update in lambdadelta

[helm.git] / matita / matita / contribs / lambdadelta / basic_2A / computation / fsb_alt.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/computation/fsb_alt.ma b/matita/matita/contribs/lambdadelta/basic_2A/computation/fsb_alt.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_2A/notation/relations/btsnalt_5.ma".
+include "basic_2A/computation/fpbg_fpbs.ma".
+include "basic_2A/computation/fsb.ma".
+
+(* "QRST" STRONGLY NORMALIZING CLOSURES *************************************)
+
+(* Note: alternative definition of fsb *)
+inductive fsba (h) (g): relation3 genv lenv term ≝
+| fsba_intro: ∀G1,L1,T1. (
+                 ∀G2,L2,T2. ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄ → fsba h g G2 L2 T2
+              ) → fsba h g G1 L1 T1.
+
+interpretation
+   "'big tree' strong normalization (closure) alternative"
+   'BTSNAlt h g G L T = (fsba h g G L T).
+
+(* Basic eliminators ********************************************************)
+
+lemma fsba_ind_alt: ∀h,g. ∀R: relation3 …. (
+                       ∀G1,L1,T1. ⦥⦥[h,g] ⦃G1, L1, T1⦄ → (
+                          ∀G2,L2,T2. ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄ → R G2 L2 T2
+                       ) → R G1 L1 T1
+                    ) →
+                    ∀G,L,T. ⦥⦥[h, g] ⦃G, L, T⦄ → R G L T.
+#h #g #R #IH #G #L #T #H elim H -G -L -T
+/4 width=1 by fsba_intro/
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma fsba_fpbs_trans: ∀h,g,G1,L1,T1. ⦥⦥[h, g] ⦃G1, L1, T1⦄ →
+                       ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ → ⦥⦥[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #L1 #T1 #H @(fsba_ind_alt … H) -G1 -L1 -T1
+/4 width=5 by fsba_intro, fpbs_fpbg_trans/
+qed-.
+
+(* Main properties **********************************************************)
+
+theorem fsb_fsba: ∀h,g,G,L,T. ⦥[h, g] ⦃G, L, T⦄  → ⦥⦥[h, g] ⦃G, L, T⦄.
+#h #g #G #L #T #H @(fsb_ind_alt … H) -G -L -T
+#G1 #L1 #T1 #_ #IH @fsba_intro
+#G2 #L2 #T2 * /3 width=5 by fsba_fpbs_trans/
+qed.
+
+(* Main inversion lemmas ****************************************************)
+
+theorem fsba_inv_fsb: ∀h,g,G,L,T. ⦥⦥[h, g] ⦃G, L, T⦄ → ⦥[h, g] ⦃G, L, T⦄.
+#h #g #G #L #T #H @(fsba_ind_alt … H) -G -L -T
+/4 width=1 by fsb_intro, fpb_fpbg/
+qed-.
+
+(* Advanced properties ******************************************************)
+
+lemma fsb_fpbs_trans: ∀h,g,G1,L1,T1. ⦥[h, g] ⦃G1, L1, T1⦄ →
+                      ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ → ⦥[h, g] ⦃G2, L2, T2⦄.
+/4 width=5 by fsba_inv_fsb, fsb_fsba, fsba_fpbs_trans/ qed-.
+
+(* Advanced eliminators *****************************************************)
+
+lemma fsb_ind_fpbg: ∀h,g. ∀R:relation3 genv lenv term.
+                    (∀G1,L1,T1. ⦥[h, g] ⦃G1, L1, T1⦄ →
+                                (∀G2,L2,T2. ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄ → R G2 L2 T2) →
+                                R G1 L1 T1
+                    ) →
+                    ∀G1,L1,T1. ⦥[h, g] ⦃G1, L1, T1⦄ → R G1 L1 T1.
+#h #g #R #IH #G1 #L1 #T1 #H @(fsba_ind_alt h g … G1 L1 T1)
+/3 width=1 by fsba_inv_fsb, fsb_fsba/
+qed-.