- commit of the "s_computation" component ...

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

diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/csx.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/csx.ma
index 61174c0ff807dc95179295dd5f898bab727d85ba..0def9956806acaeb92766c53240ae6c147c4b4c7 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/csx.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/csx.ma
@@ -18,65 +18,62 @@ include "basic_2/reduction/cnx.ma".
 (* CONTEXT-SENSITIVE EXTENDED STRONGLY NORMALIZING TERMS ********************)
 
 definition csx: ∀h. sd h → relation3 genv lenv term ≝
-                λh,g,G,L. SN … (cpx h g G L) (eq …).
+                λh,o,G,L. SN … (cpx h o G L) (eq …).
 
 interpretation
    "context-sensitive extended strong normalization (term)"
-   'SN h g G L T = (csx h g G L T).
+   'SN h o G L T = (csx h o G L T).
 
 (* Basic eliminators ********************************************************)
 
-lemma csx_ind: ∀h,g,G,L. ∀R:predicate term.
-               (∀T1. ⦃G, L⦄ ⊢ ⬊*[h, g] T1 →
-                     (∀T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 → (T1 = T2 → ⊥) → R T2) →
+lemma csx_ind: ∀h,o,G,L. ∀R:predicate term.
+               (∀T1. ⦃G, L⦄ ⊢ ⬊*[h, o] T1 →
+                     (∀T2. ⦃G, L⦄ ⊢ T1 ➡[h, o] T2 → (T1 = T2 → ⊥) → R T2) →
                      R T1
                ) →
-               ∀T. ⦃G, L⦄ ⊢ ⬊*[h, g] T → R T.
-#h #g #G #L #R #H0 #T1 #H elim H -T1 #T1 #HT1 #IHT1
-@H0 -H0 /3 width=1/ -IHT1 /4 width=1/
+               ∀T. ⦃G, L⦄ ⊢ ⬊*[h, o] T → R T.
+#h #o #G #L #R #H0 #T1 #H elim H -T1
+/5 width=1 by SN_intro/
 qed-.
 
 (* Basic properties *********************************************************)
 
 (* Basic_1: was just: sn3_pr2_intro *)
-lemma csx_intro: ∀h,g,G,L,T1.
-                 (∀T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 → (T1 = T2 → ⊥) → ⦃G, L⦄ ⊢ ⬊*[h, g] T2) →
-                 ⦃G, L⦄ ⊢ ⬊*[h, g] T1.
-/4 width=1/ qed.
-
-lemma csx_cpx_trans: ∀h,g,G,L,T1. ⦃G, L⦄ ⊢ ⬊*[h, g] T1 →
-                     ∀T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 → ⦃G, L⦄ ⊢ ⬊*[h, g] T2.
-#h #g #G #L #T1 #H elim H -T1 #T1 #HT1 #IHT1 #T2 #HLT12
-@csx_intro #T #HLT2 #HT2
-elim (term_eq_dec T1 T2) #HT12
-[ -IHT1 -HLT12 destruct /3 width=1/
-| -HT1 -HT2 /3 width=4/
+lemma csx_intro: ∀h,o,G,L,T1.
+                 (∀T2. ⦃G, L⦄ ⊢ T1 ➡[h, o] T2 → (T1 = T2 → ⊥) → ⦃G, L⦄ ⊢ ⬊*[h, o] T2) →
+                 ⦃G, L⦄ ⊢ ⬊*[h, o] T1.
+/4 width=1 by SN_intro/ qed.
+
+lemma csx_cpx_trans: ∀h,o,G,L,T1. ⦃G, L⦄ ⊢ ⬊*[h, o] T1 →
+                     ∀T2. ⦃G, L⦄ ⊢ T1 ➡[h, o] T2 → ⦃G, L⦄ ⊢ ⬊*[h, o] T2.
+#h #o #G #L #T1 #H @(csx_ind … H) -T1 #T1 #HT1 #IHT1 #T2 #HLT12
+elim (eq_term_dec T1 T2) #HT12 destruct /3 width=4 by/
 qed-.
 
 (* Basic_1: was just: sn3_nf2 *)
-lemma cnx_csx: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ 𝐍[h, g]⦃T⦄ → ⦃G, L⦄ ⊢ ⬊*[h, g] T.
-/2 width=1/ qed.
-
-lemma cnx_sort: ∀h,g,G,L,k. ⦃G, L⦄ ⊢ ⬊*[h, g] ⋆k.
-#h #g #G #L #k elim (deg_total h g k)
-#l generalize in match k; -k @(nat_ind_plus … l) -l /3 width=1/
-#l #IHl #k #Hkl lapply (deg_next_SO … Hkl) -Hkl
-#Hkl @csx_intro #X #H #HX elim (cpx_inv_sort1 … H) -H
+lemma cnx_csx: ∀h,o,G,L,T. ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃T⦄ → ⦃G, L⦄ ⊢ ⬊*[h, o] T.
+/2 width=1 by NF_to_SN/ qed.
+
+lemma csx_sort: ∀h,o,G,L,s. ⦃G, L⦄ ⊢ ⬊*[h, o] ⋆s.
+#h #o #G #L #s elim (deg_total h o s)
+#d generalize in match s; -s @(nat_ind_plus … d) -d /3 width=6 by cnx_csx, cnx_sort/
+#d #IHd #s #Hkd lapply (deg_next_SO … Hkd) -Hkd
+#Hkd @csx_intro #X #H #HX elim (cpx_inv_sort1 … H) -H
 [ #H destruct elim HX //
-| -HX * #l0 #_ #H destruct -l0 /2 width=1/
+| -HX * #d0 #_ #H destruct -d0 /2 width=1 by/
 ]
 qed.
 
 (* Basic_1: was just: sn3_cast *)
-lemma csx_cast: ∀h,g,G,L,W. ⦃G, L⦄ ⊢ ⬊*[h, g] W →
-                ∀T. ⦃G, L⦄ ⊢ ⬊*[h, g] T → ⦃G, L⦄ ⊢ ⬊*[h, g] ⓝW.T.
-#h #g #G #L #W #HW @(csx_ind … HW) -W #W #HW #IHW #T #HT @(csx_ind … HT) -T #T #HT #IHT
+lemma csx_cast: ∀h,o,G,L,W. ⦃G, L⦄ ⊢ ⬊*[h, o] W →
+                ∀T. ⦃G, L⦄ ⊢ ⬊*[h, o] T → ⦃G, L⦄ ⊢ ⬊*[h, o] ⓝW.T.
+#h #o #G #L #W #HW @(csx_ind … HW) -W #W #HW #IHW #T #HT @(csx_ind … HT) -T #T #HT #IHT
 @csx_intro #X #H1 #H2
 elim (cpx_inv_cast1 … H1) -H1
 [ * #W0 #T0 #HLW0 #HLT0 #H destruct
   elim (eq_false_inv_tpair_sn … H2) -H2
   [ /3 width=3 by csx_cpx_trans/
-  | -HLW0 * #H destruct /3 width=1/
+  | -HLW0 * #H destruct /3 width=1 by/
   ]
 |2,3: /3 width=3 by csx_cpx_trans/
 ]
@@ -84,39 +81,50 @@ qed.
 
 (* Basic forward lemmas *****************************************************)
 
-fact csx_fwd_pair_sn_aux: ∀h,g,G,L,U. ⦃G, L⦄ ⊢ ⬊*[h, g] U →
-                          ∀I,V,T. U = ②{I}V.T → ⦃G, L⦄ ⊢ ⬊*[h, g] V.
-#h #g #G #L #U #H elim H -H #U0 #_ #IH #I #V #T #H destruct
+fact csx_fwd_pair_sn_aux: ∀h,o,G,L,U. ⦃G, L⦄ ⊢ ⬊*[h, o] U →
+                          ∀I,V,T. U = ②{I}V.T → ⦃G, L⦄ ⊢ ⬊*[h, o] V.
+#h #o #G #L #U #H elim H -H #U0 #_ #IH #I #V #T #H destruct
 @csx_intro #V2 #HLV2 #HV2
-@(IH (②{I}V2.T)) -IH // /2 width=1/ -HLV2 #H destruct /2 width=1/
+@(IH (②{I}V2.T)) -IH /2 width=3 by cpx_pair_sn/ -HLV2
+#H destruct /2 width=1 by/
 qed-.
 
 (* Basic_1: was just: sn3_gen_head *)
-lemma csx_fwd_pair_sn: ∀h,g,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬊*[h, g] ②{I}V.T → ⦃G, L⦄ ⊢ ⬊*[h, g] V.
+lemma csx_fwd_pair_sn: ∀h,o,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬊*[h, o] ②{I}V.T → ⦃G, L⦄ ⊢ ⬊*[h, o] V.
 /2 width=5 by csx_fwd_pair_sn_aux/ qed-.
 
-fact csx_fwd_bind_dx_aux: ∀h,g,G,L,U. ⦃G, L⦄ ⊢ ⬊*[h, g] U →
-                          ∀a,I,V,T. U = ⓑ{a,I}V.T → ⦃G, L.ⓑ{I}V⦄ ⊢ ⬊*[h, g] T.
-#h #g #G #L #U #H elim H -H #U0 #_ #IH #a #I #V #T #H destruct
+fact csx_fwd_bind_dx_aux: ∀h,o,G,L,U. ⦃G, L⦄ ⊢ ⬊*[h, o] U →
+                          ∀a,I,V,T. U = ⓑ{a,I}V.T → ⦃G, L.ⓑ{I}V⦄ ⊢ ⬊*[h, o] T.
+#h #o #G #L #U #H elim H -H #U0 #_ #IH #a #I #V #T #H destruct
 @csx_intro #T2 #HLT2 #HT2
-@(IH (ⓑ{a,I}V.T2)) -IH // /2 width=1/ -HLT2 #H destruct /2 width=1/
+@(IH (ⓑ{a,I}V.T2)) -IH /2 width=3 by cpx_bind/ -HLT2
+#H destruct /2 width=1 by/
 qed-.
 
 (* Basic_1: was just: sn3_gen_bind *)
-lemma csx_fwd_bind_dx: ∀h,g,a,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬊*[h, g] ⓑ{a,I}V.T → ⦃G, L.ⓑ{I}V⦄ ⊢ ⬊*[h, g] T.
+lemma csx_fwd_bind_dx: ∀h,o,a,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬊*[h, o] ⓑ{a,I}V.T → ⦃G, L.ⓑ{I}V⦄ ⊢ ⬊*[h, o] T.
 /2 width=4 by csx_fwd_bind_dx_aux/ qed-.
 
-fact csx_fwd_flat_dx_aux: ∀h,g,G,L,U. ⦃G, L⦄ ⊢ ⬊*[h, g] U →
-                          ∀I,V,T. U = ⓕ{I}V.T → ⦃G, L⦄ ⊢ ⬊*[h, g] T.
-#h #g #G #L #U #H elim H -H #U0 #_ #IH #I #V #T #H destruct
+fact csx_fwd_flat_dx_aux: ∀h,o,G,L,U. ⦃G, L⦄ ⊢ ⬊*[h, o] U →
+                          ∀I,V,T. U = ⓕ{I}V.T → ⦃G, L⦄ ⊢ ⬊*[h, o] T.
+#h #o #G #L #U #H elim H -H #U0 #_ #IH #I #V #T #H destruct
 @csx_intro #T2 #HLT2 #HT2
-@(IH (ⓕ{I}V.T2)) -IH // /2 width=1/ -HLT2 #H destruct /2 width=1/
+@(IH (ⓕ{I}V.T2)) -IH /2 width=3 by cpx_flat/ -HLT2
+#H destruct /2 width=1 by/
 qed-.
 
 (* Basic_1: was just: sn3_gen_flat *)
-lemma csx_fwd_flat_dx: ∀h,g,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬊*[h, g] ⓕ{I}V.T → ⦃G, L⦄ ⊢ ⬊*[h, g] T.
+lemma csx_fwd_flat_dx: ∀h,o,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬊*[h, o] ⓕ{I}V.T → ⦃G, L⦄ ⊢ ⬊*[h, o] T.
 /2 width=5 by csx_fwd_flat_dx_aux/ qed-.
 
+lemma csx_fwd_bind: ∀h,o,a,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬊*[h, o] ⓑ{a,I}V.T →
+                    ⦃G, L⦄ ⊢ ⬊*[h, o] V ∧ ⦃G, L.ⓑ{I}V⦄ ⊢ ⬊*[h, o] T.
+/3 width=3 by csx_fwd_pair_sn, csx_fwd_bind_dx, conj/ qed-.
+
+lemma csx_fwd_flat: ∀h,o,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬊*[h, o] ⓕ{I}V.T →
+                    ⦃G, L⦄ ⊢ ⬊*[h, o] V ∧ ⦃G, L⦄ ⊢ ⬊*[h, o] T.
+/3 width=3 by csx_fwd_pair_sn, csx_fwd_flat_dx, conj/ qed-.
+
 (* Basic_1: removed theorems 14:
             sn3_cdelta
             sn3_gen_cflat sn3_cflat sn3_cpr3_trans sn3_shift sn3_change