--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "static_2/notation/relations/atomicarity_4.ma".
+include "static_2/syntax/aarity.ma".
+include "static_2/syntax/lenv.ma".
+include "static_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 G L T A = (aaa G L T A).
+
+(* Basic inversion lemmas ***************************************************)
+
+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: ∀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_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_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_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_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_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-.
+
+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: ∀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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