From: Ferruccio Guidi <ferruccio.guidi@unibo.it>
Date: Mon, 8 Aug 2011 22:41:37 +0000 (+0000)
Subject: - tps_tpr closed! (substitution is a reduction)
X-Git-Tag: make_still_working~2341
X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=commitdiff_plain;h=9271b3ca211007ca5ffac1e7644ebc02b0689d6e;p=helm.git

- tps_tpr closed! (substitution is a reduction)
- some refactoring
---

diff --git a/matita/matita/lib/lambda-delta/Makefile b/matita/matita/lib/lambda-delta/Makefile
index 648904368..8feb040db 100644
--- a/matita/matita/lib/lambda-delta/Makefile
+++ b/matita/matita/lib/lambda-delta/Makefile
@@ -3,7 +3,7 @@ XOA_DIR = ../../../components/binaries/xoa
 XOA     = xoa.native
 
 CONF    = xoa.conf.xml
-TARGETS = xoa_natation.ma  xoa_defs.ma
+TARGETS = xoa_natation.ma xoa.ma
 
 all: $(TARGETS)
 
diff --git a/matita/matita/lib/lambda-delta/ground.ma b/matita/matita/lib/lambda-delta/ground.ma
index 58d86aa5a..92a3f8167 100644
--- a/matita/matita/lib/lambda-delta/ground.ma
+++ b/matita/matita/lib/lambda-delta/ground.ma
@@ -74,12 +74,13 @@ lemma le_plus_minus_comm: ∀n,m,p. p ≤ m → m + n - p = m - p + n.
 >(commutative_plus p) <plus_minus_m_m //
 qed.
 
-lemma minus_le_minus_minus_comm: ∀m,p,n. 
-                                 p ≤ m → m - p ≤ n → n + p - m = n - (m - p).
-#m elim m -m
-[ #p #n #H >(le_O_to_eq_O … H) -H //
-| #m #IHm #p elim p -p //
-  #p #_ #n #Hpm <plus_n_Sm @IHm /2/
+lemma minus_le_minus_minus_comm: ∀b,c,a. c ≤ b → a - (b - c) = a + c - b.
+#b elim b -b
+[ #c #a #H >(le_O_to_eq_O … H) -H //
+| #b #IHb #c elim c -c //
+  #c #_ #a #Hcb
+  lapply (le_S_S_to_le … Hcb) -Hcb #Hcb
+  <plus_n_Sm normalize /2/
 ]
 qed.
 
diff --git a/matita/matita/lib/lambda-delta/reduction/cpr.ma b/matita/matita/lib/lambda-delta/reduction/cpr.ma
index 80ccfd947..cfd51af8a 100644
--- a/matita/matita/lib/lambda-delta/reduction/cpr.ma
+++ b/matita/matita/lib/lambda-delta/reduction/cpr.ma
@@ -9,7 +9,6 @@
      \ /
       V_______________________________________________________________ *)
 
-include "lambda-delta/syntax/length.ma".
 include "lambda-delta/reduction/tpr.ma".
 
 (* CONTEXT-SENSITIVE PARALLEL REDUCTION ON TERMS ****************************)
@@ -26,7 +25,8 @@ interpretation
 lemma cpr_pr: ∀T1,T2. T1 ⇒ T2 → ∀L. L ⊢ T1 ⇒ T2.
 /2/ qed.
 
-axiom cpr_pts: ∀L,T1,T2,d,e. L ⊢ T1 [d, e] ≫ T2 → L ⊢ T1 ⇒ T2.
+lemma cpr_tps: ∀L,T1,T2,d,e. L ⊢ T1 [d, e] ≫ T2 → L ⊢ T1 ⇒ T2.
+/3 width=5/ qed. 
 
 lemma cpr_refl: ∀L,T. L ⊢ T ⇒ T.
 /2/ qed.
@@ -37,13 +37,12 @@ lemma cpr_flat: ∀I,L,V1,V2,T1,T2.
 #I #L #V1 #V2 #T1 #T2 * #V #HV1 #HV2 * /3 width=5/
 qed.
 
-axiom cpr_delta: ∀L,K,V0,V,i.
-                 ↓[0, i] L ≡ K. 𝕓{Abbr} V0 → ↑[0, i + 1] V0 ≡ V → L ⊢ #i ⇒ V.
-(*
-#L #K #V0 #V #i #HLK #HV0
-@ex2_1_intro [2: // |1: skip ]
-@pts_subst [4,6,7,8: // |1,2,3: skip |5: 
-*)
+lemma cpr_delta: ∀L,K,V1,V2,V,i.
+                 ↓[0, i] L ≡ K. 𝕓{Abbr} V1 → K ⊢ V1 [0, |L| - i - 1] ≫ V2 →
+                 ↑[0, i + 1] V2 ≡ V → L ⊢ #i ⇒ V.
+#L #K #V1 #V2 #V #i #HLK #HV12 #HV2
+@ex2_1_intro [2: // | skip ] /3 width=8/ (**) (* /4/ is too slow *)
+qed.
 
 lemma cpr_cast: ∀L,V,T1,T2.
                 L ⊢ T1 ⇒ T2 → L ⊢ 𝕗{Cast} V. T1 ⇒ T2.
diff --git a/matita/matita/lib/lambda-delta/reduction/tpr.ma b/matita/matita/lib/lambda-delta/reduction/tpr.ma
index 3145ccaa5..fbf4d47c4 100644
--- a/matita/matita/lib/lambda-delta/reduction/tpr.ma
+++ b/matita/matita/lib/lambda-delta/reduction/tpr.ma
@@ -9,7 +9,7 @@
      \ /
       V_______________________________________________________________ *)
 
-include "lambda-delta/substitution/pts.ma".
+include "lambda-delta/substitution/tps.ma".
 
 (* CONTEXT-FREE PARALLEL REDUCTION ON TERMS *********************************)
 
diff --git a/matita/matita/lib/lambda-delta/reduction/tpr_lift.ma b/matita/matita/lib/lambda-delta/reduction/tpr_lift.ma
index d639f7119..e6fd96454 100644
--- a/matita/matita/lib/lambda-delta/reduction/tpr_lift.ma
+++ b/matita/matita/lib/lambda-delta/reduction/tpr_lift.ma
@@ -12,7 +12,7 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "lambda-delta/substitution/pts_lift.ma".
+include "lambda-delta/substitution/tps_lift.ma".
 include "lambda-delta/reduction/tpr.ma".
 
 (* Relocation properties ****************************************************)
@@ -41,7 +41,7 @@ lemma tpr_lift: ∀T1,T2. T1 ⇒ T2 →
   elim (lift_inv_bind1 … HX2) -HX2 #W2 #U0 #HVW2 #HTU0 #HX2 destruct -X2;
   elim (lift_total T2 (d + 1) e) #U2 #HTU2
   @tpr_delta
-  [4: @(pts_lift_le … HT2 … HTU2 HTU0 ?) /2/ |1: skip |2: /2/ |3: /2/ ] (**) (*/3. is too slow *)
+  [4: @(tps_lift_le … HT2 … HTU2 HTU0 ?) /2/ |1: skip |2: /2/ |3: /2/ ] (**) (*/3. is too slow *)
 | #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV12 #IHW12 #IHT12 #d #e #X1 #HX1 #X2 #HX2
   elim (lift_inv_flat1 … HX1) -HX1 #V0 #X #HV10 #HX #HX1 destruct -X1;
   elim (lift_inv_bind1 … HX) -HX #W0 #T0 #HW0 #HT10 #HX destruct -X;
@@ -81,7 +81,7 @@ lemma tpr_inv_lift: ∀T1,T2. T1 ⇒ T2 →
   elim (lift_inv_bind2 … HX) -HX #W1 #U1 #HWV1 #HUT1 #HX destruct -X;
   elim (IHV12 … HWV1) -IHV12 HWV1 #W2 #HWV2 #HW12
   elim (IHT12 … HUT1) -IHT12 HUT1 #U2 #HUT2 #HU12
-  elim (pts_inv_lift1_le … HT20 … HUT2 ?) -HT20 HUT2 // [3: /2 width=5/ |2: skip ] #U0 #HU20 #HUT0
+  elim (tps_inv_lift1_le … HT20 … HUT2 ?) -HT20 HUT2 // [3: /2 width=5/ |2: skip ] #U0 #HU20 #HUT0
   @ex2_1_intro  [2: /2/ |1: skip |3: /2/ ] (**) (* /3 width=5/ is slow *)
 | #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV12 #IHW12 #IHT12 #d #e #X #HX
   elim (lift_inv_flat2 … HX) -HX #V0 #Y #HV01 #HY #HX destruct -X;
diff --git a/matita/matita/lib/lambda-delta/reduction/tpr_pts.ma b/matita/matita/lib/lambda-delta/reduction/tpr_pts.ma
deleted file mode 100644
index f8a41f025..000000000
--- a/matita/matita/lib/lambda-delta/reduction/tpr_pts.ma
+++ /dev/null
@@ -1,26 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-include "lambda-delta/reduction/lpr.ma".
-
-(* CONTEXT-FREE PARALLEL REDUCTION ON TERMS *********************************)
-
-axiom tpr_pts_lpr: ∀L1,L2. L1 ⇒ L2 → ∀T1,T2. T1 ⇒ T2 →
-                   ∀d,e,U1. L1 ⊢ T1 [d, e] ≫ U1 →
-                   ∃∃U2. U1 ⇒ U2 & L2 ⊢ T2 [d, e] ≫ U2.
-
-lemma tpr_pts_bind: ∀I,V1,V2,T1,T2,U1. V1 ⇒ V2 → T1 ⇒ T2 →
-                    ⋆. 𝕓{I} V1 ⊢ T1 [0, 1] ≫ U1 →
-                    ∃∃U2. U1 ⇒ U2 & ⋆. 𝕓{I} V2 ⊢ T2 [0, 1] ≫ U2.
-/3 width=5/ qed.
diff --git a/matita/matita/lib/lambda-delta/reduction/tpr_tpr.ma b/matita/matita/lib/lambda-delta/reduction/tpr_tpr.ma
index b5743ddd1..ee22bfc21 100644
--- a/matita/matita/lib/lambda-delta/reduction/tpr_tpr.ma
+++ b/matita/matita/lib/lambda-delta/reduction/tpr_tpr.ma
@@ -10,9 +10,9 @@
       V_______________________________________________________________ *)
 
 include "lambda-delta/substitution/lift_weight.ma".
-include "lambda-delta/substitution/pts_pts.ma".
+include "lambda-delta/substitution/tps_tps.ma".
 include "lambda-delta/reduction/tpr_lift.ma".
-include "lambda-delta/reduction/tpr_pts.ma".
+include "lambda-delta/reduction/tpr_tps.ma".
 
 (* CONTEXT-FREE PARALLEL REDUCTION ON TERMS *********************************)
 
@@ -49,7 +49,7 @@ lemma tpr_conf_bind_delta:
 #V0 #V1 #T0 #T1 #V2 #T2 #T #IH #HV01 #HV02 #HT01 #HT02 #HT2
 elim (IH … HV01 … HV02) -HV01 HV02 // #V #HV1 #HV2
 elim (IH … HT01 … HT02) -HT01 HT02 IH // -V0 T0 #T0 #HT10 #HT20
-elim (tpr_pts_bind … HV2 HT20 … HT2) -HT20 HT2 /3 width=5/
+elim (tpr_tps_bind … HV2 HT20 … HT2) -HT20 HT2 /3 width=5/
 qed.
 
 lemma tpr_conf_bind_zeta:
@@ -117,7 +117,7 @@ elim (tpr_inv_abbr1 … H) -H *
 | -HV2 HVV2 #WW2 #UU2 #UU #HWW2 #HUU02 #HUU2 #H destruct -T1;
   elim (IH … HW02 … HWW2) -HW02 HWW2 // #W #HW02 #HWW2
   elim (IH … HU02 … HUU02) -HU02 HUU02 IH // #U #HU2 #HUUU2
-  elim (tpr_pts_bind … HWW2 HUUU2 … HUU2) -HUU2 HUUU2 #UUU #HUUU2 #HUUU1
+  elim (tpr_tps_bind … HWW2 HUUU2 … HUU2) -HUU2 HUUU2 #UUU #HUUU2 #HUUU1
   @ex2_1_intro
   [2: @tpr_theta [6: @HVV |7: @HWW2 |8: @HUUU2 |1,2,3,4: skip | // ]
   |1:skip
@@ -174,9 +174,9 @@ lemma tpr_conf_delta_delta:
 #V0 #V1 #T0 #T1 #TT1 #V2 #T2 #TT2 #IH #HV01 #HV02 #HT01 #HT02 #HTT1 #HTT2
 elim (IH … HV01 … HV02) -HV01 HV02 // #V #HV1 #HV2
 elim (IH … HT01 … HT02) -HT01 HT02 IH // #T #HT1 #HT2
-elim (tpr_pts_bind … HV1 HT1 … HTT1) -HT1 HTT1 #U1 #TTU1 #HTU1
-elim (tpr_pts_bind … HV2 HT2 … HTT2) -HT2 HTT2 #U2 #TTU2 #HTU2
-elim (pts_conf … HTU1 … HTU2) -HTU1 HTU2 #U #HU1 #HU2
+elim (tpr_tps_bind … HV1 HT1 … HTT1) -HT1 HTT1 #U1 #TTU1 #HTU1
+elim (tpr_tps_bind … HV2 HT2 … HTT2) -HT2 HTT2 #U2 #TTU2 #HTU2
+elim (tps_conf … HTU1 … HTU2) -HTU1 HTU2 #U #HU1 #HU2
 @ex2_1_intro [2,3: @tpr_delta |1: skip ] /width=10/ (**) (* /3 width=10/ is too slow *)
 qed.
 
@@ -191,7 +191,7 @@ lemma tpr_conf_delta_zeta:
    ∃∃X. 𝕓{Abbr} V1. TT1 ⇒ X & X2 ⇒ X.
 #X2 #V0 #V1 #T0 #T1 #TT1 #T2 #IH #_ #HT01 #HTT1 #HTX2 #HTT20
 elim (tpr_inv_lift … HT01 … HTT20) -HT01 #TT2 #HTT21 #HTT2
-lapply (pts_inv_lift1_eq … HTT1 … HTT21) -HTT1 #HTT1 destruct -T1;
+lapply (tps_inv_lift1_eq … HTT1 … HTT21) -HTT1 #HTT1 destruct -T1;
 lapply (tw_lift … HTT20) -HTT20 #HTT20
 elim (IH … HTX2 … HTT2) -HTX2 HTT2 IH /3/
 qed.
diff --git a/matita/matita/lib/lambda-delta/reduction/tpr_tps.ma b/matita/matita/lib/lambda-delta/reduction/tpr_tps.ma
new file mode 100644
index 000000000..96bc9b76e
--- /dev/null
+++ b/matita/matita/lib/lambda-delta/reduction/tpr_tps.ma
@@ -0,0 +1,26 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "lambda-delta/reduction/lpr.ma".
+
+(* CONTEXT-FREE PARALLEL REDUCTION ON TERMS *********************************)
+
+axiom tpr_tps_lpr: ∀L1,L2. L1 ⇒ L2 → ∀T1,T2. T1 ⇒ T2 →
+                   ∀d,e,U1. L1 ⊢ T1 [d, e] ≫ U1 →
+                   ∃∃U2. U1 ⇒ U2 & L2 ⊢ T2 [d, e] ≫ U2.
+
+lemma tpr_tps_bind: ∀I,V1,V2,T1,T2,U1. V1 ⇒ V2 → T1 ⇒ T2 →
+                    ⋆. 𝕓{I} V1 ⊢ T1 [0, 1] ≫ U1 →
+                    ∃∃U2. U1 ⇒ U2 & ⋆. 𝕓{I} V2 ⊢ T2 [0, 1] ≫ U2.
+/3 width=5/ qed.
diff --git a/matita/matita/lib/lambda-delta/substitution/drop.ma b/matita/matita/lib/lambda-delta/substitution/drop.ma
index d192c84dc..f5911775b 100644
--- a/matita/matita/lib/lambda-delta/substitution/drop.ma
+++ b/matita/matita/lib/lambda-delta/substitution/drop.ma
@@ -110,18 +110,6 @@ lemma drop_drop_lt: ∀L1,L2,I,V,e.
 #L1 #L2 #I #V #e #HL12 #He >(plus_minus_m_m e 1) /2/
 qed.
 
-lemma drop_fwd_drop2: ∀L1,I2,K2,V2,e. ↓[O, e] L1 ≡ K2. 𝕓{I2} V2 →
-                      ↓[O, e + 1] L1 ≡ K2.
-#L1 elim L1 -L1
-[ #I2 #K2 #V2 #e #H lapply (drop_inv_sort1 … H) -H #H destruct
-| #K1 #I1 #V1 #IHL1 #I2 #K2 #V2 #e #H
-  elim (drop_inv_O1 … H) -H * #He #H
-  [ -IHL1; destruct -e K2 I2 V2 /2/
-  | @drop_drop >(plus_minus_m_m e 1) /2/
-  ]
-]
-qed.
-
 lemma drop_leq_drop1: ∀L1,L2,d,e. L1 [d, e] ≈ L2 →
                       ∀I,K1,V,i. ↓[0, i] L1 ≡ K1. 𝕓{I} V →
                       d ≤ i → i < d + e →
@@ -145,3 +133,41 @@ lemma drop_leq_drop1: ∀L1,L2,d,e. L1 [d, e] ≈ L2 →
   elim (IHL12 … HLK1 ? ?) -IHL12 HLK1 [2: /2/ |3: /2/ ] -Hdi Hide >arith_g1 // /3/
 ]
 qed.
+
+(* Basic forvard lemmas *****************************************************)
+
+lemma drop_fwd_drop2: ∀L1,I2,K2,V2,e. ↓[O, e] L1 ≡ K2. 𝕓{I2} V2 →
+                      ↓[O, e + 1] L1 ≡ K2.
+#L1 elim L1 -L1
+[ #I2 #K2 #V2 #e #H lapply (drop_inv_sort1 … H) -H #H destruct
+| #K1 #I1 #V1 #IHL1 #I2 #K2 #V2 #e #H
+  elim (drop_inv_O1 … H) -H * #He #H
+  [ -IHL1; destruct -e K2 I2 V2 /2/
+  | @drop_drop >(plus_minus_m_m e 1) /2/
+  ]
+]
+qed.
+
+lemma drop_fwd_drop2_length: ∀L1,I2,K2,V2,e. 
+                             ↓[0, e] L1 ≡ K2. 𝕓{I2} V2 → e < |L1|.
+#L1 elim L1 -L1
+[ #I2 #K2 #V2 #e #H lapply (drop_inv_sort1 … H) -H #H destruct
+| #K1 #I1 #V1 #IHL1 #I2 #K2 #V2 #e #H
+  elim (drop_inv_O1 … H) -H * #He #H
+  [ -IHL1; destruct -e K2 I2 V2 //
+  | lapply (IHL1 … H) -IHL1 H #HeK1 whd in ⊢ (? ? %) /2/
+  ]
+]
+qed.
+
+lemma drop_fwd_O1_length: ∀L1,L2,e. ↓[0, e] L1 ≡ L2 → |L2| = |L1| - e.
+#L1 elim L1 -L1
+[ #L2 #e #H >(drop_inv_sort1 … H) -H //
+| #K1 #I1 #V1 #IHL1 #L2 #e #H
+  elim (drop_inv_O1 … H) -H * #He #H
+  [ -IHL1; destruct -e L2 //
+  | lapply (IHL1 … H) -IHL1 H #H >H -H; normalize
+    >minus_le_minus_minus_comm //
+  ]
+]
+qed.
diff --git a/matita/matita/lib/lambda-delta/substitution/drop_drop.ma b/matita/matita/lib/lambda-delta/substitution/drop_drop.ma
index 96fd0db51..135db64ca 100644
--- a/matita/matita/lib/lambda-delta/substitution/drop_drop.ma
+++ b/matita/matita/lib/lambda-delta/substitution/drop_drop.ma
@@ -76,7 +76,7 @@ lemma drop_trans_le: ∀d1,e1,L1,L. ↓[d1, e1] L1 ≡ L →
   [ -He2d IHL12 #H1 #H2 destruct -e2 L /3 width=5/
   | -HL12 HV12 #He2 #HL2
     elim (IHL12 … HL2 ?) -IHL12 HL2 L2
-    [ <minus_le_minus_minus_comm /3/ | /2/ ]
+    [ >minus_le_minus_minus_comm // /3/ | /2/ ]
   ]
 ]
 qed.
diff --git a/matita/matita/lib/lambda-delta/substitution/leq.ma b/matita/matita/lib/lambda-delta/substitution/leq.ma
index 3c7981566..b0e28d48e 100644
--- a/matita/matita/lib/lambda-delta/substitution/leq.ma
+++ b/matita/matita/lib/lambda-delta/substitution/leq.ma
@@ -9,7 +9,7 @@
      \ /
       V_______________________________________________________________ *)
 
-include "lambda-delta/syntax/lenv.ma".
+include "lambda-delta/syntax/length.ma".
 
 (* LOCAL ENVIRONMENT EQUALITY ***********************************************)
 
@@ -37,12 +37,16 @@ lemma leq_sym: ∀L1,L2,d,e. L1 [d, e] ≈ L2 → L2 [d, e] ≈ L1.
 #L1 #L2 #d #e #H elim H -H L1 L2 d e /2/
 qed.
 
-lemma leq_skip_lt: ∀L1,L2,d,e. leq L1 (d - 1) e L2 → 0 < d →
+lemma leq_skip_lt: ∀L1,L2,d,e. L1 [d - 1, e] ≈ L2 → 0 < d →
                    ∀I1,I2,V1,V2. L1. 𝕓{I1} V1 [d, e] ≈ L2. 𝕓{I2} V2.
 
 #L1 #L2 #d #e #HL12 #Hd >(plus_minus_m_m d 1) /2/ 
 qed.
 
+lemma leq_fwd_length: ∀L1,L2,d,e. L1 [d, e] ≈ L2 → |L1| = |L2|.
+#L1 #L2 #d #e #H elim H -H L1 L2 d e; normalize //
+qed.  
+
 (* Basic inversion lemmas ***************************************************)
 
 lemma leq_inv_sort1_aux: ∀L1,L2,d,e. L1 [d, e] ≈ L2 → L1 = ⋆ → L2 = ⋆.
diff --git a/matita/matita/lib/lambda-delta/substitution/pts.ma b/matita/matita/lib/lambda-delta/substitution/pts.ma
deleted file mode 100644
index ac962355e..000000000
--- a/matita/matita/lib/lambda-delta/substitution/pts.ma
+++ /dev/null
@@ -1,107 +0,0 @@
-(*
-    ||M||  This file is part of HELM, an Hypertextual, Electronic
-    ||A||  Library of Mathematics, developed at the Computer Science
-    ||T||  Department of the University of Bologna, Italy.
-    ||I||
-    ||T||
-    ||A||  This file is distributed under the terms of the
-    \   /  GNU General Public License Version 2
-     \ /
-      V_______________________________________________________________ *)
-
-include "lambda-delta/substitution/drop.ma".
-
-(* PARTIAL TELESCOPIC SUBSTITUTION ******************************************)
-
-inductive pts: lenv → term → nat → nat → term → Prop ≝
-| pts_sort : ∀L,k,d,e. pts L (⋆k) d e (⋆k)
-| pts_lref : ∀L,i,d,e. pts L (#i) d e (#i)
-| pts_subst: ∀L,K,V,U1,U2,i,d,e.
-             d ≤ i → i < d + e →
-             ↓[0, i] L ≡ K. 𝕓{Abbr} V → pts K V 0 (d + e - i - 1) U1 →
-             ↑[0, i + 1] U1 ≡ U2 → pts L (#i) d e U2
-| pts_bind : ∀L,I,V1,V2,T1,T2,d,e.
-             pts L V1 d e V2 → pts (L. 𝕓{I} V1) T1 (d + 1) e T2 →
-             pts L (𝕓{I} V1. T1) d e (𝕓{I} V2. T2)
-| pts_flat : ∀L,I,V1,V2,T1,T2,d,e.
-             pts L V1 d e V2 → pts L T1 d e T2 →
-             pts L (𝕗{I} V1. T1) d e (𝕗{I} V2. T2)
-.
-
-interpretation "partial telescopic substritution"
-   'PSubst L T1 d e T2 = (pts L T1 d e T2).
-
-(* Basic properties *********************************************************)
-
-lemma pts_leq_repl: ∀L1,T1,T2,d,e. L1 ⊢ T1 [d, e] ≫ T2 →
-                    ∀L2. L1 [d, e] ≈ L2 → L2 ⊢ T1 [d, e] ≫ T2.
-#L1 #T1 #T2 #d #e #H elim H -H L1 T1 T2 d e
-[ //
-| //
-| #L1 #K1 #V #V1 #V2 #i #d #e #Hdi #Hide #HLK1 #_ #HV12 #IHV12 #L2 #HL12
-  elim (drop_leq_drop1 … HL12 … HLK1 ? ?) -HL12 HLK1 // #K2 #HK12 #HLK2
-  @pts_subst [4,5,6,8: // |1,2,3: skip | /2/ ] (**) (* /3 width=6/ is too slow *)
-| /4/
-| /3/
-]
-qed.
-
-lemma pts_refl: ∀T,L,d,e. L ⊢ T [d, e] ≫ T.
-#T elim T -T //
-#I elim I -I /2/
-qed.
-
-lemma pts_weak: ∀L,T1,T2,d1,e1. L ⊢ T1 [d1, e1] ≫ T2 →
-                ∀d2,e2. d2 ≤ d1 → d1 + e1 ≤ d2 + e2 →
-                L ⊢ T1 [d2, e2] ≫ T2.
-#L #T1 #T #d1 #e1 #H elim H -L T1 T d1 e1
-[ //
-| //
-| #L #K #V #V1 #V2 #i #d1 #e1 #Hid1 #Hide1 #HLK #_ #HV12 #IHV12 #d2 #e2 #Hd12 #Hde12
-  lapply (transitive_le … Hd12 … Hid1) -Hd12 Hid1 #Hid2
-  lapply (lt_to_le_to_lt … Hide1 … Hde12) -Hide1 #Hide2
-  @pts_subst [4,5,6,8: // |1,2,3: skip | @IHV12 /2/ ] (**) (* /4 width=6/ is too slow *)
-| /4/
-| /4/
-]
-qed.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma pts_inv_bind1_aux: ∀d,e,L,U1,U2. L ⊢ U1 [d, e] ≫ U2 →
-                         ∀I,V1,T1. U1 = 𝕓{I} V1. T1 →
-                         ∃∃V2,T2. L ⊢ V1 [d, e] ≫ V2 & 
-                                  L. 𝕓{I} V1 ⊢ T1 [d + 1, e] ≫ T2 &
-                                  U2 =  𝕓{I} V2. T2.
-#d #e #L #U1 #U2 * -d e L U1 U2
-[ #L #k #d #e #I #V1 #T1 #H destruct
-| #L #i #d #e #I #V1 #T1 #H destruct
-| #L #K #V #U1 #U2 #i #d #e #_ #_ #_ #_ #_ #I #V1 #T1 #H destruct
-| #L #J #V1 #V2 #T1 #T2 #d #e #HV12 #HT12 #I #V #T #H destruct /2 width=5/
-| #L #J #V1 #V2 #T1 #T2 #d #e #_ #_ #I #V #T #H destruct
-]
-qed.
-
-lemma pts_inv_bind1: ∀d,e,L,I,V1,T1,U2. L ⊢ 𝕓{I} V1. T1 [d, e] ≫ U2 →
-                     ∃∃V2,T2. L ⊢ V1 [d, e] ≫ V2 & 
-                              L. 𝕓{I} V1 ⊢ T1 [d + 1, e] ≫ T2 &
-                              U2 =  𝕓{I} V2. T2.
-/2/ qed.
-
-lemma pts_inv_flat1_aux: ∀d,e,L,U1,U2. L ⊢ U1 [d, e] ≫ U2 →
-                         ∀I,V1,T1. U1 = 𝕗{I} V1. T1 →
-                         ∃∃V2,T2. L ⊢ V1 [d, e] ≫ V2 & L ⊢ T1 [d, e] ≫ T2 &
-                                  U2 =  𝕗{I} V2. T2.
-#d #e #L #U1 #U2 * -d e L U1 U2
-[ #L #k #d #e #I #V1 #T1 #H destruct
-| #L #i #d #e #I #V1 #T1 #H destruct
-| #L #K #V #U1 #U2 #i #d #e #_ #_ #_ #_ #_ #I #V1 #T1 #H destruct
-| #L #J #V1 #V2 #T1 #T2 #d #e #_ #_ #I #V #T #H destruct
-| #L #J #V1 #V2 #T1 #T2 #d #e #HV12 #HT12 #I #V #T #H destruct /2 width=5/
-]
-qed.
-
-lemma pts_inv_flat1: ∀d,e,L,I,V1,T1,U2. L ⊢ 𝕗{I} V1. T1 [d, e] ≫ U2 →
-                     ∃∃V2,T2. L ⊢ V1 [d, e] ≫ V2 & L ⊢ T1 [d, e] ≫ T2 &
-                              U2 =  𝕗{I} V2. T2.
-/2/ qed.
diff --git a/matita/matita/lib/lambda-delta/substitution/pts_lift.ma b/matita/matita/lib/lambda-delta/substitution/pts_lift.ma
deleted file mode 100644
index 8cd477738..000000000
--- a/matita/matita/lib/lambda-delta/substitution/pts_lift.ma
+++ /dev/null
@@ -1,174 +0,0 @@
-(*
-    ||M||  This file is part of HELM, an Hypertextual, Electronic
-    ||A||  Library of Mathematics, developed at the Computer Science
-    ||T||  Department of the University of Bologna, Italy.
-    ||I||
-    ||T||
-    ||A||  This file is distributed under the terms of the
-    \   /  GNU General Public License Version 2
-     \ /
-      V_______________________________________________________________ *)
-
-include "lambda-delta/substitution/lift_lift.ma".
-include "lambda-delta/substitution/drop_drop.ma".
-include "lambda-delta/substitution/pts.ma".
-
-(* PARTIAL TELESCOPIC SUBSTITUTION ******************************************)
-
-(* Relocation properties ****************************************************)
-
-lemma pts_lift_le: ∀K,T1,T2,dt,et. K ⊢ T1 [dt, et] ≫ T2 →
-                   ∀L,U1,U2,d,e. ↓[d, e] L ≡ K →
-                   ↑[d, e] T1 ≡ U1 → ↑[d, e] T2 ≡ U2 →
-                   dt + et ≤ d →
-                   L ⊢ U1 [dt, et] ≫ U2.
-#K #T1 #T2 #dt #et #H elim H -H K T1 T2 dt et
-[ #K #k #dt #et #L #U1 #U2 #d #e #_ #H1 #H2 #_
-  lapply (lift_mono … H1 … H2) -H1 H2 #H destruct -U1 //
-| #K #i #dt #et #L #U1 #U2 #d #e #_ #H1 #H2 #_
-  lapply (lift_mono … H1 … H2) -H1 H2 #H destruct -U1 //
-| #K #KV #V #V1 #V2 #i #dt #et #Hdti #Hidet #HKV #_ #HV12 #IHV12 #L #U1 #U2 #d #e #HLK #H #HVU2 #Hdetd
-  lapply (lt_to_le_to_lt … Hidet … Hdetd) #Hid
-  lapply (lift_inv_lref1_lt … H … Hid) -H #H destruct -U1;
-  elim (lift_trans_ge … HV12 … HVU2 ?) -HV12 HVU2 V2 // <minus_plus #V2 #HV12 #HVU2
-  elim (drop_trans_le … HLK … HKV ?) -HLK HKV K /2/ #X #HLK #H
-  elim (drop_inv_skip2 … H ?) -H /2/ -Hid #K #W #HKV #HVW #H destruct -X
-  @pts_subst [4,5,6,8: // |1,2,3: skip | @IHV12 /2 width=6/ ] (**) (* explicit constructor *)
-| #K #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hdetd
-  elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
-  elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct -U1 U2;
-  @pts_bind [ /2 width=6/ | @IHT12 [3,4,5: /2/ |1,2: skip | /2/ ] ] (**) (* /3 width=6/ is too slow, arith3 needed to avoid crash *)
-| #K #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hdetd
-  elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
-  elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct -U1 U2;
-  /3 width=6/
-]
-qed.
-
-lemma pts_lift_ge: ∀K,T1,T2,dt,et. K ⊢ T1 [dt, et] ≫ T2 →
-                   ∀L,U1,U2,d,e. ↓[d, e] L ≡ K →
-                   ↑[d, e] T1 ≡ U1 → ↑[d, e] T2 ≡ U2 →
-                   d ≤ dt →
-                   L ⊢ U1 [dt + e, et] ≫ U2.
-#K #T1 #T2 #dt #et #H elim H -H K T1 T2 dt et
-[ #K #k #dt #et #L #U1 #U2 #d #e #_ #H1 #H2 #_
-  lapply (lift_mono … H1 … H2) -H1 H2 #H destruct -U1 //
-| #K #i #dt #et #L #U1 #U2 #d #e #_ #H1 #H2 #_
-  lapply (lift_mono … H1 … H2) -H1 H2 #H destruct -U1 //
-| #K #KV #V #V1 #V2 #i #dt #et #Hdti #Hidet #HKV #HV1 #HV12 #_ #L #U1 #U2 #d #e #HLK #H #HVU2 #Hddt
-  <(arith_c1x ? ? ? e) in HV1 #HV1 (**) (* explicit athmetical rewrite and ?'s *)
-  lapply (transitive_le … Hddt … Hdti) -Hddt #Hid
-  lapply (lift_inv_lref1_ge … H … Hid) -H #H destruct -U1;
-  lapply (lift_trans_be … HV12 … HVU2 ? ?) -HV12 HVU2 V2 /2/ >plus_plus_comm_23 #HV1U2
-  lapply (drop_trans_ge_comm … HLK … HKV ?) -HLK HKV K // -Hid #HLKV
-  @pts_subst [4,5: /2/ |6,7,8: // |1,2,3: skip ] (**) (* /3 width=8/ is too slow *)
-| #K #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hddt
-  elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
-  elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct -U1 U2;
-  @pts_bind [ /2 width=5/ | /3 width=5/ ] (**) (* explicit constructor *)
-| #K #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hddt
-  elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
-  elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct -U1 U2;
-  /3 width=5/
-]
-qed.
-
-lemma pts_inv_lift1_le: ∀L,U1,U2,dt,et. L ⊢ U1 [dt, et] ≫ U2 →
-                        ∀K,d,e. ↓[d, e] L ≡ K → ∀T1. ↑[d, e] T1 ≡ U1 →
-                        dt + et ≤ d →
-                        ∃∃T2. K ⊢ T1 [dt, et] ≫ T2 & ↑[d, e] T2 ≡ U2.
-#L #U1 #U2 #dt #et #H elim H -H L U1 U2 dt et
-[ #L #k #dt #et #K #d #e #_ #T1 #H #_
-  lapply (lift_inv_sort2 … H) -H #H destruct -T1 /2/
-| #L #i #dt #et #K #d #e #_ #T1 #H #_
-  elim (lift_inv_lref2 … H) -H * #Hid #H destruct -T1 /3/
-| #L #KV #V #V1 #V2 #i #dt #et #Hdti #Hidet #HLKV #_ #HV12 #IHV12 #K #d #e #HLK #T1 #H #Hdetd
-  lapply (lt_to_le_to_lt … Hidet … Hdetd) #Hid
-  lapply (lift_inv_lref2_lt … H … Hid) -H #H destruct -T1;
-  elim (drop_conf_lt … HLK … HLKV ?) -HLK HLKV L // #L #W #HKL #HKVL #HWV
-  elim (IHV12 … HKVL … HWV ?) -HKVL HWV /2/ -Hdetd #W1 #HW1 #HWV1
-  elim (lift_trans_le … HWV1 … HV12 ?) -HWV1 HV12 V1 // >arith_a2 /3 width=6/
-| #L #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdetd
-  elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct -X;
-  elim (IHV12 … HLK … HWV1 ?) -IHV12 //
-  elim (IHU12 … HTU1 ?) -IHU12 HTU1 [3: /2/ |4: @drop_skip // |2: skip ] -HLK HWV1 Hdetd /3 width=5/ (**) (* just /3 width=5/ is too slow *)
-| #L #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdetd
-  elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct -X;
-  elim (IHV12 … HLK … HWV1 ?) -IHV12 HWV1 //
-  elim (IHU12 … HLK … HTU1 ?) -IHU12 HLK HTU1 // /3 width=5/
-]
-qed.
-
-lemma pts_inv_lift1_ge: ∀L,U1,U2,dt,et. L ⊢ U1 [dt, et] ≫ U2 →
-                        ∀K,d,e. ↓[d, e] L ≡ K → ∀T1. ↑[d, e] T1 ≡ U1 →
-                        d + e ≤ dt →
-                        ∃∃T2. K ⊢ T1 [dt - e, et] ≫ T2 & ↑[d, e] T2 ≡ U2.
-#L #U1 #U2 #dt #et #H elim H -H L U1 U2 dt et
-[ #L #k #dt #et #K #d #e #_ #T1 #H #_
-  lapply (lift_inv_sort2 … H) -H #H destruct -T1 /2/
-| #L #i #dt #et #K #d #e #_ #T1 #H #_
-  elim (lift_inv_lref2 … H) -H * #Hid #H destruct -T1 /3/
-| #L #KV #V #V1 #V2 #i #dt #et #Hdti #Hidet #HLKV #HV1 #HV12 #_ #K #d #e #HLK #T1 #H #Hdedt
-  lapply (transitive_le … Hdedt … Hdti) #Hdei
-  lapply (plus_le_weak … Hdedt) -Hdedt #Hedt
-  lapply (plus_le_weak … Hdei) #Hei
-  <(arith_h1 ? ? ? e ? ?) in HV1 // #HV1
-  lapply (lift_inv_lref2_ge … H … Hdei) -H #H destruct -T1;
-  lapply (drop_conf_ge … HLK … HLKV ?) -HLK HLKV L // #HKV
-  elim (lift_split … HV12 d (i - e + 1) ? ? ?) -HV12; [2,3,4: normalize /2/ ] -Hdei >arith_e2 // #V0 #HV10 #HV02
-  @ex2_1_intro
-  [2: @pts_subst [4: /2/ |6,7,8: // |1,2,3: skip |5: @arith5 // ]  
-  |1: skip
-  | //
-  ] (**) (* explicitc constructors *)
-| #L #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdetd
-  elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct -X;
-  lapply (plus_le_weak … Hdetd) #Hedt
-  elim (IHV12 … HLK … HWV1 ?) -IHV12 // #W2 #HW12 #HWV2
-  elim (IHU12 … HTU1 ?) -IHU12 HTU1 [4: @drop_skip // |2: skip |3: /2/ ]
-  <plus_minus // /3 width=5/
-| #L #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdetd
-  elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct -X;
-  elim (IHV12 … HLK … HWV1 ?) -IHV12 HWV1 //
-  elim (IHU12 … HLK … HTU1 ?) -IHU12 HLK HTU1 // /3 width=5/
-]
-qed.
-
-lemma pts_inv_lift1_eq: ∀L,U1,U2,d,e.
-                        L ⊢ U1 [d, e] ≫ U2 → ∀T1. ↑[d, e] T1 ≡ U1 → U1 = U2.
-#L #U1 #U2 #d #e #H elim H -H L U1 U2 d e
-[ //
-| //
-| #L #K #V #V1 #V2 #i #d #e #Hdi #Hide #_ #_ #_ #_ #T1 #H
-  elim (lift_inv_lref2 … H) -H * #H
-  [ lapply (le_to_lt_to_lt … Hdi … H) -Hdi H #H
-    elim (lt_refl_false … H)
-  | lapply (lt_to_le_to_lt … Hide … H) -Hide H #H
-    elim (lt_refl_false … H)
-  ]
-| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
-  elim (lift_inv_bind2 … HX) -HX #V #T #HV1 #HT1 #H destruct -X
-  >IHV12 // >IHT12 //
-| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
-  elim (lift_inv_flat2 … HX) -HX #V #T #HV1 #HT1 #H destruct -X
-  >IHV12 // >IHT12 //
-]
-qed.
-(*
-      Theorem subst0_gen_lift_ge : (u,t1,x:?; i,h,d:?) (subst0 i u (lift h d t1) x) ->
-                                   (le (plus d h) i) ->
-                                   (EX t2 | x = (lift h d t2) & (subst0 (minus i h) u t1 t2)).
-
-      Theorem subst0_gen_lift_rev_ge: (t1,v,u2:?; i,h,d:?) 
-                                      (subst0 i v t1 (lift h d u2)) ->
-                                      (le (plus d h) i) ->
-                                      (EX u1 | (subst0 (minus i h) v u1 u2) &
-		                               t1 = (lift h d u1)
-		                      ).
-
-
-      Theorem subst0_gen_lift_rev_lelt: (t1,v,u2:?; i,h,d:?)
-                                        (subst0 i v t1 (lift h d u2)) ->
-                                        (le d i) -> (lt i (plus d h)) ->
-				        (EX u1 | t1 = (lift (minus (plus d h) (S i)) (S i) u1)).
-*)
diff --git a/matita/matita/lib/lambda-delta/substitution/pts_pts.ma b/matita/matita/lib/lambda-delta/substitution/pts_pts.ma
deleted file mode 100644
index 0461a1823..000000000
--- a/matita/matita/lib/lambda-delta/substitution/pts_pts.ma
+++ /dev/null
@@ -1,47 +0,0 @@
-(*
-    ||M||  This file is part of HELM, an Hypertextual, Electronic
-    ||A||  Library of Mathematics, developed at the Computer Science
-    ||T||  Department of the University of Bologna, Italy.
-    ||I||
-    ||T||
-    ||A||  This file is distributed under the terms of the
-    \   /  GNU General Public License Version 2
-     \ /
-      V_______________________________________________________________ *)
-
-include "lambda-delta/substitution/pts_split.ma".
-
-(* PARTIAL TELESCOPIC SUBSTITUTION ******************************************)
-
-(* Main properties **********************************************************)
-
-lemma pts_trans: ∀L,T1,T,d,e. L ⊢ T1 [d, e] ≫ T → ∀T2. L ⊢ T [d, e] ≫ T2 →
-                 L ⊢ T1 [d, e] ≫ T2.
-#L #T1 #T #d #e #H elim H -L T1 T d e
-[ //
-| //
-| #L #K #V #V1 #V2 #i #d #e #Hdi #Hide #HLK #_ #HV12 #IHV12 #T2 #HVT2
-  lapply (drop_fwd_drop2 … HLK) #H
-  elim (pts_inv_lift1_up … HVT2 … H … HV12 ? ? ?) -HVT2 H HV12 // normalize [2: /2/ ] #V #HV1 #HVT2
-  @pts_subst [4,5,6,8: // |1,2,3: skip | /2/ ]
-| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
-  elim (pts_inv_bind1 … HX) -HX #V #T #HV2 #HT2 #HX destruct -X;
-  @pts_bind /2/ @IHT12 @(pts_leq_repl … HT2) /2/
-| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
-  elim (pts_inv_flat1 … HX) -HX #V #T #HV2 #HT2 #HX destruct -X /3/
-]
-qed.
-
-axiom pts_conf: ∀L,T0,d,e,T1. L ⊢ T0 [d, e] ≫ T1 → ∀T2. L ⊢ T0 [d, e] ≫ T2 →
-                  ∃∃T. L ⊢ T1 [d, e] ≫ T & L ⊢ T2 [d, e] ≫ T.
-
-(*
-      Theorem subst0_subst0: (t1,t2,u2:?; j:?) (subst0 j u2 t1 t2) ->
-                             (u1,u:?; i:?) (subst0 i u u1 u2) ->
-                             (EX t | (subst0 j u1 t1 t) & (subst0 (S (plus i j)) u t t2)).
-
-      Theorem subst0_subst0_back: (t1,t2,u2:?; j:?) (subst0 j u2 t1 t2) ->
-                                  (u1,u:?; i:?) (subst0 i u u2 u1) ->
-                                  (EX t | (subst0 j u1 t1 t) & (subst0 (S (plus i j)) u t2 t)).
-
-*)
diff --git a/matita/matita/lib/lambda-delta/substitution/pts_split.ma b/matita/matita/lib/lambda-delta/substitution/pts_split.ma
deleted file mode 100644
index 72da008ea..000000000
--- a/matita/matita/lib/lambda-delta/substitution/pts_split.ma
+++ /dev/null
@@ -1,58 +0,0 @@
-(*
-    ||M||  This file is part of HELM, an Hypertextual, Electronic
-    ||A||  Library of Mathematics, developed at the Computer Science
-    ||T||  Department of the University of Bologna, Italy.
-    ||I||
-    ||T||
-    ||A||  This file is distributed under the terms of the
-    \   /  GNU General Public License Version 2
-     \ /
-      V_______________________________________________________________ *)
-
-include "lambda-delta/substitution/pts_lift.ma".
-
-(* PARTIAL TELESCOPIC SUBSTITUTION ******************************************)
-
-(* Split properties *********************************************************)
-
-lemma pts_split_up: ∀L,T1,T2,d,e. L ⊢ T1 [d, e] ≫ T2 → ∀i. d ≤ i → i ≤ d + e →
-                    ∃∃T. L ⊢ T1 [d, i - d] ≫ T & L ⊢ T [i, d + e - i] ≫ T2.
-#L #T1 #T2 #d #e #H elim H -L T1 T2 d e
-[ /2/
-| /2/
-| #L #K #V #V1 #V2 #i #d #e #Hdi #Hide #HLK #HV1 #HV12 #IHV12 #j #Hdj #Hjde
-  elim (lt_or_ge i j) #Hij
-  [ -HV1 Hide;
-    lapply (drop_fwd_drop2 … HLK) #HLK'
-    elim (IHV12 (j - i - 1) ? ?) -IHV12; normalize /2/ -Hjde <minus_n_O >arith_b2 // #W1 #HVW1 #HWV1
-    generalize in match HVW1 generalize in match Hij -HVW1 (**) (* rewriting in the premises, rewrites in the goal too *)
-    >(plus_minus_m_m_comm … Hdj) in ⊢ (% → % → ?) -Hdj #Hij' #HVW1
-    elim (lift_total W1 0 (i + 1)) #W2 #HW12
-    lapply (pts_lift_ge … HWV1 … HLK' HW12 HV12 ?) -HWV1 HLK' HV12 // >arith_a2 /3 width=6/
-  | -IHV12 Hdi Hdj;
-    generalize in match HV1 generalize in match Hide -HV1 Hide (**) (* rewriting in the premises, rewrites in the goal too *)
-    >(plus_minus_m_m_comm … Hjde) in ⊢ (% → % → ?) -Hjde #Hide #HV1
-    @ex2_1_intro [2: @pts_lref |1: skip | /2 width=6/ ] (**) (* /3 width=6 is too slow *)
-  ]
-| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hdi #Hide
-  elim (IHV12 i ? ?) -IHV12 // #V #HV1 #HV2
-  elim (IHT12 (i + 1) ? ?) -IHT12 [2: /2 by arith4/ |3: /2/ ] (* just /2/ is too slow *)
-  -Hdi Hide >arith_c1 >arith_c1x #T #HT1 #HT2
-  @ex2_1_intro [2,3: @pts_bind | skip ] (**) (* explicit constructors *)
-  [3: @HV1 |4: @HT1 |5: // |1,2: skip | /3 width=5/ ]
-| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hdi #Hide
-  elim (IHV12 i ? ?) -IHV12 // elim (IHT12 i ? ?) -IHT12 //
-  -Hdi Hide /3 width=5/
-]
-qed.
-
-lemma pts_inv_lift1_up: ∀L,U1,U2,dt,et. L ⊢ U1 [dt, et] ≫ U2 →
-                        ∀K,d,e. ↓[d, e] L ≡ K → ∀T1. ↑[d, e] T1 ≡ U1 →
-                        d ≤ dt → dt ≤ d + e → d + e ≤ dt + et →
-                        ∃∃T2. K ⊢ T1 [d, dt + et - (d + e)] ≫ T2 & ↑[d, e] T2 ≡ U2.
-#L #U1 #U2 #dt #et #HU12 #K #d #e #HLK #T1 #HTU1 #Hddt #Hdtde #Hdedet
-elim (pts_split_up … HU12 (d + e) ? ?) -HU12 // -Hdedet #U #HU1 #HU2
-lapply (pts_weak … HU1 d e ? ?) -HU1 // <plus_minus_m_m_comm // -Hddt Hdtde #HU1
-lapply (pts_inv_lift1_eq … HU1 … HTU1) -HU1 #HU1 destruct -U1;
-elim (pts_inv_lift1_ge … HU2 … HLK … HTU1 ?) -HU2 HLK HTU1 // <minus_plus_m_m /2/
-qed.
diff --git a/matita/matita/lib/lambda-delta/substitution/tps.ma b/matita/matita/lib/lambda-delta/substitution/tps.ma
new file mode 100644
index 000000000..9fe0c29e7
--- /dev/null
+++ b/matita/matita/lib/lambda-delta/substitution/tps.ma
@@ -0,0 +1,130 @@
+(*
+    ||M||  This file is part of HELM, an Hypertextual, Electronic
+    ||A||  Library of Mathematics, developed at the Computer Science
+    ||T||  Department of the University of Bologna, Italy.
+    ||I||
+    ||T||
+    ||A||  This file is distributed under the terms of the
+    \   /  GNU General Public License Version 2
+     \ /
+      V_______________________________________________________________ *)
+
+include "lambda-delta/substitution/drop.ma".
+
+(* PARTIAL SUBSTITUTION ON TERMS ********************************************)
+
+inductive tps: lenv → term → nat → nat → term → Prop ≝
+| tps_sort : ∀L,k,d,e. tps L (⋆k) d e (⋆k)
+| tps_lref : ∀L,i,d,e. tps L (#i) d e (#i)
+| tps_subst: ∀L,K,V,U1,U2,i,d,e.
+             d ≤ i → i < d + e →
+             ↓[0, i] L ≡ K. 𝕓{Abbr} V → tps K V 0 (d + e - i - 1) U1 →
+             ↑[0, i + 1] U1 ≡ U2 → tps L (#i) d e U2
+| tps_bind : ∀L,I,V1,V2,T1,T2,d,e.
+             tps L V1 d e V2 → tps (L. 𝕓{I} V1) T1 (d + 1) e T2 →
+             tps L (𝕓{I} V1. T1) d e (𝕓{I} V2. T2)
+| tps_flat : ∀L,I,V1,V2,T1,T2,d,e.
+             tps L V1 d e V2 → tps L T1 d e T2 →
+             tps L (𝕗{I} V1. T1) d e (𝕗{I} V2. T2)
+.
+
+interpretation "partial telescopic substritution"
+   'PSubst L T1 d e T2 = (tps L T1 d e T2).
+
+(* Basic properties *********************************************************)
+
+lemma tps_leq_repl: ∀L1,T1,T2,d,e. L1 ⊢ T1 [d, e] ≫ T2 →
+                    ∀L2. L1 [d, e] ≈ L2 → L2 ⊢ T1 [d, e] ≫ T2.
+#L1 #T1 #T2 #d #e #H elim H -H L1 T1 T2 d e
+[ //
+| //
+| #L1 #K1 #V #V1 #V2 #i #d #e #Hdi #Hide #HLK1 #_ #HV12 #IHV12 #L2 #HL12
+  elim (drop_leq_drop1 … HL12 … HLK1 ? ?) -HL12 HLK1 // #K2 #HK12 #HLK2
+  @tps_subst [4,5,6,8: // |1,2,3: skip | /2/ ] (**) (* /3 width=6/ is too slow *)
+| /4/
+| /3/
+]
+qed.
+
+lemma tps_refl: ∀T,L,d,e. L ⊢ T [d, e] ≫ T.
+#T elim T -T //
+#I elim I -I /2/
+qed.
+
+lemma tps_weak: ∀L,T1,T2,d1,e1. L ⊢ T1 [d1, e1] ≫ T2 →
+                ∀d2,e2. d2 ≤ d1 → d1 + e1 ≤ d2 + e2 →
+                L ⊢ T1 [d2, e2] ≫ T2.
+#L #T1 #T #d1 #e1 #H elim H -L T1 T d1 e1
+[ //
+| //
+| #L #K #V #V1 #V2 #i #d1 #e1 #Hid1 #Hide1 #HLK #_ #HV12 #IHV12 #d2 #e2 #Hd12 #Hde12
+  lapply (transitive_le … Hd12 … Hid1) -Hd12 Hid1 #Hid2
+  lapply (lt_to_le_to_lt … Hide1 … Hde12) -Hide1 #Hide2
+  @tps_subst [4,5,6,8: // |1,2,3: skip | @IHV12 /2/ ] (**) (* /4 width=6/ is too slow *)
+| /4/
+| /4/
+]
+qed.
+
+lemma tps_weak_top: ∀L,T1,T2,d,e.
+                    L ⊢ T1 [d, e] ≫ T2 → L ⊢ T1 [d, |L| - d] ≫ T2.
+#L #T1 #T #d #e #H elim H -L T1 T d e
+[ //
+| //
+| #L #K #V #V1 #V2 #i #d #e #Hdi #_ #HLK #_ #HV12 #IHV12
+  lapply (drop_fwd_drop2_length … HLK) #Hi
+  lapply (le_to_lt_to_lt … Hdi Hi) #Hd
+  lapply (plus_minus_m_m_comm (|L|) d ?) [ /2/ ] -Hd #Hd
+  lapply (drop_fwd_O1_length … HLK) normalize #HKL
+  lapply (tps_weak … IHV12 0 (|L| - i - 1) ? ?) -IHV12 // -HKL /2 width=6/
+| normalize /2/
+| /2/
+]
+qed.
+
+lemma tps_weak_all: ∀L,T1,T2,d,e.
+                    L ⊢ T1 [d, e] ≫ T2 → L ⊢ T1 [0, |L|] ≫ T2.
+#L #T1 #T #d #e #HT12
+lapply (tps_weak … HT12 0 (d + e) ? ?) -HT12 // #HT12
+lapply (tps_weak_top … HT12) //
+qed.     
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma tps_inv_bind1_aux: ∀d,e,L,U1,U2. L ⊢ U1 [d, e] ≫ U2 →
+                         ∀I,V1,T1. U1 = 𝕓{I} V1. T1 →
+                         ∃∃V2,T2. L ⊢ V1 [d, e] ≫ V2 & 
+                                  L. 𝕓{I} V1 ⊢ T1 [d + 1, e] ≫ T2 &
+                                  U2 =  𝕓{I} V2. T2.
+#d #e #L #U1 #U2 * -d e L U1 U2
+[ #L #k #d #e #I #V1 #T1 #H destruct
+| #L #i #d #e #I #V1 #T1 #H destruct
+| #L #K #V #U1 #U2 #i #d #e #_ #_ #_ #_ #_ #I #V1 #T1 #H destruct
+| #L #J #V1 #V2 #T1 #T2 #d #e #HV12 #HT12 #I #V #T #H destruct /2 width=5/
+| #L #J #V1 #V2 #T1 #T2 #d #e #_ #_ #I #V #T #H destruct
+]
+qed.
+
+lemma tps_inv_bind1: ∀d,e,L,I,V1,T1,U2. L ⊢ 𝕓{I} V1. T1 [d, e] ≫ U2 →
+                     ∃∃V2,T2. L ⊢ V1 [d, e] ≫ V2 & 
+                              L. 𝕓{I} V1 ⊢ T1 [d + 1, e] ≫ T2 &
+                              U2 =  𝕓{I} V2. T2.
+/2/ qed.
+
+lemma tps_inv_flat1_aux: ∀d,e,L,U1,U2. L ⊢ U1 [d, e] ≫ U2 →
+                         ∀I,V1,T1. U1 = 𝕗{I} V1. T1 →
+                         ∃∃V2,T2. L ⊢ V1 [d, e] ≫ V2 & L ⊢ T1 [d, e] ≫ T2 &
+                                  U2 =  𝕗{I} V2. T2.
+#d #e #L #U1 #U2 * -d e L U1 U2
+[ #L #k #d #e #I #V1 #T1 #H destruct
+| #L #i #d #e #I #V1 #T1 #H destruct
+| #L #K #V #U1 #U2 #i #d #e #_ #_ #_ #_ #_ #I #V1 #T1 #H destruct
+| #L #J #V1 #V2 #T1 #T2 #d #e #_ #_ #I #V #T #H destruct
+| #L #J #V1 #V2 #T1 #T2 #d #e #HV12 #HT12 #I #V #T #H destruct /2 width=5/
+]
+qed.
+
+lemma tps_inv_flat1: ∀d,e,L,I,V1,T1,U2. L ⊢ 𝕗{I} V1. T1 [d, e] ≫ U2 →
+                     ∃∃V2,T2. L ⊢ V1 [d, e] ≫ V2 & L ⊢ T1 [d, e] ≫ T2 &
+                              U2 =  𝕗{I} V2. T2.
+/2/ qed.
diff --git a/matita/matita/lib/lambda-delta/substitution/tps_lift.ma b/matita/matita/lib/lambda-delta/substitution/tps_lift.ma
new file mode 100644
index 000000000..70dc0903d
--- /dev/null
+++ b/matita/matita/lib/lambda-delta/substitution/tps_lift.ma
@@ -0,0 +1,174 @@
+(*
+    ||M||  This file is part of HELM, an Hypertextual, Electronic
+    ||A||  Library of Mathematics, developed at the Computer Science
+    ||T||  Department of the University of Bologna, Italy.
+    ||I||
+    ||T||
+    ||A||  This file is distributed under the terms of the
+    \   /  GNU General Public License Version 2
+     \ /
+      V_______________________________________________________________ *)
+
+include "lambda-delta/substitution/lift_lift.ma".
+include "lambda-delta/substitution/drop_drop.ma".
+include "lambda-delta/substitution/tps.ma".
+
+(* PARTIAL SUBSTITUTION ON TERMS ********************************************)
+
+(* Relocation properties ****************************************************)
+
+lemma tps_lift_le: ∀K,T1,T2,dt,et. K ⊢ T1 [dt, et] ≫ T2 →
+                   ∀L,U1,U2,d,e. ↓[d, e] L ≡ K →
+                   ↑[d, e] T1 ≡ U1 → ↑[d, e] T2 ≡ U2 →
+                   dt + et ≤ d →
+                   L ⊢ U1 [dt, et] ≫ U2.
+#K #T1 #T2 #dt #et #H elim H -H K T1 T2 dt et
+[ #K #k #dt #et #L #U1 #U2 #d #e #_ #H1 #H2 #_
+  lapply (lift_mono … H1 … H2) -H1 H2 #H destruct -U1 //
+| #K #i #dt #et #L #U1 #U2 #d #e #_ #H1 #H2 #_
+  lapply (lift_mono … H1 … H2) -H1 H2 #H destruct -U1 //
+| #K #KV #V #V1 #V2 #i #dt #et #Hdti #Hidet #HKV #_ #HV12 #IHV12 #L #U1 #U2 #d #e #HLK #H #HVU2 #Hdetd
+  lapply (lt_to_le_to_lt … Hidet … Hdetd) #Hid
+  lapply (lift_inv_lref1_lt … H … Hid) -H #H destruct -U1;
+  elim (lift_trans_ge … HV12 … HVU2 ?) -HV12 HVU2 V2 // <minus_plus #V2 #HV12 #HVU2
+  elim (drop_trans_le … HLK … HKV ?) -HLK HKV K /2/ #X #HLK #H
+  elim (drop_inv_skip2 … H ?) -H /2/ -Hid #K #W #HKV #HVW #H destruct -X
+  @tps_subst [4,5,6,8: // |1,2,3: skip | @IHV12 /2 width=6/ ] (**) (* explicit constructor *)
+| #K #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hdetd
+  elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
+  elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct -U1 U2;
+  @tps_bind [ /2 width=6/ | @IHT12 [3,4,5: /2/ |1,2: skip | /2/ ] ] (**) (* /3 width=6/ is too slow, arith3 needed to avoid crash *)
+| #K #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hdetd
+  elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
+  elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct -U1 U2;
+  /3 width=6/
+]
+qed.
+
+lemma tps_lift_ge: ∀K,T1,T2,dt,et. K ⊢ T1 [dt, et] ≫ T2 →
+                   ∀L,U1,U2,d,e. ↓[d, e] L ≡ K →
+                   ↑[d, e] T1 ≡ U1 → ↑[d, e] T2 ≡ U2 →
+                   d ≤ dt →
+                   L ⊢ U1 [dt + e, et] ≫ U2.
+#K #T1 #T2 #dt #et #H elim H -H K T1 T2 dt et
+[ #K #k #dt #et #L #U1 #U2 #d #e #_ #H1 #H2 #_
+  lapply (lift_mono … H1 … H2) -H1 H2 #H destruct -U1 //
+| #K #i #dt #et #L #U1 #U2 #d #e #_ #H1 #H2 #_
+  lapply (lift_mono … H1 … H2) -H1 H2 #H destruct -U1 //
+| #K #KV #V #V1 #V2 #i #dt #et #Hdti #Hidet #HKV #HV1 #HV12 #_ #L #U1 #U2 #d #e #HLK #H #HVU2 #Hddt
+  <(arith_c1x ? ? ? e) in HV1 #HV1 (**) (* explicit athmetical rewrite and ?'s *)
+  lapply (transitive_le … Hddt … Hdti) -Hddt #Hid
+  lapply (lift_inv_lref1_ge … H … Hid) -H #H destruct -U1;
+  lapply (lift_trans_be … HV12 … HVU2 ? ?) -HV12 HVU2 V2 /2/ >plus_plus_comm_23 #HV1U2
+  lapply (drop_trans_ge_comm … HLK … HKV ?) -HLK HKV K // -Hid #HLKV
+  @tps_subst [4,5: /2/ |6,7,8: // |1,2,3: skip ] (**) (* /3 width=8/ is too slow *)
+| #K #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hddt
+  elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
+  elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct -U1 U2;
+  @tps_bind [ /2 width=5/ | /3 width=5/ ] (**) (* explicit constructor *)
+| #K #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hddt
+  elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
+  elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct -U1 U2;
+  /3 width=5/
+]
+qed.
+
+lemma tps_inv_lift1_le: ∀L,U1,U2,dt,et. L ⊢ U1 [dt, et] ≫ U2 →
+                        ∀K,d,e. ↓[d, e] L ≡ K → ∀T1. ↑[d, e] T1 ≡ U1 →
+                        dt + et ≤ d →
+                        ∃∃T2. K ⊢ T1 [dt, et] ≫ T2 & ↑[d, e] T2 ≡ U2.
+#L #U1 #U2 #dt #et #H elim H -H L U1 U2 dt et
+[ #L #k #dt #et #K #d #e #_ #T1 #H #_
+  lapply (lift_inv_sort2 … H) -H #H destruct -T1 /2/
+| #L #i #dt #et #K #d #e #_ #T1 #H #_
+  elim (lift_inv_lref2 … H) -H * #Hid #H destruct -T1 /3/
+| #L #KV #V #V1 #V2 #i #dt #et #Hdti #Hidet #HLKV #_ #HV12 #IHV12 #K #d #e #HLK #T1 #H #Hdetd
+  lapply (lt_to_le_to_lt … Hidet … Hdetd) #Hid
+  lapply (lift_inv_lref2_lt … H … Hid) -H #H destruct -T1;
+  elim (drop_conf_lt … HLK … HLKV ?) -HLK HLKV L // #L #W #HKL #HKVL #HWV
+  elim (IHV12 … HKVL … HWV ?) -HKVL HWV /2/ -Hdetd #W1 #HW1 #HWV1
+  elim (lift_trans_le … HWV1 … HV12 ?) -HWV1 HV12 V1 // >arith_a2 /3 width=6/
+| #L #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdetd
+  elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct -X;
+  elim (IHV12 … HLK … HWV1 ?) -IHV12 //
+  elim (IHU12 … HTU1 ?) -IHU12 HTU1 [3: /2/ |4: @drop_skip // |2: skip ] -HLK HWV1 Hdetd /3 width=5/ (**) (* just /3 width=5/ is too slow *)
+| #L #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdetd
+  elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct -X;
+  elim (IHV12 … HLK … HWV1 ?) -IHV12 HWV1 //
+  elim (IHU12 … HLK … HTU1 ?) -IHU12 HLK HTU1 // /3 width=5/
+]
+qed.
+
+lemma tps_inv_lift1_ge: ∀L,U1,U2,dt,et. L ⊢ U1 [dt, et] ≫ U2 →
+                        ∀K,d,e. ↓[d, e] L ≡ K → ∀T1. ↑[d, e] T1 ≡ U1 →
+                        d + e ≤ dt →
+                        ∃∃T2. K ⊢ T1 [dt - e, et] ≫ T2 & ↑[d, e] T2 ≡ U2.
+#L #U1 #U2 #dt #et #H elim H -H L U1 U2 dt et
+[ #L #k #dt #et #K #d #e #_ #T1 #H #_
+  lapply (lift_inv_sort2 … H) -H #H destruct -T1 /2/
+| #L #i #dt #et #K #d #e #_ #T1 #H #_
+  elim (lift_inv_lref2 … H) -H * #Hid #H destruct -T1 /3/
+| #L #KV #V #V1 #V2 #i #dt #et #Hdti #Hidet #HLKV #HV1 #HV12 #_ #K #d #e #HLK #T1 #H #Hdedt
+  lapply (transitive_le … Hdedt … Hdti) #Hdei
+  lapply (plus_le_weak … Hdedt) -Hdedt #Hedt
+  lapply (plus_le_weak … Hdei) #Hei
+  <(arith_h1 ? ? ? e ? ?) in HV1 // #HV1
+  lapply (lift_inv_lref2_ge … H … Hdei) -H #H destruct -T1;
+  lapply (drop_conf_ge … HLK … HLKV ?) -HLK HLKV L // #HKV
+  elim (lift_split … HV12 d (i - e + 1) ? ? ?) -HV12; [2,3,4: normalize /2/ ] -Hdei >arith_e2 // #V0 #HV10 #HV02
+  @ex2_1_intro
+  [2: @tps_subst [4: /2/ |6,7,8: // |1,2,3: skip |5: @arith5 // ]  
+  |1: skip
+  | //
+  ] (**) (* explicitc constructors *)
+| #L #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdetd
+  elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct -X;
+  lapply (plus_le_weak … Hdetd) #Hedt
+  elim (IHV12 … HLK … HWV1 ?) -IHV12 // #W2 #HW12 #HWV2
+  elim (IHU12 … HTU1 ?) -IHU12 HTU1 [4: @drop_skip // |2: skip |3: /2/ ]
+  <plus_minus // /3 width=5/
+| #L #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdetd
+  elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct -X;
+  elim (IHV12 … HLK … HWV1 ?) -IHV12 HWV1 //
+  elim (IHU12 … HLK … HTU1 ?) -IHU12 HLK HTU1 // /3 width=5/
+]
+qed.
+
+lemma tps_inv_lift1_eq: ∀L,U1,U2,d,e.
+                        L ⊢ U1 [d, e] ≫ U2 → ∀T1. ↑[d, e] T1 ≡ U1 → U1 = U2.
+#L #U1 #U2 #d #e #H elim H -H L U1 U2 d e
+[ //
+| //
+| #L #K #V #V1 #V2 #i #d #e #Hdi #Hide #_ #_ #_ #_ #T1 #H
+  elim (lift_inv_lref2 … H) -H * #H
+  [ lapply (le_to_lt_to_lt … Hdi … H) -Hdi H #H
+    elim (lt_refl_false … H)
+  | lapply (lt_to_le_to_lt … Hide … H) -Hide H #H
+    elim (lt_refl_false … H)
+  ]
+| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
+  elim (lift_inv_bind2 … HX) -HX #V #T #HV1 #HT1 #H destruct -X
+  >IHV12 // >IHT12 //
+| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
+  elim (lift_inv_flat2 … HX) -HX #V #T #HV1 #HT1 #H destruct -X
+  >IHV12 // >IHT12 //
+]
+qed.
+(*
+      Theorem subst0_gen_lift_ge : (u,t1,x:?; i,h,d:?) (subst0 i u (lift h d t1) x) ->
+                                   (le (plus d h) i) ->
+                                   (EX t2 | x = (lift h d t2) & (subst0 (minus i h) u t1 t2)).
+
+      Theorem subst0_gen_lift_rev_ge: (t1,v,u2:?; i,h,d:?) 
+                                      (subst0 i v t1 (lift h d u2)) ->
+                                      (le (plus d h) i) ->
+                                      (EX u1 | (subst0 (minus i h) v u1 u2) &
+		                               t1 = (lift h d u1)
+		                      ).
+
+
+      Theorem subst0_gen_lift_rev_lelt: (t1,v,u2:?; i,h,d:?)
+                                        (subst0 i v t1 (lift h d u2)) ->
+                                        (le d i) -> (lt i (plus d h)) ->
+				        (EX u1 | t1 = (lift (minus (plus d h) (S i)) (S i) u1)).
+*)
diff --git a/matita/matita/lib/lambda-delta/substitution/tps_split.ma b/matita/matita/lib/lambda-delta/substitution/tps_split.ma
new file mode 100644
index 000000000..23cdb39df
--- /dev/null
+++ b/matita/matita/lib/lambda-delta/substitution/tps_split.ma
@@ -0,0 +1,58 @@
+(*
+    ||M||  This file is part of HELM, an Hypertextual, Electronic
+    ||A||  Library of Mathematics, developed at the Computer Science
+    ||T||  Department of the University of Bologna, Italy.
+    ||I||
+    ||T||
+    ||A||  This file is distributed under the terms of the
+    \   /  GNU General Public License Version 2
+     \ /
+      V_______________________________________________________________ *)
+
+include "lambda-delta/substitution/tps_lift.ma".
+
+(* PARTIAL SUBSTITUTION ON TERMS ********************************************)
+
+(* Split properties *********************************************************)
+
+lemma tps_split_up: ∀L,T1,T2,d,e. L ⊢ T1 [d, e] ≫ T2 → ∀i. d ≤ i → i ≤ d + e →
+                    ∃∃T. L ⊢ T1 [d, i - d] ≫ T & L ⊢ T [i, d + e - i] ≫ T2.
+#L #T1 #T2 #d #e #H elim H -L T1 T2 d e
+[ /2/
+| /2/
+| #L #K #V #V1 #V2 #i #d #e #Hdi #Hide #HLK #HV1 #HV12 #IHV12 #j #Hdj #Hjde
+  elim (lt_or_ge i j) #Hij
+  [ -HV1 Hide;
+    lapply (drop_fwd_drop2 … HLK) #HLK'
+    elim (IHV12 (j - i - 1) ? ?) -IHV12; normalize /2/ -Hjde <minus_n_O >arith_b2 // #W1 #HVW1 #HWV1
+    generalize in match HVW1 generalize in match Hij -HVW1 (**) (* rewriting in the premises, rewrites in the goal too *)
+    >(plus_minus_m_m_comm … Hdj) in ⊢ (% → % → ?) -Hdj #Hij' #HVW1
+    elim (lift_total W1 0 (i + 1)) #W2 #HW12
+    lapply (tps_lift_ge … HWV1 … HLK' HW12 HV12 ?) -HWV1 HLK' HV12 // >arith_a2 /3 width=6/
+  | -IHV12 Hdi Hdj;
+    generalize in match HV1 generalize in match Hide -HV1 Hide (**) (* rewriting in the premises, rewrites in the goal too *)
+    >(plus_minus_m_m_comm … Hjde) in ⊢ (% → % → ?) -Hjde #Hide #HV1
+    @ex2_1_intro [2: @tps_lref |1: skip | /2 width=6/ ] (**) (* /3 width=6 is too slow *)
+  ]
+| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hdi #Hide
+  elim (IHV12 i ? ?) -IHV12 // #V #HV1 #HV2
+  elim (IHT12 (i + 1) ? ?) -IHT12 [2: /2 by arith4/ |3: /2/ ] (* just /2/ is too slow *)
+  -Hdi Hide >arith_c1 >arith_c1x #T #HT1 #HT2
+  @ex2_1_intro [2,3: @tps_bind | skip ] (**) (* explicit constructors *)
+  [3: @HV1 |4: @HT1 |5: // |1,2: skip | /3 width=5/ ]
+| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hdi #Hide
+  elim (IHV12 i ? ?) -IHV12 // elim (IHT12 i ? ?) -IHT12 //
+  -Hdi Hide /3 width=5/
+]
+qed.
+
+lemma tps_inv_lift1_up: ∀L,U1,U2,dt,et. L ⊢ U1 [dt, et] ≫ U2 →
+                        ∀K,d,e. ↓[d, e] L ≡ K → ∀T1. ↑[d, e] T1 ≡ U1 →
+                        d ≤ dt → dt ≤ d + e → d + e ≤ dt + et →
+                        ∃∃T2. K ⊢ T1 [d, dt + et - (d + e)] ≫ T2 & ↑[d, e] T2 ≡ U2.
+#L #U1 #U2 #dt #et #HU12 #K #d #e #HLK #T1 #HTU1 #Hddt #Hdtde #Hdedet
+elim (tps_split_up … HU12 (d + e) ? ?) -HU12 // -Hdedet #U #HU1 #HU2
+lapply (tps_weak … HU1 d e ? ?) -HU1 // <plus_minus_m_m_comm // -Hddt Hdtde #HU1
+lapply (tps_inv_lift1_eq … HU1 … HTU1) -HU1 #HU1 destruct -U1;
+elim (tps_inv_lift1_ge … HU2 … HLK … HTU1 ?) -HU2 HLK HTU1 // <minus_plus_m_m /2/
+qed.
diff --git a/matita/matita/lib/lambda-delta/substitution/tps_tps.ma b/matita/matita/lib/lambda-delta/substitution/tps_tps.ma
new file mode 100644
index 000000000..55e8be812
--- /dev/null
+++ b/matita/matita/lib/lambda-delta/substitution/tps_tps.ma
@@ -0,0 +1,47 @@
+(*
+    ||M||  This file is part of HELM, an Hypertextual, Electronic
+    ||A||  Library of Mathematics, developed at the Computer Science
+    ||T||  Department of the University of Bologna, Italy.
+    ||I||
+    ||T||
+    ||A||  This file is distributed under the terms of the
+    \   /  GNU General Public License Version 2
+     \ /
+      V_______________________________________________________________ *)
+
+include "lambda-delta/substitution/tps_split.ma".
+
+(* PARTIAL SUBSTITUTION ON TERMS ********************************************)
+
+(* Main properties **********************************************************)
+
+lemma tps_trans: ∀L,T1,T,d,e. L ⊢ T1 [d, e] ≫ T → ∀T2. L ⊢ T [d, e] ≫ T2 →
+                 L ⊢ T1 [d, e] ≫ T2.
+#L #T1 #T #d #e #H elim H -L T1 T d e
+[ //
+| //
+| #L #K #V #V1 #V2 #i #d #e #Hdi #Hide #HLK #_ #HV12 #IHV12 #T2 #HVT2
+  lapply (drop_fwd_drop2 … HLK) #H
+  elim (tps_inv_lift1_up … HVT2 … H … HV12 ? ? ?) -HVT2 H HV12 // normalize [2: /2/ ] #V #HV1 #HVT2
+  @tps_subst [4,5,6,8: // |1,2,3: skip | /2/ ]
+| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
+  elim (tps_inv_bind1 … HX) -HX #V #T #HV2 #HT2 #HX destruct -X;
+  @tps_bind /2/ @IHT12 @(tps_leq_repl … HT2) /2/
+| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
+  elim (tps_inv_flat1 … HX) -HX #V #T #HV2 #HT2 #HX destruct -X /3/
+]
+qed.
+
+axiom tps_conf: ∀L,T0,d,e,T1. L ⊢ T0 [d, e] ≫ T1 → ∀T2. L ⊢ T0 [d, e] ≫ T2 →
+                  ∃∃T. L ⊢ T1 [d, e] ≫ T & L ⊢ T2 [d, e] ≫ T.
+
+(*
+      Theorem subst0_subst0: (t1,t2,u2:?; j:?) (subst0 j u2 t1 t2) ->
+                             (u1,u:?; i:?) (subst0 i u u1 u2) ->
+                             (EX t | (subst0 j u1 t1 t) & (subst0 (S (plus i j)) u t t2)).
+
+      Theorem subst0_subst0_back: (t1,t2,u2:?; j:?) (subst0 j u2 t1 t2) ->
+                                  (u1,u:?; i:?) (subst0 i u u2 u1) ->
+                                  (EX t | (subst0 j u1 t1 t) & (subst0 (S (plus i j)) u t2 t)).
+
+*)
diff --git a/matita/matita/lib/lambda-delta/xoa.conf.xml b/matita/matita/lib/lambda-delta/xoa.conf.xml
index 3330813ae..d25c5dcca 100644
--- a/matita/matita/lib/lambda-delta/xoa.conf.xml
+++ b/matita/matita/lib/lambda-delta/xoa.conf.xml
@@ -2,7 +2,7 @@
 <helm_registry>
   <section name="matita">
     <key name="rt_base_dir">$(MATITA_RT_BASE_DIR)</key>
-<!--    
+<!--
     <key name="system">false</key>
     <key name="map_unicode_to_tex">false</key>
     <key name="do_heavy_checks">true</key>
@@ -27,6 +27,8 @@
     <key name="ex">7 6</key>
     <key name="or">3</key>
     <key name="or">4</key>
+<!--    
     <key name="and">3</key>
+-->
   </section>
 </helm_registry>
diff --git a/matita/matita/lib/lambda-delta/xoa.ma b/matita/matita/lib/lambda-delta/xoa.ma
index 582b2d25f..68a0776dc 100644
--- a/matita/matita/lib/lambda-delta/xoa.ma
+++ b/matita/matita/lib/lambda-delta/xoa.ma
@@ -125,11 +125,3 @@ inductive or4 (P0,P1,P2,P3:Prop) : Prop ≝
 
 interpretation "multiple disjunction connective (4)" 'Or P0 P1 P2 P3 = (or4 P0 P1 P2 P3).
 
-(* multiple conjunction connective (3) *)
-
-inductive and3 (P0,P1,P2:Prop) : Prop ≝
-   | and3_intro: P0 → P1 → P2 → and3 ? ? ?
-.
-
-interpretation "multiple conjunction connective (3)" 'And P0 P1 P2 = (and3 P0 P1 P2).
-
diff --git a/matita/matita/lib/lambda-delta/xoa_notation.ma b/matita/matita/lib/lambda-delta/xoa_notation.ma
index ce9460eb6..b8270b660 100644
--- a/matita/matita/lib/lambda-delta/xoa_notation.ma
+++ b/matita/matita/lib/lambda-delta/xoa_notation.ma
@@ -136,9 +136,3 @@ notation "hvbox(∨∨ term 29 P0 break | term 29 P1 break | term 29 P2 break |
  non associative with precedence 30
  for @{ 'Or $P0 $P1 $P2 $P3 }.
 
-(* multiple conjunction connective (3) *)
-
-notation "hvbox(∧∧ term 34 P0 break & term 34 P1 break & term 34 P2)"
- non associative with precedence 35
- for @{ 'And $P0 $P1 $P2 }.
-