(* *)
(**************************************************************************)
-include "basic_2/notation/relations/notreducible_4.ma".
+include "basic_2/notation/relations/notreducible_5.ma".
include "basic_2/reduction/cir.ma".
include "basic_2/reduction/crx.ma".
(* CONTEXT-SENSITIVE EXTENDED IRREDUCIBLE TERMS *****************************)
-definition cix: ∀h. sd h → lenv → predicate term ≝ λh,g,L,T. ⦃G, L⦄ ⊢ 𝐑[h, g]⦃T⦄ → ⊥.
+definition cix: ∀h. sd h → relation3 genv lenv term ≝
+ λh,g,G,L,T. ⦃G, L⦄ ⊢ 𝐑[h, g]⦃T⦄ → ⊥.
interpretation "context-sensitive extended irreducibility (term)"
- 'NotReducible h g L T = (cix h g L T).
+ 'NotReducible h g G L T = (cix h g G L T).
(* Basic inversion lemmas ***************************************************)
-lemma cix_inv_sort: ∀h,g,L,k,l. deg h g k (l+1) → ⦃G, L⦄ ⊢ 𝐈[h, g]⦃⋆k⦄ → ⊥.
+lemma cix_inv_sort: ∀h,g,G,L,k,l. deg h g k (l+1) → ⦃G, L⦄ ⊢ 𝐈[h, g]⦃⋆k⦄ → ⊥.
/3 width=2/ qed-.
-lemma cix_inv_delta: ∀h,g,I,L,K,V,i. ⇩[0, i] L ≡ K.ⓑ{I}V → ⦃G, L⦄ ⊢ 𝐈[h, g]⦃#i⦄ → ⊥.
+lemma cix_inv_delta: ∀h,g,I,G,L,K,V,i. ⇩[0, i] L ≡ K.ⓑ{I}V → ⦃G, L⦄ ⊢ 𝐈[h, g]⦃#i⦄ → ⊥.
/3 width=4/ qed-.
-lemma cix_inv_ri2: ∀h,g,I,L,V,T. ri2 I → ⦃G, L⦄ ⊢ 𝐈[h, g]⦃②{I}V.T⦄ → ⊥.
+lemma cix_inv_ri2: ∀h,g,I,G,L,V,T. ri2 I → ⦃G, L⦄ ⊢ 𝐈[h, g]⦃②{I}V.T⦄ → ⊥.
/3 width=1/ qed-.
-lemma cix_inv_ib2: ∀h,g,a,I,L,V,T. ib2 a I → ⦃G, L⦄ ⊢ 𝐈[h, g]⦃ⓑ{a,I}V.T⦄ →
- ⦃G, L⦄ ⊢ 𝐈[h, g]⦃V⦄ ∧ ⦃h, L.ⓑ{I}V⦄ ⊢ 𝐈[h, g]⦃T⦄.
+lemma cix_inv_ib2: ∀h,g,a,I,G,L,V,T. ib2 a I → ⦃G, L⦄ ⊢ 𝐈[h, g]⦃ⓑ{a,I}V.T⦄ →
+ ⦃G, L⦄ ⊢ 𝐈[h, g]⦃V⦄ ∧ ⦃G, L.ⓑ{I}V⦄ ⊢ 𝐈[h, g]⦃T⦄.
/4 width=1/ qed-.
-lemma cix_inv_bind: ∀h,g,a,I,L,V,T. ⦃G, L⦄ ⊢ 𝐈[h, g]⦃ⓑ{a,I}V.T⦄ →
- ∧∧ ⦃G, L⦄ ⊢ 𝐈[h, g]⦃V⦄ & ⦃h, L.ⓑ{I}V⦄ ⊢ 𝐈[h, g]⦃T⦄ & ib2 a I.
+lemma cix_inv_bind: ∀h,g,a,I,G,L,V,T. ⦃G, L⦄ ⊢ 𝐈[h, g]⦃ⓑ{a,I}V.T⦄ →
+ ∧∧ ⦃G, L⦄ ⊢ 𝐈[h, g]⦃V⦄ & ⦃G, L.ⓑ{I}V⦄ ⊢ 𝐈[h, g]⦃T⦄ & ib2 a I.
#h #g #a * [ elim a -a ]
-[ #L #V #T #H elim H -H /3 width=1/
-|*: #L #V #T #H elim (cix_inv_ib2 … H) -H /2 width=1/ /3 width=1/
-]
+#G #L #V #T #H [ elim H -H /3 width=1/ ]
+elim (cix_inv_ib2 … H) -H /2 width=1/ /3 width=1/
qed-.
-lemma cix_inv_appl: ∀h,g,L,V,T. ⦃G, L⦄ ⊢ 𝐈[h, g]⦃ⓐV.T⦄ →
+lemma cix_inv_appl: ∀h,g,G,L,V,T. ⦃G, L⦄ ⊢ 𝐈[h, g]⦃ⓐV.T⦄ →
∧∧ ⦃G, L⦄ ⊢ 𝐈[h, g]⦃V⦄ & ⦃G, L⦄ ⊢ 𝐈[h, g]⦃T⦄ & 𝐒⦃T⦄.
-#h #g #L #V #T #HVT @and3_intro /3 width=1/
+#h #g #G #L #V #T #HVT @and3_intro /3 width=1/
generalize in match HVT; -HVT elim T -T //
* // #a * #U #T #_ #_ #H elim H -H /2 width=1/
qed-.
-lemma cix_inv_flat: ∀h,g,I,L,V,T. ⦃G, L⦄ ⊢ 𝐈[h, g]⦃ⓕ{I}V.T⦄ →
+lemma cix_inv_flat: ∀h,g,I,G,L,V,T. ⦃G, L⦄ ⊢ 𝐈[h, g]⦃ⓕ{I}V.T⦄ →
∧∧ ⦃G, L⦄ ⊢ 𝐈[h, g]⦃V⦄ & ⦃G, L⦄ ⊢ 𝐈[h, g]⦃T⦄ & 𝐒⦃T⦄ & I = Appl.
-#h #g * #L #V #T #H
+#h #g * #G #L #V #T #H
[ elim (cix_inv_appl … H) -H /2 width=1/
| elim (cix_inv_ri2 … H) -H /2 width=1/
]
(* Basic forward lemmas *****************************************************)
-lemma cix_inv_cir: ∀h,g,L,T. ⦃G, L⦄ ⊢ 𝐈[h, g]⦃T⦄ → ⦃G, L⦄ ⊢ 𝐈⦃T⦄.
+lemma cix_inv_cir: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ 𝐈[h, g]⦃T⦄ → ⦃G, L⦄ ⊢ 𝐈⦃T⦄.
/3 width=1/ qed-.
(* Basic properties *********************************************************)
-lemma cix_sort: ∀h,g,L,k. deg h g k 0 → ⦃G, L⦄ ⊢ 𝐈[h, g]⦃⋆k⦄.
-#h #g #L #k #Hk #H elim (crx_inv_sort … H) -L #l #Hkl
+lemma cix_sort: ∀h,g,G,L,k. deg h g k 0 → ⦃G, L⦄ ⊢ 𝐈[h, g]⦃⋆k⦄.
+#h #g #G #L #k #Hk #H elim (crx_inv_sort … H) -L #l #Hkl
lapply (deg_mono … Hk Hkl) -h -k <plus_n_Sm #H destruct
qed.
-lemma tix_lref: ∀h,g,i. ⦃h, ⋆⦄ ⊢ 𝐈[h, g]⦃#i⦄.
-#h #g #i #H elim (trx_inv_atom … H) -H #k #l #_ #H destruct
+lemma tix_lref: ∀h,g,G,i. ⦃G, ⋆⦄ ⊢ 𝐈[h, g]⦃#i⦄.
+#h #g #G #i #H elim (trx_inv_atom … H) -H #k #l #_ #H destruct
qed.
-lemma cix_gref: ∀h,g,L,p. ⦃G, L⦄ ⊢ 𝐈[h, g]⦃§p⦄.
-#h #g #L #p #H elim (crx_inv_gref … H)
+lemma cix_gref: ∀h,g,G,L,p. ⦃G, L⦄ ⊢ 𝐈[h, g]⦃§p⦄.
+#h #g #G #L #p #H elim (crx_inv_gref … H)
qed.
-lemma cix_ib2: ∀h,g,a,I,L,V,T. ib2 a I → ⦃G, L⦄ ⊢ 𝐈[h, g]⦃V⦄ → ⦃h, L.ⓑ{I}V⦄ ⊢ 𝐈[h, g]⦃T⦄ →
+lemma cix_ib2: ∀h,g,a,I,G,L,V,T. ib2 a I → ⦃G, L⦄ ⊢ 𝐈[h, g]⦃V⦄ → ⦃G, L.ⓑ{I}V⦄ ⊢ 𝐈[h, g]⦃T⦄ →
⦃G, L⦄ ⊢ 𝐈[h, g]⦃ⓑ{a,I}V.T⦄.
-#h #g #a #I #L #V #T #HI #HV #HT #H
+#h #g #a #I #G #L #V #T #HI #HV #HT #H
elim (crx_inv_ib2 … HI H) -HI -H /2 width=1/
qed.
-lemma cix_appl: ∀h,g,L,V,T. ⦃G, L⦄ ⊢ 𝐈[h, g]⦃V⦄ → ⦃G, L⦄ ⊢ 𝐈[h, g]⦃T⦄ → 𝐒⦃T⦄ → ⦃G, L⦄ ⊢ 𝐈[h, g]⦃ⓐV.T⦄.
-#h #g #L #V #T #HV #HT #H1 #H2
+lemma cix_appl: ∀h,g,G,L,V,T. ⦃G, L⦄ ⊢ 𝐈[h, g]⦃V⦄ → ⦃G, L⦄ ⊢ 𝐈[h, g]⦃T⦄ → 𝐒⦃T⦄ → ⦃G, L⦄ ⊢ 𝐈[h, g]⦃ⓐV.T⦄.
+#h #g #G #L #V #T #HV #HT #H1 #H2
elim (crx_inv_appl … H2) -H2 /2 width=1/
qed.