X-Git-Url: http://matita.cs.unibo.it/gitweb/?p=helm.git;a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Fstatic%2Ffrees_fqup.ma;h=217534a142f83b7786b580e98040f7eba90986be;hp=c82690309a7b312ebb4e44ce193c4fac0adf41a3;hb=222044da28742b24584549ba86b1805a87def070;hpb=199ba569adf94f9948053352c2c0a1c6deb62bc5

diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/frees_fqup.ma b/matita/matita/contribs/lambdadelta/basic_2/static/frees_fqup.ma
index c82690309..217534a14 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/static/frees_fqup.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/static/frees_fqup.ma
@@ -13,34 +13,138 @@
 (**************************************************************************)
 
 include "basic_2/s_computation/fqup_weight.ma".
-include "basic_2/static/frees.ma".
+include "basic_2/static/lsubf_lsubr.ma".
 
 (* CONTEXT-SENSITIVE FREE VARIABLES *****************************************)
 
 (* Advanced properties ******************************************************)
 
 (* Note: this replaces lemma 1400 concluding the "big tree" theorem *)
-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 *
-[ cases L -L /3 width=2 by frees_atom, ex_intro/
-  #L #I #V *
-  [ #s #HG #HL #HT destruct
-    elim (IH G L (⋆s)) -IH /3 width=2 by frees_sort_gen, fqu_fqup, fqu_drop, lifts_sort, ex_intro/
-  | * [2: #i ] #HG #HL #HT destruct
-    [ elim (IH G L (#i)) -IH /3 width=2 by frees_lref, fqu_fqup, ex_intro/
-    | elim (IH G L V) -IH /3 width=2 by frees_zero, fqu_fqup, fqu_lref_O, ex_intro/
+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/
+| cases L -L /3 width=2 by frees_atom, ex_intro/
+  #L #I *
+  [ cases I -I #I [2: #V ] #HG #HL #HT destruct
+    [ elim (IH G L V) -IH
+      /3 width=2 by frees_pair, fqu_fqup, fqu_lref_O, ex_intro/
+    | -IH /3 width=2 by frees_unit, ex_intro/
     ]
-  | #l #HG #HL #HT destruct
-    elim (IH G L (§l)) -IH /3 width=2 by frees_gref_gen, fqu_fqup, fqu_drop, lifts_gref, ex_intro/
+  | #i #HG #HL #HT destruct
+    elim (IH G L (#i)) -IH
+    /3 width=2 by frees_lref, fqu_fqup, fqu_drop, 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 (sor_isfin_ex f1 (⫱f2))
-    /3 width=6 by frees_fwd_isfin, frees_bind, isfin_tl, ex_intro/
-  | elim (IH G L T) -IH // #f2 #HT
-    elim (sor_isfin_ex f1 f2)
-    /3 width=6 by frees_fwd_isfin, frees_flat, ex_intro/ 
+| /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 (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
+  elim (IH G L V) // #f1 #HV
+  elim (IH G L T) -IH // #f2 #HT
+  elim (sor_isfin_ex f1 f2)
+  /3 width=6 by frees_fwd_isfin, frees_flat, ex_intro/
+]
+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.
+#f1 #L #V #Hf1 #f2 #T #Hf2 #f #Hf #p #I
+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
+  lapply (lsubf_inv_refl … H02) -H02 #H02
+  lapply (sor_eq_repl_fwd2 … Hf … H02) -g2 #Hf
+  /2 width=5 by frees_bind/
+| elim (lsubf_inv_unit2 … H02) -H02 * [ #g0 #Y #_ #_ #H destruct ]
+  #z1 #g0 #z #Z #Y #X #H02 #Hz1 #Hz #H0 #H destruct
+  lapply (lsubf_inv_refl … H02) -H02 #H02
+  lapply (frees_mono … Hz1 … Hf1) -Hz1 #H1
+  lapply (sor_eq_repl_back1 … Hz … H02) -g0 #Hz
+  lapply (sor_eq_repl_back2 … Hz … H1) -z1 #Hz
+  lapply (sor_comm … Hz) -Hz #Hz
+  lapply (sor_mono … f Hz ?) // -Hz #H
+  lapply (sor_inv_sle_sn … Hf) -Hf #Hf
+  lapply (frees_eq_repl_back … Hf0 (↑f) ?) /2 width=5 by eq_next/ -z #Hf0
+  @(frees_bind … Hf1 Hf0) -Hf1 -Hf0 (**) (* constructor needed *)
+  /2 width=1 by sor_sle_dx/
+]
+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.
+#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
+lapply (lsubr_lsubf … Hf0 … Hf2) -Hf2 /2 width=5 by lsubr_unit/ #H20
+elim (pn_split f0) * #g0 #H destruct
+[ elim (lsubf_inv_push2 … H20) -H20 #g2 #I #Y #H20 #H2 #H destruct
+  lapply (lsubf_inv_refl … H20) -H20 #H20
+  lapply (sor_eq_repl_back2 … Hf … H20) -g2 #Hf
+  /2 width=5 by ex3_2_intro/
+| elim (lsubf_inv_unit2 … H20) -H20 * [ #g2 #Y #_ #_ #H destruct ]
+  #z1 #z0 #g2 #Z #Y #X #H20 #Hz1 #Hg2 #H2 #H destruct
+  lapply (lsubf_inv_refl … H20) -H20 #H0
+  lapply (frees_mono … Hz1 … Hf1) -Hz1 #H1
+  lapply (sor_eq_repl_back1 … Hg2 … H0) -z0 #Hg2
+  lapply (sor_eq_repl_back2 … Hg2 … H1) -z1 #Hg2
+  @(ex3_2_intro … Hf1 Hf0) -Hf1 -Hf0 (**) (* constructor needed *)
+  /2 width=3 by sor_comm_23_idem/
+]
+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.
+#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
+  lapply (frees_inv_sort … H) -H /2 width=1 by/
+| cases L -L
+  [ #i #HG #HL #HT #f #H destruct -IH
+    elim (frees_inv_atom … H) -H #g #Hg #H destruct /2 width=1 by/
+  | #L #I * [ cases I -I #I [ | #V ] | #i ] #HG #HL #HT #f #H destruct
+    [ elim (frees_inv_unit … H) -H #g #Hg #H destruct /2 width=1 by/
+    | elim (frees_inv_pair … H) -H #g #Hg #H destruct
+      /4 width=2 by fqu_fqup, fqu_lref_O/
+    | elim (frees_inv_lref … H) -H #g #Hg #H destruct
+      /4 width=2 by fqu_fqup/
+    ]
   ]
+| #l #HG #HL #HT #f #H destruct -IH
+  lapply (frees_inv_gref … H) -H /2 width=1 by/
+| #p #I #V #T #HG #HL #HT #f #H destruct
+  elim (frees_inv_bind_void … H) -H /3 width=7 by/
+| #I #V #T #HG #HL #HT #f #H destruct
+  elim (frees_inv_flat … H) -H /3 width=7 by/
 ]
 qed-.