X-Git-Url: http://matita.cs.unibo.it/gitweb/?p=helm.git;a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fstatic_2%2Frelocation%2Fdrops.ma;h=7b29dd1a3acd673ec879ebd243f186b8a768d09a;hp=b934de2a96af15a0bd9b7cc0b6cf71d63fbca023;hb=bd53c4e895203eb049e75434f638f26b5a161a2b;hpb=3b7b8afcb429a60d716d5226a5b6ab0d003228b1

diff --git a/matita/matita/contribs/lambdadelta/static_2/relocation/drops.ma b/matita/matita/contribs/lambdadelta/static_2/relocation/drops.ma
index b934de2a9..7b29dd1a3 100644
--- a/matita/matita/contribs/lambdadelta/static_2/relocation/drops.ma
+++ b/matita/matita/contribs/lambdadelta/static_2/relocation/drops.ma
@@ -27,11 +27,11 @@ include "static_2/relocation/lifts_bind.ma".
                         drop_refl_atom_O2 drop_drop_lt drop_skip_lt
 *)
 inductive drops (b:bool): rtmap → relation lenv ≝
-| drops_atom: ∀f. (b = Ⓣ → 𝐈⦃f⦄) → drops b (f) (⋆) (⋆)
-| drops_drop: ∀f,I,L1,L2. drops b f L1 L2 → drops b (↑f) (L1.ⓘ{I}) L2
+| drops_atom: ∀f. (b = Ⓣ → 𝐈❪f❫) → drops b (f) (⋆) (⋆)
+| drops_drop: ∀f,I,L1,L2. drops b f L1 L2 → drops b (↑f) (L1.ⓘ[I]) L2
 | drops_skip: ∀f,I1,I2,L1,L2.
               drops b f L1 L2 → ⇧*[f] I2 ≘ I1 →
-              drops b (⫯f) (L1.ⓘ{I1}) (L2.ⓘ{I2})
+              drops b (⫯f) (L1.ⓘ[I1]) (L2.ⓘ[I2])
 .
 
 interpretation "uniform slicing (local environment)"
@@ -45,7 +45,7 @@ definition d_liftable1: predicate (relation2 lenv term) ≝
                         ∀U. ⇧*[f] T ≘ U → R L U.
 
 definition d_liftable1_isuni: predicate (relation2 lenv term) ≝
-                              λR. ∀K,T. R K T → ∀b,f,L. ⇩*[b,f] L ≘ K → 𝐔⦃f⦄ →
+                              λR. ∀K,T. R K T → ∀b,f,L. ⇩*[b,f] L ≘ K → 𝐔❪f❫ →
                               ∀U. ⇧*[f] T ≘ U → R L U.
 
 definition d_deliftable1: predicate (relation2 lenv term) ≝
@@ -53,7 +53,7 @@ definition d_deliftable1: predicate (relation2 lenv term) ≝
                           ∀T. ⇧*[f] T ≘ U → R K T.
 
 definition d_deliftable1_isuni: predicate (relation2 lenv term) ≝
-                                λR. ∀L,U. R L U → ∀b,f,K. ⇩*[b,f] L ≘ K → 𝐔⦃f⦄ →
+                                λR. ∀L,U. R L U → ∀b,f,K. ⇩*[b,f] L ≘ K → 𝐔❪f❫ →
                                 ∀T. ⇧*[f] T ≘ U → R K T.
 
 definition d_liftable2_sn: ∀C:Type[0]. ∀S:rtmap → relation C.
@@ -81,13 +81,13 @@ definition d_deliftable2_bi: ∀C:Type[0]. ∀S:rtmap → relation C.
                              ∀T2. S f T2 U2 → R K T1 T2.
 
 definition co_dropable_sn: predicate (rtmap → relation lenv) ≝
-                           λR. ∀b,f,L1,K1. ⇩*[b,f] L1 ≘ K1 → 𝐔⦃f⦄ →
+                           λR. ∀b,f,L1,K1. ⇩*[b,f] L1 ≘ K1 → 𝐔❪f❫ →
                            ∀f2,L2. R f2 L1 L2 → ∀f1. f ~⊚ f1 ≘ f2 →
                            ∃∃K2. R f1 K1 K2 & ⇩*[b,f] L2 ≘ K2.
 
 definition co_dropable_dx: predicate (rtmap → relation lenv) ≝
                            λR. ∀f2,L1,L2. R f2 L1 L2 →
-                           ∀b,f,K2. ⇩*[b,f] L2 ≘ K2 → 𝐔⦃f⦄ →
+                           ∀b,f,K2. ⇩*[b,f] L2 ≘ K2 → 𝐔❪f❫ →
                            ∀f1. f ~⊚ f1 ≘ f2 →
                            ∃∃K1. ⇩*[b,f] L1 ≘ K1 & R f1 K1 K2.
 
@@ -149,7 +149,7 @@ qed-.
 (* Basic inversion lemmas ***************************************************)
 
 fact drops_inv_atom1_aux: ∀b,f,X,Y. ⇩*[b,f] X ≘ Y → X = ⋆ →
-                          Y = ⋆ ∧ (b = Ⓣ → 𝐈⦃f⦄).
+                          Y = ⋆ ∧ (b = Ⓣ → 𝐈❪f❫).
 #b #f #X #Y * -f -X -Y
 [ /3 width=1 by conj/
 | #f #I #L1 #L2 #_ #H destruct
@@ -159,10 +159,10 @@ qed-.
 
 (* Basic_1: includes: drop_gen_sort *)
 (* Basic_2A1: includes: drop_inv_atom1 *)
-lemma drops_inv_atom1: ∀b,f,Y. ⇩*[b,f] ⋆ ≘ Y → Y = ⋆ ∧ (b = Ⓣ → 𝐈⦃f⦄).
+lemma drops_inv_atom1: ∀b,f,Y. ⇩*[b,f] ⋆ ≘ Y → Y = ⋆ ∧ (b = Ⓣ → 𝐈❪f❫).
 /2 width=3 by drops_inv_atom1_aux/ qed-.
 
-fact drops_inv_drop1_aux: ∀b,f,X,Y. ⇩*[b,f] X ≘ Y → ∀g,I,K. X = K.ⓘ{I} → f = ↑g →
+fact drops_inv_drop1_aux: ∀b,f,X,Y. ⇩*[b,f] X ≘ Y → ∀g,I,K. X = K.ⓘ[I] → f = ↑g →
                           ⇩*[b,g] K ≘ Y.
 #b #f #X #Y * -f -X -Y
 [ #f #Hf #g #J #K #H destruct
@@ -173,11 +173,11 @@ qed-.
 
 (* Basic_1: includes: drop_gen_drop *)
 (* Basic_2A1: includes: drop_inv_drop1_lt drop_inv_drop1 *)
-lemma drops_inv_drop1: ∀b,f,I,K,Y. ⇩*[b,↑f] K.ⓘ{I} ≘ Y → ⇩*[b,f] K ≘ Y.
+lemma drops_inv_drop1: ∀b,f,I,K,Y. ⇩*[b,↑f] K.ⓘ[I] ≘ Y → ⇩*[b,f] K ≘ Y.
 /2 width=6 by drops_inv_drop1_aux/ qed-.
 
-fact drops_inv_skip1_aux: ∀b,f,X,Y. ⇩*[b,f] X ≘ Y → ∀g,I1,K1. X = K1.ⓘ{I1} → f = ⫯g →
-                          ∃∃I2,K2. ⇩*[b,g] K1 ≘ K2 & ⇧*[g] I2 ≘ I1 & Y = K2.ⓘ{I2}.
+fact drops_inv_skip1_aux: ∀b,f,X,Y. ⇩*[b,f] X ≘ Y → ∀g,I1,K1. X = K1.ⓘ[I1] → f = ⫯g →
+                          ∃∃I2,K2. ⇩*[b,g] K1 ≘ K2 & ⇧*[g] I2 ≘ I1 & Y = K2.ⓘ[I2].
 #b #f #X #Y * -f -X -Y
 [ #f #Hf #g #J1 #K1 #H destruct
 | #f #I #L1 #L2 #_ #g #J1 #K1 #_ #H2 elim (discr_next_push … H2)
@@ -188,12 +188,12 @@ qed-.
 
 (* Basic_1: includes: drop_gen_skip_l *)
 (* Basic_2A1: includes: drop_inv_skip1 *)
-lemma drops_inv_skip1: ∀b,f,I1,K1,Y. ⇩*[b,⫯f] K1.ⓘ{I1} ≘ Y →
-                       ∃∃I2,K2. ⇩*[b,f] K1 ≘ K2 & ⇧*[f] I2 ≘ I1 & Y = K2.ⓘ{I2}.
+lemma drops_inv_skip1: ∀b,f,I1,K1,Y. ⇩*[b,⫯f] K1.ⓘ[I1] ≘ Y →
+                       ∃∃I2,K2. ⇩*[b,f] K1 ≘ K2 & ⇧*[f] I2 ≘ I1 & Y = K2.ⓘ[I2].
 /2 width=5 by drops_inv_skip1_aux/ qed-.
 
-fact drops_inv_skip2_aux: ∀b,f,X,Y. ⇩*[b,f] X ≘ Y → ∀g,I2,K2. Y = K2.ⓘ{I2} → f = ⫯g →
-                          ∃∃I1,K1. ⇩*[b,g] K1 ≘ K2 & ⇧*[g] I2 ≘ I1 & X = K1.ⓘ{I1}.
+fact drops_inv_skip2_aux: ∀b,f,X,Y. ⇩*[b,f] X ≘ Y → ∀g,I2,K2. Y = K2.ⓘ[I2] → f = ⫯g →
+                          ∃∃I1,K1. ⇩*[b,g] K1 ≘ K2 & ⇧*[g] I2 ≘ I1 & X = K1.ⓘ[I1].
 #b #f #X #Y * -f -X -Y
 [ #f #Hf #g #J2 #K2 #H destruct
 | #f #I #L1 #L2 #_ #g #J2 #K2 #_ #H2 elim (discr_next_push … H2)
@@ -204,14 +204,14 @@ qed-.
 
 (* Basic_1: includes: drop_gen_skip_r *)
 (* Basic_2A1: includes: drop_inv_skip2 *)
-lemma drops_inv_skip2: ∀b,f,I2,X,K2. ⇩*[b,⫯f] X ≘ K2.ⓘ{I2} →
-                       ∃∃I1,K1. ⇩*[b,f] K1 ≘ K2 & ⇧*[f] I2 ≘ I1 & X = K1.ⓘ{I1}.
+lemma drops_inv_skip2: ∀b,f,I2,X,K2. ⇩*[b,⫯f] X ≘ K2.ⓘ[I2] →
+                       ∃∃I1,K1. ⇩*[b,f] K1 ≘ K2 & ⇧*[f] I2 ≘ I1 & X = K1.ⓘ[I1].
 /2 width=5 by drops_inv_skip2_aux/ qed-.
 
 (* Basic forward lemmas *****************************************************)
 
-fact drops_fwd_drop2_aux: ∀b,f2,X,Y. ⇩*[b,f2] X ≘ Y → ∀I,K. Y = K.ⓘ{I} →
-                          ∃∃f1,f. 𝐈⦃f1⦄ & f2 ⊚ ↑f1 ≘ f & ⇩*[b,f] X ≘ K.
+fact drops_fwd_drop2_aux: ∀b,f2,X,Y. ⇩*[b,f2] X ≘ Y → ∀I,K. Y = K.ⓘ[I] →
+                          ∃∃f1,f. 𝐈❪f1❫ & f2 ⊚ ↑f1 ≘ f & ⇩*[b,f] X ≘ K.
 #b #f2 #X #Y #H elim H -f2 -X -Y
 [ #f2 #Hf2 #J #K #H destruct
 | #f2 #I #L1 #L2 #_ #IHL #J #K #H elim (IHL … H) -IHL
@@ -221,14 +221,14 @@ fact drops_fwd_drop2_aux: ∀b,f2,X,Y. ⇩*[b,f2] X ≘ Y → ∀I,K. Y = K.ⓘ{
 ]
 qed-.
 
-lemma drops_fwd_drop2: ∀b,f2,I,X,K. ⇩*[b,f2] X ≘ K.ⓘ{I} →
-                       ∃∃f1,f. 𝐈⦃f1⦄ & f2 ⊚ ↑f1 ≘ f & ⇩*[b,f] X ≘ K.
+lemma drops_fwd_drop2: ∀b,f2,I,X,K. ⇩*[b,f2] X ≘ K.ⓘ[I] →
+                       ∃∃f1,f. 𝐈❪f1❫ & f2 ⊚ ↑f1 ≘ f & ⇩*[b,f] X ≘ K.
 /2 width=4 by drops_fwd_drop2_aux/ qed-.
 
 (* Properties with test for identity ****************************************)
 
 (* Basic_2A1: includes: drop_refl *)
-lemma drops_refl: ∀b,L,f. 𝐈⦃f⦄ → ⇩*[b,f] L ≘ L.
+lemma drops_refl: ∀b,L,f. 𝐈❪f❫ → ⇩*[b,f] L ≘ L.
 #b #L elim L -L /2 width=1 by drops_atom/
 #L #I #IHL #f #Hf elim (isid_inv_gen … Hf) -Hf
 /3 width=1 by drops_skip, liftsb_refl/
@@ -238,15 +238,15 @@ qed.
 
 (* Basic_1: includes: drop_gen_refl *)
 (* Basic_2A1: includes: drop_inv_O2 *)
-lemma drops_fwd_isid: ∀b,f,L1,L2. ⇩*[b,f] L1 ≘ L2 → 𝐈⦃f⦄ → L1 = L2.
+lemma drops_fwd_isid: ∀b,f,L1,L2. ⇩*[b,f] L1 ≘ L2 → 𝐈❪f❫ → L1 = L2.
 #b #f #L1 #L2 #H elim H -f -L1 -L2 //
 [ #f #I #L1 #L2 #_ #_ #H elim (isid_inv_next … H) //
 | /5 width=5 by isid_inv_push, liftsb_fwd_isid, eq_f2, sym_eq/
 ]
 qed-.
 
-lemma drops_after_fwd_drop2: ∀b,f2,I,X,K. ⇩*[b,f2] X ≘ K.ⓘ{I} →
-                             ∀f1,f. 𝐈⦃f1⦄ → f2 ⊚ ↑f1 ≘ f → ⇩*[b,f] X ≘ K.
+lemma drops_after_fwd_drop2: ∀b,f2,I,X,K. ⇩*[b,f2] X ≘ K.ⓘ[I] →
+                             ∀f1,f. 𝐈❪f1❫ → f2 ⊚ ↑f1 ≘ f → ⇩*[b,f] X ≘ K.
 #b #f2 #I #X #K #H #f1 #f #Hf1 #Hf elim (drops_fwd_drop2 … H) -H
 #g1 #g #Hg1 #Hg #HK lapply (after_mono_eq … Hg … Hf ??) -Hg -Hf
 /3 width=5 by drops_eq_repl_back, isid_inv_eq_repl, eq_next/
@@ -254,14 +254,14 @@ qed-.
 
 (* Forward lemmas with test for finite colength *****************************)
 
-lemma drops_fwd_isfin: ∀f,L1,L2. ⇩*[Ⓣ,f] L1 ≘ L2 → 𝐅⦃f⦄.
+lemma drops_fwd_isfin: ∀f,L1,L2. ⇩*[Ⓣ,f] L1 ≘ L2 → 𝐅❪f❫.
 #f #L1 #L2 #H elim H -f -L1 -L2
 /3 width=1 by isfin_next, isfin_push, isfin_isid/
 qed-.
 
 (* Properties with test for uniformity **************************************)
 
-lemma drops_isuni_ex: ∀f. 𝐔⦃f⦄ → ∀L. ∃K. ⇩*[Ⓕ,f] L ≘ K.
+lemma drops_isuni_ex: ∀f. 𝐔❪f❫ → ∀L. ∃K. ⇩*[Ⓕ,f] L ≘ K.
 #f #H elim H -f /4 width=2 by drops_refl, drops_TF, ex_intro/
 #f #_ #g #H #IH destruct * /2 width=2 by ex_intro/
 #L #I elim (IH L) -IH /3 width=2 by drops_drop, ex_intro/
@@ -269,9 +269,9 @@ qed-.
 
 (* Inversion lemmas with test for uniformity ********************************)
 
-lemma drops_inv_isuni: ∀f,L1,L2. ⇩*[Ⓣ,f] L1 ≘ L2 → 𝐔⦃f⦄ →
-                       (𝐈⦃f⦄ ∧ L1 = L2) ∨
-                       ∃∃g,I,K. ⇩*[Ⓣ,g] K ≘ L2 & 𝐔⦃g⦄ & L1 = K.ⓘ{I} & f = ↑g.
+lemma drops_inv_isuni: ∀f,L1,L2. ⇩*[Ⓣ,f] L1 ≘ L2 → 𝐔❪f❫ →
+                       (𝐈❪f❫ ∧ L1 = L2) ∨
+                       ∃∃g,I,K. ⇩*[Ⓣ,g] K ≘ L2 & 𝐔❪g❫ & L1 = K.ⓘ[I] & f = ↑g.
 #f #L1 #L2 * -f -L1 -L2
 [ /4 width=1 by or_introl, conj/
 | /4 width=7 by isuni_inv_next, ex4_3_intro, or_intror/
@@ -280,9 +280,9 @@ lemma drops_inv_isuni: ∀f,L1,L2. ⇩*[Ⓣ,f] L1 ≘ L2 → 𝐔⦃f⦄ →
 qed-.
 
 (* Basic_2A1: was: drop_inv_O1_pair1 *)
-lemma drops_inv_bind1_isuni: ∀b,f,I,K,L2. 𝐔⦃f⦄ → ⇩*[b,f] K.ⓘ{I} ≘ L2 →
-                             (𝐈⦃f⦄ ∧ L2 = K.ⓘ{I}) ∨
-                             ∃∃g. 𝐔⦃g⦄ & ⇩*[b,g] K ≘ L2 & f = ↑g.
+lemma drops_inv_bind1_isuni: ∀b,f,I,K,L2. 𝐔❪f❫ → ⇩*[b,f] K.ⓘ[I] ≘ L2 →
+                             (𝐈❪f❫ ∧ L2 = K.ⓘ[I]) ∨
+                             ∃∃g. 𝐔❪g❫ & ⇩*[b,g] K ≘ L2 & f = ↑g.
 #b #f #I #K #L2 #Hf #H elim (isuni_split … Hf) -Hf * #g #Hg #H0 destruct
 [ lapply (drops_inv_skip1 … H) -H * #Z #Y #HY #HZ #H destruct
   <(drops_fwd_isid … HY Hg) -Y >(liftsb_fwd_isid … HZ Hg) -Z
@@ -292,9 +292,9 @@ lemma drops_inv_bind1_isuni: ∀b,f,I,K,L2. 𝐔⦃f⦄ → ⇩*[b,f] K.ⓘ{I} 
 qed-.
 
 (* Basic_2A1: was: drop_inv_O1_pair2 *)
-lemma drops_inv_bind2_isuni: ∀b,f,I,K,L1. 𝐔⦃f⦄ → ⇩*[b,f] L1 ≘ K.ⓘ{I} →
-                             (𝐈⦃f⦄ ∧ L1 = K.ⓘ{I}) ∨
-                             ∃∃g,I1,K1. 𝐔⦃g⦄ & ⇩*[b,g] K1 ≘ K.ⓘ{I} & L1 = K1.ⓘ{I1} & f = ↑g.
+lemma drops_inv_bind2_isuni: ∀b,f,I,K,L1. 𝐔❪f❫ → ⇩*[b,f] L1 ≘ K.ⓘ[I] →
+                             (𝐈❪f❫ ∧ L1 = K.ⓘ[I]) ∨
+                             ∃∃g,I1,K1. 𝐔❪g❫ & ⇩*[b,g] K1 ≘ K.ⓘ[I] & L1 = K1.ⓘ[I1] & f = ↑g.
 #b #f #I #K *
 [ #Hf #H elim (drops_inv_atom1 … H) -H #H destruct
 | #L1 #I1 #Hf #H elim (drops_inv_bind1_isuni … Hf H) -Hf -H *
@@ -304,16 +304,16 @@ lemma drops_inv_bind2_isuni: ∀b,f,I,K,L1. 𝐔⦃f⦄ → ⇩*[b,f] L1 ≘ K.
 ]
 qed-.
 
-lemma drops_inv_bind2_isuni_next: ∀b,f,I,K,L1. 𝐔⦃f⦄ → ⇩*[b,↑f] L1 ≘ K.ⓘ{I} →
-                                  ∃∃I1,K1. ⇩*[b,f] K1 ≘ K.ⓘ{I} & L1 = K1.ⓘ{I1}.
+lemma drops_inv_bind2_isuni_next: ∀b,f,I,K,L1. 𝐔❪f❫ → ⇩*[b,↑f] L1 ≘ K.ⓘ[I] →
+                                  ∃∃I1,K1. ⇩*[b,f] K1 ≘ K.ⓘ[I] & L1 = K1.ⓘ[I1].
 #b #f #I #K #L1 #Hf #H elim (drops_inv_bind2_isuni … H) -H /2 width=3 by isuni_next/ -Hf *
 [ #H elim (isid_inv_next … H) -H //
 | /2 width=4 by ex2_2_intro/
 ]
 qed-.
 
-fact drops_inv_TF_aux: ∀f,L1,L2. ⇩*[Ⓕ,f] L1 ≘ L2 → 𝐔⦃f⦄ →
-                       ∀I,K. L2 = K.ⓘ{I} → ⇩*[Ⓣ,f] L1 ≘ K.ⓘ{I}.
+fact drops_inv_TF_aux: ∀f,L1,L2. ⇩*[Ⓕ,f] L1 ≘ L2 → 𝐔❪f❫ →
+                       ∀I,K. L2 = K.ⓘ[I] → ⇩*[Ⓣ,f] L1 ≘ K.ⓘ[I].
 #f #L1 #L2 #H elim H -f -L1 -L2
 [ #f #_ #_ #J #K #H destruct
 | #f #I #L1 #L2 #_ #IH #Hf #J #K #H destruct
@@ -326,16 +326,16 @@ fact drops_inv_TF_aux: ∀f,L1,L2. ⇩*[Ⓕ,f] L1 ≘ L2 → 𝐔⦃f⦄ →
 qed-.
 
 (* Basic_2A1: includes: drop_inv_FT *)
-lemma drops_inv_TF: ∀f,I,L,K. ⇩*[Ⓕ,f] L ≘ K.ⓘ{I} → 𝐔⦃f⦄ → ⇩*[Ⓣ,f] L ≘ K.ⓘ{I}.
+lemma drops_inv_TF: ∀f,I,L,K. ⇩*[Ⓕ,f] L ≘ K.ⓘ[I] → 𝐔❪f❫ → ⇩*[Ⓣ,f] L ≘ K.ⓘ[I].
 /2 width=3 by drops_inv_TF_aux/ qed-.
 
 (* Basic_2A1: includes: drop_inv_gen *)
-lemma drops_inv_gen: ∀b,f,I,L,K. ⇩*[b,f] L ≘ K.ⓘ{I} → 𝐔⦃f⦄ → ⇩*[Ⓣ,f] L ≘ K.ⓘ{I}.
+lemma drops_inv_gen: ∀b,f,I,L,K. ⇩*[b,f] L ≘ K.ⓘ[I] → 𝐔❪f❫ → ⇩*[Ⓣ,f] L ≘ K.ⓘ[I].
 * /2 width=1 by drops_inv_TF/
 qed-.
 
 (* Basic_2A1: includes: drop_inv_T *)
-lemma drops_inv_F: ∀b,f,I,L,K. ⇩*[Ⓕ,f] L ≘ K.ⓘ{I} → 𝐔⦃f⦄ → ⇩*[b,f] L ≘ K.ⓘ{I}.
+lemma drops_inv_F: ∀b,f,I,L,K. ⇩*[Ⓕ,f] L ≘ K.ⓘ[I] → 𝐔❪f❫ → ⇩*[b,f] L ≘ K.ⓘ[I].
 * /2 width=1 by drops_inv_TF/
 qed-.
 
@@ -343,13 +343,13 @@ qed-.
 
 (* Basic_1: was: drop_S *)
 (* Basic_2A1: was: drop_fwd_drop2 *)
-lemma drops_isuni_fwd_drop2: ∀b,f,I,X,K. 𝐔⦃f⦄ → ⇩*[b,f] X ≘ K.ⓘ{I} → ⇩*[b,↑f] X ≘ K.
+lemma drops_isuni_fwd_drop2: ∀b,f,I,X,K. 𝐔❪f❫ → ⇩*[b,f] X ≘ K.ⓘ[I] → ⇩*[b,↑f] X ≘ K.
 /3 width=7 by drops_after_fwd_drop2, after_isid_isuni/ qed-.
 
 (* Inversion lemmas with uniform relocations ********************************)
 
 lemma drops_inv_atom2: ∀b,L,f. ⇩*[b,f] L ≘ ⋆ →
-                       ∃∃n,f1. ⇩*[b,𝐔❴n❵] L ≘ ⋆ & 𝐔❴n❵ ⊚ f1 ≘ f.
+                       ∃∃n,f1. ⇩*[b,𝐔❨n❩] L ≘ ⋆ & 𝐔❨n❩ ⊚ f1 ≘ f.
 #b #L elim L -L
 [ /3 width=4 by drops_atom, after_isid_sn, ex2_2_intro/
 | #L #I #IH #f #H elim (pn_split f) * #g #H0 destruct
@@ -361,7 +361,7 @@ lemma drops_inv_atom2: ∀b,L,f. ⇩*[b,f] L ≘ ⋆ →
 qed-.
 
 lemma drops_inv_succ: ∀L1,L2,i. ⇩*[↑i] L1 ≘ L2 →
-                      ∃∃I,K. ⇩*[i] K ≘ L2 & L1 = K.ⓘ{I}.
+                      ∃∃I,K. ⇩*[i] K ≘ L2 & L1 = K.ⓘ[I].
 #L1 #L2 #i #H elim (drops_inv_isuni … H) -H // *
 [ #H elim (isid_inv_next … H) -H //
 | /2 width=4 by ex2_2_intro/
@@ -370,7 +370,7 @@ qed-.
 
 (* Properties with uniform relocations **************************************)
 
-lemma drops_F_uni: ∀L,i. ⇩*[Ⓕ,𝐔❴i❵] L ≘ ⋆ ∨ ∃∃I,K. ⇩*[i] L ≘ K.ⓘ{I}.
+lemma drops_F_uni: ∀L,i. ⇩*[Ⓕ,𝐔❨i❩] L ≘ ⋆ ∨ ∃∃I,K. ⇩*[i] L ≘ K.ⓘ[I].
 #L elim L -L /2 width=1 by or_introl/
 #L #I #IH * /4 width=3 by drops_refl, ex1_2_intro, or_intror/
 #i elim (IH i) -IH /3 width=1 by drops_drop, or_introl/
@@ -378,7 +378,7 @@ lemma drops_F_uni: ∀L,i. ⇩*[Ⓕ,𝐔❴i❵] L ≘ ⋆ ∨ ∃∃I,K. ⇩*[i
 qed-.
 
 (* Basic_2A1: includes: drop_split *)
-lemma drops_split_trans: ∀b,f,L1,L2. ⇩*[b,f] L1 ≘ L2 → ∀f1,f2. f1 ⊚ f2 ≘ f → 𝐔⦃f1⦄ →
+lemma drops_split_trans: ∀b,f,L1,L2. ⇩*[b,f] L1 ≘ L2 → ∀f1,f2. f1 ⊚ f2 ≘ f → 𝐔❪f1❫ →
                          ∃∃L. ⇩*[b,f1] L1 ≘ L & ⇩*[b,f2] L ≘ L2.
 #b #f #L1 #L2 #H elim H -f -L1 -L2
 [ #f #H0f #f1 #f2 #Hf #Hf1 @(ex2_intro … (⋆)) @drops_atom
@@ -398,7 +398,7 @@ lemma drops_split_trans: ∀b,f,L1,L2. ⇩*[b,f] L1 ≘ L2 → ∀f1,f2. f1 ⊚
 ]
 qed-.
 
-lemma drops_split_div: ∀b,f1,L1,L. ⇩*[b,f1] L1 ≘ L → ∀f2,f. f1 ⊚ f2 ≘ f → 𝐔⦃f2⦄ →
+lemma drops_split_div: ∀b,f1,L1,L. ⇩*[b,f1] L1 ≘ L → ∀f2,f. f1 ⊚ f2 ≘ f → 𝐔❪f2❫ →
                        ∃∃L2. ⇩*[Ⓕ,f2] L ≘ L2 & ⇩*[Ⓕ,f] L1 ≘ L2.
 #b #f1 #L1 #L #H elim H -f1 -L1 -L
 [ #f1 #Hf1 #f2 #f #Hf #Hf2 @(ex2_intro … (⋆)) @drops_atom #H destruct
@@ -418,13 +418,13 @@ qed-.
 
 (* Properties with application **********************************************)
 
-lemma drops_tls_at: ∀f,i1,i2. @⦃i1,f⦄ ≘ i2 →
+lemma drops_tls_at: ∀f,i1,i2. @❪i1,f❫ ≘ i2 →
                     ∀b,L1,L2. ⇩*[b,⫱*[i2]f] L1 ≘ L2 →
                     ⇩*[b,⫯⫱*[↑i2]f] L1 ≘ L2.
 /3 width=3 by drops_eq_repl_fwd, at_inv_tls/ qed-.
 
-lemma drops_split_trans_bind2: ∀b,f,I,L,K0. ⇩*[b,f] L ≘ K0.ⓘ{I} → ∀i. @⦃O,f⦄ ≘ i →
-                               ∃∃J,K. ⇩*[i]L ≘ K.ⓘ{J} & ⇩*[b,⫱*[↑i]f] K ≘ K0 & ⇧*[⫱*[↑i]f] I ≘ J.
+lemma drops_split_trans_bind2: ∀b,f,I,L,K0. ⇩*[b,f] L ≘ K0.ⓘ[I] → ∀i. @❪O,f❫ ≘ i →
+                               ∃∃J,K. ⇩*[i]L ≘ K.ⓘ[J] & ⇩*[b,⫱*[↑i]f] K ≘ K0 & ⇧*[⫱*[↑i]f] I ≘ J.
 #b #f #I #L #K0 #H #i #Hf
 elim (drops_split_trans … H) -H [ |5: @(after_uni_dx … Hf) |2,3: skip ] /2 width=1 by after_isid_dx/ #Y #HLY #H
 lapply (drops_tls_at … Hf … H) -H #H