X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fground_2%2Frelocation%2Frtmap_coafter.ma;h=0eb098301ceca87f2f6a1daf1b8705e3eb2139b7;hb=bd53c4e895203eb049e75434f638f26b5a161a2b;hp=cb5ede22b61de36de2ab149c6d9711148486000a;hpb=0c7cb3c503c0fcab104ad89ebc88683dc9830d06;p=helm.git

diff --git a/matita/matita/contribs/lambdadelta/ground_2/relocation/rtmap_coafter.ma b/matita/matita/contribs/lambdadelta/ground_2/relocation/rtmap_coafter.ma
index cb5ede22b..0eb098301 100644
--- a/matita/matita/contribs/lambdadelta/ground_2/relocation/rtmap_coafter.ma
+++ b/matita/matita/contribs/lambdadelta/ground_2/relocation/rtmap_coafter.ma
@@ -20,30 +20,30 @@ include "ground_2/relocation/rtmap_after.ma".
 
 coinductive coafter: relation3 rtmap rtmap rtmap ≝
 | coafter_refl: ∀f1,f2,f,g1,g2,g. coafter f1 f2 f →
-                ↑f1 = g1 → ↑f2 = g2 → ↑f = g → coafter g1 g2 g
+                ⫯f1 = g1 → ⫯f2 = g2 → ⫯f = g → coafter g1 g2 g
 | coafter_push: ∀f1,f2,f,g1,g2,g. coafter f1 f2 f →
-                ↑f1 = g1 → ⫯f2 = g2 → ⫯f = g → coafter g1 g2 g
+                ⫯f1 = g1 → ↑f2 = g2 → ↑f = g → coafter g1 g2 g
 | coafter_next: ∀f1,f2,f,g1,g. coafter f1 f2 f →
-                ⫯f1 = g1 → ↑f = g → coafter g1 f2 g
+                ↑f1 = g1 → ⫯f = g → coafter g1 f2 g
 .
 
 interpretation "relational co-composition (rtmap)"
    'RCoAfter f1 f2 f = (coafter f1 f2 f).
 
 definition H_coafter_inj: predicate rtmap ≝
-                          λf1. 𝐓⦃f1⦄ →
-                          ∀f,f21,f22. f1 ~⊚ f21 ≡ f → f1 ~⊚ f22 ≡ f → f21 ≗ f22.
+                          λf1. 𝐓❪f1❫ →
+                          ∀f,f21,f22. f1 ~⊚ f21 ≘ f → f1 ~⊚ f22 ≘ f → f21 ≡ f22.
 
 definition H_coafter_fwd_isid2: predicate rtmap ≝
-                                λf1. ∀f2,f. f1 ~⊚ f2 ≡ f → 𝐓⦃f1⦄ → 𝐈⦃f⦄ → 𝐈⦃f2⦄.
+                                λf1. ∀f2,f. f1 ~⊚ f2 ≘ f → 𝐓❪f1❫ → 𝐈❪f❫ → 𝐈❪f2❫.
 
 definition H_coafter_isfin2_fwd: predicate rtmap ≝
-                                 λf1. ∀f2. 𝐅⦃f2⦄ → 𝐓⦃f1⦄ → ∀f. f1 ~⊚ f2 ≡ f →  𝐅⦃f⦄.
+                                 λf1. ∀f2. 𝐅❪f2❫ → 𝐓❪f1❫ → ∀f. f1 ~⊚ f2 ≘ f →  𝐅❪f❫.
 
 (* Basic inversion lemmas ***************************************************)
 
-lemma coafter_inv_ppx: ∀g1,g2,g. g1 ~⊚ g2 ≡ g → ∀f1,f2. ↑f1 = g1 → ↑f2 = g2 →
-                       ∃∃f. f1 ~⊚ f2 ≡ f & ↑f = g.
+lemma coafter_inv_ppx: ∀g1,g2,g. g1 ~⊚ g2 ≘ g → ∀f1,f2. ⫯f1 = g1 → ⫯f2 = g2 →
+                       ∃∃f. f1 ~⊚ f2 ≘ f & ⫯f = g.
 #g1 #g2 #g * -g1 -g2 -g #f1 #f2 #f #g1
 [ #g2 #g #Hf #H1 #H2 #H #x1 #x2 #Hx1 #Hx2 destruct
   >(injective_push … Hx1) >(injective_push … Hx2) -x2 -x1
@@ -55,8 +55,8 @@ lemma coafter_inv_ppx: ∀g1,g2,g. g1 ~⊚ g2 ≡ g → ∀f1,f2. ↑f1 = g1 →
 ]
 qed-.
 
-lemma coafter_inv_pnx: ∀g1,g2,g. g1 ~⊚ g2 ≡ g → ∀f1,f2. ↑f1 = g1 → ⫯f2 = g2 →
-                       ∃∃f. f1 ~⊚ f2 ≡ f & ⫯f = g.
+lemma coafter_inv_pnx: ∀g1,g2,g. g1 ~⊚ g2 ≘ g → ∀f1,f2. ⫯f1 = g1 → ↑f2 = g2 →
+                       ∃∃f. f1 ~⊚ f2 ≘ f & ↑f = g.
 #g1 #g2 #g * -g1 -g2 -g #f1 #f2 #f #g1
 [ #g2 #g #_ #_ #H2 #_ #x1 #x2 #_ #Hx2 destruct
   elim (discr_next_push … Hx2)
@@ -68,8 +68,8 @@ lemma coafter_inv_pnx: ∀g1,g2,g. g1 ~⊚ g2 ≡ g → ∀f1,f2. ↑f1 = g1 →
 ]
 qed-.
 
-lemma coafter_inv_nxx: ∀g1,f2,g. g1 ~⊚ f2 ≡ g → ∀f1. ⫯f1 = g1 →
-                       ∃∃f. f1 ~⊚ f2 ≡ f & ↑f = g.
+lemma coafter_inv_nxx: ∀g1,f2,g. g1 ~⊚ f2 ≘ g → ∀f1. ↑f1 = g1 →
+                       ∃∃f. f1 ~⊚ f2 ≘ f & ⫯f = g.
 #g1 #f2 #g * -g1 -f2 -g #f1 #f2 #f #g1
 [ #g2 #g #_ #H1 #_ #_ #x1 #Hx1 destruct
   elim (discr_next_push … Hx1)
@@ -83,77 +83,85 @@ qed-.
 
 (* Advanced inversion lemmas ************************************************)
 
-lemma coafter_inv_ppp: ∀g1,g2,g. g1 ~⊚ g2 ≡ g →
-                       ∀f1,f2,f. ↑f1 = g1 → ↑f2 = g2 → ↑f = g → f1 ~⊚ f2 ≡ f.
+lemma coafter_inv_ppp: ∀g1,g2,g. g1 ~⊚ g2 ≘ g →
+                       ∀f1,f2,f. ⫯f1 = g1 → ⫯f2 = g2 → ⫯f = g → f1 ~⊚ f2 ≘ f.
 #g1 #g2 #g #Hg #f1 #f2 #f #H1 #H2 #H
 elim (coafter_inv_ppx … Hg … H1 H2) -g1 -g2 #x #Hf #Hx destruct
 <(injective_push … Hx) -f //
 qed-.
 
-lemma coafter_inv_ppn: ∀g1,g2,g. g1 ~⊚ g2 ≡ g →
-                       ∀f1,f2,f. ↑f1 = g1 → ↑f2 = g2 → ⫯f = g → ⊥.
+lemma coafter_inv_ppn: ∀g1,g2,g. g1 ~⊚ g2 ≘ g →
+                       ∀f1,f2,f. ⫯f1 = g1 → ⫯f2 = g2 → ↑f = g → ⊥.
 #g1 #g2 #g #Hg #f1 #f2 #f #H1 #H2 #H
 elim (coafter_inv_ppx … Hg … H1 H2) -g1 -g2 #x #Hf #Hx destruct
 elim (discr_push_next … Hx)
 qed-.
 
-lemma coafter_inv_pnn: ∀g1,g2,g. g1 ~⊚ g2 ≡ g →
-                       ∀f1,f2,f. ↑f1 = g1 → ⫯f2 = g2 → ⫯f = g → f1 ~⊚ f2 ≡ f.
+lemma coafter_inv_pnn: ∀g1,g2,g. g1 ~⊚ g2 ≘ g →
+                       ∀f1,f2,f. ⫯f1 = g1 → ↑f2 = g2 → ↑f = g → f1 ~⊚ f2 ≘ f.
 #g1 #g2 #g #Hg #f1 #f2 #f #H1 #H2 #H
 elim (coafter_inv_pnx … Hg … H1 H2) -g1 -g2 #x #Hf #Hx destruct
 <(injective_next … Hx) -f //
 qed-.
 
-lemma coafter_inv_pnp: ∀g1,g2,g. g1 ~⊚ g2 ≡ g →
-                       ∀f1,f2,f. ↑f1 = g1 → ⫯f2 = g2 → ↑f = g → ⊥.
+lemma coafter_inv_pnp: ∀g1,g2,g. g1 ~⊚ g2 ≘ g →
+                       ∀f1,f2,f. ⫯f1 = g1 → ↑f2 = g2 → ⫯f = g → ⊥.
 #g1 #g2 #g #Hg #f1 #f2 #f #H1 #H2 #H
 elim (coafter_inv_pnx … Hg … H1 H2) -g1 -g2 #x #Hf #Hx destruct
 elim (discr_next_push … Hx)
 qed-.
 
-lemma coafter_inv_nxp: ∀g1,f2,g. g1 ~⊚ f2 ≡ g →
-                       ∀f1,f. ⫯f1 = g1 → ↑f = g → f1 ~⊚ f2 ≡ f.
+lemma coafter_inv_nxp: ∀g1,f2,g. g1 ~⊚ f2 ≘ g →
+                       ∀f1,f. ↑f1 = g1 → ⫯f = g → f1 ~⊚ f2 ≘ f.
 #g1 #f2 #g #Hg #f1 #f #H1 #H
 elim (coafter_inv_nxx … Hg … H1) -g1 #x #Hf #Hx destruct
 <(injective_push … Hx) -f //
 qed-.
 
-lemma coafter_inv_nxn: ∀g1,f2,g. g1 ~⊚ f2 ≡ g →
-                       ∀f1,f. ⫯f1 = g1 → ⫯f = g → ⊥.
+lemma coafter_inv_nxn: ∀g1,f2,g. g1 ~⊚ f2 ≘ g →
+                       ∀f1,f. ↑f1 = g1 → ↑f = g → ⊥.
 #g1 #f2 #g #Hg #f1 #f #H1 #H
 elim (coafter_inv_nxx … Hg … H1) -g1 #x #Hf #Hx destruct
 elim (discr_push_next … Hx)
 qed-.
 
-lemma coafter_inv_pxp: ∀g1,g2,g. g1 ~⊚ g2 ≡ g →
-                       ∀f1,f. ↑f1 = g1 → ↑f = g →
-                       ∃∃f2. f1 ~⊚ f2 ≡ f & ↑f2 = g2.
+lemma coafter_inv_pxp: ∀g1,g2,g. g1 ~⊚ g2 ≘ g →
+                       ∀f1,f. ⫯f1 = g1 → ⫯f = g →
+                       ∃∃f2. f1 ~⊚ f2 ≘ f & ⫯f2 = g2.
 #g1 #g2 #g #Hg #f1 #f #H1 #H elim (pn_split g2) * #f2 #H2
 [ lapply (coafter_inv_ppp … Hg … H1 H2 H) -g1 -g /2 width=3 by ex2_intro/
 | elim (coafter_inv_pnp … Hg … H1 H2 H)
 ]
 qed-.
 
-lemma coafter_inv_pxn: ∀g1,g2,g. g1 ~⊚ g2 ≡ g →
-                       ∀f1,f. ↑f1 = g1 → ⫯f = g →
-                       ∃∃f2. f1 ~⊚ f2 ≡ f & ⫯f2 = g2.
+lemma coafter_inv_pxn: ∀g1,g2,g. g1 ~⊚ g2 ≘ g →
+                       ∀f1,f. ⫯f1 = g1 → ↑f = g →
+                       ∃∃f2. f1 ~⊚ f2 ≘ f & ↑f2 = g2.
 #g1 #g2 #g #Hg #f1 #f #H1 #H elim (pn_split g2) * #f2 #H2
 [ elim (coafter_inv_ppn … Hg … H1 H2 H)
 | lapply (coafter_inv_pnn … Hg … H1 … H) -g1 -g /2 width=3 by ex2_intro/
 ]
 qed-.
 
-lemma coafter_inv_xxn: ∀g1,g2,g. g1 ~⊚ g2 ≡ g → ∀f. ⫯f = g →
-                       ∃∃f1,f2. f1 ~⊚ f2 ≡ f & ↑f1 = g1 & ⫯f2 = g2.
+lemma coafter_inv_xxn: ∀g1,g2,g. g1 ~⊚ g2 ≘ g → ∀f. ↑f = g →
+                       ∃∃f1,f2. f1 ~⊚ f2 ≘ f & ⫯f1 = g1 & ↑f2 = g2.
 #g1 #g2 #g #Hg #f #H elim (pn_split g1) * #f1 #H1
 [ elim (coafter_inv_pxn … Hg … H1 H) -g /2 width=5 by ex3_2_intro/
 | elim (coafter_inv_nxn … Hg … H1 H)
 ]
 qed-.
 
-lemma coafter_inv_xxp: ∀g1,g2,g. g1 ~⊚ g2 ≡ g → ∀f. ↑f = g →
-                       (∃∃f1,f2. f1 ~⊚ f2 ≡ f & ↑f1 = g1 & ↑f2 = g2) ∨
-                       ∃∃f1. f1 ~⊚ g2 ≡ f & ⫯f1 = g1.
+lemma coafter_inv_xnn: ∀g1,g2,g. g1 ~⊚ g2 ≘ g →
+                       ∀f2,f. ↑f2 = g2 → ↑f = g →
+                       ∃∃f1. f1 ~⊚ f2 ≘ f & ⫯f1 = g1.
+#g1 #g2 #g #Hg #f2 #f #H2 destruct #H
+elim (coafter_inv_xxn … Hg … H) -g
+#z1 #z2 #Hf #H1 #H2 destruct /2 width=3 by ex2_intro/
+qed-.
+
+lemma coafter_inv_xxp: ∀g1,g2,g. g1 ~⊚ g2 ≘ g → ∀f. ⫯f = g →
+                       (∃∃f1,f2. f1 ~⊚ f2 ≘ f & ⫯f1 = g1 & ⫯f2 = g2) ∨
+                       ∃∃f1. f1 ~⊚ g2 ≘ f & ↑f1 = g1.
 #g1 #g2 #g #Hg #f #H elim (pn_split g1) * #f1 #H1
 [ elim (coafter_inv_pxp … Hg … H1 H) -g
   /3 width=5 by or_introl, ex3_2_intro/
@@ -161,9 +169,9 @@ lemma coafter_inv_xxp: ∀g1,g2,g. g1 ~⊚ g2 ≡ g → ∀f. ↑f = g →
 ]
 qed-.
 
-lemma coafter_inv_pxx: ∀g1,g2,g. g1 ~⊚ g2 ≡ g → ∀f1. ↑f1 = g1 →
-                       (∃∃f2,f. f1 ~⊚ f2 ≡ f & ↑f2 = g2 & ↑f = g) ∨
-                       (∃∃f2,f. f1 ~⊚ f2 ≡ f & ⫯f2 = g2 & ⫯f = g).
+lemma coafter_inv_pxx: ∀g1,g2,g. g1 ~⊚ g2 ≘ g → ∀f1. ⫯f1 = g1 →
+                       (∃∃f2,f. f1 ~⊚ f2 ≘ f & ⫯f2 = g2 & ⫯f = g) ∨
+                       (∃∃f2,f. f1 ~⊚ f2 ≘ f & ↑f2 = g2 & ↑f = g).
 #g1 #g2 #g #Hg #f1 #H1 elim (pn_split g2) * #f2 #H2
 [ elim (coafter_inv_ppx … Hg … H1 H2) -g1
   /3 width=5 by or_introl, ex3_2_intro/
@@ -174,7 +182,7 @@ qed-.
 
 (* Basic properties *********************************************************)
 
-corec lemma coafter_eq_repl_back2: ∀f1,f. eq_repl_back (λf2. f2 ~⊚ f1 ≡ f).
+corec lemma coafter_eq_repl_back2: ∀f1,f. eq_repl_back (λf2. f2 ~⊚ f1 ≘ f).
 #f1 #f #f2 * -f2 -f1 -f
 #f21 #f1 #f #g21 [1,2: #g1 ] #g #Hf #H21 [1,2: #H1 ] #H #g22 #H0
 [ cases (eq_inv_px …  H0 …  H21) -g21 /3 width=7 by coafter_refl/
@@ -183,11 +191,11 @@ corec lemma coafter_eq_repl_back2: ∀f1,f. eq_repl_back (λf2. f2 ~⊚ f1 ≡ f
 ]
 qed-.
 
-lemma coafter_eq_repl_fwd2: ∀f1,f. eq_repl_fwd (λf2. f2 ~⊚ f1 ≡ f).
+lemma coafter_eq_repl_fwd2: ∀f1,f. eq_repl_fwd (λf2. f2 ~⊚ f1 ≘ f).
 #f1 #f @eq_repl_sym /2 width=3 by coafter_eq_repl_back2/
 qed-.
 
-corec lemma coafter_eq_repl_back1: ∀f2,f. eq_repl_back (λf1. f2 ~⊚ f1 ≡ f).
+corec lemma coafter_eq_repl_back1: ∀f2,f. eq_repl_back (λf1. f2 ~⊚ f1 ≘ f).
 #f2 #f #f1 * -f2 -f1 -f
 #f2 #f11 #f #g2 [1,2: #g11 ] #g #Hf #H2 [1,2: #H11 ] #H #g2 #H0
 [ cases (eq_inv_px …  H0 …  H11) -g11 /3 width=7 by coafter_refl/
@@ -196,11 +204,11 @@ corec lemma coafter_eq_repl_back1: ∀f2,f. eq_repl_back (λf1. f2 ~⊚ f1 ≡ f
 ]
 qed-.
 
-lemma coafter_eq_repl_fwd1: ∀f2,f. eq_repl_fwd (λf1. f2 ~⊚ f1 ≡ f).
+lemma coafter_eq_repl_fwd1: ∀f2,f. eq_repl_fwd (λf1. f2 ~⊚ f1 ≘ f).
 #f2 #f @eq_repl_sym /2 width=3 by coafter_eq_repl_back1/
 qed-.
 
-corec lemma coafter_eq_repl_back0: ∀f1,f2. eq_repl_back (λf. f2 ~⊚ f1 ≡ f).
+corec lemma coafter_eq_repl_back0: ∀f1,f2. eq_repl_back (λf. f2 ~⊚ f1 ≘ f).
 #f2 #f1 #f * -f2 -f1 -f
 #f2 #f1 #f01 #g2 [1,2: #g1 ] #g01 #Hf01 #H2 [1,2: #H1 ] #H01 #g02 #H0
 [ cases (eq_inv_px …  H0 …  H01) -g01 /3 width=7 by coafter_refl/
@@ -209,13 +217,13 @@ corec lemma coafter_eq_repl_back0: ∀f1,f2. eq_repl_back (λf. f2 ~⊚ f1 ≡ f
 ]
 qed-.
 
-lemma coafter_eq_repl_fwd0: ∀f2,f1. eq_repl_fwd (λf. f2 ~⊚ f1 ≡ f).
+lemma coafter_eq_repl_fwd0: ∀f2,f1. eq_repl_fwd (λf. f2 ~⊚ f1 ≘ f).
 #f2 #f1 @eq_repl_sym /2 width=3 by coafter_eq_repl_back0/
 qed-.
 
 (* Main inversion lemmas ****************************************************)
 
-corec theorem coafter_mono: ∀f1,f2,x,y. f1 ~⊚ f2 ≡ x → f1 ~⊚ f2 ≡ y → x ≗ y.
+corec theorem coafter_mono: ∀f1,f2,x,y. f1 ~⊚ f2 ≘ x → f1 ~⊚ f2 ≘ y → x ≡ y.
 #f1 #f2 #x #y * -f1 -f2 -x
 #f1 #f2 #x #g1 [1,2: #g2 ] #g #Hx #H1 [1,2: #H2 ] #H0x #Hy
 [ cases (coafter_inv_ppx … Hy … H1 H2) -g1 -g2 /3 width=8 by eq_push/
@@ -224,52 +232,74 @@ corec theorem coafter_mono: ∀f1,f2,x,y. f1 ~⊚ f2 ≡ x → f1 ~⊚ f2 ≡ y
 ]
 qed-.
 
-lemma coafter_mono_eq: ∀f1,f2,f. f1 ~⊚ f2 ≡ f → ∀g1,g2,g. g1 ~⊚ g2 ≡ g →
-                       f1 ≗ g1 → f2 ≗ g2 → f ≗ g.
+lemma coafter_mono_eq: ∀f1,f2,f. f1 ~⊚ f2 ≘ f → ∀g1,g2,g. g1 ~⊚ g2 ≘ g →
+                       f1 ≡ g1 → f2 ≡ g2 → f ≡ g.
 /4 width=4 by coafter_mono, coafter_eq_repl_back1, coafter_eq_repl_back2/ qed-.
 
-(* Inversion lemmas with pushs **********************************************)
-
-lemma coafter_fwd_pushs: ∀n,g2,g1,g. g2 ~⊚ g1 ≡ g → @⦃0, g2⦄ ≡ n →
-                         ∃f. ↑*[n]f = g.
-#n elim n -n /2 width=2 by ex_intro/
-#n #IH #g2 #g1 #g #Hg #Hg2
-cases (at_inv_pxn … Hg2) -Hg2 [ |*: // ] #f2 #Hf2 #H2
-cases (coafter_inv_nxx … Hg … H2) -Hg -H2 #f #Hf #H0 destruct
-elim (IH … Hf Hf2) -g1 -g2 -f2 /2 width=2 by ex_intro/
+(* Forward lemmas with pushs ************************************************)
+
+lemma coafter_fwd_pushs: ∀j,i,g2,f1,g. g2 ~⊚ ⫯*[i]f1 ≘ g → @❪i, g2❫ ≘ j →
+                         ∃f. ⫯*[j] f = g.
+#j elim j -j
+[ #i #g2 #f1 #g #Hg #H
+  elim (at_inv_xxp … H) -H [|*: // ] #f2 #H1 #H2 destruct
+  /2 width=2 by ex_intro/
+| #j #IH * [| #i ] #g2 #f1 #g #Hg #H
+  [ elim (at_inv_pxn … H) -H [|*: // ] #f2 #Hij #H destruct
+    elim (coafter_inv_nxx … Hg) -Hg [|*: // ] #f #Hf #H destruct
+    elim (IH … Hf Hij) -f1 -f2 -IH /2 width=2 by ex_intro/
+  | elim (at_inv_nxn … H) -H [1,4: * |*: // ] #f2 #Hij #H destruct
+    [ elim (coafter_inv_ppx … Hg) -Hg [|*: // ] #f #Hf #H destruct
+      elim (IH … Hf Hij) -f1 -f2 -i /2 width=2 by ex_intro/
+    | elim (coafter_inv_nxx … Hg) -Hg [|*: // ] #f #Hf #H destruct
+      elim (IH … Hf Hij) -f1 -f2 -i /2 width=2 by ex_intro/
+    ]
+  ]
+]
 qed-.
 
 (* Inversion lemmas with tail ***********************************************)
 
-lemma coafter_inv_tl1: ∀g2,g1,g. g2 ~⊚ ⫱g1 ≡ g →
-                       ∃∃f. ↑g2 ~⊚ g1 ≡ f & ⫱f = g.
+lemma coafter_inv_tl1: ∀g2,g1,g. g2 ~⊚ ⫱g1 ≘ g →
+                       ∃∃f. ⫯g2 ~⊚ g1 ≘ f & ⫱f = g.
 #g2 #g1 #g elim (pn_split g1) * #f1 #H1 #H destruct
 [ /3 width=7 by coafter_refl, ex2_intro/
-| @(ex2_intro … (⫯g)) /2 width=7 by coafter_push/ (**) (* full auto fails *)
+| @(ex2_intro … (↑g)) /2 width=7 by coafter_push/ (**) (* full auto fails *)
 ]
 qed-.
 
-lemma coafter_inv_tl0: ∀g2,g1,g. g2 ~⊚ g1 ≡ ⫱g →
-                       ∃∃f1. ↑g2 ~⊚ f1 ≡ g & ⫱f1 = g1.
+lemma coafter_inv_tl0: ∀g2,g1,g. g2 ~⊚ g1 ≘ ⫱g →
+                       ∃∃f1. ⫯g2 ~⊚ f1 ≘ g & ⫱f1 = g1.
 #g2 #g1 #g elim (pn_split g) * #f #H0 #H destruct
 [ /3 width=7 by coafter_refl, ex2_intro/
-| @(ex2_intro … (⫯g1)) /2 width=7 by coafter_push/ (**) (* full auto fails *)
+| @(ex2_intro … (↑g1)) /2 width=7 by coafter_push/ (**) (* full auto fails *)
 ]
 qed-.
 
-(* Properties on tls ********************************************************)
-
-lemma coafter_tls: ∀n,f1,f2,f. @⦃0, f1⦄ ≡ n →
-                   f1 ~⊚ f2 ≡ f → ⫱*[n]f1 ~⊚ f2 ≡ ⫱*[n]f.
-#n elim n -n //
-#n #IH #f1 #f2 #f #Hf1 #Hf
-cases (at_inv_pxn … Hf1) -Hf1 [ |*: // ] #g1 #Hg1 #H1
-cases (coafter_inv_nxx … Hf … H1) -Hf #g #Hg #H0 destruct
-<tls_xn <tls_xn /2 width=1 by/
+(* Properties with iterated tail ********************************************)
+
+lemma coafter_tls: ∀j,i,f1,f2,f. @❪i, f1❫ ≘ j →
+                   f1 ~⊚ f2 ≘ f → ⫱*[j]f1 ~⊚ ⫱*[i]f2 ≘ ⫱*[j]f.
+#j elim j -j [ #i | #j #IH * [| #i ] ] #f1 #f2 #f #Hf1 #Hf
+[ elim (at_inv_xxp … Hf1) -Hf1 [ |*: // ] #g1 #Hg1 #H1 destruct //
+| elim (at_inv_pxn … Hf1) -Hf1 [ |*: // ] #g1 #Hg1 #H1
+  elim (coafter_inv_nxx … Hf … H1) -Hf #g #Hg #H0 destruct
+  lapply (IH … Hg1 Hg) -IH -Hg1 -Hg //
+| elim (at_inv_nxn … Hf1) -Hf1 [1,4: * |*: // ] #g1 #Hg1 #H1
+  [ elim (coafter_inv_pxx … Hf … H1) -Hf * #g2 #g #Hg #H2 #H0 destruct
+    lapply (IH … Hg1 Hg) -IH -Hg1 -Hg #H //
+  | elim (coafter_inv_nxx … Hf … H1) -Hf #g #Hg #H0 destruct
+    lapply (IH … Hg1 Hg) -IH -Hg1 -Hg #H //
+  ]
+]
 qed.
 
-lemma coafter_tls_succ: ∀g2,g1,g. g2 ~⊚ g1 ≡ g →
-                        ∀n. @⦃0, g2⦄ ≡ n → ⫱*[⫯n]g2 ~⊚ ⫱g1 ≡ ⫱*[⫯n]g.
+lemma coafter_tls_O: ∀n,f1,f2,f. @❪0, f1❫ ≘ n →
+                     f1 ~⊚ f2 ≘ f → ⫱*[n]f1 ~⊚ f2 ≘ ⫱*[n]f.
+/2 width=1 by coafter_tls/ qed.
+
+lemma coafter_tls_succ: ∀g2,g1,g. g2 ~⊚ g1 ≘ g →
+                        ∀n. @❪0, g2❫ ≘ n → ⫱*[↑n]g2 ~⊚ ⫱g1 ≘ ⫱*[↑n]g.
 #g2 #g1 #g #Hg #n #Hg2
 lapply (coafter_tls … Hg2 … Hg) -Hg #Hg
 lapply (at_pxx_tls … Hg2) -Hg2 #H
@@ -278,62 +308,65 @@ elim (coafter_inv_pxx … Hg … H2) -Hg * #f1 #f #Hf #H1 #H0 destruct
 <tls_S <tls_S <H2 <H0 -g2 -g -n //
 qed.
 
-lemma coafter_fwd_xpx_pushs: ∀g2,f1,g,n. g2 ~⊚ ↑f1 ≡ g → @⦃0, g2⦄ ≡ n →
-                             ∃f. ↑*[⫯n]f = g.
-#g2 #g1 #g #n #Hg #Hg2
+lemma coafter_fwd_xpx_pushs: ∀g2,f1,g,i,j. @❪i, g2❫ ≘ j → g2 ~⊚ ⫯*[↑i]f1 ≘ g →
+                             ∃∃f.  ⫱*[↑j]g2 ~⊚ f1 ≘ f & ⫯*[↑j]f = g.
+#g2 #g1 #g #i #j #Hg2 <pushs_xn #Hg
 elim (coafter_fwd_pushs … Hg Hg2) #f #H0 destruct
-lapply (coafter_tls … Hg2 Hg) -Hg <tls_pushs #Hf
-lapply (at_pxx_tls … Hg2) -Hg2 #H
-elim (at_inv_pxp … H) -H [ |*: // ] #f2 #H2
-elim (coafter_inv_pxx … Hf … H2) -Hf -H2 * #f1 #g #_ #H1 #H0 destruct
-[ /2 width=2 by ex_intro/
-| elim (discr_next_push … H1)
-] 
+lapply (coafter_tls … Hg2 Hg) -Hg <tls_pushs <tls_pushs #Hf
+lapply (at_inv_tls … Hg2) -Hg2 #H
+lapply (coafter_eq_repl_fwd2 … Hf … H) -H -Hf #Hf
+elim (coafter_inv_ppx … Hf) [|*: // ] -Hf #g #Hg #H destruct
+/2 width=3 by ex2_intro/
 qed-.
 
-lemma coafter_fwd_xnx_pushs: ∀g2,f1,g,n. g2 ~⊚ ⫯f1 ≡ g → @⦃0, g2⦄ ≡ n →
-                             ∃f. ↑*[n] ⫯f = g.
-#g2 #g1 #g #n #Hg #Hg2
+lemma coafter_fwd_xnx_pushs: ∀g2,f1,g,i,j. @❪i, g2❫ ≘ j → g2 ~⊚ ⫯*[i]↑f1 ≘ g →
+                             ∃∃f. ⫱*[↑j]g2 ~⊚ f1 ≘ f & ⫯*[j] ↑f = g.
+#g2 #g1 #g #i #j #Hg2 #Hg
 elim (coafter_fwd_pushs … Hg Hg2) #f #H0 destruct
-lapply (coafter_tls … Hg2 Hg) -Hg <tls_pushs #Hf
-lapply (at_pxx_tls … Hg2) -Hg2 #H
-elim (at_inv_pxp … H) -H [ |*: // ] #f2 #H2
-elim (coafter_inv_pxx … Hf … H2) -Hf -H2 * #f1 #g #_ #H1 #H0 destruct
-[ elim (discr_push_next … H1)
-| /2 width=2 by ex_intro/
-] 
+lapply (coafter_tls … Hg2 Hg) -Hg <tls_pushs <tls_pushs #Hf
+lapply (at_inv_tls … Hg2) -Hg2 #H
+lapply (coafter_eq_repl_fwd2 … Hf … H) -H -Hf #Hf
+elim (coafter_inv_pnx … Hf) [|*: // ] -Hf #g #Hg #H destruct
+/2 width=3 by ex2_intro/
 qed-.
 
-(* Properties on isid *******************************************************)
+(* Properties with test for identity ****************************************)
 
-corec lemma coafter_isid_sn: ∀f1. 𝐈⦃f1⦄ → ∀f2. f1 ~⊚ f2 ≡ f2.
+corec lemma coafter_isid_sn: ∀f1. 𝐈❪f1❫ → ∀f2. f1 ~⊚ f2 ≘ f2.
 #f1 * -f1 #f1 #g1 #Hf1 #H1 #f2 cases (pn_split f2) * #g2 #H2
 /3 width=7 by coafter_push, coafter_refl/
 qed.
 
-corec lemma coafter_isid_dx: ∀f2,f. 𝐈⦃f2⦄ → 𝐈⦃f⦄ → ∀f1. f1 ~⊚ f2 ≡ f.
+corec lemma coafter_isid_dx: ∀f2,f. 𝐈❪f2❫ → 𝐈❪f❫ → ∀f1. f1 ~⊚ f2 ≘ f.
 #f2 #f * -f2 #f2 #g2 #Hf2 #H2 * -f #f #g #Hf #H #f1 cases (pn_split f1) * #g1 #H1
 [ /3 width=7 by coafter_refl/
 | @(coafter_next … H1 … H) /3 width=3 by isid_push/
 ]
 qed.
 
-(* Inversion lemmas on isid *************************************************)
+(* Inversion lemmas with test for identity **********************************)
 
-lemma coafter_isid_inv_sn: ∀f1,f2,f. f1 ~⊚ f2 ≡ f → 𝐈⦃f1⦄ → f2 ≗ f.
+lemma coafter_isid_inv_sn: ∀f1,f2,f. f1 ~⊚ f2 ≘ f → 𝐈❪f1❫ → f2 ≡ f.
 /3 width=6 by coafter_isid_sn, coafter_mono/ qed-.
 
-lemma coafter_isid_inv_dx: ∀f1,f2,f. f1 ~⊚ f2 ≡ f → 𝐈⦃f2⦄ → 𝐈⦃f⦄.
+lemma coafter_isid_inv_dx: ∀f1,f2,f. f1 ~⊚ f2 ≘ f → 𝐈❪f2❫ → 𝐈❪f❫.
 /4 width=4 by eq_id_isid, coafter_isid_dx, coafter_mono/ qed-.
-(*
-(* Properties on isuni ******************************************************)
 
-lemma coafter_isid_isuni: ∀f1,f2. 𝐈⦃f2⦄ → 𝐔⦃f1⦄ → f1 ~⊚ ⫯f2 ≡ ⫯f1.
+(* Properties with test for uniform relocations *****************************)
+
+lemma coafter_isuni_isid: ∀f2. 𝐈❪f2❫ → ∀f1. 𝐔❪f1❫ → f1 ~⊚ f2 ≘ f2.
+#f #Hf #g #H elim H -g
+/3 width=5 by coafter_isid_sn, coafter_eq_repl_back0, coafter_next, eq_push_inv_isid/
+qed.
+
+
+(*
+lemma coafter_isid_isuni: ∀f1,f2. 𝐈❪f2❫ → 𝐔❪f1❫ → f1 ~⊚ ↑f2 ≘ ↑f1.
 #f1 #f2 #Hf2 #H elim H -H
 /5 width=7 by coafter_isid_dx, coafter_eq_repl_back2, coafter_next, coafter_push, eq_push_inv_isid/
 qed.
 
-lemma coafter_uni_next2: ∀f2. 𝐔⦃f2⦄ → ∀f1,f. ⫯f2 ~⊚ f1 ≡ f → f2 ~⊚ ⫯f1 ≡ f.
+lemma coafter_uni_next2: ∀f2. 𝐔❪f2❫ → ∀f1,f. ↑f2 ~⊚ f1 ≘ f → f2 ~⊚ ↑f1 ≘ f.
 #f2 #H elim H -f2
 [ #f2 #Hf2 #f1 #f #Hf
   elim (coafter_inv_nxx … Hf) -Hf [2,3: // ] #g #Hg #H0 destruct
@@ -343,18 +376,24 @@ lemma coafter_uni_next2: ∀f2. 𝐔⦃f2⦄ → ∀f1,f. ⫯f2 ~⊚ f1 ≡ f 
   /3 width=5 by coafter_next/
 ]
 qed.
+*)
 
-(* Properties on uni ********************************************************)
+(* Properties with uniform relocations **************************************)
 
-lemma coafter_uni: ∀n1,n2. 𝐔❴n1❵ ~⊚ 𝐔❴n2❵ ≡ 𝐔❴n1+n2❵.
+lemma coafter_uni_sn: ∀i,f. 𝐔❨i❩ ~⊚ f ≘ ⫯*[i] f.
+#i elim i -i /2 width=5 by coafter_isid_sn, coafter_next/
+qed.
+
+(*
+lemma coafter_uni: ∀n1,n2. 𝐔❨n1❩ ~⊚ 𝐔❨n2❩ ≘ 𝐔❨n1+n2❩.
 @nat_elim2
 /4 width=5 by coafter_uni_next2, coafter_isid_sn, coafter_isid_dx, coafter_next/
 qed.
 
 (* Forward lemmas on at *****************************************************)
 
-lemma coafter_at_fwd: ∀i,i1,f. @⦃i1, f⦄ ≡ i → ∀f2,f1. f2 ~⊚ f1 ≡ f →
-                    ∃∃i2. @⦃i1, f1⦄ ≡ i2 & @⦃i2, f2⦄ ≡ i.
+lemma coafter_at_fwd: ∀i,i1,f. @❪i1, f❫ ≘ i → ∀f2,f1. f2 ~⊚ f1 ≘ f →
+                      ∃∃i2. @❪i1, f1❫ ≘ i2 & @❪i2, f2❫ ≘ i.
 #i elim i -i [2: #i #IH ] #i1 #f #Hf #f2 #f1 #Hf21
 [ elim (at_inv_xxn … Hf) -Hf [1,3:* |*: // ]
   [1: #g #j1 #Hg #H0 #H |2,4: #g #Hg #H ]
@@ -370,8 +409,8 @@ lemma coafter_at_fwd: ∀i,i1,f. @⦃i1, f⦄ ≡ i → ∀f2,f1. f2 ~⊚ f1 ≡
 /3 width=9 by at_refl, at_push, at_next, ex2_intro/
 qed-.
 
-lemma coafter_fwd_at: ∀i,i2,i1,f1,f2. @⦃i1, f1⦄ ≡ i2 → @⦃i2, f2⦄ ≡ i →
-                    ∀f. f2 ~⊚ f1 ≡ f → @⦃i1, f⦄ ≡ i.
+lemma coafter_fwd_at: ∀i,i2,i1,f1,f2. @❪i1, f1❫ ≘ i2 → @❪i2, f2❫ ≘ i →
+                      ∀f. f2 ~⊚ f1 ≘ f → @❪i1, f❫ ≘ i.
 #i elim i -i [2: #i #IH ] #i2 #i1 #f1 #f2 #Hf1 #Hf2 #f #Hf
 [ elim (at_inv_xxn … Hf2) -Hf2 [1,3: * |*: // ]
   #g2 [ #j2 ] #Hg2 [ #H22 ] #H20
@@ -388,14 +427,14 @@ lemma coafter_fwd_at: ∀i,i2,i1,f1,f2. @⦃i1, f1⦄ ≡ i2 → @⦃i2, f2⦄ 
 ]
 qed-.
 
-lemma coafter_fwd_at2: ∀f,i1,i. @⦃i1, f⦄ ≡ i → ∀f1,i2. @⦃i1, f1⦄ ≡ i2 →
-                     ∀f2. f2 ~⊚ f1 ≡ f → @⦃i2, f2⦄ ≡ i.
+lemma coafter_fwd_at2: ∀f,i1,i. @❪i1, f❫ ≘ i → ∀f1,i2. @❪i1, f1❫ ≘ i2 →
+                       ∀f2. f2 ~⊚ f1 ≘ f → @❪i2, f2❫ ≘ i.
 #f #i1 #i #Hf #f1 #i2 #Hf1 #f2 #H elim (coafter_at_fwd … Hf … H) -f
 #j1 #H #Hf2 <(at_mono … Hf1 … H) -i1 -i2 //
 qed-.
 
-lemma coafter_fwd_at1: ∀i,i2,i1,f,f2. @⦃i1, f⦄ ≡ i → @⦃i2, f2⦄ ≡ i →
-                     ∀f1. f2 ~⊚ f1 ≡ f → @⦃i1, f1⦄ ≡ i2.
+lemma coafter_fwd_at1: ∀i,i2,i1,f,f2. @❪i1, f❫ ≘ i → @❪i2, f2❫ ≘ i →
+                       ∀f1. f2 ~⊚ f1 ≘ f → @❪i1, f1❫ ≘ i2.
 #i elim i -i [2: #i #IH ] #i2 #i1 #f #f2 #Hf #Hf2 #f1 #Hf1
 [ elim (at_inv_xxn … Hf) -Hf [1,3: * |*: // ]
   #g [ #j1 ] #Hg [ #H01 ] #H00
@@ -414,8 +453,8 @@ qed-.
 
 (* Properties with at *******************************************************)
 
-lemma coafter_uni_dx: ∀i2,i1,f2. @⦃i1, f2⦄ ≡ i2 →
-                    ∀f. f2 ~⊚ 𝐔❴i1❵ ≡ f → 𝐔❴i2❵ ~⊚ ⫱*[i2] f2 ≡ f.
+lemma coafter_uni_dx: ∀i2,i1,f2. @❪i1, f2❫ ≘ i2 →
+                      ∀f. f2 ~⊚ 𝐔❨i1❩ ≘ f → 𝐔❨i2❩ ~⊚ ⫱*[i2] f2 ≘ f.
 #i2 elim i2 -i2
 [ #i1 #f2 #Hf2 #f #Hf
   elim (at_inv_xxp … Hf2) -Hf2 // #g2 #H1 #H2 destruct
@@ -433,8 +472,8 @@ lemma coafter_uni_dx: ∀i2,i1,f2. @⦃i1, f2⦄ ≡ i2 →
 ]
 qed.
 
-lemma coafter_uni_sn: ∀i2,i1,f2. @⦃i1, f2⦄ ≡ i2 →
-                    ∀f. 𝐔❴i2❵ ~⊚ ⫱*[i2] f2 ≡ f → f2 ~⊚ 𝐔❴i1❵ ≡ f.
+lemma coafter_uni_sn: ∀i2,i1,f2. @❪i1, f2❫ ≘ i2 →
+                      ∀f. 𝐔❨i2❩ ~⊚ ⫱*[i2] f2 ≘ f → f2 ~⊚ 𝐔❨i1❩ ≘ f.
 #i2 elim i2 -i2
 [ #i1 #f2 #Hf2 #f #Hf
   elim (at_inv_xxp … Hf2) -Hf2 // #g2 #H1 #H2 destruct
@@ -449,8 +488,8 @@ lemma coafter_uni_sn: ∀i2,i1,f2. @⦃i1, f2⦄ ≡ i2 →
 ]
 qed-.
 
-lemma coafter_uni_succ_dx: ∀i2,i1,f2. @⦃i1, f2⦄ ≡ i2 →
-                         ∀f. f2 ~⊚ 𝐔❴⫯i1❵ ≡ f → 𝐔❴⫯i2❵ ~⊚ ⫱*[⫯i2] f2 ≡ f.
+lemma coafter_uni_succ_dx: ∀i2,i1,f2. @❪i1, f2❫ ≘ i2 →
+                           ∀f. f2 ~⊚ 𝐔❨↑i1❩ ≘ f → 𝐔❨↑i2❩ ~⊚ ⫱*[↑i2] f2 ≘ f.
 #i2 elim i2 -i2
 [ #i1 #f2 #Hf2 #f #Hf
   elim (at_inv_xxp … Hf2) -Hf2 // #g2 #H1 #H2 destruct
@@ -469,8 +508,8 @@ lemma coafter_uni_succ_dx: ∀i2,i1,f2. @⦃i1, f2⦄ ≡ i2 →
 ]
 qed.
 
-lemma coafter_uni_succ_sn: ∀i2,i1,f2. @⦃i1, f2⦄ ≡ i2 →
-                         ∀f. 𝐔❴⫯i2❵ ~⊚ ⫱*[⫯i2] f2 ≡ f → f2 ~⊚ 𝐔❴⫯i1❵ ≡ f.
+lemma coafter_uni_succ_sn: ∀i2,i1,f2. @❪i1, f2❫ ≘ i2 →
+                           ∀f. 𝐔❨↑i2❩ ~⊚ ⫱*[↑i2] f2 ≘ f → f2 ~⊚ 𝐔❨↑i1❩ ≘ f.
 #i2 elim i2 -i2
 [ #i1 #f2 #Hf2 #f #Hf
   elim (at_inv_xxp … Hf2) -Hf2 // #g2 #H1 #H2 destruct
@@ -486,43 +525,43 @@ lemma coafter_uni_succ_sn: ∀i2,i1,f2. @⦃i1, f2⦄ ≡ i2 →
 ]
 qed-.
 
-lemma coafter_uni_one_dx: ∀f2,f. ↑f2 ~⊚ 𝐔❴⫯O❵ ≡ f → 𝐔❴⫯O❵ ~⊚ f2 ≡ f.
-#f2 #f #H @(coafter_uni_succ_dx … (↑f2)) /2 width=3 by at_refl/
+lemma coafter_uni_one_dx: ∀f2,f. ⫯f2 ~⊚ 𝐔❨↑O❩ ≘ f → 𝐔❨↑O❩ ~⊚ f2 ≘ f.
+#f2 #f #H @(coafter_uni_succ_dx … (⫯f2)) /2 width=3 by at_refl/
 qed.
 
-lemma coafter_uni_one_sn: ∀f1,f. 𝐔❴⫯O❵ ~⊚ f1 ≡ f → ↑f1 ~⊚ 𝐔❴⫯O❵ ≡ f.
+lemma coafter_uni_one_sn: ∀f1,f. 𝐔❨↑O❩ ~⊚ f1 ≘ f → ⫯f1 ~⊚ 𝐔❨↑O❩ ≘ f.
 /3 width=3 by coafter_uni_succ_sn, at_refl/ qed-.
 *)
 (* Forward lemmas with istot ************************************************)
 (*
-lemma coafter_istot_fwd: ∀f2,f1,f. f2 ~⊚ f1 ≡ f → 𝐓⦃f2⦄ → 𝐓⦃f1⦄ → 𝐓⦃f⦄.
+lemma coafter_istot_fwd: ∀f2,f1,f. f2 ~⊚ f1 ≘ f → 𝐓❪f2❫ → 𝐓❪f1❫ → 𝐓❪f❫.
 #f2 #f1 #f #Hf #Hf2 #Hf1 #i1 elim (Hf1 i1) -Hf1
 #i2 #Hf1 elim (Hf2 i2) -Hf2
 /3 width=7 by coafter_fwd_at, ex_intro/
 qed-.
 
-lemma coafter_fwd_istot_dx: ∀f2,f1,f. f2 ~⊚ f1 ≡ f → 𝐓⦃f⦄ → 𝐓⦃f1⦄.
+lemma coafter_fwd_istot_dx: ∀f2,f1,f. f2 ~⊚ f1 ≘ f → 𝐓❪f❫ → 𝐓❪f1❫.
 #f2 #f1 #f #H #Hf #i1 elim (Hf i1) -Hf
 #i2 #Hf elim (coafter_at_fwd … Hf … H) -f /2 width=2 by ex_intro/
 qed-.
 
-lemma coafter_fwd_istot_sn: ∀f2,f1,f. f2 ~⊚ f1 ≡ f → 𝐓⦃f⦄ → 𝐓⦃f2⦄.
+lemma coafter_fwd_istot_sn: ∀f2,f1,f. f2 ~⊚ f1 ≘ f → 𝐓❪f❫ → 𝐓❪f2❫.
 #f2 #f1 #f #H #Hf #i1 elim (Hf i1) -Hf
 #i #Hf elim (coafter_at_fwd … Hf … H) -f
 #i2 #Hf1 #Hf2 lapply (at_increasing … Hf1) -f1
 #Hi12 elim (at_le_ex … Hf2 … Hi12) -i2 /2 width=2 by ex_intro/
 qed-.
 
-lemma coafter_inv_istot: ∀f2,f1,f. f2 ~⊚ f1 ≡ f → 𝐓⦃f⦄ → 𝐓⦃f2⦄ ∧ 𝐓⦃f1⦄.
+lemma coafter_inv_istot: ∀f2,f1,f. f2 ~⊚ f1 ≘ f → 𝐓❪f❫ → 𝐓❪f2❫ ∧ 𝐓❪f1❫.
 /3 width=4 by coafter_fwd_istot_sn, coafter_fwd_istot_dx, conj/ qed-.
 
-lemma coafter_at1_fwd: ∀f1,i1,i2. @⦃i1, f1⦄ ≡ i2 → ∀f2. 𝐓⦃f2⦄ → ∀f. f2 ~⊚ f1 ≡ f →
-                     ∃∃i. @⦃i2, f2⦄ ≡ i & @⦃i1, f⦄ ≡ i.
+lemma coafter_at1_fwd: ∀f1,i1,i2. @❪i1, f1❫ ≘ i2 → ∀f2. 𝐓❪f2❫ → ∀f. f2 ~⊚ f1 ≘ f →
+                     ∃∃i. @❪i2, f2❫ ≘ i & @❪i1, f❫ ≘ i.
 #f1 #i1 #i2 #Hf1 #f2 #Hf2 #f #Hf elim (Hf2 i2) -Hf2
 /3 width=8 by coafter_fwd_at, ex2_intro/
 qed-.
 
-lemma coafter_fwd_isid_sn: ∀f2,f1,f. 𝐓⦃f⦄ → f2 ~⊚ f1 ≡ f → f1 ≗ f → 𝐈⦃f2⦄.
+lemma coafter_fwd_isid_sn: ∀f2,f1,f. 𝐓❪f❫ → f2 ~⊚ f1 ≘ f → f1 ≡ f → 𝐈❪f2❫.
 #f2 #f1 #f #H #Hf elim (coafter_inv_istot … Hf H) -H
 #Hf2 #Hf1 #H @isid_at_total // -Hf2
 #i2 #i #Hf2 elim (Hf1 i2) -Hf1
@@ -531,14 +570,14 @@ lemma coafter_fwd_isid_sn: ∀f2,f1,f. 𝐓⦃f⦄ → f2 ~⊚ f1 ≡ f → f1 
 /3 width=7 by at_eq_repl_back, at_mono, at_id_le/
 qed-.
 
-lemma coafter_fwd_isid_dx: ∀f2,f1,f.  𝐓⦃f⦄ → f2 ~⊚ f1 ≡ f → f2 ≗ f → 𝐈⦃f1⦄.
+lemma coafter_fwd_isid_dx: ∀f2,f1,f.  𝐓❪f❫ → f2 ~⊚ f1 ≘ f → f2 ≡ f → 𝐈❪f1❫.
 #f2 #f1 #f #H #Hf elim (coafter_inv_istot … Hf H) -H
 #Hf2 #Hf1 #H2 @isid_at_total // -Hf1
 #i1 #i2 #Hi12 elim (coafter_at1_fwd … Hi12 … Hf) -f1
 /3 width=8 by at_inj, at_eq_repl_back/
 qed-.
 *)
-corec fact coafter_inj_O_aux: ∀f1. @⦃0, f1⦄ ≡ 0 → H_coafter_inj f1.
+corec fact coafter_inj_O_aux: ∀f1. @❪0, f1❫ ≘ 0 → H_coafter_inj f1.
 #f1 #H1f1 #H2f1 #f #f21 #f22 #H1f #H2f
 cases (at_inv_pxp … H1f1) -H1f1 [ |*: // ] #g1 #H1
 lapply (istot_inv_push … H2f1 … H1) -H2f1 #H2g1
@@ -553,8 +592,8 @@ cases (coafter_inv_pxx … H1f … H1) -H1f * #g21 #g #H1g #H21 #H
 /2 width=1 by coafter_tls, istot_tls, at_pxx_tls/
 qed-.
 
-fact coafter_inj_aux: (∀f1. @⦃0, f1⦄ ≡ 0 → H_coafter_inj f1) →
-                      ∀i2,f1. @⦃0, f1⦄ ≡ i2 → H_coafter_inj f1.
+fact coafter_inj_aux: (∀f1. @❪0, f1❫ ≘ 0 → H_coafter_inj f1) →
+                      ∀i2,f1. @❪0, f1❫ ≘ i2 → H_coafter_inj f1.
 #H0 #i2 elim i2 -i2 /2 width=1 by/ -H0
 #i2 #IH #f1 #H1f1 #H2f1 #f #f21 #f22 #H1f #H2f
 elim (at_inv_pxn … H1f1) -H1f1 [ |*: // ] #g1 #H1g1 #H1
@@ -567,7 +606,7 @@ theorem coafter_inj: ∀f1. H_coafter_inj f1.
 #f1 #H cases (H 0) /3 width=7 by coafter_inj_aux, coafter_inj_O_aux/
 qed-.
 
-corec fact coafter_fwd_isid2_O_aux: ∀f1. @⦃0, f1⦄ ≡ 0 →
+corec fact coafter_fwd_isid2_O_aux: ∀f1. @❪0, f1❫ ≘ 0 →
                                     H_coafter_fwd_isid2 f1.
 #f1 #H1f1 #f2 #f #H #H2f1 #Hf
 cases (at_inv_pxp … H1f1) -H1f1 [ |*: // ] #g1 #H1
@@ -575,14 +614,14 @@ lapply (istot_inv_push … H2f1 … H1) -H2f1 #H2g1
 cases (H2g1 0) #n #Hn
 cases (coafter_inv_pxx … H … H1) -H * #g2 #g #H #H2 #H0
 [ lapply (isid_inv_push … Hf … H0) -Hf #Hg
-  @(isid_push … H2)
-  /3 width=7 by coafter_tls, istot_tls, at_pxx_tls, isid_tls/
+  @(isid_push … H2) -H2
+  /3 width=7 by coafter_tls_O, at_pxx_tls, istot_tls, isid_tls/
 | cases (isid_inv_next … Hf … H0)
 ]
 qed-.
 
-fact coafter_fwd_isid2_aux: (∀f1. @⦃0, f1⦄ ≡ 0 → H_coafter_fwd_isid2 f1) →
-                            ∀i2,f1. @⦃0, f1⦄ ≡ i2 → H_coafter_fwd_isid2 f1.
+fact coafter_fwd_isid2_aux: (∀f1. @❪0, f1❫ ≘ 0 → H_coafter_fwd_isid2 f1) →
+                            ∀i2,f1. @❪0, f1❫ ≘ i2 → H_coafter_fwd_isid2 f1.
 #H0 #i2 elim i2 -i2 /2 width=1 by/ -H0
 #i2 #IH #f1 #H1f1 #f2 #f #H #H2f1 #Hf
 elim (at_inv_pxn … H1f1) -H1f1 [ |*: // ] #g1 #Hg1 #H1
@@ -595,7 +634,7 @@ lemma coafter_fwd_isid2: ∀f1. H_coafter_fwd_isid2 f1.
 /3 width=7 by coafter_fwd_isid2_aux, coafter_fwd_isid2_O_aux/
 qed-.
 
-fact coafter_isfin2_fwd_O_aux: ∀f1. @⦃0, f1⦄ ≡ 0 →
+fact coafter_isfin2_fwd_O_aux: ∀f1. @❪0, f1❫ ≘ 0 →
                                H_coafter_isfin2_fwd f1.
 #f1 #Hf1 #f2 #H
 generalize in match Hf1; generalize in match f1; -f1
@@ -607,11 +646,11 @@ lapply (istot_inv_push … Hf1 … H1) -Hf1 #Hg1
 elim (Hg1 0) #n #Hn
 [ elim (coafter_inv_ppx … Hf) | elim (coafter_inv_pnx … Hf)
 ] -Hf [1,6: |*: // ] #g #Hg #H0 destruct
-/5 width=6 by isfin_next, isfin_push, isfin_inv_tls, istot_tls, at_pxx_tls, coafter_tls/
+/5 width=6 by isfin_next, isfin_push, isfin_inv_tls, istot_tls, at_pxx_tls, coafter_tls_O/
 qed-.
 
-fact coafter_isfin2_fwd_aux: (∀f1. @⦃0, f1⦄ ≡ 0 → H_coafter_isfin2_fwd f1) →
-                             ∀i2,f1. @⦃0, f1⦄ ≡ i2 → H_coafter_isfin2_fwd f1.
+fact coafter_isfin2_fwd_aux: (∀f1. @❪0, f1❫ ≘ 0 → H_coafter_isfin2_fwd f1) →
+                             ∀i2,f1. @❪0, f1❫ ≘ i2 → H_coafter_isfin2_fwd f1.
 #H0 #i2 elim i2 -i2 /2 width=1 by/ -H0
 #i2 #IH #f1 #H1f1 #f2 #Hf2 #H2f1 #f #Hf
 elim (at_inv_pxn … H1f1) -H1f1 [ |*: // ] #g1 #Hg1 #H1
@@ -625,8 +664,8 @@ lemma coafter_isfin2_fwd: ∀f1. H_coafter_isfin2_fwd f1.
 /3 width=7 by coafter_isfin2_fwd_aux, coafter_isfin2_fwd_O_aux/
 qed-.
 
-lemma coafter_inv_sor: ∀f. 𝐅⦃f⦄ → ∀f2. 𝐓⦃f2⦄ → ∀f1. f2 ~⊚ f1 ≡ f → ∀fa,fb. fa ⋓ fb ≡ f →
-                       ∃∃f1a,f1b. f2 ~⊚ f1a ≡ fa & f2 ~⊚ f1b ≡ fb & f1a ⋓ f1b ≡ f1.
+lemma coafter_inv_sor: ∀f. 𝐅❪f❫ → ∀f2. 𝐓❪f2❫ → ∀f1. f2 ~⊚ f1 ≘ f → ∀fa,fb. fa ⋓ fb ≘ f →
+                       ∃∃f1a,f1b. f2 ~⊚ f1a ≘ fa & f2 ~⊚ f1b ≘ fb & f1a ⋓ f1b ≘ f1.
 @isfin_ind
 [ #f #Hf #f2 #Hf2 #f1 #H1f #fa #fb #H2f
   elim (sor_inv_isid3 … H2f) -H2f //
@@ -649,13 +688,13 @@ qed-.
 
 (* Properties with istot ****************************************************)
 
-lemma coafter_sor: ∀f. 𝐅⦃f⦄ → ∀f2. 𝐓⦃f2⦄ → ∀f1. f2 ~⊚ f1 ≡ f → ∀f1a,f1b. f1a ⋓ f1b ≡ f1 →
-                   ∃∃fa,fb. f2 ~⊚ f1a ≡ fa & f2 ~⊚ f1b ≡ fb & fa ⋓ fb ≡ f.
+lemma coafter_sor: ∀f. 𝐅❪f❫ → ∀f2. 𝐓❪f2❫ → ∀f1. f2 ~⊚ f1 ≘ f → ∀f1a,f1b. f1a ⋓ f1b ≘ f1 →
+                   ∃∃fa,fb. f2 ~⊚ f1a ≘ fa & f2 ~⊚ f1b ≘ fb & fa ⋓ fb ≘ f.
 @isfin_ind
 [ #f #Hf #f2 #Hf2 #f1 #Hf #f1a #f1b #Hf1
   lapply (coafter_fwd_isid2 … Hf ??) -Hf // #H2f1
   elim (sor_inv_isid3 … Hf1) -Hf1 //
-  /3 width=5 by coafter_isid_dx, sor_refl, ex3_2_intro/
+  /3 width=5 by coafter_isid_dx, sor_idem, ex3_2_intro/
 | #f #_ #IH #f2 #Hf2 #f1 #H1 #f1a #f1b #H2
   elim (coafter_inv_xxp … H1) -H1 [1,3: * |*: // ]
   [ #g2 #g1 #Hf #Hgf2 #Hgf1
@@ -679,9 +718,9 @@ qed-.
 
 (* Properties with after ****************************************************)
 (*
-corec theorem coafter_trans1: ∀f0,f3,f4. f0 ~⊚ f3 ≡ f4 →
-                            ∀f1,f2. f1 ~⊚ f2 ≡ f0 →
-                            ∀f. f2 ~⊚ f3 ≡ f → f1 ~⊚ f ≡ f4.
+corec theorem coafter_trans1: ∀f0,f3,f4. f0 ~⊚ f3 ≘ f4 →
+                            ∀f1,f2. f1 ~⊚ f2 ≘ f0 →
+                            ∀f. f2 ~⊚ f3 ≘ f → f1 ~⊚ f ≘ f4.
 #f0 #f3 #f4 * -f0 -f3 -f4 #f0 #f3 #f4 #g0 [1,2: #g3 ] #g4
 [ #Hf4 #H0 #H3 #H4 #g1 #g2 #Hg0 #g #Hg
   cases (coafter_inv_xxp … Hg0 … H0) -g0
@@ -703,9 +742,9 @@ corec theorem coafter_trans1: ∀f0,f3,f4. f0 ~⊚ f3 ≡ f4 →
 ]
 qed-.
 
-corec theorem coafter_trans2: ∀f1,f0,f4. f1 ~⊚ f0 ≡ f4 →
-                            ∀f2, f3. f2 ~⊚ f3 ≡ f0 →
-                            ∀f. f1 ~⊚ f2 ≡ f → f ~⊚ f3 ≡ f4.
+corec theorem coafter_trans2: ∀f1,f0,f4. f1 ~⊚ f0 ≘ f4 →
+                            ∀f2, f3. f2 ~⊚ f3 ≘ f0 →
+                            ∀f. f1 ~⊚ f2 ≘ f → f ~⊚ f3 ≘ f4.
 #f1 #f0 #f4 * -f1 -f0 -f4 #f1 #f0 #f4 #g1 [1,2: #g0 ] #g4
 [ #Hf4 #H1 #H0 #H4 #g2 #g3 #Hg0 #g #Hg
   cases (coafter_inv_xxp … Hg0 … H0) -g0