From 926796df5884453d8f0cf9f294d7776d469ef45b Mon Sep 17 00:00:00 2001
From: Ferruccio Guidi <ferruccio.guidi@unibo.it>
Date: Mon, 6 Jun 2016 20:20:44 +0000
Subject: [PATCH] work in progress on frees_drops

---
 .../lambdadelta/basic_2/relocation/drops.ma   |  12 ++
 .../lambdadelta/basic_2/relocation/lifts.ma   |  30 +++-
 .../lambdadelta/basic_2/static/frees_drops.ma | 163 +++++++++++++++++-
 .../ground_2/relocation/rtmap_at.ma           |   5 +
 .../ground_2/relocation/rtmap_coafter.ma      |  77 ++++++++-
 .../ground_2/relocation/rtmap_isfin.ma        |   8 +-
 6 files changed, 276 insertions(+), 19 deletions(-)

diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/drops.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/drops.ma
index 8d22bcf00..9a4cfca83 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/relocation/drops.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/drops.ma
@@ -282,6 +282,18 @@ lemma drops_inv_TF: ∀f,I,L,K,V. ⬇*[Ⓕ, f] L ≡ K.ⓑ{I}V → 𝐔⦃f⦄ 
 
 (* Advanced inversion lemmas ************************************************)
 
+lemma drops_inv_atom2: ∀b,L,f. ⬇*[b,f] L ≡ ⋆ →
+                       ∃∃n,f1. ⬇*[b,𝐔❴n❵] L ≡ ⋆ & 𝐔❴n❵ ⊚ f1 ≡ f.
+#b #L elim L -L
+[ /3 width=4 by drops_atom, after_isid_sn, ex2_2_intro/
+| #L #I #V #IH #f #H elim (pn_split f) * #g #H0 destruct
+  [ elim (drops_inv_skip1 … H) -H #K #W #_ #_ #H destruct
+  | lapply (drops_inv_drop1 … H) -H #HL
+    elim (IH … HL) -IH -HL /3 width=8 by drops_drop, after_next, ex2_2_intro/
+  ]
+]
+qed-.
+
 (* Basic_2A1: includes: drop_inv_gen *)
 lemma drops_inv_gen: ∀b,f,I,L,K,V. ⬇*[b, f] L ≡ K.ⓑ{I}V → 𝐔⦃f⦄ →
                      ⬇*[Ⓣ, f] L ≡ K.ⓑ{I}V.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/lifts.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/lifts.ma
index a621b819f..be42c0ebf 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/relocation/lifts.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/lifts.ma
@@ -210,6 +210,28 @@ lemma lifts_inv_flat2: ∀f:rtmap. ∀I,V2,T2,X. ⬆*[f] X ≡ ⓕ{I}V2.T2 →
 
 (* Advanced inversion lemmas ************************************************)
 
+lemma lifts_inv_atom1: ∀f,I,Y. ⬆*[f] ⓪{I} ≡ Y →
+                       ∨∨ ∃∃s. I = Sort s & Y = ⋆s
+                        | ∃∃i,j. @⦃i, f⦄ ≡ j & I = LRef i & Y = #j
+                        | ∃∃l. I = GRef l & Y = §l.
+#f * #n #Y #H
+[ lapply (lifts_inv_sort1 … H)
+| elim (lifts_inv_lref1 … H)
+| lapply (lifts_inv_gref1 … H)
+] -H /3 width=5 by or3_intro0, or3_intro1, or3_intro2, ex3_2_intro, ex2_intro/
+qed-.
+
+lemma lifts_inv_atom2: ∀f,I,X. ⬆*[f] X ≡ ⓪{I} →
+                       ∨∨ ∃∃s. X = ⋆s & I = Sort s
+                        | ∃∃i,j. @⦃i, f⦄ ≡ j & X = #i & I = LRef j
+                        | ∃∃l. X = §l & I = GRef l.
+#f * #n #X #H
+[ lapply (lifts_inv_sort2 … H)
+| elim (lifts_inv_lref2 … H)
+| lapply (lifts_inv_gref2 … H)
+] -H /3 width=5 by or3_intro0, or3_intro1, or3_intro2, ex3_2_intro, ex2_intro/
+qed-.
+
 (* Basic_2A1: includes: lift_inv_pair_xy_x *)
 lemma lifts_inv_pair_xy_x: ∀f,I,V,T. ⬆*[f] ②{I}V.T ≡ V → ⊥.
 #f #J #V elim V -V
@@ -249,14 +271,6 @@ qed-.
 
 (* Basic forward lemmas *****************************************************)
 
-lemma lifts_fwd_atom2: ∀f,I,X. ⬆*[f] X ≡ ⓪{I} → ∃J. X = ⓪{J}. 
-#f * #n #X #H
-[ lapply (lifts_inv_sort2 … H)
-| elim (lifts_inv_lref2 … H)
-| lapply (lifts_inv_gref2 … H)
-] -H /2 width=2 by ex_intro/
-qed-.
-
 (* Basic_2A1: includes: lift_inv_O2 *)
 lemma lifts_fwd_isid: ∀f,T1,T2. ⬆*[f] T1 ≡ T2 → 𝐈⦃f⦄ → T1 = T2.
 #f #T1 #T2 #H elim H -f -T1 -T2
diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/frees_drops.ma b/matita/matita/contribs/lambdadelta/basic_2/static/frees_drops.ma
index fd4ec0ce1..aa4f34998 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/static/frees_drops.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/static/frees_drops.ma
@@ -14,19 +14,19 @@
 
 include "ground_2/relocation/rtmap_pushs.ma".
 include "ground_2/relocation/rtmap_coafter.ma".
-include "basic_2/relocation/lifts_lifts.ma".
-include "basic_2/relocation/drops.ma".
+include "basic_2/relocation/drops_drops.ma".
 include "basic_2/static/frees.ma".
 
 (* CONTEXT-SENSITIVE FREE VARIABLES *****************************************)
 
 (* Advanced properties ******************************************************)
 
-lemma frees_lref_atom: ∀L,i. ⬇*[Ⓕ, 𝐔❴i❵] L ≡ ⋆ → L ⊢ 𝐅*⦃#i⦄ ≡ 𝐈𝐝.
-#L elim L -L /2 width=1 by frees_atom/
+lemma frees_lref_atom: ∀b,L,i. ⬇*[b, 𝐔❴i❵] L ≡ ⋆ →
+                       ∀f. 𝐈⦃f⦄ → L ⊢ 𝐅*⦃#i⦄ ≡ f.
+#b #L elim L -L /2 width=1 by frees_atom/
 #L #I #V #IH *
 [ #H lapply (drops_fwd_isid … H ?) -H // #H destruct
-| /4 width=3 by frees_lref, drops_inv_drop1/
+| /5 width=3 by frees_eq_repl_back, frees_lref, drops_inv_drop1, eq_push_inv_isid/
 ]
 qed.
 
@@ -58,6 +58,91 @@ qed-.
 
 (* Properties with generic slicing for local environments *******************)
 
+axiom coafter_inv_xpx: ∀g2,f1,g. g2 ~⊚ ↑f1 ≡ g → ∀n. @⦃0, g2⦄ ≡ n →
+                       ∃∃f2,f. f2 ~⊚ f1 ≡ f & ⫱*[n]g2 = ↑f2 & ⫱*[n]g = ↑f.
+(*
+#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_rew_S <tls_rew_S <H2 <H0 -g2 -g -n //
+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_rew_S <tls_rew_S <H2 <H0 -g2 -g -n //
+qed.
+
+lemma frees_lifts: ∀b,f1,K,T. K ⊢ 𝐅*⦃T⦄ ≡ f1 →
+                   ∀f,L. ⬇*[b, f] L ≡ K → ∀U. ⬆*[f] T ≡ U →
+                   ∀f2. f ~⊚ f1 ≡ f2 → L ⊢ 𝐅*⦃U⦄ ≡ f2.
+#b #f1 #K #T #H lapply (frees_fwd_isfin … H) elim H -f1 -K -T
+[ #f1 #I #Hf1 #_ #f #L #H1 #U #H2 #f2 #H3
+  lapply (coafter_isid_inv_dx … H3 … Hf1) -f1 #Hf2
+  elim (lifts_inv_atom1 … H2) -H2 *
+  /2 width=1 by frees_sort_gen, frees_gref_gen/
+  #i #j #Hij #H #H0 destruct
+  elim (drops_inv_atom2 … H1) -H1 #n #g #H1 #Hf
+  elim (after_at_fwd … Hij … Hf) -f #x #_ #Hj -g -i
+  lapply (at_inv_uni … Hj) -Hj #H destruct
+  /3 width=8 by frees_lref_atom, drops_trans/
+| #f1 #I #K #V #s #_ #IH #Hf1 #f #L #H1 #U #H2 #f2 #H3
+  lapply (isfin_fwd_push … Hf1 ??) -Hf1 [3: |*: // ] #Hf1
+  lapply (lifts_inv_sort1 … H2) -H2 #H destruct
+  lapply (at_total 0 f) #H
+  elim (drops_split_trans … H1) -H1
+  [5: @(after_uni_dx … H) /2 width=1 by after_isid_dx/ |2,3: skip
+  |4: // ] #X #HLX #HXK
+  lapply (drops_inv_tls_at … H … HXK) -HXK #HXK
+  elim (drops_inv_skip2 … HXK) -HXK
+  #Y #W #HYK #HVW #H0 destruct
+(*  
+    
+  elim (coafter_inv_xpx … H3 ??) -H3 [ |*: // ] #g2 #g #Hg #H2 #H0 
+  lapply (IH … Hg) -IH -Hg
+  [1,5: // | skip
+  | 
+  |6: #H 
+*)
+
+  lapply (coafter_tls_succ … H3 ??) -H3 [3: |*: // ] #H3
+  lapply (IH … HYK … H3) -IH -H3 -HYK
+  [1,3: // | skip ]
+  #H lapply (frees_sort … H)
+   
+   ]
+
+  
+  elim (coafter_inv_xxp … H3) -H3 [1,3: * |*: // ]
+  [ #g #g1 #Hf2 #H #H0 destruct
+    elim (drops_inv_skip1 … H1) -H1 #K #V #HLK #_ #H destruct
+  | #g #Hf2 #H destruct
+    lapply (drops_inv_drop1 … H1) -H1
+  ] /3 width=4 by frees_sort/
+
+|
+|
+|
+| #f1V #f1T #f1 #p #I #K #V #T #_ #_ #H1f1 #IHV #IHT #H2f1 #f #L #H1 #Y #H2 #f2 #H3
+  elim (sor_inv_isfin3 … H1f1) // #Hf1V #H
+  lapply (isfin_inv_tl … H) -H
+  elim (lifts_inv_bind1 … H2) -H2 #W #U #HVW #HTU #H destruct
+  elim (coafter_sor … H3 … H1f1) /2 width=5 by coafter_isfin2_fwd/ -H3 -H1f1 #f2V #f2T #Hf2V #H
+  elim (coafter_inv_tl1 … H) -H /4 width=5 by frees_bind, drops_skip/
+| #f1V #f1T #f1 #I #K #V #T #_ #_ #H1f1 #IHV #IHT #H2f1 #f #L #H1 #Y #H2 #f2 #H3
+  elim (sor_inv_isfin3 … H1f1) //
+  elim (lifts_inv_flat1 … H2) -H2 #W #U #HVW #HTU #H destruct
+  elim (coafter_sor … H3 … H1f1)
+  /3 width=5 by coafter_isfin2_fwd, frees_flat/
+]
+
 (* Inversion lemmas with generic slicing for local environments *************)
 
 lemma frees_inv_lifts: ∀b,f2,L,U. L ⊢ 𝐅*⦃U⦄ ≡ f2 →
@@ -67,8 +152,7 @@ lemma frees_inv_lifts: ∀b,f2,L,U. L ⊢ 𝐅*⦃U⦄ ≡ f2 →
 [ #f2 #I #Hf2 #_ #f #K #H1 #T #H2 #f1 #H3
   lapply (coafter_fwd_isid2 … H3 … Hf2) -H3 // -Hf2 #Hf1
   elim (drops_inv_atom1 … H1) -H1 #H #_ destruct
-  elim (lifts_fwd_atom2 … H2) -H2
-  /2 width=3 by frees_atom/
+  elim (lifts_inv_atom2 … H2) -H2 * /2 width=3 by frees_atom/
 | #f2 #I #L #W #s #_ #IH #Hf2 #f #Y #H1 #T #H2 #f1 #H3
   lapply (isfin_fwd_push … Hf2 ??) -Hf2 [3: |*: // ] #Hf2
   lapply (lifts_inv_sort2 … H2) -H2 #H destruct
@@ -117,3 +201,68 @@ lemma frees_inv_lifts: ∀b,f2,L,U. L ⊢ 𝐅*⦃U⦄ ≡ f2 →
   elim (coafter_inv_sor … H3 … H1f2) -H3 -H1f2 /3 width=5 by frees_flat/
 ]
 qed-.
+
+lemma frees_inv_drops: ∀f2,L,U. L ⊢ 𝐅*⦃U⦄ ≡ f2 →
+                       ∀f,K. ⬇*[Ⓣ, f] L ≡ K → ∀f1. f ~⊚ f1 ≡ f2 →
+                       ∃∃T. K ⊢ 𝐅*⦃T⦄ ≡ f1 & ⬆*[f] T ≡ U.
+#f2 #L #U #H lapply (frees_fwd_isfin … H) elim H -f2 -L -U
+[ #f2 #I #Hf2 #_ #f #K #H1 #f1 #H2
+  lapply (coafter_fwd_isid2 … H2 ??) -H2 // -Hf2 #Hf1
+  elim (drops_inv_atom1 … H1) -H1 #H #Hf destruct
+  /4 width=3 by frees_atom, lifts_refl, ex2_intro/
+| #f2 #I #L #W #s #_ #IH #Hf2 #f #Y #H1 #f1 #H2
+  lapply (isfin_fwd_push … Hf2 ??) -Hf2 [3: |*: // ] #Hf2
+  elim (coafter_inv_xxp … H2) -H2 [1,3: * |*: // ]
+  [ #g #g1 #Hf2 #H #H0 destruct
+    elim (drops_inv_skip1 … H1) -H1 #K #V #HLK #_ #H destruct
+  | #g #Hf2 #H destruct
+    lapply (drops_inv_drop1 … H1) -H1 #HLK
+  ]
+  elim (IH … HLK … Hf2) -L // -f2 #X #Hg1 #HX
+  lapply (lifts_inv_sort2 … HX) -HX #H destruct
+  /3 width=3 by frees_sort, lifts_sort, ex2_intro/
+| #f2 #I #L #W #_ #IH #Hf2 #f #Y #H1 #f1 #H2
+  lapply (isfin_inv_next … Hf2 ??) -Hf2 [3: |*: // ] #Hf2
+  elim (coafter_inv_xxn … H2) -H2 [ |*: // ] #g #g1 #Hf2 #H0 #H destruct
+  elim (drops_inv_skip1 … H1) -H1 #K #V #HLK #HVW #H destruct
+  elim (IH … HLK … Hf2) -L // -f2 #X #Hg1 #HX
+  lapply (lifts_inj … HX … HVW) -W #H destruct
+  /3 width=3 by frees_zero, lifts_lref, ex2_intro/
+| #f2 #I #L #W #j #_ #IH #Hf2 #f #Y #H1 #f1 #H2
+  lapply (isfin_fwd_push … Hf2 ??) -Hf2 [3: |*: // ] #Hf2
+  elim (coafter_inv_xxp … H2) -H2 [1,3: * |*: // ]
+  [ #g #g1 #Hf2 #H #H0 destruct
+    elim (drops_inv_skip1 … H1) -H1 #K #V #HLK #_ #H destruct
+  | #g #Hf2 #H destruct
+    lapply (drops_inv_drop1 … H1) -H1 #HLK
+  ]
+  elim (IH … HLK … Hf2) -L // -f2 #X #Hg1 #HX
+  elim (lifts_inv_lref2 … HX) -HX #i #Hij #H destruct
+  /4 width=7 by frees_lref, lifts_lref, at_S1, at_next, ex2_intro/
+| #f2 #I #L #W #l #_ #IH #Hf2 #f #Y #H1 #f1 #H2
+  lapply (isfin_fwd_push … Hf2 ??) -Hf2 [3: |*: // ] #Hf2
+  elim (coafter_inv_xxp … H2) -H2 [1,3: * |*: // ]
+  [ #g #g1 #Hf2 #H #H0 destruct
+    elim (drops_inv_skip1 … H1) -H1 #K #V #HLK #_ #H destruct
+  | #g #Hf2 #H destruct
+    lapply (drops_inv_drop1 … H1) -H1 #HLK
+  ]
+  elim (IH … HLK … Hf2) -L // -f2 #X #Hg1 #HX
+  lapply (lifts_inv_gref2 … HX) -HX #H destruct
+  /3 width=3 by frees_gref, lifts_gref, ex2_intro/
+| #f2W #f2U #f2 #p #I #L #W #U #_ #_ #H1f2 #IHW #IHU #H2f2 #f #K #H1 #f1 #H2
+  elim (sor_inv_isfin3 … H1f2) // #H1f2W #H
+  lapply (isfin_inv_tl … H) -H #H1f2U
+  elim (coafter_inv_sor … H2 … H1f2) -H2 -H1f2 // #f1W #f1U #H2f2W #H #Hf1
+  elim (coafter_inv_tl0 … H) -H #g1 #H2f2U #H destruct
+  elim (IHW … H1 … H2f2W) -IHW -H2f2W // -H1f2W #V #Hf1W #HVW
+  elim (IHU … H2f2U) -IHU -H2f2U
+  /3 width=5 by frees_bind, drops_skip, lifts_bind, ex2_intro/
+| #f2W #f2U #f2 #I #L #W #U #_ #_ #H1f2 #IHW #IHU #H2f2 #f #K #H1 #f1 #H2
+  elim (sor_inv_isfin3 … H1f2) // #H1f2W #H1f2U
+  elim (coafter_inv_sor … H2 … H1f2) -H2 -H1f2 // #f1W #f1U #H2f2W #H2f2U #Hf1
+  elim (IHW … H1 … H2f2W) -IHW -H2f2W // -H1f2W
+  elim (IHU … H1 … H2f2U) -L -H2f2U
+  /3 width=5 by frees_flat, lifts_flat, ex2_intro/
+]
+qed-.
diff --git a/matita/matita/contribs/lambdadelta/ground_2/relocation/rtmap_at.ma b/matita/matita/contribs/lambdadelta/ground_2/relocation/rtmap_at.ma
index 90e73d347..4af66a1c2 100644
--- a/matita/matita/contribs/lambdadelta/ground_2/relocation/rtmap_at.ma
+++ b/matita/matita/contribs/lambdadelta/ground_2/relocation/rtmap_at.ma
@@ -385,3 +385,8 @@ theorem at_div_id_sn: ∀f. H_at_div 𝐈𝐝 f f 𝐈𝐝.
 lemma at_uni: ∀n,i. @⦃i,𝐔❴n❵⦄ ≡ n+i.
 #n elim n -n /2 width=5 by at_next/
 qed.
+
+(* Inversion lemmas with uniform relocations ********************************)
+
+lemma at_inv_uni: ∀n,i,j. @⦃i,𝐔❴n❵⦄ ≡ j → j = n+i.
+/2 width=4 by at_mono/ qed-.
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 c58db6ec0..5ac0c06f7 100644
--- a/matita/matita/contribs/lambdadelta/ground_2/relocation/rtmap_coafter.ma
+++ b/matita/matita/contribs/lambdadelta/ground_2/relocation/rtmap_coafter.ma
@@ -37,6 +37,9 @@ definition H_coafter_inj: predicate rtmap ≝
 definition H_coafter_fwd_isid2: predicate rtmap ≝
                                 λf1. ∀f2,f. f1 ~⊚ f2 ≡ f → 𝐓⦃f1⦄ → 𝐈⦃f⦄ → 𝐈⦃f2⦄.
 
+definition H_coafter_isfin2_fwd: predicate rtmap ≝
+                                 λf1. ∀f2. 𝐅⦃f2⦄ → 𝐓⦃f1⦄ → ∀f. f1 ~⊚ f2 ≡ f →  𝐅⦃f⦄.
+
 (* Basic inversion lemmas ***************************************************)
 
 lemma coafter_inv_ppx: ∀g1,g2,g. g1 ~⊚ g2 ≡ g → ∀f1,f2. ↑f1 = g1 → ↑f2 = g2 →
@@ -277,11 +280,19 @@ lemma coafter_mono_eq: ∀f1,f2,f. f1 ~⊚ f2 ≡ f → ∀g1,g2,g. g1 ~⊚ g2 
 
 (* Inversion lemmas with tail ***********************************************)
 
+lemma coafter_inv_tl1: ∀g2,g1,g. g2 ~⊚ ⫱g1 ≡ g →
+                       ∃∃f. ↑g2 ~⊚ g1 ≡ f & ⫱f = g.
+#g2 #g1 #g elim (pn_split g1) * #f1 #H1 #H destruct
+[ /3 width=7 by coafter_refl, ex2_intro/
+| @(ex2_intro … (⫯g)) /2 width=7 by coafter_push/ (**) (* full auto fails *)
+]
+qed-.
+
 lemma coafter_inv_tl0: ∀g2,g1,g. g2 ~⊚ g1 ≡ ⫱g →
                        ∃∃f1. ↑g2 ~⊚ f1 ≡ g & ⫱f1 = g1.
-#g1 #g2 #g elim (pn_split g) * #f #H0 #H destruct
+#g2 #g1 #g elim (pn_split g) * #f #H0 #H destruct
 [ /3 width=7 by coafter_refl, ex2_intro/
-| @(ex2_intro … (⫯g2)) /2 width=7 by coafter_push/ (**) (* full auto fails *)
+| @(ex2_intro … (⫯g1)) /2 width=7 by coafter_push/ (**) (* full auto fails *)
 ]
 qed-.
 
@@ -586,6 +597,36 @@ 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: ∀f1. @⦃0, f1⦄ ≡ 0 →
+                               H_coafter_isfin2_fwd f1.
+#f1 #Hf1 #f2 #H
+generalize in match Hf1; generalize in match f1; -f1
+@(isfin_ind … H) -f2
+[ /3 width=4 by coafter_isid_inv_dx, isfin_isid/ ]
+#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 (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/
+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.
+#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
+elim (coafter_inv_nxx … Hf … H1) -Hf #g #Hg #H0
+lapply (IH … Hg1 … Hg) -i2 -Hg
+/2 width=4 by istot_inv_next, isfin_push/ (**) (* full auto fails *)
+qed-.
+
+lemma coafter_isfin2_fwd: ∀f1. H_coafter_isfin2_fwd f1.
+#f1 #f2 #Hf2 #Hf1 cases (Hf1 0)
+/3 width=7 by coafter_isfin2_fwd_aux, coafter_isfin2_fwd_O_aux/
+qed-.
+
 lemma coafter_inv_sor: ∀f. 𝐅⦃f⦄ → ∀f2. 𝐓⦃f2⦄ → ∀f1. f2 ~⊚ f1 ≡ f → ∀fa,fb. fa ⋓ fb ≡ f →
                        ∃∃f1a,f1b. f2 ~⊚ f1a ≡ fa & f2 ~⊚ f1b ≡ fb & f1a ⋓ f1b ≡ f1.
 @isfin_ind
@@ -606,4 +647,34 @@ lemma coafter_inv_sor: ∀f. 𝐅⦃f⦄ → ∀f2. 𝐓⦃f2⦄ → ∀f1. f2 ~
   elim (IH … Hg2 … H1f … H2f) -f -Hg2
   /3 width=11 by sor_np, sor_pn, sor_nn, ex3_2_intro, coafter_refl, coafter_push/
 ]
-qed-.  
+qed-.
+
+(* Properties with istot ****************************************************)
+
+lemma coafter_sor: ∀f. 𝐅⦃f⦄ → ∀f2. 𝐓⦃f2⦄ → ∀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
+  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/
+| #f #_ #IH #f2 #Hf2 #f1 #H1 #f1a #f1b #H2
+  elim (coafter_inv_xxp … H1) -H1 [1,3: * |*: // ]
+  [ #g2 #g1 #Hf #Hgf2 #Hgf1
+    elim (sor_inv_xxp … H2) -H2 [ |*: // ] #ga #gb #Hg1
+    lapply (istot_inv_push … Hf2 … Hgf2) -Hf2 #Hg2
+    elim (IH … Hf … Hg1) // -f1 -g1 -IH -Hg2
+    /3 width=11 by coafter_refl, sor_pp, ex3_2_intro/
+  | #g2 #Hf #Hgf2
+    lapply (istot_inv_next … Hf2 … Hgf2) -Hf2 #Hg2
+    elim (IH … Hf … H2) // -f1 -IH -Hg2
+    /3 width=11 by coafter_next, sor_pp, ex3_2_intro/
+  ]
+| #f #_ #IH #f2 #Hf2 #f1 #H1 #f1a #f1b #H2
+  elim (coafter_inv_xxn … H1) -H1 [ |*: // ] #g2 #g1 #Hf #Hgf2 #Hgf1
+  lapply (istot_inv_push … Hf2 … Hgf2) -Hf2 #Hg2
+  elim (sor_inv_xxn … H2) -H2 [1,3,4: * |*: // ] #ga #gb #Hg1
+  elim (IH … Hf … Hg1) // -f1 -g1 -IH -Hg2
+  /3 width=11 by coafter_refl, coafter_push, sor_np, sor_pn, sor_nn, ex3_2_intro/
+]
+qed-.
diff --git a/matita/matita/contribs/lambdadelta/ground_2/relocation/rtmap_isfin.ma b/matita/matita/contribs/lambdadelta/ground_2/relocation/rtmap_isfin.ma
index 8a127b359..b73e34cea 100644
--- a/matita/matita/contribs/lambdadelta/ground_2/relocation/rtmap_isfin.ma
+++ b/matita/matita/contribs/lambdadelta/ground_2/relocation/rtmap_isfin.ma
@@ -74,5 +74,11 @@ qed.
 (* Inversion lemmas with tail ***********************************************)
 
 lemma isfin_inv_tl: ∀f. 𝐅⦃⫱f⦄ → 𝐅⦃f⦄.
-#f elim (pn_split f) * /2 width=1 by isfin_next, isfin_push/   
+#f elim (pn_split f) * /2 width=1 by isfin_next, isfin_push/
+qed-.
+
+(* Inversion lemmas with tls ********************************************************)
+
+lemma isfin_inv_tls: ∀n,f. 𝐅⦃⫱*[n]f⦄ → 𝐅⦃f⦄.
+#n elim n -n /3 width=1 by isfin_inv_tl/
 qed-.
-- 
2.39.5