(* *)
(**************************************************************************)
-include "basic_2/notation/relations/snalt_4.ma".
+include "basic_2/notation/relations/snalt_5.ma".
include "basic_2/computation/cpxs.ma".
include "basic_2/computation/csn.ma".
(* CONTEXT-SENSITIVE EXTENDED STRONGLY NORMALIZING TERMS ********************)
(* alternative definition of csn *)
-definition csna: ∀h. sd h → lenv → predicate term ≝
- λh,g,L. SN … (cpxs h g L) (eq …).
+definition csna: ∀h. sd h → relation3 genv lenv term ≝
+ λh,g,G,L. SN … (cpxs h g G L) (eq …).
interpretation
"context-sensitive extended strong normalization (term) alternative"
- 'SNAlt h g L T = (csna h g L T).
+ 'SNAlt h g G L T = (csna h g G L T).
(* Basic eliminators ********************************************************)
-lemma csna_ind: ∀h,g,L. ∀R:predicate term.
+lemma csna_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) → R T1
) →
∀T. ⦃G, L⦄ ⊢ ⬊⬊*[h, g] T → R T.
-#h #g #L #R #H0 #T1 #H elim H -T1 #T1 #HT1 #IHT1
+#h #g #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 just: sn3_intro *)
-lemma csna_intro: ∀h,g,L,T1.
+lemma csna_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.
-fact csna_intro_aux: ∀h,g,L,T1. (
+fact csna_intro_aux: ∀h,g,G,L,T1. (
∀T,T2. ⦃G, L⦄ ⊢ T ➡*[h, g] T2 → T1 = T → (T1 = T2 → ⊥) → ⦃G, L⦄ ⊢ ⬊⬊*[h, g] T2
) → ⦃G, L⦄ ⊢ ⬊⬊*[h, g] T1.
/4 width=3/ qed-.
(* Basic_1: was just: sn3_pr3_trans (old version) *)
-lemma csna_cpxs_trans: ∀h,g,L,T1. ⦃G, L⦄ ⊢ ⬊⬊*[h, g] T1 →
+lemma csna_cpxs_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 #L #T1 #H elim H -T1 #T1 #HT1 #IHT1 #T2 #HLT12
+#h #g #G #L #T1 #H elim H -T1 #T1 #HT1 #IHT1 #T2 #HLT12
@csna_intro #T #HLT2 #HT2
elim (term_eq_dec T1 T2) #HT12
[ -IHT1 -HLT12 destruct /3 width=1/
qed.
(* Basic_1: was just: sn3_pr2_intro (old version) *)
-lemma csna_intro_cpx: ∀h,g,L,T1. (
+lemma csna_intro_cpx: ∀h,g,G,L,T1. (
∀T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 → (T1 = T2 → ⊥) → ⦃G, L⦄ ⊢ ⬊⬊*[h, g] T2
) → ⦃G, L⦄ ⊢ ⬊⬊*[h, g] T1.
-#h #g #L #T1 #H
+#h #g #G #L #T1 #H
@csna_intro_aux #T #T2 #H @(cpxs_ind_dx … H) -T
[ -H #H destruct #H
elim H //
(* Main properties **********************************************************)
-theorem csn_csna: ∀h,g,L,T. ⦃G, L⦄ ⊢ ⬊*[h, g] T → ⦃G, L⦄ ⊢ ⬊⬊*[h, g] T.
-#h #g #L #T #H @(csn_ind … H) -T /4 width=1/
+theorem csn_csna: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ ⬊*[h, g] T → ⦃G, L⦄ ⊢ ⬊⬊*[h, g] T.
+#h #g #G #L #T #H @(csn_ind … H) -T /4 width=1/
qed.
-theorem csna_csn: ∀h,g,L,T. ⦃G, L⦄ ⊢ ⬊⬊*[h, g] T → ⦃G, L⦄ ⊢ ⬊*[h, g] T.
-#h #g #L #T #H @(csna_ind … H) -T /4 width=1/
+theorem csna_csn: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ ⬊⬊*[h, g] T → ⦃G, L⦄ ⊢ ⬊*[h, g] T.
+#h #g #G #L #T #H @(csna_ind … H) -T /4 width=1/
qed.
(* Basic_1: was just: sn3_pr3_trans *)
-lemma csn_cpxs_trans: ∀h,g,L,T1. ⦃G, L⦄ ⊢ ⬊*[h, g] T1 →
+lemma csn_cpxs_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 #L #T1 #HT1 #T2 #H @(cpxs_ind … H) -T2 // /2 width=3 by csn_cpx_trans/
+#h #g #G #L #T1 #HT1 #T2 #H @(cpxs_ind … H) -T2 // /2 width=3 by csn_cpx_trans/
qed-.
(* Main eliminators *********************************************************)
-lemma csn_ind_alt: ∀h,g,L. ∀R:predicate term.
+lemma csn_ind_alt: ∀h,g,G,L. ∀R:predicate term.
(∀T1. ⦃G, L⦄ ⊢ ⬊*[h, g] T1 →
(∀T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 → (T1 = T2 → ⊥) → R T2) → R T1
) →
∀T. ⦃G, L⦄ ⊢ ⬊*[h, g] T → R T.
-#h #g #L #R #H0 #T1 #H @(csna_ind … (csn_csna … H)) -T1 #T1 #HT1 #IHT1
+#h #g #G #L #R #H0 #T1 #H @(csna_ind … (csn_csna … H)) -T1 #T1 #HT1 #IHT1
@H0 -H0 /2 width=1/ -HT1 /3 width=1/
qed-.