(* *)
(**************************************************************************)
-include "basic_2/grammar/aarity.ma".
-include "basic_2/substitution/ldrop.ma".
-
-(* ATONIC ARITY ASSIGNMENT ON TERMS *****************************************)
-
-inductive aaa: lenv → term → predicate aarity ≝
-| aaa_sort: ∀L,k. aaa L (⋆k) ⓪
-| aaa_lref: ∀I,L,K,V,B,i. ⇩[0, i] L ≡ K. ⓑ{I} V → aaa K V B → aaa L (#i) B
-| aaa_abbr: ∀a,L,V,T,B,A.
- aaa L V B → aaa (L. ⓓV) T A → aaa L (ⓓ{a}V. T) A
-| aaa_abst: ∀a,L,V,T,B,A.
- aaa L V B → aaa (L. ⓛV) T A → aaa L (ⓛ{a}V. T) (②B. A)
-| aaa_appl: ∀L,V,T,B,A. aaa L V B → aaa L T (②B. A) → aaa L (ⓐV. T) A
-| aaa_cast: ∀L,V,T,A. aaa L V A → aaa L T A → aaa L (ⓝV. T) A
+include "basic_2/notation/relations/atomicarity_4.ma".
+include "basic_2/syntax/aarity.ma".
+include "basic_2/syntax/lenv.ma".
+include "basic_2/syntax/genv.ma".
+
+(* ATONIC ARITY ASSIGNMENT FOR TERMS ****************************************)
+
+(* activate genv *)
+inductive aaa: relation4 genv lenv term aarity ≝
+| aaa_sort: ∀G,L,s. aaa G L (⋆s) (⓪)
+| aaa_zero: ∀I,G,L,V,B. aaa G L V B → aaa G (L.ⓑ{I}V) (#0) B
+| aaa_lref: ∀I,G,L,A,i. aaa G L (#i) A → aaa G (L.ⓘ{I}) (#↑i) A
+| aaa_abbr: ∀p,G,L,V,T,B,A.
+ aaa G L V B → aaa G (L.ⓓV) T A → aaa G L (ⓓ{p}V.T) A
+| aaa_abst: ∀p,G,L,V,T,B,A.
+ aaa G L V B → aaa G (L.ⓛV) T A → aaa G L (ⓛ{p}V.T) (②B.A)
+| aaa_appl: ∀G,L,V,T,B,A. aaa G L V B → aaa G L T (②B.A) → aaa G L (ⓐV.T) A
+| aaa_cast: ∀G,L,V,T,A. aaa G L V A → aaa G L T A → aaa G L (ⓝV.T) A
.
interpretation "atomic arity assignment (term)"
- 'AtomicArity L T A = (aaa L T A).
+ 'AtomicArity G L T A = (aaa G L T A).
(* Basic inversion lemmas ***************************************************)
-fact aaa_inv_sort_aux: ∀L,T,A. L ⊢ T ⁝ A → ∀k. T = ⋆k → A = ⓪.
-#L #T #A * -L -T -A
-[ //
-| #I #L #K #V #B #i #_ #_ #k #H destruct
-| #a #L #V #T #B #A #_ #_ #k #H destruct
-| #a #L #V #T #B #A #_ #_ #k #H destruct
-| #L #V #T #B #A #_ #_ #k #H destruct
-| #L #V #T #A #_ #_ #k #H destruct
+fact aaa_inv_sort_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀s. T = ⋆s → A = ⓪.
+#G #L #T #A * -G -L -T -A //
+[ #I #G #L #V #B #_ #s #H destruct
+| #I #G #L #A #i #_ #s #H destruct
+| #p #G #L #V #T #B #A #_ #_ #s #H destruct
+| #p #G #L #V #T #B #A #_ #_ #s #H destruct
+| #G #L #V #T #B #A #_ #_ #s #H destruct
+| #G #L #V #T #A #_ #_ #s #H destruct
]
-qed.
-
-lemma aaa_inv_sort: ∀L,A,k. L ⊢ ⋆k ⁝ A → A = ⓪.
-/2 width=5/ qed-.
-
-fact aaa_inv_lref_aux: ∀L,T,A. L ⊢ T ⁝ A → ∀i. T = #i →
- ∃∃I,K,V. ⇩[0, i] L ≡ K. ⓑ{I} V & K ⊢ V ⁝ A.
-#L #T #A * -L -T -A
-[ #L #k #i #H destruct
-| #I #L #K #V #B #j #HLK #HB #i #H destruct /2 width=5/
-| #a #L #V #T #B #A #_ #_ #i #H destruct
-| #a #L #V #T #B #A #_ #_ #i #H destruct
-| #L #V #T #B #A #_ #_ #i #H destruct
-| #L #V #T #A #_ #_ #i #H destruct
+qed-.
+
+lemma aaa_inv_sort: ∀G,L,A,s. ⦃G, L⦄ ⊢ ⋆s ⁝ A → A = ⓪.
+/2 width=6 by aaa_inv_sort_aux/ qed-.
+
+fact aaa_inv_zero_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → T = #0 →
+ ∃∃I,K,V. L = K.ⓑ{I}V & ⦃G, K⦄ ⊢ V ⁝ A.
+#G #L #T #A * -G -L -T -A /2 width=5 by ex2_3_intro/
+[ #G #L #s #H destruct
+| #I #G #L #A #i #_ #H destruct
+| #p #G #L #V #T #B #A #_ #_ #H destruct
+| #p #G #L #V #T #B #A #_ #_ #H destruct
+| #G #L #V #T #B #A #_ #_ #H destruct
+| #G #L #V #T #A #_ #_ #H destruct
]
-qed.
-
-lemma aaa_inv_lref: ∀L,A,i. L ⊢ #i ⁝ A →
- ∃∃I,K,V. ⇩[0, i] L ≡ K. ⓑ{I} V & K ⊢ V ⁝ A.
-/2 width=3/ qed-.
-
-fact aaa_inv_gref_aux: ∀L,T,A. L ⊢ T ⁝ A → ∀p. T = §p → ⊥.
-#L #T #A * -L -T -A
-[ #L #k #q #H destruct
-| #I #L #K #V #B #i #HLK #HB #q #H destruct
-| #a #L #V #T #B #A #_ #_ #q #H destruct
-| #a #L #V #T #B #A #_ #_ #q #H destruct
-| #L #V #T #B #A #_ #_ #q #H destruct
-| #L #V #T #A #_ #_ #q #H destruct
+qed-.
+
+lemma aaa_inv_zero: ∀G,L,A. ⦃G, L⦄ ⊢ #0 ⁝ A →
+ ∃∃I,K,V. L = K.ⓑ{I}V & ⦃G, K⦄ ⊢ V ⁝ A.
+/2 width=3 by aaa_inv_zero_aux/ qed-.
+
+fact aaa_inv_lref_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀i. T = #(↑i) →
+ ∃∃I,K. L = K.ⓘ{I} & ⦃G, K⦄ ⊢ #i ⁝ A.
+#G #L #T #A * -G -L -T -A
+[ #G #L #s #j #H destruct
+| #I #G #L #V #B #_ #j #H destruct
+| #I #G #L #A #i #HA #j #H destruct /2 width=4 by ex2_2_intro/
+| #p #G #L #V #T #B #A #_ #_ #j #H destruct
+| #p #G #L #V #T #B #A #_ #_ #j #H destruct
+| #G #L #V #T #B #A #_ #_ #j #H destruct
+| #G #L #V #T #A #_ #_ #j #H destruct
]
-qed.
-
-lemma aaa_inv_gref: ∀L,A,p. L ⊢ §p ⁝ A → ⊥.
-/2 width=6/ qed-.
-
-fact aaa_inv_abbr_aux: ∀L,T,A. L ⊢ T ⁝ A → ∀a,W,U. T = ⓓ{a}W. U →
- ∃∃B. L ⊢ W ⁝ B & L. ⓓW ⊢ U ⁝ A.
-#L #T #A * -L -T -A
-[ #L #k #a #W #U #H destruct
-| #I #L #K #V #B #i #_ #_ #a #W #U #H destruct
-| #b #L #V #T #B #A #HV #HT #a #W #U #H destruct /2 width=2/
-| #b #L #V #T #B #A #_ #_ #a #W #U #H destruct
-| #L #V #T #B #A #_ #_ #a #W #U #H destruct
-| #L #V #T #A #_ #_ #a #W #U #H destruct
+qed-.
+
+lemma aaa_inv_lref: ∀G,L,A,i. ⦃G, L⦄ ⊢ #↑i ⁝ A →
+ ∃∃I,K. L = K.ⓘ{I} & ⦃G, K⦄ ⊢ #i ⁝ A.
+/2 width=3 by aaa_inv_lref_aux/ qed-.
+
+fact aaa_inv_gref_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀l. T = §l → ⊥.
+#G #L #T #A * -G -L -T -A
+[ #G #L #s #k #H destruct
+| #I #G #L #V #B #_ #k #H destruct
+| #I #G #L #A #i #_ #k #H destruct
+| #p #G #L #V #T #B #A #_ #_ #k #H destruct
+| #p #G #L #V #T #B #A #_ #_ #k #H destruct
+| #G #L #V #T #B #A #_ #_ #k #H destruct
+| #G #L #V #T #A #_ #_ #k #H destruct
]
-qed.
-
-lemma aaa_inv_abbr: ∀a,L,V,T,A. L ⊢ ⓓ{a}V. T ⁝ A →
- ∃∃B. L ⊢ V ⁝ B & L. ⓓV ⊢ T ⁝ A.
-/2 width=4/ qed-.
-
-fact aaa_inv_abst_aux: ∀L,T,A. L ⊢ T ⁝ A → ∀a,W,U. T = ⓛ{a}W. U →
- ∃∃B1,B2. L ⊢ W ⁝ B1 & L. ⓛW ⊢ U ⁝ B2 & A = ②B1. B2.
-#L #T #A * -L -T -A
-[ #L #k #a #W #U #H destruct
-| #I #L #K #V #B #i #_ #_ #a #W #U #H destruct
-| #b #L #V #T #B #A #_ #_ #a #W #U #H destruct
-| #b #L #V #T #B #A #HV #HT #a #W #U #H destruct /2 width=5/
-| #L #V #T #B #A #_ #_ #a #W #U #H destruct
-| #L #V #T #A #_ #_ #a #W #U #H destruct
+qed-.
+
+lemma aaa_inv_gref: ∀G,L,A,l. ⦃G, L⦄ ⊢ §l ⁝ A → ⊥.
+/2 width=7 by aaa_inv_gref_aux/ qed-.
+
+fact aaa_inv_abbr_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀p,W,U. T = ⓓ{p}W.U →
+ ∃∃B. ⦃G, L⦄ ⊢ W ⁝ B & ⦃G, L.ⓓW⦄ ⊢ U ⁝ A.
+#G #L #T #A * -G -L -T -A
+[ #G #L #s #q #W #U #H destruct
+| #I #G #L #V #B #_ #q #W #U #H destruct
+| #I #G #L #A #i #_ #q #W #U #H destruct
+| #p #G #L #V #T #B #A #HV #HT #q #W #U #H destruct /2 width=2 by ex2_intro/
+| #p #G #L #V #T #B #A #_ #_ #q #W #U #H destruct
+| #G #L #V #T #B #A #_ #_ #q #W #U #H destruct
+| #G #L #V #T #A #_ #_ #q #W #U #H destruct
]
-qed.
-
-lemma aaa_inv_abst: ∀a,L,W,T,A. L ⊢ ⓛ{a}W. T ⁝ A →
- ∃∃B1,B2. L ⊢ W ⁝ B1 & L. ⓛW ⊢ T ⁝ B2 & A = ②B1. B2.
-/2 width=4/ qed-.
-
-fact aaa_inv_appl_aux: ∀L,T,A. L ⊢ T ⁝ A → ∀W,U. T = ⓐW. U →
- ∃∃B. L ⊢ W ⁝ B & L ⊢ U ⁝ ②B. A.
-#L #T #A * -L -T -A
-[ #L #k #W #U #H destruct
-| #I #L #K #V #B #i #_ #_ #W #U #H destruct
-| #a #L #V #T #B #A #_ #_ #W #U #H destruct
-| #a #L #V #T #B #A #_ #_ #W #U #H destruct
-| #L #V #T #B #A #HV #HT #W #U #H destruct /2 width=3/
-| #L #V #T #A #_ #_ #W #U #H destruct
+qed-.
+
+lemma aaa_inv_abbr: ∀p,G,L,V,T,A. ⦃G, L⦄ ⊢ ⓓ{p}V.T ⁝ A →
+ ∃∃B. ⦃G, L⦄ ⊢ V ⁝ B & ⦃G, L.ⓓV⦄ ⊢ T ⁝ A.
+/2 width=4 by aaa_inv_abbr_aux/ qed-.
+
+fact aaa_inv_abst_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀p,W,U. T = ⓛ{p}W.U →
+ ∃∃B1,B2. ⦃G, L⦄ ⊢ W ⁝ B1 & ⦃G, L.ⓛW⦄ ⊢ U ⁝ B2 & A = ②B1.B2.
+#G #L #T #A * -G -L -T -A
+[ #G #L #s #q #W #U #H destruct
+| #I #G #L #V #B #_ #q #W #U #H destruct
+| #I #G #L #A #i #_ #q #W #U #H destruct
+| #p #G #L #V #T #B #A #_ #_ #q #W #U #H destruct
+| #p #G #L #V #T #B #A #HV #HT #q #W #U #H destruct /2 width=5 by ex3_2_intro/
+| #G #L #V #T #B #A #_ #_ #q #W #U #H destruct
+| #G #L #V #T #A #_ #_ #q #W #U #H destruct
]
-qed.
-
-lemma aaa_inv_appl: ∀L,V,T,A. L ⊢ ⓐV. T ⁝ A →
- ∃∃B. L ⊢ V ⁝ B & L ⊢ T ⁝ ②B. A.
-/2 width=3/ qed-.
-
-fact aaa_inv_cast_aux: ∀L,T,A. L ⊢ T ⁝ A → ∀W,U. T = ⓝW. U →
- L ⊢ W ⁝ A ∧ L ⊢ U ⁝ A.
-#L #T #A * -L -T -A
-[ #L #k #W #U #H destruct
-| #I #L #K #V #B #i #_ #_ #W #U #H destruct
-| #a #L #V #T #B #A #_ #_ #W #U #H destruct
-| #a #L #V #T #B #A #_ #_ #W #U #H destruct
-| #L #V #T #B #A #_ #_ #W #U #H destruct
-| #L #V #T #A #HV #HT #W #U #H destruct /2 width=1/
+qed-.
+
+lemma aaa_inv_abst: ∀p,G,L,W,T,A. ⦃G, L⦄ ⊢ ⓛ{p}W.T ⁝ A →
+ ∃∃B1,B2. ⦃G, L⦄ ⊢ W ⁝ B1 & ⦃G, L.ⓛW⦄ ⊢ T ⁝ B2 & A = ②B1.B2.
+/2 width=4 by aaa_inv_abst_aux/ qed-.
+
+fact aaa_inv_appl_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀W,U. T = ⓐW.U →
+ ∃∃B. ⦃G, L⦄ ⊢ W ⁝ B & ⦃G, L⦄ ⊢ U ⁝ ②B.A.
+#G #L #T #A * -G -L -T -A
+[ #G #L #s #W #U #H destruct
+| #I #G #L #V #B #_ #W #U #H destruct
+| #I #G #L #A #i #_ #W #U #H destruct
+| #p #G #L #V #T #B #A #_ #_ #W #U #H destruct
+| #p #G #L #V #T #B #A #_ #_ #W #U #H destruct
+| #G #L #V #T #B #A #HV #HT #W #U #H destruct /2 width=3 by ex2_intro/
+| #G #L #V #T #A #_ #_ #W #U #H destruct
]
-qed.
+qed-.
+
+lemma aaa_inv_appl: ∀G,L,V,T,A. ⦃G, L⦄ ⊢ ⓐV.T ⁝ A →
+ ∃∃B. ⦃G, L⦄ ⊢ V ⁝ B & ⦃G, L⦄ ⊢ T ⁝ ②B.A.
+/2 width=3 by aaa_inv_appl_aux/ qed-.
+
+fact aaa_inv_cast_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀W,U. T = ⓝW.U →
+ ⦃G, L⦄ ⊢ W ⁝ A ∧ ⦃G, L⦄ ⊢ U ⁝ A.
+#G #L #T #A * -G -L -T -A
+[ #G #L #s #W #U #H destruct
+| #I #G #L #V #B #_ #W #U #H destruct
+| #I #G #L #A #i #_ #W #U #H destruct
+| #p #G #L #V #T #B #A #_ #_ #W #U #H destruct
+| #p #G #L #V #T #B #A #_ #_ #W #U #H destruct
+| #G #L #V #T #B #A #_ #_ #W #U #H destruct
+| #G #L #V #T #A #HV #HT #W #U #H destruct /2 width=1 by conj/
+]
+qed-.
-lemma aaa_inv_cast: ∀L,W,T,A. L ⊢ ⓝW. T ⁝ A →
- L ⊢ W ⁝ A ∧ L ⊢ T ⁝ A.
-/2 width=3/ qed-.
+lemma aaa_inv_cast: ∀G,L,W,T,A. ⦃G, L⦄ ⊢ ⓝW.T ⁝ A →
+ ⦃G, L⦄ ⊢ W ⁝ A ∧ ⦃G, L⦄ ⊢ T ⁝ A.
+/2 width=3 by aaa_inv_cast_aux/ qed-.
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