update in ground_2, static_2, basic_2, apps_2, alpha_1

[helm.git] / matita / matita / contribs / lambdadelta / static_2 / static / frees_fqup.ma

diff --git a/matita/matita/contribs/lambdadelta/static_2/static/frees_fqup.ma b/matita/matita/contribs/lambdadelta/static_2/static/frees_fqup.ma
index abaf4ff15945e715dcbe7d2c48e23b758488048f..05500252c002af036f93651bdd89355ed036fc53 100644 (file)
--- a/matita/matita/contribs/lambdadelta/static_2/static/frees_fqup.ma
+++ b/matita/matita/contribs/lambdadelta/static_2/static/frees_fqup.ma
@@ -20,7 +20,7 @@ include "static_2/static/lsubf_lsubr.ma".
 (* Advanced properties ******************************************************)
 
 (* Note: this replaces lemma 1400 concluding the "big tree" theorem *)
-lemma frees_total: ∀L,T. ∃f. L ⊢ 𝐅*⦃T⦄ ≘ f.
+lemma frees_total: ∀L,T. ∃f. L ⊢ 𝐅+❪T❫ ≘ f.
 #L #T @(fqup_wf_ind_eq (Ⓣ) … (⋆) L T) -L -T
 #G0 #L0 #T0 #IH #G #L * *
 [ /3 width=2 by frees_sort, ex_intro/
@@ -38,7 +38,7 @@ lemma frees_total: ∀L,T. ∃f. L ⊢ 𝐅*⦃T⦄ ≘ f.
 | /3 width=2 by frees_gref, ex_intro/
 | #p #I #V #T #HG #HL #HT destruct
   elim (IH G L V) // #f1 #HV
-  elim (IH G (L.ⓑ{I}V) T) -IH // #f2 #HT
+  elim (IH G (L.ⓑ[I]V) T) -IH // #f2 #HT
   elim (sor_isfin_ex f1 (⫱f2))
   /3 width=6 by frees_fwd_isfin, frees_bind, isfin_tl, ex_intro/
 | #I #V #T #HG #HL #HT destruct
@@ -51,10 +51,11 @@ qed-.
 
 (* Advanced main properties *************************************************)
 
-theorem frees_bind_void: ∀f1,L,V. L ⊢ 𝐅*⦃V⦄ ≘ f1 → ∀f2,T. L.ⓧ ⊢ 𝐅*⦃T⦄ ≘ f2 →
-                         ∀f. f1 ⋓ ⫱f2 ≘ f → ∀p,I. L ⊢ 𝐅*⦃ⓑ{p,I}V.T⦄ ≘ f.
+theorem frees_bind_void:
+        ∀f1,L,V. L ⊢ 𝐅+❪V❫ ≘ f1 → ∀f2,T. L.ⓧ ⊢ 𝐅+❪T❫ ≘ f2 →
+        ∀f. f1 ⋓ ⫱f2 ≘ f → ∀p,I. L ⊢ 𝐅+❪ⓑ[p,I]V.T❫ ≘ f.
 #f1 #L #V #Hf1 #f2 #T #Hf2 #f #Hf #p #I
-elim (frees_total (L.ⓑ{I}V) T) #f0 #Hf0
+elim (frees_total (L.ⓑ[I]V) T) #f0 #Hf0
 lapply (lsubr_lsubf … Hf2 … Hf0) -Hf2 /2 width=5 by lsubr_unit/ #H02
 elim (pn_split f2) * #g2 #H destruct
 [ elim (lsubf_inv_push2 … H02) -H02 #g0 #Z #Y #H02 #H0 #H destruct
@@ -78,8 +79,9 @@ qed-.
 
 (* Advanced inversion lemmas ************************************************)
 
-lemma frees_inv_bind_void: ∀f,p,I,L,V,T. L ⊢ 𝐅*⦃ⓑ{p,I}V.T⦄ ≘ f →
-                           ∃∃f1,f2. L ⊢ 𝐅*⦃V⦄ ≘ f1 & L.ⓧ ⊢ 𝐅*⦃T⦄ ≘ f2 & f1 ⋓ ⫱f2 ≘ f.
+lemma frees_inv_bind_void:
+      ∀f,p,I,L,V,T. L ⊢ 𝐅+❪ⓑ[p,I]V.T❫ ≘ f →
+      ∃∃f1,f2. L ⊢ 𝐅+❪V❫ ≘ f1 & L.ⓧ ⊢ 𝐅+❪T❫ ≘ f2 & f1 ⋓ ⫱f2 ≘ f.
 #f #p #I #L #V #T #H
 elim (frees_inv_bind … H) -H #f1 #f2 #Hf1 #Hf2 #Hf
 elim (frees_total (L.ⓧ) T) #f0 #Hf0
@@ -100,31 +102,31 @@ elim (pn_split f0) * #g0 #H destruct
 ]
 qed-.
 
-lemma frees_ind_void: ∀Q:relation3 ….
-                      (
-                         ∀f,L,s. 𝐈⦃f⦄ →  Q L (⋆s) f
-                      ) → (
-                         ∀f,i. 𝐈⦃f⦄ →  Q (⋆) (#i) (⫯*[i]↑f)
-                      ) → (
-                         ∀f,I,L,V.
-                         L ⊢ 𝐅*⦃V⦄ ≘ f →  Q L V f→ Q (L.ⓑ{I}V) (#O) (↑f) 
-                      ) → (
-                         ∀f,I,L. 𝐈⦃f⦄ →  Q (L.ⓤ{I}) (#O) (↑f)
-                      ) → (
-                         ∀f,I,L,i.
-                         L ⊢ 𝐅*⦃#i⦄ ≘ f →  Q L (#i) f → Q (L.ⓘ{I}) (#(↑i)) (⫯f)
-                      ) → (
-                         ∀f,L,l. 𝐈⦃f⦄ →  Q L (§l) f
-                      ) → (
-                         ∀f1,f2,f,p,I,L,V,T.
-                         L ⊢ 𝐅*⦃V⦄ ≘ f1 → L.ⓧ ⊢𝐅*⦃T⦄≘ f2 → f1 ⋓ ⫱f2 ≘ f →
-                         Q L V f1 → Q (L.ⓧ) T f2 → Q L (ⓑ{p,I}V.T) f
-                      ) → (
-                         ∀f1,f2,f,I,L,V,T.
-                         L ⊢ 𝐅*⦃V⦄ ≘ f1 → L ⊢𝐅*⦃T⦄ ≘ f2 → f1 ⋓ f2 ≘ f →
-                         Q L V f1 → Q L T f2 → Q L (ⓕ{I}V.T) f
-                      ) →
-                      ∀L,T,f. L ⊢ 𝐅*⦃T⦄ ≘ f →  Q L T f.
+lemma frees_ind_void (Q:relation3 …):
+      (
+        ∀f,L,s. 𝐈❪f❫ →  Q L (⋆s) f
+      ) → (
+        ∀f,i. 𝐈❪f❫ →  Q (⋆) (#i) (⫯*[i]↑f)
+      ) → (
+        ∀f,I,L,V.
+        L ⊢ 𝐅+❪V❫ ≘ f →  Q L V f→ Q (L.ⓑ[I]V) (#O) (↑f)
+      ) → (
+        ∀f,I,L. 𝐈❪f❫ →  Q (L.ⓤ[I]) (#O) (↑f)
+      ) → (
+        ∀f,I,L,i.
+        L ⊢ 𝐅+❪#i❫ ≘ f →  Q L (#i) f → Q (L.ⓘ[I]) (#(↑i)) (⫯f)
+      ) → (
+        ∀f,L,l. 𝐈❪f❫ →  Q L (§l) f
+      ) → (
+        ∀f1,f2,f,p,I,L,V,T.
+        L ⊢ 𝐅+❪V❫ ≘ f1 → L.ⓧ ⊢𝐅+❪T❫≘ f2 → f1 ⋓ ⫱f2 ≘ f →
+        Q L V f1 → Q (L.ⓧ) T f2 → Q L (ⓑ[p,I]V.T) f
+      ) → (
+        ∀f1,f2,f,I,L,V,T.
+        L ⊢ 𝐅+❪V❫ ≘ f1 → L ⊢𝐅+❪T❫ ≘ f2 → f1 ⋓ f2 ≘ f →
+        Q L V f1 → Q L T f2 → Q L (ⓕ[I]V.T) f
+      ) →
+      ∀L,T,f. L ⊢ 𝐅+❪T❫ ≘ f →  Q L T f.
 #Q #IH1 #IH2 #IH3 #IH4 #IH5 #IH6 #IH7 #IH8 #L #T
 @(fqup_wf_ind_eq (Ⓕ) … (⋆) L T) -L -T #G0 #L0 #T0 #IH #G #L * *
 [ #s #HG #HL #HT #f #H destruct -IH