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[helm.git] / matita / matita / contribs / lambdadelta / basic_2 / computation / fqus.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 "basic_2/notation/relations/suptermstar_6.ma".
-include "basic_2/substitution/fquq.ma".
-include "basic_2/multiple/fqup.ma".
-
-(* STAR-ITERATED SUPCLOSURE *************************************************)
-
-definition fqus: tri_relation genv lenv term ≝ tri_TC … fquq.
-
-interpretation "star-iterated structural successor (closure)"
-   'SupTermStar G1 L1 T1 G2 L2 T2 = (fqus G1 L1 T1 G2 L2 T2).
-
-(* Basic eliminators ********************************************************)
-
-lemma fqus_ind: ∀G1,L1,T1. ∀R:relation3 …. R G1 L1 T1 →
-                (∀G,G2,L,L2,T,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐⸮ ⦃G2, L2, T2⦄ → R G L T → R G2 L2 T2) →
-                ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ → R G2 L2 T2.
-#G1 #L1 #T1 #R #IH1 #IH2 #G2 #L2 #T2 #H
-@(tri_TC_star_ind … IH1 IH2 G2 L2 T2 H) //
-qed-.
-
-lemma fqus_ind_dx: ∀G2,L2,T2. ∀R:relation3 …. R G2 L2 T2 →
-                   (∀G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐* ⦃G2, L2, T2⦄ → R G L T → R G1 L1 T1) →
-                   ∀G1,L1,T1. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ → R G1 L1 T1.
-#G2 #L2 #T2 #R #IH1 #IH2 #G1 #L1 #T1 #H
-@(tri_TC_star_ind_dx … IH1 IH2 G1 L1 T1 H) //
-qed-.
-
-(* Basic properties *********************************************************)
-
-lemma fqus_refl: tri_reflexive … fqus.
-/2 width=1 by tri_inj/ qed.
-
-lemma fquq_fqus: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄.
-/2 width=1 by tri_inj/ qed.
-
-lemma fqus_strap1: ∀G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
-                   ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄.
-/2 width=5 by tri_step/ qed-.
-
-lemma fqus_strap2: ∀G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐* ⦃G2, L2, T2⦄ →
-                   ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄.
-/2 width=5 by tri_TC_strap/ qed-.
-
-lemma fqus_drop: ∀G1,G2,K1,K2,T1,T2. ⦃G1, K1, T1⦄ ⊐* ⦃G2, K2, T2⦄ →
-                  ∀L1,U1,k. ⬇[k] L1 ≡ K1 → ⬆[0, k] T1 ≡ U1 →
-                  ⦃G1, L1, U1⦄ ⊐* ⦃G2, K2, T2⦄.
-#G1 #G2 #K1 #K2 #T1 #T2 #H @(fqus_ind … H) -G2 -K2 -T2
-/3 width=5 by fqus_strap1, fquq_fqus, fquq_drop/
-qed-.
-
-lemma fqup_fqus: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄.
-#G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
-/3 width=5 by fqus_strap1, fquq_fqus, fqu_fquq/
-qed.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma fqus_fwd_fw: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ → ♯{G2, L2, T2} ≤ ♯{G1, L1, T1}.
-#G1 #G2 #L1 #L2 #T1 #T2 #H @(fqus_ind … H) -L2 -T2
-/3 width=3 by fquq_fwd_fw, transitive_le/
-qed-.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma fqup_inv_step_sn: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
-                        ∃∃G,L,T. ⦃G1, L1, T1⦄ ⊐ ⦃G, L, T⦄ & ⦃G, L, T⦄ ⊐* ⦃G2, L2, T2⦄.
-#G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind_dx … H) -G1 -L1 -T1 /2 width=5 by ex2_3_intro/
-#G1 #G #L1 #L #T1 #T #H1 #_ * /4 width=9 by fqus_strap2, fqu_fquq, ex2_3_intro/
-qed-.