(* *)
(**************************************************************************)
-include "basic_2/notation/relations/lazycosn_5.ma".
+include "basic_2/notation/relations/cosn_5.ma".
include "basic_2/computation/lsx.ma".
(* SN EXTENDED STRONGLY CONORMALIZING LOCAL ENVIRONMENTS ********************)
inductive lcosx (h) (g) (G): relation2 ynat lenv ≝
| lcosx_sort: ∀d. lcosx h g G d (⋆)
| lcosx_skip: ∀I,L,T. lcosx h g G 0 L → lcosx h g G 0 (L.ⓑ{I}T)
-| lcosx_pair: â\88\80I,L,T,d. G â\8a¢ â\8b\95â¬\8a*[h, g, T, d] L â\86\92
+| lcosx_pair: ∀I,L,T,d. G ⊢ ⬊*[h, g, T, d] L →
lcosx h g G d L → lcosx h g G (⫯d) (L.ⓑ{I}T)
.
interpretation
"sn extended strong conormalization (local environment)"
- 'LazyCoSN h g d G L = (lcosx h g G d L).
+ 'CoSN h g d G L = (lcosx h g G d L).
(* Basic properties *********************************************************)
-lemma lcosx_O: ∀h,g,G,L. G ⊢ ⧤⬊*[h, g, 0] L.
+lemma lcosx_O: ∀h,g,G,L. G ⊢ ~⬊*[h, g, 0] L.
#h #g #G #L elim L /2 width=1 by lcosx_skip/
qed.
-lemma lcosx_ldrop_trans_lt: ∀h,g,G,L,d. G ⊢ ⧤⬊*[h, g, d] L →
- â\88\80I,K,V,i. â\87©[i] L ≡ K.ⓑ{I}V → i < d →
- G ⊢ ⧤⬊*[h, g, ⫰(d-i)] K ∧ G ⊢ ⋕⬊*[h, g, V, ⫰(d-i)] K.
+lemma lcosx_drop_trans_lt: ∀h,g,G,L,d. G ⊢ ~⬊*[h, g, d] L →
+ â\88\80I,K,V,i. â¬\87[i] L ≡ K.ⓑ{I}V → i < d →
+ G ⊢ ~⬊*[h, g, ⫰(d-i)] K ∧ G ⊢ ⬊*[h, g, V, ⫰(d-i)] K.
#h #g #G #L #d #H elim H -L -d
-[ #d #J #K #V #i #H elim (ldrop_inv_atom1 … H) -H #H destruct
+[ #d #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 #d #HT #HL #IHL #J #K #V #i #H #Hid
- elim (ldrop_inv_O1_pair1 … H) -H * #Hi #HLK destruct
+ elim (drop_inv_O1_pair1 … H) -H * #Hi #HLK destruct
[ >ypred_succ /2 width=1 by conj/
| lapply (ylt_pred … Hid ?) -Hid /2 width=1 by ylt_inj/ >ypred_succ #Hid
elim (IHL … HLK ?) -IHL -HLK <yminus_inj >yminus_SO2 //
(* Basic inversion lemmas ***************************************************)
-fact lcosx_inv_succ_aux: ∀h,g,G,L,x. G ⊢ ⧤⬊*[h, g, x] L → ∀d. x = ⫯d →
+fact lcosx_inv_succ_aux: ∀h,g,G,L,x. G ⊢ ~⬊*[h, g, x] L → ∀d. x = ⫯d →
L = ⋆ ∨
- ∃∃I,K,V. L = K.ⓑ{I}V & G ⊢ ⧤⬊*[h, g, d] K &
- G â\8a¢ â\8b\95â¬\8a*[h, g, V, d] K.
+ ∃∃I,K,V. L = K.ⓑ{I}V & G ⊢ ~⬊*[h, g, d] K &
+ G ⊢ ⬊*[h, g, V, d] K.
#h #g #G #L #d * -L -d /2 width=1 by or_introl/
[ #I #L #T #_ #x #H elim (ysucc_inv_O_sn … H)
| #I #L #T #d #HT #HL #x #H <(ysucc_inj … H) -x
]
qed-.
-lemma lcosx_inv_succ: ∀h,g,G,L,d. G ⊢ ⧤⬊*[h, g, ⫯d] L → L = ⋆ ∨
- ∃∃I,K,V. L = K.ⓑ{I}V & G ⊢ ⧤⬊*[h, g, d] K &
- G â\8a¢ â\8b\95â¬\8a*[h, g, V, d] K.
+lemma lcosx_inv_succ: ∀h,g,G,L,d. G ⊢ ~⬊*[h, g, ⫯d] L → L = ⋆ ∨
+ ∃∃I,K,V. L = K.ⓑ{I}V & G ⊢ ~⬊*[h, g, d] K &
+ G ⊢ ⬊*[h, g, V, d] K.
/2 width=3 by lcosx_inv_succ_aux/ qed-.
-lemma lcosx_inv_pair: ∀h,g,I,G,L,T,d. G ⊢ ⧤⬊*[h, g, ⫯d] L.ⓑ{I}T →
- G ⊢ ⧤⬊*[h, g, d] L ∧ G ⊢ ⋕⬊*[h, g, T, d] L.
+lemma lcosx_inv_pair: ∀h,g,I,G,L,T,d. G ⊢ ~⬊*[h, g, ⫯d] L.ⓑ{I}T →
+ G ⊢ ~⬊*[h, g, d] L ∧ G ⊢ ⬊*[h, g, T, d] L.
#h #g #I #G #L #T #d #H elim (lcosx_inv_succ … H) -H
[ #H destruct
| * #Z #Y #X #H destruct /2 width=1 by conj/