#L elim L -L /2 width=2 by ex_intro, lveq_atom/
+#L #I * #n #IH cases I -I /3 width=2 by ex_intro, lveq_pair_dx/
+* /4 width=2 by ex_intro, lveq_void_sn, lveq_void_dx/
+qed-.
+
+lemma lveq_sym: bi_symmetric … lveq.
+#n1 #n2 #L1 #L2 #H elim H -L1 -L2 -n1 -n2
+/2 width=2 by lveq_atom, lveq_pair_sn, lveq_pair_dx, lveq_void_sn, lveq_void_dx/
+qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+fact lveq_inv_atom_aux: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 →
+                        ⋆ = L1 → ⋆ = L2 → ∧∧ 0 = n1 & 0 = n2.
+#L1 #L2 #n1 #n2 * -L1 -L2 -n1 -n2
+[ /2 width=1 by conj/
+|2,3: #I1 #I2 #K1 #K2 #V #n #_ #H1 #H2 destruct
+|4,5: #K1 #K2 #n1 #n2 #_ #H1 #H2 destruct
+]
+qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma lveq_inv_atom: ∀n1,n2. ⋆ ≋ⓧ*[n1, n2] ⋆ → 0 = n1 ∧ 0 = n2.
+/2 width=5 by lveq_inv_atom_aux/ qed-.
+
+fact lveq_inv_void_succ_sn_aux: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 →
+                                ∀K1,m1. L1 = K1.ⓧ → n1 = ⫯m1 → K1 ≋ ⓧ*[m1, n2] L2.
+#L1 #L2 #n1 #n2 #H elim H -L1 -L2 -n1 -n2
+[ #K2 #m2 #H destruct
+| #I1 #I2 #L1 #L2 #V #n #_ #_ #K1 #m1 #H1 #H2 destruct
+| #I1 #I2 #L1 #L2 #V #n #_ #_ #K1 #m1 #H1 #H2 destruct
+| #L1 #L2 #n1 #n2 #HL12 #_ #K1 #m1 #H1 #H2 destruct //
+| #L1 #L2 #n1 #n2 #_ #IH #K1 #m1 #H1 #H2 destruct
+  /3 width=1 by lveq_void_dx/
+]
+qed-.
+
+lemma lveq_inv_void_succ_sn: ∀L1,L2,n1,n2. L1.ⓧ ≋ⓧ*[⫯n1, n2] L2 → L1 ≋ ⓧ*[n1, n2] L2.
+/2 width=5 by lveq_inv_void_succ_sn_aux/ qed-.
+
+lemma lveq_inv_void_succ_dx: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, ⫯n2] L2.ⓧ → L1 ≋ ⓧ*[n1, n2] L2.
+/4 width=5 by lveq_inv_void_succ_sn_aux, lveq_sym/ qed-.
+
+(* Advanced forward lemmas **************************************************)
+
+fact lveq_fwd_pair_sn_aux: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 →
+                           ∀I,K1,V. K1.ⓑ{I}V = L1 → 0 = n1.
+#L1 #L2 #n1 #n2 #H elim H -L1 -L2 -n1 -n2 //
+#K1 #K2 #n1 #n2 #_ #IH #J #L1 #V #H destruct /2 width=4 by/
+qed-.
+
+lemma lveq_fwd_pair_sn: ∀I,K1,L2,V,n1,n2. K1.ⓑ{I}V ≋ⓧ*[n1, n2] L2 → 0 = n1.
+/2 width=8 by lveq_fwd_pair_sn_aux/ qed-.
+
+lemma lveq_fwd_pair_dx: ∀I,L1,K2,V,n1,n2. L1 ≋ⓧ*[n1, n2] K2.ⓑ{I}V → 0 = n2.
+/3 width=6 by lveq_fwd_pair_sn, lveq_sym/ qed-.