- commit of the "s_computation" component ...

[helm.git] / matita / matita / contribs / lambdadelta / basic_2 / computation / lsx.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/lsx.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/lsx.ma
index c842a9448fa67aa7ec6feffc2250fae4752c378d..7bb8aec4af5f8e77b80b6f1a404f46bf5df9f8d4 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/lsx.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/lsx.ma
@@ -12,52 +12,98 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "basic_2/notation/relations/lazysn_5.ma".
-include "basic_2/substitution/lleq.ma".
-include "basic_2/computation/lpxs.ma".
+include "basic_2/notation/relations/sn_6.ma".
+include "basic_2/multiple/lleq.ma".
+include "basic_2/reduction/lpx.ma".
 
 (* SN EXTENDED STRONGLY NORMALIZING LOCAL ENVIRONMENTS **********************)
 
-definition lsx: ∀h. sd h → relation3 term genv lenv ≝
-                λh,g,T,G. SN … (lpxs h g G) (lleq 0 T).
+definition lsx: ∀h. sd h → relation4 ynat term genv lenv ≝
+                λh,o,l,T,G. SN … (lpx h o G) (lleq l T).
 
 interpretation
    "extended strong normalization (local environment)"
-   'LazySN h g T G L = (lsx h g T G L).
+   'SN h o l T G L = (lsx h o T l G L).
 
 (* Basic eliminators ********************************************************)
 
-lemma lsx_ind: ∀h,g,T,G. ∀R:predicate lenv.
-               (â\88\80L1. G â\8a¢ â\8b\95â¬\8a*[h, g, T] L1 →
-                     (∀L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 → (L1 ⋕[T, 0] L2 → ⊥) → R L2) →
+lemma lsx_ind: ∀h,o,G,T,l. ∀R:predicate lenv.
+               (â\88\80L1. G â\8a¢ â¬\8a*[h, o, T, l] L1 →
+                     (∀L2. ⦃G, L1⦄ ⊢ ➡[h, o] L2 → (L1 ≡[T, l] L2 → ⊥) → R L2) →
                      R L1
                ) →
-               â\88\80L. G â\8a¢ â\8b\95â¬\8a*[h, g, T] L → R L.
-#h #g #T #G #R #H0 #L1 #H elim H -L1
+               â\88\80L. G â\8a¢ â¬\8a*[h, o, T, l] L → R L.
+#h #o #G #T #l #R #H0 #L1 #H elim H -L1
 /5 width=1 by lleq_sym, SN_intro/
 qed-.
 
 (* Basic properties *********************************************************)
 
-lemma lsx_intro: ∀h,g,T,G,L1.
-                 (∀L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 → (L1 ⋕[T, 0] L2 → ⊥) → G ⊢ ⋕⬊*[h, g, T] L2) →
-                 G â\8a¢ â\8b\95â¬\8a*[h, g, T] L1.
+lemma lsx_intro: ∀h,o,G,L1,T,l.
+                 (∀L2. ⦃G, L1⦄ ⊢ ➡[h, o] L2 → (L1 ≡[T, l] L2 → ⊥) → G ⊢ ⬊*[h, o, T, l] L2) →
+                 G â\8a¢ â¬\8a*[h, o, T, l] L1.
 /5 width=1 by lleq_sym, SN_intro/ qed.
 
-lemma lsx_atom: ∀h,g,T,G. G ⊢ ⋕⬊*[h, g, T] ⋆.
-#h #g #T #G @lsx_intro
-#X #H #HT lapply (lpxs_inv_atom1 … H) -H
+lemma lsx_atom: ∀h,o,G,T,l. G ⊢ ⬊*[h, o, T, l] ⋆.
+#h #o #G #T #l @lsx_intro
+#X #H #HT lapply (lpx_inv_atom1 … H) -H
 #H destruct elim HT -HT //
 qed.
 
-lemma lsx_sort: ∀h,g,G,L,k. G ⊢ ⋕⬊*[h, g, ⋆k] L.
-#h #g #G #L1 #k @lsx_intro
+lemma lsx_sort: ∀h,o,G,L,l,s. G ⊢ ⬊*[h, o, ⋆s, l] L.
+#h #o #G #L1 #l #s @lsx_intro
 #L2 #HL12 #H elim H -H
-/3 width=4 by lpxs_fwd_length, lleq_sort/
+/3 width=4 by lpx_fwd_length, lleq_sort/
 qed.
 
-lemma lsx_gref: ∀h,g,G,L,p. G ⊢ ⋕⬊*[h, g, §p] L.
-#h #g #G #L1 #p @lsx_intro
+lemma lsx_gref: ∀h,o,G,L,l,p. G ⊢ ⬊*[h, o, §p, l] L.
+#h #o #G #L1 #l #p @lsx_intro
 #L2 #HL12 #H elim H -H
-/3 width=4 by lpxs_fwd_length, lleq_gref/
+/3 width=4 by lpx_fwd_length, lleq_gref/
 qed.
+
+lemma lsx_ge_up: ∀h,o,G,L,T,U,lt,l,k. lt ≤ yinj l + yinj k →
+                 ⬆[l, k] T ≡ U → G ⊢ ⬊*[h, o, U, lt] L → G ⊢ ⬊*[h, o, U, l] L.
+#h #o #G #L #T #U #lt #l #k #Hltlm #HTU #H @(lsx_ind … H) -L
+/5 width=7 by lsx_intro, lleq_ge_up/
+qed-.
+
+lemma lsx_ge: ∀h,o,G,L,T,l1,l2. l1 ≤ l2 →
+              G ⊢ ⬊*[h, o, T, l1] L → G ⊢ ⬊*[h, o, T, l2] L.
+#h #o #G #L #T #l1 #l2 #Hl12 #H @(lsx_ind … H) -L
+/5 width=7 by lsx_intro, lleq_ge/
+qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma lsx_fwd_bind_sn: ∀h,o,a,I,G,L,V,T,l. G ⊢ ⬊*[h, o, ⓑ{a,I}V.T, l] L →
+                       G ⊢ ⬊*[h, o, V, l] L.
+#h #o #a #I #G #L #V #T #l #H @(lsx_ind … H) -L
+#L1 #_ #IHL1 @lsx_intro
+#L2 #HL12 #HV @IHL1 /3 width=4 by lleq_fwd_bind_sn/
+qed-.
+
+lemma lsx_fwd_flat_sn: ∀h,o,I,G,L,V,T,l. G ⊢ ⬊*[h, o, ⓕ{I}V.T, l] L →
+                       G ⊢ ⬊*[h, o, V, l] L.
+#h #o #I #G #L #V #T #l #H @(lsx_ind … H) -L
+#L1 #_ #IHL1 @lsx_intro
+#L2 #HL12 #HV @IHL1 /3 width=3 by lleq_fwd_flat_sn/
+qed-.
+
+lemma lsx_fwd_flat_dx: ∀h,o,I,G,L,V,T,l. G ⊢ ⬊*[h, o, ⓕ{I}V.T, l] L →
+                       G ⊢ ⬊*[h, o, T, l] L.
+#h #o #I #G #L #V #T #l #H @(lsx_ind … H) -L
+#L1 #_ #IHL1 @lsx_intro
+#L2 #HL12 #HV @IHL1 /3 width=3 by lleq_fwd_flat_dx/
+qed-.
+
+lemma lsx_fwd_pair_sn: ∀h,o,I,G,L,V,T,l. G ⊢ ⬊*[h, o, ②{I}V.T, l] L →
+                       G ⊢ ⬊*[h, o, V, l] L.
+#h #o * /2 width=4 by lsx_fwd_bind_sn, lsx_fwd_flat_sn/
+qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma lsx_inv_flat: ∀h,o,I,G,L,V,T,l. G ⊢ ⬊*[h, o, ⓕ{I}V.T, l] L →
+                    G ⊢ ⬊*[h, o, V, l] L ∧ G ⊢ ⬊*[h, o, T, l] L.
+/3 width=3 by lsx_fwd_flat_sn, lsx_fwd_flat_dx, conj/ qed-.