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diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/fpbg_fpbs.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/fpbg_fpbs.ma
new file mode 100644 (file)
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+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/fpbg_fpbs.ma
@@ -0,0 +1,88 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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/computation/lpxs_lleq.ma".
+include "basic_2/computation/fpbs_lift.ma".
+include "basic_2/computation/fpbg_fleq.ma".
+
+(* "QRST" PROPER PARALLEL COMPUTATION FOR CLOSURES **************************)
+
+(* Properties on "qrst" parallel reduction on closures **********************)
+
+lemma fpbg_fpb_trans: ∀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⦄.
+#h #g #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2 elim (fpb_fpbu … H2) -H2
+/3 width=5 by fpbg_fleq_trans, fpbg_strap1, fpbu_fpbc/
+qed-.
+
+lemma fpb_fpbg_trans: ∀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⦄.
+#h #g #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 elim (fpb_fpbu … H1) -H1
+/3 width=5 by fleq_fpbg_trans, fpbg_strap2, fpbu_fpbc/
+qed-.
+
+(* Properties on "qrst" parallel compuutation on closures *******************)
+
+lemma fpbs_fpbg_trans: ∀h,g,G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G, L, T⦄ →
+                       ∀G2,L2,T2. ⦃G, L, T⦄ >≡[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G #L1 #L #T1 #T #H @(fpbs_ind … H) -G -L -T /3 width=5 by fpb_fpbg_trans/
+qed-.
+
+(* Note: this is used in the closure proof *)
+lemma fpbg_fpbs_trans: ∀h,g,G,G2,L,L2,T,T2. ⦃G, L, T⦄ ≥[h, g] ⦃G2, L2, T2⦄ →
+                       ∀G1,L1,T1. ⦃G1, L1, T1⦄ >≡[h, g] ⦃G, L, T⦄ → ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
+#h #g #G #G2 #L #L2 #T #T2 #H @(fpbs_ind_dx … H) -G -L -T /3 width=5 by fpbg_fpb_trans/
+qed-.
+
+lemma fpbu_fpbs_fpbg: ∀h,g,G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G, L, T⦄ → 
+                      ∀G2,L2,T2. ⦃G, L, T⦄ ≥[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
+/3 width=5 by fpbg_fpbs_trans, fpbu_fpbg/ qed.
+
+(* Note: this is used in the closure proof *)
+lemma fqup_fpbg: ∀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 elim (fqup_inv_step_sn … H) -H
+/3 width=5 by fqus_fpbs, fpbu_fqu, fpbu_fpbs_fpbg/
+qed.
+
+lemma cpxs_fpbg: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 →
+                 (T1 = T2 → ⊥) → ⦃G, L, T1⦄ >≡[h, g] ⦃G, L, T2⦄.
+#h #g #G #L #T1 #T2 #H #H0 elim (cpxs_neq_inv_step_sn … H … H0) -H -H0
+/4 width=5 by cpxs_fpbs, fpbu_cpx, fpbu_fpbs_fpbg/
+qed.
+
+lemma lstas_fpbg: ∀h,g,G,L,T1,T2,l2. ⦃G, L⦄ ⊢ T1 •*[h, l2] T2 → (T1 = T2 → ⊥) →
+                  ∀l1. l2 ≤ l1 → ⦃G, L⦄ ⊢ T1 ▪[h, g] l1 → ⦃G, L, T1⦄ >≡[h, g] ⦃G, L, T2⦄.
+/3 width=5 by lstas_cpxs, cpxs_fpbg/ qed.
+
+lemma lpxs_fpbg: ∀h,g,G,L1,L2,T. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 →
+                 (L1 ≡[T, 0] L2 → ⊥) → ⦃G, L1, T⦄ >≡[h, g] ⦃G, L2, T⦄.
+#h #g #G #L1 #L2 #T #H #H0 elim (lpxs_nlleq_inv_step_sn … H … H0) -H -H0
+/4 width=5 by fpbu_fpbs_fpbg, fpbu_lpx, lpxs_lleq_fpbs/
+qed.
+
+lemma fpbs_fpbg: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ →
+                 ⦃G1, L1, T1⦄ ≡[0] ⦃G2, L2, T2⦄ ∨
+                 ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpbs_ind … H) -G2 -L2 -T2
+[ /2 width=1 by or_introl/
+| #G #G2 #L #L2 #T #T2 #_ #H2 * #H1 elim (fpb_fpbu … H2) -H2 #H2
+  [ /3 width=5 by fleq_trans, or_introl/
+  | /5 width=5 by fpbc_fpbg, fleq_fpbc_trans, fpbu_fpbc, or_intror/
+  | /3 width=5 by fpbg_fleq_trans, or_intror/
+  | /4 width=5 by fpbg_strap1, fpbu_fpbc, or_intror/
+  ]
+]
+qed-.

diff --git a/matita/matita/contribs/lambdadelta/basic_2/examples/ex_fpbg_refl.ma b/matita/matita/contribs/lambdadelta/basic_2/examples/ex_fpbg_refl.ma
new file mode 100644 (file)
index 0000000..a0f03d6
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/examples/ex_fpbg_refl.ma
@@ -0,0 +1,54 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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/computation/fpbg_fpbs.ma".
+
+(* EXAMPLES *****************************************************************)
+
+(* Reflexivity of proper qrst-computation: the term ApplOmega ***************)
+
+definition ApplDelta: term → nat → term ≝ λW,k. +ⓛW.ⓐ⋆k.ⓐ#0.#0.
+
+definition ApplOmega1: term → nat → term ≝ λW,k. ⓐ(ApplDelta W k).(ApplDelta W k).
+
+definition ApplOmega2: term → nat → term ≝ λW,k. +ⓓⓝW.(ApplDelta W k).ⓐ⋆k.ⓐ#0.#0.
+
+definition ApplOmega3: term → nat → term ≝ λW,k. ⓐ⋆k.(ApplOmega1 W k).
+
+(* Basic properties *********************************************************)
+
+lemma ApplDelta_lift: ∀W1,W2,k,d,e. ⬆[d, e] W1 ≡ W2 →
+                      ⬆[d, e] (ApplDelta W1 k) ≡ (ApplDelta W2 k).
+/5 width=1 by lift_flat, lift_bind, lift_lref_lt/ qed.
+
+lemma cpr_ApplOmega_12: ∀G,L,W,k. ⦃G, L⦄ ⊢ ApplOmega1 W k ➡ ApplOmega2 W k.
+/2 width=1 by cpr_beta/ qed.
+
+lemma cpr_ApplOmega_23: ∀G,L,W,k. ⦃G, L⦄ ⊢ ApplOmega2 W k ➡ ApplOmega3 W k.
+#G #L #W1 #k elim (lift_total W1 0 1) #W2 #HW12
+@(cpr_zeta … (ApplOmega3 W2 k)) /4 width=1 by ApplDelta_lift, lift_flat/
+@cpr_flat // @cpr_flat @(cpr_delta … (ApplDelta W1 k) ? 0)
+[3,5,8,10: /2 width=2 by ApplDelta_lift/ |4,9: /2 width=1 by cpr_eps/ |*: skip ]
+qed.
+
+lemma cpxs_ApplOmega_13: ∀h,g,G,L,W,k. ⦃G, L⦄ ⊢ ApplOmega1 W k ➡*[h,g] ApplOmega3 W k.
+/4 width=3 by cpxs_strap1, cpr_cpx/ qed.
+
+lemma fqup_ApplOmega_13: ∀G,L,W,k. ⦃G, L, ApplOmega3 W k⦄ ⊐+ ⦃G, L, ApplOmega1 W k⦄.
+/2 width=1 by/ qed.
+
+(* Main properties **********************************************************)
+
+theorem fpbg_refl: ∀h,g,G,L,W,k. ⦃G, L, ApplOmega1 W k⦄ >≡[h,g] ⦃G, L, ApplOmega1 W k⦄.
+/3 width=5 by fpbs_fpbg_trans, fqup_fpbg, cpxs_fpbs/ qed.

diff --git a/matita/matita/contribs/lambdadelta/basic_2/web/basic_2.ldw.xml b/matita/matita/contribs/lambdadelta/basic_2/web/basic_2.ldw.xml
index e6c828efbefc20207e99b25b96b8f12342b6d3d2..ca4970bcb5d75dc7e4f059d2440716e53e83fdc1 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/web/basic_2.ldw.xml
+++ b/matita/matita/contribs/lambdadelta/basic_2/web/basic_2.ldw.xml
@@ -44,16 +44,16 @@
          for context-sensitive computation on terms.
    </news>
    <news date="2014 June 9.">
-         "Big tree" strong normalization
+         strong qrst-normalization
          for simply typed terms.
    </news>
    <news date="2014 April 16.">
          lazy equivalence on local environments
-        serves as irrelevant step in "big tree" computation on closures
+        addded as q-step to rst-computation on closures
          (anniversary milestone).
    </news>
    <news date="2014 January 20.">
-         Parametrized slicing for local environments
+         Parametrized slicing of local environments
         comprises both versions of this operation
         (one from basic_1, the other used in basic_2 till now).
    </news>
@@ -61,10 +61,10 @@
          Passive support for global environments.
    </news>
    <news date="2013 July 27.">
-         Reaxiomatized β-reductum as in extended β-reduction
+         Reaxiomatized β-reductum as in rt-reduction.
    </news>
    <news date="2013 July 20.">
-         Context-sensitive extended strong normalization
+         Context-sensitive strong rt-normalization
          for simply typed terms.
    </news>
    <news date="2013 April 16.">
@@ -74,7 +74,7 @@
    </news>
    <news date="2013 March 16.">
          Mutual recursive preservation of stratified native validity
-         for "big tree" computation on closures.
+         for rst-computation on closures.
    </news>
    <news date="2012 October 16.">
          Confluence for context-free parallel reduction on closures.