X-Git-Url: http://matita.cs.unibo.it/gitweb/?p=helm.git;a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Fstatic%2Ffrees.ma;h=b35ea99b828235e63e509e378d657cf5802ce5ed;hp=206ec63e4b8a99a4c9d27e9417b680003970730c;hb=222044da28742b24584549ba86b1805a87def070;hpb=47a745462a714af9d65cea7b61af56524bd98fa1

diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/frees.ma b/matita/matita/contribs/lambdadelta/basic_2/static/frees.ma
index 206ec63e4..b35ea99b8 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/static/frees.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/static/frees.ma
@@ -20,17 +20,17 @@ include "basic_2/syntax/lenv.ma".
 
 inductive frees: relation3 lenv term rtmap ≝
 | frees_sort: ∀f,L,s. 𝐈⦃f⦄ → frees L (⋆s) f
-| frees_atom: ∀f,i. 𝐈⦃f⦄ → frees (⋆) (#i) (↑*[i]⫯f)
+| frees_atom: ∀f,i. 𝐈⦃f⦄ → frees (⋆) (#i) (⫯*[i]↑f)
 | frees_pair: ∀f,I,L,V. frees L V f →
-              frees (L.ⓑ{I}V) (#0) (⫯f)
-| frees_unit: ∀f,I,L. 𝐈⦃f⦄ → frees (L.ⓤ{I}) (#0) (⫯f)
+              frees (L.ⓑ{I}V) (#0) (↑f)
+| frees_unit: ∀f,I,L. 𝐈⦃f⦄ → frees (L.ⓤ{I}) (#0) (↑f)
 | frees_lref: ∀f,I,L,i. frees L (#i) f →
-              frees (L.ⓘ{I}) (#⫯i) (↑f)
+              frees (L.ⓘ{I}) (#↑i) (⫯f)
 | frees_gref: ∀f,L,l. 𝐈⦃f⦄ → frees L (§l) f
 | frees_bind: ∀f1,f2,f,p,I,L,V,T. frees L V f1 → frees (L.ⓑ{I}V) T f2 →
-              f1 ⋓ ⫱f2 ≡ f → frees L (ⓑ{p,I}V.T) f
+              f1 ⋓ ⫱f2 ≘ f → frees L (ⓑ{p,I}V.T) f
 | frees_flat: ∀f1,f2,f,I,L,V,T. frees L V f1 → frees L T f2 →
-              f1 ⋓ f2 ≡ f → frees L (ⓕ{I}V.T) f
+              f1 ⋓ f2 ≘ f → frees L (ⓕ{I}V.T) f
 .
 
 interpretation
@@ -39,7 +39,7 @@ interpretation
 
 (* 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 +50,11 @@ 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 +67,11 @@ 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 +84,11 @@ 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 +101,11 @@ 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 +118,11 @@ 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 +133,11 @@ 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 +150,12 @@ 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 +168,13 @@ 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 +195,11 @@ 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 +207,7 @@ 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-.