renaming in basic_2

[helm.git] / matita / matita / contribs / lambdadelta / basic_2 / static / aaa.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/aaa.ma b/matita/matita/contribs/lambdadelta/basic_2/static/aaa.ma
index e0039e238db451c5c3b556759b0dce7006f304f6..6ea5a39c40b289f7ee55ddc69b2bcec459187602 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/static/aaa.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/static/aaa.ma
@@ -13,20 +13,21 @@
 (**************************************************************************)
 
 include "basic_2/notation/relations/atomicarity_4.ma".
-include "basic_2/grammar/aarity.ma".
-include "basic_2/grammar/genv.ma".
-include "basic_2/substitution/ldrop.ma".
+include "basic_2/syntax/aarity.ma".
+include "basic_2/syntax/lenv.ma".
+include "basic_2/syntax/genv.ma".
 
-(* ATONIC ARITY ASSIGNMENT ON TERMS *****************************************)
+(* ATONIC ARITY ASSIGNMENT FOR TERMS ****************************************)
 
 (* activate genv *)
 inductive aaa: relation4 genv lenv term aarity ≝
-| aaa_sort: ∀G,L,k. aaa G L (⋆k) (⓪)
-| aaa_lref: ∀I,G,L,K,V,B,i. ⇩[i] L ≡ K. ⓑ{I}V → aaa G K V B → aaa G L (#i) B
-| aaa_abbr: ∀a,G,L,V,T,B,A.
-            aaa G L V B → aaa G (L.ⓓV) T A → aaa G L (ⓓ{a}V.T) A
-| aaa_abst: ∀a,G,L,V,T,B,A.
-            aaa G L V B → aaa G (L.ⓛV) T A → aaa G L (ⓛ{a}V.T) (②B.A)
+| 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
 .
@@ -36,89 +37,110 @@ interpretation "atomic arity assignment (term)"
 
 (* Basic inversion lemmas ***************************************************)
 
-fact aaa_inv_sort_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀k. T = ⋆k → A = ⓪.
-#G #L #T #A * -G -L -T -A
-[ //
-| #I #G #L #K #V #B #i #_ #_ #k #H destruct
-| #a #G #L #V #T #B #A #_ #_ #k #H destruct
-| #a #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
+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,k. ⦃G, L⦄ ⊢ ⋆k ⁝ A → A = ⓪.
+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_lref_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀i. T = #i →
-                       ∃∃I,K,V. ⇩[i] L ≡ K.ⓑ{I} V & ⦃G, K⦄ ⊢ V ⁝ A.
+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 #k #i #H destruct
-| #I #G #L #K #V #B #j #HLK #HB #i #H destruct /2 width=5 by ex2_3_intro/
-| #a #G #L #V #T #B #A #_ #_ #i #H destruct
-| #a #G #L #V #T #B #A #_ #_ #i #H destruct
-| #G #L #V #T #B #A #_ #_ #i #H destruct
-| #G #L #V #T #A #_ #_ #i #H destruct
+[ #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,V. ⇩[i] L ≡ K. ⓑ{I} V & ⦃G, K⦄ ⊢ V ⁝ A.
+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 → ∀p. T = §p → ⊥.
+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 #k #q #H destruct
-| #I #G #L #K #V #B #i #HLK #HB #q #H destruct
-| #a #G #L #V #T #B #A #_ #_ #q #H destruct
-| #a #G #L #V #T #B #A #_ #_ #q #H destruct
-| #G #L #V #T #B #A #_ #_ #q #H destruct
-| #G #L #V #T #A #_ #_ #q #H destruct
+[ #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,p. ⦃G, L⦄ ⊢ §p ⁝ A → ⊥.
+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 → ∀a,W,U. T = ⓓ{a}W. U →
+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 #k #a #W #U #H destruct
-| #I #G #L #K #V #B #i #_ #_ #a #W #U #H destruct
-| #b #G #L #V #T #B #A #HV #HT #a #W #U #H destruct /2 width=2 by ex2_intro/
-| #b #G #L #V #T #B #A #_ #_ #a #W #U #H destruct
-| #G #L #V #T #B #A #_ #_ #a #W #U #H destruct
-| #G #L #V #T #A #_ #_ #a #W #U #H destruct
+[ #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: ∀a,G,L,V,T,A. ⦃G, L⦄ ⊢ ⓓ{a}V. T ⁝ A →
+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 → ∀a,W,U. T = ⓛ{a}W. U →
+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 #k #a #W #U #H destruct
-| #I #G #L #K #V #B #i #_ #_ #a #W #U #H destruct
-| #b #G #L #V #T #B #A #_ #_ #a #W #U #H destruct
-| #b #G #L #V #T #B #A #HV #HT #a #W #U #H destruct /2 width=5 by ex3_2_intro/
-| #G #L #V #T #B #A #_ #_ #a #W #U #H destruct
-| #G #L #V #T #A #_ #_ #a #W #U #H destruct
+[ #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: ∀a,G,L,W,T,A. ⦃G, L⦄ ⊢ ⓛ{a}W. T ⁝ A →
+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 #k #W #U #H destruct
-| #I #G #L #K #V #B #i #_ #_ #W #U #H destruct
-| #a #G #L #V #T #B #A #_ #_ #W #U #H destruct
-| #a #G #L #V #T #B #A #_ #_ #W #U #H destruct
+[ #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
 ]
@@ -128,18 +150,19 @@ 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 →
+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 #k #W #U #H destruct
-| #I #G #L #K #V #B #i #_ #_ #W #U #H destruct
-| #a #G #L #V #T #B #A #_ #_ #W #U #H destruct
-| #a #G #L #V #T #B #A #_ #_ #W #U #H destruct
+[ #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 →
+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-.