X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fground_2%2Frelocation%2Frtmap_coafter.ma;h=2d1d860f920d593b6985c5f427da321d73147e08;hb=528f8ea107f689d07d060e1d31ba32bf65b4e6ba;hp=5ac0c06f75173c8f70e35e0cd8194ef0143d3546;hpb=926796df5884453d8f0cf9f294d7776d469ef45b;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 5ac0c06f7..2d1d860f9 100644
--- a/matita/matita/contribs/lambdadelta/ground_2/relocation/rtmap_coafter.ma
+++ b/matita/matita/contribs/lambdadelta/ground_2/relocation/rtmap_coafter.ma
@@ -14,7 +14,7 @@
 
 include "ground_2/notation/relations/rcoafter_3.ma".
 include "ground_2/relocation/rtmap_sor.ma".
-include "ground_2/relocation/rtmap_istot.ma".
+include "ground_2/relocation/rtmap_after.ma".
 
 (* RELOCATION MAP ***********************************************************)
 
@@ -151,6 +151,14 @@ lemma coafter_inv_xxn: ∀g1,g2,g. g1 ~⊚ g2 ≡ g → ∀f. ⫯f = g →
 ]
 qed-.
 
+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.
@@ -213,56 +221,6 @@ 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 properties **********************************************************)
-(*
-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
-  #f1 #f2 #Hf0 #H1 #H2
-  cases (coafter_inv_ppx … Hg … H2 H3) -g2 -g3
-  #f #Hf #H /3 width=7 by coafter_refl/
-| #Hf4 #H0 #H3 #H4 #g1 #g2 #Hg0 #g #Hg
-  cases (coafter_inv_xxp … Hg0 … H0) -g0
-  #f1 #f2 #Hf0 #H1 #H2
-  cases (coafter_inv_pnx … Hg … H2 H3) -g2 -g3
-  #f #Hf #H /3 width=7 by coafter_push/
-| #Hf4 #H0 #H4 #g1 #g2 #Hg0 #g #Hg
-  cases (coafter_inv_xxn … Hg0 … H0) -g0 *
-  [ #f1 #f2 #Hf0 #H1 #H2
-    cases (coafter_inv_nxx … Hg … H2) -g2
-    #f #Hf #H /3 width=7 by coafter_push/
-  | #f1 #Hf0 #H1 /3 width=6 by coafter_next/
-  ]
-]
-qed-.
-
-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
-  #f2 #f3 #Hf0 #H2 #H3
-  cases (coafter_inv_ppx … Hg … H1 H2) -g1 -g2
-  #f #Hf #H /3 width=7 by coafter_refl/
-| #Hf4 #H1 #H0 #H4 #g2 #g3 #Hg0 #g #Hg
-  cases (coafter_inv_xxn … Hg0 … H0) -g0 *
-  [ #f2 #f3 #Hf0 #H2 #H3
-    cases (coafter_inv_ppx … Hg … H1 H2) -g1 -g2
-    #f #Hf #H /3 width=7 by coafter_push/
-  | #f2 #Hf0 #H2
-    cases (coafter_inv_pnx … Hg … H1 H2) -g1 -g2
-    #f #Hf #H /3 width=6 by coafter_next/
-  ]
-| #Hf4 #H1 #H4 #f2 #f3 #Hf0 #g #Hg
-  cases (coafter_inv_nxx … Hg … H1) -g1
-  #f #Hg #H /3 width=6 by coafter_next/
-]
-qed-.
-*)
 (* Main inversion lemmas ****************************************************)
 
 corec theorem coafter_mono: ∀f1,f2,x,y. f1 ~⊚ f2 ≡ x → f1 ~⊚ f2 ≡ y → x ≗ y.
@@ -278,6 +236,28 @@ lemma coafter_mono_eq: ∀f1,f2,f. f1 ~⊚ f2 ≡ f → ∀g1,g2,g. g1 ~⊚ g2 
                        f1 ≗ g1 → f2 ≗ g2 → f ≗ g.
 /4 width=4 by coafter_mono, coafter_eq_repl_back1, coafter_eq_repl_back2/ qed-.
 
+(* 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 →
@@ -296,17 +276,61 @@ lemma coafter_inv_tl0: ∀g2,g1,g. g2 ~⊚ g1 ≡ ⫱g →
 ]
 qed-.
 
-(* Properties on tls ********************************************************)
+(* 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: ∀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 /2 width=1 by/
+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
+elim (at_inv_pxp … H) -H [ |*: // ] #f2 #H2
+elim (coafter_inv_pxx … Hg … H2) -Hg * #f1 #f #Hf #H1 #H0 destruct
+<tls_S <tls_S <H2 <H0 -g2 -g -n //
 qed.
 
-(* Properties on isid *******************************************************)
+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 <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,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 <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 with test for identity ****************************************)
 
 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
@@ -320,16 +344,23 @@ corec lemma coafter_isid_dx: ∀f2,f. 𝐈⦃f2⦄ → 𝐈⦃f⦄ → ∀f1. f1
 ]
 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.
 /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⦄.
 /4 width=4 by eq_id_isid, coafter_isid_dx, coafter_mono/ qed-.
-(*
-(* Properties on isuni ******************************************************)
 
+(* 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/
@@ -345,9 +376,15 @@ 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_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/
@@ -356,7 +393,7 @@ 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.
+                      ∃∃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 ]
@@ -373,7 +410,7 @@ lemma coafter_at_fwd: ∀i,i1,f. @⦃i1, f⦄ ≡ i → ∀f2,f1. f2 ~⊚ f1 ≡
 qed-.
 
 lemma coafter_fwd_at: ∀i,i2,i1,f1,f2. @⦃i1, f1⦄ ≡ i2 → @⦃i2, f2⦄ ≡ i →
-                    ∀f. f2 ~⊚ f1 ≡ f → @⦃i1, f⦄ ≡ 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
@@ -391,13 +428,13 @@ 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.
+                       ∀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.
+                       ∀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
@@ -417,7 +454,7 @@ qed-.
 (* Properties with at *******************************************************)
 
 lemma coafter_uni_dx: ∀i2,i1,f2. @⦃i1, f2⦄ ≡ i2 →
-                    ∀f. f2 ~⊚ 𝐔❴i1❵ ≡ f → 𝐔❴i2❵ ~⊚ ⫱*[i2] f2 ≡ f.
+                      ∀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
@@ -436,7 +473,7 @@ 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.
+                      ∀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
@@ -452,7 +489,7 @@ 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.
+                           ∀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
@@ -472,7 +509,7 @@ 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.
+                           ∀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
@@ -577,8 +614,8 @@ 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-.
@@ -609,11 +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.
+                             ∀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
@@ -657,7 +694,7 @@ lemma coafter_sor: ∀f. 𝐅⦃f⦄ → ∀f2. 𝐓⦃f2⦄ → ∀f1. f2 ~⊚
 [ #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
@@ -678,3 +715,54 @@ lemma coafter_sor: ∀f. 𝐅⦃f⦄ → ∀f2. 𝐓⦃f2⦄ → ∀f1. f2 ~⊚
   /3 width=11 by coafter_refl, coafter_push, sor_np, sor_pn, sor_nn, ex3_2_intro/
 ]
 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.
+#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
+  #f1 #f2 #Hf0 #H1 #H2
+  cases (coafter_inv_ppx … Hg … H2 H3) -g2 -g3
+  #f #Hf #H /3 width=7 by coafter_refl/
+| #Hf4 #H0 #H3 #H4 #g1 #g2 #Hg0 #g #Hg
+  cases (coafter_inv_xxp … Hg0 … H0) -g0
+  #f1 #f2 #Hf0 #H1 #H2
+  cases (coafter_inv_pnx … Hg … H2 H3) -g2 -g3
+  #f #Hf #H /3 width=7 by coafter_push/
+| #Hf4 #H0 #H4 #g1 #g2 #Hg0 #g #Hg
+  cases (coafter_inv_xxn … Hg0 … H0) -g0 *
+  [ #f1 #f2 #Hf0 #H1 #H2
+    cases (coafter_inv_nxx … Hg … H2) -g2
+    #f #Hf #H /3 width=7 by coafter_push/
+  | #f1 #Hf0 #H1 /3 width=6 by coafter_next/
+  ]
+]
+qed-.
+
+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
+  #f2 #f3 #Hf0 #H2 #H3
+  cases (coafter_inv_ppx … Hg … H1 H2) -g1 -g2
+  #f #Hf #H /3 width=7 by coafter_refl/
+| #Hf4 #H1 #H0 #H4 #g2 #g3 #Hg0 #g #Hg
+  cases (coafter_inv_xxn … Hg0 … H0) -g0 *
+  [ #f2 #f3 #Hf0 #H2 #H3
+    cases (coafter_inv_ppx … Hg … H1 H2) -g1 -g2
+    #f #Hf #H /3 width=7 by coafter_push/
+  | #f2 #Hf0 #H2
+    cases (coafter_inv_pnx … Hg … H1 H2) -g1 -g2
+    #f #Hf #H /3 width=6 by coafter_next/
+  ]
+| #Hf4 #H1 #H4 #f2 #f3 #Hf0 #g #Hg
+  cases (coafter_inv_nxx … Hg … H1) -g1
+  #f #Hg #H /3 width=6 by coafter_next/
+]
+qed-.
+*)