precommit for rtmap ...

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

diff --git a/matita/matita/contribs/lambdadelta/ground_2/relocation/rtmap_istot.ma b/matita/matita/contribs/lambdadelta/ground_2/relocation/rtmap_istot.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_2/notation/relations/istotal_1.ma".
+include "ground_2/relocation/rtmap_at.ma".
+
+(* RELOCATION MAP ***********************************************************)
+
+definition istot: predicate rtmap ≝ λf. ∀i. ∃j. @⦃i, f⦄ ≡ j.
+
+interpretation "test for totality (rtmap)"
+   'IsTotal f = (istot f).
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma istot_inv_push: ∀g. 𝐓⦃g⦄ → ∀f. ↑f = g → 𝐓⦃f⦄.
+#g #Hg #f #H #i elim (Hg (⫯i)) -Hg
+#j #Hg elim (at_inv_npx … Hg … H) -Hg -H /2 width=3 by ex_intro/
+qed-.
+
+lemma istot_inv_next: ∀g. 𝐓⦃g⦄ → ∀f. ⫯f = g → 𝐓⦃f⦄.
+#g #Hg #f #H #i elim (Hg i) -Hg
+#j #Hg elim (at_inv_xnx … Hg … H) -Hg -H /2 width=2 by ex_intro/
+qed-.
+
+(* Advanced forward lemmas on at ********************************************)
+
+let corec at_ext: ∀f1,f2. 𝐓⦃f1⦄ → 𝐓⦃f2⦄ →
+                  (∀i,i1,i2. @⦃i, f1⦄ ≡ i1 → @⦃i, f2⦄ ≡ i2 → i1 = i2) → f1 ≗ f2 ≝ ?.
+#f1 cases (pn_split f1) * #g1 #H1
+#f2 cases (pn_split f2) * #g2 #H2
+#Hf1 #Hf2 #Hi
+[ @(eq_push … H1 H2) @at_ext -at_ext /2 width=3 by istot_inv_push/ -Hf1 -Hf2
+  #i #i1 #i2 #Hg1 #Hg2 lapply (Hi (⫯i) (⫯i1) (⫯i2) ??) /2 width=7 by at_push/
+| cases (Hf2 0) -Hf1 -Hf2 -at_ext
+  #j2 #Hf2 cases (at_increasing_strict … Hf2 … H2) -H2
+  lapply (Hi 0 0 j2 … Hf2) /2 width=2 by at_refl/ -Hi -Hf2 -H1
+  #H2 #H cases (lt_le_false … H) -H //
+| cases (Hf1 0) -Hf1 -Hf2 -at_ext
+  #j1 #Hf1 cases (at_increasing_strict … Hf1 … H1) -H1
+  lapply (Hi 0 j1 0 Hf1 ?) /2 width=2 by at_refl/ -Hi -Hf1 -H2
+  #H1 #H cases (lt_le_false … H) -H //
+| @(eq_next … H1 H2) @at_ext -at_ext /2 width=3 by istot_inv_next/ -Hf1 -Hf2
+  #i #i1 #i2 #Hg1 #Hg2 lapply (Hi i (⫯i1) (⫯i2) ??) /2 width=5 by at_next/
+]
+qed-.
+
+(* Main properties on at ****************************************************)
+
+lemma at_dec: ∀f,i1,i2. 𝐓⦃f⦄ → Decidable (@⦃i1, f⦄ ≡ i2).
+#f #i1 #i2 #Hf lapply (Hf i1) -Hf *
+#j2 #Hf elim (eq_nat_dec i2 j2)
+[ #H destruct /2 width=1 by or_introl/
+| /4 width=6 by at_mono, or_intror/
+]
+qed-.
+
+lemma is_at_dec_le: ∀f,i2,i. 𝐓⦃f⦄ → (∀i1. i1 + i ≤ i2 → @⦃i1, f⦄ ≡ i2 → ⊥) → Decidable (∃i1. @⦃i1, f⦄ ≡ i2).
+#f #i2 #i #Hf elim i -i
+[ #Ht @or_intror * /3 width=3 by at_increasing/
+| #i #IH #Ht elim (at_dec f (i2-i) i2) /3 width=2 by ex_intro, or_introl/
+  #Hi2 @IH -IH #i1 #H #Hi elim (le_to_or_lt_eq … H) -H /2 width=3 by/
+  #H destruct -Ht /2 width=1 by/
+]
+qed-.
+
+lemma is_at_dec: ∀f,i2. 𝐓⦃f⦄ → Decidable (∃i1. @⦃i1, f⦄ ≡ i2).
+#f #i2 #Hf @(is_at_dec_le ?? (⫯i2)) /2 width=4 by lt_le_false/
+qed-.
+
+(* Advanced properties on isid **********************************************)
+
+lemma isid_at_total: ∀f. 𝐓⦃f⦄ → (∀i1,i2. @⦃i1, f⦄ ≡ i2 → i1 = i2) → 𝐈⦃f⦄.
+#f #H1f #H2f @isid_at
+#i lapply (H1f i) -H1f *
+#j #Hf >(H2f … Hf) in ⊢ (???%); -H2f //
+qed.