From 9271b3ca211007ca5ffac1e7644ebc02b0689d6e Mon Sep 17 00:00:00 2001
From: Ferruccio Guidi <ferruccio.guidi@unibo.it>
Date: Mon, 8 Aug 2011 22:41:37 +0000
Subject: [PATCH] - tps_tpr closed! (substitution is a reduction) - some
 refactoring

---
 matita/matita/lib/lambda-delta/Makefile       |  2 +-
 matita/matita/lib/lambda-delta/ground.ma      | 13 ++--
 .../matita/lib/lambda-delta/reduction/cpr.ma  | 17 +++--
 .../matita/lib/lambda-delta/reduction/tpr.ma  |  2 +-
 .../lib/lambda-delta/reduction/tpr_lift.ma    |  6 +-
 .../lib/lambda-delta/reduction/tpr_tpr.ma     | 16 ++---
 .../reduction/{tpr_pts.ma => tpr_tps.ma}      |  4 +-
 .../lib/lambda-delta/substitution/drop.ma     | 50 ++++++++++----
 .../lambda-delta/substitution/drop_drop.ma    |  2 +-
 .../lib/lambda-delta/substitution/leq.ma      |  8 ++-
 .../substitution/{pts.ma => tps.ma}           | 69 ++++++++++++-------
 .../substitution/{pts_lift.ma => tps_lift.ma} | 24 +++----
 .../{pts_split.ma => tps_split.ma}            | 22 +++---
 .../substitution/{pts_pts.ma => tps_tps.ma}   | 18 ++---
 matita/matita/lib/lambda-delta/xoa.conf.xml   |  4 +-
 matita/matita/lib/lambda-delta/xoa.ma         |  8 ---
 .../matita/lib/lambda-delta/xoa_notation.ma   |  6 --
 17 files changed, 156 insertions(+), 115 deletions(-)
 rename matita/matita/lib/lambda-delta/reduction/{tpr_pts.ma => tpr_tps.ma} (91%)
 rename matita/matita/lib/lambda-delta/substitution/{pts.ma => tps.ma} (59%)
 rename matita/matita/lib/lambda-delta/substitution/{pts_lift.ma => tps_lift.ma} (91%)
 rename matita/matita/lib/lambda-delta/substitution/{pts_split.ma => tps_split.ma} (77%)
 rename matita/matita/lib/lambda-delta/substitution/{pts_pts.ma => tps_tps.ma} (73%)

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_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_pts.ma b/matita/matita/lib/lambda-delta/reduction/tpr_tps.ma
similarity index 91%
rename from matita/matita/lib/lambda-delta/reduction/tpr_pts.ma
rename to matita/matita/lib/lambda-delta/reduction/tpr_tps.ma
index f8a41f025..96bc9b76e 100644
--- a/matita/matita/lib/lambda-delta/reduction/tpr_pts.ma
+++ b/matita/matita/lib/lambda-delta/reduction/tpr_tps.ma
@@ -16,11 +16,11 @@ include "lambda-delta/reduction/lpr.ma".
 
 (* CONTEXT-FREE PARALLEL REDUCTION ON TERMS *********************************)
 
-axiom tpr_pts_lpr: ∀L1,L2. L1 ⇒ L2 → ∀T1,T2. T1 ⇒ T2 →
+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_pts_bind: ∀I,V1,V2,T1,T2,U1. V1 ⇒ V2 → T1 ⇒ T2 →
+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/tps.ma
similarity index 59%
rename from matita/matita/lib/lambda-delta/substitution/pts.ma
rename to matita/matita/lib/lambda-delta/substitution/tps.ma
index ac962355e..9fe0c29e7 100644
--- a/matita/matita/lib/lambda-delta/substitution/pts.ma
+++ b/matita/matita/lib/lambda-delta/substitution/tps.ma
@@ -11,47 +11,47 @@
 
 include "lambda-delta/substitution/drop.ma".
 
-(* PARTIAL TELESCOPIC SUBSTITUTION ******************************************)
+(* PARTIAL SUBSTITUTION ON TERMS ********************************************)
 
-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.
+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 → 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)
+             ↓[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 = (pts L T1 d e T2).
+   'PSubst L T1 d e T2 = (tps L T1 d e T2).
 
 (* Basic properties *********************************************************)
 
-lemma pts_leq_repl: ∀L1,T1,T2,d,e. L1 ⊢ T1 [d, e] ≫ T2 →
+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
-  @pts_subst [4,5,6,8: // |1,2,3: skip | /2/ ] (**) (* /3 width=6/ is too slow *)
+  @tps_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.
+lemma tps_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 →
+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
@@ -60,15 +60,38 @@ lemma pts_weak: ∀L,T1,T2,d1,e1. L ⊢ T1 [d1, e1] ≫ T2 →
 | #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 *)
+  @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 pts_inv_bind1_aux: ∀d,e,L,U1,U2. L ⊢ U1 [d, e] ≫ U2 →
+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 &
@@ -82,13 +105,13 @@ lemma pts_inv_bind1_aux: ∀d,e,L,U1,U2. L ⊢ U1 [d, e] ≫ U2 →
 ]
 qed.
 
-lemma pts_inv_bind1: ∀d,e,L,I,V1,T1,U2. L ⊢ 𝕓{I} V1. T1 [d, e] ≫ U2 →
+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 pts_inv_flat1_aux: ∀d,e,L,U1,U2. L ⊢ U1 [d, e] ≫ U2 →
+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.
@@ -101,7 +124,7 @@ lemma pts_inv_flat1_aux: ∀d,e,L,U1,U2. L ⊢ U1 [d, e] ≫ U2 →
 ]
 qed.
 
-lemma pts_inv_flat1: ∀d,e,L,I,V1,T1,U2. L ⊢ 𝕗{I} V1. T1 [d, e] ≫ U2 →
+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/pts_lift.ma b/matita/matita/lib/lambda-delta/substitution/tps_lift.ma
similarity index 91%
rename from matita/matita/lib/lambda-delta/substitution/pts_lift.ma
rename to matita/matita/lib/lambda-delta/substitution/tps_lift.ma
index 8cd477738..70dc0903d 100644
--- a/matita/matita/lib/lambda-delta/substitution/pts_lift.ma
+++ b/matita/matita/lib/lambda-delta/substitution/tps_lift.ma
@@ -11,13 +11,13 @@
 
 include "lambda-delta/substitution/lift_lift.ma".
 include "lambda-delta/substitution/drop_drop.ma".
-include "lambda-delta/substitution/pts.ma".
+include "lambda-delta/substitution/tps.ma".
 
-(* PARTIAL TELESCOPIC SUBSTITUTION ******************************************)
+(* PARTIAL SUBSTITUTION ON TERMS ********************************************)
 
 (* Relocation properties ****************************************************)
 
-lemma pts_lift_le: ∀K,T1,T2,dt,et. K ⊢ T1 [dt, et] ≫ T2 →
+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 →
@@ -33,11 +33,11 @@ lemma pts_lift_le: ∀K,T1,T2,dt,et. K ⊢ T1 [dt, et] ≫ T2 →
   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 *)
+  @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;
-  @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 *)
+  @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;
@@ -45,7 +45,7 @@ lemma pts_lift_le: ∀K,T1,T2,dt,et. K ⊢ T1 [dt, et] ≫ T2 →
 ]
 qed.
 
-lemma pts_lift_ge: ∀K,T1,T2,dt,et. K ⊢ T1 [dt, et] ≫ T2 →
+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 →
@@ -61,11 +61,11 @@ lemma pts_lift_ge: ∀K,T1,T2,dt,et. K ⊢ T1 [dt, et] ≫ T2 →
   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 *)
+  @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;
-  @pts_bind [ /2 width=5/ | /3 width=5/ ] (**) (* explicit constructor *)
+  @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;
@@ -73,7 +73,7 @@ lemma pts_lift_ge: ∀K,T1,T2,dt,et. K ⊢ T1 [dt, et] ≫ T2 →
 ]
 qed.
 
-lemma pts_inv_lift1_le: ∀L,U1,U2,dt,et. L ⊢ U1 [dt, et] ≫ U2 →
+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.
@@ -99,7 +99,7 @@ lemma pts_inv_lift1_le: ∀L,U1,U2,dt,et. L ⊢ U1 [dt, et] ≫ U2 →
 ]
 qed.
 
-lemma pts_inv_lift1_ge: ∀L,U1,U2,dt,et. L ⊢ U1 [dt, et] ≫ U2 →
+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.
@@ -117,7 +117,7 @@ lemma pts_inv_lift1_ge: ∀L,U1,U2,dt,et. L ⊢ U1 [dt, et] ≫ U2 →
   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 // ]  
+  [2: @tps_subst [4: /2/ |6,7,8: // |1,2,3: skip |5: @arith5 // ]  
   |1: skip
   | //
   ] (**) (* explicitc constructors *)
@@ -134,7 +134,7 @@ lemma pts_inv_lift1_ge: ∀L,U1,U2,dt,et. L ⊢ U1 [dt, et] ≫ U2 →
 ]
 qed.
 
-lemma pts_inv_lift1_eq: ∀L,U1,U2,d,e.
+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
 [ //
diff --git a/matita/matita/lib/lambda-delta/substitution/pts_split.ma b/matita/matita/lib/lambda-delta/substitution/tps_split.ma
similarity index 77%
rename from matita/matita/lib/lambda-delta/substitution/pts_split.ma
rename to matita/matita/lib/lambda-delta/substitution/tps_split.ma
index 72da008ea..23cdb39df 100644
--- a/matita/matita/lib/lambda-delta/substitution/pts_split.ma
+++ b/matita/matita/lib/lambda-delta/substitution/tps_split.ma
@@ -9,13 +9,13 @@
      \ /
       V_______________________________________________________________ *)
 
-include "lambda-delta/substitution/pts_lift.ma".
+include "lambda-delta/substitution/tps_lift.ma".
 
-(* PARTIAL TELESCOPIC SUBSTITUTION ******************************************)
+(* PARTIAL SUBSTITUTION ON TERMS ********************************************)
 
 (* Split properties *********************************************************)
 
-lemma pts_split_up: ∀L,T1,T2,d,e. L ⊢ T1 [d, e] ≫ T2 → ∀i. d ≤ i → i ≤ d + e →
+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/
@@ -28,17 +28,17 @@ lemma pts_split_up: ∀L,T1,T2,d,e. L ⊢ T1 [d, e] ≫ T2 → ∀i. d ≤ i →
     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/
+    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: @pts_lref |1: skip | /2 width=6/ ] (**) (* /3 width=6 is too slow *)
+    @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: @pts_bind | skip ] (**) (* explicit constructors *)
+  @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 //
@@ -46,13 +46,13 @@ lemma pts_split_up: ∀L,T1,T2,d,e. L ⊢ T1 [d, e] ≫ T2 → ∀i. d ≤ i →
 ]
 qed.
 
-lemma pts_inv_lift1_up: ∀L,U1,U2,dt,et. L ⊢ U1 [dt, et] ≫ U2 →
+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 (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/
+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/pts_pts.ma b/matita/matita/lib/lambda-delta/substitution/tps_tps.ma
similarity index 73%
rename from matita/matita/lib/lambda-delta/substitution/pts_pts.ma
rename to matita/matita/lib/lambda-delta/substitution/tps_tps.ma
index 0461a1823..55e8be812 100644
--- a/matita/matita/lib/lambda-delta/substitution/pts_pts.ma
+++ b/matita/matita/lib/lambda-delta/substitution/tps_tps.ma
@@ -9,30 +9,30 @@
      \ /
       V_______________________________________________________________ *)
 
-include "lambda-delta/substitution/pts_split.ma".
+include "lambda-delta/substitution/tps_split.ma".
 
-(* PARTIAL TELESCOPIC SUBSTITUTION ******************************************)
+(* PARTIAL SUBSTITUTION ON TERMS ********************************************)
 
 (* Main properties **********************************************************)
 
-lemma pts_trans: ∀L,T1,T,d,e. L ⊢ T1 [d, e] ≫ T → ∀T2. L ⊢ T [d, e] ≫ T2 →
+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 (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/ ]
+  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 (pts_inv_bind1 … HX) -HX #V #T #HV2 #HT2 #HX destruct -X;
-  @pts_bind /2/ @IHT12 @(pts_leq_repl … HT2) /2/
+  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 (pts_inv_flat1 … HX) -HX #V #T #HV2 #HT2 #HX destruct -X /3/
+  elim (tps_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 →
+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.
 
 (*
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 }.
-
-- 
2.39.2