update in ground_2, static_2, basic_2, apps_2, alpha_1

[helm.git] / matita / matita / contribs / lambdadelta / ground_2 / relocation / rtmap_coafter.ma

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 5b0e3c1e9e1872e08c631e5758887140013c8911..0eb098301ceca87f2f6a1daf1b8705e3eb2139b7 100644 (file)
--- a/matita/matita/contribs/lambdadelta/ground_2/relocation/rtmap_coafter.ma
+++ b/matita/matita/contribs/lambdadelta/ground_2/relocation/rtmap_coafter.ma
@@ -31,14 +31,14 @@ interpretation "relational co-composition (rtmap)"
    'RCoAfter f1 f2 f = (coafter f1 f2 f).
 
 definition H_coafter_inj: predicate rtmap ≝
-                          λf1. ð\9d\90\93â¦\83f1â¦\84 →
+                          λf1. ð\9d\90\93â\9dªf1â\9d« →
                           ∀f,f21,f22. f1 ~⊚ f21 ≘ f → f1 ~⊚ f22 ≘ f → f21 ≡ f22.
 
 definition H_coafter_fwd_isid2: predicate rtmap ≝
-                                λf1. â\88\80f2,f. f1 ~â\8a\9a f2 â\89\98 f â\86\92 ð\9d\90\93â¦\83f1â¦\84 â\86\92 ð\9d\90\88â¦\83fâ¦\84 â\86\92 ð\9d\90\88â¦\83f2â¦\84.
+                                λf1. â\88\80f2,f. f1 ~â\8a\9a f2 â\89\98 f â\86\92 ð\9d\90\93â\9dªf1â\9d« â\86\92 ð\9d\90\88â\9dªfâ\9d« â\86\92 ð\9d\90\88â\9dªf2â\9d«.
 
 definition H_coafter_isfin2_fwd: predicate rtmap ≝
-                                 λf1. â\88\80f2. ð\9d\90\85â¦\83f2â¦\84 â\86\92 ð\9d\90\93â¦\83f1â¦\84 â\86\92 â\88\80f. f1 ~â\8a\9a f2 â\89\98 f â\86\92  ð\9d\90\85â¦\83fâ¦\84.
+                                 λf1. â\88\80f2. ð\9d\90\85â\9dªf2â\9d« â\86\92 ð\9d\90\93â\9dªf1â\9d« â\86\92 â\88\80f. f1 ~â\8a\9a f2 â\89\98 f â\86\92  ð\9d\90\85â\9dªfâ\9d«.
 
 (* Basic inversion lemmas ***************************************************)
 
@@ -238,7 +238,7 @@ lemma coafter_mono_eq: ∀f1,f2,f. f1 ~⊚ f2 ≘ f → ∀g1,g2,g. g1 ~⊚ g2 
 
 (* Forward lemmas with pushs ************************************************)
 
-lemma coafter_fwd_pushs: â\88\80j,i,g2,f1,g. g2 ~â\8a\9a ⫯*[i]f1 â\89\98 g â\86\92 @â¦\83i, g2â¦\84 ≘ j →
+lemma coafter_fwd_pushs: â\88\80j,i,g2,f1,g. g2 ~â\8a\9a ⫯*[i]f1 â\89\98 g â\86\92 @â\9dªi, g2â\9d« ≘ j →
                          ∃f. ⫯*[j] f = g.
 #j elim j -j
 [ #i #g2 #f1 #g #Hg #H
@@ -278,7 +278,7 @@ qed-.
 
 (* Properties with iterated tail ********************************************)
 
-lemma coafter_tls: â\88\80j,i,f1,f2,f. @â¦\83i, f1â¦\84 ≘ j →
+lemma coafter_tls: â\88\80j,i,f1,f2,f. @â\9dªi, f1â\9d« ≘ 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 //
@@ -294,12 +294,12 @@ lemma coafter_tls: ∀j,i,f1,f2,f. @⦃i, f1⦄ ≘ j →
 ]
 qed.
 
-lemma coafter_tls_O: â\88\80n,f1,f2,f. @â¦\830, f1â¦\84 ≘ n →
+lemma coafter_tls_O: â\88\80n,f1,f2,f. @â\9dª0, f1â\9d« ≘ 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 →
-                        â\88\80n. @â¦\830, g2â¦\84 ≘ n → ⫱*[↑n]g2 ~⊚ ⫱g1 ≘ ⫱*[↑n]g.
+                        â\88\80n. @â\9dª0, g2â\9d« ≘ 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
@@ -308,7 +308,7 @@ 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: â\88\80g2,f1,g,i,j. @â¦\83i, g2â¦\84 ≘ j → g2 ~⊚ ⫯*[↑i]f1 ≘ g →
+lemma coafter_fwd_xpx_pushs: â\88\80g2,f1,g,i,j. @â\9dªi, g2â\9d« ≘ 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
@@ -319,7 +319,7 @@ elim (coafter_inv_ppx … Hf) [|*: // ] -Hf #g #Hg #H destruct
 /2 width=3 by ex2_intro/
 qed-.
 
-lemma coafter_fwd_xnx_pushs: â\88\80g2,f1,g,i,j. @â¦\83i, g2â¦\84 ≘ j → g2 ~⊚ ⫯*[i]↑f1 ≘ g →
+lemma coafter_fwd_xnx_pushs: â\88\80g2,f1,g,i,j. @â\9dªi, g2â\9d« ≘ 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
@@ -332,12 +332,12 @@ qed-.
 
 (* Properties with test for identity ****************************************)
 
-corec lemma coafter_isid_sn: â\88\80f1. ð\9d\90\88â¦\83f1â¦\84 → ∀f2. f1 ~⊚ f2 ≘ f2.
+corec lemma coafter_isid_sn: â\88\80f1. ð\9d\90\88â\9dªf1â\9d« → ∀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: â\88\80f2,f. ð\9d\90\88â¦\83f2â¦\84 â\86\92 ð\9d\90\88â¦\83fâ¦\84 → ∀f1. f1 ~⊚ f2 ≘ f.
+corec lemma coafter_isid_dx: â\88\80f2,f. ð\9d\90\88â\9dªf2â\9d« â\86\92 ð\9d\90\88â\9dªfâ\9d« → ∀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/
@@ -346,27 +346,27 @@ qed.
 
 (* Inversion lemmas with test for identity **********************************)
 
-lemma coafter_isid_inv_sn: â\88\80f1,f2,f. f1 ~â\8a\9a f2 â\89\98 f â\86\92 ð\9d\90\88â¦\83f1â¦\84 → f2 ≡ f.
+lemma coafter_isid_inv_sn: â\88\80f1,f2,f. f1 ~â\8a\9a f2 â\89\98 f â\86\92 ð\9d\90\88â\9dªf1â\9d« → f2 ≡ f.
 /3 width=6 by coafter_isid_sn, coafter_mono/ qed-.
 
-lemma coafter_isid_inv_dx: â\88\80f1,f2,f. f1 ~â\8a\9a f2 â\89\98 f â\86\92 ð\9d\90\88â¦\83f2â¦\84 â\86\92 ð\9d\90\88â¦\83fâ¦\84.
+lemma coafter_isid_inv_dx: â\88\80f1,f2,f. f1 ~â\8a\9a f2 â\89\98 f â\86\92 ð\9d\90\88â\9dªf2â\9d« â\86\92 ð\9d\90\88â\9dªfâ\9d«.
 /4 width=4 by eq_id_isid, coafter_isid_dx, coafter_mono/ qed-.
 
 (* Properties with test for uniform relocations *****************************)
 
-lemma coafter_isuni_isid: â\88\80f2. ð\9d\90\88â¦\83f2â¦\84 â\86\92 â\88\80f1. ð\9d\90\94â¦\83f1â¦\84 → f1 ~⊚ f2 ≘ f2.
+lemma coafter_isuni_isid: â\88\80f2. ð\9d\90\88â\9dªf2â\9d« â\86\92 â\88\80f1. ð\9d\90\94â\9dªf1â\9d« → 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: â\88\80f1,f2. ð\9d\90\88â¦\83f2â¦\84 â\86\92 ð\9d\90\94â¦\83f1â¦\84 → f1 ~⊚ ↑f2 ≘ ↑f1.
+lemma coafter_isid_isuni: â\88\80f1,f2. ð\9d\90\88â\9dªf2â\9d« â\86\92 ð\9d\90\94â\9dªf1â\9d« → 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: â\88\80f2. ð\9d\90\94â¦\83f2â¦\84 → ∀f1,f. ↑f2 ~⊚ f1 ≘ f → f2 ~⊚ ↑f1 ≘ f.
+lemma coafter_uni_next2: â\88\80f2. ð\9d\90\94â\9dªf2â\9d« → ∀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
@@ -380,20 +380,20 @@ qed.
 
 (* Properties with uniform relocations **************************************)
 
-lemma coafter_uni_sn: â\88\80i,f. ð\9d\90\94â\9d´iâ\9dµ ~⊚ f ≘ ⫯*[i] f.
+lemma coafter_uni_sn: â\88\80i,f. ð\9d\90\94â\9d¨iâ\9d© ~⊚ f ≘ ⫯*[i] f.
 #i elim i -i /2 width=5 by coafter_isid_sn, coafter_next/
 qed.
 
 (*
-lemma coafter_uni: â\88\80n1,n2. ð\9d\90\94â\9d´n1â\9dµ ~â\8a\9a ð\9d\90\94â\9d´n2â\9dµ â\89\98 ð\9d\90\94â\9d´n1+n2â\9dµ.
+lemma coafter_uni: â\88\80n1,n2. ð\9d\90\94â\9d¨n1â\9d© ~â\8a\9a ð\9d\90\94â\9d¨n2â\9d© â\89\98 ð\9d\90\94â\9d¨n1+n2â\9d©.
 @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: â\88\80i,i1,f. @â¦\83i1, fâ¦\84 ≘ i → ∀f2,f1. f2 ~⊚ f1 ≘ f →
-                      â\88\83â\88\83i2. @â¦\83i1, f1â¦\84 â\89\98 i2 & @â¦\83i2, f2â¦\84 ≘ i.
+lemma coafter_at_fwd: â\88\80i,i1,f. @â\9dªi1, fâ\9d« ≘ i → ∀f2,f1. f2 ~⊚ f1 ≘ f →
+                      â\88\83â\88\83i2. @â\9dªi1, f1â\9d« â\89\98 i2 & @â\9dªi2, f2â\9d« ≘ 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 ]
@@ -409,8 +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: â\88\80i,i2,i1,f1,f2. @â¦\83i1, f1â¦\84 â\89\98 i2 â\86\92 @â¦\83i2, f2â¦\84 ≘ i →
-                      â\88\80f. f2 ~â\8a\9a f1 â\89\98 f â\86\92 @â¦\83i1, fâ¦\84 ≘ i.
+lemma coafter_fwd_at: â\88\80i,i2,i1,f1,f2. @â\9dªi1, f1â\9d« â\89\98 i2 â\86\92 @â\9dªi2, f2â\9d« ≘ i →
+                      â\88\80f. f2 ~â\8a\9a f1 â\89\98 f â\86\92 @â\9dªi1, fâ\9d« ≘ 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
@@ -427,14 +427,14 @@ lemma coafter_fwd_at: ∀i,i2,i1,f1,f2. @⦃i1, f1⦄ ≘ i2 → @⦃i2, f2⦄ 
 ]
 qed-.
 
-lemma coafter_fwd_at2: â\88\80f,i1,i. @â¦\83i1, fâ¦\84 â\89\98 i â\86\92 â\88\80f1,i2. @â¦\83i1, f1â¦\84 ≘ i2 →
-                       â\88\80f2. f2 ~â\8a\9a f1 â\89\98 f â\86\92 @â¦\83i2, f2â¦\84 ≘ i.
+lemma coafter_fwd_at2: â\88\80f,i1,i. @â\9dªi1, fâ\9d« â\89\98 i â\86\92 â\88\80f1,i2. @â\9dªi1, f1â\9d« ≘ i2 →
+                       â\88\80f2. f2 ~â\8a\9a f1 â\89\98 f â\86\92 @â\9dªi2, f2â\9d« ≘ 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: â\88\80i,i2,i1,f,f2. @â¦\83i1, fâ¦\84 â\89\98 i â\86\92 @â¦\83i2, f2â¦\84 ≘ i →
-                       â\88\80f1. f2 ~â\8a\9a f1 â\89\98 f â\86\92 @â¦\83i1, f1â¦\84 ≘ i2.
+lemma coafter_fwd_at1: â\88\80i,i2,i1,f,f2. @â\9dªi1, fâ\9d« â\89\98 i â\86\92 @â\9dªi2, f2â\9d« ≘ i →
+                       â\88\80f1. f2 ~â\8a\9a f1 â\89\98 f â\86\92 @â\9dªi1, f1â\9d« ≘ 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
@@ -453,8 +453,8 @@ qed-.
 
 (* Properties with at *******************************************************)
 
-lemma coafter_uni_dx: â\88\80i2,i1,f2. @â¦\83i1, f2â¦\84 ≘ i2 →
-                      â\88\80f. f2 ~â\8a\9a ð\9d\90\94â\9d´i1â\9dµ â\89\98 f â\86\92 ð\9d\90\94â\9d´i2â\9dµ ~⊚ ⫱*[i2] f2 ≘ f.
+lemma coafter_uni_dx: â\88\80i2,i1,f2. @â\9dªi1, f2â\9d« ≘ i2 →
+                      â\88\80f. f2 ~â\8a\9a ð\9d\90\94â\9d¨i1â\9d© â\89\98 f â\86\92 ð\9d\90\94â\9d¨i2â\9d© ~⊚ ⫱*[i2] f2 ≘ f.
 #i2 elim i2 -i2
 [ #i1 #f2 #Hf2 #f #Hf
   elim (at_inv_xxp … Hf2) -Hf2 // #g2 #H1 #H2 destruct
@@ -472,8 +472,8 @@ lemma coafter_uni_dx: ∀i2,i1,f2. @⦃i1, f2⦄ ≘ i2 →
 ]
 qed.
 
-lemma coafter_uni_sn: â\88\80i2,i1,f2. @â¦\83i1, f2â¦\84 ≘ i2 →
-                      â\88\80f. ð\9d\90\94â\9d´i2â\9dµ ~â\8a\9a ⫱*[i2] f2 â\89\98 f â\86\92 f2 ~â\8a\9a ð\9d\90\94â\9d´i1â\9dµ ≘ f.
+lemma coafter_uni_sn: â\88\80i2,i1,f2. @â\9dªi1, f2â\9d« ≘ i2 →
+                      â\88\80f. ð\9d\90\94â\9d¨i2â\9d© ~â\8a\9a ⫱*[i2] f2 â\89\98 f â\86\92 f2 ~â\8a\9a ð\9d\90\94â\9d¨i1â\9d© ≘ f.
 #i2 elim i2 -i2
 [ #i1 #f2 #Hf2 #f #Hf
   elim (at_inv_xxp … Hf2) -Hf2 // #g2 #H1 #H2 destruct
@@ -488,8 +488,8 @@ lemma coafter_uni_sn: ∀i2,i1,f2. @⦃i1, f2⦄ ≘ i2 →
 ]
 qed-.
 
-lemma coafter_uni_succ_dx: â\88\80i2,i1,f2. @â¦\83i1, f2â¦\84 ≘ i2 →
-                           â\88\80f. f2 ~â\8a\9a ð\9d\90\94â\9d´â\86\91i1â\9dµ â\89\98 f â\86\92 ð\9d\90\94â\9d´â\86\91i2â\9dµ ~⊚ ⫱*[↑i2] f2 ≘ f.
+lemma coafter_uni_succ_dx: â\88\80i2,i1,f2. @â\9dªi1, f2â\9d« ≘ i2 →
+                           â\88\80f. f2 ~â\8a\9a ð\9d\90\94â\9d¨â\86\91i1â\9d© â\89\98 f â\86\92 ð\9d\90\94â\9d¨â\86\91i2â\9d© ~⊚ ⫱*[↑i2] f2 ≘ f.
 #i2 elim i2 -i2
 [ #i1 #f2 #Hf2 #f #Hf
   elim (at_inv_xxp … Hf2) -Hf2 // #g2 #H1 #H2 destruct
@@ -508,8 +508,8 @@ lemma coafter_uni_succ_dx: ∀i2,i1,f2. @⦃i1, f2⦄ ≘ i2 →
 ]
 qed.
 
-lemma coafter_uni_succ_sn: â\88\80i2,i1,f2. @â¦\83i1, f2â¦\84 ≘ i2 →
-                           â\88\80f. ð\9d\90\94â\9d´â\86\91i2â\9dµ ~â\8a\9a ⫱*[â\86\91i2] f2 â\89\98 f â\86\92 f2 ~â\8a\9a ð\9d\90\94â\9d´â\86\91i1â\9dµ ≘ f.
+lemma coafter_uni_succ_sn: â\88\80i2,i1,f2. @â\9dªi1, f2â\9d« ≘ i2 →
+                           â\88\80f. ð\9d\90\94â\9d¨â\86\91i2â\9d© ~â\8a\9a ⫱*[â\86\91i2] f2 â\89\98 f â\86\92 f2 ~â\8a\9a ð\9d\90\94â\9d¨â\86\91i1â\9d© ≘ f.
 #i2 elim i2 -i2
 [ #i1 #f2 #Hf2 #f #Hf
   elim (at_inv_xxp … Hf2) -Hf2 // #g2 #H1 #H2 destruct
@@ -525,43 +525,43 @@ lemma coafter_uni_succ_sn: ∀i2,i1,f2. @⦃i1, f2⦄ ≘ i2 →
 ]
 qed-.
 
-lemma coafter_uni_one_dx: â\88\80f2,f. ⫯f2 ~â\8a\9a ð\9d\90\94â\9d´â\86\91Oâ\9dµ â\89\98 f â\86\92 ð\9d\90\94â\9d´â\86\91Oâ\9dµ ~⊚ f2 ≘ f.
+lemma coafter_uni_one_dx: â\88\80f2,f. ⫯f2 ~â\8a\9a ð\9d\90\94â\9d¨â\86\91Oâ\9d© â\89\98 f â\86\92 ð\9d\90\94â\9d¨â\86\91Oâ\9d© ~⊚ f2 ≘ f.
 #f2 #f #H @(coafter_uni_succ_dx … (⫯f2)) /2 width=3 by at_refl/
 qed.
 
-lemma coafter_uni_one_sn: â\88\80f1,f. ð\9d\90\94â\9d´â\86\91Oâ\9dµ ~â\8a\9a f1 â\89\98 f â\86\92 ⫯f1 ~â\8a\9a ð\9d\90\94â\9d´â\86\91Oâ\9dµ ≘ f.
+lemma coafter_uni_one_sn: â\88\80f1,f. ð\9d\90\94â\9d¨â\86\91Oâ\9d© ~â\8a\9a f1 â\89\98 f â\86\92 ⫯f1 ~â\8a\9a ð\9d\90\94â\9d¨â\86\91Oâ\9d© ≘ f.
 /3 width=3 by coafter_uni_succ_sn, at_refl/ qed-.
 *)
 (* Forward lemmas with istot ************************************************)
 (*
-lemma coafter_istot_fwd: â\88\80f2,f1,f. f2 ~â\8a\9a f1 â\89\98 f â\86\92 ð\9d\90\93â¦\83f2â¦\84 â\86\92 ð\9d\90\93â¦\83f1â¦\84 â\86\92 ð\9d\90\93â¦\83fâ¦\84.
+lemma coafter_istot_fwd: â\88\80f2,f1,f. f2 ~â\8a\9a f1 â\89\98 f â\86\92 ð\9d\90\93â\9dªf2â\9d« â\86\92 ð\9d\90\93â\9dªf1â\9d« â\86\92 ð\9d\90\93â\9dªfâ\9d«.
 #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: â\88\80f2,f1,f. f2 ~â\8a\9a f1 â\89\98 f â\86\92 ð\9d\90\93â¦\83fâ¦\84 â\86\92 ð\9d\90\93â¦\83f1â¦\84.
+lemma coafter_fwd_istot_dx: â\88\80f2,f1,f. f2 ~â\8a\9a f1 â\89\98 f â\86\92 ð\9d\90\93â\9dªfâ\9d« â\86\92 ð\9d\90\93â\9dªf1â\9d«.
 #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: â\88\80f2,f1,f. f2 ~â\8a\9a f1 â\89\98 f â\86\92 ð\9d\90\93â¦\83fâ¦\84 â\86\92 ð\9d\90\93â¦\83f2â¦\84.
+lemma coafter_fwd_istot_sn: â\88\80f2,f1,f. f2 ~â\8a\9a f1 â\89\98 f â\86\92 ð\9d\90\93â\9dªfâ\9d« â\86\92 ð\9d\90\93â\9dªf2â\9d«.
 #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: â\88\80f2,f1,f. f2 ~â\8a\9a f1 â\89\98 f â\86\92 ð\9d\90\93â¦\83fâ¦\84 â\86\92 ð\9d\90\93â¦\83f2â¦\84 â\88§ ð\9d\90\93â¦\83f1â¦\84.
+lemma coafter_inv_istot: â\88\80f2,f1,f. f2 ~â\8a\9a f1 â\89\98 f â\86\92 ð\9d\90\93â\9dªfâ\9d« â\86\92 ð\9d\90\93â\9dªf2â\9d« â\88§ ð\9d\90\93â\9dªf1â\9d«.
 /3 width=4 by coafter_fwd_istot_sn, coafter_fwd_istot_dx, conj/ qed-.
 
-lemma coafter_at1_fwd: â\88\80f1,i1,i2. @â¦\83i1, f1â¦\84 â\89\98 i2 â\86\92 â\88\80f2. ð\9d\90\93â¦\83f2â¦\84 → ∀f. f2 ~⊚ f1 ≘ f →
-                     â\88\83â\88\83i. @â¦\83i2, f2â¦\84 â\89\98 i & @â¦\83i1, fâ¦\84 ≘ i.
+lemma coafter_at1_fwd: â\88\80f1,i1,i2. @â\9dªi1, f1â\9d« â\89\98 i2 â\86\92 â\88\80f2. ð\9d\90\93â\9dªf2â\9d« → ∀f. f2 ~⊚ f1 ≘ f →
+                     â\88\83â\88\83i. @â\9dªi2, f2â\9d« â\89\98 i & @â\9dªi1, fâ\9d« ≘ 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: â\88\80f2,f1,f. ð\9d\90\93â¦\83fâ¦\84 â\86\92 f2 ~â\8a\9a f1 â\89\98 f â\86\92 f1 â\89¡ f â\86\92 ð\9d\90\88â¦\83f2â¦\84.
+lemma coafter_fwd_isid_sn: â\88\80f2,f1,f. ð\9d\90\93â\9dªfâ\9d« â\86\92 f2 ~â\8a\9a f1 â\89\98 f â\86\92 f1 â\89¡ f â\86\92 ð\9d\90\88â\9dªf2â\9d«.
 #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
@@ -570,14 +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: â\88\80f2,f1,f.  ð\9d\90\93â¦\83fâ¦\84 â\86\92 f2 ~â\8a\9a f1 â\89\98 f â\86\92 f2 â\89¡ f â\86\92 ð\9d\90\88â¦\83f1â¦\84.
+lemma coafter_fwd_isid_dx: â\88\80f2,f1,f.  ð\9d\90\93â\9dªfâ\9d« â\86\92 f2 ~â\8a\9a f1 â\89\98 f â\86\92 f2 â\89¡ f â\86\92 ð\9d\90\88â\9dªf1â\9d«.
 #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: â\88\80f1. @â¦\830, f1â¦\84 ≘ 0 → H_coafter_inj f1.
+corec fact coafter_inj_O_aux: â\88\80f1. @â\9dª0, f1â\9d« ≘ 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
@@ -592,8 +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: (â\88\80f1. @â¦\830, f1â¦\84 ≘ 0 → H_coafter_inj f1) →
-                      â\88\80i2,f1. @â¦\830, f1â¦\84 ≘ i2 → H_coafter_inj f1.
+fact coafter_inj_aux: (â\88\80f1. @â\9dª0, f1â\9d« ≘ 0 → H_coafter_inj f1) →
+                      â\88\80i2,f1. @â\9dª0, f1â\9d« ≘ 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
@@ -606,7 +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: â\88\80f1. @â¦\830, f1â¦\84 ≘ 0 →
+corec fact coafter_fwd_isid2_O_aux: â\88\80f1. @â\9dª0, f1â\9d« ≘ 0 →
                                     H_coafter_fwd_isid2 f1.
 #f1 #H1f1 #f2 #f #H #H2f1 #Hf
 cases (at_inv_pxp … H1f1) -H1f1 [ |*: // ] #g1 #H1
@@ -620,8 +620,8 @@ cases (coafter_inv_pxx … H … H1) -H * #g2 #g #H #H2 #H0
 ]
 qed-.
 
-fact coafter_fwd_isid2_aux: (â\88\80f1. @â¦\830, f1â¦\84 ≘ 0 → H_coafter_fwd_isid2 f1) →
-                            â\88\80i2,f1. @â¦\830, f1â¦\84 ≘ i2 → H_coafter_fwd_isid2 f1.
+fact coafter_fwd_isid2_aux: (â\88\80f1. @â\9dª0, f1â\9d« ≘ 0 → H_coafter_fwd_isid2 f1) →
+                            â\88\80i2,f1. @â\9dª0, f1â\9d« ≘ 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
@@ -634,7 +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: â\88\80f1. @â¦\830, f1â¦\84 ≘ 0 →
+fact coafter_isfin2_fwd_O_aux: â\88\80f1. @â\9dª0, f1â\9d« ≘ 0 →
                                H_coafter_isfin2_fwd f1.
 #f1 #Hf1 #f2 #H
 generalize in match Hf1; generalize in match f1; -f1
@@ -649,8 +649,8 @@ elim (Hg1 0) #n #Hn
 /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: (â\88\80f1. @â¦\830, f1â¦\84 ≘ 0 → H_coafter_isfin2_fwd f1) →
-                             â\88\80i2,f1. @â¦\830, f1â¦\84 ≘ i2 → H_coafter_isfin2_fwd f1.
+fact coafter_isfin2_fwd_aux: (â\88\80f1. @â\9dª0, f1â\9d« ≘ 0 → H_coafter_isfin2_fwd f1) →
+                             â\88\80i2,f1. @â\9dª0, f1â\9d« ≘ 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
@@ -664,7 +664,7 @@ 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: â\88\80f. ð\9d\90\85â¦\83fâ¦\84 â\86\92 â\88\80f2. ð\9d\90\93â¦\83f2â¦\84 → ∀f1. f2 ~⊚ f1 ≘ f → ∀fa,fb. fa ⋓ fb ≘ f →
+lemma coafter_inv_sor: â\88\80f. ð\9d\90\85â\9dªfâ\9d« â\86\92 â\88\80f2. ð\9d\90\93â\9dªf2â\9d« → ∀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
@@ -688,7 +688,7 @@ qed-.
 
 (* Properties with istot ****************************************************)
 
-lemma coafter_sor: â\88\80f. ð\9d\90\85â¦\83fâ¦\84 â\86\92 â\88\80f2. ð\9d\90\93â¦\83f2â¦\84 → ∀f1. f2 ~⊚ f1 ≘ f → ∀f1a,f1b. f1a ⋓ f1b ≘ f1 →
+lemma coafter_sor: â\88\80f. ð\9d\90\85â\9dªfâ\9d« â\86\92 â\88\80f2. ð\9d\90\93â\9dªf2â\9d« → ∀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