(* alternative definition of csx *)
definition csxa: ∀h. sd h → relation3 genv lenv term ≝
- λh,g,G,L. SN … (cpxs h g G L) (eq …).
+ λh,o,G,L. SN … (cpxs h o G L) (eq …).
interpretation
"context-sensitive extended strong normalization (term) alternative"
- 'SNAlt h g G L T = (csxa h g G L T).
+ 'SNAlt h o G L T = (csxa h o G L T).
(* Basic eliminators ********************************************************)
-lemma csxa_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
+lemma csxa_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 *********************************************************)
+lemma csx_intro_cpxs: ∀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 cpx_cpxs, csx_intro/ qed.
+
(* Basic_1: was just: sn3_intro *)
-lemma csxa_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 csxa_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.
-fact csxa_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-.
+fact csxa_intro_aux: ∀h,o,G,L,T1. (
+ ∀T,T2. ⦃G, L⦄ ⊢ T ➡*[h, o] T2 → T1 = T → (T1 = T2 → ⊥) → ⦃G, L⦄ ⊢ ⬊⬊*[h, o] T2
+ ) → ⦃G, L⦄ ⊢ ⬊⬊*[h, o] T1.
+/4 width=3 by csxa_intro/ qed-.
(* Basic_1: was just: sn3_pr3_trans (old version) *)
-lemma csxa_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 #G #L #T1 #H elim H -T1 #T1 #HT1 #IHT1 #T2 #HLT12
+lemma csxa_cpxs_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 elim H -T1 #T1 #HT1 #IHT1 #T2 #HLT12
@csxa_intro #T #HLT2 #HT2
-elim (term_eq_dec T1 T2) #HT12
-[ -IHT1 -HLT12 destruct /3 width=1/
-| -HT1 -HT2 /3 width=4/
+elim (eq_term_dec T1 T2) #HT12
+[ -IHT1 -HLT12 destruct /3 width=1 by/
+| -HT1 -HT2 /3 width=4 by/
qed.
(* Basic_1: was just: sn3_pr2_intro (old version) *)
-lemma csxa_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 #G #L #T1 #H
+lemma csxa_intro_cpx: ∀h,o,G,L,T1. (
+ ∀T2. ⦃G, L⦄ ⊢ T1 ➡[h, o] T2 → (T1 = T2 → ⊥) → ⦃G, L⦄ ⊢ ⬊⬊*[h, o] T2
+ ) → ⦃G, L⦄ ⊢ ⬊⬊*[h, o] T1.
+#h #o #G #L #T1 #H
@csxa_intro_aux #T #T2 #H @(cpxs_ind_dx … H) -T
[ -H #H destruct #H
elim H //
| #T0 #T #HLT1 #HLT2 #IHT #HT10 #HT12 destruct
- elim (term_eq_dec T0 T) #HT0
- [ -HLT1 -HLT2 -H /3 width=1/
- | -IHT -HT12 /4 width=3/
+ elim (eq_term_dec T0 T) #HT0
+ [ -HLT1 -HLT2 -H /3 width=1 by/
+ | -IHT -HT12 /4 width=3 by csxa_cpxs_trans/
]
]
qed.
(* Main properties **********************************************************)
-theorem csx_csxa: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ ⬊*[h, g] T → ⦃G, L⦄ ⊢ ⬊⬊*[h, g] T.
-#h #g #G #L #T #H @(csx_ind … H) -T /4 width=1/
+theorem csx_csxa: ∀h,o,G,L,T. ⦃G, L⦄ ⊢ ⬊*[h, o] T → ⦃G, L⦄ ⊢ ⬊⬊*[h, o] T.
+#h #o #G #L #T #H @(csx_ind … H) -T /4 width=1 by csxa_intro_cpx/
qed.
-theorem csxa_csx: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ ⬊⬊*[h, g] T → ⦃G, L⦄ ⊢ ⬊*[h, g] T.
-#h #g #G #L #T #H @(csxa_ind … H) -T /4 width=1/
+theorem csxa_csx: ∀h,o,G,L,T. ⦃G, L⦄ ⊢ ⬊⬊*[h, o] T → ⦃G, L⦄ ⊢ ⬊*[h, o] T.
+#h #o #G #L #T #H @(csxa_ind … H) -T /4 width=1 by cpx_cpxs, csx_intro/
qed.
(* Basic_1: was just: sn3_pr3_trans *)
-lemma csx_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 #G #L #T1 #HT1 #T2 #H @(cpxs_ind … H) -T2 // /2 width=3 by csx_cpx_trans/
+lemma csx_cpxs_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 #HT1 #T2 #H @(cpxs_ind … H) -T2 /2 width=3 by csx_cpx_trans/
qed-.
(* Main eliminators *********************************************************)
-lemma csx_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
+lemma csx_ind_alt: ∀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 @(csxa_ind … (csx_csxa … H)) -T1 #T1 #HT1 #IHT1
-@H0 -H0 /2 width=1/ -HT1 /3 width=1/
+ ∀T. ⦃G, L⦄ ⊢ ⬊*[h, o] T → R T.
+#h #o #G #L #R #H0 #T1 #H @(csxa_ind … (csx_csxa … H)) -T1 /4 width=1 by csxa_csx/
qed-.