update in ground_2 static_2 basic_2

[helm.git] / matita / matita / contribs / lambdadelta / basic_2 / dynamic / cnv.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv.ma
index bde05a33cabcc6d4ee858568f74563abdaeb164d..0bd1aa9157b5a6e9ef6a82bdc6d84dbea4ed37ef 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/cnv.ma
@@ -12,7 +12,10 @@
 (*                                                                        *)
 (**************************************************************************)
 
+include "static_2/syntax/ac.ma".
 include "basic_2/notation/relations/exclaim_5.ma".
+include "basic_2/notation/relations/exclaim_4.ma".
+include "basic_2/notation/relations/exclaimstar_4.ma".
 include "basic_2/rt_computation/cpms.ma".
 
 (* CONTEXT-SENSITIVE NATIVE VALIDITY FOR TERMS ******************************)
@@ -24,19 +27,26 @@ inductive cnv (a) (h): relation3 genv lenv term ≝
 | cnv_zero: ∀I,G,K,V. cnv a h G K V → cnv a h G (K.ⓑ{I}V) (#0)
 | cnv_lref: ∀I,G,K,i. cnv a h G K (#i) → cnv a h G (K.ⓘ{I}) (#↑i)
 | cnv_bind: ∀p,I,G,L,V,T. cnv a h G L V → cnv a h G (L.ⓑ{I}V) T → cnv a h G L (ⓑ{p,I}V.T)
-| cnv_appl: ∀n,p,G,L,V,W0,T,U0. (a = Ⓣ → n ≤ 1) → cnv a h G L V → cnv a h G L T →
-            ⦃G, L⦄ ⊢ V ➡*[1, h] W0 → ⦃G, L⦄ ⊢ T ➡*[n, h] ⓛ{p}W0.U0 → cnv a h G L (ⓐV.T)
+| cnv_appl: ∀n,p,G,L,V,W0,T,U0. appl a n → cnv a h G L V → cnv a h G L T →
+            ⦃G,L⦄ ⊢ V ➡*[1,h] W0 → ⦃G,L⦄ ⊢ T ➡*[n,h] ⓛ{p}W0.U0 → cnv a h G L (ⓐV.T)
 | cnv_cast: ∀G,L,U,T,U0. cnv a h G L U → cnv a h G L T →
-            ⦃G, L⦄ ⊢ U ➡*[h] U0 → ⦃G, L⦄ ⊢ T ➡*[1, h] U0 → cnv a h G L (ⓝU.T)
+            ⦃G,L⦄ ⊢ U ➡*[h] U0 → ⦃G,L⦄ ⊢ T ➡*[1,h] U0 → cnv a h G L (ⓝU.T)
 .
 
 interpretation "context-sensitive native validity (term)"
    'Exclaim a h G L T = (cnv a h G L T).
 
+interpretation "context-sensitive restricted native validity (term)"
+   'Exclaim h G L T = (cnv (ac_eq (S O)) h G L T).
+
+interpretation "context-sensitive extended native validity (term)"
+   'ExclaimStar h G L T = (cnv ac_top h G L T).
+
 (* Basic inversion lemmas ***************************************************)
 
-fact cnv_inv_zero_aux (a) (h): ∀G,L,X. ⦃G, L⦄ ⊢ X ![a, h] → X = #0 →
-                               ∃∃I,K,V. ⦃G, K⦄ ⊢ V ![a, h] & L = K.ⓑ{I}V.
+fact cnv_inv_zero_aux (a) (h):
+     ∀G,L,X. ⦃G,L⦄ ⊢ X ![a,h] → X = #0 →
+     ∃∃I,K,V. ⦃G,K⦄ ⊢ V ![a,h] & L = K.ⓑ{I}V.
 #a #h #G #L #X * -G -L -X
 [ #G #L #s #H destruct
 | #I #G #K #V #HV #_ /2 width=5 by ex2_3_intro/
@@ -47,12 +57,14 @@ fact cnv_inv_zero_aux (a) (h): ∀G,L,X. ⦃G, L⦄ ⊢ X ![a, h] → X = #0 →
 ]
 qed-.
 
-lemma cnv_inv_zero (a) (h): ∀G,L. ⦃G, L⦄ ⊢ #0 ![a, h] →
-                            ∃∃I,K,V. ⦃G, K⦄ ⊢ V ![a, h] & L = K.ⓑ{I}V.
+lemma cnv_inv_zero (a) (h):
+      ∀G,L. ⦃G,L⦄ ⊢ #0 ![a,h] →
+      ∃∃I,K,V. ⦃G,K⦄ ⊢ V ![a,h] & L = K.ⓑ{I}V.
 /2 width=3 by cnv_inv_zero_aux/ qed-.
 
-fact cnv_inv_lref_aux (a) (h): ∀G,L,X. ⦃G, L⦄ ⊢ X ![a, h] → ∀i. X = #(↑i) →
-                              ∃∃I,K. ⦃G, K⦄ ⊢ #i ![a, h] & L = K.ⓘ{I}.
+fact cnv_inv_lref_aux (a) (h):
+     ∀G,L,X. ⦃G,L⦄ ⊢ X ![a,h] → ∀i. X = #(↑i) →
+     ∃∃I,K. ⦃G,K⦄ ⊢ #i ![a,h] & L = K.ⓘ{I}.
 #a #h #G #L #X * -G -L -X
 [ #G #L #s #j #H destruct
 | #I #G #K #V #_ #j #H destruct
@@ -63,11 +75,12 @@ fact cnv_inv_lref_aux (a) (h): ∀G,L,X. ⦃G, L⦄ ⊢ X ![a, h] → ∀i. X =
 ]
 qed-.
 
-lemma cnv_inv_lref (a) (h): ∀G,L,i. ⦃G, L⦄ ⊢ #↑i ![a, h] →
-                            ∃∃I,K. ⦃G, K⦄ ⊢ #i ![a, h] & L = K.ⓘ{I}.
+lemma cnv_inv_lref (a) (h):
+      ∀G,L,i. ⦃G,L⦄ ⊢ #↑i ![a,h] →
+      ∃∃I,K. ⦃G,K⦄ ⊢ #i ![a,h] & L = K.ⓘ{I}.
 /2 width=3 by cnv_inv_lref_aux/ qed-.
 
-fact cnv_inv_gref_aux (a) (h): ∀G,L,X. ⦃G, L⦄ ⊢ X ![a, h] → ∀l. X = §l → ⊥.
+fact cnv_inv_gref_aux (a) (h): ∀G,L,X. ⦃G,L⦄ ⊢ X ![a,h] → ∀l. X = §l → ⊥.
 #a #h #G #L #X * -G -L -X
 [ #G #L #s #l #H destruct
 | #I #G #K #V #_ #l #H destruct
@@ -79,13 +92,13 @@ fact cnv_inv_gref_aux (a) (h): ∀G,L,X. ⦃G, L⦄ ⊢ X ![a, h] → ∀l. X =
 qed-.
 
 (* Basic_2A1: uses: snv_inv_gref *)
-lemma cnv_inv_gref (a) (h): ∀G,L,l. ⦃G, L⦄ ⊢ §l ![a, h] → ⊥.
+lemma cnv_inv_gref (a) (h): ∀G,L,l. ⦃G,L⦄ ⊢ §l ![a,h] → ⊥.
 /2 width=8 by cnv_inv_gref_aux/ qed-.
 
-fact cnv_inv_bind_aux (a) (h): ∀G,L,X. ⦃G, L⦄ ⊢ X ![a, h] →
-                               ∀p,I,V,T. X = ⓑ{p,I}V.T →
-                               ∧∧ ⦃G, L⦄ ⊢ V ![a, h]
-                                & ⦃G, L.ⓑ{I}V⦄ ⊢ T ![a, h].
+fact cnv_inv_bind_aux (a) (h):
+     ∀G,L,X. ⦃G,L⦄ ⊢ X ![a,h] →
+     ∀p,I,V,T. X = ⓑ{p,I}V.T →
+     ∧∧ ⦃G,L⦄ ⊢ V ![a,h] & ⦃G,L.ⓑ{I}V⦄ ⊢ T ![a,h].
 #a #h #G #L #X * -G -L -X
 [ #G #L #s #q #Z #X1 #X2 #H destruct
 | #I #G #K #V #_ #q #Z #X1 #X2 #H destruct
@@ -97,14 +110,15 @@ fact cnv_inv_bind_aux (a) (h): ∀G,L,X. ⦃G, L⦄ ⊢ X ![a, h] →
 qed-.
 
 (* Basic_2A1: uses: snv_inv_bind *)
-lemma cnv_inv_bind (a) (h): ∀p,I,G,L,V,T. ⦃G, L⦄ ⊢ ⓑ{p,I}V.T ![a, h] →
-                            ∧∧ ⦃G, L⦄ ⊢ V ![a, h]
-                             & ⦃G, L.ⓑ{I}V⦄ ⊢ T ![a, h].
+lemma cnv_inv_bind (a) (h):
+      ∀p,I,G,L,V,T. ⦃G,L⦄ ⊢ ⓑ{p,I}V.T ![a,h] →
+      ∧∧ ⦃G,L⦄ ⊢ V ![a,h] & ⦃G,L.ⓑ{I}V⦄ ⊢ T ![a,h].
 /2 width=4 by cnv_inv_bind_aux/ qed-.
 
-fact cnv_inv_appl_aux (a) (h): ∀G,L,X. ⦃G, L⦄ ⊢ X ![a, h] → ∀V,T. X = ⓐV.T →
-                               ∃∃n,p,W0,U0. a = Ⓣ → n ≤ 1 & ⦃G, L⦄ ⊢ V ![a, h] & ⦃G, L⦄ ⊢ T ![a, h] &
-                                            ⦃G, L⦄ ⊢ V ➡*[1, h] W0 & ⦃G, L⦄ ⊢ T ➡*[n, h] ⓛ{p}W0.U0.
+fact cnv_inv_appl_aux (a) (h):
+     ∀G,L,X. ⦃G,L⦄ ⊢ X ![a,h] → ∀V,T. X = ⓐV.T →
+     ∃∃n,p,W0,U0. appl a n & ⦃G,L⦄ ⊢ V ![a,h] & ⦃G,L⦄ ⊢ T ![a,h] &
+                  ⦃G,L⦄ ⊢ V ➡*[1,h] W0 & ⦃G,L⦄ ⊢ T ➡*[n,h] ⓛ{p}W0.U0.
 #a #h #G #L #X * -L -X
 [ #G #L #s #X1 #X2 #H destruct
 | #I #G #K #V #_ #X1 #X2 #H destruct
@@ -116,14 +130,16 @@ fact cnv_inv_appl_aux (a) (h): ∀G,L,X. ⦃G, L⦄ ⊢ X ![a, h] → ∀V,T. X
 qed-.
 
 (* Basic_2A1: uses: snv_inv_appl *)
-lemma cnv_inv_appl (a) (h): ∀G,L,V,T. ⦃G, L⦄ ⊢ ⓐV.T ![a, h] →
-                            ∃∃n,p,W0,U0. a = Ⓣ → n ≤ 1 & ⦃G, L⦄ ⊢ V ![a, h] & ⦃G, L⦄ ⊢ T ![a, h] &
-                                         ⦃G, L⦄ ⊢ V ➡*[1, h] W0 & ⦃G, L⦄ ⊢ T ➡*[n, h] ⓛ{p}W0.U0.
+lemma cnv_inv_appl (a) (h):
+      ∀G,L,V,T. ⦃G,L⦄ ⊢ ⓐV.T ![a,h] →
+      ∃∃n,p,W0,U0. appl a n & ⦃G,L⦄ ⊢ V ![a,h] & ⦃G,L⦄ ⊢ T ![a,h] &
+                   ⦃G,L⦄ ⊢ V ➡*[1,h] W0 & ⦃G,L⦄ ⊢ T ➡*[n,h] ⓛ{p}W0.U0.
 /2 width=3 by cnv_inv_appl_aux/ qed-.
 
-fact cnv_inv_cast_aux (a) (h): ∀G,L,X. ⦃G, L⦄ ⊢ X ![a, h] → ∀U,T. X = ⓝU.T →
-                               ∃∃U0. ⦃G, L⦄ ⊢ U ![a, h] & ⦃G, L⦄ ⊢ T ![a, h] &
-                                     ⦃G, L⦄ ⊢ U ➡*[h] U0 & ⦃G, L⦄ ⊢ T ➡*[1, h] U0.
+fact cnv_inv_cast_aux (a) (h):
+     ∀G,L,X. ⦃G,L⦄ ⊢ X ![a,h] → ∀U,T. X = ⓝU.T →
+     ∃∃U0. ⦃G,L⦄ ⊢ U ![a,h] & ⦃G,L⦄ ⊢ T ![a,h] &
+           ⦃G,L⦄ ⊢ U ➡*[h] U0 & ⦃G,L⦄ ⊢ T ➡*[1,h] U0.
 #a #h #G #L #X * -G -L -X
 [ #G #L #s #X1 #X2 #H destruct
 | #I #G #K #V #_ #X1 #X2 #H destruct
@@ -134,24 +150,32 @@ fact cnv_inv_cast_aux (a) (h): ∀G,L,X. ⦃G, L⦄ ⊢ X ![a, h] → ∀U,T. X
 ]
 qed-.
 
-(* Basic_2A1: uses: snv_inv_appl *)
-lemma cnv_inv_cast (a) (h): ∀G,L,U,T. ⦃G, L⦄ ⊢ ⓝU.T ![a, h] →
-                            ∃∃U0. ⦃G, L⦄ ⊢ U ![a, h] & ⦃G, L⦄ ⊢ T ![a, h] &
-                                  ⦃G, L⦄ ⊢ U ➡*[h] U0 & ⦃G, L⦄ ⊢ T ➡*[1, h] U0.
+(* Basic_2A1: uses: snv_inv_cast *)
+lemma cnv_inv_cast (a) (h):
+      ∀G,L,U,T. ⦃G,L⦄ ⊢ ⓝU.T ![a,h] →
+      ∃∃U0. ⦃G,L⦄ ⊢ U ![a,h] & ⦃G,L⦄ ⊢ T ![a,h] &
+            ⦃G,L⦄ ⊢ U ➡*[h] U0 & ⦃G,L⦄ ⊢ T ➡*[1,h] U0.
 /2 width=3 by cnv_inv_cast_aux/ qed-.
 
 (* Basic forward lemmas *****************************************************)
 
 lemma cnv_fwd_flat (a) (h) (I) (G) (L):
-                   ∀V,T. ⦃G, L⦄ ⊢ ⓕ{I}V.T ![a,h] →
-                   ∧∧ ⦃G, L⦄ ⊢ V ![a,h] & ⦃G, L⦄ ⊢ T ![a,h].
+      ∀V,T. ⦃G,L⦄ ⊢ ⓕ{I}V.T ![a,h] →
+      ∧∧ ⦃G,L⦄ ⊢ V ![a,h] & ⦃G,L⦄ ⊢ T ![a,h].
 #a #h * #G #L #V #T #H
 [ elim (cnv_inv_appl … H) #n #p #W #U #_ #HV #HT #_ #_
 | elim (cnv_inv_cast … H) #U #HV #HT #_ #_
 ] -H /2 width=1 by conj/
 qed-.
 
-(* Basic_2A1: removed theorems 6:
-              snv_fwd_da snv_fwd_lstas snv_cast_scpes
+lemma cnv_fwd_pair_sn (a) (h) (I) (G) (L):
+      ∀V,T. ⦃G,L⦄ ⊢ ②{I}V.T ![a,h] → ⦃G,L⦄ ⊢ V ![a,h].
+#a #h * [ #p ] #I #G #L #V #T #H
+[ elim (cnv_inv_bind … H) -H #HV #_
+| elim (cnv_fwd_flat … H) -H #HV #_
+] //
+qed-.
+
+(* Basic_2A1: removed theorems 3:
               shnv_cast shnv_inv_cast snv_shnv_cast
 *)