- lambda: parallel reduction to obtain diamond property

[helm.git] / matita / matita / contribs / lambda / parallel_reduction.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 "multiplicity.ma".
+
+(* PARALLEL REDUCTION (SINGLE STEP) *****************************************)
+
+(* Note: the application "(A B)" is represented by "@B.A"
+         as for labelled sequential reduction
+*)
+inductive pred: relation term ≝
+| pred_vref: ∀i. pred (#i) (#i)
+| pred_abst: ∀A,C. pred A C → pred (𝛌.A) (𝛌.C) 
+| pred_appl: ∀B,D,A,C. pred B D → pred A C → pred (@B.A) (@D.C)
+| pred_beta: ∀B,D,A,C. pred B D → pred A C → pred (@B.𝛌.A) ([⬐D]C)
+.
+
+interpretation "parallel reduction"
+    'ParRed M N = (pred M N).
+
+notation "hvbox( M break ⥤ break term 46 N )"
+   non associative with precedence 45
+   for @{ 'ParRed $M $N }.
+
+lemma pred_refl: reflexive … pred.
+#M elim M -M // /2 width=1/
+qed.
+
+lemma pred_inv_vref: ∀M,N. M ⥤ N → ∀i. #i = M → #i = N.
+#M #N * -M -N //
+[ #A #C #_ #i #H destruct
+| #B #D #A #C #_ #_ #i #H destruct
+| #B #D #A #C #_ #_ #i #H destruct
+]
+qed-.
+
+lemma pred_inv_abst: ∀M,N. M ⥤ N → ∀A. 𝛌.A = M →
+                     ∃∃C. A ⥤ C & 𝛌.C = N.
+#M #N * -M -N
+[ #i #A0 #H destruct
+| #A #C #HAC #A0 #H destruct /2 width=3/
+| #B #D #A #C #_ #_ #A0 #H destruct
+| #B #D #A #C #_ #_ #A0 #H destruct
+]
+qed-.
+
+lemma pred_lift: liftable pred.
+#h #M1 #M2 #H elim H -M1 -M2 normalize // /2 width=1/
+#D #D #A #C #_ #_ #IHBD #IHAC #d <dsubst_lift_le // /2 width=1/
+qed.
+
+lemma pred_inv_lift: deliftable pred.
+#h #N1 #N2 #H elim H -N1 -N2 /2 width=3/
+[ #C1 #C2 #_ #IHC12 #d #M1 #H
+  elim (lift_inv_abst … H) -H #A1 #HAC1 #H
+  elim (IHC12 … HAC1) -C1 #A2 #HA12 #HAC2 destruct
+  @(ex2_1_intro … (𝛌.A2)) // /2 width=1/
+| #D1 #D2 #C1 #C2 #_ #_ #IHD12 #IHC12 #d #M1 #H
+  elim (lift_inv_appl … H) -H #B1 #A1 #HBD1 #HAC1 #H
+  elim (IHD12 … HBD1) -D1 #B2 #HB12 #HBD2
+  elim (IHC12 … HAC1) -C1 #A2 #HA12 #HAC2 destruct
+  @(ex2_1_intro … (@B2.A2)) // /2 width=1/
+| #D1 #D2 #C1 #C2 #_ #_ #IHD12 #IHC12 #d #M1 #H
+  elim (lift_inv_appl … H) -H #B1 #M #HBD1 #HM #H1
+  elim (lift_inv_abst … HM) -HM #A1 #HAC1 #H
+  elim (IHD12 … HBD1) -D1 #B2 #HB12 #HBD2
+  elim (IHC12 … HAC1) -C1 #A2 #HA12 #HAC2 destruct
+  @(ex2_1_intro … ([⬐B2]A2)) /2 width=1/
+]
+qed-.