(* SN EXTENDED STRONGLY CONORMALIZING LOCAL ENVIRONMENTS ********************)
-inductive lcosx (h) (g) (G): relation2 ynat lenv ≝
-| lcosx_sort: ∀l. lcosx h g G l (⋆)
-| lcosx_skip: ∀I,L,T. lcosx h g G 0 L → lcosx h g G 0 (L.ⓑ{I}T)
-| lcosx_pair: ∀I,L,T,l. G ⊢ ⬊*[h, g, T, l] L →
- lcosx h g G l L → lcosx h g G (⫯l) (L.ⓑ{I}T)
+inductive lcosx (h) (o) (G): relation2 ynat lenv ≝
+| lcosx_sort: ∀l. lcosx h o G l (⋆)
+| lcosx_skip: ∀I,L,T. lcosx h o G 0 L → lcosx h o G 0 (L.ⓑ{I}T)
+| lcosx_pair: ∀I,L,T,l. G ⊢ ⬊*[h, o, T, l] L →
+ lcosx h o G l L → lcosx h o G (⫯l) (L.ⓑ{I}T)
.
interpretation
"sn extended strong conormalization (local environment)"
- 'CoSN h g l G L = (lcosx h g G l L).
+ 'CoSN h o l G L = (lcosx h o G l L).
(* Basic properties *********************************************************)
-lemma lcosx_O: ∀h,g,G,L. G ⊢ ~⬊*[h, g, 0] L.
-#h #g #G #L elim L /2 width=1 by lcosx_skip/
+lemma lcosx_O: ∀h,o,G,L. G ⊢ ~⬊*[h, o, 0] L.
+#h #o #G #L elim L /2 width=1 by lcosx_skip/
qed.
-lemma lcosx_drop_trans_lt: ∀h,g,G,L,l. G ⊢ ~⬊*[h, g, l] L →
+lemma lcosx_drop_trans_lt: ∀h,o,G,L,l. G ⊢ ~⬊*[h, o, l] L →
∀I,K,V,i. ⬇[i] L ≡ K.ⓑ{I}V → i < l →
- G ⊢ ~⬊*[h, g, ⫰(l-i)] K ∧ G ⊢ ⬊*[h, g, V, ⫰(l-i)] K.
-#h #g #G #L #l #H elim H -L -l
+ G ⊢ ~⬊*[h, o, ⫰(l-i)] K ∧ G ⊢ ⬊*[h, o, V, ⫰(l-i)] K.
+#h #o #G #L #l #H elim H -L -l
[ #l #J #K #V #i #H elim (drop_inv_atom1 … H) -H #H destruct
| #I #L #T #_ #_ #J #K #V #i #_ #H elim (ylt_yle_false … H) -H //
| #I #L #T #l #HT #HL #IHL #J #K #V #i #H #Hil
(* Basic inversion lemmas ***************************************************)
-fact lcosx_inv_succ_aux: ∀h,g,G,L,x. G ⊢ ~⬊*[h, g, x] L → ∀l. x = ⫯l →
+fact lcosx_inv_succ_aux: ∀h,o,G,L,x. G ⊢ ~⬊*[h, o, x] L → ∀l. x = ⫯l →
L = ⋆ ∨
- ∃∃I,K,V. L = K.ⓑ{I}V & G ⊢ ~⬊*[h, g, l] K &
- G ⊢ ⬊*[h, g, V, l] K.
-#h #g #G #L #l * -L -l /2 width=1 by or_introl/
+ ∃∃I,K,V. L = K.ⓑ{I}V & G ⊢ ~⬊*[h, o, l] K &
+ G ⊢ ⬊*[h, o, V, l] K.
+#h #o #G #L #l * -L -l /2 width=1 by or_introl/
[ #I #L #T #_ #x #H elim (ysucc_inv_O_sn … H)
| #I #L #T #l #HT #HL #x #H <(ysucc_inv_inj … H) -x
/3 width=6 by ex3_3_intro, or_intror/
]
qed-.
-lemma lcosx_inv_succ: ∀h,g,G,L,l. G ⊢ ~⬊*[h, g, ⫯l] L → L = ⋆ ∨
- ∃∃I,K,V. L = K.ⓑ{I}V & G ⊢ ~⬊*[h, g, l] K &
- G ⊢ ⬊*[h, g, V, l] K.
+lemma lcosx_inv_succ: ∀h,o,G,L,l. G ⊢ ~⬊*[h, o, ⫯l] L → L = ⋆ ∨
+ ∃∃I,K,V. L = K.ⓑ{I}V & G ⊢ ~⬊*[h, o, l] K &
+ G ⊢ ⬊*[h, o, V, l] K.
/2 width=3 by lcosx_inv_succ_aux/ qed-.
-lemma lcosx_inv_pair: ∀h,g,I,G,L,T,l. G ⊢ ~⬊*[h, g, ⫯l] L.ⓑ{I}T →
- G ⊢ ~⬊*[h, g, l] L ∧ G ⊢ ⬊*[h, g, T, l] L.
-#h #g #I #G #L #T #l #H elim (lcosx_inv_succ … H) -H
+lemma lcosx_inv_pair: ∀h,o,I,G,L,T,l. G ⊢ ~⬊*[h, o, ⫯l] L.ⓑ{I}T →
+ G ⊢ ~⬊*[h, o, l] L ∧ G ⊢ ⬊*[h, o, T, l] L.
+#h #o #I #G #L #T #l #H elim (lcosx_inv_succ … H) -H
[ #H destruct
| * #Z #Y #X #H destruct /2 width=1 by conj/
]