- ground_2: rtmap: disjointness relation

[helm.git] / matita / matita / contribs / lambdadelta / ground_2 / relocation / rtmap_sdj.ma

diff --git a/matita/matita/contribs/lambdadelta/ground_2/relocation/rtmap_sdj.ma b/matita/matita/contribs/lambdadelta/ground_2/relocation/rtmap_sdj.ma
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+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.tcs.unibo.it                            *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "ground_2/notation/relations/parallel_2.ma".
+include "ground_2/relocation/rtmap_isid.ma".
+
+(* RELOCATION MAP ***********************************************************)
+
+coinductive sdj: relation rtmap ≝
+| sdj_pp: ∀f1,f2,g1,g2. sdj f1 f2 → ↑f1 = g1 → ↑f2 = g2 → sdj g1 g2
+| sdj_np: ∀f1,f2,g1,g2. sdj f1 f2 → ⫯f1 = g1 → ↑f2 = g2 → sdj g1 g2
+| sdj_pn: ∀f1,f2,g1,g2. sdj f1 f2 → ↑f1 = g1 → ⫯f2 = g2 → sdj g1 g2
+.
+
+interpretation "disjointness (rtmap)"
+   'Parallel f1 f2 = (sdj f1 f2).
+
+(* Basic properties *********************************************************)
+
+axiom sdj_eq_repl_back1: ∀f2. eq_repl_back … (λf1. f1 ∥ f2).
+
+lemma sdj_eq_repl_fwd1: ∀f2. eq_repl_fwd … (λf1. f1 ∥ f2).
+#f2 @eq_repl_sym /2 width=3 by sdj_eq_repl_back1/
+qed-.
+
+axiom sdj_eq_repl_back2: ∀f1. eq_repl_back … (λf2. f1 ∥ f2).
+
+lemma sdj_eq_repl_fwd2: ∀f1. eq_repl_fwd … (λf2. f1 ∥ f2).
+#f1 @eq_repl_sym /2 width=3 by sdj_eq_repl_back2/
+qed-.
+
+corec lemma sdj_sym: symmetric … sdj.
+#f1 #f2 * -f1 -f2
+#f1 #f2 #g1 #g2 #Hf #H1 #H2
+[ @(sdj_pp … H2 H1) | @(sdj_pn … H2 H1) | @(sdj_np … H2 H1) ] -g2 -g1
+/2 width=1 by/
+qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma sdj_inv_pp: ∀g1,g2. g1 ∥ g2 → ∀f1,f2. ↑f1 = g1 → ↑f2 = g2 → f1 ∥ f2.
+#g1 #g2 * -g1 -g2
+#f1 #f2 #g1 #g2 #H #H1 #H2 #x1 #x2 #Hx1 #Hx2 destruct
+[ lapply (injective_push … Hx1) -Hx1
+  lapply (injective_push … Hx2) -Hx2 //
+| elim (discr_push_next … Hx1)
+| elim (discr_push_next … Hx2)
+]
+qed-.
+
+lemma sdj_inv_np: ∀g1,g2. g1 ∥ g2 → ∀f1,f2. ⫯f1 = g1 → ↑f2 = g2 → f1 ∥ f2.
+#g1 #g2 * -g1 -g2
+#f1 #f2 #g1 #g2 #H #H1 #H2 #x1 #x2 #Hx1 #Hx2 destruct
+[ elim (discr_next_push … Hx1)
+| lapply (injective_next … Hx1) -Hx1
+  lapply (injective_push … Hx2) -Hx2 //
+| elim (discr_push_next … Hx2)
+]
+qed-.
+
+lemma sdj_inv_pn: ∀g1,g2. g1 ∥ g2 → ∀f1,f2. ↑f1 = g1 → ⫯f2 = g2 → f1 ∥ f2.
+#g1 #g2 * -g1 -g2
+#f1 #f2 #g1 #g2 #H #H1 #H2 #x1 #x2 #Hx1 #Hx2 destruct
+[ elim (discr_next_push … Hx2)
+| elim (discr_push_next … Hx1)
+| lapply (injective_push … Hx1) -Hx1
+  lapply (injective_next … Hx2) -Hx2 //
+]
+qed-.
+
+lemma sdj_inv_nn: ∀g1,g2. g1 ∥ g2 → ∀f1,f2. ⫯f1 = g1 → ⫯f2 = g2 → ⊥.
+#g1 #g2 * -g1 -g2
+#f1 #f2 #g1 #g2 #H #H1 #H2 #x1 #x2 #Hx1 #Hx2 destruct
+[ elim (discr_next_push … Hx1)
+| elim (discr_next_push … Hx2)
+| elim (discr_next_push … Hx1)
+]
+qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma sdj_inv_nx: ∀g1,g2. g1 ∥ g2 → ∀f1. ⫯f1 = g1 →
+                  ∃∃f2. f1 ∥ f2 & ↑f2 = g2.
+#g1 #g2 elim (pn_split g2) * #f2 #H2 #H #f1 #H1
+[ lapply (sdj_inv_np … H … H1 H2) -H /2 width=3 by ex2_intro/
+| elim (sdj_inv_nn … H … H1 H2)
+]
+qed-.
+
+lemma sdj_inv_xn: ∀g1,g2. g1 ∥ g2 → ∀f2. ⫯f2 = g2 →
+                  ∃∃f1. f1 ∥ f2 & ↑f1 = g1.
+#g1 #g2 elim (pn_split g1) * #f1 #H1 #H #f2 #H2
+[ lapply (sdj_inv_pn … H … H1 H2) -H /2 width=3 by ex2_intro/
+| elim (sdj_inv_nn … H … H1 H2)
+]
+qed-.
+
+lemma sdj_inv_xp: ∀g1,g2. g1 ∥ g2 → ∀f2. ↑f2 = g2 →
+                  ∨∨ ∃∃f1. f1 ∥ f2 & ↑f1 = g1
+                   | ∃∃f1. f1 ∥ f2 & ⫯f1 = g1.
+#g1 #g2 elim (pn_split g1) * #f1 #H1 #H #f2 #H2
+[ lapply (sdj_inv_pp … H … H1 H2) | lapply (sdj_inv_np … H … H1 H2) ] -H -H2
+/3 width=3 by ex2_intro, or_introl, or_intror/
+qed-.
+
+lemma sdj_inv_px: ∀g1,g2. g1 ∥ g2 → ∀f1. ↑f1 = g1 →
+                  ∨∨ ∃∃f2. f1 ∥ f2 & ↑f2 = g2
+                   | ∃∃f2. f1 ∥ f2 & ⫯f2 = g2.
+#g1 #g2 elim (pn_split g2) * #f2 #H2 #H #f1 #H1
+[ lapply (sdj_inv_pp … H … H1 H2) | lapply (sdj_inv_pn … H … H1 H2) ] -H -H1
+/3 width=3 by ex2_intro, or_introl, or_intror/
+qed-.
+
+(* Properties with isid *****************************************************)
+
+corec lemma sdj_isid_dx: ∀f2. 𝐈⦃f2⦄ → ∀f1. f1 ∥ f2.
+#f2 * -f2
+#f2 #g2 #Hf2 #H2 #f1 cases (pn_split f1) *
+/3 width=5 by sdj_np, sdj_pp/
+qed.
+
+corec lemma sdj_isid_sn: ∀f1. 𝐈⦃f1⦄ → ∀f2. f1 ∥ f2.
+#f1 * -f1
+#f1 #g1 #Hf1 #H1 #f2 cases (pn_split f2) *
+/3 width=5 by sdj_pn, sdj_pp/
+qed.
+
+(* Inversion lemmas with isid ***********************************************)
+
+corec lemma sdj_inv_refl: ∀f. f ∥ f →  𝐈⦃f⦄.
+#f cases (pn_split f) * #g #Hg #H
+[ lapply (sdj_inv_pp … H … Hg Hg) -H /3 width=3 by isid_push/
+| elim (sdj_inv_nn … H … Hg Hg)
+]
+qed-.