final na,e for big-tree rediction and computation

[helm.git] / matita / matita / contribs / lambdadelta / basic_2 / computation / fpbs.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/fpbs.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/fpbs.ma
new file mode 100644 (file)
index 0000000..361642d
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/fpbs.ma
@@ -0,0 +1,82 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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/btpredstar_8.ma".
+include "basic_2/substitution/fsupp.ma".
+include "basic_2/reduction/fpb.ma".
+include "basic_2/computation/cprs.ma".
+include "basic_2/computation/lprs.ma".
+
+(* "BIG TREE" PARALLEL COMPUTATION FOR CLOSURES *****************************)
+
+definition fpbs: ∀h. sd h → tri_relation genv lenv term ≝
+                 λh,g. tri_TC … (fpb h g).
+
+interpretation "'big tree' parallel computation (closure)"
+   'BTPRedStar h g G1 L1 T1 G2 L2 T2 = (fpbs h g G1 L1 T1 G2 L2 T2).
+
+(* Basic eliminators ********************************************************)
+
+lemma fpbs_ind: ∀h,g,G1,L1,T1. ∀R:relation3 genv lenv term. R G1 L1 T1 →
+                (∀L,G2,G,L2,T,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G, L, T⦄ → ⦃G, L, T⦄ ≽[h, g] ⦃G2, L2, T2⦄ → R G L T → R G2 L2 T2) →
+                ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ → R G2 L2 T2.
+/3 width=8 by tri_TC_star_ind/ qed-.
+
+lemma fpbs_ind_dx: ∀h,g,G2,L2,T2. ∀R:relation3 genv lenv term. R G2 L2 T2 →
+                   (∀G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ≽[h, g] ⦃G, L, T⦄ → ⦃G, L, T⦄ ≥[h, g] ⦃G2, L2, T2⦄ → R G L T → R G1 L1 T1) →
+                   ∀G1,L1,T1. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ → R G1 L1 T1.
+/3 width=8 by tri_TC_star_ind_dx/ qed-.
+
+(* Basic properties *********************************************************)
+
+lemma fpbs_refl: ∀h,g. tri_reflexive … (fpbs h g).
+/2 width=1 by tri_inj/ qed.
+
+lemma fpb_fpbs: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≽[h, g] ⦃G2, L2, T2⦄ →
+                ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
+/2 width=1 by tri_inj/ qed.
+
+lemma fpbs_strap1: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G, L, T⦄ →
+                   ⦃G, L, T⦄ ≽[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
+/2 width=5 by tri_step/ qed-.
+
+lemma fpbs_strap2: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ≽[h, g] ⦃G, L, T⦄ →
+                   ⦃G, L, T⦄ ≥[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
+/2 width=5 by tri_TC_strap/ qed-.
+
+(* Note: this is a general property of bi_TC *)
+lemma fsupp_fpbs: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃+ ⦃G2, L2, T2⦄ →
+                  ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fsupp_ind … L2 T2 H) -G2 -L2 -T2
+/3 width=5 by fpb_fsup, tri_step, fpb_fpbs/
+qed.
+
+lemma cprs_fpbs: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡* T2 → ⦃G, L, T1⦄ ≥[h, g] ⦃G, L, T2⦄.
+#h #g #G #L #T1 #T2 #H @(cprs_ind … H) -T2 
+/3 width=5 by fpb_cpr, fpbs_strap1/
+qed.
+
+lemma lprs_fpbs: ∀h,g,G,L1,L2,T. ⦃G, L1⦄ ⊢ ➡* L2 → ⦃G, L1, T⦄ ≥[h, g] ⦃G, L2, T⦄.
+#h #g #G #L1 #L2 #T #H @(lprs_ind … H) -L2
+/3 width=5 by fpb_lpr, fpbs_strap1/
+qed.
+
+lemma cpr_lpr_fpbs: ∀h,g,G,L1,L2,T1,T2. ⦃G, L1⦄ ⊢ T1 ➡ T2 → ⦃G, L1⦄ ⊢ ➡ L2 →
+                    ⦃G, L1, T1⦄ ≥[h, g] ⦃G, L2, T2⦄.
+/4 width=5 by fpbs_strap1, fpb_lpr, fpb_cpr/ qed.
+
+lemma ssta_fpbs: ∀h,g,G,L,T,U,l.
+                 ⦃G, L⦄ ⊢ T ▪[h, g] l+1 → ⦃G, L⦄ ⊢ T •[h, g] U →
+                 ⦃G, L, T⦄ ≥[h, g] ⦃G, L, U⦄.
+/3 width=2 by fpb_fpbs, fpb_ssta/ qed.