some renaming ...

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

diff --git a/matita/matita/contribs/lambda/labelled_sequential_reduction.ma b/matita/matita/contribs/lambda/labelled_sequential_reduction.ma
index c5bf4ba999564949a784eced2a19890315e5c2d6..40d5188469b701c6eb37f4044f62688dd0999be1 100644 (file)
--- a/matita/matita/contribs/lambda/labelled_sequential_reduction.ma
+++ b/matita/matita/contribs/lambda/labelled_sequential_reduction.ma
@@ -12,33 +12,73 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "redex_pointer.ma".
+include "pointer.ma".
 include "multiplicity.ma".
 
-(* LABELLED SEQUENTIAL REDUCTION (ONE STEP) *********************************)
+(* LABELLED SEQUENTIAL REDUCTION (SINGLE STEP) ******************************)
 
 (* Note: the application "(A B)" is represented by "@B.A" following:
          F. Kamareddine and R.P. Nederpelt: "A useful λ-notation".
          Theoretical Computer Science 155(1), Elsevier (1996), pp. 85-109.
 *)
-inductive lsred: rpointer → relation term ≝
-| lsred_beta   : ∀A,D. lsred (◊) (@D.𝛌.A) ([⬐D]A)
-| lsred_abst   : ∀p,A,C. lsred p A C → lsred p (𝛌.A) (𝛌.C) 
-| lsred_appl_sn: ∀p,B,D,A. lsred p B D → lsred (true::p) (@B.A) (@D.A)
-| lsred_appl_dx: ∀p,B,A,C. lsred p A C → lsred (false::p) (@B.A) (@B.C)
+inductive lsred: ptr → relation term ≝
+| lsred_beta   : ∀B,A. lsred (◊) (@B.𝛌.A) ([↙B]A)
+| lsred_abst   : ∀p,A1,A2. lsred p A1 A2 → lsred (sn::p) (𝛌.A1) (𝛌.A2) 
+| lsred_appl_sn: ∀p,B1,B2,A. lsred p B1 B2 → lsred (sn::p) (@B1.A) (@B2.A)
+| lsred_appl_dx: ∀p,B,A1,A2. lsred p A1 A2 → lsred (dx::p) (@B.A1) (@B.A2)
 .
 
 interpretation "labelled sequential reduction"
-    'LablSeqRed M p N = (lsred p M N).
+   'SeqRed M p N = (lsred p M N).
 
 (* Note: we do not use → since it is reserved by CIC *)
-notation "hvbox( M break â\87\80 [ term 46 p ] break term 46 N )"
+notation "hvbox( M break â\86¦ [ term 46 p ] break term 46 N )"
    non associative with precedence 45
-   for @{ 'LablSeqRed $M $p $N }.
+   for @{ 'SeqRed $M $p $N }.
 
-theorem lsred_fwd_mult: ∀p,M,N. M ⇀[p] N → #{N} < #{M} * #{M}.
+lemma lsred_inv_vref: ∀p,M,N. M ↦[p] N → ∀i. #i = M → ⊥.
+#p #M #N * -p -M -N
+[ #B #A #i #H destruct
+| #p #A1 #A2 #_ #i #H destruct
+| #p #B1 #B2 #A #_ #i #H destruct
+| #p #B #A1 #A2 #_ #i #H destruct
+]
+qed-.
+
+lemma lsred_inv_nil: ∀p,M,N. M ↦[p] N → ◊ = p →
+                     ∃∃B,A. @B. 𝛌.A = M & [↙B] A = N.
+#p #M #N * -p -M -N
+[ #B #A #_ destruct /2 width=4/
+| #p #A1 #A2 #_ #H destruct
+| #p #B1 #B2 #A #_ #H destruct
+| #p #B #A1 #A2 #_ #H destruct
+]
+qed-.
+
+lemma lsred_inv_sn: ∀p,M,N. M ↦[p] N → ∀q. sn::q = p →
+                    (∃∃A1,A2. A1 ↦[q] A2 & 𝛌.A1 = M & 𝛌.A2 = N) ∨
+                    ∃∃B1,B2,A. B1 ↦[q] B2 & @B1.A = M & @B2.A = N.
+#p #M #N * -p -M -N
+[ #B #A #q #H destruct
+| #p #A1 #A2 #HA12 #q #H destruct /3 width=5/
+| #p #B1 #B2 #A #HB12 #q #H destruct /3 width=6/
+| #p #B #A1 #A2 #_ #q #H destruct
+]
+qed-.
+
+lemma lsred_inv_dx: ∀p,M,N. M ↦[p] N → ∀q. dx::q = p →
+                    ∃∃B,A1,A2. A1 ↦[q] A2 & @B.A1 = M & @B.A2 = N.
+#p #M #N * -p -M -N
+[ #B #A #q #H destruct
+| #p #A1 #A2 #_ #q #H destruct
+| #p #B1 #B2 #A #_ #q #H destruct
+| #p #B #A1 #A2 #HA12 #q #H destruct /2 width=6/
+]
+qed-.
+
+lemma lsred_fwd_mult: ∀p,M,N. M ↦[p] N → #{N} < #{M} * #{M}.
 #p #M #N #H elim H -p -M -N
-[ #A #D @(le_to_lt_to_lt … (#{A}*#{D})) //
+[ #B #A @(le_to_lt_to_lt … (#{A}*#{B})) //
   normalize /3 width=1 by lt_minus_to_plus_r, lt_times/ (**) (* auto: too slow without trace *) 
 | //
 | #p #B #D #A #_ #IHBD
@@ -49,3 +89,48 @@ theorem lsred_fwd_mult: ∀p,M,N. M ⇀[p] N → #{N} < #{M} * #{M}.
 @(transitive_le … (#{B}*#{B}+#{A}*#{A})) [ /2 width=1/ ]
 >distributive_times_plus normalize /2 width=1/
 qed-.
+
+lemma lsred_lift: ∀p. liftable (lsred p).
+#p #h #M1 #M2 #H elim H -p -M1 -M2 normalize /2 width=1/
+#B #A #d <dsubst_lift_le //
+qed.
+
+lemma lsred_inv_lift: ∀p. deliftable_sn (lsred p).
+#p #h #N1 #N2 #H elim H -p -N1 -N2
+[ #D #C #d #M1 #H
+  elim (lift_inv_appl … H) -H #B #M #H0 #HM #H destruct
+  elim (lift_inv_abst … HM) -HM #A #H0 #H destruct /3 width=3/
+| #p #C1 #C2 #_ #IHC12 #d #M1 #H
+  elim (lift_inv_abst … H) -H #A1 #HAC1 #H
+  elim (IHC12 … HAC1) -C1 #A2 #HA12 #HAC2 destruct
+  @(ex2_intro … (𝛌.A2)) // /2 width=1/
+| #p #D1 #D2 #C1 #_ #IHD12 #d #M1 #H
+  elim (lift_inv_appl … H) -H #B1 #A #HBD1 #H1 #H2
+  elim (IHD12 … HBD1) -D1 #B2 #HB12 #HBD2 destruct
+  @(ex2_intro … (@B2.A)) // /2 width=1/
+| #p #D1 #C1 #C2 #_ #IHC12 #d #M1 #H
+  elim (lift_inv_appl … H) -H #B #A1 #H1 #HAC1 #H2
+  elim (IHC12 … HAC1) -C1 #A2 #HA12 #HAC2 destruct
+  @(ex2_intro … (@B.A2)) // /2 width=1/
+]
+qed-.
+
+lemma lsred_dsubst: ∀p. dsubstable_dx (lsred p).
+#p #D1 #M1 #M2 #H elim H -p -M1 -M2 normalize /2 width=1/
+#D2 #A #d >dsubst_dsubst_ge //
+qed.
+
+theorem lsred_mono: ∀p. singlevalued … (lsred p).
+#p #M #N1 #H elim H -p -M -N1
+[ #B #A #N2 #H elim (lsred_inv_nil … H ?) -H // #D #C #H #HN2 destruct //
+| #p #A1 #A2 #_ #IHA12 #N2 #H elim (lsred_inv_sn … H ??) -H [4: // |2: skip ] * (**) (* simplify line *)
+  [ #C1 #C2 #HC12 #H #HN2 destruct /3 width=1/
+  | #D1 #D2 #C #_ #H destruct
+  ]
+| #p #B1 #B2 #A #_ #IHB12 #N2 #H elim (lsred_inv_sn … H ??) -H [4: // |2: skip ] * (**) (* simplify line *)
+  [ #C1 #C2 #_ #H destruct
+  | #D1 #D2 #C #HD12 #H #HN2 destruct /3 width=1/
+  ]
+| #p #B #A1 #A2 #_ #IHA12 #N2 #H elim (lsred_inv_dx … H ??) -H [3: // |2: skip ] #D #C1 #C2 #HC12 #H #HN2 destruct /3 width=1/ (**) (* simplify line *)
+]
+qed-.