X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Fmultiple%2Ffrees.ma;h=fe9292e9de49005aaa36479f85834c6d65573483;hb=c60524dec7ace912c416a90d6b926bee8553250b;hp=f7269bc09050092caa706acdf83e06b3d6789da7;hpb=f10cfe417b6b8ec1c7ac85c6ecf5fb1b3fdf37db;p=helm.git

diff --git a/matita/matita/contribs/lambdadelta/basic_2/multiple/frees.ma b/matita/matita/contribs/lambdadelta/basic_2/multiple/frees.ma
index f7269bc09..fe9292e9d 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/multiple/frees.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/multiple/frees.ma
@@ -20,75 +20,75 @@ include "basic_2/substitution/drop.ma".
 (* CONTEXT-SENSITIVE FREE VARIABLES *****************************************)
 
 inductive frees: relation4 ynat lenv term nat ≝
-| frees_eq: ∀L,U,d,i. (∀T. ⬆[i, 1] T ≡ U → ⊥) → frees d L U i
-| frees_be: ∀I,L,K,U,W,d,i,j. d ≤ yinj j → j < i →
+| frees_eq: ∀L,U,l,i. (∀T. ⬆[i, 1] T ≡ U → ⊥) → frees l L U i
+| frees_be: ∀I,L,K,U,W,l,i,j. l ≤ yinj j → j < i →
             (∀T. ⬆[j, 1] T ≡ U → ⊥) → ⬇[j]L ≡ K.ⓑ{I}W →
-            frees 0 K W (i-j-1) → frees d L U i.
+            frees 0 K W (i-j-1) → frees l L U i.
 
 interpretation
    "context-sensitive free variables (term)"
-   'FreeStar L i d U = (frees d L U i).
+   'FreeStar L i l U = (frees l L U i).
 
 definition frees_trans: predicate (relation3 lenv term term) ≝
                         λR. ∀L,U1,U2,i. R L U1 U2 → L ⊢ i ϵ 𝐅*[0]⦃U2⦄ → L ⊢ i ϵ 𝐅*[0]⦃U1⦄.
 
 (* Basic inversion lemmas ***************************************************)
 
-lemma frees_inv: ∀L,U,d,i. L ⊢ i ϵ 𝐅*[d]⦃U⦄ →
+lemma frees_inv: ∀L,U,l,i. L ⊢ i ϵ 𝐅*[l]⦃U⦄ →
                  (∀T. ⬆[i, 1] T ≡ U → ⊥) ∨
-                 ∃∃I,K,W,j. d ≤ yinj j & j < i & (∀T. ⬆[j, 1] T ≡ U → ⊥) &
+                 ∃∃I,K,W,j. l ≤ yinj j & j < i & (∀T. ⬆[j, 1] T ≡ U → ⊥) &
                             ⬇[j]L ≡ K.ⓑ{I}W & K ⊢ (i-j-1) ϵ 𝐅*[yinj 0]⦃W⦄.
-#L #U #d #i * -L -U -d -i /4 width=9 by ex5_4_intro, or_intror, or_introl/
+#L #U #l #i * -L -U -l -i /4 width=9 by ex5_4_intro, or_intror, or_introl/
 qed-.
 
-lemma frees_inv_sort: ∀L,d,i,k. L ⊢ i ϵ 𝐅*[d]⦃⋆k⦄ → ⊥.
-#L #d #i #k #H elim (frees_inv … H) -H [|*] /2 width=2 by/
+lemma frees_inv_sort: ∀L,l,i,k. L ⊢ i ϵ 𝐅*[l]⦃⋆k⦄ → ⊥.
+#L #l #i #k #H elim (frees_inv … H) -H [|*] /2 width=2 by/
 qed-.
 
-lemma frees_inv_gref: ∀L,d,i,p. L ⊢ i ϵ 𝐅*[d]⦃§p⦄ → ⊥.
-#L #d #i #p #H elim (frees_inv … H) -H [|*] /2 width=2 by/
+lemma frees_inv_gref: ∀L,l,i,p. L ⊢ i ϵ 𝐅*[l]⦃§p⦄ → ⊥.
+#L #l #i #p #H elim (frees_inv … H) -H [|*] /2 width=2 by/
 qed-.
 
-lemma frees_inv_lref: ∀L,d,j,i. L ⊢ i ϵ 𝐅*[d]⦃#j⦄ →
+lemma frees_inv_lref: ∀L,l,j,i. L ⊢ i ϵ 𝐅*[l]⦃#j⦄ →
                       j = i ∨
-                      ∃∃I,K,W. d ≤ yinj j & j < i & ⬇[j] L ≡ K.ⓑ{I}W & K ⊢ (i-j-1) ϵ 𝐅*[yinj 0]⦃W⦄.
-#L #d #x #i #H elim (frees_inv … H) -H
+                      ∃∃I,K,W. l ≤ yinj j & j < i & ⬇[j] L ≡ K.ⓑ{I}W & K ⊢ (i-j-1) ϵ 𝐅*[yinj 0]⦃W⦄.
+#L #l #x #i #H elim (frees_inv … H) -H
 [ /4 width=2 by nlift_inv_lref_be_SO, or_introl/
-| * #I #K #W #j #Hdj #Hji #Hnx #HLK #HW
+| * #I #K #W #j #Hlj #Hji #Hnx #HLK #HW
   >(nlift_inv_lref_be_SO … Hnx) -x /3 width=5 by ex4_3_intro, or_intror/
 ]
 qed-.
 
-lemma frees_inv_lref_free: ∀L,d,j,i. L ⊢ i ϵ 𝐅*[d]⦃#j⦄ → |L| ≤ j → j = i.
-#L #d #j #i #H #Hj elim (frees_inv_lref … H) -H //
+lemma frees_inv_lref_free: ∀L,l,j,i. L ⊢ i ϵ 𝐅*[l]⦃#j⦄ → |L| ≤ j → j = i.
+#L #l #j #i #H #Hj elim (frees_inv_lref … H) -H //
 * #I #K #W #_ #_ #HLK lapply (drop_fwd_length_lt2 … HLK) -I
 #H elim (lt_refl_false j) /2 width=3 by lt_to_le_to_lt/
 qed-.
 
-lemma frees_inv_lref_skip: ∀L,d,j,i. L ⊢ i ϵ 𝐅*[d]⦃#j⦄ → yinj j < d → j = i.
-#L #d #j #i #H #Hjd elim (frees_inv_lref … H) -H //
-* #I #K #W #Hdj elim (ylt_yle_false … Hdj) -Hdj //
+lemma frees_inv_lref_skip: ∀L,l,j,i. L ⊢ i ϵ 𝐅*[l]⦃#j⦄ → yinj j < l → j = i.
+#L #l #j #i #H #Hjl elim (frees_inv_lref … H) -H //
+* #I #K #W #Hlj elim (ylt_yle_false … Hlj) -Hlj //
 qed-. 
 
-lemma frees_inv_lref_ge: ∀L,d,j,i. L ⊢ i ϵ 𝐅*[d]⦃#j⦄ → i ≤ j → j = i.
-#L #d #j #i #H #Hij elim (frees_inv_lref … H) -H //
-* #I #K #W #_ #Hji elim (lt_refl_false j) -I -L -K -W -d /2 width=3 by lt_to_le_to_lt/
+lemma frees_inv_lref_ge: ∀L,l,j,i. L ⊢ i ϵ 𝐅*[l]⦃#j⦄ → i ≤ j → j = i.
+#L #l #j #i #H #Hij elim (frees_inv_lref … H) -H //
+* #I #K #W #_ #Hji elim (lt_refl_false j) -I -L -K -W -l /2 width=3 by lt_to_le_to_lt/
 qed-.
 
-lemma frees_inv_lref_lt: ∀L,d,j,i.L ⊢ i ϵ 𝐅*[d]⦃#j⦄ → j < i →
-                         ∃∃I,K,W. d ≤ yinj j & ⬇[j] L ≡ K.ⓑ{I}W & K ⊢ (i-j-1) ϵ 𝐅*[yinj 0]⦃W⦄.
-#L #d #j #i #H #Hji elim (frees_inv_lref … H) -H
+lemma frees_inv_lref_lt: ∀L,l,j,i.L ⊢ i ϵ 𝐅*[l]⦃#j⦄ → j < i →
+                         ∃∃I,K,W. l ≤ yinj j & ⬇[j] L ≡ K.ⓑ{I}W & K ⊢ (i-j-1) ϵ 𝐅*[yinj 0]⦃W⦄.
+#L #l #j #i #H #Hji elim (frees_inv_lref … H) -H
 [ #H elim (lt_refl_false j) //
 | * /2 width=5 by ex3_3_intro/
 ]
 qed-.
 
-lemma frees_inv_bind: ∀a,I,L,W,U,d,i. L ⊢ i ϵ 𝐅*[d]⦃ⓑ{a,I}W.U⦄ →
-                      L ⊢ i ϵ 𝐅*[d]⦃W⦄ ∨ L.ⓑ{I}W ⊢ i+1 ϵ 𝐅*[⫯d]⦃U⦄ .
-#a #J #L #V #U #d #i #H elim (frees_inv … H) -H
+lemma frees_inv_bind: ∀a,I,L,W,U,l,i. L ⊢ i ϵ 𝐅*[l]⦃ⓑ{a,I}W.U⦄ →
+                      L ⊢ i ϵ 𝐅*[l]⦃W⦄ ∨ L.ⓑ{I}W ⊢ i+1 ϵ 𝐅*[⫯l]⦃U⦄ .
+#a #J #L #V #U #l #i #H elim (frees_inv … H) -H
 [ #HnX elim (nlift_inv_bind … HnX) -HnX
   /4 width=2 by frees_eq, or_intror, or_introl/
-| * #I #K #W #j #Hdj #Hji #HnX #HLK #HW elim (nlift_inv_bind … HnX) -HnX
+| * #I #K #W #j #Hlj #Hji #HnX #HLK #HW elim (nlift_inv_bind … HnX) -HnX
   [ /4 width=9 by frees_be, or_introl/
   | #HnT @or_intror @(frees_be … HnT) -HnT
     [4,5,6: /2 width=1 by drop_drop, yle_succ, lt_minus_to_plus/
@@ -99,58 +99,58 @@ lemma frees_inv_bind: ∀a,I,L,W,U,d,i. L ⊢ i ϵ 𝐅*[d]⦃ⓑ{a,I}W.U⦄ →
 ]
 qed-.
 
-lemma frees_inv_flat: ∀I,L,W,U,d,i. L ⊢ i ϵ 𝐅*[d]⦃ⓕ{I}W.U⦄ →
-                      L ⊢ i ϵ 𝐅*[d]⦃W⦄ ∨ L ⊢ i ϵ 𝐅*[d]⦃U⦄ .
-#J #L #V #U #d #i #H elim (frees_inv … H) -H
+lemma frees_inv_flat: ∀I,L,W,U,l,i. L ⊢ i ϵ 𝐅*[l]⦃ⓕ{I}W.U⦄ →
+                      L ⊢ i ϵ 𝐅*[l]⦃W⦄ ∨ L ⊢ i ϵ 𝐅*[l]⦃U⦄ .
+#J #L #V #U #l #i #H elim (frees_inv … H) -H
 [ #HnX elim (nlift_inv_flat … HnX) -HnX
   /4 width=2 by frees_eq, or_intror, or_introl/
-| * #I #K #W #j #Hdj #Hji #HnX #HLK #HW elim (nlift_inv_flat … HnX) -HnX
+| * #I #K #W #j #Hlj #Hji #HnX #HLK #HW elim (nlift_inv_flat … HnX) -HnX
   /4 width=9 by frees_be, or_intror, or_introl/
 ]
 qed-.
 
 (* Basic properties *********************************************************)
 
-lemma frees_lref_eq: ∀L,d,i. L ⊢ i ϵ 𝐅*[d]⦃#i⦄.
+lemma frees_lref_eq: ∀L,l,i. L ⊢ i ϵ 𝐅*[l]⦃#i⦄.
 /3 width=7 by frees_eq, lift_inv_lref2_be/ qed.
 
-lemma frees_lref_be: ∀I,L,K,W,d,i,j. d ≤ yinj j → j < i → ⬇[j]L ≡ K.ⓑ{I}W →
-                     K ⊢ i-j-1 ϵ 𝐅*[0]⦃W⦄ → L ⊢ i ϵ 𝐅*[d]⦃#j⦄.
+lemma frees_lref_be: ∀I,L,K,W,l,i,j. l ≤ yinj j → j < i → ⬇[j]L ≡ K.ⓑ{I}W →
+                     K ⊢ i-j-1 ϵ 𝐅*[0]⦃W⦄ → L ⊢ i ϵ 𝐅*[l]⦃#j⦄.
 /3 width=9 by frees_be, lift_inv_lref2_be/ qed.
 
-lemma frees_bind_sn: ∀a,I,L,W,U,d,i. L ⊢ i ϵ 𝐅*[d]⦃W⦄ →
-                     L ⊢ i ϵ 𝐅*[d]⦃ⓑ{a,I}W.U⦄.
-#a #I #L #W #U #d #i #H elim (frees_inv … H) -H [|*]
+lemma frees_bind_sn: ∀a,I,L,W,U,l,i. L ⊢ i ϵ 𝐅*[l]⦃W⦄ →
+                     L ⊢ i ϵ 𝐅*[l]⦃ⓑ{a,I}W.U⦄.
+#a #I #L #W #U #l #i #H elim (frees_inv … H) -H [|*]
 /4 width=9 by frees_be, frees_eq, nlift_bind_sn/
 qed.
 
-lemma frees_bind_dx: ∀a,I,L,W,U,d,i. L.ⓑ{I}W ⊢ i+1 ϵ 𝐅*[⫯d]⦃U⦄ →
-                     L ⊢ i ϵ 𝐅*[d]⦃ⓑ{a,I}W.U⦄.
-#a #J #L #V #U #d #i #H elim (frees_inv … H) -H
+lemma frees_bind_dx: ∀a,I,L,W,U,l,i. L.ⓑ{I}W ⊢ i+1 ϵ 𝐅*[⫯l]⦃U⦄ →
+                     L ⊢ i ϵ 𝐅*[l]⦃ⓑ{a,I}W.U⦄.
+#a #J #L #V #U #l #i #H elim (frees_inv … H) -H
 [ /4 width=9 by frees_eq, nlift_bind_dx/
-| * #I #K #W #j #Hdj #Hji #HnU #HLK #HW
-  elim (yle_inv_succ1 … Hdj) -Hdj <yminus_SO2 #Hyj #H
+| * #I #K #W #j #Hlj #Hji #HnU #HLK #HW
+  elim (yle_inv_succ1 … Hlj) -Hlj <yminus_SO2 #Hyj #H
   lapply (ylt_O … H) -H #Hj
   >(plus_minus_m_m j 1) in HnU; // <minus_le_minus_minus_comm in HW;
   /4 width=9 by frees_be, nlift_bind_dx, drop_inv_drop1_lt, lt_plus_to_minus/
 ]
 qed.
 
-lemma frees_flat_sn: ∀I,L,W,U,d,i. L ⊢ i ϵ 𝐅*[d]⦃W⦄ →
-                     L ⊢ i ϵ 𝐅*[d]⦃ⓕ{I}W.U⦄.
-#I #L #W #U #d #i #H elim (frees_inv … H) -H [|*]
+lemma frees_flat_sn: ∀I,L,W,U,l,i. L ⊢ i ϵ 𝐅*[l]⦃W⦄ →
+                     L ⊢ i ϵ 𝐅*[l]⦃ⓕ{I}W.U⦄.
+#I #L #W #U #l #i #H elim (frees_inv … H) -H [|*]
 /4 width=9 by frees_be, frees_eq, nlift_flat_sn/
 qed.
 
-lemma frees_flat_dx: ∀I,L,W,U,d,i. L ⊢ i ϵ 𝐅*[d]⦃U⦄ →
-                     L ⊢ i ϵ 𝐅*[d]⦃ⓕ{I}W.U⦄.
-#I #L #W #U #d #i #H elim (frees_inv … H) -H [|*]
+lemma frees_flat_dx: ∀I,L,W,U,l,i. L ⊢ i ϵ 𝐅*[l]⦃U⦄ →
+                     L ⊢ i ϵ 𝐅*[l]⦃ⓕ{I}W.U⦄.
+#I #L #W #U #l #i #H elim (frees_inv … H) -H [|*]
 /4 width=9 by frees_be, frees_eq, nlift_flat_dx/
 qed.
 
-lemma frees_weak: ∀L,U,d1,i. L ⊢ i ϵ 𝐅*[d1]⦃U⦄ →
-                  ∀d2. d2 ≤ d1 → L ⊢ i ϵ 𝐅*[d2]⦃U⦄.
-#L #U #d1 #i #H elim H -L -U -d1 -i
+lemma frees_weak: ∀L,U,l1,i. L ⊢ i ϵ 𝐅*[l1]⦃U⦄ →
+                  ∀l2. l2 ≤ l1 → L ⊢ i ϵ 𝐅*[l2]⦃U⦄.
+#L #U #l1 #i #H elim H -L -U -l1 -i
 /3 width=9 by frees_be, frees_eq, yle_trans/
 qed-.