- nat.ma: we added a general induction principle

[helm.git] / matita / matita / contribs / lambda / parallel_reduction.ma

diff --git a/matita/matita/contribs/lambda/parallel_reduction.ma b/matita/matita/contribs/lambda/parallel_reduction.ma
index 4cb204bb9dea651a629bc2274d306952d552c716..7aa47be7f496bd164c52dfff3fa469c4cab1988a 100644 (file)
--- a/matita/matita/contribs/lambda/parallel_reduction.ma
+++ b/matita/matita/contribs/lambda/parallel_reduction.ma
@@ -12,7 +12,8 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "multiplicity.ma".
+include "size.ma".
+include "labelled_sequential_reduction.ma".
 
 (* PARALLEL REDUCTION (SINGLE STEP) *****************************************)
 
@@ -55,6 +56,17 @@ lemma pred_inv_abst: ∀M,N. M ⥤ N → ∀A. 𝛌.A = M →
 ]
 qed-.
 
+lemma pred_inv_appl: ∀M,N. M ⥤ N → ∀B,A. @B.A = M →
+                     (∃∃D,C. B ⥤ D & A ⥤ C & @D.C = N) ∨
+                     ∃∃A0,D,C0. B ⥤ D & A0 ⥤ C0 & 𝛌.A0 = A & [⬐D]C0 = N.
+#M #N * -M -N
+[ #i #B0 #A0 #H destruct
+| #A #C #_ #B0 #A0 #H destruct
+| #B #D #A #C #HBD #HAC #B0 #A0 #H destruct /3 width=5/
+| #B #D #A #C #HBD #HAC #B0 #A0 #H destruct /3 width=7/
+]
+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/
@@ -79,3 +91,66 @@ lemma pred_inv_lift: deliftable pred.
   @(ex2_1_intro … ([⬐B2]A2)) /2 width=1/
 ]
 qed-.
+
+lemma pred_dsubst: dsubstable pred.
+#N1 #N2 #HN12 #M1 #M2 #H elim H -M1 -M2
+[ #i #d elim (lt_or_eq_or_gt i d) #Hid
+  [ >(dsubst_vref_lt … Hid) >(dsubst_vref_lt … Hid) //
+  | destruct >dsubst_vref_eq >dsubst_vref_eq /2 width=1/
+  | >(dsubst_vref_gt … Hid) >(dsubst_vref_gt … Hid) //
+  ]
+| normalize /2 width=1/
+| normalize /2 width=1/
+| normalize #B #D #A #C #_ #_ #IHBD #IHAC #d
+  >dsubst_dsubst_ge // /2 width=1/
+]
+qed.
+
+lemma pred_conf1_vref: ∀i. confluent1 … pred (#i).
+#i #M1 #H1 #M2 #H2
+<(pred_inv_vref … H1) -H1 [3: // |2: skip ] (**) (* simplify line *)
+<(pred_inv_vref … H2) -H2 [3: // |2: skip ] (**) (* simplify line *)
+/2 width=3/
+qed-.
+
+lemma pred_conf1_abst: ∀A. confluent1 … pred A → confluent1 … pred (𝛌.A).
+#A #IH #M1 #H1 #M2 #H2
+elim (pred_inv_abst … H1 ??) -H1 [3: // |2: skip ] #A1 #HA1 #H destruct (**) (* simplify line *)
+elim (pred_inv_abst … H2 ??) -H2 [3: // |2: skip ] #A2 #HA2 #H destruct (**) (* simplify line *)
+elim (IH … HA1 … HA2) -A /3 width=3/
+qed-.
+
+lemma pred_conf1_appl_beta: ∀B,B1,B2,C,C2,M1.
+                            (∀M0. |M0| < |B|+|𝛌.C|+1 → confluent1 ? pred M0) → (**) (* ? needed in place of … *)
+                            B ⥤ B1 → B ⥤ B2 → 𝛌.C ⥤ M1 → C ⥤ C2 →
+                            ∃∃M. @B1.M1 ⥤ M & [⬐B2]C2 ⥤ M.
+#B #B1 #B2 #C #C2 #M1 #IH #HB1 #HB2 #H1 #HC2
+elim (pred_inv_abst … H1 ??) -H1 [3: // |2: skip ] #C1 #HC1 #H destruct (**) (* simplify line *)
+elim (IH B … HB1 … HB2) -HB1 -HB2 //
+elim (IH C … HC1 … HC2) normalize // -B -C /3 width=5/
+qed-.
+
+theorem pred_conf: confluent … pred.
+#M @(f_ind … size … M) -M #n #IH * normalize
+[ /2 width=3 by pred_conf1_vref/
+| /3 width=4 by pred_conf1_abst/
+| #B #A #H #M1 #H1 #M2 #H2 destruct
+  elim (pred_inv_appl … H1 ???) -H1 [5: // |2,3: skip ] * (**) (* simplify line *)
+  elim (pred_inv_appl … H2 ???) -H2 [5,10: // |2,3,7,8: skip ] * (**) (* simplify line *) 
+  [ #B2 #A2 #HB2 #HA2 #H2 #B1 #A1 #HB1 #HA1 #H1 destruct
+    elim (IH A … HA1 … HA2) -HA1 -HA2 //
+    elim (IH B … HB1 … HB2) // -A -B /3 width=5/
+  | #C #B2 #C2 #HB2 #HC2 #H2 #HM2 #B1 #N #HB1 #H #HM1 destruct
+    @(pred_conf1_appl_beta … IH) // (**) (* /2 width=7 by pred_conf1_appl_beta/ does not work *)
+  | #B2 #N #B2 #H #HM2 #C #B1 #C1 #HB1 #HC1 #H1 #HM1 destruct
+    @ex2_1_commute @(pred_conf1_appl_beta … IH) //
+  | #C #B2 #C2 #HB2 #HC2 #H2 #HM2 #C0 #B1 #C1 #HB1 #HC1 #H1 #HM1 destruct
+    elim (IH B … HB1 … HB2) -HB1 -HB2 //
+    elim (IH C … HC1 … HC2) normalize // -B -C /3 width=5/
+  ]
+]
+qed-.
+
+lemma lsred_pred: ∀p,M,N. M ⇀[p] N → M ⥤ N.
+#p #M #N #H elim H -p -M -N /2 width=1/
+qed.