X-Git-Url: http://matita.cs.unibo.it/gitweb/?p=helm.git;a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fground%2Frelocation%2Frtmap_coafter.ma;h=aede6b3a31512f86a2e2afbf0adec99cd4e0bc1d;hp=e74866dac69eb1b77631eb88233dab2c52cdb9c2;hb=8fdf1af656038d0245eba64ff2531bbe94ce0e9e;hpb=77c9255de3c5f7780aeacd745703a1cc76328a68

diff --git a/matita/matita/contribs/lambdadelta/ground/relocation/rtmap_coafter.ma b/matita/matita/contribs/lambdadelta/ground/relocation/rtmap_coafter.ma
index e74866dac..aede6b3a3 100644
--- a/matita/matita/contribs/lambdadelta/ground/relocation/rtmap_coafter.ma
+++ b/matita/matita/contribs/lambdadelta/ground/relocation/rtmap_coafter.ma
@@ -14,6 +14,7 @@
 
 include "ground/notation/relations/rcoafter_3.ma".
 include "ground/relocation/rtmap_sor.ma".
+include "ground/relocation/rtmap_nat.ma".
 include "ground/relocation/rtmap_after.ma".
 
 (* RELOCATION MAP ***********************************************************)
@@ -238,21 +239,21 @@ lemma coafter_mono_eq: ∀f1,f2,f. f1 ~⊚ f2 ≘ f → ∀g1,g2,g. g1 ~⊚ g2 
 
 (* 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
+lemma coafter_fwd_pushs: ∀k,l,g2,f1,g. g2 ~⊚ ⫯*[l]f1 ≘ g → @↑❪l, g2❫ ≘ k →
+                         ∃∃f. ⫱*[k]g2 ~⊚ f1 ≘ f & ⫯*[k] f = g.
+#k @(nat_ind_succ … k) -k
+[ #l #g2 #f1 #g #Hg #H
+  elim (rm_nat_inv_xxp … H) -H [|*: // ] #f2 #H1 #H2 destruct
+  /2 width=3 by ex2_intro/
+| #k #IH * [| #l ] #g2 #f1 #g #Hg #H
+  [ elim (rm_nat_inv_pxn … H) -H [|*: // ] #f2 #Hlk #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 (IH … Hf Hlk) -IH -Hf -Hlk /2 width=3 by ex2_intro/
+  | elim (rm_nat_inv_nxn … H) -H [1,4: * |*: // ] #f2 #Hlk #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 (IH … Hf Hlk) -IH -Hf -Hlk /2 width=3 by ex2_intro/
     | elim (coafter_inv_nxx … Hg) -Hg [|*: // ] #f #Hf #H destruct
-      elim (IH … Hf Hij) -f1 -f2 -i /2 width=2 by ex_intro/
+      elim (IH … Hf Hlk) -IH -Hf -Hlk /2 width=3 by ex2_intro/
     ]
   ]
 ]
@@ -278,14 +279,14 @@ qed-.
 
 (* 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
+lemma coafter_tls: ∀l2,l1,f1,f2,f. @↑❪l1, f1❫ ≘ l2 →
+                   f1 ~⊚ f2 ≘ f → ⫱*[l2]f1 ~⊚ ⫱*[l1]f2 ≘ ⫱*[l2]f.
+#l2 @(nat_ind_succ … l2) -l2 [ #l1 | #l2 #IH * [| #l1 ] ] #f1 #f2 #f #Hf1 #Hf
+[ elim (rm_nat_inv_xxp … Hf1) -Hf1 [ |*: // ] #g1 #Hg1 #H1 destruct //
+| elim (rm_nat_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 (rm_nat_inv_xxn … Hf1) -Hf1 [1,3: * |*: // ] #g1 [ #k1 ] #Hg1 [ #H ] #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
@@ -294,24 +295,25 @@ lemma coafter_tls: ∀j,i,f1,f2,f. @❪i, f1❫ ≘ j →
 ]
 qed.
 
-lemma coafter_tls_O: ∀n,f1,f2,f. @❪0, f1❫ ≘ n →
-                     f1 ~⊚ f2 ≘ f → ⫱*[n]f1 ~⊚ f2 ≘ ⫱*[n]f.
+lemma coafter_tls_O: ∀k,f1,f2,f. @↑❪𝟎, f1❫ ≘ k →
+                     f1 ~⊚ f2 ≘ f → ⫱*[k]f1 ~⊚ f2 ≘ ⫱*[k]f.
 /2 width=1 by coafter_tls/ qed.
 
+(* Note: this does not require ↑ first and second j *)
 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
+                        ∀j. @❪𝟏, g2❫ ≘ j → ⫱*[j]g2 ~⊚ ⫱g1 ≘ ⫱*[j]g.
+#g2 #g1 #g #Hg #j #Hg2
+lapply (rm_nat_pred_bi … Hg2) -Hg2 #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 //
+elim (coafter_inv_pxx … Hg … H2) -Hg * #f1 #f #Hf #H1 #H0
+>(npsucc_pred j) <tls_S <tls_S //
 qed.
-
-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
+(*
+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(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
@@ -329,7 +331,7 @@ 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.
@@ -352,248 +354,25 @@ lemma coafter_isid_inv_sn: ∀f1,f2,f. f1 ~⊚ f2 ≘ f → 𝐈❪f1❫ → f2
 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 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.
-#f2 #H elim H -f2
-[ #f2 #Hf2 #f1 #f #Hf
-  elim (coafter_inv_nxx … Hf) -Hf [2,3: // ] #g #Hg #H0 destruct
-  /4 width=7 by coafter_isid_inv_sn, coafter_isid_sn, coafter_eq_repl_back0, eq_next/
-| #f2 #_ #g2 #H2 #IH #f1 #f #Hf
-  elim (coafter_inv_nxx … Hf) -Hf [2,3: // ] #g #Hg #H0 destruct
-  /3 width=5 by coafter_next/
-]
-qed.
-*)
-
-(* 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/
-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.
-#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 ]
-| elim (at_inv_xxp … Hf) -Hf //
-  #g #H1 #H
-]
-[2: elim (coafter_inv_xxn … Hf21 … H) -f *
-    [ #g2 #g1 #Hg21 #H2 #H1 | #g2 #Hg21 #H2 ]
-|*: elim (coafter_inv_xxp … Hf21 … H) -f
-    #g2 #g1 #Hg21 #H2 #H1
-]
-[4: -Hg21 |*: elim (IH … Hg … Hg21) -g -IH ]
-/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.
-#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
-  [ elim (at_inv_xxn … Hf1 … H22) -i2 *
-    #g1 [ #j1 ] #Hg1 [ #H11 ] #H10
-    [ elim (coafter_inv_ppx … Hf … H20 H10) -f1 -f2 /3 width=7 by at_push/
-    | elim (coafter_inv_pnx … Hf … H20 H10) -f1 -f2 /3 width=6 by at_next/
-    ]
-  | elim (coafter_inv_nxx … Hf … H20) -f2 /3 width=7 by at_next/
-  ]
-| elim (at_inv_xxp … Hf2) -Hf2 // #g2 #H22 #H20
-  elim (at_inv_xxp … Hf1 … H22) -i2 #g1 #H11 #H10
-  elim (coafter_inv_ppx … Hf … H20 H10) -f1 -f2 /2 width=2 by at_refl/
-]
-qed-.
-
-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.
-#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
-  elim (at_inv_xxn … Hf2) -Hf2 [1,3,5,7: * |*: // ]
-  #g2 [1,3: #j2 ] #Hg2 [1,2: #H22 ] #H20
-  [ elim (coafter_inv_pxp … Hf1 … H20 H00) -f2 -f /3 width=7 by at_push/
-  | elim (coafter_inv_pxn … Hf1 … H20 H00) -f2 -f /3 width=5 by at_next/
-  | elim (coafter_inv_nxp … Hf1 … H20 H00)
-  | /4 width=9 by coafter_inv_nxn, at_next/
-  ]
-| elim (at_inv_xxp … Hf) -Hf // #g #H01 #H00
-  elim (at_inv_xxp … Hf2) -Hf2 // #g2 #H21 #H20
-  elim (coafter_inv_pxp … Hf1 … H20 H00) -f2 -f /3 width=2 by at_refl/
-]
-qed-.
-
-(* Properties with at *******************************************************)
-
-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
-  lapply (coafter_isid_inv_dx … Hf ?) -Hf
-  /3 width=3 by coafter_isid_sn, coafter_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 (coafter_inv_pnx … Hf) -Hf [ |*: // ] #g #Hg #H destruct
-    /3 width=5 by coafter_next/
-  | #g2 #Hg2 #H2 destruct
-    elim (coafter_inv_nxx … Hf) -Hf [2,3: // ] #g #Hg #H destruct
-    /3 width=5 by coafter_next/
-  ]
-]
-qed.
-
-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
-  lapply (coafter_isid_inv_sn … Hf ?) -Hf
-  /3 width=3 by coafter_isid_dx, coafter_eq_repl_back0/
-| #i2 #IH #i1 #f2 #Hf2 #f #Hf
-  elim (coafter_inv_nxx … Hf) -Hf [2,3: // ] #g #Hg #H destruct
-  elim (at_inv_xxn … Hf2) -Hf2 [1,3: * |*: // ]
-  [ #g2 #j1 #Hg2 #H1 #H2 destruct /3 width=7 by coafter_push/
-  | #g2 #Hg2 #H2 destruct /3 width=5 by coafter_next/
-  ]
-]
-qed-.
-
-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
-  elim (coafter_inv_pnx … Hf) -Hf [ |*: // ] #g #Hg #H
-  lapply (coafter_isid_inv_dx … Hg ?) -Hg
-  /4 width=5 by coafter_isid_sn, coafter_eq_repl_back0, coafter_next/
-| #i2 #IH #i1 #f2 #Hf2 #f #Hf
-  elim (at_inv_xxn … Hf2) -Hf2 [1,3: * |*: // ]
-  [ #g2 #j1 #Hg2 #H1 #H2 destruct
-    elim (coafter_inv_pnx … Hf) -Hf [ |*: // ] #g #Hg #H destruct
-    /3 width=5 by coafter_next/
-  | #g2 #Hg2 #H2 destruct
-    elim (coafter_inv_nxx … Hf) -Hf [2,3: // ] #g #Hg #H destruct
-    /3 width=5 by coafter_next/
-  ]
-]
-qed.
-
-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
-  elim (coafter_inv_nxx … Hf) -Hf [ |*: // ] #g #Hg #H destruct
-  lapply (coafter_isid_inv_sn … Hg ?) -Hg
-  /4 width=7 by coafter_isid_dx, coafter_eq_repl_back0, coafter_push/
-| #i2 #IH #i1 #f2 #Hf2 #f #Hf
-  elim (coafter_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 coafter_push/
-  | #g2 #Hg2 #H2 destruct <tls_xn in Hg; /3 width=5 by coafter_next/
-  ]
-]
-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/
-qed.
-
-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❫.
-#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❫.
-#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❫.
-#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❫.
-/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.
-#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❫.
-#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
-#i0 #Hf1 lapply (at_increasing … Hf1)
-#Hi20 lapply (coafter_fwd_at2 … i0 … Hf1 … Hf) -Hf
-/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❫.
-#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. @❪𝟏, f1❫ ≘ 𝟏 → 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
-cases (H2g1 0) #n #Hn
+cases (H2g1 (𝟏)) #n #Hn
 cases (coafter_inv_pxx … H1f … H1) -H1f * #g21 #g #H1g #H21 #H
 [ cases (coafter_inv_pxp … H2f … H1 H) -f1 -f #g22 #H2g #H22
   @(eq_push … H21 H22) -f21 -f22
 | cases (coafter_inv_pxn … H2f … H1 H) -f1 -f #g22 #H2g #H22
   @(eq_next … H21 H22) -f21 -f22
 ]
-@(coafter_inj_O_aux (⫱*[n]g1) … (⫱*[n]g)) -coafter_inj_O_aux
+@(coafter_inj_O_aux (⫱*[↓n]g1) … (⫱*[↓n]g)) -coafter_inj_O_aux
 /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. @❪𝟏, f1❫ ≘ 𝟏 → H_coafter_inj f1) →
+                      ∀i2,f1. @❪𝟏, 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
@@ -603,15 +382,15 @@ lapply (coafter_inv_nxp … H2f … H1 H) -f #H2g
 qed-.
 
 theorem coafter_inj: ∀f1. H_coafter_inj f1.
-#f1 #H cases (H 0) /3 width=7 by coafter_inj_aux, coafter_inj_O_aux/
+#f1 #H cases (H (𝟏)) /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. @❪𝟏, f1❫ ≘ 𝟏 →
                                     H_coafter_fwd_isid2 f1.
 #f1 #H1f1 #f2 #f #H #H2f1 #Hf
 cases (at_inv_pxp … H1f1) -H1f1 [ |*: // ] #g1 #H1
 lapply (istot_inv_push … H2f1 … H1) -H2f1 #H2g1
-cases (H2g1 0) #n #Hn
+cases (H2g1 (𝟏)) #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) -H2
@@ -620,8 +399,8 @@ cases (coafter_inv_pxx … H … H1) -H * #g2 #g #H #H2 #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. @❪𝟏, f1❫ ≘ 𝟏 → H_coafter_fwd_isid2 f1) →
+                            ∀i2,f1. @❪𝟏, 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
@@ -630,11 +409,11 @@ elim (coafter_inv_nxx … H … H1) -H #g #Hg #H0
 qed-.
 
 lemma coafter_fwd_isid2: ∀f1. H_coafter_fwd_isid2 f1.
-#f1 #f2 #f #Hf #H cases (H 0)
+#f1 #f2 #f #Hf #H cases (H (𝟏))
 /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. @❪𝟏, f1❫ ≘ 𝟏 →
                                H_coafter_isfin2_fwd f1.
 #f1 #Hf1 #f2 #H
 generalize in match Hf1; generalize in match f1; -f1
@@ -643,14 +422,14 @@ generalize in match Hf1; generalize in match f1; -f1
 #f2 #_ #IH #f1 #H #Hf1 #f #Hf
 elim (at_inv_pxp … H) -H [ |*: // ] #g1 #H1
 lapply (istot_inv_push … Hf1 … H1) -Hf1 #Hg1
-elim (Hg1 0) #n #Hn
+elim (Hg1 (𝟏)) #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_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. @❪𝟏, f1❫ ≘ 𝟏 → H_coafter_isfin2_fwd f1) →
+                             ∀i2,f1. @❪𝟏, 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
@@ -660,7 +439,7 @@ lapply (IH … Hg1 … Hg) -i2 -Hg
 qed-.
 
 lemma coafter_isfin2_fwd: ∀f1. H_coafter_isfin2_fwd f1.
-#f1 #f2 #Hf2 #Hf1 cases (Hf1 0)
+#f1 #f2 #Hf2 #Hf1 cases (Hf1 (𝟏))
 /3 width=7 by coafter_isfin2_fwd_aux, coafter_isfin2_fwd_O_aux/
 qed-.
 
@@ -715,54 +494,3 @@ 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-.
-*)