X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fground_2%2Frelocation%2Frtmap_after.ma;h=b2ec8605cd6dcabd3d60a18283b2236726ae46c0;hb=bd53c4e895203eb049e75434f638f26b5a161a2b;hp=09bab5e109f5d2665cd9377585216bf4ef7aff61;hpb=5275f55f5ec528edbb223834f3ec2cf1d3ce9b84;p=helm.git

diff --git a/matita/matita/contribs/lambdadelta/ground_2/relocation/rtmap_after.ma b/matita/matita/contribs/lambdadelta/ground_2/relocation/rtmap_after.ma
index 09bab5e10..b2ec8605c 100644
--- a/matita/matita/contribs/lambdadelta/ground_2/relocation/rtmap_after.ma
+++ b/matita/matita/contribs/lambdadelta/ground_2/relocation/rtmap_after.ma
@@ -19,24 +19,24 @@ include "ground_2/relocation/rtmap_istot.ma".
 
 coinductive after: relation3 rtmap rtmap rtmap ≝
 | after_refl: ∀f1,f2,f,g1,g2,g.
-              after f1 f2 f → ↑f1 = g1 → ↑f2 = g2 → ↑f = g → after g1 g2 g
+              after f1 f2 f → ⫯f1 = g1 → ⫯f2 = g2 → ⫯f = g → after g1 g2 g
 | after_push: ∀f1,f2,f,g1,g2,g.
-              after f1 f2 f → ↑f1 = g1 → ⫯f2 = g2 → ⫯f = g → after g1 g2 g
+              after f1 f2 f → ⫯f1 = g1 → ↑f2 = g2 → ↑f = g → after g1 g2 g
 | after_next: ∀f1,f2,f,g1,g.
-              after f1 f2 f → ⫯f1 = g1 → ⫯f = g → after g1 f2 g
+              after f1 f2 f → ↑f1 = g1 → ↑f = g → after g1 f2 g
 .
 
 interpretation "relational composition (rtmap)"
    'RAfter f1 f2 f = (after f1 f2 f).
 
 definition H_after_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.
 
 (* Basic inversion lemmas ***************************************************)
 
-lemma after_inv_ppx: ∀g1,g2,g. g1 ⊚ g2 ≡ g → ∀f1,f2. ↑f1 = g1 → ↑f2 = g2 →
-                     ∃∃f. f1 ⊚ f2 ≡ f & ↑f = g.
+lemma after_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
@@ -48,8 +48,8 @@ lemma after_inv_ppx: ∀g1,g2,g. g1 ⊚ g2 ≡ g → ∀f1,f2. ↑f1 = g1 → 
 ]
 qed-.
 
-lemma after_inv_pnx: ∀g1,g2,g. g1 ⊚ g2 ≡ g → ∀f1,f2. ↑f1 = g1 → ⫯f2 = g2 →
-                     ∃∃f. f1 ⊚ f2 ≡ f & ⫯f = g.
+lemma after_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)
@@ -61,8 +61,8 @@ lemma after_inv_pnx: ∀g1,g2,g. g1 ⊚ g2 ≡ g → ∀f1,f2. ↑f1 = g1 → 
 ]
 qed-.
 
-lemma after_inv_nxx: ∀g1,f2,g. g1 ⊚ f2 ≡ g → ∀f1. ⫯f1 = g1 →
-                     ∃∃f. f1 ⊚ f2 ≡ f & ⫯f = g.
+lemma after_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)
@@ -76,71 +76,71 @@ qed-.
 
 (* Advanced inversion lemmas ************************************************)
 
-lemma after_inv_ppp: ∀g1,g2,g. g1 ⊚ g2 ≡ g →
-                     ∀f1,f2,f. ↑f1 = g1 → ↑f2 = g2 → ↑f = g → f1 ⊚ f2 ≡ f.
+lemma after_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 (after_inv_ppx … Hg … H1 H2) -g1 -g2
 #x #Hf #Hx destruct <(injective_push … Hx) -f //
 qed-.
 
-lemma after_inv_ppn: ∀g1,g2,g. g1 ⊚ g2 ≡ g →
-                     ∀f1,f2,f. ↑f1 = g1 → ↑f2 = g2 → ⫯f = g → ⊥.
+lemma after_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 (after_inv_ppx … Hg … H1 H2) -g1 -g2
 #x #Hf #Hx destruct elim (discr_push_next … Hx)
 qed-.
 
-lemma after_inv_pnn: ∀g1,g2,g. g1 ⊚ g2 ≡ g →
-                    ∀f1,f2,f. ↑f1 = g1 → ⫯f2 = g2 → ⫯f = g → f1 ⊚ f2 ≡ f.
+lemma after_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 (after_inv_pnx … Hg … H1 H2) -g1 -g2
 #x #Hf #Hx destruct <(injective_next … Hx) -f //
 qed-.
 
-lemma after_inv_pnp: ∀g1,g2,g. g1 ⊚ g2 ≡ g →
-                     ∀f1,f2,f. ↑f1 = g1 → ⫯f2 = g2 → ↑f = g → ⊥.
+lemma after_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 (after_inv_pnx … Hg … H1 H2) -g1 -g2
 #x #Hf #Hx destruct elim (discr_next_push … Hx)
 qed-.
 
-lemma after_inv_nxn: ∀g1,f2,g. g1 ⊚ f2 ≡ g →
-                     ∀f1,f. ⫯f1 = g1 → ⫯f = g → f1 ⊚ f2 ≡ f.
+lemma after_inv_nxn: ∀g1,f2,g. g1 ⊚ f2 ≘ g →
+                     ∀f1,f. ↑f1 = g1 → ↑f = g → f1 ⊚ f2 ≘ f.
 #g1 #f2 #g #Hg #f1 #f #H1 #H elim (after_inv_nxx … Hg … H1) -g1
 #x #Hf #Hx destruct <(injective_next … Hx) -f //
 qed-.
 
-lemma after_inv_nxp: ∀g1,f2,g. g1 ⊚ f2 ≡ g →
-                     ∀f1,f. ⫯f1 = g1 → ↑f = g → ⊥.
+lemma after_inv_nxp: ∀g1,f2,g. g1 ⊚ f2 ≘ g →
+                     ∀f1,f. ↑f1 = g1 → ⫯f = g → ⊥.
 #g1 #f2 #g #Hg #f1 #f #H1 #H elim (after_inv_nxx … Hg … H1) -g1
 #x #Hf #Hx destruct elim (discr_next_push … Hx)
 qed-.
 
-lemma after_inv_pxp: ∀g1,g2,g. g1 ⊚ g2 ≡ g →
-                     ∀f1,f. ↑f1 = g1 → ↑f = g →
-                     ∃∃f2. f1 ⊚ f2 ≡ f & ↑f2 = g2.
+lemma after_inv_pxp: ∀g1,g2,g. g1 ⊚ g2 ≘ g →
+                     ∀f1,f. ⫯f1 = g1 → ⫯f = g →
+                     ∃∃f2. f1 ⊚ f2 ≘ f & ⫯f2 = g2.
 #g1 * * [2: #m2] #g2 #g #Hg #f1 #f #H1 #H
 [ elim (after_inv_pnp … Hg … H1 … H) -g1 -g -f1 -f //
 | lapply (after_inv_ppp … Hg … H1 … H) -g1 -g /2 width=3 by ex2_intro/
 ]
 qed-.
 
-lemma after_inv_pxn: ∀g1,g2,g. g1 ⊚ g2 ≡ g →
-                     ∀f1,f. ↑f1 = g1 → ⫯f = g →
-                     ∃∃f2. f1 ⊚ f2 ≡ f & ⫯f2 = g2.
+lemma after_inv_pxn: ∀g1,g2,g. g1 ⊚ g2 ≘ g →
+                     ∀f1,f. ⫯f1 = g1 → ↑f = g →
+                     ∃∃f2. f1 ⊚ f2 ≘ f & ↑f2 = g2.
 #g1 * * [2: #m2] #g2 #g #Hg #f1 #f #H1 #H
 [ lapply (after_inv_pnn … Hg … H1 … H) -g1 -g /2 width=3 by ex2_intro/
 | elim (after_inv_ppn … Hg … H1 … H) -g1 -g -f1 -f //
 ]
 qed-.
 
-lemma after_inv_xxp: ∀g1,g2,g. g1 ⊚ g2 ≡ g → ∀f. ↑f = g →
-                     ∃∃f1,f2. f1 ⊚ f2 ≡ f & ↑f1 = g1 & ↑f2 = g2.
+lemma after_inv_xxp: ∀g1,g2,g. g1 ⊚ g2 ≘ g → ∀f. ⫯f = g →
+                     ∃∃f1,f2. f1 ⊚ f2 ≘ f & ⫯f1 = g1 & ⫯f2 = g2.
 * * [2: #m1 ] #g1 #g2 #g #Hg #f #H
 [ elim (after_inv_nxp … Hg … H) -g2 -g -f //
 | elim (after_inv_pxp … Hg … H) -g /2 width=5 by ex3_2_intro/
 ]
 qed-.
 
-lemma after_inv_xxn: ∀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 after_inv_xxn: ∀g1,g2,g. g1 ⊚ g2 ≘ g → ∀f. ↑f = g →
+                     (∃∃f1,f2. f1 ⊚ f2 ≘ f & ⫯f1 = g1 & ↑f2 = g2) ∨
+                     ∃∃f1. f1 ⊚ g2 ≘ f & ↑f1 = g1.
 * * [2: #m1 ] #g1 #g2 #g #Hg #f #H
 [ /4 width=5 by after_inv_nxn, or_intror, ex2_intro/
 | elim (after_inv_pxn … Hg … H) -g
@@ -148,9 +148,9 @@ lemma after_inv_xxn: ∀g1,g2,g. g1 ⊚ g2 ≡ g → ∀f. ⫯f = g →
 ]
 qed-.
 
-lemma after_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 after_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 * * [2: #m2 ] #g2 #g #Hg #f1 #H
 [  elim (after_inv_pnx … Hg … H) -g1
   /3 width=5 by or_intror, ex3_2_intro/
@@ -161,20 +161,20 @@ qed-.
 
 (* Basic properties *********************************************************)
 
-corec lemma after_eq_repl_back_2: ∀f1,f. eq_repl_back (λf2. f2 ⊚ f1 ≡ f).
+corec lemma after_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 after_refl/
 | cases (eq_inv_px …  H0 …  H21) -g21 /3 width=7 by after_push/
-| cases (eq_inv_nx …  H0 …  H21) -g21 /3 width=5 by after_next/ 
+| cases (eq_inv_nx …  H0 …  H21) -g21 /3 width=5 by after_next/
 ]
 qed-.
 
-lemma after_eq_repl_fwd_2: ∀f1,f. eq_repl_fwd (λf2. f2 ⊚ f1 ≡ f).
-#f1 #f @eq_repl_sym /2 width=3 by after_eq_repl_back_2/
+lemma after_eq_repl_fwd2: ∀f1,f. eq_repl_fwd (λf2. f2 ⊚ f1 ≘ f).
+#f1 #f @eq_repl_sym /2 width=3 by after_eq_repl_back2/
 qed-.
 
-corec lemma after_eq_repl_back_1: ∀f2,f. eq_repl_back (λf1. f2 ⊚ f1 ≡ f).
+corec lemma after_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 after_refl/
@@ -183,28 +183,28 @@ corec lemma after_eq_repl_back_1: ∀f2,f. eq_repl_back (λf1. f2 ⊚ f1 ≡ f).
 ]
 qed-.
 
-lemma after_eq_repl_fwd_1: ∀f2,f. eq_repl_fwd (λf1. f2 ⊚ f1 ≡ f).
-#f2 #f @eq_repl_sym /2 width=3 by after_eq_repl_back_1/
+lemma after_eq_repl_fwd1: ∀f2,f. eq_repl_fwd (λf1. f2 ⊚ f1 ≘ f).
+#f2 #f @eq_repl_sym /2 width=3 by after_eq_repl_back1/
 qed-.
 
-corec lemma after_eq_repl_back_0: ∀f1,f2. eq_repl_back (λf. f2 ⊚ f1 ≡ f).
+corec lemma after_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 after_refl/
 | cases (eq_inv_nx …  H0 …  H01) -g01 /3 width=7 by after_push/
-| cases (eq_inv_nx …  H0 …  H01) -g01 /3 width=5 by after_next/ 
+| cases (eq_inv_nx …  H0 …  H01) -g01 /3 width=5 by after_next/
 ]
 qed-.
 
-lemma after_eq_repl_fwd_0: ∀f2,f1. eq_repl_fwd (λf. f2 ⊚ f1 ≡ f).
-#f2 #f1 @eq_repl_sym /2 width=3 by after_eq_repl_back_0/
+lemma after_eq_repl_fwd0: ∀f2,f1. eq_repl_fwd (λf. f2 ⊚ f1 ≘ f).
+#f2 #f1 @eq_repl_sym /2 width=3 by after_eq_repl_back0/
 qed-.
 
 (* Main properties **********************************************************)
 
-corec theorem after_trans1: ∀f0,f3,f4. f0 ⊚ f3 ≡ f4 →
-                            ∀f1,f2. f1 ⊚ f2 ≡ f0 →
-                            ∀f. f2 ⊚ f3 ≡ f → f1 ⊚ f ≡ f4.
+corec theorem after_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 (after_inv_xxp … Hg0 … H0) -g0
@@ -226,9 +226,9 @@ corec theorem after_trans1: ∀f0,f3,f4. f0 ⊚ f3 ≡ f4 →
 ]
 qed-.
 
-corec theorem after_trans2: ∀f1,f0,f4. f1 ⊚ f0 ≡ f4 →
-                            ∀f2, f3. f2 ⊚ f3 ≡ f0 →
-                            ∀f. f1 ⊚ f2 ≡ f → f ⊚ f3 ≡ f4.
+corec theorem after_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 (after_inv_xxp … Hg0 … H0) -g0
@@ -252,7 +252,7 @@ qed-.
 
 (* Main inversion lemmas ****************************************************)
 
-corec theorem after_mono: ∀f1,f2,x,y. f1 ⊚ f2 ≡ x → f1 ⊚ f2 ≡ y → x ≗ y.
+corec theorem after_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 (after_inv_ppx … Hy … H1 H2) -g1 -g2 /3 width=8 by eq_push/
@@ -261,70 +261,72 @@ corec theorem after_mono: ∀f1,f2,x,y. f1 ⊚ f2 ≡ x → f1 ⊚ f2 ≡ y →
 ]
 qed-.
 
-lemma after_mono_eq: ∀f1,f2,f. f1 ⊚ f2 ≡ f → ∀g1,g2,g. g1 ⊚ g2 ≡ g →
-                     f1 ≗ g1 → f2 ≗ g2 → f ≗ g.
-/4 width=4 by after_mono, after_eq_repl_back_1, after_eq_repl_back_2/ qed-.
+lemma after_mono_eq: ∀f1,f2,f. f1 ⊚ f2 ≘ f → ∀g1,g2,g. g1 ⊚ g2 ≘ g →
+                     f1 ≡ g1 → f2 ≡ g2 → f ≡ g.
+/4 width=4 by after_mono, after_eq_repl_back1, after_eq_repl_back2/ qed-.
 
 (* Properties on tls ********************************************************)
 
-lemma after_tls: ∀n,f1,f2,f. @⦃0, f1⦄ ≡ n → 
-                 f1 ⊚ f2 ≡ f → ⫱*[n]f1 ⊚ f2 ≡ ⫱*[n]f.
+lemma after_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 (after_inv_nxx … Hf … H1) -Hf /2 width=1 by/
+#n #IH #f1 #f2 #f #Hf1 #Hf
+cases (at_inv_pxn … Hf1) -Hf1 [ |*: // ] #g1 #Hg1 #H1
+cases (after_inv_nxx … Hf … H1) -Hf #g #Hg #H0 destruct
+<tls_xn <tls_xn /2 width=1 by/
 qed.
 
-(* Inversion lemmas on isid *************************************************)
+(* Properties on isid *******************************************************)
 
-corec lemma isid_after_sn: ∀f1. 𝐈⦃f1⦄ → ∀f2. f1 ⊚ f2 ≡ f2.
+corec lemma after_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 after_push, after_refl/
-qed-.
+qed.
 
-corec lemma isid_after_dx: ∀f2. 𝐈⦃f2⦄ → ∀f1. f1 ⊚ f2 ≡ f1.
+corec lemma after_isid_dx: ∀f2. 𝐈❪f2❫ → ∀f1. f1 ⊚ f2 ≘ f1.
 #f2 * -f2 #f2 #g2 #Hf2 #H2 #f1 cases (pn_split f1) * #g1 #H1
 [ /3 width=7 by after_refl/
 | @(after_next … H1 H1) /3 width=3 by isid_push/
 ]
-qed-.
+qed.
 
-lemma after_isid_inv_sn: ∀f1,f2,f. f1 ⊚ f2 ≡ f →  𝐈⦃f1⦄ → f2 ≗ f.
-/3 width=6 by isid_after_sn, after_mono/
-qed-.
+(* Inversion lemmas on isid *************************************************)
 
-lemma after_isid_inv_dx: ∀f1,f2,f. f1 ⊚ f2 ≡ f →  𝐈⦃f2⦄ → f1 ≗ f.
-/3 width=6 by isid_after_dx, after_mono/
-qed-.
+lemma after_isid_inv_sn: ∀f1,f2,f. f1 ⊚ f2 ≘ f → 𝐈❪f1❫ → f2 ≡ f.
+/3 width=6 by after_isid_sn, after_mono/ qed-.
+
+lemma after_isid_inv_dx: ∀f1,f2,f. f1 ⊚ f2 ≘ f → 𝐈❪f2❫ → f1 ≡ f.
+/3 width=6 by after_isid_dx, after_mono/ qed-.
 
-corec lemma after_fwd_isid1: ∀f1,f2,f. f1 ⊚ f2 ≡ f → 𝐈⦃f⦄ → 𝐈⦃f1⦄.
+corec lemma after_fwd_isid1: ∀f1,f2,f. f1 ⊚ f2 ≘ f → 𝐈❪f❫ → 𝐈❪f1❫.
 #f1 #f2 #f * -f1 -f2 -f
 #f1 #f2 #f #g1 [1,2: #g2 ] #g #Hf #H1 [1,2: #H2 ] #H0 #H
 [ /4 width=6 by isid_inv_push, isid_push/ ]
 cases (isid_inv_next … H … H0)
 qed-.
 
-corec lemma after_fwd_isid2: ∀f1,f2,f. f1 ⊚ f2 ≡ f → 𝐈⦃f⦄ → 𝐈⦃f2⦄.
+corec lemma after_fwd_isid2: ∀f1,f2,f. f1 ⊚ f2 ≘ f → 𝐈❪f❫ → 𝐈❪f2❫.
 #f1 #f2 #f * -f1 -f2 -f
 #f1 #f2 #f #g1 [1,2: #g2 ] #g #Hf #H1 [1,2: #H2 ] #H0 #H
 [ /4 width=6 by isid_inv_push, isid_push/ ]
 cases (isid_inv_next … H … H0)
 qed-.
 
-lemma after_inv_isid3: ∀f1,f2,f. f1 ⊚ f2 ≡ f → 𝐈⦃f⦄ → 𝐈⦃f1⦄ ∧ 𝐈⦃f2⦄.
+lemma after_inv_isid3: ∀f1,f2,f. f1 ⊚ f2 ≘ f → 𝐈❪f❫ → 𝐈❪f1❫ ∧ 𝐈❪f2❫.
 /3 width=4 by after_fwd_isid2, after_fwd_isid1, conj/ qed-.
 
 (* Properties on isuni ******************************************************)
 
-lemma after_isid_isuni: ∀f1,f2. 𝐈⦃f2⦄ → 𝐔⦃f1⦄ → f1 ⊚ ⫯f2 ≡ ⫯f1.
+lemma after_isid_isuni: ∀f1,f2. 𝐈❪f2❫ → 𝐔❪f1❫ → f1 ⊚ ↑f2 ≘ ↑f1.
 #f1 #f2 #Hf2 #H elim H -H
-/5 width=7 by isid_after_dx, after_eq_repl_back_2, after_next, after_push, eq_push_inv_isid/
+/5 width=7 by after_isid_dx, after_eq_repl_back2, after_next, after_push, eq_push_inv_isid/
 qed.
 
-lemma after_uni_next2: ∀f2. 𝐔⦃f2⦄ → ∀f1,f. ⫯f2 ⊚ f1 ≡ f → f2 ⊚ ⫯f1 ≡ f.
+lemma after_uni_next2: ∀f2. 𝐔❪f2❫ → ∀f1,f. ↑f2 ⊚ f1 ≘ f → f2 ⊚ ↑f1 ≘ f.
 #f2 #H elim H -f2
 [ #f2 #Hf2 #f1 #f #Hf
   elim (after_inv_nxx … Hf) -Hf [2,3: // ] #g #Hg #H0 destruct
-  /4 width=7 by after_isid_inv_sn, isid_after_sn, after_eq_repl_back_0, eq_next/
+  /4 width=7 by after_isid_inv_sn, after_isid_sn, after_eq_repl_back0, eq_next/
 | #f2 #_ #g2 #H2 #IH #f1 #f #Hf
   elim (after_inv_nxx … Hf) -Hf [2,3: // ] #g #Hg #H0 destruct
   /3 width=5 by after_next/
@@ -333,15 +335,15 @@ qed.
 
 (* Properties on uni ********************************************************)
 
-lemma after_uni: ∀n1,n2. 𝐔❴n1❵ ⊚ 𝐔❴n2❵ ≡ 𝐔❴n1+n2❵.
-@nat_elim2
-/4 width=5 by after_uni_next2, isid_after_sn, isid_after_dx, after_next/
+lemma after_uni: ∀n1,n2. 𝐔❨n1❩ ⊚ 𝐔❨n2❩ ≘ 𝐔❨n1+n2❩.
+@nat_elim2 [3: #n #m <plus_n_Sm ] (**) (* full auto fails *)
+/4 width=5 by after_uni_next2, after_isid_dx, after_isid_sn, after_next/
 qed.
 
 (* Forward lemmas on at *****************************************************)
 
-lemma after_at_fwd: ∀i,i1,f. @⦃i1, f⦄ ≡ i → ∀f2,f1. f2 ⊚ f1 ≡ f →
-                    ∃∃i2. @⦃i1, f1⦄ ≡ i2 & @⦃i2, f2⦄ ≡ i.
+lemma after_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 ]
@@ -357,8 +359,8 @@ lemma after_at_fwd: ∀i,i1,f. @⦃i1, f⦄ ≡ i → ∀f2,f1. f2 ⊚ f1 ≡ f
 /3 width=9 by at_refl, at_push, at_next, ex2_intro/
 qed-.
 
-lemma after_fwd_at: ∀i,i2,i1,f1,f2. @⦃i1, f1⦄ ≡ i2 → @⦃i2, f2⦄ ≡ i →
-                    ∀f. f2 ⊚ f1 ≡ f → @⦃i1, f⦄ ≡ i.
+lemma after_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
@@ -375,14 +377,14 @@ lemma after_fwd_at: ∀i,i2,i1,f1,f2. @⦃i1, f1⦄ ≡ i2 → @⦃i2, f2⦄ ≡
 ]
 qed-.
 
-lemma after_fwd_at2: ∀f,i1,i. @⦃i1, f⦄ ≡ i → ∀f1,i2. @⦃i1, f1⦄ ≡ i2 →
-                     ∀f2. f2 ⊚ f1 ≡ f → @⦃i2, f2⦄ ≡ i.
+lemma after_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 (after_at_fwd … Hf … H) -f
 #j1 #H #Hf2 <(at_mono … Hf1 … H) -i1 -i2 //
 qed-.
 
-lemma after_fwd_at1: ∀i,i2,i1,f,f2. @⦃i1, f⦄ ≡ i → @⦃i2, f2⦄ ≡ i →
-                     ∀f1. f2 ⊚ f1 ≡ f → @⦃i1, f1⦄ ≡ i2.
+lemma after_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
@@ -401,32 +403,32 @@ qed-.
 
 (* Properties with at *******************************************************)
 
-lemma after_isuni_dx: ∀i2,i1,f2. @⦃i1, f2⦄ ≡ i2 →
-                      ∀f. f2 ⊚ 𝐔❴i1❵ ≡ f → 𝐔❴i2❵ ⊚ ⫱*[i2] f2 ≡ f.
+lemma after_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
   lapply (after_isid_inv_dx … Hf ?) -Hf
-  /3 width=3 by isid_after_sn, after_eq_repl_back_0/
+  /3 width=3 by after_isid_sn, after_eq_repl_back0/
 | #i2 #IH #i1 #f2 #Hf2 #f #Hf
   elim (at_inv_xxn … Hf2) -Hf2 [1,3: * |*: // ]
   [ #g2 #j1 #Hg2 #H1 #H2 destruct
     elim (after_inv_pnx … Hf) -Hf [ |*: // ] #g #Hg #H destruct
-    /3 width=5 by after_next/
+    <tls_xn /3 width=5 by after_next/
   | #g2 #Hg2 #H2 destruct
     elim (after_inv_nxx … Hf) -Hf [2,3: // ] #g #Hg #H destruct
-    /3 width=5 by after_next/
+    <tls_xn /3 width=5 by after_next/
   ]
 ]
 qed.
 
-lemma after_isuni_sn: ∀i2,i1,f2. @⦃i1, f2⦄ ≡ i2 →
-                      ∀f. 𝐔❴i2❵ ⊚ ⫱*[i2] f2 ≡ f → f2 ⊚ 𝐔❴i1❵ ≡ f.
+lemma after_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
   lapply (after_isid_inv_sn … Hf ?) -Hf
-  /3 width=3 by isid_after_dx, after_eq_repl_back_0/
+  /3 width=3 by after_isid_dx, after_eq_repl_back0/
 | #i2 #IH #i1 #f2 #Hf2 #f #Hf
   elim (after_inv_nxx … Hf) -Hf [2,3: // ] #g #Hg #H destruct
   elim (at_inv_xxn … Hf2) -Hf2 [1,3: * |*: // ]
@@ -436,36 +438,80 @@ lemma after_isuni_sn: ∀i2,i1,f2. @⦃i1, f2⦄ ≡ i2 →
 ]
 qed-.
 
+lemma after_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
+  elim (after_inv_pnx … Hf) -Hf [ |*: // ] #g #Hg #H
+  lapply (after_isid_inv_dx … Hg ?) -Hg
+  /4 width=5 by after_isid_sn, after_eq_repl_back0, after_next/
+| #i2 #IH #i1 #f2 #Hf2 #f #Hf
+  elim (at_inv_xxn … Hf2) -Hf2 [1,3: * |*: // ]
+  [ #g2 #j1 #Hg2 #H1 #H2 destruct
+    elim (after_inv_pnx … Hf) -Hf [ |*: // ] #g #Hg #H destruct
+    <tls_xn /3 width=5 by after_next/
+  | #g2 #Hg2 #H2 destruct
+    elim (after_inv_nxx … Hf) -Hf [2,3: // ] #g #Hg #H destruct
+    <tls_xn /3 width=5 by after_next/
+  ]
+]
+qed.
+
+lemma after_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
+  elim (after_inv_nxx … Hf) -Hf [ |*: // ] #g #Hg #H destruct
+  lapply (after_isid_inv_sn … Hg ?) -Hg
+  /4 width=7 by after_isid_dx, after_eq_repl_back0, after_push/
+| #i2 #IH #i1 #f2 #Hf2 #f #Hf
+  elim (after_inv_nxx … Hf) -Hf [2,3: // ] #g #Hg #H destruct
+  elim (at_inv_xxn … Hf2) -Hf2 [1,3: * |*: // ]
+  [ #g2 #j1 #Hg2 #H1 #H2 destruct <tls_xn in Hg; /3 width=7 by after_push/
+  | #g2 #Hg2 #H2 destruct <tls_xn in Hg; /3 width=5 by after_next/
+  ]
+]
+qed-.
+
+lemma after_uni_one_dx: ∀f2,f. ⫯f2 ⊚ 𝐔❨↑O❩ ≘ f → 𝐔❨↑O❩ ⊚ f2 ≘ f.
+#f2 #f #H @(after_uni_succ_dx … (⫯f2)) /2 width=3 by at_refl/
+qed.
+
+lemma after_uni_one_sn: ∀f1,f. 𝐔❨↑O❩ ⊚ f1 ≘ f → ⫯f1 ⊚ 𝐔❨↑O❩ ≘ f.
+/3 width=3 by after_uni_succ_sn, at_refl/ qed-.
+
 (* Forward lemmas on istot **************************************************)
 
-lemma after_istot_fwd: ∀f2,f1,f. f2 ⊚ f1 ≡ f → 𝐓⦃f2⦄ → 𝐓⦃f1⦄ → 𝐓⦃f⦄.
+lemma after_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 after_fwd_at, ex_intro/
 qed-.
 
-lemma after_fwd_istot_dx: ∀f2,f1,f. f2 ⊚ f1 ≡ f → 𝐓⦃f⦄ → 𝐓⦃f1⦄.
+lemma after_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 (after_at_fwd … Hf … H) -f /2 width=2 by ex_intro/
 qed-.
 
-lemma after_fwd_istot_sn: ∀f2,f1,f. f2 ⊚ f1 ≡ f → 𝐓⦃f⦄ → 𝐓⦃f2⦄.
+lemma after_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 (after_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 after_inv_istot: ∀f2,f1,f. f2 ⊚ f1 ≡ f → 𝐓⦃f⦄ → 𝐓⦃f2⦄ ∧ 𝐓⦃f1⦄.
+lemma after_inv_istot: ∀f2,f1,f. f2 ⊚ f1 ≘ f → 𝐓❪f❫ → 𝐓❪f2❫ ∧ 𝐓❪f1❫.
 /3 width=4 by after_fwd_istot_sn, after_fwd_istot_dx, conj/ qed-.
 
-lemma after_at1_fwd: ∀f1,i1,i2. @⦃i1, f1⦄ ≡ i2 → ∀f2. 𝐓⦃f2⦄ → ∀f. f2 ⊚ f1 ≡ f →
-                     ∃∃i. @⦃i2, f2⦄ ≡ i & @⦃i1, f⦄ ≡ i.
+lemma after_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 after_fwd_at, ex2_intro/
 qed-.
 
-lemma after_fwd_isid_sn: ∀f2,f1,f. 𝐓⦃f⦄ → f2 ⊚ f1 ≡ f → f1 ≗ f → 𝐈⦃f2⦄.
+lemma after_fwd_isid_sn: ∀f2,f1,f. 𝐓❪f❫ → f2 ⊚ f1 ≘ f → f1 ≡ f → 𝐈❪f2❫.
 #f2 #f1 #f #H #Hf elim (after_inv_istot … Hf H) -H
 #Hf2 #Hf1 #H @isid_at_total // -Hf2
 #i2 #i #Hf2 elim (Hf1 i2) -Hf1
@@ -474,14 +520,14 @@ lemma after_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 after_fwd_isid_dx: ∀f2,f1,f.  𝐓⦃f⦄ → f2 ⊚ f1 ≡ f → f2 ≗ f → 𝐈⦃f1⦄.
+lemma after_fwd_isid_dx: ∀f2,f1,f.  𝐓❪f❫ → f2 ⊚ f1 ≘ f → f2 ≡ f → 𝐈❪f1❫.
 #f2 #f1 #f #H #Hf elim (after_inv_istot … Hf H) -H
 #Hf2 #Hf1 #H2 @isid_at_total // -Hf1
 #i1 #i2 #Hi12 elim (after_at1_fwd … Hi12 … Hf) -f1
 /3 width=8 by at_inj, at_eq_repl_back/
 qed-.
 
-corec fact after_inj_O_aux: ∀f1. @⦃0, f1⦄ ≡ 0 → H_after_inj f1.
+corec fact after_inj_O_aux: ∀f1. @❪0, f1❫ ≘ 0 → H_after_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
@@ -496,8 +542,8 @@ cases (after_inv_pxx … H1f … H1) -H1f * #g21 #g #H1g #H21 #H
 /2 width=1 by after_tls, istot_tls, at_pxx_tls/
 qed-.
 
-fact after_inj_aux: (∀f1. @⦃0, f1⦄ ≡ 0 → H_after_inj f1) →
-                    ∀i2,f1. @⦃0, f1⦄ ≡ i2 → H_after_inj f1.
+fact after_inj_aux: (∀f1. @❪0, f1❫ ≘ 0 → H_after_inj f1) →
+                    ∀i2,f1. @❪0, f1❫ ≘ i2 → H_after_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