lift functions and identity map

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

diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/lsx_alt.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/lsx_alt.ma
index cd7c83be9831e84679703de212810aedc767ab3b..05637c0e72c4bcd66888495961e0255ae3ea4e65 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/lsx_alt.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/lsx_alt.ma
@@ -13,7 +13,6 @@
 (**************************************************************************)
 
 include "basic_2/notation/relations/snalt_6.ma".
-include "basic_2/substitution/lleq_lleq.ma".
 include "basic_2/computation/lpxs_lleq.ma".
 include "basic_2/computation/lsx.ma".
 
@@ -21,57 +20,57 @@ include "basic_2/computation/lsx.ma".
 
 (* alternative definition of lsx *)
 definition lsxa: ∀h. sd h → relation4 ynat term genv lenv ≝
-                 λh,g,d,T,G. SN … (lpxs h g G) (lleq d T).
+                 λh,g,l,T,G. SN … (lpxs h g G) (lleq l T).
 
 interpretation
    "extended strong normalization (local environment) alternative"
-   'SNAlt h g d T G L = (lsxa h g T d G L).
+   'SNAlt h g l T G L = (lsxa h g T l G L).
 
 (* Basic eliminators ********************************************************)
 
-lemma lsxa_ind: ∀h,g,G,T,d. ∀R:predicate lenv.
-                (∀L1. G ⊢ ⬊⬊*[h, g, T, d] L1 →
-                      (â\88\80L2. â¦\83G, L1â¦\84 â\8a¢ â\9e¡*[h, g] L2 â\86\92 (L1 â\8b\95[T, d] L2 → ⊥) → R L2) →
+lemma lsxa_ind: ∀h,g,G,T,l. ∀R:predicate lenv.
+                (∀L1. G ⊢ ⬊⬊*[h, g, T, l] L1 →
+                      (â\88\80L2. â¦\83G, L1â¦\84 â\8a¢ â\9e¡*[h, g] L2 â\86\92 (L1 â\89¡[T, l] L2 → ⊥) → R L2) →
                       R L1
                 ) →
-                ∀L. G ⊢ ⬊⬊*[h, g, T, d] L → R L.
-#h #g #G #T #d #R #H0 #L1 #H elim H -L1
+                ∀L. G ⊢ ⬊⬊*[h, g, T, l] L → R L.
+#h #g #G #T #l #R #H0 #L1 #H elim H -L1
 /5 width=1 by lleq_sym, SN_intro/
 qed-.
 
 (* Basic properties *********************************************************)
 
-lemma lsxa_intro: ∀h,g,G,L1,T,d.
-                  (â\88\80L2. â¦\83G, L1â¦\84 â\8a¢ â\9e¡*[h, g] L2 â\86\92 (L1 â\8b\95[T, d] L2 â\86\92 â\8a¥) â\86\92 G â\8a¢ â¬\8aâ¬\8a*[h, g, T, d] L2) →
-                  G ⊢ ⬊⬊*[h, g, T, d] L1.
+lemma lsxa_intro: ∀h,g,G,L1,T,l.
+                  (â\88\80L2. â¦\83G, L1â¦\84 â\8a¢ â\9e¡*[h, g] L2 â\86\92 (L1 â\89¡[T, l] L2 â\86\92 â\8a¥) â\86\92 G â\8a¢ â¬\8aâ¬\8a*[h, g, T, l] L2) →
+                  G ⊢ ⬊⬊*[h, g, T, l] L1.
 /5 width=1 by lleq_sym, SN_intro/ qed.
 
-fact lsxa_intro_aux: ∀h,g,G,L1,T,d.
-                     (â\88\80L,L2. â¦\83G, Lâ¦\84 â\8a¢ â\9e¡*[h, g] L2 â\86\92 L1 â\8b\95[T, d] L â\86\92 (L1 â\8b\95[T, d] L2 â\86\92 â\8a¥) â\86\92 G â\8a¢ â¬\8aâ¬\8a*[h, g, T, d] L2) →
-                     G ⊢ ⬊⬊*[h, g, T, d] L1.
+fact lsxa_intro_aux: ∀h,g,G,L1,T,l.
+                     (â\88\80L,L2. â¦\83G, Lâ¦\84 â\8a¢ â\9e¡*[h, g] L2 â\86\92 L1 â\89¡[T, l] L â\86\92 (L1 â\89¡[T, l] L2 â\86\92 â\8a¥) â\86\92 G â\8a¢ â¬\8aâ¬\8a*[h, g, T, l] L2) →
+                     G ⊢ ⬊⬊*[h, g, T, l] L1.
 /4 width=3 by lsxa_intro/ qed-.
 
-lemma lsxa_lleq_trans: ∀h,g,T,G,L1,d. G ⊢ ⬊⬊*[h, g, T, d] L1 →
-                       â\88\80L2. L1 â\8b\95[T, d] L2 â\86\92 G â\8a¢ â¬\8aâ¬\8a*[h, g, T, d] L2.
-#h #g #T #G #L1 #d #H @(lsxa_ind … H) -L1
+lemma lsxa_lleq_trans: ∀h,g,T,G,L1,l. G ⊢ ⬊⬊*[h, g, T, l] L1 →
+                       â\88\80L2. L1 â\89¡[T, l] L2 â\86\92 G â\8a¢ â¬\8aâ¬\8a*[h, g, T, l] L2.
+#h #g #T #G #L1 #l #H @(lsxa_ind … H) -L1
 #L1 #_ #IHL1 #L2 #HL12 @lsxa_intro
 #K2 #HLK2 #HnLK2 elim (lleq_lpxs_trans … HLK2 … HL12) -HLK2
 /5 width=4 by lleq_canc_sn, lleq_trans/
 qed-.
 
-lemma lsxa_lpxs_trans: ∀h,g,T,G,L1,d. G ⊢ ⬊⬊*[h, g, T, d] L1 →
-                       ∀L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 → G ⊢ ⬊⬊*[h, g, T, d] L2.
-#h #g #T #G #L1 #d #H @(lsxa_ind … H) -L1 #L1 #HL1 #IHL1 #L2 #HL12
-elim (lleq_dec T L1 L2 d) /3 width=4 by lsxa_lleq_trans/
+lemma lsxa_lpxs_trans: ∀h,g,T,G,L1,l. G ⊢ ⬊⬊*[h, g, T, l] L1 →
+                       ∀L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 → G ⊢ ⬊⬊*[h, g, T, l] L2.
+#h #g #T #G #L1 #l #H @(lsxa_ind … H) -L1 #L1 #HL1 #IHL1 #L2 #HL12
+elim (lleq_dec T L1 L2 l) /3 width=4 by lsxa_lleq_trans/
 qed-.
 
-lemma lsxa_intro_lpx: ∀h,g,G,L1,T,d.
-                      (â\88\80L2. â¦\83G, L1â¦\84 â\8a¢ â\9e¡[h, g] L2 â\86\92 (L1 â\8b\95[T, d] L2 â\86\92 â\8a¥) â\86\92 G â\8a¢ â¬\8aâ¬\8a*[h, g, T, d] L2) →
-                      G ⊢ ⬊⬊*[h, g, T, d] L1.
-#h #g #G #L1 #T #d #IH @lsxa_intro_aux
+lemma lsxa_intro_lpx: ∀h,g,G,L1,T,l.
+                      (â\88\80L2. â¦\83G, L1â¦\84 â\8a¢ â\9e¡[h, g] L2 â\86\92 (L1 â\89¡[T, l] L2 â\86\92 â\8a¥) â\86\92 G â\8a¢ â¬\8aâ¬\8a*[h, g, T, l] L2) →
+                      G ⊢ ⬊⬊*[h, g, T, l] L1.
+#h #g #G #L1 #T #l #IH @lsxa_intro_aux
 #L #L2 #H @(lpxs_ind_dx … H) -L
 [ #H destruct #H elim H //
-| #L0 #L elim (lleq_dec T L1 L d) /3 width=1 by/
+| #L0 #L elim (lleq_dec T L1 L l) /3 width=1 by/
   #HnT #HL0 #HL2 #_ #HT #_ elim (lleq_lpx_trans … HL0 … HT) -L0
   #L0 #HL10 #HL0 @(lsxa_lpxs_trans … HL2) -HL2
   /5 width=3 by lsxa_lleq_trans, lleq_trans/
@@ -80,37 +79,37 @@ qed-.
 
 (* Main properties **********************************************************)
 
-theorem lsx_lsxa: ∀h,g,G,L,T,d. G ⊢ ⬊*[h, g, T, d] L → G ⊢ ⬊⬊*[h, g, T, d] L.
-#h #g #G #L #T #d #H @(lsx_ind … H) -L
+theorem lsx_lsxa: ∀h,g,G,L,T,l. G ⊢ ⬊*[h, g, T, l] L → G ⊢ ⬊⬊*[h, g, T, l] L.
+#h #g #G #L #T #l #H @(lsx_ind … H) -L
 /4 width=1 by lsxa_intro_lpx/
 qed.
 
 (* Main inversion lemmas ****************************************************)
 
-theorem lsxa_inv_lsx: ∀h,g,G,L,T,d. G ⊢ ⬊⬊*[h, g, T, d] L → G ⊢ ⬊*[h, g, T, d] L.
-#h #g #G #L #T #d #H @(lsxa_ind … H) -L
+theorem lsxa_inv_lsx: ∀h,g,G,L,T,l. G ⊢ ⬊⬊*[h, g, T, l] L → G ⊢ ⬊*[h, g, T, l] L.
+#h #g #G #L #T #l #H @(lsxa_ind … H) -L
 /4 width=1 by lsx_intro, lpx_lpxs/
 qed-.
 
 (* Advanced properties ******************************************************)
 
-lemma lsx_intro_alt: ∀h,g,G,L1,T,d.
-                     (â\88\80L2. â¦\83G, L1â¦\84 â\8a¢ â\9e¡*[h, g] L2 â\86\92 (L1 â\8b\95[T, d] L2 â\86\92 â\8a¥) â\86\92 G â\8a¢ â¬\8a*[h, g, T, d] L2) →
-                     G ⊢ ⬊*[h, g, T, d] L1.
+lemma lsx_intro_alt: ∀h,g,G,L1,T,l.
+                     (â\88\80L2. â¦\83G, L1â¦\84 â\8a¢ â\9e¡*[h, g] L2 â\86\92 (L1 â\89¡[T, l] L2 â\86\92 â\8a¥) â\86\92 G â\8a¢ â¬\8a*[h, g, T, l] L2) →
+                     G ⊢ ⬊*[h, g, T, l] L1.
 /6 width=1 by lsxa_inv_lsx, lsx_lsxa, lsxa_intro/ qed.
 
-lemma lsx_lpxs_trans: ∀h,g,G,L1,T,d. G ⊢ ⬊*[h, g, T, d] L1 →
-                      ∀L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 → G ⊢ ⬊*[h, g, T, d] L2.
+lemma lsx_lpxs_trans: ∀h,g,G,L1,T,l. G ⊢ ⬊*[h, g, T, l] L1 →
+                      ∀L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 → G ⊢ ⬊*[h, g, T, l] L2.
 /4 width=3 by lsxa_inv_lsx, lsx_lsxa, lsxa_lpxs_trans/ qed-.
 
 (* Advanced eliminators *****************************************************)
 
-lemma lsx_ind_alt: ∀h,g,G,T,d. ∀R:predicate lenv.
-                   (∀L1. G ⊢ ⬊*[h, g, T, d] L1 →
-                         (â\88\80L2. â¦\83G, L1â¦\84 â\8a¢ â\9e¡*[h, g] L2 â\86\92 (L1 â\8b\95[T, d] L2 → ⊥) → R L2) →
+lemma lsx_ind_alt: ∀h,g,G,T,l. ∀R:predicate lenv.
+                   (∀L1. G ⊢ ⬊*[h, g, T, l] L1 →
+                         (â\88\80L2. â¦\83G, L1â¦\84 â\8a¢ â\9e¡*[h, g] L2 â\86\92 (L1 â\89¡[T, l] L2 → ⊥) → R L2) →
                          R L1
                    ) →
-                   ∀L. G ⊢ ⬊*[h, g, T, d] L → R L.
-#h #g #G #T #d #R #IH #L #H @(lsxa_ind h g G T d … L)
+                   ∀L. G ⊢ ⬊*[h, g, T, l] L → R L.
+#h #g #G #T #l #R #IH #L #H @(lsxa_ind h g G T l … L)
 /4 width=1 by lsxa_inv_lsx, lsx_lsxa/
 qed-.