more additions and corrections for the article

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

diff --git a/matita/matita/contribs/lambdadelta/static_2/static/frees.ma b/matita/matita/contribs/lambdadelta/static_2/static/frees.ma
index 054003f831a0538deb8b716e7e22b3de61ae5321..a312c25256552dbf99cd59ef32a61603f7a5cded 100644 (file)
--- a/matita/matita/contribs/lambdadelta/static_2/static/frees.ma
+++ b/matita/matita/contribs/lambdadelta/static_2/static/frees.ma
@@ -13,7 +13,7 @@
 (**************************************************************************)
 
 include "ground_2/relocation/rtmap_sor.ma".
-include "static_2/notation/relations/freestar_3.ma".
+include "static_2/notation/relations/freeplus_3.ma".
 include "static_2/syntax/lenv.ma".
 
 (* CONTEXT-SENSITIVE FREE VARIABLES *****************************************)
@@ -35,11 +35,11 @@ inductive frees: relation3 lenv term rtmap ≝
 
 interpretation
    "context-sensitive free variables (term)"
-   'FreeStar L T f = (frees L T f).
+   'FreePlus L T f = (frees L T f).
 
 (* Basic inversion lemmas ***************************************************)
 
-fact frees_inv_sort_aux: ∀f,L,X. L ⊢ 𝐅*⦃X⦄ ≘ f → ∀x. X = ⋆x → 𝐈⦃f⦄.
+fact frees_inv_sort_aux: ∀f,L,X. L ⊢ 𝐅+⦃X⦄ ≘ f → ∀x. X = ⋆x → 𝐈⦃f⦄.
 #L #X #f #H elim H -f -L -X //
 [ #f #i #_ #x #H destruct
 | #f #_ #L #V #_ #_ #x #H destruct
@@ -50,11 +50,12 @@ fact frees_inv_sort_aux: ∀f,L,X. L ⊢ 𝐅*⦃X⦄ ≘ f → ∀x. X = ⋆x 
 ]
 qed-.
 
-lemma frees_inv_sort: ∀f,L,s. L ⊢ 𝐅*⦃⋆s⦄ ≘ f → 𝐈⦃f⦄.
+lemma frees_inv_sort: ∀f,L,s. L ⊢ 𝐅+⦃⋆s⦄ ≘ f → 𝐈⦃f⦄.
 /2 width=5 by frees_inv_sort_aux/ qed-.
 
-fact frees_inv_atom_aux: ∀f,L,X. L ⊢ 𝐅*⦃X⦄ ≘ f → ∀i. L = ⋆ → X = #i →
-                         ∃∃g. 𝐈⦃g⦄ & f = ⫯*[i]↑g.
+fact frees_inv_atom_aux:
+     ∀f,L,X. L ⊢ 𝐅+⦃X⦄ ≘ f → ∀i. L = ⋆ → X = #i →
+     ∃∃g. 𝐈⦃g⦄ & f = ⫯*[i]↑g.
 #f #L #X #H elim H -f -L -X
 [ #f #L #s #_ #j #_ #H destruct
 | #f #i #Hf #j #_ #H destruct /2 width=3 by ex2_intro/
@@ -67,11 +68,12 @@ fact frees_inv_atom_aux: ∀f,L,X. L ⊢ 𝐅*⦃X⦄ ≘ f → ∀i. L = ⋆ 
 ]
 qed-.
 
-lemma frees_inv_atom: ∀f,i. ⋆ ⊢ 𝐅*⦃#i⦄ ≘ f → ∃∃g. 𝐈⦃g⦄ & f = ⫯*[i]↑g.
+lemma frees_inv_atom: ∀f,i. ⋆ ⊢ 𝐅+⦃#i⦄ ≘ f → ∃∃g. 𝐈⦃g⦄ & f = ⫯*[i]↑g.
 /2 width=5 by frees_inv_atom_aux/ qed-.
 
-fact frees_inv_pair_aux: ∀f,L,X. L ⊢ 𝐅*⦃X⦄ ≘ f → ∀I,K,V. L = K.ⓑ{I}V → X = #0 →
-                         ∃∃g. K ⊢ 𝐅*⦃V⦄ ≘ g & f = ↑g.
+fact frees_inv_pair_aux:
+     ∀f,L,X. L ⊢ 𝐅+⦃X⦄ ≘ f → ∀I,K,V. L = K.ⓑ{I}V → X = #0 →
+     ∃∃g. K ⊢ 𝐅+⦃V⦄ ≘ g & f = ↑g.
 #f #L #X * -f -L -X
 [ #f #L #s #_ #Z #Y #X #_ #H destruct
 | #f #i #_ #Z #Y #X #H destruct
@@ -84,11 +86,12 @@ fact frees_inv_pair_aux: ∀f,L,X. L ⊢ 𝐅*⦃X⦄ ≘ f → ∀I,K,V. L = K.
 ]
 qed-.
 
-lemma frees_inv_pair: ∀f,I,K,V. K.ⓑ{I}V ⊢ 𝐅*⦃#0⦄ ≘ f → ∃∃g. K ⊢ 𝐅*⦃V⦄ ≘ g & f = ↑g.
+lemma frees_inv_pair: ∀f,I,K,V. K.ⓑ{I}V ⊢ 𝐅+⦃#0⦄ ≘ f → ∃∃g. K ⊢ 𝐅+⦃V⦄ ≘ g & f = ↑g.
 /2 width=6 by frees_inv_pair_aux/ qed-.
 
-fact frees_inv_unit_aux: ∀f,L,X. L ⊢ 𝐅*⦃X⦄ ≘ f → ∀I,K. L = K.ⓤ{I} → X = #0 →
-                         ∃∃g. 𝐈⦃g⦄ & f = ↑g.
+fact frees_inv_unit_aux:
+     ∀f,L,X. L ⊢ 𝐅+⦃X⦄ ≘ f → ∀I,K. L = K.ⓤ{I} → X = #0 →
+     ∃∃g. 𝐈⦃g⦄ & f = ↑g.
 #f #L #X * -f -L -X
 [ #f #L #s #_ #Z #Y #_ #H destruct
 | #f #i #_ #Z #Y #H destruct
@@ -101,11 +104,12 @@ fact frees_inv_unit_aux: ∀f,L,X. L ⊢ 𝐅*⦃X⦄ ≘ f → ∀I,K. L = K.
 ]
 qed-.
 
-lemma frees_inv_unit: ∀f,I,K. K.ⓤ{I} ⊢ 𝐅*⦃#0⦄ ≘ f → ∃∃g. 𝐈⦃g⦄ & f = ↑g.
+lemma frees_inv_unit: ∀f,I,K. K.ⓤ{I} ⊢ 𝐅+⦃#0⦄ ≘ f → ∃∃g. 𝐈⦃g⦄ & f = ↑g.
 /2 width=7 by frees_inv_unit_aux/ qed-.
 
-fact frees_inv_lref_aux: ∀f,L,X. L ⊢ 𝐅*⦃X⦄ ≘ f → ∀I,K,j. L = K.ⓘ{I} → X = #(↑j) →
-                         ∃∃g. K ⊢ 𝐅*⦃#j⦄ ≘ g & f = ⫯g.
+fact frees_inv_lref_aux:
+     ∀f,L,X. L ⊢ 𝐅+⦃X⦄ ≘ f → ∀I,K,j. L = K.ⓘ{I} → X = #(↑j) →
+     ∃∃g. K ⊢ 𝐅+⦃#j⦄ ≘ g & f = ⫯g.
 #f #L #X * -f -L -X
 [ #f #L #s #_ #Z #Y #j #_ #H destruct
 | #f #i #_ #Z #Y #j #H destruct
@@ -118,11 +122,12 @@ fact frees_inv_lref_aux: ∀f,L,X. L ⊢ 𝐅*⦃X⦄ ≘ f → ∀I,K,j. L = K.
 ]
 qed-.
 
-lemma frees_inv_lref: ∀f,I,K,i. K.ⓘ{I} ⊢ 𝐅*⦃#(↑i)⦄ ≘ f →
-                      ∃∃g. K ⊢ 𝐅*⦃#i⦄ ≘ g & f = ⫯g.
+lemma frees_inv_lref:
+      ∀f,I,K,i. K.ⓘ{I} ⊢ 𝐅+⦃#(↑i)⦄ ≘ f →
+      ∃∃g. K ⊢ 𝐅+⦃#i⦄ ≘ g & f = ⫯g.
 /2 width=6 by frees_inv_lref_aux/ qed-.
 
-fact frees_inv_gref_aux: ∀f,L,X. L ⊢ 𝐅*⦃X⦄ ≘ f → ∀x. X = §x → 𝐈⦃f⦄.
+fact frees_inv_gref_aux: ∀f,L,X. L ⊢ 𝐅+⦃X⦄ ≘ f → ∀x. X = §x → 𝐈⦃f⦄.
 #f #L #X #H elim H -f -L -X //
 [ #f #i #_ #x #H destruct
 | #f #_ #L #V #_ #_ #x #H destruct
@@ -133,11 +138,12 @@ fact frees_inv_gref_aux: ∀f,L,X. L ⊢ 𝐅*⦃X⦄ ≘ f → ∀x. X = §x 
 ]
 qed-.
 
-lemma frees_inv_gref: ∀f,L,l. L ⊢ 𝐅*⦃§l⦄ ≘ f → 𝐈⦃f⦄.
+lemma frees_inv_gref: ∀f,L,l. L ⊢ 𝐅+⦃§l⦄ ≘ f → 𝐈⦃f⦄.
 /2 width=5 by frees_inv_gref_aux/ qed-.
 
-fact frees_inv_bind_aux: ∀f,L,X. L ⊢ 𝐅*⦃X⦄ ≘ f → ∀p,I,V,T. X = ⓑ{p,I}V.T →
-                         ∃∃f1,f2. L ⊢ 𝐅*⦃V⦄ ≘ f1 & L.ⓑ{I}V ⊢ 𝐅*⦃T⦄ ≘ f2 & f1 ⋓ ⫱f2 ≘ f.
+fact frees_inv_bind_aux:
+     ∀f,L,X. L ⊢ 𝐅+⦃X⦄ ≘ f → ∀p,I,V,T. X = ⓑ{p,I}V.T →
+     ∃∃f1,f2. L ⊢ 𝐅+⦃V⦄ ≘ f1 & L.ⓑ{I}V ⊢ 𝐅+⦃T⦄ ≘ f2 & f1 ⋓ ⫱f2 ≘ f.
 #f #L #X * -f -L -X
 [ #f #L #s #_ #q #J #W #U #H destruct
 | #f #i #_ #q #J #W #U #H destruct
@@ -150,12 +156,13 @@ fact frees_inv_bind_aux: ∀f,L,X. L ⊢ 𝐅*⦃X⦄ ≘ f → ∀p,I,V,T. X =
 ]
 qed-.
 
-lemma frees_inv_bind: ∀f,p,I,L,V,T. L ⊢ 𝐅*⦃ⓑ{p,I}V.T⦄ ≘ f →
-                      ∃∃f1,f2. L ⊢ 𝐅*⦃V⦄ ≘ f1 & L.ⓑ{I}V ⊢ 𝐅*⦃T⦄ ≘ f2 & f1 ⋓ ⫱f2 ≘ f.
+lemma frees_inv_bind:
+      ∀f,p,I,L,V,T. L ⊢ 𝐅+⦃ⓑ{p,I}V.T⦄ ≘ f →
+      ∃∃f1,f2. L ⊢ 𝐅+⦃V⦄ ≘ f1 & L.ⓑ{I}V ⊢ 𝐅+⦃T⦄ ≘ f2 & f1 ⋓ ⫱f2 ≘ f.
 /2 width=4 by frees_inv_bind_aux/ qed-.
 
-fact frees_inv_flat_aux: ∀f,L,X. L ⊢ 𝐅*⦃X⦄ ≘ f → ∀I,V,T. X = ⓕ{I}V.T →
-                         ∃∃f1,f2. L ⊢ 𝐅*⦃V⦄ ≘ f1 & L ⊢ 𝐅*⦃T⦄ ≘ f2 & f1 ⋓ f2 ≘ f.
+fact frees_inv_flat_aux: ∀f,L,X. L ⊢ 𝐅+⦃X⦄ ≘ f → ∀I,V,T. X = ⓕ{I}V.T →
+                         ∃∃f1,f2. L ⊢ 𝐅+⦃V⦄ ≘ f1 & L ⊢ 𝐅+⦃T⦄ ≘ f2 & f1 ⋓ f2 ≘ f.
 #f #L #X * -f -L -X
 [ #f #L #s #_ #J #W #U #H destruct
 | #f #i #_ #J #W #U #H destruct
@@ -168,13 +175,14 @@ fact frees_inv_flat_aux: ∀f,L,X. L ⊢ 𝐅*⦃X⦄ ≘ f → ∀I,V,T. X = 
 ]
 qed-.
 
-lemma frees_inv_flat: ∀f,I,L,V,T. L ⊢ 𝐅*⦃ⓕ{I}V.T⦄ ≘ f →
-                      ∃∃f1,f2. L ⊢ 𝐅*⦃V⦄ ≘ f1 & L ⊢ 𝐅*⦃T⦄ ≘ f2 & f1 ⋓ f2 ≘ f.
+lemma frees_inv_flat:
+      ∀f,I,L,V,T. L ⊢ 𝐅+⦃ⓕ{I}V.T⦄ ≘ f →
+      ∃∃f1,f2. L ⊢ 𝐅+⦃V⦄ ≘ f1 & L ⊢ 𝐅+⦃T⦄ ≘ f2 & f1 ⋓ f2 ≘ f.
 /2 width=4 by frees_inv_flat_aux/ qed-.
 
 (* Basic properties ********************************************************)
 
-lemma frees_eq_repl_back: ∀L,T. eq_repl_back … (λf. L ⊢ 𝐅*⦃T⦄ ≘ f).
+lemma frees_eq_repl_back: ∀L,T. eq_repl_back … (λf. L ⊢ 𝐅+⦃T⦄ ≘ f).
 #L #T #f1 #H elim H -f1 -L -T
 [ /3 width=3 by frees_sort, isid_eq_repl_back/
 | #f1 #i #Hf1 #g2 #H
@@ -195,11 +203,11 @@ lemma frees_eq_repl_back: ∀L,T. eq_repl_back … (λf. L ⊢ 𝐅*⦃T⦄ ≘
 ]
 qed-.
 
-lemma frees_eq_repl_fwd: ∀L,T. eq_repl_fwd … (λf. L ⊢ 𝐅*⦃T⦄ ≘ f).
+lemma frees_eq_repl_fwd: ∀L,T. eq_repl_fwd … (λf. L ⊢ 𝐅+⦃T⦄ ≘ f).
 #L #T @eq_repl_sym /2 width=3 by frees_eq_repl_back/
 qed-.
 
-lemma frees_lref_push: ∀f,i. ⋆ ⊢ 𝐅*⦃#i⦄ ≘ f → ⋆ ⊢ 𝐅*⦃#↑i⦄ ≘ ⫯f.
+lemma frees_lref_push: ∀f,i. ⋆ ⊢ 𝐅+⦃#i⦄ ≘ f → ⋆ ⊢ 𝐅+⦃#↑i⦄ ≘ ⫯f.
 #f #i #H
 elim (frees_inv_atom … H) -H #g #Hg #H destruct
 /2 width=1 by frees_atom/
@@ -207,7 +215,7 @@ qed.
 
 (* Forward lemmas with test for finite colength *****************************)
 
-lemma frees_fwd_isfin: ∀f,L,T. L ⊢ 𝐅*⦃T⦄ ≘ f → 𝐅⦃f⦄.
+lemma frees_fwd_isfin: ∀f,L,T. L ⊢ 𝐅+⦃T⦄ ≘ f → 𝐅⦃f⦄.
 #f #L #T #H elim H -f -L -T
 /4 width=5 by sor_isfin, isfin_isid, isfin_tl, isfin_pushs, isfin_push, isfin_next/
 qed-.