(* *)
(**************************************************************************)
-include "basic_2/notation/relations/notreducible_2.ma".
+include "basic_2/notation/relations/notreducible_3.ma".
include "basic_2/reduction/crr.ma".
(* CONTEXT-SENSITIVE IRREDUCIBLE TERMS **************************************)
-definition cir: lenv → predicate term ≝ λL,T. L ⊢ 𝐑⦃T⦄ → ⊥.
+definition cir: relation3 genv lenv term ≝ λG,L,T. ⦃G, L⦄ ⊢ 𝐑⦃T⦄ → ⊥.
interpretation "context-sensitive irreducibility (term)"
- 'NotReducible L T = (cir L T).
+ 'NotReducible G L T = (cir G L T).
(* Basic inversion lemmas ***************************************************)
-lemma cir_inv_delta: ∀L,K,V,i. ⇩[0, i] L ≡ K.ⓓV → L ⊢ 𝐈⦃#i⦄ → ⊥.
+lemma cir_inv_delta: ∀G,L,K,V,i. ⇩[0, i] L ≡ K.ⓓV → ⦃G, L⦄ ⊢ 𝐈⦃#i⦄ → ⊥.
/3 width=3/ qed-.
-lemma cir_inv_ri2: ∀I,L,V,T. ri2 I → L ⊢ 𝐈⦃②{I}V.T⦄ → ⊥.
+lemma cir_inv_ri2: ∀I,G,L,V,T. ri2 I → ⦃G, L⦄ ⊢ 𝐈⦃②{I}V.T⦄ → ⊥.
/3 width=1/ qed-.
-lemma cir_inv_ib2: ∀a,I,L,V,T. ib2 a I → L ⊢ 𝐈⦃ⓑ{a,I}V.T⦄ →
- L ⊢ 𝐈⦃V⦄ ∧ L.ⓑ{I}V ⊢ 𝐈⦃T⦄.
+lemma cir_inv_ib2: ∀a,I,G,L,V,T. ib2 a I → ⦃G, L⦄ ⊢ 𝐈⦃ⓑ{a,I}V.T⦄ →
+ ⦃G, L⦄ ⊢ 𝐈⦃V⦄ ∧ ⦃G, L.ⓑ{I}V⦄ ⊢ 𝐈⦃T⦄.
/4 width=1/ qed-.
-lemma cir_inv_bind: ∀a,I,L,V,T. L ⊢ 𝐈⦃ⓑ{a,I}V.T⦄ →
- ∧∧ L ⊢ 𝐈⦃V⦄ & L.ⓑ{I}V ⊢ 𝐈⦃T⦄ & ib2 a I.
+lemma cir_inv_bind: ∀a,I,G,L,V,T. ⦃G, L⦄ ⊢ 𝐈⦃ⓑ{a,I}V.T⦄ →
+ ∧∧ ⦃G, L⦄ ⊢ 𝐈⦃V⦄ & ⦃G, L.ⓑ{I}V⦄ ⊢ 𝐈⦃T⦄ & ib2 a I.
#a * [ elim a -a ]
-[ #L #V #T #H elim H -H /3 width=1/
-|*: #L #V #T #H elim (cir_inv_ib2 … H) -H /2 width=1/ /3 width=1/
-]
+#G #L #V #T #H [ elim H -H /3 width=1/ ]
+elim (cir_inv_ib2 … H) -H /2 width=1/ /3 width=1/
qed-.
-lemma cir_inv_appl: ∀L,V,T. L ⊢ 𝐈⦃ⓐV.T⦄ → ∧∧ L ⊢ 𝐈⦃V⦄ & L ⊢ 𝐈⦃T⦄ & 𝐒⦃T⦄.
-#L #V #T #HVT @and3_intro /3 width=1/
+lemma cir_inv_appl: ∀G,L,V,T. ⦃G, L⦄ ⊢ 𝐈⦃ⓐV.T⦄ →
+ ∧∧ ⦃G, L⦄ ⊢ 𝐈⦃V⦄ & ⦃G, L⦄ ⊢ 𝐈⦃T⦄ & 𝐒⦃T⦄.
+#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 cir_inv_flat: ∀I,L,V,T. L ⊢ 𝐈⦃ⓕ{I}V.T⦄ →
- ∧∧ L ⊢ 𝐈⦃V⦄ & L ⊢ 𝐈⦃T⦄ & 𝐒⦃T⦄ & I = Appl.
-* #L #V #T #H
+lemma cir_inv_flat: ∀I,G,L,V,T. ⦃G, L⦄ ⊢ 𝐈⦃ⓕ{I}V.T⦄ →
+ ∧∧ ⦃G, L⦄ ⊢ 𝐈⦃V⦄ & ⦃G, L⦄ ⊢ 𝐈⦃T⦄ & 𝐒⦃T⦄ & I = Appl.
+* #G #L #V #T #H
[ elim (cir_inv_appl … H) -H /2 width=1/
| elim (cir_inv_ri2 … H) -H /2 width=1/
]
(* Basic properties *********************************************************)
-lemma cir_sort: ∀L,k. L ⊢ 𝐈⦃⋆k⦄.
-/2 width=3 by crr_inv_sort/ qed.
+lemma cir_sort: ∀G,L,k. ⦃G, L⦄ ⊢ 𝐈⦃⋆k⦄.
+/2 width=4 by crr_inv_sort/ qed.
-lemma cir_gref: ∀L,p. L ⊢ 𝐈⦃§p⦄.
-/2 width=3 by crr_inv_gref/ qed.
+lemma cir_gref: ∀G,L,p. ⦃G, L⦄ ⊢ 𝐈⦃§p⦄.
+/2 width=4 by crr_inv_gref/ qed.
-lemma tir_atom: ∀I. ⋆ ⊢ 𝐈⦃⓪{I}⦄.
-/2 width=2 by trr_inv_atom/ qed.
+lemma tir_atom: ∀G,I. ⦃G, ⋆⦄ ⊢ 𝐈⦃⓪{I}⦄.
+/2 width=3 by trr_inv_atom/ qed.
-lemma cir_ib2: ∀a,I,L,V,T. ib2 a I → L ⊢ 𝐈⦃V⦄ → L.ⓑ{I}V ⊢ 𝐈⦃T⦄ → L ⊢ 𝐈⦃ⓑ{a,I}V.T⦄.
-#a #I #L #V #T #HI #HV #HT #H
+lemma cir_ib2: ∀a,I,G,L,V,T.
+ ib2 a I → ⦃G, L⦄ ⊢ 𝐈⦃V⦄ → ⦃G, L.ⓑ{I}V⦄ ⊢ 𝐈⦃T⦄ → ⦃G, L⦄ ⊢ 𝐈⦃ⓑ{a,I}V.T⦄.
+#a #I #G #L #V #T #HI #HV #HT #H
elim (crr_inv_ib2 … HI H) -HI -H /2 width=1/
qed.
-lemma cir_appl: ∀L,V,T. L ⊢ 𝐈⦃V⦄ → L ⊢ 𝐈⦃T⦄ → 𝐒⦃T⦄ → L ⊢ 𝐈⦃ⓐV.T⦄.
-#L #V #T #HV #HT #H1 #H2
+lemma cir_appl: ∀G,L,V,T. ⦃G, L⦄ ⊢ 𝐈⦃V⦄ → ⦃G, L⦄ ⊢ 𝐈⦃T⦄ → 𝐒⦃T⦄ → ⦃G, L⦄ ⊢ 𝐈⦃ⓐV.T⦄.
+#G #L #V #T #HV #HT #H1 #H2
elim (crr_inv_appl … H2) -H2 /2 width=1/
qed.