milestone update in ground

[helm.git] / matita / matita / contribs / lambdadelta / ground / relocation / gr_tl_eq.ma

diff --git a/matita/matita/contribs/lambdadelta/ground/relocation/gr_tl_eq.ma b/matita/matita/contribs/lambdadelta/ground/relocation/gr_tl_eq.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 "ground/relocation/gr_eq.ma".
+include "ground/relocation/gr_tl.ma".
+
+(* TAIL FOR GENERIC RELOCATION MAPS ***********************************************************)
+
+(* Properties with gr_eq *)
+
+(*** eq_refl *)
+corec lemma gr_eq_refl: reflexive … gr_eq.
+#f cases (gr_map_split_tl f) #Hf
+[ @(gr_eq_push … Hf Hf) | @(gr_eq_next … Hf Hf) ] -Hf //
+qed.
+
+(*** tl_eq_repl *)
+lemma gr_tl_eq_repl:
+      gr_eq_repl … (λf1,f2. ⫱f1 ≡ ⫱f2).
+#f1 #f2 * -f1 -f2 //
+qed.
+
+(* Inversion lemmas with gr_eq ***************************************************)
+
+(*** eq_inv_gen *)
+lemma gr_eq_inv_gen (g1) (g2):
+      g1 ≡ g2 →
+      ∨∨ ∧∧ ⫱g1 ≡ ⫱g2 & ⫯⫱g1 = g1 & ⫯⫱g2 = g2
+       | ∧∧ ⫱g1 ≡ ⫱g2 & ↑⫱g1 = g1 & ↑⫱g2 = g2.
+#g1 #g2 * -g1 -g2 #f1 #f2 #g1 #g2 #f * *
+/3 width=1 by and3_intro, or_introl, or_intror/
+qed-.
+
+(* Advanced Inversion lemmas with gr_eq *)
+
+(*** gr_eq_inv_px *)
+lemma gr_eq_inv_push_sn (g1) (g2):
+      g1 ≡ g2 → ∀f1. ⫯f1 = g1 →
+      ∧∧ f1 ≡ ⫱g2 & ⫯⫱g2 = g2.
+#g1 #g2 #H #f1 #Hf1
+elim (gr_eq_inv_gen … H) -H * #Hg #Hg1 #Hg2 destruct
+[ /2 width=1 by conj/
+| elim (eq_inv_gr_next_push … Hg1)
+]
+qed-.
+
+(*** gr_eq_inv_nx *)
+lemma gr_eq_inv_next_sn (g1) (g2):
+      g1 ≡ g2 → ∀f1. ↑f1 = g1 →
+      ∧∧ f1 ≡ ⫱g2 & ↑⫱g2 = g2.
+#g1 #g2 #H #f1 #Hf1
+elim (gr_eq_inv_gen … H) -H * #Hg #Hg1 #Hg2 destruct
+[ elim (eq_inv_gr_push_next … Hg1)
+| /2 width=1 by conj/
+]
+qed-.
+
+(*** gr_eq_inv_xp *)
+lemma gr_eq_inv_push_dx (g1) (g2):
+      g1 ≡ g2 → ∀f2. ⫯f2 = g2 →
+      ∧∧ ⫱g1 ≡ f2 & ⫯⫱g1 = g1.
+#g1 #g2 #H #f2 #Hf2
+elim (gr_eq_inv_gen … H) -H * #Hg #Hg1 #Hg2 destruct
+[ /2 width=1 by conj/
+| elim (eq_inv_gr_next_push … Hg2)
+]
+qed-.
+
+(*** gr_eq_inv_xn *)
+lemma gr_eq_inv_next_dx (g1) (g2):
+      g1 ≡ g2 → ∀f2. ↑f2 = g2 →
+      ∧∧ ⫱g1 ≡ f2 & ↑⫱g1 = g1.
+#g1 #g2 #H #f2 #Hf2
+elim (gr_eq_inv_gen … H) -H * #Hg #Hg1 #Hg2 destruct
+[ elim (eq_inv_gr_push_next … Hg2)
+| /2 width=1 by conj/
+]
+qed-.
+
+(*** gr_eq_inv_pp *)
+lemma gr_eq_inv_push_bi (g1) (g2):
+      g1 ≡ g2 → ∀f1,f2. ⫯f1 = g1 → ⫯f2 = g2 → f1 ≡ f2.
+#g1 #g2 #H #f1 #f2 #H1
+elim (gr_eq_inv_push_sn … H … H1) -g1 #Hx2 * #H
+lapply (eq_inv_gr_push_bi … H) -H //
+qed-.
+
+(*** gr_eq_inv_nn *)
+lemma gr_eq_inv_next_bi (g1) (g2):
+      g1 ≡ g2 → ∀f1,f2. ↑f1 = g1 → ↑f2 = g2 → f1 ≡ f2.
+#g1 #g2 #H #f1 #f2 #H1
+elim (gr_eq_inv_next_sn … H … H1) -g1 #Hx2 * #H
+lapply (eq_inv_gr_next_bi … H) -H //
+qed-.
+
+(*** gr_eq_inv_pn *)
+lemma gr_eq_inv_push_next (g1) (g2):
+      g1 ≡ g2 → ∀f1,f2. ⫯f1 = g1 → ↑f2 = g2 → ⊥.
+#g1 #g2 #H #f1 #f2 #H1
+elim (gr_eq_inv_push_sn … H … H1) -g1 #Hx2 * #H
+elim (eq_inv_gr_next_push … H)
+qed-.
+
+(*** gr_eq_inv_np *)
+lemma gr_eq_inv_next_push (g1) (g2):
+      g1 ≡ g2 → ∀f1,f2. ↑f1 = g1 → ⫯f2 = g2 → ⊥.
+#g1 #g2 #H #f1 #f2 #H1
+elim (gr_eq_inv_next_sn … H … H1) -g1 #Hx2 * #H
+elim (eq_inv_gr_push_next … H)
+qed-.