partial commit of the components before "conversion"

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

diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/csn.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/csn.ma
index 9a315478497d0a6d244a78974271292776d7021c..92719458e6e6863deca27de332c0a07bf387fe66 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/csn.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/csn.ma
@@ -12,93 +12,99 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "basic_2/reduction/cnf.ma".
+include "basic_2/reduction/cnx.ma".
 
-(* CONTEXT-SENSITIVE STRONGLY NORMALIZING TERMS *****************************)
+(* CONTEXT-SENSITIVE EXTENDED STRONGLY NORMALIZING TERMS ********************)
 
-definition csn: lenv → predicate term ≝ λL. SN … (cpr L) (eq …).
+definition csn: ∀h. sd h → lenv → predicate term ≝
+                λh,g,L. SN … (cpx h g L) (eq …).
 
 interpretation
-   "context-sensitive strong normalization (term)"
-   'SN L T = (csn L T).
+   "context-sensitive extended strong normalization (term)"
+   'SN h g L T = (csn h g L T).
 
 (* Basic eliminators ********************************************************)
 
-lemma csn_ind: ∀L. ∀R:predicate term.
-               (∀T1. L ⊢ ⬊* T1 →
-                     (∀T2. L ⊢ T1 ➡ T2 → (T1 = T2 → ⊥) → R T2) →
+lemma csn_ind: ∀h,g,L. ∀R:predicate term.
+               (∀T1. ⦃h, L⦄ ⊢ ⬊*[g] T1 →
+                     (∀T2. ⦃h, L⦄ ⊢ T1 ➡[g] T2 → (T1 = T2 → ⊥) → R T2) →
                      R T1
                ) →
-               ∀T. L ⊢ ⬊* T → R T.
-#L #R #H0 #T1 #H elim H -T1 #T1 #HT1 #IHT1
+               ∀T. ⦃h, L⦄ ⊢ ⬊*[g] T → R T.
+#h #g #L #R #H0 #T1 #H elim H -T1 #T1 #HT1 #IHT1
 @H0 -H0 /3 width=1/ -IHT1 /4 width=1/
 qed-.
 
 (* Basic properties *********************************************************)
 
-(* Basic_1: was: sn3_pr2_intro *)
-lemma csn_intro: ∀L,T1.
-                 (∀T2. L ⊢ T1 ➡ T2 → (T1 = T2 → ⊥) → L ⊢ ⬊* T2) → L ⊢ ⬊* T1.
+(* Basic_1: was just: sn3_pr2_intro *)
+lemma csn_intro: ∀h,g,L,T1.
+                 (∀T2. ⦃h, L⦄ ⊢ T1 ➡[g] T2 → (T1 = T2 → ⊥) → ⦃h, L⦄ ⊢ ⬊*[g] T2) →
+                 ⦃h, L⦄ ⊢ ⬊*[g] T1.
 /4 width=1/ qed.
 
-(* Basic_1: was: sn3_nf2 *)
-lemma csn_cnf: ∀L,T. L ⊢ 𝐍⦃T⦄ → L ⊢ ⬊* T.
+(* Basic_1: was just: sn3_nf2 *)
+lemma cnx_csn: ∀h,g,L,T. ⦃h, L⦄ ⊢ 𝐍[g]⦃T⦄ → ⦃h, L⦄ ⊢ ⬊*[g] T.
 /2 width=1/ qed.
 
-lemma csn_cpr_trans: ∀L,T1. L ⊢ ⬊* T1 → ∀T2. L ⊢ T1 ➡ T2 → L ⊢ ⬊* T2.
-#L #T1 #H elim H -T1 #T1 #HT1 #IHT1 #T2 #HLT12
+lemma csn_cpx_trans: ∀h,g,L,T1. ⦃h, L⦄ ⊢ ⬊*[g] T1 →
+                     ∀T2. ⦃h, L⦄ ⊢ T1 ➡[g] T2 → ⦃h, L⦄ ⊢ ⬊*[g] T2.
+#h #g #L #T1 #H elim H -T1 #T1 #HT1 #IHT1 #T2 #HLT12
 @csn_intro #T #HLT2 #HT2
 elim (term_eq_dec T1 T2) #HT12
 [ -IHT1 -HLT12 destruct /3 width=1/
 | -HT1 -HT2 /3 width=4/
 qed-.
 
-(* Basic_1: was: sn3_cast *)
-lemma csn_cast: ∀L,W. L ⊢ ⬊* W → ∀T. L ⊢ ⬊* T → L ⊢ ⬊* ⓝW. T.
-#L #W #HW elim HW -W #W #_ #IHW #T #HT @(csn_ind … HT) -T #T #HT #IHT
+(* Basic_1: was just: sn3_cast *)
+lemma csn_cast: ∀h,g,L,W. ⦃h, L⦄ ⊢ ⬊*[g] W →
+                ∀T. ⦃h, L⦄ ⊢ ⬊*[g] T → ⦃h, L⦄ ⊢ ⬊*[g] ⓝW.T.
+#h #g #L #W #HW elim HW -W #W #_ #IHW #T #HT @(csn_ind … HT) -T #T #HT #IHT
 @csn_intro #X #H1 #H2
-elim (cpr_inv_cast1 … H1) -H1
+elim (cpx_inv_cast1 … H1) -H1
 [ * #W0 #T0 #HLW0 #HLT0 #H destruct
   elim (eq_false_inv_tpair_sn … H2) -H2
-  [ /3 width=3 by csn_cpr_trans/
+  [ /3 width=3 by csn_cpx_trans/
   | -HLW0 * #H destruct /3 width=1/
   ]
-| /3 width=3 by csn_cpr_trans/
+| /3 width=3 by csn_cpx_trans/
 ]
 qed.
 
 (* Basic forward lemmas *****************************************************)
 
-fact csn_fwd_pair_sn_aux: ∀L,U. L ⊢ ⬊* U → ∀I,V,T. U = ②{I} V. T → L ⊢ ⬊* V.
-#L #U #H elim H -H #U0 #_ #IH #I #V #T #H destruct
+fact csn_fwd_pair_sn_aux: ∀h,g,L,U. ⦃h, L⦄ ⊢ ⬊*[g] U →
+                          ∀I,V,T. U = ②{I}V.T → ⦃h, L⦄ ⊢ ⬊*[g] V.
+#h #g #L #U #H elim H -H #U0 #_ #IH #I #V #T #H destruct
 @csn_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=1/ -HLV2 #H destruct /2 width=1/
 qed-.
 
-(* Basic_1: was: sn3_gen_head *)
-lemma csn_fwd_pair_sn: ∀I,L,V,T. L ⊢ ⬊* ②{I} V. T → L ⊢ ⬊* V.
+(* Basic_1: was just: sn3_gen_head *)
+lemma csn_fwd_pair_sn: ∀h,g,I,L,V,T. ⦃h, L⦄ ⊢ ⬊*[g] ②{I}V.T → ⦃h, L⦄ ⊢ ⬊*[g] V.
 /2 width=5 by csn_fwd_pair_sn_aux/ qed-.
 
-fact csn_fwd_bind_dx_aux: ∀L,U. L ⊢ ⬊* U →
-                          ∀a,I,V,T. U = ⓑ{a,I} V. T → L. ⓑ{I} V ⊢ ⬊* T.
-#L #U #H elim H -H #U0 #_ #IH #a #I #V #T #H destruct
+fact csn_fwd_bind_dx_aux: ∀h,g,L,U. ⦃h, L⦄ ⊢ ⬊*[g] U →
+                          ∀a,I,V,T. U = ⓑ{a,I}V.T → ⦃h, L.ⓑ{I}V⦄ ⊢ ⬊*[g] T.
+#h #g #L #U #H elim H -H #U0 #_ #IH #a #I #V #T #H destruct
 @csn_intro #T2 #HLT2 #HT2
 @(IH (ⓑ{a,I} V. T2)) -IH // /2 width=1/ -HLT2 #H destruct /2 width=1/
 qed-.
 
-(* Basic_1: was: sn3_gen_bind *)
-lemma csn_fwd_bind_dx: ∀a,I,L,V,T. L ⊢ ⬊* ⓑ{a,I} V. T → L. ⓑ{I} V ⊢ ⬊* T.
+(* Basic_1: was just: sn3_gen_bind *)
+lemma csn_fwd_bind_dx: ∀h,g,a,I,L,V,T. ⦃h, L⦄ ⊢ ⬊*[g] ⓑ{a,I}V.T → ⦃h, L.ⓑ{I}V⦄ ⊢ ⬊*[g] T.
 /2 width=4 by csn_fwd_bind_dx_aux/ qed-.
 
-fact csn_fwd_flat_dx_aux: ∀L,U. L ⊢ ⬊* U → ∀I,V,T. U = ⓕ{I} V. T → L ⊢ ⬊* T.
-#L #U #H elim H -H #U0 #_ #IH #I #V #T #H destruct
+fact csn_fwd_flat_dx_aux: ∀h,g,L,U. ⦃h, L⦄ ⊢ ⬊*[g] U →
+                          ∀I,V,T. U = ⓕ{I}V.T → ⦃h, L⦄ ⊢ ⬊*[g] T.
+#h #g #L #U #H elim H -H #U0 #_ #IH #I #V #T #H destruct
 @csn_intro #T2 #HLT2 #HT2
-@(IH (ⓕ{I} V. T2)) -IH // /2 width=1/ -HLT2 #H destruct /2 width=1/
-qed.
+@(IH (ⓕ{I}V.T2)) -IH // /2 width=1/ -HLT2 #H destruct /2 width=1/
+qed-.
 
-(* Basic_1: was: sn3_gen_flat *)
-lemma csn_fwd_flat_dx: ∀I,L,V,T. L ⊢ ⬊* ⓕ{I} V. T → L ⊢ ⬊* T.
-/2 width=5/ qed-.
+(* Basic_1: was just: sn3_gen_flat *)
+lemma csn_fwd_flat_dx: ∀h,g,I,L,V,T. ⦃h, L⦄ ⊢ ⬊*[g] ⓕ{I}V.T → ⦃h, L⦄ ⊢ ⬊*[g] T.
+/2 width=5 by csn_fwd_flat_dx_aux/ qed-.
 
 (* Basic_1: removed theorems 14:
             sn3_cdelta