X-Git-Url: http://matita.cs.unibo.it/gitweb/?p=helm.git;a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fground_2%2Frelocation%2Frtmap_after.ma;h=b2ec8605cd6dcabd3d60a18283b2236726ae46c0;hp=cc067a24d4284c57021d0c5231e88169b1d79f72;hb=bd53c4e895203eb049e75434f638f26b5a161a2b;hpb=3b7b8afcb429a60d716d5226a5b6ab0d003228b1

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 cc067a24d..b2ec8605c 100644
--- a/matita/matita/contribs/lambdadelta/ground_2/relocation/rtmap_after.ma
+++ b/matita/matita/contribs/lambdadelta/ground_2/relocation/rtmap_after.ma
@@ -30,7 +30,7 @@ interpretation "relational composition (rtmap)"
    'RAfter f1 f2 f = (after f1 f2 f).
 
 definition H_after_inj: predicate rtmap ≝
-                        λf1. 𝐓⦃f1⦄ →
+                        λf1. 𝐓❪f1❫ →
                         ∀f,f21,f22. f1 ⊚ f21 ≘ f → f1 ⊚ f22 ≘ f → f21 ≡ f22.
 
 (* Basic inversion lemmas ***************************************************)
@@ -267,7 +267,7 @@ lemma after_mono_eq: ∀f1,f2,f. f1 ⊚ f2 ≘ f → ∀g1,g2,g. g1 ⊚ g2 ≘ g
 
 (* Properties on tls ********************************************************)
 
-lemma after_tls: ∀n,f1,f2,f. @⦃0, f1⦄ ≘ n →
+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
@@ -278,12 +278,12 @@ qed.
 
 (* Properties on isid *******************************************************)
 
-corec lemma after_isid_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.
 
-corec lemma after_isid_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/
@@ -292,37 +292,37 @@ qed.
 
 (* Inversion lemmas on isid *************************************************)
 
-lemma after_isid_inv_sn: ∀f1,f2,f. f1 ⊚ f2 ≘ f → 𝐈⦃f1⦄ → f2 ≡ f.
+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.
+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 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
@@ -335,15 +335,15 @@ qed.
 
 (* Properties on uni ********************************************************)
 
-lemma after_uni: ∀n1,n2. 𝐔❴n1❵ ⊚ 𝐔❴n2❵ ≘ 𝐔❴n1+n2❵.
+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 ]
@@ -359,8 +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
@@ -377,14 +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
@@ -403,8 +403,8 @@ qed-.
 
 (* Properties with at *******************************************************)
 
-lemma after_uni_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
@@ -422,8 +422,8 @@ lemma after_uni_dx: ∀i2,i1,f2. @⦃i1, f2⦄ ≘ i2 →
 ]
 qed.
 
-lemma after_uni_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
@@ -438,8 +438,8 @@ lemma after_uni_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.
+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
@@ -458,8 +458,8 @@ lemma after_uni_succ_dx: ∀i2,i1,f2. @⦃i1, f2⦄ ≘ i2 →
 ]
 qed.
 
-lemma after_uni_succ_sn: ∀i2,i1,f2. @⦃i1, f2⦄ ≘ i2 →
-                         ∀f. 𝐔❴↑i2❵ ⊚ ⫱*[↑i2] f2 ≘ f → f2 ⊚ 𝐔❴↑i1❵ ≘ f.
+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
@@ -475,43 +475,43 @@ lemma after_uni_succ_sn: ∀i2,i1,f2. @⦃i1, f2⦄ ≘ i2 →
 ]
 qed-.
 
-lemma after_uni_one_dx: ∀f2,f. ⫯f2 ⊚ 𝐔❴↑O❵ ≘ f → 𝐔❴↑O❵ ⊚ f2 ≘ f.
+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.
+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
@@ -520,14 +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
@@ -542,8 +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