update in ground

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

diff --git a/matita/matita/contribs/lambdadelta/ground/relocation/pr_sor_fcla.ma b/matita/matita/contribs/lambdadelta/ground/relocation/pr_sor_fcla.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/xoa/ex_3_1.ma".
+include "ground/xoa/ex_4_2.ma".
+include "ground/arith/nat_plus.ma".
+include "ground/arith/nat_le_max.ma".
+include "ground/relocation/pr_fcla_eq.ma".
+include "ground/relocation/pr_sor_isi.ma".
+
+(* RELATIONAL UNION FOR PARTIAL RELOCATION MAPS *****************************)
+
+(* Constructions with pr_fcla ***********************************************)
+
+(*** sor_fcla_ex *)
+lemma pr_sor_fcla_bi:
+      ∀f1,n1. 𝐂❪f1❫ ≘ n1 → ∀f2,n2. 𝐂❪f2❫ ≘ n2 →
+      ∃∃f,n. f1 ⋓ f2 ≘ f & 𝐂❪f❫ ≘ n & (n1 ∨ n2) ≤ n & n ≤ n1 + n2.
+#f1 #n1 #Hf1 elim Hf1 -f1 -n1 /3 width=6 by pr_sor_isi_sn, ex4_2_intro/
+#f1 #n1 #Hf1 #IH #f2 #n2 * -f2 -n2 /3 width=6 by pr_fcla_push, pr_fcla_next, ex4_2_intro, pr_sor_isi_dx/
+#f2 #n2 #Hf2 elim (IH … Hf2) -IH -Hf2 -Hf1 [2,4: #f #n <nplus_succ_dx ] (* * full auto fails *)
+[ /3 width=7 by pr_fcla_next, pr_sor_push_next, nle_max_sn_succ_dx, nle_succ_bi, ex4_2_intro/
+| /4 width=7 by pr_fcla_next, pr_sor_next_bi, nle_succ_dx, nle_succ_bi, ex4_2_intro/
+| /3 width=7 by pr_fcla_push, pr_sor_push_bi, ex4_2_intro/
+| /3 width=7 by pr_fcla_next, pr_sor_next_push, nle_max_sn_succ_sn, nle_succ_bi, ex4_2_intro/
+]
+qed-.
+
+(* Destructions with pr_fcla ************************************************)
+
+(*** sor_fcla *)
+lemma pr_sor_inv_fcla_bi:
+      ∀f1,n1. 𝐂❪f1❫ ≘ n1 → ∀f2,n2. 𝐂❪f2❫ ≘ n2 → ∀f. f1 ⋓ f2 ≘ f →
+      ∃∃n. 𝐂❪f❫ ≘ n & (n1 ∨ n2) ≤ n & n ≤ n1 + n2.
+#f1 #n1 #Hf1 #f2 #n2 #Hf2 #f #Hf elim (pr_sor_fcla_bi … Hf1 … Hf2) -Hf1 -Hf2
+/4 width=6 by pr_sor_mono, pr_fcla_eq_repl_back, ex3_intro/
+qed-.
+
+(* Destructions with pr_fcla ************************************************)
+
+(*** sor_fwd_fcla_sn_ex *)
+lemma pr_sor_des_fcla_sn:
+      ∀f,n. 𝐂❪f❫ ≘ n → ∀f1,f2. f1 ⋓ f2 ≘ f →
+      ∃∃n1. 𝐂❪f1❫ ≘ n1 & n1 ≤ n.
+#f #n #H elim H -f -n
+[ /4 width=4 by pr_sor_des_isi_sn, pr_fcla_isi, ex2_intro/
+| #f #n #_ #IH #f1 #f2 #H
+  elim (pr_sor_inv_push … H) -H [ |*: // ] #g1 #g2 #Hf #H1 #H2 destruct
+  elim (IH … Hf) -f /3 width=3 by pr_fcla_push, ex2_intro/
+| #f #n #_ #IH #f1 #f2 #H
+  elim (pr_sor_inv_next … H) -H [1,3,4: * |*: // ] #g1 #g2 #Hf #H1 #H2 destruct
+  elim (IH … Hf) -f /3 width=3 by pr_fcla_push, pr_fcla_next, nle_succ_bi, nle_succ_dx, ex2_intro/
+]
+qed-.
+
+(*** sor_fwd_fcla_dx_ex *)
+lemma pr_sor_des_fcla_dx:
+      ∀f,n. 𝐂❪f❫ ≘ n → ∀f1,f2. f1 ⋓ f2 ≘ f →
+      ∃∃n2. 𝐂❪f2❫ ≘ n2 & n2 ≤ n.
+/3 width=4 by pr_sor_des_fcla_sn, pr_sor_comm/ qed-.