update in ground

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

diff --git a/matita/matita/contribs/lambdadelta/ground/relocation/pr_sand.ma b/matita/matita/contribs/lambdadelta/ground/relocation/pr_sand.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/notation/relations/rintersection_3.ma".
+include "ground/relocation/pr_tl.ma".
+
+(* RELATIONAL INTERSECTION FOR PARTIAL RELOCATION MAPS **********************)
+
+(*** sand *)
+coinductive pr_sand: relation3 pr_map pr_map pr_map ≝
+(*** sand_pp *)
+| pr_sand_push_bi (f1) (f2) (f) (g1) (g2) (g):
+  pr_sand f1 f2 f → ⫯f1 = g1 → ⫯f2 = g2 → ⫯f = g → pr_sand g1 g2 g
+(*** sand_np *)
+| pr_sand_next_push (f1) (f2) (f) (g1) (g2) (g):
+  pr_sand f1 f2 f → ↑f1 = g1 → ⫯f2 = g2 → ⫯f = g → pr_sand g1 g2 g
+(*** sand_pn *)
+| pr_sand_push_next (f1) (f2) (f) (g1) (g2) (g):
+  pr_sand f1 f2 f → ⫯f1 = g1 → ↑f2 = g2 → ⫯f = g → pr_sand g1 g2 g
+(*** sand_nn *)
+| pr_sand_next_bi (f1) (f2) (f) (g1) (g2) (g):
+  pr_sand f1 f2 f → ↑f1 = g1 → ↑f2 = g2 → ↑f = g → pr_sand g1 g2 g
+.
+
+interpretation
+  "relational intersection (partial relocation maps)"
+  'RIntersection f1 f2 f = (pr_sand f1 f2 f).
+
+(* Basic constructions ******************************************************)
+
+(*** sand_refl *)
+corec lemma pr_sand_idem:
+            ∀f. f ⋒ f ≘ f.
+#f cases (pr_map_split_tl f) #H
+[ @(pr_sand_push_bi … H H H)
+| @(pr_sand_next_bi … H H H)
+] -H //
+qed.
+
+(*** sand_sym *)
+corec lemma pr_sand_comm:
+            ∀f1,f2,f. f1 ⋒ f2 ≘ f → f2 ⋒ f1 ≘ f.
+#f1 #f2 #f * -f1 -f2 -f
+#f1 #f2 #f #g1 #g2 #g #Hf * * * -g1 -g2 -g
+[ @pr_sand_push_bi
+| @pr_sand_push_next
+| @pr_sand_next_push
+| @pr_sand_next_bi
+] /2 width=7 by/
+qed-.
+
+(* Basic inversions *********************************************************)
+
+(*** sand_inv_ppx *)
+lemma pr_sand_inv_push_bi:
+      ∀g1,g2,g. g1 ⋒ g2 ≘ g → ∀f1,f2. ⫯f1 = g1 → ⫯f2 = g2 →
+      ∃∃f. f1 ⋒ f2 ≘ f & ⫯f = g.
+#g1 #g2 #g * -g1 -g2 -g
+#f1 #f2 #f #g1 #g2 #g #Hf #H1 #H2 #H0 #x1 #x2 #Hx1 #Hx2 destruct
+try (>(eq_inv_pr_push_bi … Hx1) -x1) try (>(eq_inv_pr_next_bi … Hx1) -x1)
+try elim (eq_inv_pr_push_next … Hx1) try elim (eq_inv_pr_next_push … Hx1)
+try (>(eq_inv_pr_push_bi … Hx2) -x2) try (>(eq_inv_pr_next_bi … Hx2) -x2)
+try elim (eq_inv_pr_push_next … Hx2) try elim (eq_inv_pr_next_push … Hx2)
+/2 width=3 by ex2_intro/
+qed-.
+
+(*** sand_inv_npx *)
+lemma pr_sand_inv_next_push:
+      ∀g1,g2,g. g1 ⋒ g2 ≘ g → ∀f1,f2. ↑f1 = g1 → ⫯f2 = g2 →
+      ∃∃f. f1 ⋒ f2 ≘ f & ⫯f = g.
+#g1 #g2 #g * -g1 -g2 -g
+#f1 #f2 #f #g1 #g2 #g #Hf #H1 #H2 #H0 #x1 #x2 #Hx1 #Hx2 destruct
+try (>(eq_inv_pr_push_bi … Hx1) -x1) try (>(eq_inv_pr_next_bi … Hx1) -x1)
+try elim (eq_inv_pr_push_next … Hx1) try elim (eq_inv_pr_next_push … Hx1)
+try (>(eq_inv_pr_push_bi … Hx2) -x2) try (>(eq_inv_pr_next_bi … Hx2) -x2)
+try elim (eq_inv_pr_push_next … Hx2) try elim (eq_inv_pr_next_push … Hx2)
+/2 width=3 by ex2_intro/
+qed-.
+
+(*** sand_inv_pnx *)
+lemma pr_sand_inv_push_next:
+      ∀g1,g2,g. g1 ⋒ g2 ≘ g → ∀f1,f2. ⫯f1 = g1 → ↑f2 = g2 →
+      ∃∃f. f1 ⋒ f2 ≘ f & ⫯f = g.
+#g1 #g2 #g * -g1 -g2 -g
+#f1 #f2 #f #g1 #g2 #g #Hf #H1 #H2 #H0 #x1 #x2 #Hx1 #Hx2 destruct
+try (>(eq_inv_pr_push_bi … Hx1) -x1) try (>(eq_inv_pr_next_bi … Hx1) -x1)
+try elim (eq_inv_pr_push_next … Hx1) try elim (eq_inv_pr_next_push … Hx1)
+try (>(eq_inv_pr_push_bi … Hx2) -x2) try (>(eq_inv_pr_next_bi … Hx2) -x2)
+try elim (eq_inv_pr_push_next … Hx2) try elim (eq_inv_pr_next_push … Hx2)
+/2 width=3 by ex2_intro/
+qed-.
+
+(*** sand_inv_nnx *)
+lemma pr_sand_inv_next_bi:
+      ∀g1,g2,g. g1 ⋒ g2 ≘ g → ∀f1,f2. ↑f1 = g1 → ↑f2 = g2 →
+      ∃∃f. f1 ⋒ f2 ≘ f & ↑f = g.
+#g1 #g2 #g * -g1 -g2 -g
+#f1 #f2 #f #g1 #g2 #g #Hf #H1 #H2 #H0 #x1 #x2 #Hx1 #Hx2 destruct
+try (>(eq_inv_pr_push_bi … Hx1) -x1) try (>(eq_inv_pr_next_bi … Hx1) -x1)
+try elim (eq_inv_pr_push_next … Hx1) try elim (eq_inv_pr_next_push … Hx1)
+try (>(eq_inv_pr_push_bi … Hx2) -x2) try (>(eq_inv_pr_next_bi … Hx2) -x2)
+try elim (eq_inv_pr_push_next … Hx2) try elim (eq_inv_pr_next_push … Hx2)
+/2 width=3 by ex2_intro/
+qed-.