From 5275f55f5ec528edbb223834f3ec2cf1d3ce9b84 Mon Sep 17 00:00:00 2001
From: Ferruccio Guidi <ferruccio.guidi@unibo.it>
Date: Fri, 8 Apr 2016 20:49:11 +0000
Subject: [PATCH] - commit of the "s_computation" component ... - additions in
 ground_2/relocation - milestonee added in basic_2 - minor corrections

---
 .../lambdadelta/basic_2/computation/cpre.ma   |   4 +-
 .../lambdadelta/basic_2/computation/cprs.ma   |   4 +-
 .../basic_2/computation/cprs_cprs.ma          |   6 +-
 .../lambdadelta/basic_2/computation/cpxe.ma   |  10 +-
 .../lambdadelta/basic_2/computation/cpxs.ma   | 140 +++++++++---------
 .../basic_2/computation/cpxs_aaa.ma           |   6 +-
 .../basic_2/computation/cpxs_cpxs.ma          | 120 +++++++--------
 .../basic_2/computation/cpxs_lift.ma          |  60 ++++----
 .../basic_2/computation/cpxs_lleq.ma          |  16 +-
 .../basic_2/computation/cpxs_lreq.ma          |   2 +-
 .../basic_2/computation/cpxs_tsts.ma          |  42 +++---
 .../basic_2/computation/cpxs_tsts_vector.ma   |  62 ++++----
 .../lambdadelta/basic_2/computation/csx.ma    |  74 ++++-----
 .../basic_2/computation/csx_aaa.ma            |  30 ++--
 .../basic_2/computation/csx_alt.ma            |  70 ++++-----
 .../basic_2/computation/csx_fpbs.ma           |  12 +-
 .../basic_2/computation/csx_lift.ma           |  62 ++++----
 .../basic_2/computation/csx_lleq.ma           |  10 +-
 .../basic_2/computation/csx_lpx.ma            |  44 +++---
 .../basic_2/computation/csx_lpxs.ma           |   6 +-
 .../basic_2/computation/csx_tsts_vector.ma    |  60 ++++----
 .../basic_2/computation/csx_vector.ma         |  14 +-
 .../lambdadelta/basic_2/computation/fpbg.ma   |  18 +--
 .../basic_2/computation/fpbg_fleq.ma          |  36 ++---
 .../basic_2/computation/fpbg_fpbg.ma          |   2 +-
 .../basic_2/computation/fpbg_fpbs.ma          |  46 +++---
 .../basic_2/computation/fpbg_lift.ma          |   4 +-
 .../lambdadelta/basic_2/computation/fpbs.ma   | 128 ++++++++--------
 .../basic_2/computation/fpbs_aaa.ma           |   4 +-
 .../basic_2/computation/fpbs_alt.ma           |  38 ++---
 .../basic_2/computation/fpbs_fpb.ma           |  12 +-
 .../basic_2/computation/fpbs_fpbs.ma          |   2 +-
 .../basic_2/computation/fpbs_lift.ma          |  16 +-
 .../lambdadelta/basic_2/computation/fsb.ma    |  22 +--
 .../basic_2/computation/fsb_aaa.ma            |  32 ++--
 .../basic_2/computation/fsb_alt.ma            |  46 +++---
 .../basic_2/computation/fsb_csx.ma            |  24 +--
 .../lambdadelta/basic_2/computation/gcp.ma    |   4 +-
 .../basic_2/computation/gcp_aaa.ma            |  18 +--
 .../lambdadelta/basic_2/computation/gcp_cr.ma |  74 ++++-----
 .../lambdadelta/basic_2/computation/lcosx.ma  |  42 +++---
 .../basic_2/computation/lcosx_cpx.ma          |  13 +-
 .../basic_2/computation/lprs_cprs.ma          |  10 +-
 .../lambdadelta/basic_2/computation/lpxs.ma   |  38 ++---
 .../basic_2/computation/lpxs_aaa.ma           |   6 +-
 .../basic_2/computation/lpxs_cpxs.ma          |  94 ++++++------
 .../basic_2/computation/lpxs_drop.ma          |   6 +-
 .../basic_2/computation/lpxs_lleq.ma          |  70 ++++-----
 .../basic_2/computation/lpxs_lpxs.ma          |   2 +-
 .../basic_2/computation/lsubc_drop.ma         |  26 ++--
 .../basic_2/computation/lsubc_drops.ma        |   2 +-
 .../lambdadelta/basic_2/computation/lsx.ma    |  72 ++++-----
 .../basic_2/computation/lsx_alt.ma            |  74 ++++-----
 .../basic_2/computation/lsx_csx.ma            |  18 +--
 .../basic_2/computation/lsx_drop.ma           |  54 +++----
 .../basic_2/computation/lsx_lpx.ma            |  28 ++--
 .../basic_2/computation/lsx_lpxs.ma           |  28 ++--
 .../lambdadelta/basic_2/computation/scpds.ma  |  24 +--
 .../basic_2/computation/scpds_aaa.ma          |   4 +-
 .../basic_2/computation/scpds_lift.ma         |  10 +-
 .../basic_2/computation/scpds_scpds.ma        |  50 +++----
 .../basic_2/etc_new/drops/drops.etc           |  24 ---
 .../lambdadelta/basic_2/relocation/drops.ma   |  20 +++
 .../lambdadelta/basic_2/relocation/lifts.ma   |   6 +-
 .../{computation => s_computation}/fqup.ma    |  37 +----
 .../basic_2/s_computation/fqup_drops.ma       |  46 ++++++
 .../fqup_fqup.ma                              |   2 +-
 .../basic_2/s_computation/fqup_weight.ma      |  42 ++++++
 .../{computation => s_computation}/fqus.ma    |  30 +---
 .../basic_2/s_computation/fqus_drops.ma       |  26 ++++
 .../fqus_fqup.ma}                             |  42 ++++--
 .../fqus_fqus.ma                              |   2 +-
 .../basic_2/s_computation/fqus_weight.ma      |  25 ++++
 .../basic_2/{relocation => static}/frees.ma   |   0
 .../{relocation => static}/frees_frees.ma     |   0
 .../{relocation => static}/frees_lreq.ma      |   0
 .../{relocation => static}/frees_weight.ma    |   0
 .../basic_2/{relocation => static}/freq.ma    |   0
 .../{relocation => static}/freq_freq.ma       |   0
 .../lambdadelta/basic_2/web/basic_2.ldw.xml   |   6 +
 .../lambdadelta/basic_2/web/basic_2_src.tbl   |  10 +-
 .../ground_2/relocation/rtmap_after.ma        |  18 +++
 .../lambdadelta/ground_2/web/ground_2.ldw.xml |   1 +
 83 files changed, 1250 insertions(+), 1138 deletions(-)
 rename matita/matita/contribs/lambdadelta/basic_2/{computation => s_computation}/fqup.ma (69%)
 create mode 100644 matita/matita/contribs/lambdadelta/basic_2/s_computation/fqup_drops.ma
 rename matita/matita/contribs/lambdadelta/basic_2/{computation => s_computation}/fqup_fqup.ma (96%)
 create mode 100644 matita/matita/contribs/lambdadelta/basic_2/s_computation/fqup_weight.ma
 rename matita/matita/contribs/lambdadelta/basic_2/{computation => s_computation}/fqus.ma (67%)
 create mode 100644 matita/matita/contribs/lambdadelta/basic_2/s_computation/fqus_drops.ma
 rename matita/matita/contribs/lambdadelta/basic_2/{computation/fqus_alt.ma => s_computation/fqus_fqup.ma} (61%)
 rename matita/matita/contribs/lambdadelta/basic_2/{computation => s_computation}/fqus_fqus.ma (96%)
 create mode 100644 matita/matita/contribs/lambdadelta/basic_2/s_computation/fqus_weight.ma
 rename matita/matita/contribs/lambdadelta/basic_2/{relocation => static}/frees.ma (100%)
 rename matita/matita/contribs/lambdadelta/basic_2/{relocation => static}/frees_frees.ma (100%)
 rename matita/matita/contribs/lambdadelta/basic_2/{relocation => static}/frees_lreq.ma (100%)
 rename matita/matita/contribs/lambdadelta/basic_2/{relocation => static}/frees_weight.ma (100%)
 rename matita/matita/contribs/lambdadelta/basic_2/{relocation => static}/freq.ma (100%)
 rename matita/matita/contribs/lambdadelta/basic_2/{relocation => static}/freq_freq.ma (100%)

diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/cpre.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/cpre.ma
index 65a6146a7..d94fb6fba 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/cpre.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/cpre.ma
@@ -27,8 +27,8 @@ interpretation "evaluation for context-sensitive parallel reduction (term)"
 (* Basic properties *********************************************************)
 
 (* Basic_1: was just: nf2_sn3 *)
-lemma csx_cpre: ∀h,g,G,L,T1. ⦃G, L⦄ ⊢ ⬊*[h, g] T1 → ∃T2. ⦃G, L⦄ ⊢ T1 ➡* 𝐍⦃T2⦄.
-#h #g #G #L #T1 #H @(csx_ind … H) -T1
+lemma csx_cpre: ∀h,o,G,L,T1. ⦃G, L⦄ ⊢ ⬊*[h, o] T1 → ∃T2. ⦃G, L⦄ ⊢ T1 ➡* 𝐍⦃T2⦄.
+#h #o #G #L #T1 #H @(csx_ind … H) -T1
 #T1 #_ #IHT1 elim (cnr_dec G L T1) /3 width=3 by ex_intro, conj/
 * #T #H1T1 #H2T1 elim (IHT1 … H2T1) -IHT1 -H2T1 /2 width=2 by cpr_cpx/
 #T2 * /4 width=3 by cprs_strap2, ex_intro, conj/
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/cprs.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/cprs.ma
index 6881d9c92..638bcde72 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/cprs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/cprs.ma
@@ -113,8 +113,8 @@ qed.
 (* Basic inversion lemmas ***************************************************)
 
 (* Basic_1: was: pr3_gen_sort *)
-lemma cprs_inv_sort1: ∀G,L,U2,k. ⦃G, L⦄ ⊢ ⋆k ➡* U2 → U2 = ⋆k.
-#G #L #U2 #k #H @(cprs_ind … H) -U2 //
+lemma cprs_inv_sort1: ∀G,L,U2,s. ⦃G, L⦄ ⊢ ⋆s ➡* U2 → U2 = ⋆s.
+#G #L #U2 #s #H @(cprs_ind … H) -U2 //
 #U2 #U #_ #HU2 #IHU2 destruct
 >(cpr_inv_sort1 … HU2) -HU2 //
 qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/cprs_cprs.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/cprs_cprs.ma
index e86df253e..a6d78f115 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/cprs_cprs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/cprs_cprs.ma
@@ -104,7 +104,7 @@ qed-.
 
 (* Basic_1: was just: pr3_pr2_pr2_t *)
 (* Basic_1: includes: pr3_pr0_pr2_t *)
-lemma lpr_cpr_trans: ∀G. s_r_transitive … (cpr G) (λ_. lpr G).
+lemma lpr_cpr_trans: ∀G. c_r_transitive … (cpr G) (λ_. lpr G).
 #G #L2 #T1 #T2 #HT12 elim HT12 -G -L2 -T1 -T2
 [ /2 width=3 by/
 | #G #L2 #K2 #V0 #V2 #W2 #i #HLK2 #_ #HVW2 #IHV02 #L1 #HL12
@@ -124,8 +124,8 @@ lemma cpr_bind2: ∀G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡ V2 → ∀I,T1,T2. ⦃G, L.
 (* Advanced properties ******************************************************)
 
 (* Basic_1: was only: pr3_pr2_pr3_t pr3_wcpr0_t *)
-lemma lpr_cprs_trans: ∀G. s_rs_transitive … (cpr G) (λ_. lpr G).
-#G @s_r_trans_LTC1 /2 width=3 by lpr_cpr_trans/ (**) (* full auto fails *)
+lemma lpr_cprs_trans: ∀G. c_rs_transitive … (cpr G) (λ_. lpr G).
+#G @c_r_trans_LTC1 /2 width=3 by lpr_cpr_trans/ (**) (* full auto fails *)
 qed-.
 
 (* Basic_1: was: pr3_strip *)
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/cpxe.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/cpxe.ma
index 6d1ffd2b8..00fd04247 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/cpxe.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/cpxe.ma
@@ -19,16 +19,16 @@ include "basic_2/computation/csx.ma".
 (* EVALUATION FOR CONTEXT-SENSITIVE EXTENDED PARALLEL REDUCTION ON TERMS ****)
 
 definition cpxe: ∀h. sd h → relation4 genv lenv term term ≝
-                 λh,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 ∧ ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃T2⦄.
+                 λh,o,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h, o] T2 ∧ ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃T2⦄.
 
 interpretation "evaluation for context-sensitive extended parallel reduction (term)"
-   'PRedEval h g G L T1 T2 = (cpxe h g G L T1 T2).
+   'PRedEval h o G L T1 T2 = (cpxe h o G L T1 T2).
 
 (* Basic properties *********************************************************)
 
-lemma csx_cpxe: ∀h,g,G,L,T1. ⦃G, L⦄ ⊢ ⬊*[h, g] T1 → ∃T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] 𝐍⦃T2⦄.
-#h #g #G #L #T1 #H @(csx_ind … H) -T1
-#T1 #_ #IHT1 elim (cnx_dec h g G L T1) /3 width=3 by ex_intro, conj/
+lemma csx_cpxe: ∀h,o,G,L,T1. ⦃G, L⦄ ⊢ ⬊*[h, o] T1 → ∃T2. ⦃G, L⦄ ⊢ T1 ➡*[h, o] 𝐍⦃T2⦄.
+#h #o #G #L #T1 #H @(csx_ind … H) -T1
+#T1 #_ #IHT1 elim (cnx_dec h o G L T1) /3 width=3 by ex_intro, conj/
 * #T #H1T1 #H2T1 elim (IHT1 … H1T1 H2T1) -IHT1 -H2T1
 #T2 * /4 width=3 by cpxs_strap2, ex_intro, conj/
 qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs.ma
index 75c56d8a4..734529a9e 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs.ma
@@ -19,127 +19,127 @@ include "basic_2/computation/cprs.ma".
 (* CONTEXT-SENSITIVE EXTENDED PARALLEL COMPUTATION ON TERMS *****************)
 
 definition cpxs: ∀h. sd h → relation4 genv lenv term term ≝
-                 λh,g,G. LTC … (cpx h g G).
+                 λh,o,G. LTC … (cpx h o G).
 
 interpretation "extended context-sensitive parallel computation (term)"
-   'PRedStar h g G L T1 T2 = (cpxs h g G L T1 T2).
+   'PRedStar h o G L T1 T2 = (cpxs h o G L T1 T2).
 
 (* Basic eliminators ********************************************************)
 
-lemma cpxs_ind: ∀h,g,G,L,T1. ∀R:predicate term. R T1 →
-                (∀T,T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T → ⦃G, L⦄ ⊢ T ➡[h, g] T2 → R T → R T2) →
-                ∀T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 → R T2.
-#h #g #L #G #T1 #R #HT1 #IHT1 #T2 #HT12
+lemma cpxs_ind: ∀h,o,G,L,T1. ∀R:predicate term. R T1 →
+                (∀T,T2. ⦃G, L⦄ ⊢ T1 ➡*[h, o] T → ⦃G, L⦄ ⊢ T ➡[h, o] T2 → R T → R T2) →
+                ∀T2. ⦃G, L⦄ ⊢ T1 ➡*[h, o] T2 → R T2.
+#h #o #L #G #T1 #R #HT1 #IHT1 #T2 #HT12
 @(TC_star_ind … HT1 IHT1 … HT12) //
 qed-.
 
-lemma cpxs_ind_dx: ∀h,g,G,L,T2. ∀R:predicate term. R T2 →
-                   (∀T1,T. ⦃G, L⦄ ⊢ T1 ➡[h, g] T → ⦃G, L⦄ ⊢ T ➡*[h, g] T2 → R T → R T1) →
-                   ∀T1. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 → R T1.
-#h #g #G #L #T2 #R #HT2 #IHT2 #T1 #HT12
+lemma cpxs_ind_dx: ∀h,o,G,L,T2. ∀R:predicate term. R T2 →
+                   (∀T1,T. ⦃G, L⦄ ⊢ T1 ➡[h, o] T → ⦃G, L⦄ ⊢ T ➡*[h, o] T2 → R T → R T1) →
+                   ∀T1. ⦃G, L⦄ ⊢ T1 ➡*[h, o] T2 → R T1.
+#h #o #G #L #T2 #R #HT2 #IHT2 #T1 #HT12
 @(TC_star_ind_dx … HT2 IHT2 … HT12) //
 qed-.
 
 (* Basic properties *********************************************************)
 
-lemma cpxs_refl: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ T ➡*[h, g] T.
+lemma cpxs_refl: ∀h,o,G,L,T. ⦃G, L⦄ ⊢ T ➡*[h, o] T.
 /2 width=1 by inj/ qed.
 
-lemma cpx_cpxs: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 → ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2.
+lemma cpx_cpxs: ∀h,o,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h, o] T2 → ⦃G, L⦄ ⊢ T1 ➡*[h, o] T2.
 /2 width=1 by inj/ qed.
 
-lemma cpxs_strap1: ∀h,g,G,L,T1,T. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T →
-                   ∀T2. ⦃G, L⦄ ⊢ T ➡[h, g] T2 → ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2.
+lemma cpxs_strap1: ∀h,o,G,L,T1,T. ⦃G, L⦄ ⊢ T1 ➡*[h, o] T →
+                   ∀T2. ⦃G, L⦄ ⊢ T ➡[h, o] T2 → ⦃G, L⦄ ⊢ T1 ➡*[h, o] T2.
 normalize /2 width=3 by step/ qed.
 
-lemma cpxs_strap2: ∀h,g,G,L,T1,T. ⦃G, L⦄ ⊢ T1 ➡[h, g] T →
-                   ∀T2. ⦃G, L⦄ ⊢ T ➡*[h, g] T2 → ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2.
+lemma cpxs_strap2: ∀h,o,G,L,T1,T. ⦃G, L⦄ ⊢ T1 ➡[h, o] T →
+                   ∀T2. ⦃G, L⦄ ⊢ T ➡*[h, o] T2 → ⦃G, L⦄ ⊢ T1 ➡*[h, o] T2.
 normalize /2 width=3 by TC_strap/ qed.
 
-lemma lsubr_cpxs_trans: ∀h,g,G. lsub_trans … (cpxs h g G) lsubr.
+lemma lsubr_cpxs_trans: ∀h,o,G. lsub_trans … (cpxs h o G) lsubr.
 /3 width=5 by lsubr_cpx_trans, LTC_lsub_trans/
 qed-.
 
-lemma cprs_cpxs: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡* T2 → ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2.
-#h #g #G #L #T1 #T2 #H @(cprs_ind … H) -T2 /3 width=3 by cpxs_strap1, cpr_cpx/
+lemma cprs_cpxs: ∀h,o,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡* T2 → ⦃G, L⦄ ⊢ T1 ➡*[h, o] T2.
+#h #o #G #L #T1 #T2 #H @(cprs_ind … H) -T2 /3 width=3 by cpxs_strap1, cpr_cpx/
 qed.
 
-lemma cpxs_sort: ∀h,g,G,L,k,d1. deg h g k d1 →
-                 ∀d2. d2 ≤ d1 → ⦃G, L⦄ ⊢ ⋆k ➡*[h, g] ⋆((next h)^d2 k).
-#h #g #G #L #k #d1 #Hkd1 #d2 @(nat_ind_plus … d2) -d2 /2 width=1 by cpx_cpxs/
+lemma cpxs_sort: ∀h,o,G,L,s,d1. deg h o s d1 →
+                 ∀d2. d2 ≤ d1 → ⦃G, L⦄ ⊢ ⋆s ➡*[h, o] ⋆((next h)^d2 s).
+#h #o #G #L #s #d1 #Hkd1 #d2 @(nat_ind_plus … d2) -d2 /2 width=1 by cpx_cpxs/
 #d2 #IHd2 #Hd21 >iter_SO
-@(cpxs_strap1 … (⋆(iter d2 ℕ (next h) k)))
+@(cpxs_strap1 … (⋆(iter d2 ℕ (next h) s)))
 [ /3 width=3 by lt_to_le/
-| @(cpx_st … (d1-d2-1)) <plus_minus_m_m
+| @(cpx_st … (d1-d2-1)) <plus_minus_k_k
   /2 width=1 by deg_iter, monotonic_le_minus_r/
 ]
 qed.
 
-lemma cpxs_bind_dx: ∀h,g,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 →
-                    ∀I,T1,T2. ⦃G, L. ⓑ{I}V1⦄ ⊢ T1 ➡*[h, g] T2 →
-                    ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡*[h, g] ⓑ{a,I}V2.T2.
-#h #g #G #L #V1 #V2 #HV12 #I #T1 #T2 #HT12 #a @(cpxs_ind_dx … HT12) -T1
+lemma cpxs_bind_dx: ∀h,o,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h, o] V2 →
+                    ∀I,T1,T2. ⦃G, L. ⓑ{I}V1⦄ ⊢ T1 ➡*[h, o] T2 →
+                    ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡*[h, o] ⓑ{a,I}V2.T2.
+#h #o #G #L #V1 #V2 #HV12 #I #T1 #T2 #HT12 #a @(cpxs_ind_dx … HT12) -T1
 /3 width=3 by cpxs_strap2, cpx_cpxs, cpx_pair_sn, cpx_bind/
 qed.
 
-lemma cpxs_flat_dx: ∀h,g,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 →
-                    ∀T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 →
-                    ∀I. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ➡*[h, g] ⓕ{I}V2.T2.
-#h #g #G #L #V1 #V2 #HV12 #T1 #T2 #HT12 @(cpxs_ind … HT12) -T2
+lemma cpxs_flat_dx: ∀h,o,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h, o] V2 →
+                    ∀T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h, o] T2 →
+                    ∀I. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ➡*[h, o] ⓕ{I}V2.T2.
+#h #o #G #L #V1 #V2 #HV12 #T1 #T2 #HT12 @(cpxs_ind … HT12) -T2
 /3 width=5 by cpxs_strap1, cpx_cpxs, cpx_pair_sn, cpx_flat/
 qed.
 
-lemma cpxs_flat_sn: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 →
-                    ∀V1,V2. ⦃G, L⦄ ⊢ V1 ➡*[h, g] V2 →
-                    ∀I. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ➡*[h, g] ⓕ{I}V2.T2.
-#h #g #G #L #T1 #T2 #HT12 #V1 #V2 #H @(cpxs_ind … H) -V2
+lemma cpxs_flat_sn: ∀h,o,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h, o] T2 →
+                    ∀V1,V2. ⦃G, L⦄ ⊢ V1 ➡*[h, o] V2 →
+                    ∀I. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ➡*[h, o] ⓕ{I}V2.T2.
+#h #o #G #L #T1 #T2 #HT12 #V1 #V2 #H @(cpxs_ind … H) -V2
 /3 width=5 by cpxs_strap1, cpx_cpxs, cpx_pair_sn, cpx_flat/
 qed.
 
-lemma cpxs_pair_sn: ∀h,g,I,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡*[h, g] V2 →
-                    ∀T. ⦃G, L⦄ ⊢ ②{I}V1.T ➡*[h, g] ②{I}V2.T.
-#h #g #I #G #L #V1 #V2 #H @(cpxs_ind … H) -V2
+lemma cpxs_pair_sn: ∀h,o,I,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡*[h, o] V2 →
+                    ∀T. ⦃G, L⦄ ⊢ ②{I}V1.T ➡*[h, o] ②{I}V2.T.
+#h #o #I #G #L #V1 #V2 #H @(cpxs_ind … H) -V2
 /3 width=3 by cpxs_strap1, cpx_pair_sn/
 qed.
 
-lemma cpxs_zeta: ∀h,g,G,L,V,T1,T,T2. ⬆[0, 1] T2 ≡ T →
-                 ⦃G, L.ⓓV⦄ ⊢ T1 ➡*[h, g] T → ⦃G, L⦄ ⊢ +ⓓV.T1 ➡*[h, g] T2.
-#h #g #G #L #V #T1 #T #T2 #HT2 #H @(cpxs_ind_dx … H) -T1
+lemma cpxs_zeta: ∀h,o,G,L,V,T1,T,T2. ⬆[0, 1] T2 ≡ T →
+                 ⦃G, L.ⓓV⦄ ⊢ T1 ➡*[h, o] T → ⦃G, L⦄ ⊢ +ⓓV.T1 ➡*[h, o] T2.
+#h #o #G #L #V #T1 #T #T2 #HT2 #H @(cpxs_ind_dx … H) -T1
 /3 width=3 by cpxs_strap2, cpx_cpxs, cpx_bind, cpx_zeta/
 qed.
 
-lemma cpxs_eps: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 →
-                ∀V. ⦃G, L⦄ ⊢ ⓝV.T1 ➡*[h, g] T2.
-#h #g #G #L #T1 #T2 #H @(cpxs_ind … H) -T2
+lemma cpxs_eps: ∀h,o,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h, o] T2 →
+                ∀V. ⦃G, L⦄ ⊢ ⓝV.T1 ➡*[h, o] T2.
+#h #o #G #L #T1 #T2 #H @(cpxs_ind … H) -T2
 /3 width=3 by cpxs_strap1, cpx_cpxs, cpx_eps/
 qed.
 
-lemma cpxs_ct: ∀h,g,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡*[h, g] V2 →
-               ∀T. ⦃G, L⦄ ⊢ ⓝV1.T ➡*[h, g] V2.
-#h #g #G #L #V1 #V2 #H @(cpxs_ind … H) -V2
+lemma cpxs_ct: ∀h,o,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡*[h, o] V2 →
+               ∀T. ⦃G, L⦄ ⊢ ⓝV1.T ➡*[h, o] V2.
+#h #o #G #L #V1 #V2 #H @(cpxs_ind … H) -V2
 /3 width=3 by cpxs_strap1, cpx_cpxs, cpx_ct/
 qed.
 
-lemma cpxs_beta_dx: ∀h,g,a,G,L,V1,V2,W1,W2,T1,T2.
-                    ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 → ⦃G, L.ⓛW1⦄ ⊢ T1 ➡*[h, g] T2 → ⦃G, L⦄ ⊢ W1 ➡[h, g] W2 →
-                    ⦃G, L⦄ ⊢ ⓐV1.ⓛ{a}W1.T1 ➡*[h, g] ⓓ{a}ⓝW2.V2.T2.
-#h #g #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 * -T2
+lemma cpxs_beta_dx: ∀h,o,a,G,L,V1,V2,W1,W2,T1,T2.
+                    ⦃G, L⦄ ⊢ V1 ➡[h, o] V2 → ⦃G, L.ⓛW1⦄ ⊢ T1 ➡*[h, o] T2 → ⦃G, L⦄ ⊢ W1 ➡[h, o] W2 →
+                    ⦃G, L⦄ ⊢ ⓐV1.ⓛ{a}W1.T1 ➡*[h, o] ⓓ{a}ⓝW2.V2.T2.
+#h #o #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 * -T2
 /4 width=7 by cpx_cpxs, cpxs_strap1, cpxs_bind_dx, cpxs_flat_dx, cpx_beta/
 qed.
 
-lemma cpxs_theta_dx: ∀h,g,a,G,L,V1,V,V2,W1,W2,T1,T2.
-                     ⦃G, L⦄ ⊢ V1 ➡[h, g] V → ⬆[0, 1] V ≡ V2 → ⦃G, L.ⓓW1⦄ ⊢ T1 ➡*[h, g] T2 →
-                     ⦃G, L⦄ ⊢ W1 ➡[h, g] W2 → ⦃G, L⦄ ⊢ ⓐV1.ⓓ{a}W1.T1 ➡*[h, g] ⓓ{a}W2.ⓐV2.T2.
-#h #g #a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 * -T2 
+lemma cpxs_theta_dx: ∀h,o,a,G,L,V1,V,V2,W1,W2,T1,T2.
+                     ⦃G, L⦄ ⊢ V1 ➡[h, o] V → ⬆[0, 1] V ≡ V2 → ⦃G, L.ⓓW1⦄ ⊢ T1 ➡*[h, o] T2 →
+                     ⦃G, L⦄ ⊢ W1 ➡[h, o] W2 → ⦃G, L⦄ ⊢ ⓐV1.ⓓ{a}W1.T1 ➡*[h, o] ⓓ{a}W2.ⓐV2.T2.
+#h #o #a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 * -T2 
 /4 width=9 by cpx_cpxs, cpxs_strap1, cpxs_bind_dx, cpxs_flat_dx, cpx_theta/
 qed.
 
 (* Basic inversion lemmas ***************************************************)
 
-lemma cpxs_inv_sort1: ∀h,g,G,L,U2,k. ⦃G, L⦄ ⊢ ⋆k ➡*[h, g] U2 →
-                      ∃∃n,d. deg h g k (n+d) & U2 = ⋆((next h)^n k).
-#h #g #G #L #U2 #k #H @(cpxs_ind … H) -U2
-[ elim (deg_total h g k) #d #Hkd
+lemma cpxs_inv_sort1: ∀h,o,G,L,U2,s. ⦃G, L⦄ ⊢ ⋆s ➡*[h, o] U2 →
+                      ∃∃n,d. deg h o s (n+d) & U2 = ⋆((next h)^n s).
+#h #o #G #L #U2 #s #H @(cpxs_ind … H) -U2
+[ elim (deg_total h o s) #d #Hkd
   @(ex2_2_intro … 0 … Hkd) -Hkd //
 | #U #U2 #_ #HU2 * #n #d #Hknd #H destruct
   elim (cpx_inv_sort1 … HU2) -HU2
@@ -150,11 +150,11 @@ lemma cpxs_inv_sort1: ∀h,g,G,L,U2,k. ⦃G, L⦄ ⊢ ⋆k ➡*[h, g] U2 →
 ]
 qed-.
 
-lemma cpxs_inv_cast1: ∀h,g,G,L,W1,T1,U2. ⦃G, L⦄ ⊢ ⓝW1.T1 ➡*[h, g] U2 →
-                      ∨∨ ∃∃W2,T2. ⦃G, L⦄ ⊢ W1 ➡*[h, g] W2 & ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 & U2 = ⓝW2.T2
-                       | ⦃G, L⦄ ⊢ T1 ➡*[h, g] U2
-                       | ⦃G, L⦄ ⊢ W1 ➡*[h, g] U2.
-#h #g #G #L #W1 #T1 #U2 #H @(cpxs_ind … H) -U2 /3 width=5 by or3_intro0, ex3_2_intro/
+lemma cpxs_inv_cast1: ∀h,o,G,L,W1,T1,U2. ⦃G, L⦄ ⊢ ⓝW1.T1 ➡*[h, o] U2 →
+                      ∨∨ ∃∃W2,T2. ⦃G, L⦄ ⊢ W1 ➡*[h, o] W2 & ⦃G, L⦄ ⊢ T1 ➡*[h, o] T2 & U2 = ⓝW2.T2
+                       | ⦃G, L⦄ ⊢ T1 ➡*[h, o] U2
+                       | ⦃G, L⦄ ⊢ W1 ➡*[h, o] U2.
+#h #o #G #L #W1 #T1 #U2 #H @(cpxs_ind … H) -U2 /3 width=5 by or3_intro0, ex3_2_intro/
 #U2 #U #_ #HU2 * /3 width=3 by cpxs_strap1, or3_intro1, or3_intro2/ *
 #W #T #HW1 #HT1 #H destruct
 elim (cpx_inv_cast1 … HU2) -HU2 /3 width=3 by cpxs_strap1, or3_intro1, or3_intro2/ *
@@ -163,15 +163,15 @@ lapply (cpxs_strap1 … HW1 … HW2) -W
 lapply (cpxs_strap1 … HT1 … HT2) -T /3 width=5 by or3_intro0, ex3_2_intro/
 qed-.
 
-lemma cpxs_inv_cnx1: ∀h,g,G,L,T,U. ⦃G, L⦄ ⊢ T ➡*[h, g] U → ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃T⦄ → T = U.
-#h #g #G #L #T #U #H @(cpxs_ind_dx … H) -T //
+lemma cpxs_inv_cnx1: ∀h,o,G,L,T,U. ⦃G, L⦄ ⊢ T ➡*[h, o] U → ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃T⦄ → T = U.
+#h #o #G #L #T #U #H @(cpxs_ind_dx … H) -T //
 #T0 #T #H1T0 #_ #IHT #H2T0
 lapply (H2T0 … H1T0) -H1T0 #H destruct /2 width=1 by/
 qed-.
 
-lemma cpxs_neq_inv_step_sn: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 → (T1 = T2 → ⊥) →
-                            ∃∃T. ⦃G, L⦄ ⊢ T1 ➡[h, g] T & T1 = T → ⊥ & ⦃G, L⦄ ⊢ T ➡*[h, g] T2.
-#h #g #G #L #T1 #T2 #H @(cpxs_ind_dx … H) -T1
+lemma cpxs_neq_inv_step_sn: ∀h,o,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h, o] T2 → (T1 = T2 → ⊥) →
+                            ∃∃T. ⦃G, L⦄ ⊢ T1 ➡[h, o] T & T1 = T → ⊥ & ⦃G, L⦄ ⊢ T ➡*[h, o] T2.
+#h #o #G #L #T1 #T2 #H @(cpxs_ind_dx … H) -T1
 [ #H elim H -H //
 | #T1 #T #H1 #H2 #IH2 #H12 elim (eq_term_dec T1 T) #H destruct
   [ -H1 -H2 /3 width=1 by/
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs_aaa.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs_aaa.ma
index e4afd5b50..e3d5e8af1 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs_aaa.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs_aaa.ma
@@ -19,9 +19,9 @@ include "basic_2/computation/cpxs.ma".
 
 (* Properties about atomic arity assignment on terms ************************)
 
-lemma cpxs_aaa_conf: ∀h,g,G,L,T1,A. ⦃G, L⦄ ⊢ T1 ⁝ A →
-                     ∀T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 → ⦃G, L⦄ ⊢ T2 ⁝ A.
-#h #g #G #L #T1 #A #HT1 #T2 #HT12
+lemma cpxs_aaa_conf: ∀h,o,G,L,T1,A. ⦃G, L⦄ ⊢ T1 ⁝ A →
+                     ∀T2. ⦃G, L⦄ ⊢ T1 ➡*[h, o] T2 → ⦃G, L⦄ ⊢ T2 ⁝ A.
+#h #o #G #L #T1 #A #HT1 #T2 #HT12
 @(TC_Conf3 … HT1 ? HT12) -A -T1 -T2 /2 width=5 by cpx_aaa_conf/
 qed-.
 
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs_cpxs.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs_cpxs.ma
index e6f9c381f..02436e7a4 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs_cpxs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs_cpxs.ma
@@ -19,62 +19,62 @@ include "basic_2/computation/cpxs_lift.ma".
 
 (* Main properties **********************************************************)
 
-theorem cpxs_trans: ∀h,g,G,L. Transitive … (cpxs h g G L).
+theorem cpxs_trans: ∀h,o,G,L. Transitive … (cpxs h o G L).
 normalize /2 width=3 by trans_TC/ qed-.
 
-theorem cpxs_bind: ∀h,g,a,I,G,L,V1,V2,T1,T2. ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ➡*[h, g] T2 →
-                   ⦃G, L⦄ ⊢ V1 ➡*[h, g] V2 →
-                   ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡*[h, g] ⓑ{a,I}V2.T2.
-#h #g #a #I #G #L #V1 #V2 #T1 #T2 #HT12 #H @(cpxs_ind … H) -V2
+theorem cpxs_bind: ∀h,o,a,I,G,L,V1,V2,T1,T2. ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ➡*[h, o] T2 →
+                   ⦃G, L⦄ ⊢ V1 ➡*[h, o] V2 →
+                   ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡*[h, o] ⓑ{a,I}V2.T2.
+#h #o #a #I #G #L #V1 #V2 #T1 #T2 #HT12 #H @(cpxs_ind … H) -V2
 /3 width=5 by cpxs_trans, cpxs_bind_dx/
 qed.
 
-theorem cpxs_flat: ∀h,g,I,G,L,V1,V2,T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 →
-                   ⦃G, L⦄ ⊢ V1 ➡*[h, g] V2 →
-                   ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ➡*[h, g] ⓕ{I}V2.T2.
-#h #g #I #G #L #V1 #V2 #T1 #T2 #HT12 #H @(cpxs_ind … H) -V2
+theorem cpxs_flat: ∀h,o,I,G,L,V1,V2,T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h, o] T2 →
+                   ⦃G, L⦄ ⊢ V1 ➡*[h, o] V2 →
+                   ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ➡*[h, o] ⓕ{I}V2.T2.
+#h #o #I #G #L #V1 #V2 #T1 #T2 #HT12 #H @(cpxs_ind … H) -V2
 /3 width=5 by cpxs_trans, cpxs_flat_dx/
 qed.
 
-theorem cpxs_beta_rc: ∀h,g,a,G,L,V1,V2,W1,W2,T1,T2.
-                      ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 → ⦃G, L.ⓛW1⦄ ⊢ T1 ➡*[h, g] T2 → ⦃G, L⦄ ⊢ W1 ➡*[h, g] W2 →
-                      ⦃G, L⦄ ⊢ ⓐV1.ⓛ{a}W1.T1 ➡*[h, g] ⓓ{a}ⓝW2.V2.T2.
-#h #g #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 #HT12 #H @(cpxs_ind … H) -W2
+theorem cpxs_beta_rc: ∀h,o,a,G,L,V1,V2,W1,W2,T1,T2.
+                      ⦃G, L⦄ ⊢ V1 ➡[h, o] V2 → ⦃G, L.ⓛW1⦄ ⊢ T1 ➡*[h, o] T2 → ⦃G, L⦄ ⊢ W1 ➡*[h, o] W2 →
+                      ⦃G, L⦄ ⊢ ⓐV1.ⓛ{a}W1.T1 ➡*[h, o] ⓓ{a}ⓝW2.V2.T2.
+#h #o #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 #HT12 #H @(cpxs_ind … H) -W2
 /4 width=5 by cpxs_trans, cpxs_beta_dx, cpxs_bind_dx, cpx_pair_sn/
 qed.
 
-theorem cpxs_beta: ∀h,g,a,G,L,V1,V2,W1,W2,T1,T2.
-                   ⦃G, L.ⓛW1⦄ ⊢ T1 ➡*[h, g] T2 → ⦃G, L⦄ ⊢ W1 ➡*[h, g] W2 → ⦃G, L⦄ ⊢ V1 ➡*[h, g] V2 →
-                   ⦃G, L⦄ ⊢ ⓐV1.ⓛ{a}W1.T1 ➡*[h, g] ⓓ{a}ⓝW2.V2.T2.
-#h #g #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HT12 #HW12 #H @(cpxs_ind … H) -V2
+theorem cpxs_beta: ∀h,o,a,G,L,V1,V2,W1,W2,T1,T2.
+                   ⦃G, L.ⓛW1⦄ ⊢ T1 ➡*[h, o] T2 → ⦃G, L⦄ ⊢ W1 ➡*[h, o] W2 → ⦃G, L⦄ ⊢ V1 ➡*[h, o] V2 →
+                   ⦃G, L⦄ ⊢ ⓐV1.ⓛ{a}W1.T1 ➡*[h, o] ⓓ{a}ⓝW2.V2.T2.
+#h #o #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HT12 #HW12 #H @(cpxs_ind … H) -V2
 /4 width=5 by cpxs_trans, cpxs_beta_rc, cpxs_bind_dx, cpx_flat/
 qed.
 
-theorem cpxs_theta_rc: ∀h,g,a,G,L,V1,V,V2,W1,W2,T1,T2.
-                       ⦃G, L⦄ ⊢ V1 ➡[h, g] V → ⬆[0, 1] V ≡ V2 →
-                       ⦃G, L.ⓓW1⦄ ⊢ T1 ➡*[h, g] T2 → ⦃G, L⦄ ⊢ W1 ➡*[h, g] W2 →
-                       ⦃G, L⦄ ⊢ ⓐV1.ⓓ{a}W1.T1 ➡*[h, g] ⓓ{a}W2.ⓐV2.T2.
-#h #g #a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 #HT12 #H @(cpxs_ind … H) -W2
+theorem cpxs_theta_rc: ∀h,o,a,G,L,V1,V,V2,W1,W2,T1,T2.
+                       ⦃G, L⦄ ⊢ V1 ➡[h, o] V → ⬆[0, 1] V ≡ V2 →
+                       ⦃G, L.ⓓW1⦄ ⊢ T1 ➡*[h, o] T2 → ⦃G, L⦄ ⊢ W1 ➡*[h, o] W2 →
+                       ⦃G, L⦄ ⊢ ⓐV1.ⓓ{a}W1.T1 ➡*[h, o] ⓓ{a}W2.ⓐV2.T2.
+#h #o #a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 #HT12 #H @(cpxs_ind … H) -W2
 /3 width=5 by cpxs_trans, cpxs_theta_dx, cpxs_bind_dx/
 qed.
 
-theorem cpxs_theta: ∀h,g,a,G,L,V1,V,V2,W1,W2,T1,T2.
-                    ⬆[0, 1] V ≡ V2 → ⦃G, L⦄ ⊢ W1 ➡*[h, g] W2 →
-                    ⦃G, L.ⓓW1⦄ ⊢ T1 ➡*[h, g] T2 → ⦃G, L⦄ ⊢ V1 ➡*[h, g] V →
-                    ⦃G, L⦄ ⊢ ⓐV1.ⓓ{a}W1.T1 ➡*[h, g] ⓓ{a}W2.ⓐV2.T2.
-#h #g #a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV2 #HW12 #HT12 #H @(TC_ind_dx … V1 H) -V1
+theorem cpxs_theta: ∀h,o,a,G,L,V1,V,V2,W1,W2,T1,T2.
+                    ⬆[0, 1] V ≡ V2 → ⦃G, L⦄ ⊢ W1 ➡*[h, o] W2 →
+                    ⦃G, L.ⓓW1⦄ ⊢ T1 ➡*[h, o] T2 → ⦃G, L⦄ ⊢ V1 ➡*[h, o] V →
+                    ⦃G, L⦄ ⊢ ⓐV1.ⓓ{a}W1.T1 ➡*[h, o] ⓓ{a}W2.ⓐV2.T2.
+#h #o #a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV2 #HW12 #HT12 #H @(TC_ind_dx … V1 H) -V1
 /3 width=5 by cpxs_trans, cpxs_theta_rc, cpxs_flat_dx/
 qed.
 
 (* Advanced inversion lemmas ************************************************)
 
-lemma cpxs_inv_appl1: ∀h,g,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓐV1.T1 ➡*[h, g] U2 →
-                      ∨∨ ∃∃V2,T2.       ⦃G, L⦄ ⊢ V1 ➡*[h, g] V2 & ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 &
+lemma cpxs_inv_appl1: ∀h,o,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓐV1.T1 ➡*[h, o] U2 →
+                      ∨∨ ∃∃V2,T2.       ⦃G, L⦄ ⊢ V1 ➡*[h, o] V2 & ⦃G, L⦄ ⊢ T1 ➡*[h, o] T2 &
                                         U2 = ⓐV2. T2
-                       | ∃∃a,W,T.       ⦃G, L⦄ ⊢ T1 ➡*[h, g] ⓛ{a}W.T & ⦃G, L⦄ ⊢ ⓓ{a}ⓝW.V1.T ➡*[h, g] U2
-                       | ∃∃a,V0,V2,V,T. ⦃G, L⦄ ⊢ V1 ➡*[h, g] V0 & ⬆[0,1] V0 ≡ V2 &
-                                        ⦃G, L⦄ ⊢ T1 ➡*[h, g] ⓓ{a}V.T & ⦃G, L⦄ ⊢ ⓓ{a}V.ⓐV2.T ➡*[h, g] U2.
-#h #g #G #L #V1 #T1 #U2 #H @(cpxs_ind … H) -U2 [ /3 width=5 by or3_intro0, ex3_2_intro/ ]
+                       | ∃∃a,W,T.       ⦃G, L⦄ ⊢ T1 ➡*[h, o] ⓛ{a}W.T & ⦃G, L⦄ ⊢ ⓓ{a}ⓝW.V1.T ➡*[h, o] U2
+                       | ∃∃a,V0,V2,V,T. ⦃G, L⦄ ⊢ V1 ➡*[h, o] V0 & ⬆[0,1] V0 ≡ V2 &
+                                        ⦃G, L⦄ ⊢ T1 ➡*[h, o] ⓓ{a}V.T & ⦃G, L⦄ ⊢ ⓓ{a}V.ⓐV2.T ➡*[h, o] U2.
+#h #o #G #L #V1 #T1 #U2 #H @(cpxs_ind … H) -U2 [ /3 width=5 by or3_intro0, ex3_2_intro/ ]
 #U #U2 #_ #HU2 * *
 [ #V0 #T0 #HV10 #HT10 #H destruct
   elim (cpx_inv_appl1 … HU2) -HU2 *
@@ -93,8 +93,8 @@ qed-.
 
 (* Properties on sn extended parallel reduction for local environments ******)
 
-lemma lpx_cpx_trans: ∀h,g,G. s_r_transitive … (cpx h g G) (λ_.lpx h g G).
-#h #g #G #L2 #T1 #T2 #HT12 elim HT12 -G -L2 -T1 -T2
+lemma lpx_cpx_trans: ∀h,o,G. c_r_transitive … (cpx h o G) (λ_.lpx h o G).
+#h #o #G #L2 #T1 #T2 #HT12 elim HT12 -G -L2 -T1 -T2
 [ /2 width=3 by/
 | /3 width=2 by cpx_cpxs, cpx_st/
 | #I #G #L2 #K2 #V0 #V2 #W2 #i #HLK2 #_ #HVW2 #IHV02 #L1 #HL12
@@ -108,28 +108,28 @@ lemma lpx_cpx_trans: ∀h,g,G. s_r_transitive … (cpx h g G) (λ_.lpx h g G).
 ]
 qed-.
 
-lemma cpx_bind2: ∀h,g,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 →
-                 ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ➡[h, g] T2 →
-                 ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡*[h, g] ⓑ{a,I}V2.T2.
+lemma cpx_bind2: ∀h,o,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h, o] V2 →
+                 ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ➡[h, o] T2 →
+                 ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡*[h, o] ⓑ{a,I}V2.T2.
 /4 width=5 by lpx_cpx_trans, cpxs_bind_dx, lpx_pair/ qed.
 
 (* Advanced properties ******************************************************)
 
-lemma lpx_cpxs_trans: ∀h,g,G. s_rs_transitive … (cpx h g G) (λ_.lpx h g G).
-#h #g #G @s_r_trans_LTC1 /2 width=3 by lpx_cpx_trans/ (**) (* full auto fails *)
+lemma lpx_cpxs_trans: ∀h,o,G. c_rs_transitive … (cpx h o G) (λ_.lpx h o G).
+#h #o #G @c_r_trans_LTC1 /2 width=3 by lpx_cpx_trans/ (**) (* full auto fails *)
 qed-.
 
-lemma cpxs_bind2_dx: ∀h,g,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 →
-                     ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ➡*[h, g] T2 →
-                     ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡*[h, g] ⓑ{a,I}V2.T2.
+lemma cpxs_bind2_dx: ∀h,o,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h, o] V2 →
+                     ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ➡*[h, o] T2 →
+                     ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡*[h, o] ⓑ{a,I}V2.T2.
 /4 width=5 by lpx_cpxs_trans, cpxs_bind_dx, lpx_pair/ qed.
 
 (* Properties on supclosure *************************************************)
 
-lemma fqu_cpxs_trans_neq: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
-                          ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡*[h, g] U2 → (T2 = U2 → ⊥) →
-                          ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐ ⦃G2, L2, U2⦄.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
+lemma fqu_cpxs_trans_neq: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
+                          ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡*[h, o] U2 → (T2 = U2 → ⊥) →
+                          ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡*[h, o] U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐ ⦃G2, L2, U2⦄.
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
 [ #I #G #L #V1 #V2 #HV12 #_ elim (lift_total V2 0 1)
   #U2 #HVU2 @(ex3_intro … U2)
   [1,3: /3 width=7 by fqu_drop, cpxs_delta, drop_pair, drop_drop/
@@ -148,27 +148,27 @@ lemma fqu_cpxs_trans_neq: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2,
   [1,3: /2 width=4 by fqu_flat_dx, cpxs_flat/
   | #H0 destruct /2 width=1 by/
   ]
-| #G #L #K #T1 #U1 #m #HLK #HTU1 #T2 #HT12 #H elim (lift_total T2 0 (m+1))
+| #G #L #K #T1 #U1 #k #HLK #HTU1 #T2 #HT12 #H elim (lift_total T2 0 (k+1))
   #U2 #HTU2 @(ex3_intro … U2)
   [1,3: /2 width=10 by cpxs_lift, fqu_drop/
   | #H0 destruct /3 width=5 by lift_inj/
 ]
 qed-.
 
-lemma fquq_cpxs_trans_neq: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
-                           ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡*[h, g] U2 → (T2 = U2 → ⊥) →
-                           ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐⸮ ⦃G2, L2, U2⦄.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H12 #U2 #HTU2 #H elim (fquq_inv_gen … H12) -H12
+lemma fquq_cpxs_trans_neq: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
+                           ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡*[h, o] U2 → (T2 = U2 → ⊥) →
+                           ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡*[h, o] U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐⸮ ⦃G2, L2, U2⦄.
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H12 #U2 #HTU2 #H elim (fquq_inv_gen … H12) -H12
 [ #H12 elim (fqu_cpxs_trans_neq … H12 … HTU2 H) -T2
   /3 width=4 by fqu_fquq, ex3_intro/
 | * #HG #HL #HT destruct /3 width=4 by ex3_intro/
 ]
 qed-.
 
-lemma fqup_cpxs_trans_neq: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
-                           ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡*[h, g] U2 → (T2 = U2 → ⊥) →
-                           ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐+ ⦃G2, L2, U2⦄.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind_dx … H) -G1 -L1 -T1
+lemma fqup_cpxs_trans_neq: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
+                           ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡*[h, o] U2 → (T2 = U2 → ⊥) →
+                           ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡*[h, o] U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐+ ⦃G2, L2, U2⦄.
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind_dx … H) -G1 -L1 -T1
 [ #G1 #L1 #T1 #H12 #U2 #HTU2 #H elim (fqu_cpxs_trans_neq … H12 … HTU2 H) -T2
   /3 width=4 by fqu_fqup, ex3_intro/
 | #G #G1 #L #L1 #T #T1 #H1 #_ #IH12 #U2 #HTU2 #H elim (IH12 … HTU2 H) -T2
@@ -177,10 +177,10 @@ lemma fqup_cpxs_trans_neq: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2
 ]
 qed-.
 
-lemma fqus_cpxs_trans_neq: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
-                           ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡*[h, g] U2 → (T2 = U2 → ⊥) →
-                           ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐* ⦃G2, L2, U2⦄.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H12 #U2 #HTU2 #H elim (fqus_inv_gen … H12) -H12
+lemma fqus_cpxs_trans_neq: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
+                           ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡*[h, o] U2 → (T2 = U2 → ⊥) →
+                           ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡*[h, o] U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐* ⦃G2, L2, U2⦄.
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H12 #U2 #HTU2 #H elim (fqus_inv_gen … H12) -H12
 [ #H12 elim (fqup_cpxs_trans_neq … H12 … HTU2 H) -T2
   /3 width=4 by fqup_fqus, ex3_intro/
 | * #HG #HL #HT destruct /3 width=4 by ex3_intro/
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs_lift.ma
index 494eee301..1d5a3e778 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs_lift.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs_lift.ma
@@ -20,19 +20,19 @@ include "basic_2/computation/cpxs.ma".
 
 (* Advanced properties ******************************************************)
 
-lemma cpxs_delta: ∀h,g,I,G,L,K,V,V2,i.
-                  ⬇[i] L ≡ K.ⓑ{I}V → ⦃G, K⦄ ⊢ V ➡*[h, g] V2 →
-                  ∀W2. ⬆[0, i+1] V2 ≡ W2 → ⦃G, L⦄ ⊢ #i ➡*[h, g] W2.
-#h #g #I #G #L #K #V #V2 #i #HLK #H elim H -V2
+lemma cpxs_delta: ∀h,o,I,G,L,K,V,V2,i.
+                  ⬇[i] L ≡ K.ⓑ{I}V → ⦃G, K⦄ ⊢ V ➡*[h, o] V2 →
+                  ∀W2. ⬆[0, i+1] V2 ≡ W2 → ⦃G, L⦄ ⊢ #i ➡*[h, o] W2.
+#h #o #I #G #L #K #V #V2 #i #HLK #H elim H -V2
 [ /3 width=9 by cpx_cpxs, cpx_delta/
 | #V1 lapply (drop_fwd_drop2 … HLK) -HLK
   elim (lift_total V1 0 (i+1)) /4 width=12 by cpx_lift, cpxs_strap1/
 ]
 qed.
 
-lemma lstas_cpxs: ∀h,g,G,L,T1,T2,d2. ⦃G, L⦄ ⊢ T1 •*[h, d2] T2 →
-                  ∀d1. ⦃G, L⦄ ⊢ T1 ▪[h, g] d1 → d2 ≤ d1 → ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2.
-#h #g #G #L #T1 #T2 #d2 #H elim H -G -L -T1 -T2 -d2 //
+lemma lstas_cpxs: ∀h,o,G,L,T1,T2,d2. ⦃G, L⦄ ⊢ T1 •*[h, d2] T2 →
+                  ∀d1. ⦃G, L⦄ ⊢ T1 ▪[h, o] d1 → d2 ≤ d1 → ⦃G, L⦄ ⊢ T1 ➡*[h, o] T2.
+#h #o #G #L #T1 #T2 #d2 #H elim H -G -L -T1 -T2 -d2 //
 [ /3 width=3 by cpxs_sort, da_inv_sort/
 | #G #L #K #V1 #V2 #W2 #i #d2 #HLK #_ #HVW2 #IHV12 #d1 #H #Hd21
   elim (da_inv_lref … H) -H * #K0 #V0 [| #d0 ] #HLK0
@@ -50,11 +50,11 @@ qed.
 
 (* Advanced inversion lemmas ************************************************)
 
-lemma cpxs_inv_lref1: ∀h,g,G,L,T2,i. ⦃G, L⦄ ⊢ #i ➡*[h, g] T2 →
+lemma cpxs_inv_lref1: ∀h,o,G,L,T2,i. ⦃G, L⦄ ⊢ #i ➡*[h, o] T2 →
                       T2 = #i ∨
-                      ∃∃I,K,V1,T1. ⬇[i] L ≡ K.ⓑ{I}V1 & ⦃G, K⦄ ⊢ V1 ➡*[h, g] T1 &
+                      ∃∃I,K,V1,T1. ⬇[i] L ≡ K.ⓑ{I}V1 & ⦃G, K⦄ ⊢ V1 ➡*[h, o] T1 &
                                    ⬆[0, i+1] T1 ≡ T2.
-#h #g #G #L #T2 #i #H @(cpxs_ind … H) -T2 /2 width=1 by or_introl/
+#h #o #G #L #T2 #i #H @(cpxs_ind … H) -T2 /2 width=1 by or_introl/
 #T #T2 #_ #HT2 *
 [ #H destruct
   elim (cpx_inv_lref1 … HT2) -HT2 /2 width=1 by or_introl/
@@ -68,57 +68,57 @@ qed-.
 
 (* Relocation properties ****************************************************)
 
-lemma cpxs_lift: ∀h,g,G. d_liftable (cpxs h g G).
+lemma cpxs_lift: ∀h,o,G. d_liftable (cpxs h o G).
 /3 width=10 by cpx_lift, cpxs_strap1, d_liftable_LTC/ qed.
 
-lemma cpxs_inv_lift1: ∀h,g,G. d_deliftable_sn (cpxs h g G).
+lemma cpxs_inv_lift1: ∀h,o,G. d_deliftable_sn (cpxs h o G).
 /3 width=6 by d_deliftable_sn_LTC, cpx_inv_lift1/
 qed-.
 
 (* Properties on supclosure *************************************************)
 
-lemma fqu_cpxs_trans: ∀h,g,G1,G2,L1,L2,T2,U2. ⦃G2, L2⦄ ⊢ T2 ➡*[h, g] U2 →
+lemma fqu_cpxs_trans: ∀h,o,G1,G2,L1,L2,T2,U2. ⦃G2, L2⦄ ⊢ T2 ➡*[h, o] U2 →
                       ∀T1. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
-                      ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] U1 & ⦃G1, L1, U1⦄ ⊐ ⦃G2, L2, U2⦄.
-#h #g #G1 #G2 #L1 #L2 #T2 #U2 #H @(cpxs_ind_dx … H) -T2 /2 width=3 by ex2_intro/
+                      ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡*[h, o] U1 & ⦃G1, L1, U1⦄ ⊐ ⦃G2, L2, U2⦄.
+#h #o #G1 #G2 #L1 #L2 #T2 #U2 #H @(cpxs_ind_dx … H) -T2 /2 width=3 by ex2_intro/
 #T #T2 #HT2 #_ #IHTU2 #T1 #HT1 elim (fqu_cpx_trans … HT1 … HT2) -T
 #T #HT1 #HT2 elim (IHTU2 … HT2) -T2 /3 width=3 by cpxs_strap2, ex2_intro/
 qed-.
 
-lemma fquq_cpxs_trans: ∀h,g,G1,G2,L1,L2,T2,U2. ⦃G2, L2⦄ ⊢ T2 ➡*[h, g] U2 →
+lemma fquq_cpxs_trans: ∀h,o,G1,G2,L1,L2,T2,U2. ⦃G2, L2⦄ ⊢ T2 ➡*[h, o] U2 →
                        ∀T1. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
-                       ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] U1 & ⦃G1, L1, U1⦄ ⊐⸮ ⦃G2, L2, U2⦄.
-#h #g #G1 #G2 #L1 #L2 #T2 #U2 #HTU2 #T1 #H elim (fquq_inv_gen … H) -H
+                       ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡*[h, o] U1 & ⦃G1, L1, U1⦄ ⊐⸮ ⦃G2, L2, U2⦄.
+#h #o #G1 #G2 #L1 #L2 #T2 #U2 #HTU2 #T1 #H elim (fquq_inv_gen … H) -H
 [ #HT12 elim (fqu_cpxs_trans … HTU2 … HT12) /3 width=3 by fqu_fquq, ex2_intro/
 | * #H1 #H2 #H3 destruct /2 width=3 by ex2_intro/
 ]
 qed-.
 
-lemma fquq_lstas_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
+lemma fquq_lstas_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
                         ∀U2,d1. ⦃G2, L2⦄ ⊢ T2 •*[h, d1] U2 →
-                        ∀d2. ⦃G2, L2⦄ ⊢ T2 ▪[h, g] d2 → d1 ≤ d2 →
-                        ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] U1 & ⦃G1, L1, U1⦄ ⊐⸮ ⦃G2, L2, U2⦄.
+                        ∀d2. ⦃G2, L2⦄ ⊢ T2 ▪[h, o] d2 → d1 ≤ d2 →
+                        ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡*[h, o] U1 & ⦃G1, L1, U1⦄ ⊐⸮ ⦃G2, L2, U2⦄.
 /3 width=5 by fquq_cpxs_trans, lstas_cpxs/ qed-.
 
-lemma fqup_cpxs_trans: ∀h,g,G1,G2,L1,L2,T2,U2. ⦃G2, L2⦄ ⊢ T2 ➡*[h, g] U2 →
+lemma fqup_cpxs_trans: ∀h,o,G1,G2,L1,L2,T2,U2. ⦃G2, L2⦄ ⊢ T2 ➡*[h, o] U2 →
                        ∀T1. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
-                       ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] U1 & ⦃G1, L1, U1⦄ ⊐+ ⦃G2, L2, U2⦄.
-#h #g #G1 #G2 #L1 #L2 #T2 #U2 #H @(cpxs_ind_dx … H) -T2 /2 width=3 by ex2_intro/
+                       ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡*[h, o] U1 & ⦃G1, L1, U1⦄ ⊐+ ⦃G2, L2, U2⦄.
+#h #o #G1 #G2 #L1 #L2 #T2 #U2 #H @(cpxs_ind_dx … H) -T2 /2 width=3 by ex2_intro/
 #T #T2 #HT2 #_ #IHTU2 #T1 #HT1 elim (fqup_cpx_trans … HT1 … HT2) -T
 #U1 #HTU1 #H2 elim (IHTU2 … H2) -T2 /3 width=3 by cpxs_strap2, ex2_intro/
 qed-.
 
-lemma fqus_cpxs_trans: ∀h,g,G1,G2,L1,L2,T2,U2. ⦃G2, L2⦄ ⊢ T2 ➡*[h, g] U2 →
+lemma fqus_cpxs_trans: ∀h,o,G1,G2,L1,L2,T2,U2. ⦃G2, L2⦄ ⊢ T2 ➡*[h, o] U2 →
                        ∀T1. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
-                       ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] U1 & ⦃G1, L1, U1⦄ ⊐* ⦃G2, L2, U2⦄.
-#h #g #G1 #G2 #L1 #L2 #T2 #U2 #HTU2 #T1 #H elim (fqus_inv_gen … H) -H
+                       ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡*[h, o] U1 & ⦃G1, L1, U1⦄ ⊐* ⦃G2, L2, U2⦄.
+#h #o #G1 #G2 #L1 #L2 #T2 #U2 #HTU2 #T1 #H elim (fqus_inv_gen … H) -H
 [ #HT12 elim (fqup_cpxs_trans … HTU2 … HT12) /3 width=3 by fqup_fqus, ex2_intro/
 | * #H1 #H2 #H3 destruct /2 width=3 by ex2_intro/
 ]
 qed-.
 
-lemma fqus_lstas_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
+lemma fqus_lstas_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
                         ∀U2,d1. ⦃G2, L2⦄ ⊢ T2 •*[h, d1] U2 →
-                        ∀d2. ⦃G2, L2⦄ ⊢ T2 ▪[h, g] d2 → d1 ≤ d2 →
-                        ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] U1 & ⦃G1, L1, U1⦄ ⊐* ⦃G2, L2, U2⦄.
+                        ∀d2. ⦃G2, L2⦄ ⊢ T2 ▪[h, o] d2 → d1 ≤ d2 →
+                        ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡*[h, o] U1 & ⦃G1, L1, U1⦄ ⊐* ⦃G2, L2, U2⦄.
 /3 width=6 by fqus_cpxs_trans, lstas_cpxs/ qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs_lleq.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs_lleq.ma
index 2bcff48e4..be3e0f0b5 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs_lleq.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs_lleq.ma
@@ -19,21 +19,21 @@ include "basic_2/computation/cpxs.ma".
 
 (* Properties on lazy equivalence for local environments ********************)
 
-lemma lleq_cpxs_trans: ∀h,g,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ➡*[h, g] T2 →
-                       ∀L1. L1 ≡[T1, 0] L2 → ⦃G, L1⦄ ⊢ T1 ➡*[h, g] T2.
-#h #g #G #L2 #T1 #T2 #H @(cpxs_ind_dx … H) -T1
+lemma lleq_cpxs_trans: ∀h,o,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ➡*[h, o] T2 →
+                       ∀L1. L1 ≡[T1, 0] L2 → ⦃G, L1⦄ ⊢ T1 ➡*[h, o] T2.
+#h #o #G #L2 #T1 #T2 #H @(cpxs_ind_dx … H) -T1
 /4 width=6 by cpx_lleq_conf_dx, lleq_cpx_trans, cpxs_strap2/
 qed-.
 
-lemma cpxs_lleq_conf: ∀h,g,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ➡*[h, g] T2 →
-                      ∀L1. L2 ≡[T1, 0] L1 → ⦃G, L1⦄ ⊢ T1 ➡*[h, g] T2.
+lemma cpxs_lleq_conf: ∀h,o,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ➡*[h, o] T2 →
+                      ∀L1. L2 ≡[T1, 0] L1 → ⦃G, L1⦄ ⊢ T1 ➡*[h, o] T2.
 /3 width=3 by lleq_cpxs_trans, lleq_sym/ qed-.
 
-lemma cpxs_lleq_conf_dx: ∀h,g,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ➡*[h, g] T2 →
+lemma cpxs_lleq_conf_dx: ∀h,o,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ➡*[h, o] T2 →
                          ∀L1. L1 ≡[T1, 0] L2 → L1 ≡[T2, 0] L2.
-#h #g #G #L2 #T1 #T2 #H @(cpxs_ind … H) -T2 /3 width=6 by cpx_lleq_conf_dx/
+#h #o #G #L2 #T1 #T2 #H @(cpxs_ind … H) -T2 /3 width=6 by cpx_lleq_conf_dx/
 qed-.
 
-lemma cpxs_lleq_conf_sn: ∀h,g,G,L1,T1,T2. ⦃G, L1⦄ ⊢ T1 ➡*[h, g] T2 →
+lemma cpxs_lleq_conf_sn: ∀h,o,G,L1,T1,T2. ⦃G, L1⦄ ⊢ T1 ➡*[h, o] T2 →
                          ∀L2. L1 ≡[T1, 0] L2 → L1 ≡[T2, 0] L2.
 /4 width=6 by cpxs_lleq_conf_dx, lleq_sym/ qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs_lreq.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs_lreq.ma
index 4abb0b91e..ec7a863c7 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs_lreq.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs_lreq.ma
@@ -19,6 +19,6 @@ include "basic_2/computation/cpxs.ma".
 
 (* Properties on equivalence for local environments *************************)
 
-lemma lreq_cpxs_trans: ∀h,g,G. lsub_trans … (cpxs h g G) (lreq 0 (∞)).
+lemma lreq_cpxs_trans: ∀h,o,G. lsub_trans … (cpxs h o G) (lreq 0 (∞)).
 /3 width=5 by lreq_cpx_trans, LTC_lsub_trans/
 qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs_tsts.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs_tsts.ma
index dbed5a334..ace2532d9 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs_tsts.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs_tsts.ma
@@ -19,17 +19,17 @@ include "basic_2/computation/lpxs_cpxs.ma".
 
 (* Forward lemmas involving same top term structure *************************)
 
-lemma cpxs_fwd_cnx: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃T⦄ → ∀U. ⦃G, L⦄ ⊢ T ➡*[h, g] U → T ≂ U.
-#h #g #G #L #T #HT #U #H
+lemma cpxs_fwd_cnx: ∀h,o,G,L,T. ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃T⦄ → ∀U. ⦃G, L⦄ ⊢ T ➡*[h, o] U → T ≂ U.
+#h #o #G #L #T #HT #U #H
 >(cpxs_inv_cnx1 … H HT) -G -L -T //
 qed-.
 
-lemma cpxs_fwd_sort: ∀h,g,G,L,U,k. ⦃G, L⦄ ⊢ ⋆k ➡*[h, g] U →
-                     ⋆k ≂ U ∨ ⦃G, L⦄ ⊢ ⋆(next h k) ➡*[h, g] U.
-#h #g #G #L #U #k #H
-elim (cpxs_inv_sort1 … H) -H #n #d generalize in match k; -k @(nat_ind_plus … n) -n
-[ #k #_ #H -d destruct /2 width=1 by or_introl/
-| #n #IHn #k >plus_plus_comm_23 #Hnd #H destruct
+lemma cpxs_fwd_sort: ∀h,o,G,L,U,s. ⦃G, L⦄ ⊢ ⋆s ➡*[h, o] U →
+                     ⋆s ≂ U ∨ ⦃G, L⦄ ⊢ ⋆(next h s) ➡*[h, o] U.
+#h #o #G #L #U #s #H
+elim (cpxs_inv_sort1 … H) -H #n #d generalize in match s; -s @(nat_ind_plus … n) -n
+[ #s #_ #H -d destruct /2 width=1 by or_introl/
+| #n #IHn #s >plus_plus_comm_23 #Hnd #H destruct
   lapply (deg_next_SO … Hnd) -Hnd #Hnd
   elim (IHn … Hnd) -IHn
   [ #H lapply (tsts_inv_atom1 … H) -H #H >H -H /2 width=1 by or_intror/
@@ -41,9 +41,9 @@ elim (cpxs_inv_sort1 … H) -H #n #d generalize in match k; -k @(nat_ind_plus 
 qed-.
 
 (* Basic_1: was just: pr3_iso_beta *)
-lemma cpxs_fwd_beta: ∀h,g,a,G,L,V,W,T,U. ⦃G, L⦄ ⊢ ⓐV.ⓛ{a}W.T ➡*[h, g] U →
-                     ⓐV.ⓛ{a}W.T ≂ U ∨ ⦃G, L⦄ ⊢ ⓓ{a}ⓝW.V.T ➡*[h, g] U.
-#h #g #a #G #L #V #W #T #U #H
+lemma cpxs_fwd_beta: ∀h,o,a,G,L,V,W,T,U. ⦃G, L⦄ ⊢ ⓐV.ⓛ{a}W.T ➡*[h, o] U →
+                     ⓐV.ⓛ{a}W.T ≂ U ∨ ⦃G, L⦄ ⊢ ⓓ{a}ⓝW.V.T ➡*[h, o] U.
+#h #o #a #G #L #V #W #T #U #H
 elim (cpxs_inv_appl1 … H) -H *
 [ #V0 #T0 #_ #_ #H destruct /2 width=1 by tsts_pair, or_introl/
 | #b #W0 #T0 #HT0 #HU
@@ -56,21 +56,21 @@ elim (cpxs_inv_appl1 … H) -H *
 qed-.
 
 (* Note: probably this is an inversion lemma *)
-lemma cpxs_fwd_delta: ∀h,g,I,G,L,K,V1,i. ⬇[i] L ≡ K.ⓑ{I}V1 →
+lemma cpxs_fwd_delta: ∀h,o,I,G,L,K,V1,i. ⬇[i] L ≡ K.ⓑ{I}V1 →
                       ∀V2. ⬆[0, i + 1] V1 ≡ V2 →
-                      ∀U. ⦃G, L⦄ ⊢ #i ➡*[h, g] U →
-                      #i ≂ U ∨ ⦃G, L⦄ ⊢ V2 ➡*[h, g] U.
-#h #g #I #G #L #K #V1 #i #HLK #V2 #HV12 #U #H
+                      ∀U. ⦃G, L⦄ ⊢ #i ➡*[h, o] U →
+                      #i ≂ U ∨ ⦃G, L⦄ ⊢ V2 ➡*[h, o] U.
+#h #o #I #G #L #K #V1 #i #HLK #V2 #HV12 #U #H
 elim (cpxs_inv_lref1 … H) -H /2 width=1 by or_introl/
 * #I0 #K0 #V0 #U0 #HLK0 #HVU0 #HU0
 lapply (drop_mono … HLK0 … HLK) -HLK0 #H destruct
 /4 width=10 by cpxs_lift, drop_fwd_drop2, or_intror/
 qed-.
 
-lemma cpxs_fwd_theta: ∀h,g,a,G,L,V1,V,T,U. ⦃G, L⦄ ⊢ ⓐV1.ⓓ{a}V.T ➡*[h, g] U →
+lemma cpxs_fwd_theta: ∀h,o,a,G,L,V1,V,T,U. ⦃G, L⦄ ⊢ ⓐV1.ⓓ{a}V.T ➡*[h, o] U →
                       ∀V2. ⬆[0, 1] V1 ≡ V2 → ⓐV1.ⓓ{a}V.T ≂ U ∨
-                      ⦃G, L⦄ ⊢ ⓓ{a}V.ⓐV2.T ➡*[h, g] U.
-#h #g #a #G #L #V1 #V #T #U #H #V2 #HV12
+                      ⦃G, L⦄ ⊢ ⓓ{a}V.ⓐV2.T ➡*[h, o] U.
+#h #o #a #G #L #V1 #V #T #U #H #V2 #HV12
 elim (cpxs_inv_appl1 … H) -H *
 [ -HV12 #V0 #T0 #_ #_ #H destruct /2 width=1 by tsts_pair, or_introl/
 | #b #W #T0 #HT0 #HU
@@ -99,9 +99,9 @@ elim (cpxs_inv_appl1 … H) -H *
 ]
 qed-.
 
-lemma cpxs_fwd_cast: ∀h,g,G,L,W,T,U. ⦃G, L⦄ ⊢ ⓝW.T ➡*[h, g] U →
-                     ∨∨ ⓝW. T ≂ U | ⦃G, L⦄ ⊢ T ➡*[h, g] U | ⦃G, L⦄ ⊢ W ➡*[h, g] U.
-#h #g #G #L #W #T #U #H
+lemma cpxs_fwd_cast: ∀h,o,G,L,W,T,U. ⦃G, L⦄ ⊢ ⓝW.T ➡*[h, o] U →
+                     ∨∨ ⓝW. T ≂ U | ⦃G, L⦄ ⊢ T ➡*[h, o] U | ⦃G, L⦄ ⊢ W ➡*[h, o] U.
+#h #o #G #L #W #T #U #H
 elim (cpxs_inv_cast1 … H) -H /2 width=1 by or3_intro1, or3_intro2/ *
 #W0 #T0 #_ #_ #H destruct /2 width=1 by tsts_pair, or3_intro0/
 qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs_tsts_vector.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs_tsts_vector.ma
index c26d55921..e82935af4 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs_tsts_vector.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs_tsts_vector.ma
@@ -21,9 +21,9 @@ include "basic_2/computation/cpxs_tsts.ma".
 (* Vector form of forward lemmas involving same top term structure **********)
 
 (* Basic_1: was just: nf2_iso_appls_lref *)
-lemma cpxs_fwd_cnx_vector: ∀h,g,G,L,T.  𝐒⦃T⦄ → ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃T⦄ →
-                           ∀Vs,U. ⦃G, L⦄ ⊢ ⒶVs.T ➡*[h, g] U → ⒶVs.T ≂ U.
-#h #g #G #L #T #H1T #H2T #Vs elim Vs -Vs [ @(cpxs_fwd_cnx … H2T) ] (**) (* /2 width=3 by cpxs_fwd_cnx/ does not work *)
+lemma cpxs_fwd_cnx_vector: ∀h,o,G,L,T.  𝐒⦃T⦄ → ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃T⦄ →
+                           ∀Vs,U. ⦃G, L⦄ ⊢ ⒶVs.T ➡*[h, o] U → ⒶVs.T ≂ U.
+#h #o #G #L #T #H1T #H2T #Vs elim Vs -Vs [ @(cpxs_fwd_cnx … H2T) ] (**) (* /2 width=3 by cpxs_fwd_cnx/ does not work *)
 #V #Vs #IHVs #U #H
 elim (cpxs_inv_appl1 … H) -H *
 [ -IHVs #V0 #T0 #_ #_ #H destruct /2 width=1 by tsts_pair/
@@ -36,9 +36,9 @@ elim (cpxs_inv_appl1 … H) -H *
 ]
 qed-.
 
-lemma cpxs_fwd_sort_vector: ∀h,g,G,L,k,Vs,U. ⦃G, L⦄ ⊢ ⒶVs.⋆k ➡*[h, g] U →
-                            ⒶVs.⋆k ≂ U ∨ ⦃G, L⦄ ⊢ ⒶVs.⋆(next h k) ➡*[h, g] U.
-#h #g #G #L #k #Vs elim Vs -Vs /2 width=1 by cpxs_fwd_sort/
+lemma cpxs_fwd_sort_vector: ∀h,o,G,L,s,Vs,U. ⦃G, L⦄ ⊢ ⒶVs.⋆s ➡*[h, o] U →
+                            ⒶVs.⋆s ≂ U ∨ ⦃G, L⦄ ⊢ ⒶVs.⋆(next h s) ➡*[h, o] U.
+#h #o #G #L #s #Vs elim Vs -Vs /2 width=1 by cpxs_fwd_sort/
 #V #Vs #IHVs #U #H
 elim (cpxs_inv_appl1 … H) -H *
 [ -IHVs #V1 #T1 #_ #_ #H destruct /2 width=1 by tsts_pair, or_introl/
@@ -61,9 +61,9 @@ qed-.
 
 
 (* Basic_1: was just: pr3_iso_appls_beta *)
-lemma cpxs_fwd_beta_vector: ∀h,g,a,G,L,Vs,V,W,T,U. ⦃G, L⦄ ⊢ ⒶVs.ⓐV.ⓛ{a}W.T ➡*[h, g] U →
-                            ⒶVs. ⓐV. ⓛ{a}W. T ≂ U ∨ ⦃G, L⦄ ⊢ ⒶVs.ⓓ{a}ⓝW.V.T ➡*[h, g] U.
-#h #g #a #G #L #Vs elim Vs -Vs /2 width=1 by cpxs_fwd_beta/
+lemma cpxs_fwd_beta_vector: ∀h,o,a,G,L,Vs,V,W,T,U. ⦃G, L⦄ ⊢ ⒶVs.ⓐV.ⓛ{a}W.T ➡*[h, o] U →
+                            ⒶVs. ⓐV. ⓛ{a}W. T ≂ U ∨ ⦃G, L⦄ ⊢ ⒶVs.ⓓ{a}ⓝW.V.T ➡*[h, o] U.
+#h #o #a #G #L #Vs elim Vs -Vs /2 width=1 by cpxs_fwd_beta/
 #V0 #Vs #IHVs #V #W #T #U #H
 elim (cpxs_inv_appl1 … H) -H *
 [ -IHVs #V1 #T1 #_ #_ #H destruct /2 width=1 by tsts_pair, or_introl/
@@ -84,11 +84,11 @@ elim (cpxs_inv_appl1 … H) -H *
 ]
 qed-.
 
-lemma cpxs_fwd_delta_vector: ∀h,g,I,G,L,K,V1,i. ⬇[i] L ≡ K.ⓑ{I}V1 →
+lemma cpxs_fwd_delta_vector: ∀h,o,I,G,L,K,V1,i. ⬇[i] L ≡ K.ⓑ{I}V1 →
                              ∀V2. ⬆[0, i + 1] V1 ≡ V2 →
-                             ∀Vs,U. ⦃G, L⦄ ⊢ ⒶVs.#i ➡*[h, g] U →
-                             ⒶVs.#i ≂ U ∨ ⦃G, L⦄ ⊢ ⒶVs.V2 ➡*[h, g] U.
-#h #g #I #G #L #K #V1 #i #HLK #V2 #HV12 #Vs elim Vs -Vs /2 width=5 by cpxs_fwd_delta/
+                             ∀Vs,U. ⦃G, L⦄ ⊢ ⒶVs.#i ➡*[h, o] U →
+                             ⒶVs.#i ≂ U ∨ ⦃G, L⦄ ⊢ ⒶVs.V2 ➡*[h, o] U.
+#h #o #I #G #L #K #V1 #i #HLK #V2 #HV12 #Vs elim Vs -Vs /2 width=5 by cpxs_fwd_delta/
 #V #Vs #IHVs #U #H -K -V1
 elim (cpxs_inv_appl1 … H) -H *
 [ -IHVs #V0 #T0 #_ #_ #H destruct /2 width=1 by tsts_pair, or_introl/
@@ -110,38 +110,38 @@ elim (cpxs_inv_appl1 … H) -H *
 qed-.
 
 (* Basic_1: was just: pr3_iso_appls_abbr *)
-lemma cpxs_fwd_theta_vector: ∀h,g,G,L,V1s,V2s. ⬆[0, 1] V1s ≡ V2s →
-                             ∀a,V,T,U. ⦃G, L⦄ ⊢ ⒶV1s.ⓓ{a}V.T ➡*[h, g] U →
-                             ⒶV1s. ⓓ{a}V. T ≂ U ∨ ⦃G, L⦄ ⊢ ⓓ{a}V.ⒶV2s.T ➡*[h, g] U.
-#h #g #G #L #V1s #V2s * -V1s -V2s /3 width=1 by or_intror/
-#V1s #V2s #V1a #V2a #HV12a #HV12s #a
+lemma cpxs_fwd_theta_vector: ∀h,o,G,L,V1c,V2c. ⬆[0, 1] V1c ≡ V2c →
+                             ∀a,V,T,U. ⦃G, L⦄ ⊢ ⒶV1c.ⓓ{a}V.T ➡*[h, o] U →
+                             ⒶV1c. ⓓ{a}V. T ≂ U ∨ ⦃G, L⦄ ⊢ ⓓ{a}V.ⒶV2c.T ➡*[h, o] U.
+#h #o #G #L #V1c #V2c * -V1c -V2c /3 width=1 by or_intror/
+#V1c #V2c #V1a #V2a #HV12a #HV12c #a
 generalize in match HV12a; -HV12a
 generalize in match V2a; -V2a
 generalize in match V1a; -V1a
-elim HV12s -V1s -V2s /2 width=1 by cpxs_fwd_theta/
-#V1s #V2s #V1b #V2b #HV12b #_ #IHV12s #V1a #V2a #HV12a #V #T #U #H
+elim HV12c -V1c -V2c /2 width=1 by cpxs_fwd_theta/
+#V1c #V2c #V1b #V2b #HV12b #_ #IHV12c #V1a #V2a #HV12a #V #T #U #H
 elim (cpxs_inv_appl1 … H) -H *
-[ -IHV12s -HV12a -HV12b #V0 #T0 #_ #_ #H destruct /2 width=1 by tsts_pair, or_introl/
+[ -IHV12c -HV12a -HV12b #V0 #T0 #_ #_ #H destruct /2 width=1 by tsts_pair, or_introl/
 | #b #W0 #T0 #HT0 #HU
-  elim (IHV12s … HV12b … HT0) -IHV12s -HT0 #HT0
+  elim (IHV12c … HV12b … HT0) -IHV12c -HT0 #HT0
   [ -HV12a -HV12b -HU
     elim (tsts_inv_pair1 … HT0) #V1 #T1 #H destruct
-  | @or_intror -V1s (**) (* explicit constructor *)
+  | @or_intror -V1c (**) (* explicit constructor *)
     @(cpxs_trans … HU) -U
     elim (cpxs_inv_abbr1 … HT0) -HT0 *
     [ -HV12a -HV12b #V1 #T1 #_ #_ #H destruct
     | -V1b #X #HT1 #H #H0 destruct
       elim (lift_inv_bind1 … H) -H #W1 #T1 #HW01 #HT01 #H destruct
-      @(cpxs_trans … (+ⓓV.ⓐV2a.ⓛ{b}W1.T1)) [ /3 width=1 by cpxs_flat_dx, cpxs_bind_dx/ ] -T -V2b -V2s
+      @(cpxs_trans … (+ⓓV.ⓐV2a.ⓛ{b}W1.T1)) [ /3 width=1 by cpxs_flat_dx, cpxs_bind_dx/ ] -T -V2b -V2c
       @(cpxs_strap2 … (ⓐV1a.ⓛ{b}W0.T0))
       /4 width=7 by cpxs_beta_dx, cpx_zeta, lift_bind, lift_flat/
     ]
   ]
 | #b #V0a #Va #V0 #T0 #HV10a #HV0a #HT0 #HU
-  elim (IHV12s … HV12b … HT0) -HV12b -IHV12s -HT0 #HT0
+  elim (IHV12c … HV12b … HT0) -HV12b -IHV12c -HT0 #HT0
   [ -HV12a -HV10a -HV0a -HU
     elim (tsts_inv_pair1 … HT0) #V1 #T1 #H destruct
-  | @or_intror -V1s -V1b (**) (* explicit constructor *)
+  | @or_intror -V1c -V1b (**) (* explicit constructor *)
     @(cpxs_trans … HU) -U
     elim (cpxs_inv_abbr1 … HT0) -HT0 *
     [ #V1 #T1 #HV1 #HT1 #H destruct
@@ -150,7 +150,7 @@ elim (cpxs_inv_appl1 … H) -H *
     | #X #HT1 #H #H0 destruct
       elim (lift_inv_bind1 … H) -H #V1 #T1 #HW01 #HT01 #H destruct
       lapply (cpxs_lift … HV10a (L.ⓓV0) (Ⓕ) … HV12a … HV0a) -V0a [ /2 width=1 by drop_drop/ ] #HV2a
-      @(cpxs_trans … (+ⓓV.ⓐV2a.ⓓ{b}V1.T1)) [ /3 width=1 by cpxs_flat_dx, cpxs_bind_dx/ ] -T -V2b -V2s
+      @(cpxs_trans … (+ⓓV.ⓐV2a.ⓓ{b}V1.T1)) [ /3 width=1 by cpxs_flat_dx, cpxs_bind_dx/ ] -T -V2b -V2c
       @(cpxs_strap2 … (ⓐV1a.ⓓ{b}V0.T0)) [ /4 width=7 by cpx_zeta, lift_bind, lift_flat/ ] -V -V1 -T1
       @(cpxs_strap2 … (ⓓ{b}V0.ⓐV2a.T0)) /3 width=3 by cpxs_pair_sn, cpxs_bind_dx, cpr_cpx, cpr_theta/
     ]
@@ -159,11 +159,11 @@ elim (cpxs_inv_appl1 … H) -H *
 qed-.
 
 (* Basic_1: was just: pr3_iso_appls_cast *)
-lemma cpxs_fwd_cast_vector: ∀h,g,G,L,Vs,W,T,U. ⦃G, L⦄ ⊢ ⒶVs.ⓝW.T ➡*[h, g] U →
+lemma cpxs_fwd_cast_vector: ∀h,o,G,L,Vs,W,T,U. ⦃G, L⦄ ⊢ ⒶVs.ⓝW.T ➡*[h, o] U →
                             ∨∨ ⒶVs. ⓝW. T ≂ U
-                             | ⦃G, L⦄ ⊢ ⒶVs.T ➡*[h, g] U
-                             | ⦃G, L⦄ ⊢ ⒶVs.W ➡*[h, g] U.
-#h #g #G #L #Vs elim Vs -Vs /2 width=1 by cpxs_fwd_cast/
+                             | ⦃G, L⦄ ⊢ ⒶVs.T ➡*[h, o] U
+                             | ⦃G, L⦄ ⊢ ⒶVs.W ➡*[h, o] U.
+#h #o #G #L #Vs elim Vs -Vs /2 width=1 by cpxs_fwd_cast/
 #V #Vs #IHVs #W #T #U #H
 elim (cpxs_inv_appl1 … H) -H *
 [ -IHVs #V0 #T0 #_ #_ #H destruct /2 width=1 by tsts_pair, or3_intro0/
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/csx.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/csx.ma
index 9b83b6084..0def99568 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/csx.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/csx.ma
@@ -18,46 +18,46 @@ include "basic_2/reduction/cnx.ma".
 (* CONTEXT-SENSITIVE EXTENDED STRONGLY NORMALIZING TERMS ********************)
 
 definition csx: ∀h. sd h → relation3 genv lenv term ≝
-                λh,g,G,L. SN … (cpx h g G L) (eq …).
+                λh,o,G,L. SN … (cpx h o G L) (eq …).
 
 interpretation
    "context-sensitive extended strong normalization (term)"
-   'SN h g G L T = (csx h g G L T).
+   'SN h o G L T = (csx h o G L T).
 
 (* Basic eliminators ********************************************************)
 
-lemma csx_ind: ∀h,g,G,L. ∀R:predicate term.
-               (∀T1. ⦃G, L⦄ ⊢ ⬊*[h, g] T1 →
-                     (∀T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 → (T1 = T2 → ⊥) → R T2) →
+lemma csx_ind: ∀h,o,G,L. ∀R:predicate term.
+               (∀T1. ⦃G, L⦄ ⊢ ⬊*[h, o] T1 →
+                     (∀T2. ⦃G, L⦄ ⊢ T1 ➡[h, o] T2 → (T1 = T2 → ⊥) → R T2) →
                      R T1
                ) →
-               ∀T. ⦃G, L⦄ ⊢ ⬊*[h, g] T → R T.
-#h #g #G #L #R #H0 #T1 #H elim H -T1
+               ∀T. ⦃G, L⦄ ⊢ ⬊*[h, o] T → R T.
+#h #o #G #L #R #H0 #T1 #H elim H -T1
 /5 width=1 by SN_intro/
 qed-.
 
 (* Basic properties *********************************************************)
 
 (* Basic_1: was just: sn3_pr2_intro *)
-lemma csx_intro: ∀h,g,G,L,T1.
-                 (∀T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 → (T1 = T2 → ⊥) → ⦃G, L⦄ ⊢ ⬊*[h, g] T2) →
-                 ⦃G, L⦄ ⊢ ⬊*[h, g] T1.
+lemma csx_intro: ∀h,o,G,L,T1.
+                 (∀T2. ⦃G, L⦄ ⊢ T1 ➡[h, o] T2 → (T1 = T2 → ⊥) → ⦃G, L⦄ ⊢ ⬊*[h, o] T2) →
+                 ⦃G, L⦄ ⊢ ⬊*[h, o] T1.
 /4 width=1 by SN_intro/ qed.
 
-lemma csx_cpx_trans: ∀h,g,G,L,T1. ⦃G, L⦄ ⊢ ⬊*[h, g] T1 →
-                     ∀T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 → ⦃G, L⦄ ⊢ ⬊*[h, g] T2.
-#h #g #G #L #T1 #H @(csx_ind … H) -T1 #T1 #HT1 #IHT1 #T2 #HLT12
+lemma csx_cpx_trans: ∀h,o,G,L,T1. ⦃G, L⦄ ⊢ ⬊*[h, o] T1 →
+                     ∀T2. ⦃G, L⦄ ⊢ T1 ➡[h, o] T2 → ⦃G, L⦄ ⊢ ⬊*[h, o] T2.
+#h #o #G #L #T1 #H @(csx_ind … H) -T1 #T1 #HT1 #IHT1 #T2 #HLT12
 elim (eq_term_dec T1 T2) #HT12 destruct /3 width=4 by/
 qed-.
 
 (* Basic_1: was just: sn3_nf2 *)
-lemma cnx_csx: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃T⦄ → ⦃G, L⦄ ⊢ ⬊*[h, g] T.
+lemma cnx_csx: ∀h,o,G,L,T. ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃T⦄ → ⦃G, L⦄ ⊢ ⬊*[h, o] T.
 /2 width=1 by NF_to_SN/ qed.
 
-lemma csx_sort: ∀h,g,G,L,k. ⦃G, L⦄ ⊢ ⬊*[h, g] ⋆k.
-#h #g #G #L #k elim (deg_total h g k)
-#d generalize in match k; -k @(nat_ind_plus … d) -d /3 width=6 by cnx_csx, cnx_sort/
-#d #IHd #k #Hkd lapply (deg_next_SO … Hkd) -Hkd
+lemma csx_sort: ∀h,o,G,L,s. ⦃G, L⦄ ⊢ ⬊*[h, o] ⋆s.
+#h #o #G #L #s elim (deg_total h o s)
+#d generalize in match s; -s @(nat_ind_plus … d) -d /3 width=6 by cnx_csx, cnx_sort/
+#d #IHd #s #Hkd lapply (deg_next_SO … Hkd) -Hkd
 #Hkd @csx_intro #X #H #HX elim (cpx_inv_sort1 … H) -H
 [ #H destruct elim HX //
 | -HX * #d0 #_ #H destruct -d0 /2 width=1 by/
@@ -65,9 +65,9 @@ lemma csx_sort: ∀h,g,G,L,k. ⦃G, L⦄ ⊢ ⬊*[h, g] ⋆k.
 qed.
 
 (* Basic_1: was just: sn3_cast *)
-lemma csx_cast: ∀h,g,G,L,W. ⦃G, L⦄ ⊢ ⬊*[h, g] W →
-                ∀T. ⦃G, L⦄ ⊢ ⬊*[h, g] T → ⦃G, L⦄ ⊢ ⬊*[h, g] ⓝW.T.
-#h #g #G #L #W #HW @(csx_ind … HW) -W #W #HW #IHW #T #HT @(csx_ind … HT) -T #T #HT #IHT
+lemma csx_cast: ∀h,o,G,L,W. ⦃G, L⦄ ⊢ ⬊*[h, o] W →
+                ∀T. ⦃G, L⦄ ⊢ ⬊*[h, o] T → ⦃G, L⦄ ⊢ ⬊*[h, o] ⓝW.T.
+#h #o #G #L #W #HW @(csx_ind … HW) -W #W #HW #IHW #T #HT @(csx_ind … HT) -T #T #HT #IHT
 @csx_intro #X #H1 #H2
 elim (cpx_inv_cast1 … H1) -H1
 [ * #W0 #T0 #HLW0 #HLT0 #H destruct
@@ -81,48 +81,48 @@ qed.
 
 (* Basic forward lemmas *****************************************************)
 
-fact csx_fwd_pair_sn_aux: ∀h,g,G,L,U. ⦃G, L⦄ ⊢ ⬊*[h, g] U →
-                          ∀I,V,T. U = ②{I}V.T → ⦃G, L⦄ ⊢ ⬊*[h, g] V.
-#h #g #G #L #U #H elim H -H #U0 #_ #IH #I #V #T #H destruct
+fact csx_fwd_pair_sn_aux: ∀h,o,G,L,U. ⦃G, L⦄ ⊢ ⬊*[h, o] U →
+                          ∀I,V,T. U = ②{I}V.T → ⦃G, L⦄ ⊢ ⬊*[h, o] V.
+#h #o #G #L #U #H elim H -H #U0 #_ #IH #I #V #T #H destruct
 @csx_intro #V2 #HLV2 #HV2
 @(IH (②{I}V2.T)) -IH /2 width=3 by cpx_pair_sn/ -HLV2
 #H destruct /2 width=1 by/
 qed-.
 
 (* Basic_1: was just: sn3_gen_head *)
-lemma csx_fwd_pair_sn: ∀h,g,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬊*[h, g] ②{I}V.T → ⦃G, L⦄ ⊢ ⬊*[h, g] V.
+lemma csx_fwd_pair_sn: ∀h,o,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬊*[h, o] ②{I}V.T → ⦃G, L⦄ ⊢ ⬊*[h, o] V.
 /2 width=5 by csx_fwd_pair_sn_aux/ qed-.
 
-fact csx_fwd_bind_dx_aux: ∀h,g,G,L,U. ⦃G, L⦄ ⊢ ⬊*[h, g] U →
-                          ∀a,I,V,T. U = ⓑ{a,I}V.T → ⦃G, L.ⓑ{I}V⦄ ⊢ ⬊*[h, g] T.
-#h #g #G #L #U #H elim H -H #U0 #_ #IH #a #I #V #T #H destruct
+fact csx_fwd_bind_dx_aux: ∀h,o,G,L,U. ⦃G, L⦄ ⊢ ⬊*[h, o] U →
+                          ∀a,I,V,T. U = ⓑ{a,I}V.T → ⦃G, L.ⓑ{I}V⦄ ⊢ ⬊*[h, o] T.
+#h #o #G #L #U #H elim H -H #U0 #_ #IH #a #I #V #T #H destruct
 @csx_intro #T2 #HLT2 #HT2
 @(IH (ⓑ{a,I}V.T2)) -IH /2 width=3 by cpx_bind/ -HLT2
 #H destruct /2 width=1 by/
 qed-.
 
 (* Basic_1: was just: sn3_gen_bind *)
-lemma csx_fwd_bind_dx: ∀h,g,a,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬊*[h, g] ⓑ{a,I}V.T → ⦃G, L.ⓑ{I}V⦄ ⊢ ⬊*[h, g] T.
+lemma csx_fwd_bind_dx: ∀h,o,a,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬊*[h, o] ⓑ{a,I}V.T → ⦃G, L.ⓑ{I}V⦄ ⊢ ⬊*[h, o] T.
 /2 width=4 by csx_fwd_bind_dx_aux/ qed-.
 
-fact csx_fwd_flat_dx_aux: ∀h,g,G,L,U. ⦃G, L⦄ ⊢ ⬊*[h, g] U →
-                          ∀I,V,T. U = ⓕ{I}V.T → ⦃G, L⦄ ⊢ ⬊*[h, g] T.
-#h #g #G #L #U #H elim H -H #U0 #_ #IH #I #V #T #H destruct
+fact csx_fwd_flat_dx_aux: ∀h,o,G,L,U. ⦃G, L⦄ ⊢ ⬊*[h, o] U →
+                          ∀I,V,T. U = ⓕ{I}V.T → ⦃G, L⦄ ⊢ ⬊*[h, o] T.
+#h #o #G #L #U #H elim H -H #U0 #_ #IH #I #V #T #H destruct
 @csx_intro #T2 #HLT2 #HT2
 @(IH (ⓕ{I}V.T2)) -IH /2 width=3 by cpx_flat/ -HLT2
 #H destruct /2 width=1 by/
 qed-.
 
 (* Basic_1: was just: sn3_gen_flat *)
-lemma csx_fwd_flat_dx: ∀h,g,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬊*[h, g] ⓕ{I}V.T → ⦃G, L⦄ ⊢ ⬊*[h, g] T.
+lemma csx_fwd_flat_dx: ∀h,o,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬊*[h, o] ⓕ{I}V.T → ⦃G, L⦄ ⊢ ⬊*[h, o] T.
 /2 width=5 by csx_fwd_flat_dx_aux/ qed-.
 
-lemma csx_fwd_bind: ∀h,g,a,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬊*[h, g] ⓑ{a,I}V.T →
-                    ⦃G, L⦄ ⊢ ⬊*[h, g] V ∧ ⦃G, L.ⓑ{I}V⦄ ⊢ ⬊*[h, g] T.
+lemma csx_fwd_bind: ∀h,o,a,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬊*[h, o] ⓑ{a,I}V.T →
+                    ⦃G, L⦄ ⊢ ⬊*[h, o] V ∧ ⦃G, L.ⓑ{I}V⦄ ⊢ ⬊*[h, o] T.
 /3 width=3 by csx_fwd_pair_sn, csx_fwd_bind_dx, conj/ qed-.
 
-lemma csx_fwd_flat: ∀h,g,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬊*[h, g] ⓕ{I}V.T →
-                    ⦃G, L⦄ ⊢ ⬊*[h, g] V ∧ ⦃G, L⦄ ⊢ ⬊*[h, g] T.
+lemma csx_fwd_flat: ∀h,o,I,G,L,V,T. ⦃G, L⦄ ⊢ ⬊*[h, o] ⓕ{I}V.T →
+                    ⦃G, L⦄ ⊢ ⬊*[h, o] V ∧ ⦃G, L⦄ ⊢ ⬊*[h, o] T.
 /3 width=3 by csx_fwd_pair_sn, csx_fwd_flat_dx, conj/ qed-.
 
 (* Basic_1: removed theorems 14:
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/csx_aaa.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/csx_aaa.ma
index 6d8aa3ed9..a9d3e4cae 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/csx_aaa.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/csx_aaa.ma
@@ -20,39 +20,39 @@ include "basic_2/computation/csx_tsts_vector.ma".
 
 (* Main properties on atomic arity assignment *******************************)
 
-theorem aaa_csx: ∀h,g,G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ⦃G, L⦄ ⊢ ⬊*[h, g] T.
-#h #g #G #L #T #A #H
-@(gcr_aaa … (csx_gcp h g) (csx_gcr h g) … H)
+theorem aaa_csx: ∀h,o,G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ⦃G, L⦄ ⊢ ⬊*[h, o] T.
+#h #o #G #L #T #A #H
+@(gcr_aaa … (csx_gcp h o) (csx_gcr h o) … H)
 qed.
 
 (* Advanced eliminators *****************************************************)
 
-fact aaa_ind_csx_aux: ∀h,g,G,L,A. ∀R:predicate term.
+fact aaa_ind_csx_aux: ∀h,o,G,L,A. ∀R:predicate term.
                       (∀T1. ⦃G, L⦄ ⊢ T1 ⁝ A →
-                            (∀T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 → (T1 = T2 → ⊥) → R T2) → R T1
+                            (∀T2. ⦃G, L⦄ ⊢ T1 ➡[h, o] T2 → (T1 = T2 → ⊥) → R T2) → R T1
                       ) →
-                      ∀T. ⦃G, L⦄ ⊢ ⬊*[h, g] T → ⦃G, L⦄ ⊢ T ⁝ A → R T.
-#h #g #G #L #A #R #IH #T #H @(csx_ind … H) -T /4 width=5 by cpx_aaa_conf/
+                      ∀T. ⦃G, L⦄ ⊢ ⬊*[h, o] T → ⦃G, L⦄ ⊢ T ⁝ A → R T.
+#h #o #G #L #A #R #IH #T #H @(csx_ind … H) -T /4 width=5 by cpx_aaa_conf/
 qed-.
 
-lemma aaa_ind_csx: ∀h,g,G,L,A. ∀R:predicate term.
+lemma aaa_ind_csx: ∀h,o,G,L,A. ∀R:predicate term.
                    (∀T1. ⦃G, L⦄ ⊢ T1 ⁝ A →
-                         (∀T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 → (T1 = T2 → ⊥) → R T2) → R T1
+                         (∀T2. ⦃G, L⦄ ⊢ T1 ➡[h, o] T2 → (T1 = T2 → ⊥) → R T2) → R T1
                    ) →
                    ∀T. ⦃G, L⦄ ⊢ T ⁝ A → R T.
 /5 width=9 by aaa_ind_csx_aux, aaa_csx/ qed-.
 
-fact aaa_ind_csx_alt_aux: ∀h,g,G,L,A. ∀R:predicate term.
+fact aaa_ind_csx_alt_aux: ∀h,o,G,L,A. ∀R:predicate term.
                           (∀T1. ⦃G, L⦄ ⊢ T1 ⁝ A →
-                                (∀T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 → (T1 = T2 → ⊥) → R T2) → R T1
+                                (∀T2. ⦃G, L⦄ ⊢ T1 ➡*[h, o] T2 → (T1 = T2 → ⊥) → R T2) → R T1
                           ) →
-                          ∀T. ⦃G, L⦄ ⊢ ⬊*[h, g] T → ⦃G, L⦄ ⊢ T ⁝ A → R T.
-#h #g #G #L #A #R #IH #T #H @(csx_ind_alt … H) -T /4 width=5 by cpxs_aaa_conf/
+                          ∀T. ⦃G, L⦄ ⊢ ⬊*[h, o] T → ⦃G, L⦄ ⊢ T ⁝ A → R T.
+#h #o #G #L #A #R #IH #T #H @(csx_ind_alt … H) -T /4 width=5 by cpxs_aaa_conf/
 qed-.
 
-lemma aaa_ind_csx_alt: ∀h,g,G,L,A. ∀R:predicate term.
+lemma aaa_ind_csx_alt: ∀h,o,G,L,A. ∀R:predicate term.
                        (∀T1. ⦃G, L⦄ ⊢ T1 ⁝ A →
-                             (∀T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 → (T1 = T2 → ⊥) → R T2) → R T1
+                             (∀T2. ⦃G, L⦄ ⊢ T1 ➡*[h, o] T2 → (T1 = T2 → ⊥) → R T2) → R T1
                        ) →
                        ∀T. ⦃G, L⦄ ⊢ T ⁝ A → R T.
 /5 width=9 by aaa_ind_csx_alt_aux, aaa_csx/ qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/csx_alt.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/csx_alt.ma
index a6f46c787..0b9b29799 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/csx_alt.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/csx_alt.ma
@@ -20,44 +20,44 @@ include "basic_2/computation/csx.ma".
 
 (* alternative definition of csx *)
 definition csxa: ∀h. sd h → relation3 genv lenv term ≝
-                 λh,g,G,L. SN … (cpxs h g G L) (eq …).
+                 λh,o,G,L. SN … (cpxs h o G L) (eq …).
 
 interpretation
    "context-sensitive extended strong normalization (term) alternative"
-   'SNAlt h g G L T = (csxa h g G L T).
+   'SNAlt h o G L T = (csxa h o G L T).
 
 (* Basic eliminators ********************************************************)
 
-lemma csxa_ind: ∀h,g,G,L. ∀R:predicate term.
-                (∀T1. ⦃G, L⦄ ⊢ ⬊⬊*[h, g] T1 →
-                      (∀T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 → (T1 = T2 → ⊥) → R T2) → R T1
+lemma csxa_ind: ∀h,o,G,L. ∀R:predicate term.
+                (∀T1. ⦃G, L⦄ ⊢ ⬊⬊*[h, o] T1 →
+                      (∀T2. ⦃G, L⦄ ⊢ T1 ➡*[h, o] T2 → (T1 = T2 → ⊥) → R T2) → R T1
                 ) →
-                ∀T. ⦃G, L⦄ ⊢ ⬊⬊*[h, g] T → R T.
-#h #g #G #L #R #H0 #T1 #H elim H -T1 /5 width=1 by SN_intro/
+                ∀T. ⦃G, L⦄ ⊢ ⬊⬊*[h, o] T → R T.
+#h #o #G #L #R #H0 #T1 #H elim H -T1 /5 width=1 by SN_intro/
 qed-.
 
 (* Basic properties *********************************************************)
 
-lemma csx_intro_cpxs: ∀h,g,G,L,T1.
-                         (∀T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 → (T1 = T2 → ⊥) → ⦃G, L⦄ ⊢ ⬊*[h, g] T2) →
-                      ⦃G, L⦄ ⊢ ⬊*[h, g] T1.
+lemma csx_intro_cpxs: ∀h,o,G,L,T1.
+                         (∀T2. ⦃G, L⦄ ⊢ T1 ➡*[h, o] T2 → (T1 = T2 → ⊥) → ⦃G, L⦄ ⊢ ⬊*[h, o] T2) →
+                      ⦃G, L⦄ ⊢ ⬊*[h, o] T1.
 /4 width=1 by cpx_cpxs, csx_intro/ qed.
 
 (* Basic_1: was just: sn3_intro *)
-lemma csxa_intro: ∀h,g,G,L,T1.
-                  (∀T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 → (T1 = T2 → ⊥) → ⦃G, L⦄ ⊢ ⬊⬊*[h, g] T2) →
-                  ⦃G, L⦄ ⊢ ⬊⬊*[h, g] T1.
+lemma csxa_intro: ∀h,o,G,L,T1.
+                  (∀T2. ⦃G, L⦄ ⊢ T1 ➡*[h, o] T2 → (T1 = T2 → ⊥) → ⦃G, L⦄ ⊢ ⬊⬊*[h, o] T2) →
+                  ⦃G, L⦄ ⊢ ⬊⬊*[h, o] T1.
 /4 width=1 by SN_intro/ qed.
 
-fact csxa_intro_aux: ∀h,g,G,L,T1. (
-                        ∀T,T2. ⦃G, L⦄ ⊢ T ➡*[h, g] T2 → T1 = T → (T1 = T2 → ⊥) → ⦃G, L⦄ ⊢ ⬊⬊*[h, g] T2
-                     ) → ⦃G, L⦄ ⊢ ⬊⬊*[h, g] T1.
+fact csxa_intro_aux: ∀h,o,G,L,T1. (
+                        ∀T,T2. ⦃G, L⦄ ⊢ T ➡*[h, o] T2 → T1 = T → (T1 = T2 → ⊥) → ⦃G, L⦄ ⊢ ⬊⬊*[h, o] T2
+                     ) → ⦃G, L⦄ ⊢ ⬊⬊*[h, o] T1.
 /4 width=3 by csxa_intro/ qed-.
 
 (* Basic_1: was just: sn3_pr3_trans (old version) *)
-lemma csxa_cpxs_trans: ∀h,g,G,L,T1. ⦃G, L⦄ ⊢ ⬊⬊*[h, g] T1 →
-                       ∀T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 → ⦃G, L⦄ ⊢ ⬊⬊*[h, g] T2.
-#h #g #G #L #T1 #H elim H -T1 #T1 #HT1 #IHT1 #T2 #HLT12
+lemma csxa_cpxs_trans: ∀h,o,G,L,T1. ⦃G, L⦄ ⊢ ⬊⬊*[h, o] T1 →
+                       ∀T2. ⦃G, L⦄ ⊢ T1 ➡*[h, o] T2 → ⦃G, L⦄ ⊢ ⬊⬊*[h, o] T2.
+#h #o #G #L #T1 #H elim H -T1 #T1 #HT1 #IHT1 #T2 #HLT12
 @csxa_intro #T #HLT2 #HT2
 elim (eq_term_dec T1 T2) #HT12
 [ -IHT1 -HLT12 destruct /3 width=1 by/
@@ -65,10 +65,10 @@ elim (eq_term_dec T1 T2) #HT12
 qed.
 
 (* Basic_1: was just: sn3_pr2_intro (old version) *)
-lemma csxa_intro_cpx: ∀h,g,G,L,T1. (
-                         ∀T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 → (T1 = T2 → ⊥) → ⦃G, L⦄ ⊢ ⬊⬊*[h, g] T2
-                      ) → ⦃G, L⦄ ⊢ ⬊⬊*[h, g] T1.
-#h #g #G #L #T1 #H
+lemma csxa_intro_cpx: ∀h,o,G,L,T1. (
+                         ∀T2. ⦃G, L⦄ ⊢ T1 ➡[h, o] T2 → (T1 = T2 → ⊥) → ⦃G, L⦄ ⊢ ⬊⬊*[h, o] T2
+                      ) → ⦃G, L⦄ ⊢ ⬊⬊*[h, o] T1.
+#h #o #G #L #T1 #H
 @csxa_intro_aux #T #T2 #H @(cpxs_ind_dx … H) -T
 [ -H #H destruct #H
   elim H //
@@ -82,26 +82,26 @@ qed.
 
 (* Main properties **********************************************************)
 
-theorem csx_csxa: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ ⬊*[h, g] T → ⦃G, L⦄ ⊢ ⬊⬊*[h, g] T.
-#h #g #G #L #T #H @(csx_ind … H) -T /4 width=1 by csxa_intro_cpx/
+theorem csx_csxa: ∀h,o,G,L,T. ⦃G, L⦄ ⊢ ⬊*[h, o] T → ⦃G, L⦄ ⊢ ⬊⬊*[h, o] T.
+#h #o #G #L #T #H @(csx_ind … H) -T /4 width=1 by csxa_intro_cpx/
 qed.
 
-theorem csxa_csx: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ ⬊⬊*[h, g] T → ⦃G, L⦄ ⊢ ⬊*[h, g] T.
-#h #g #G #L #T #H @(csxa_ind … H) -T /4 width=1 by cpx_cpxs, csx_intro/
+theorem csxa_csx: ∀h,o,G,L,T. ⦃G, L⦄ ⊢ ⬊⬊*[h, o] T → ⦃G, L⦄ ⊢ ⬊*[h, o] T.
+#h #o #G #L #T #H @(csxa_ind … H) -T /4 width=1 by cpx_cpxs, csx_intro/
 qed.
 
 (* Basic_1: was just: sn3_pr3_trans *)
-lemma csx_cpxs_trans: ∀h,g,G,L,T1. ⦃G, L⦄ ⊢ ⬊*[h, g] T1 →
-                      ∀T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 → ⦃G, L⦄ ⊢ ⬊*[h, g] T2.
-#h #g #G #L #T1 #HT1 #T2 #H @(cpxs_ind … H) -T2 /2 width=3 by csx_cpx_trans/
+lemma csx_cpxs_trans: ∀h,o,G,L,T1. ⦃G, L⦄ ⊢ ⬊*[h, o] T1 →
+                      ∀T2. ⦃G, L⦄ ⊢ T1 ➡*[h, o] T2 → ⦃G, L⦄ ⊢ ⬊*[h, o] T2.
+#h #o #G #L #T1 #HT1 #T2 #H @(cpxs_ind … H) -T2 /2 width=3 by csx_cpx_trans/
 qed-.
 
 (* Main eliminators *********************************************************)
 
-lemma csx_ind_alt: ∀h,g,G,L. ∀R:predicate term.
-                   (∀T1. ⦃G, L⦄ ⊢ ⬊*[h, g] T1 →
-                         (∀T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 → (T1 = T2 → ⊥) → R T2) → R T1
+lemma csx_ind_alt: ∀h,o,G,L. ∀R:predicate term.
+                   (∀T1. ⦃G, L⦄ ⊢ ⬊*[h, o] T1 →
+                         (∀T2. ⦃G, L⦄ ⊢ T1 ➡*[h, o] T2 → (T1 = T2 → ⊥) → R T2) → R T1
                    ) →
-                   ∀T. ⦃G, L⦄ ⊢ ⬊*[h, g] T → R T.
-#h #g #G #L #R #H0 #T1 #H @(csxa_ind … (csx_csxa … H)) -T1 /4 width=1 by csxa_csx/
+                   ∀T. ⦃G, L⦄ ⊢ ⬊*[h, o] T → R T.
+#h #o #G #L #R #H0 #T1 #H @(csxa_ind … (csx_csxa … H)) -T1 /4 width=1 by csxa_csx/
 qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/csx_fpbs.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/csx_fpbs.ma
index 6b7fcd333..c756c4ab8 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/csx_fpbs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/csx_fpbs.ma
@@ -20,14 +20,14 @@ include "basic_2/computation/csx_lpx.ma".
 
 (* Advanced properties ******************************************************)
 
-lemma csx_fpb_conf: ∀h,g,G1,L1,T1. ⦃G1, L1⦄ ⊢ ⬊*[h, g] T1 →
-                    ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≽[h, g] ⦃G2, L2, T2⦄ → ⦃G2, L2⦄ ⊢ ⬊*[h, g] T2.
-#h #g #G1 #L1 #T1 #HT1 #G2 #L2 #T2 *
+lemma csx_fpb_conf: ∀h,o,G1,L1,T1. ⦃G1, L1⦄ ⊢ ⬊*[h, o] T1 →
+                    ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≽[h, o] ⦃G2, L2, T2⦄ → ⦃G2, L2⦄ ⊢ ⬊*[h, o] T2.
+#h #o #G1 #L1 #T1 #HT1 #G2 #L2 #T2 *
 /2 width=5 by csx_cpx_trans, csx_fquq_conf, csx_lpx_conf, csx_lleq_conf/
 qed-.
 
-lemma csx_fpbs_conf: ∀h,g,G1,L1,T1. ⦃G1, L1⦄ ⊢ ⬊*[h, g] T1 →
-                     ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ → ⦃G2, L2⦄ ⊢ ⬊*[h, g] T2.
-#h #g #G1 #L1 #T1 #HT1 #G2 #L2 #T2 #H @(fpbs_ind … H) -G2 -L2 -T2
+lemma csx_fpbs_conf: ∀h,o,G1,L1,T1. ⦃G1, L1⦄ ⊢ ⬊*[h, o] T1 →
+                     ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄ → ⦃G2, L2⦄ ⊢ ⬊*[h, o] T2.
+#h #o #G1 #L1 #T1 #HT1 #G2 #L2 #T2 #H @(fpbs_ind … H) -G2 -L2 -T2
 /2 width=5 by csx_fpb_conf/
 qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/csx_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/csx_lift.ma
index fabd9b1b9..354b59a21 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/csx_lift.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/csx_lift.ma
@@ -21,9 +21,9 @@ include "basic_2/computation/csx.ma".
 (* Relocation properties ****************************************************)
 
 (* Basic_1: was just: sn3_lift *)
-lemma csx_lift: ∀h,g,G,L2,L1,T1,s,l,m. ⦃G, L1⦄ ⊢ ⬊*[h, g] T1 →
-                ∀T2. ⬇[s, l, m] L2 ≡ L1 → ⬆[l, m] T1 ≡ T2 → ⦃G, L2⦄ ⊢ ⬊*[h, g] T2.
-#h #g #G #L2 #L1 #T1 #s #l #m #H elim H -T1 #T1 #_ #IHT1 #T2 #HL21 #HT12
+lemma csx_lift: ∀h,o,G,L2,L1,T1,c,l,k. ⦃G, L1⦄ ⊢ ⬊*[h, o] T1 →
+                ∀T2. ⬇[c, l, k] L2 ≡ L1 → ⬆[l, k] T1 ≡ T2 → ⦃G, L2⦄ ⊢ ⬊*[h, o] T2.
+#h #o #G #L2 #L1 #T1 #c #l #k #H elim H -T1 #T1 #_ #IHT1 #T2 #HL21 #HT12
 @csx_intro #T #HLT2 #HT2
 elim (cpx_inv_lift1 … HLT2 … HL21 … HT12) -HLT2 #T0 #HT0 #HLT10
 @(IHT1 … HLT10) // -L1 -L2 #H destruct
@@ -31,11 +31,11 @@ elim (cpx_inv_lift1 … HLT2 … HL21 … HT12) -HLT2 #T0 #HT0 #HLT10
 qed.
 
 (* Basic_1: was just: sn3_gen_lift *)
-lemma csx_inv_lift: ∀h,g,G,L2,L1,T1,s,l,m. ⦃G, L1⦄ ⊢ ⬊*[h, g] T1 →
-                    ∀T2. ⬇[s, l, m] L1 ≡ L2 → ⬆[l, m] T2 ≡ T1 → ⦃G, L2⦄ ⊢ ⬊*[h, g] T2.
-#h #g #G #L2 #L1 #T1 #s #l #m #H elim H -T1 #T1 #_ #IHT1 #T2 #HL12 #HT21
+lemma csx_inv_lift: ∀h,o,G,L2,L1,T1,c,l,k. ⦃G, L1⦄ ⊢ ⬊*[h, o] T1 →
+                    ∀T2. ⬇[c, l, k] L1 ≡ L2 → ⬆[l, k] T2 ≡ T1 → ⦃G, L2⦄ ⊢ ⬊*[h, o] T2.
+#h #o #G #L2 #L1 #T1 #c #l #k #H elim H -T1 #T1 #_ #IHT1 #T2 #HL12 #HT21
 @csx_intro #T #HLT2 #HT2
-elim (lift_total T l m) #T0 #HT0
+elim (lift_total T l k) #T0 #HT0
 lapply (cpx_lift … HLT2 … HL12 … HT21 … HT0) -HLT2 #HLT10
 @(IHT1 … HLT10) // -L1 -L2 #H destruct
 >(lift_inj … HT0 … HT21) in HT2; -T1 /2 width=1 by/
@@ -44,9 +44,9 @@ qed.
 (* Advanced inversion lemmas ************************************************)
 
 (* Basic_1: was: sn3_gen_def *)
-lemma csx_inv_lref_bind: ∀h,g,I,G,L,K,V,i. ⬇[i] L ≡ K.ⓑ{I}V →
-                         ⦃G, L⦄ ⊢ ⬊*[h, g] #i → ⦃G, K⦄ ⊢ ⬊*[h, g] V.
-#h #g #I #G #L #K #V #i #HLK #Hi
+lemma csx_inv_lref_bind: ∀h,o,I,G,L,K,V,i. ⬇[i] L ≡ K.ⓑ{I}V →
+                         ⦃G, L⦄ ⊢ ⬊*[h, o] #i → ⦃G, K⦄ ⊢ ⬊*[h, o] V.
+#h #o #I #G #L #K #V #i #HLK #Hi
 elim (lift_total V 0 (i+1))
 /4 width=9 by csx_inv_lift, csx_cpx_trans, cpx_delta, drop_fwd_drop2/
 qed-.
@@ -54,8 +54,8 @@ qed-.
 (* Advanced properties ******************************************************)
 
 (* Basic_1: was just: sn3_abbr *)
-lemma csx_lref_bind: ∀h,g,I,G,L,K,V,i. ⬇[i] L ≡ K.ⓑ{I}V → ⦃G, K⦄ ⊢ ⬊*[h, g] V → ⦃G, L⦄ ⊢ ⬊*[h, g] #i.
-#h #g #I #G #L #K #V #i #HLK #HV
+lemma csx_lref_bind: ∀h,o,I,G,L,K,V,i. ⬇[i] L ≡ K.ⓑ{I}V → ⦃G, K⦄ ⊢ ⬊*[h, o] V → ⦃G, L⦄ ⊢ ⬊*[h, o] #i.
+#h #o #I #G #L #K #V #i #HLK #HV
 @csx_intro #X #H #Hi
 elim (cpx_inv_lref1 … H) -H
 [ #H destruct elim Hi //
@@ -65,10 +65,10 @@ elim (cpx_inv_lref1 … H) -H
 ]
 qed.
 
-lemma csx_appl_simple: ∀h,g,G,L,V. ⦃G, L⦄ ⊢ ⬊*[h, g] V → ∀T1.
-                       (∀T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 → (T1 = T2 → ⊥) → ⦃G, L⦄ ⊢ ⬊*[h, g] ⓐV.T2) →
-                       𝐒⦃T1⦄ → ⦃G, L⦄ ⊢ ⬊*[h, g] ⓐV.T1.
-#h #g #G #L #V #H @(csx_ind … H) -V #V #_ #IHV #T1 #IHT1 #HT1
+lemma csx_appl_simple: ∀h,o,G,L,V. ⦃G, L⦄ ⊢ ⬊*[h, o] V → ∀T1.
+                       (∀T2. ⦃G, L⦄ ⊢ T1 ➡[h, o] T2 → (T1 = T2 → ⊥) → ⦃G, L⦄ ⊢ ⬊*[h, o] ⓐV.T2) →
+                       𝐒⦃T1⦄ → ⦃G, L⦄ ⊢ ⬊*[h, o] ⓐV.T1.
+#h #o #G #L #V #H @(csx_ind … H) -V #V #_ #IHV #T1 #IHT1 #HT1
 @csx_intro #X #H1 #H2
 elim (cpx_inv_appl1_simple … H1) // -H1
 #V0 #T0 #HLV0 #HLT10 #H destruct
@@ -79,29 +79,29 @@ elim (eq_false_inv_tpair_dx … H2) -H2
 ]
 qed.
 
-lemma csx_fqu_conf: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
-                    ⦃G1, L1⦄ ⊢ ⬊*[h, g] T1 → ⦃G2, L2⦄ ⊢ ⬊*[h, g] T2.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
+lemma csx_fqu_conf: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
+                    ⦃G1, L1⦄ ⊢ ⬊*[h, o] T1 → ⦃G2, L2⦄ ⊢ ⬊*[h, o] T2.
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
 /2 width=8 by csx_inv_lref_bind, csx_inv_lift, csx_fwd_flat_dx, csx_fwd_bind_dx, csx_fwd_pair_sn/
 qed-.
 
-lemma csx_fquq_conf: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
-                     ⦃G1, L1⦄ ⊢ ⬊*[h, g] T1 → ⦃G2, L2⦄ ⊢ ⬊*[h, g] T2.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H12 #H elim (fquq_inv_gen … H12) -H12
+lemma csx_fquq_conf: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
+                     ⦃G1, L1⦄ ⊢ ⬊*[h, o] T1 → ⦃G2, L2⦄ ⊢ ⬊*[h, o] T2.
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H12 #H elim (fquq_inv_gen … H12) -H12
 [ /2 width=5 by csx_fqu_conf/
 | * #HG #HL #HT destruct //
 ]
 qed-.
 
-lemma csx_fqup_conf: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
-                     ⦃G1, L1⦄ ⊢ ⬊*[h, g] T1 → ⦃G2, L2⦄ ⊢ ⬊*[h, g] T2.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
+lemma csx_fqup_conf: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
+                     ⦃G1, L1⦄ ⊢ ⬊*[h, o] T1 → ⦃G2, L2⦄ ⊢ ⬊*[h, o] T2.
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
 /3 width=5 by csx_fqu_conf/
 qed-.
 
-lemma csx_fqus_conf: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
-                     ⦃G1, L1⦄ ⊢ ⬊*[h, g] T1 → ⦃G2, L2⦄ ⊢ ⬊*[h, g] T2.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H12 #H elim (fqus_inv_gen … H12) -H12
+lemma csx_fqus_conf: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
+                     ⦃G1, L1⦄ ⊢ ⬊*[h, o] T1 → ⦃G2, L2⦄ ⊢ ⬊*[h, o] T2.
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H12 #H elim (fqus_inv_gen … H12) -H12
 [ /2 width=5 by csx_fqup_conf/
 | * #HG #HL #HT destruct //
 ]
@@ -109,10 +109,10 @@ qed-.
 
 (* Main properties **********************************************************)
 
-theorem csx_gcp: ∀h,g. gcp (cpx h g) (eq …) (csx h g).
-#h #g @mk_gcp
+theorem csx_gcp: ∀h,o. gcp (cpx h o) (eq …) (csx h o).
+#h #o @mk_gcp
 [ normalize /3 width=13 by cnx_lift/
-| #G #L elim (deg_total h g 0) /3 width=8 by cnx_sort_iter, ex_intro/
+| #G #L elim (deg_total h o 0) /3 width=8 by cnx_sort_iter, ex_intro/
 | /2 width=8 by csx_lift/
 | /2 width=3 by csx_fwd_flat_dx/
 ]
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/csx_lleq.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/csx_lleq.ma
index 765861e3b..71009c473 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/csx_lleq.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/csx_lleq.ma
@@ -19,12 +19,12 @@ include "basic_2/computation/csx.ma".
 
 (* Properties on lazy equivalence for local environments ********************)
 
-lemma csx_lleq_conf: ∀h,g,G,L1,T. ⦃G, L1⦄ ⊢ ⬊*[h, g] T →
-                     ∀L2. L1 ≡[T, 0] L2 → ⦃G, L2⦄ ⊢ ⬊*[h, g] T.
-#h #g #G #L1 #T #H @(csx_ind … H) -T
+lemma csx_lleq_conf: ∀h,o,G,L1,T. ⦃G, L1⦄ ⊢ ⬊*[h, o] T →
+                     ∀L2. L1 ≡[T, 0] L2 → ⦃G, L2⦄ ⊢ ⬊*[h, o] T.
+#h #o #G #L1 #T #H @(csx_ind … H) -T
 /4 width=6 by csx_intro, cpx_lleq_conf_dx, lleq_cpx_trans/
 qed-.
 
-lemma csx_lleq_trans: ∀h,g,G,L1,L2,T.
-                      L1 ≡[T, 0] L2 → ⦃G, L2⦄ ⊢ ⬊*[h, g] T → ⦃G, L1⦄ ⊢ ⬊*[h, g] T.
+lemma csx_lleq_trans: ∀h,o,G,L1,L2,T.
+                      L1 ≡[T, 0] L2 → ⦃G, L2⦄ ⊢ ⬊*[h, o] T → ⦃G, L1⦄ ⊢ ⬊*[h, o] T.
 /3 width=3 by csx_lleq_conf, lleq_sym/ qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/csx_lpx.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/csx_lpx.ma
index b87a08bb4..dd01e6a24 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/csx_lpx.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/csx_lpx.ma
@@ -21,15 +21,15 @@ include "basic_2/computation/csx_lift.ma".
 
 (* Advanced properties ******************************************************)
 
-lemma csx_lpx_conf: ∀h,g,G,L1,L2. ⦃G, L1⦄ ⊢ ➡[h, g] L2 →
-                    ∀T. ⦃G, L1⦄ ⊢ ⬊*[h, g] T → ⦃G, L2⦄ ⊢ ⬊*[h, g] T.
-#h #g #G #L1 #L2 #HL12 #T #H @(csx_ind_alt … H) -T
+lemma csx_lpx_conf: ∀h,o,G,L1,L2. ⦃G, L1⦄ ⊢ ➡[h, o] L2 →
+                    ∀T. ⦃G, L1⦄ ⊢ ⬊*[h, o] T → ⦃G, L2⦄ ⊢ ⬊*[h, o] T.
+#h #o #G #L1 #L2 #HL12 #T #H @(csx_ind_alt … H) -T
 /4 width=3 by csx_intro, lpx_cpx_trans/
 qed-.
 
-lemma csx_abst: ∀h,g,a,G,L,W. ⦃G, L⦄ ⊢ ⬊*[h, g] W →
-                ∀T. ⦃G, L.ⓛW⦄ ⊢ ⬊*[h, g] T → ⦃G, L⦄ ⊢ ⬊*[h, g] ⓛ{a}W.T.
-#h #g #a #G #L #W #HW @(csx_ind … HW) -W #W #_ #IHW #T #HT @(csx_ind … HT) -T #T #HT #IHT
+lemma csx_abst: ∀h,o,a,G,L,W. ⦃G, L⦄ ⊢ ⬊*[h, o] W →
+                ∀T. ⦃G, L.ⓛW⦄ ⊢ ⬊*[h, o] T → ⦃G, L⦄ ⊢ ⬊*[h, o] ⓛ{a}W.T.
+#h #o #a #G #L #W #HW @(csx_ind … HW) -W #W #_ #IHW #T #HT @(csx_ind … HT) -T #T #HT #IHT
 @csx_intro #X #H1 #H2
 elim (cpx_inv_abst1 … H1) -H1
 #W0 #T0 #HLW0 #HLT0 #H destruct
@@ -40,9 +40,9 @@ elim (eq_false_inv_tpair_sn … H2) -H2
 ]
 qed.
 
-lemma csx_abbr: ∀h,g,a,G,L,V. ⦃G, L⦄ ⊢ ⬊*[h, g] V →
-                ∀T. ⦃G, L.ⓓV⦄ ⊢ ⬊*[h, g] T → ⦃G, L⦄ ⊢ ⬊*[h, g] ⓓ{a}V. T.
-#h #g #a #G #L #V #HV elim HV -V #V #_ #IHV #T #HT @(csx_ind_alt … HT) -T #T #HT #IHT
+lemma csx_abbr: ∀h,o,a,G,L,V. ⦃G, L⦄ ⊢ ⬊*[h, o] V →
+                ∀T. ⦃G, L.ⓓV⦄ ⊢ ⬊*[h, o] T → ⦃G, L⦄ ⊢ ⬊*[h, o] ⓓ{a}V. T.
+#h #o #a #G #L #V #HV elim HV -V #V #_ #IHV #T #HT @(csx_ind_alt … HT) -T #T #HT #IHT
 @csx_intro #X #H1 #H2
 elim (cpx_inv_abbr1 … H1) -H1 *
 [ #V1 #T1 #HLV1 #HLT1 #H destruct
@@ -55,9 +55,9 @@ elim (cpx_inv_abbr1 … H1) -H1 *
 ]
 qed.
 
-fact csx_appl_beta_aux: ∀h,g,a,G,L,U1. ⦃G, L⦄ ⊢ ⬊*[h, g] U1 →
-                        ∀V,W,T1. U1 = ⓓ{a}ⓝW.V.T1 → ⦃G, L⦄ ⊢ ⬊*[h, g] ⓐV.ⓛ{a}W.T1.
-#h #g #a #G #L #X #H @(csx_ind … H) -X
+fact csx_appl_beta_aux: ∀h,o,a,G,L,U1. ⦃G, L⦄ ⊢ ⬊*[h, o] U1 →
+                        ∀V,W,T1. U1 = ⓓ{a}ⓝW.V.T1 → ⦃G, L⦄ ⊢ ⬊*[h, o] ⓐV.ⓛ{a}W.T1.
+#h #o #a #G #L #X #H @(csx_ind … H) -X
 #X #HT1 #IHT1 #V #W #T1 #H1 destruct
 @csx_intro #X #H1 #H2
 elim (cpx_inv_appl1 … H1) -H1 *
@@ -73,12 +73,12 @@ elim (cpx_inv_appl1 … H1) -H1 *
 qed-.
 
 (* Basic_1: was just: sn3_beta *)
-lemma csx_appl_beta: ∀h,g,a,G,L,V,W,T. ⦃G, L⦄ ⊢ ⬊*[h, g] ⓓ{a}ⓝW.V.T → ⦃G, L⦄ ⊢ ⬊*[h, g] ⓐV.ⓛ{a}W.T.
+lemma csx_appl_beta: ∀h,o,a,G,L,V,W,T. ⦃G, L⦄ ⊢ ⬊*[h, o] ⓓ{a}ⓝW.V.T → ⦃G, L⦄ ⊢ ⬊*[h, o] ⓐV.ⓛ{a}W.T.
 /2 width=3 by csx_appl_beta_aux/ qed.
 
-fact csx_appl_theta_aux: ∀h,g,a,G,L,U. ⦃G, L⦄ ⊢ ⬊*[h, g] U → ∀V1,V2. ⬆[0, 1] V1 ≡ V2 →
-                         ∀V,T. U = ⓓ{a}V.ⓐV2.T → ⦃G, L⦄ ⊢ ⬊*[h, g] ⓐV1.ⓓ{a}V.T.
-#h #g #a #G #L #X #H @(csx_ind_alt … H) -X #X #HVT #IHVT #V1 #V2 #HV12 #V #T #H destruct
+fact csx_appl_theta_aux: ∀h,o,a,G,L,U. ⦃G, L⦄ ⊢ ⬊*[h, o] U → ∀V1,V2. ⬆[0, 1] V1 ≡ V2 →
+                         ∀V,T. U = ⓓ{a}V.ⓐV2.T → ⦃G, L⦄ ⊢ ⬊*[h, o] ⓐV1.ⓓ{a}V.T.
+#h #o #a #G #L #X #H @(csx_ind_alt … H) -X #X #HVT #IHVT #V1 #V2 #HV12 #V #T #H destruct
 lapply (csx_fwd_pair_sn … HVT) #HV
 lapply (csx_fwd_bind_dx … HVT) -HVT #HVT
 @csx_intro #X #HL #H
@@ -113,15 +113,15 @@ elim (cpx_inv_appl1 … HL) -HL *
 ]
 qed-.
 
-lemma csx_appl_theta: ∀h,g,a,V1,V2. ⬆[0, 1] V1 ≡ V2 →
-                      ∀G,L,V,T. ⦃G, L⦄ ⊢ ⬊*[h, g] ⓓ{a}V.ⓐV2.T → ⦃G, L⦄ ⊢ ⬊*[h, g] ⓐV1.ⓓ{a}V.T.
+lemma csx_appl_theta: ∀h,o,a,V1,V2. ⬆[0, 1] V1 ≡ V2 →
+                      ∀G,L,V,T. ⦃G, L⦄ ⊢ ⬊*[h, o] ⓓ{a}V.ⓐV2.T → ⦃G, L⦄ ⊢ ⬊*[h, o] ⓐV1.ⓓ{a}V.T.
 /2 width=5 by csx_appl_theta_aux/ qed.
 
 (* Basic_1: was just: sn3_appl_appl *)
-lemma csx_appl_simple_tsts: ∀h,g,G,L,V. ⦃G, L⦄ ⊢ ⬊*[h, g] V → ∀T1. ⦃G, L⦄ ⊢ ⬊*[h, g] T1 →
-                            (∀T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 → (T1 ≂ T2 → ⊥) → ⦃G, L⦄ ⊢ ⬊*[h, g] ⓐV.T2) →
-                            𝐒⦃T1⦄ → ⦃G, L⦄ ⊢ ⬊*[h, g] ⓐV.T1.
-#h #g #G #L #V #H @(csx_ind … H) -V #V #_ #IHV #T1 #H @(csx_ind … H) -T1 #T1 #H1T1 #IHT1 #H2T1 #H3T1
+lemma csx_appl_simple_tsts: ∀h,o,G,L,V. ⦃G, L⦄ ⊢ ⬊*[h, o] V → ∀T1. ⦃G, L⦄ ⊢ ⬊*[h, o] T1 →
+                            (∀T2. ⦃G, L⦄ ⊢ T1 ➡*[h, o] T2 → (T1 ≂ T2 → ⊥) → ⦃G, L⦄ ⊢ ⬊*[h, o] ⓐV.T2) →
+                            𝐒⦃T1⦄ → ⦃G, L⦄ ⊢ ⬊*[h, o] ⓐV.T1.
+#h #o #G #L #V #H @(csx_ind … H) -V #V #_ #IHV #T1 #H @(csx_ind … H) -T1 #T1 #H1T1 #IHT1 #H2T1 #H3T1
 @csx_intro #X #HL #H
 elim (cpx_inv_appl1_simple … HL) -HL //
 #V0 #T0 #HLV0 #HLT10 #H0 destruct
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/csx_lpxs.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/csx_lpxs.ma
index d49b6e7ae..9d06a4142 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/csx_lpxs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/csx_lpxs.ma
@@ -19,7 +19,7 @@ include "basic_2/computation/lpxs.ma".
 
 (* Properties on sn extended parallel computation for local environments ****)
 
-lemma csx_lpxs_conf: ∀h,g,G,L1,L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 →
-                     ∀T. ⦃G, L1⦄ ⊢ ⬊*[h, g] T → ⦃G, L2⦄ ⊢ ⬊*[h, g] T.
-#h #g #G #L1 #L2 #H @(lpxs_ind … H) -L2 /3 by lpxs_strap1, csx_lpx_conf/
+lemma csx_lpxs_conf: ∀h,o,G,L1,L2. ⦃G, L1⦄ ⊢ ➡*[h, o] L2 →
+                     ∀T. ⦃G, L1⦄ ⊢ ⬊*[h, o] T → ⦃G, L2⦄ ⊢ ⬊*[h, o] T.
+#h #o #G #L1 #L2 #H @(lpxs_ind … H) -L2 /3 by lpxs_strap1, csx_lpx_conf/
 qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/csx_tsts_vector.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/csx_tsts_vector.ma
index d420563d7..8e3aa6e3f 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/csx_tsts_vector.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/csx_tsts_vector.ma
@@ -22,9 +22,9 @@ include "basic_2/computation/csx_vector.ma".
 (* Advanced properties ******************************************************)
 
 (* Basic_1: was just: sn3_appls_lref *)
-lemma csx_applv_cnx: ∀h,g,G,L,T. 𝐒⦃T⦄ → ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃T⦄ →
-                     ∀Vs. ⦃G, L⦄ ⊢ ⬊*[h, g] Vs → ⦃G, L⦄ ⊢ ⬊*[h, g] ⒶVs.T.
-#h #g #G #L #T #H1T #H2T #Vs elim Vs -Vs [ #_ @(cnx_csx … H2T) ] (**) (* /2 width=1/ does not work *)
+lemma csx_applv_cnx: ∀h,o,G,L,T. 𝐒⦃T⦄ → ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃T⦄ →
+                     ∀Vs. ⦃G, L⦄ ⊢ ⬊*[h, o] Vs → ⦃G, L⦄ ⊢ ⬊*[h, o] ⒶVs.T.
+#h #o #G #L #T #H1T #H2T #Vs elim Vs -Vs [ #_ @(cnx_csx … H2T) ] (**) (* /2 width=1/ does not work *)
 #V #Vs #IHV #H
 elim (csxv_inv_cons … H) -H #HV #HVs
 @csx_appl_simple_tsts /2 width=1 by applv_simple/ -IHV -HV -HVs
@@ -33,10 +33,10 @@ lapply (cpxs_fwd_cnx_vector … H) -H // -H1T -H2T #H
 elim (H0) -H0 //
 qed.
 
-lemma csx_applv_sort: ∀h,g,G,L,k,Vs. ⦃G, L⦄ ⊢ ⬊*[h, g] Vs → ⦃G, L⦄ ⊢ ⬊*[h, g] ⒶVs.⋆k.
-#h #g #G #L #k elim (deg_total h g k)
-#d generalize in match k; -k @(nat_ind_plus … d) -d [ /3 width=6 by csx_applv_cnx, cnx_sort, simple_atom/ ]
-#d #IHd #k #Hkd lapply (deg_next_SO … Hkd) -Hkd
+lemma csx_applv_sort: ∀h,o,G,L,s,Vs. ⦃G, L⦄ ⊢ ⬊*[h, o] Vs → ⦃G, L⦄ ⊢ ⬊*[h, o] ⒶVs.⋆s.
+#h #o #G #L #s elim (deg_total h o s)
+#d generalize in match s; -s @(nat_ind_plus … d) -d [ /3 width=6 by csx_applv_cnx, cnx_sort, simple_atom/ ]
+#d #IHd #s #Hkd lapply (deg_next_SO … Hkd) -Hkd
 #Hkd #Vs elim Vs -Vs /2 width=1 by/
 #V #Vs #IHVs #HVVs
 elim (csxv_inv_cons … HVVs) #HV #HVs
@@ -44,14 +44,14 @@ elim (csxv_inv_cons … HVVs) #HV #HVs
 #X #H #H0
 elim (cpxs_fwd_sort_vector … H) -H #H
 [ elim H0 -H0 //
-| -H0 @(csx_cpxs_trans … (Ⓐ(V@Vs).⋆(next h k))) /2 width=1 by cpxs_flat_dx/
+| -H0 @(csx_cpxs_trans … (Ⓐ(V@Vs).⋆(next h s))) /2 width=1 by cpxs_flat_dx/
 ]
 qed.
 
 (* Basic_1: was just: sn3_appls_beta *)
-lemma csx_applv_beta: ∀h,g,a,G,L,Vs,V,W,T. ⦃G, L⦄ ⊢ ⬊*[h, g] ⒶVs.ⓓ{a}ⓝW.V.T →
-                      ⦃G, L⦄ ⊢ ⬊*[h, g] ⒶVs. ⓐV.ⓛ{a}W.T.
-#h #g #a #G #L #Vs elim Vs -Vs /2 width=1 by csx_appl_beta/
+lemma csx_applv_beta: ∀h,o,a,G,L,Vs,V,W,T. ⦃G, L⦄ ⊢ ⬊*[h, o] ⒶVs.ⓓ{a}ⓝW.V.T →
+                      ⦃G, L⦄ ⊢ ⬊*[h, o] ⒶVs. ⓐV.ⓛ{a}W.T.
+#h #o #a #G #L #Vs elim Vs -Vs /2 width=1 by csx_appl_beta/
 #V0 #Vs #IHV #V #W #T #H1T
 lapply (csx_fwd_pair_sn … H1T) #HV0
 lapply (csx_fwd_flat_dx … H1T) #H2T
@@ -63,10 +63,10 @@ elim (cpxs_fwd_beta_vector … H) -H #H
 ]
 qed.
 
-lemma csx_applv_delta: ∀h,g,I,G,L,K,V1,i. ⬇[i] L ≡ K.ⓑ{I}V1 →
+lemma csx_applv_delta: ∀h,o,I,G,L,K,V1,i. ⬇[i] L ≡ K.ⓑ{I}V1 →
                        ∀V2. ⬆[0, i + 1] V1 ≡ V2 →
-                       ∀Vs. ⦃G, L⦄ ⊢ ⬊*[h, g] (ⒶVs.V2) → ⦃G, L⦄ ⊢ ⬊*[h, g] (ⒶVs.#i).
-#h #g #I #G #L #K #V1 #i #HLK #V2 #HV12 #Vs elim Vs -Vs
+                       ∀Vs. ⦃G, L⦄ ⊢ ⬊*[h, o] (ⒶVs.V2) → ⦃G, L⦄ ⊢ ⬊*[h, o] (ⒶVs.#i).
+#h #o #I #G #L #K #V1 #i #HLK #V2 #HV12 #Vs elim Vs -Vs
 [ /4 width=12 by csx_inv_lift, csx_lref_bind, drop_fwd_drop2/
 | #V #Vs #IHV #H1T
   lapply (csx_fwd_pair_sn … H1T) #HV
@@ -81,28 +81,28 @@ lemma csx_applv_delta: ∀h,g,I,G,L,K,V1,i. ⬇[i] L ≡ K.ⓑ{I}V1 →
 qed.
 
 (* Basic_1: was just: sn3_appls_abbr *)
-lemma csx_applv_theta: ∀h,g,a,G,L,V1s,V2s. ⬆[0, 1] V1s ≡ V2s →
-                       ∀V,T. ⦃G, L⦄ ⊢ ⬊*[h, g] ⓓ{a}V.ⒶV2s.T →
-                       ⦃G, L⦄ ⊢ ⬊*[h, g] ⒶV1s.ⓓ{a}V.T.
-#h #g #a #G #L #V1s #V2s * -V1s -V2s /2 width=1 by/
-#V1s #V2s #V1 #V2 #HV12 #H
+lemma csx_applv_theta: ∀h,o,a,G,L,V1c,V2c. ⬆[0, 1] V1c ≡ V2c →
+                       ∀V,T. ⦃G, L⦄ ⊢ ⬊*[h, o] ⓓ{a}V.ⒶV2c.T →
+                       ⦃G, L⦄ ⊢ ⬊*[h, o] ⒶV1c.ⓓ{a}V.T.
+#h #o #a #G #L #V1c #V2c * -V1c -V2c /2 width=1 by/
+#V1c #V2c #V1 #V2 #HV12 #H
 generalize in match HV12; -HV12 generalize in match V2; -V2 generalize in match V1; -V1
-elim H -V1s -V2s /2 width=3 by csx_appl_theta/
-#V1s #V2s #V1 #V2 #HV12 #HV12s #IHV12s #W1 #W2 #HW12 #V #T #H
+elim H -V1c -V2c /2 width=3 by csx_appl_theta/
+#V1c #V2c #V1 #V2 #HV12 #HV12c #IHV12c #W1 #W2 #HW12 #V #T #H
 lapply (csx_appl_theta … HW12 … H) -H -HW12 #H
 lapply (csx_fwd_pair_sn … H) #HW1
 lapply (csx_fwd_flat_dx … H) #H1
-@csx_appl_simple_tsts /2 width=3 by simple_flat/ -IHV12s -HW1 -H1 #X #H1 #H2
-elim (cpxs_fwd_theta_vector … (V2@V2s) … H1) -H1 /2 width=1 by liftv_cons/ -HV12s -HV12
+@csx_appl_simple_tsts /2 width=3 by simple_flat/ -IHV12c -HW1 -H1 #X #H1 #H2
+elim (cpxs_fwd_theta_vector … (V2@V2c) … H1) -H1 /2 width=1 by liftv_cons/ -HV12c -HV12
 [ -H #H elim H2 -H2 //
 | -H2 /3 width=5 by csx_cpxs_trans, cpxs_flat_dx/
 ]
 qed.
 
 (* Basic_1: was just: sn3_appls_cast *)
-lemma csx_applv_cast: ∀h,g,G,L,Vs,W,T. ⦃G, L⦄ ⊢ ⬊*[h, g] ⒶVs.W → ⦃G, L⦄ ⊢ ⬊*[h, g] ⒶVs.T →
-                      ⦃G, L⦄ ⊢ ⬊*[h, g] ⒶVs.ⓝW.T.
-#h #g #G #L #Vs elim Vs -Vs /2 width=1 by csx_cast/
+lemma csx_applv_cast: ∀h,o,G,L,Vs,W,T. ⦃G, L⦄ ⊢ ⬊*[h, o] ⒶVs.W → ⦃G, L⦄ ⊢ ⬊*[h, o] ⒶVs.T →
+                      ⦃G, L⦄ ⊢ ⬊*[h, o] ⒶVs.ⓝW.T.
+#h #o #G #L #Vs elim Vs -Vs /2 width=1 by csx_cast/
 #V #Vs #IHV #W #T #H1W #H1T
 lapply (csx_fwd_pair_sn … H1W) #HV
 lapply (csx_fwd_flat_dx … H1W) #H2W
@@ -116,13 +116,13 @@ elim (cpxs_fwd_cast_vector … H) -H #H
 ]
 qed.
 
-theorem csx_gcr: ∀h,g. gcr (cpx h g) (eq …) (csx h g) (csx h g).
-#h #g @mk_gcr //
+theorem csx_gcr: ∀h,o. gcr (cpx h o) (eq …) (csx h o) (csx h o).
+#h #o @mk_gcr //
 [ /3 width=1 by csx_applv_cnx/
 |2,3,6: /2 width=1 by csx_applv_beta, csx_applv_sort, csx_applv_cast/
 | /2 width=7 by csx_applv_delta/
-| #G #L #V1s #V2s #HV12s #a #V #T #H #HV
-  @(csx_applv_theta … HV12s) -HV12s
+| #G #L #V1c #V2c #HV12c #a #V #T #H #HV
+  @(csx_applv_theta … HV12c) -HV12c
   @csx_abbr //
 ]
 qed.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/csx_vector.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/csx_vector.ma
index 5b4d465cb..95269945f 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/csx_vector.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/csx_vector.ma
@@ -18,23 +18,23 @@ include "basic_2/computation/csx.ma".
 (* CONTEXT-SENSITIVE EXTENDED STRONGLY NORMALIZING TERM VECTORS *************)
 
 definition csxv: ∀h. sd h → relation3 genv lenv (list term) ≝
-                 λh,g,G,L. all … (csx h g G L).
+                 λh,o,G,L. all … (csx h o G L).
 
 interpretation
    "context-sensitive strong normalization (term vector)"
-   'SN h g G L Ts = (csxv h g G L Ts).
+   'SN h o G L Ts = (csxv h o G L Ts).
 
 (* Basic inversion lemmas ***************************************************)
 
-lemma csxv_inv_cons: ∀h,g,G,L,T,Ts. ⦃G, L⦄ ⊢ ⬊*[h, g] T @ Ts →
-                     ⦃G, L⦄ ⊢ ⬊*[h, g] T ∧ ⦃G, L⦄ ⊢ ⬊*[h, g] Ts.
+lemma csxv_inv_cons: ∀h,o,G,L,T,Ts. ⦃G, L⦄ ⊢ ⬊*[h, o] T @ Ts →
+                     ⦃G, L⦄ ⊢ ⬊*[h, o] T ∧ ⦃G, L⦄ ⊢ ⬊*[h, o] Ts.
 normalize // qed-.
 
 (* Basic forward lemmas *****************************************************)
 
-lemma csx_fwd_applv: ∀h,g,G,L,T,Vs. ⦃G, L⦄ ⊢ ⬊*[h, g] Ⓐ Vs.T →
-                     ⦃G, L⦄ ⊢ ⬊*[h, g] Vs ∧ ⦃G, L⦄ ⊢ ⬊*[h, g] T.
-#h #g #G #L #T #Vs elim Vs -Vs /2 width=1 by conj/
+lemma csx_fwd_applv: ∀h,o,G,L,T,Vs. ⦃G, L⦄ ⊢ ⬊*[h, o] Ⓐ Vs.T →
+                     ⦃G, L⦄ ⊢ ⬊*[h, o] Vs ∧ ⦃G, L⦄ ⊢ ⬊*[h, o] T.
+#h #o #G #L #T #Vs elim Vs -Vs /2 width=1 by conj/
 #V #Vs #IHVs #HVs
 lapply (csx_fwd_pair_sn … HVs) #HV
 lapply (csx_fwd_flat_dx … HVs) -HVs #HVs
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/fpbg.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/fpbg.ma
index 5c8a6bd20..c1e2b94c1 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/fpbg.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/fpbg.ma
@@ -19,21 +19,21 @@ include "basic_2/computation/fpbs.ma".
 (* "QRST" PROPER PARALLEL COMPUTATION FOR CLOSURES **************************)
 
 definition fpbg: ∀h. sd h → tri_relation genv lenv term ≝
-                 λh,g,G1,L1,T1,G2,L2,T2.
-                 ∃∃G,L,T. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G, L, T⦄ & ⦃G, L, T⦄ ≥[h, g] ⦃G2, L2, T2⦄.
+                 λh,o,G1,L1,T1,G2,L2,T2.
+                 ∃∃G,L,T. ⦃G1, L1, T1⦄ ≻[h, o] ⦃G, L, T⦄ & ⦃G, L, T⦄ ≥[h, o] ⦃G2, L2, T2⦄.
 
 interpretation "'qrst' proper parallel computation (closure)"
-   'LazyBTPRedStarProper h g G1 L1 T1 G2 L2 T2 = (fpbg h g G1 L1 T1 G2 L2 T2).
+   'LazyBTPRedStarProper h o G1 L1 T1 G2 L2 T2 = (fpbg h o G1 L1 T1 G2 L2 T2).
 
 (* Basic properties *********************************************************)
 
-lemma fpb_fpbg: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G2, L2, T2⦄ →
-                ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
+lemma fpb_fpbg: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻[h, o] ⦃G2, L2, T2⦄ →
+                ⦃G1, L1, T1⦄ >≡[h, o] ⦃G2, L2, T2⦄.
 /2 width=5 by ex2_3_intro/ qed.
 
-lemma fpbg_fpbq_trans: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2.
-                       ⦃G1, L1, T1⦄ >≡[h, g] ⦃G, L, T⦄ → ⦃G, L, T⦄ ≽[h, g] ⦃G2, L2, T2⦄ →
-                       ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
-#h #g #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 *
+lemma fpbg_fpbq_trans: ∀h,o,G1,G,G2,L1,L,L2,T1,T,T2.
+                       ⦃G1, L1, T1⦄ >≡[h, o] ⦃G, L, T⦄ → ⦃G, L, T⦄ ≽[h, o] ⦃G2, L2, T2⦄ →
+                       ⦃G1, L1, T1⦄ >≡[h, o] ⦃G2, L2, T2⦄.
+#h #o #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 *
 /3 width=9 by fpbs_strap1, ex2_3_intro/
 qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/fpbg_fleq.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/fpbg_fleq.ma
index 71ed202d7..ed03b2549 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/fpbg_fleq.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/fpbg_fleq.ma
@@ -20,34 +20,34 @@ include "basic_2/computation/fpbg.ma".
 
 (* Properties on lazy equivalence for closures ******************************)
 
-lemma fpbg_fleq_trans: ∀h,g,G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ >≡[h, g] ⦃G, L, T⦄ →
-                       ∀G2,L2,T2. ⦃G, L, T⦄ ≡[0] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
+lemma fpbg_fleq_trans: ∀h,o,G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ >≡[h, o] ⦃G, L, T⦄ →
+                       ∀G2,L2,T2. ⦃G, L, T⦄ ≡[0] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >≡[h, o] ⦃G2, L2, T2⦄.
 /3 width=5 by fpbg_fpbq_trans, fleq_fpbq/ qed-.
 
-lemma fleq_fpbg_trans: ∀h,g,G,G2,L,L2,T,T2. ⦃G, L, T⦄ >≡[h, g] ⦃G2, L2, T2⦄ →
-                       ∀G1,L1,T1. ⦃G1, L1, T1⦄ ≡[0] ⦃G, L, T⦄ → ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
-#h #g #G #G2 #L #L2 #T #T2 * #G0 #L0 #T0 #H0 #H02 #G1 #L1 #T1 #H1
+lemma fleq_fpbg_trans: ∀h,o,G,G2,L,L2,T,T2. ⦃G, L, T⦄ >≡[h, o] ⦃G2, L2, T2⦄ →
+                       ∀G1,L1,T1. ⦃G1, L1, T1⦄ ≡[0] ⦃G, L, T⦄ → ⦃G1, L1, T1⦄ >≡[h, o] ⦃G2, L2, T2⦄.
+#h #o #G #G2 #L #L2 #T #T2 * #G0 #L0 #T0 #H0 #H02 #G1 #L1 #T1 #H1
 elim (fleq_fpb_trans …  H1 … H0) -G -L -T
 /4 width=9 by fpbs_strap2, fleq_fpbq, ex2_3_intro/
 qed-.
 
 (* alternative definition of fpbs *******************************************)
 
-lemma fleq_fpbs: ∀h,g,G1,G2,L1,L2,T1,T2.
-                 ⦃G1, L1, T1⦄ ≡[0] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 * /2 width=1 by lleq_fpbs/
+lemma fleq_fpbs: ∀h,o,G1,G2,L1,L2,T1,T2.
+                 ⦃G1, L1, T1⦄ ≡[0] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 * /2 width=1 by lleq_fpbs/
 qed.
 
-lemma fpbg_fwd_fpbs: ∀h,g,G1,G2,L1,L2,T1,T2.
-                     ⦃G1, L1, T1⦄ >≡[h,g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 *
+lemma fpbg_fwd_fpbs: ∀h,o,G1,G2,L1,L2,T1,T2.
+                     ⦃G1, L1, T1⦄ >≡[h,o] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 *
 /3 width=5 by fpbs_strap2, fpb_fpbq/
 qed-.
 
-lemma fpbs_fpbg: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ →
+lemma fpbs_fpbg: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄ →
                  ⦃G1, L1, T1⦄ ≡[0] ⦃G2, L2, T2⦄ ∨
-                 ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpbs_ind … H) -G2 -L2 -T2
+                 ⦃G1, L1, T1⦄ >≡[h, o] ⦃G2, L2, T2⦄.
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpbs_ind … H) -G2 -L2 -T2
 [ /2 width=1 by or_introl/
 | #G #G2 #L #L2 #T #T2 #_ #H2 * #H1 @(fpbq_ind_alt … H2) -H2 #H2
   [ /3 width=5 by fleq_trans, or_introl/
@@ -61,10 +61,10 @@ qed-.
 
 (* Advanced properties of "qrst" parallel computation on closures ***********)
 
-lemma fpbs_fpb_trans: ∀h,g,F1,F2,K1,K2,T1,T2. ⦃F1, K1, T1⦄ ≥[h, g] ⦃F2, K2, T2⦄ →
-                      ∀G2,L2,U2. ⦃F2, K2, T2⦄ ≻[h, g] ⦃G2, L2, U2⦄ →
-                      ∃∃G1,L1,U1. ⦃F1, K1, T1⦄ ≻[h, g] ⦃G1, L1, U1⦄ & ⦃G1, L1, U1⦄ ≥[h, g] ⦃G2, L2, U2⦄.
-#h #g #F1 #F2 #K1 #K2 #T1 #T2 #H elim (fpbs_fpbg … H) -H
+lemma fpbs_fpb_trans: ∀h,o,F1,F2,K1,K2,T1,T2. ⦃F1, K1, T1⦄ ≥[h, o] ⦃F2, K2, T2⦄ →
+                      ∀G2,L2,U2. ⦃F2, K2, T2⦄ ≻[h, o] ⦃G2, L2, U2⦄ →
+                      ∃∃G1,L1,U1. ⦃F1, K1, T1⦄ ≻[h, o] ⦃G1, L1, U1⦄ & ⦃G1, L1, U1⦄ ≥[h, o] ⦃G2, L2, U2⦄.
+#h #o #F1 #F2 #K1 #K2 #T1 #T2 #H elim (fpbs_fpbg … H) -H
 [ #H12 #G2 #L2 #U2 #H2 elim (fleq_fpb_trans … H12 … H2) -F2 -K2 -T2
   /3 width=5 by fleq_fpbs, ex2_3_intro/
 | * #H1 #H2 #H3 #H4 #H5 #H6 #H7 #H8 #H9
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/fpbg_fpbg.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/fpbg_fpbg.ma
index bbbcc2487..28748f278 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/fpbg_fpbg.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/fpbg_fpbg.ma
@@ -18,5 +18,5 @@ include "basic_2/computation/fpbg_fpbs.ma".
 
 (* Main properties **********************************************************)
 
-theorem fpbg_trans: ∀h,g. tri_transitive … (fpbg h g).
+theorem fpbg_trans: ∀h,o. tri_transitive … (fpbg h o).
 /3 width=5 by fpbg_fpbs_trans, fpbg_fwd_fpbs/ qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/fpbg_fpbs.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/fpbg_fpbs.ma
index e39f3a5a3..df666294c 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/fpbg_fpbs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/fpbg_fpbs.ma
@@ -20,49 +20,49 @@ include "basic_2/computation/fpbg_fleq.ma".
 
 (* Properties on "qrst" parallel reduction on closures **********************)
 
-lemma fpb_fpbg_trans: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2.
-                      ⦃G1, L1, T1⦄ ≻[h, g] ⦃G, L, T⦄ → ⦃G, L, T⦄ >≡[h, g] ⦃G2, L2, T2⦄ →
-                      ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
+lemma fpb_fpbg_trans: ∀h,o,G1,G,G2,L1,L,L2,T1,T,T2.
+                      ⦃G1, L1, T1⦄ ≻[h, o] ⦃G, L, T⦄ → ⦃G, L, T⦄ >≡[h, o] ⦃G2, L2, T2⦄ →
+                      ⦃G1, L1, T1⦄ >≡[h, o] ⦃G2, L2, T2⦄.
 /3 width=5 by fpbg_fwd_fpbs, ex2_3_intro/ qed-.
 
-lemma fpbq_fpbg_trans: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2.
-                       ⦃G1, L1, T1⦄ ≽[h, g] ⦃G, L, T⦄ → ⦃G, L, T⦄ >≡[h, g] ⦃G2, L2, T2⦄ →
-                       ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
-#h #g #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2 @(fpbq_ind_alt … H1) -H1
+lemma fpbq_fpbg_trans: ∀h,o,G1,G,G2,L1,L,L2,T1,T,T2.
+                       ⦃G1, L1, T1⦄ ≽[h, o] ⦃G, L, T⦄ → ⦃G, L, T⦄ >≡[h, o] ⦃G2, L2, T2⦄ →
+                       ⦃G1, L1, T1⦄ >≡[h, o] ⦃G2, L2, T2⦄.
+#h #o #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2 @(fpbq_ind_alt … H1) -H1
 /2 width=5 by fleq_fpbg_trans, fpb_fpbg_trans/
 qed-.
 
 (* Properties on "qrst" parallel compuutation on closures *******************)
 
-lemma fpbs_fpbg_trans: ∀h,g,G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G, L, T⦄ →
-                       ∀G2,L2,T2. ⦃G, L, T⦄ >≡[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
-#h #g #G1 #G #L1 #L #T1 #T #H @(fpbs_ind … H) -G -L -T /3 width=5 by fpbq_fpbg_trans/
+lemma fpbs_fpbg_trans: ∀h,o,G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ≥[h, o] ⦃G, L, T⦄ →
+                       ∀G2,L2,T2. ⦃G, L, T⦄ >≡[h, o] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >≡[h, o] ⦃G2, L2, T2⦄.
+#h #o #G1 #G #L1 #L #T1 #T #H @(fpbs_ind … H) -G -L -T /3 width=5 by fpbq_fpbg_trans/
 qed-.
 
 (* Note: this is used in the closure proof *)
-lemma fpbg_fpbs_trans: ∀h,g,G,G2,L,L2,T,T2. ⦃G, L, T⦄ ≥[h, g] ⦃G2, L2, T2⦄ →
-                       ∀G1,L1,T1. ⦃G1, L1, T1⦄ >≡[h, g] ⦃G, L, T⦄ → ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
-#h #g #G #G2 #L #L2 #T #T2 #H @(fpbs_ind_dx … H) -G -L -T /3 width=5 by fpbg_fpbq_trans/
+lemma fpbg_fpbs_trans: ∀h,o,G,G2,L,L2,T,T2. ⦃G, L, T⦄ ≥[h, o] ⦃G2, L2, T2⦄ →
+                       ∀G1,L1,T1. ⦃G1, L1, T1⦄ >≡[h, o] ⦃G, L, T⦄ → ⦃G1, L1, T1⦄ >≡[h, o] ⦃G2, L2, T2⦄.
+#h #o #G #G2 #L #L2 #T #T2 #H @(fpbs_ind_dx … H) -G -L -T /3 width=5 by fpbg_fpbq_trans/
 qed-.
 
 (* Note: this is used in the closure proof *)
-lemma fqup_fpbg: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim (fqup_inv_step_sn … H) -H
+lemma fqup_fpbg: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >≡[h, o] ⦃G2, L2, T2⦄.
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H elim (fqup_inv_step_sn … H) -H
 /3 width=5 by fqus_fpbs, fpb_fqu, ex2_3_intro/
 qed.
 
-lemma cpxs_fpbg: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 →
-                 (T1 = T2 → ⊥) → ⦃G, L, T1⦄ >≡[h, g] ⦃G, L, T2⦄.
-#h #g #G #L #T1 #T2 #H #H0 elim (cpxs_neq_inv_step_sn … H … H0) -H -H0
+lemma cpxs_fpbg: ∀h,o,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h, o] T2 →
+                 (T1 = T2 → ⊥) → ⦃G, L, T1⦄ >≡[h, o] ⦃G, L, T2⦄.
+#h #o #G #L #T1 #T2 #H #H0 elim (cpxs_neq_inv_step_sn … H … H0) -H -H0
 /4 width=5 by cpxs_fpbs, fpb_cpx, ex2_3_intro/
 qed.
 
-lemma lstas_fpbg: ∀h,g,G,L,T1,T2,d2. ⦃G, L⦄ ⊢ T1 •*[h, d2] T2 → (T1 = T2 → ⊥) →
-                  ∀d1. d2 ≤ d1 → ⦃G, L⦄ ⊢ T1 ▪[h, g] d1 → ⦃G, L, T1⦄ >≡[h, g] ⦃G, L, T2⦄.
+lemma lstas_fpbg: ∀h,o,G,L,T1,T2,d2. ⦃G, L⦄ ⊢ T1 •*[h, d2] T2 → (T1 = T2 → ⊥) →
+                  ∀d1. d2 ≤ d1 → ⦃G, L⦄ ⊢ T1 ▪[h, o] d1 → ⦃G, L, T1⦄ >≡[h, o] ⦃G, L, T2⦄.
 /3 width=5 by lstas_cpxs, cpxs_fpbg/ qed.
 
-lemma lpxs_fpbg: ∀h,g,G,L1,L2,T. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 →
-                 (L1 ≡[T, 0] L2 → ⊥) → ⦃G, L1, T⦄ >≡[h, g] ⦃G, L2, T⦄.
-#h #g #G #L1 #L2 #T #H #H0 elim (lpxs_nlleq_inv_step_sn … H … H0) -H -H0
+lemma lpxs_fpbg: ∀h,o,G,L1,L2,T. ⦃G, L1⦄ ⊢ ➡*[h, o] L2 →
+                 (L1 ≡[T, 0] L2 → ⊥) → ⦃G, L1, T⦄ >≡[h, o] ⦃G, L2, T⦄.
+#h #o #G #L1 #L2 #T #H #H0 elim (lpxs_nlleq_inv_step_sn … H … H0) -H -H0
 /4 width=5 by fpb_lpx, lpxs_lleq_fpbs, ex2_3_intro/
 qed.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/fpbg_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/fpbg_lift.ma
index d34676e48..a0f4edbb8 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/fpbg_lift.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/fpbg_lift.ma
@@ -19,6 +19,6 @@ include "basic_2/computation/fpbg.ma".
 
 (* Advanced properties ******************************************************)
 
-lemma sta_fpbg: ∀h,g,G,L,T1,T2,d. ⦃G, L⦄ ⊢ T1 ▪[h, g] d+1 →
-                ⦃G, L⦄ ⊢ T1 •*[h, 1] T2 → ⦃G, L, T1⦄ >≡[h, g] ⦃G, L, T2⦄.
+lemma sta_fpbg: ∀h,o,G,L,T1,T2,d. ⦃G, L⦄ ⊢ T1 ▪[h, o] d+1 →
+                ⦃G, L⦄ ⊢ T1 •*[h, 1] T2 → ⦃G, L, T1⦄ >≡[h, o] ⦃G, L, T2⦄.
 /4 width=2 by fpb_fpbg, sta_fpb/ qed.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/fpbs.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/fpbs.ma
index 79670f2a2..125efb684 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/fpbs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/fpbs.ma
@@ -21,141 +21,141 @@ include "basic_2/computation/lpxs.ma".
 (* "QRST" PARALLEL COMPUTATION FOR CLOSURES *********************************)
 
 definition fpbs: ∀h. sd h → tri_relation genv lenv term ≝
-                 λh,g. tri_TC … (fpbq h g).
+                 λh,o. tri_TC … (fpbq h o).
 
 interpretation "'qrst' parallel computation (closure)"
-   'BTPRedStar h g G1 L1 T1 G2 L2 T2 = (fpbs h g G1 L1 T1 G2 L2 T2).
+   'BTPRedStar h o G1 L1 T1 G2 L2 T2 = (fpbs h o G1 L1 T1 G2 L2 T2).
 
 (* Basic eliminators ********************************************************)
 
-lemma fpbs_ind: ∀h,g,G1,L1,T1. ∀R:relation3 genv lenv term. R G1 L1 T1 →
-                (∀G,G2,L,L2,T,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G, L, T⦄ → ⦃G, L, T⦄ ≽[h, g] ⦃G2, L2, T2⦄ → R G L T → R G2 L2 T2) →
-                ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ → R G2 L2 T2.
+lemma fpbs_ind: ∀h,o,G1,L1,T1. ∀R:relation3 genv lenv term. R G1 L1 T1 →
+                (∀G,G2,L,L2,T,T2. ⦃G1, L1, T1⦄ ≥[h, o] ⦃G, L, T⦄ → ⦃G, L, T⦄ ≽[h, o] ⦃G2, L2, T2⦄ → R G L T → R G2 L2 T2) →
+                ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄ → R G2 L2 T2.
 /3 width=8 by tri_TC_star_ind/ qed-.
 
-lemma fpbs_ind_dx: ∀h,g,G2,L2,T2. ∀R:relation3 genv lenv term. R G2 L2 T2 →
-                   (∀G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ≽[h, g] ⦃G, L, T⦄ → ⦃G, L, T⦄ ≥[h, g] ⦃G2, L2, T2⦄ → R G L T → R G1 L1 T1) →
-                   ∀G1,L1,T1. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ → R G1 L1 T1.
+lemma fpbs_ind_dx: ∀h,o,G2,L2,T2. ∀R:relation3 genv lenv term. R G2 L2 T2 →
+                   (∀G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ≽[h, o] ⦃G, L, T⦄ → ⦃G, L, T⦄ ≥[h, o] ⦃G2, L2, T2⦄ → R G L T → R G1 L1 T1) →
+                   ∀G1,L1,T1. ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄ → R G1 L1 T1.
 /3 width=8 by tri_TC_star_ind_dx/ qed-.
 
 (* Basic properties *********************************************************)
 
-lemma fpbs_refl: ∀h,g. tri_reflexive … (fpbs h g).
+lemma fpbs_refl: ∀h,o. tri_reflexive … (fpbs h o).
 /2 width=1 by tri_inj/ qed.
 
-lemma fpbq_fpbs: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≽[h, g] ⦃G2, L2, T2⦄ →
-                 ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
+lemma fpbq_fpbs: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≽[h, o] ⦃G2, L2, T2⦄ →
+                 ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
 /2 width=1 by tri_inj/ qed.
 
-lemma fpbs_strap1: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G, L, T⦄ →
-                   ⦃G, L, T⦄ ≽[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
+lemma fpbs_strap1: ∀h,o,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ≥[h, o] ⦃G, L, T⦄ →
+                   ⦃G, L, T⦄ ≽[h, o] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
 /2 width=5 by tri_step/ qed-.
 
-lemma fpbs_strap2: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ≽[h, g] ⦃G, L, T⦄ →
-                   ⦃G, L, T⦄ ≥[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
+lemma fpbs_strap2: ∀h,o,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ≽[h, o] ⦃G, L, T⦄ →
+                   ⦃G, L, T⦄ ≥[h, o] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
 /2 width=5 by tri_TC_strap/ qed-.
 
-lemma fqup_fpbs: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
-                 ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2 
+lemma fqup_fpbs: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
+                 ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2 
 /4 width=5 by fqu_fquq, fpbq_fquq, tri_step/
 qed.
 
-lemma fqus_fpbs: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
-                 ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqus_ind … H) -G2 -L2 -T2 
+lemma fqus_fpbs: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
+                 ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqus_ind … H) -G2 -L2 -T2 
 /3 width=5 by fpbq_fquq, tri_step/
 qed.
 
-lemma cpxs_fpbs: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 → ⦃G, L, T1⦄ ≥[h, g] ⦃G, L, T2⦄.
-#h #g #G #L #T1 #T2 #H @(cpxs_ind … H) -T2
+lemma cpxs_fpbs: ∀h,o,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h, o] T2 → ⦃G, L, T1⦄ ≥[h, o] ⦃G, L, T2⦄.
+#h #o #G #L #T1 #T2 #H @(cpxs_ind … H) -T2
 /3 width=5 by fpbq_cpx, fpbs_strap1/
 qed.
 
-lemma lpxs_fpbs: ∀h,g,G,L1,L2,T. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 → ⦃G, L1, T⦄ ≥[h, g] ⦃G, L2, T⦄.
-#h #g #G #L1 #L2 #T #H @(lpxs_ind … H) -L2
+lemma lpxs_fpbs: ∀h,o,G,L1,L2,T. ⦃G, L1⦄ ⊢ ➡*[h, o] L2 → ⦃G, L1, T⦄ ≥[h, o] ⦃G, L2, T⦄.
+#h #o #G #L1 #L2 #T #H @(lpxs_ind … H) -L2
 /3 width=5 by fpbq_lpx, fpbs_strap1/
 qed.
 
-lemma lleq_fpbs: ∀h,g,G,L1,L2,T. L1 ≡[T, 0] L2 → ⦃G, L1, T⦄ ≥[h, g] ⦃G, L2, T⦄.
+lemma lleq_fpbs: ∀h,o,G,L1,L2,T. L1 ≡[T, 0] L2 → ⦃G, L1, T⦄ ≥[h, o] ⦃G, L2, T⦄.
 /3 width=1 by fpbq_fpbs, fpbq_lleq/ qed.
 
-lemma cprs_fpbs: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡* T2 → ⦃G, L, T1⦄ ≥[h, g] ⦃G, L, T2⦄.
+lemma cprs_fpbs: ∀h,o,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡* T2 → ⦃G, L, T1⦄ ≥[h, o] ⦃G, L, T2⦄.
 /3 width=1 by cprs_cpxs, cpxs_fpbs/ qed.
 
-lemma lprs_fpbs: ∀h,g,G,L1,L2,T. ⦃G, L1⦄ ⊢ ➡* L2 → ⦃G, L1, T⦄ ≥[h, g] ⦃G, L2, T⦄.
+lemma lprs_fpbs: ∀h,o,G,L1,L2,T. ⦃G, L1⦄ ⊢ ➡* L2 → ⦃G, L1, T⦄ ≥[h, o] ⦃G, L2, T⦄.
 /3 width=1 by lprs_lpxs, lpxs_fpbs/ qed.
 
-lemma fpbs_fqus_trans: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G, L, T⦄ →
-                       ⦃G, L, T⦄ ⊐* ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
-#h #g #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H @(fqus_ind … H) -G2 -L2 -T2
+lemma fpbs_fqus_trans: ∀h,o,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ≥[h, o] ⦃G, L, T⦄ →
+                       ⦃G, L, T⦄ ⊐* ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
+#h #o #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H @(fqus_ind … H) -G2 -L2 -T2
 /3 width=5 by fpbs_strap1, fpbq_fquq/
 qed-.
 
-lemma fpbs_fqup_trans: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G, L, T⦄ →
-                       ⦃G, L, T⦄ ⊐+ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
+lemma fpbs_fqup_trans: ∀h,o,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ≥[h, o] ⦃G, L, T⦄ →
+                       ⦃G, L, T⦄ ⊐+ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
 /3 width=5 by fpbs_fqus_trans, fqup_fqus/ qed-.
 
-lemma fpbs_cpxs_trans: ∀h,g,G1,G,L1,L,T1,T,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G, L, T⦄ →
-                       ⦃G, L⦄ ⊢ T ➡*[h, g] T2 → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G, L, T2⦄.
-#h #g #G1 #G #L1 #L #T1 #T #T2 #H1 #H @(cpxs_ind … H) -T2
+lemma fpbs_cpxs_trans: ∀h,o,G1,G,L1,L,T1,T,T2. ⦃G1, L1, T1⦄ ≥[h, o] ⦃G, L, T⦄ →
+                       ⦃G, L⦄ ⊢ T ➡*[h, o] T2 → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G, L, T2⦄.
+#h #o #G1 #G #L1 #L #T1 #T #T2 #H1 #H @(cpxs_ind … H) -T2
 /3 width=5 by fpbs_strap1, fpbq_cpx/
 qed-.
 
-lemma fpbs_lpxs_trans: ∀h,g,G1,G,L1,L,L2,T1,T. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G, L, T⦄ →
-                       ⦃G, L⦄ ⊢ ➡*[h, g] L2 → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G, L2, T⦄.
-#h #g #G1 #G #L1 #L #L2 #T1 #T #H1 #H @(lpxs_ind … H) -L2
+lemma fpbs_lpxs_trans: ∀h,o,G1,G,L1,L,L2,T1,T. ⦃G1, L1, T1⦄ ≥[h, o] ⦃G, L, T⦄ →
+                       ⦃G, L⦄ ⊢ ➡*[h, o] L2 → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G, L2, T⦄.
+#h #o #G1 #G #L1 #L #L2 #T1 #T #H1 #H @(lpxs_ind … H) -L2
 /3 width=5 by fpbs_strap1, fpbq_lpx/
 qed-.
 
-lemma fpbs_lleq_trans: ∀h,g,G1,G,L1,L,L2,T1,T. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G, L, T⦄ →
-                       L ≡[T, 0] L2 → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G, L2, T⦄.
+lemma fpbs_lleq_trans: ∀h,o,G1,G,L1,L,L2,T1,T. ⦃G1, L1, T1⦄ ≥[h, o] ⦃G, L, T⦄ →
+                       L ≡[T, 0] L2 → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G, L2, T⦄.
 /3 width=5 by fpbs_strap1, fpbq_lleq/ qed-.
 
-lemma fqus_fpbs_trans: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G, L, T⦄ ≥[h, g] ⦃G2, L2, T2⦄ →
-                       ⦃G1, L1, T1⦄ ⊐* ⦃G, L, T⦄ → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
-#h #g #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H @(fqus_ind_dx … H) -G1 -L1 -T1
+lemma fqus_fpbs_trans: ∀h,o,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G, L, T⦄ ≥[h, o] ⦃G2, L2, T2⦄ →
+                       ⦃G1, L1, T1⦄ ⊐* ⦃G, L, T⦄ → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
+#h #o #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H @(fqus_ind_dx … H) -G1 -L1 -T1
 /3 width=5 by fpbs_strap2, fpbq_fquq/
 qed-.
 
-lemma cpxs_fpbs_trans: ∀h,g,G1,G2,L1,L2,T1,T,T2. ⦃G1, L1, T⦄ ≥[h, g] ⦃G2, L2, T2⦄ →
-                       ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] T → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
-#h #g #G1 #G2 #L1 #L2 #T1 #T #T2 #H1 #H @(cpxs_ind_dx … H) -T1
+lemma cpxs_fpbs_trans: ∀h,o,G1,G2,L1,L2,T1,T,T2. ⦃G1, L1, T⦄ ≥[h, o] ⦃G2, L2, T2⦄ →
+                       ⦃G1, L1⦄ ⊢ T1 ➡*[h, o] T → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
+#h #o #G1 #G2 #L1 #L2 #T1 #T #T2 #H1 #H @(cpxs_ind_dx … H) -T1
 /3 width=5 by fpbs_strap2, fpbq_cpx/
 qed-.
 
-lemma lpxs_fpbs_trans: ∀h,g,G1,G2,L1,L,L2,T1,T2. ⦃G1, L, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ →
-                       ⦃G1, L1⦄ ⊢ ➡*[h, g] L → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
-#h #g #G1 #G2 #L1 #L #L2 #T1 #T2 #H1 #H @(lpxs_ind_dx … H) -L1
+lemma lpxs_fpbs_trans: ∀h,o,G1,G2,L1,L,L2,T1,T2. ⦃G1, L, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄ →
+                       ⦃G1, L1⦄ ⊢ ➡*[h, o] L → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
+#h #o #G1 #G2 #L1 #L #L2 #T1 #T2 #H1 #H @(lpxs_ind_dx … H) -L1
 /3 width=5 by fpbs_strap2, fpbq_lpx/
 qed-.
 
-lemma lleq_fpbs_trans: ∀h,g,G1,G2,L1,L,L2,T1,T2. ⦃G1, L, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ →
-                       L1 ≡[T1, 0] L → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
+lemma lleq_fpbs_trans: ∀h,o,G1,G2,L1,L,L2,T1,T2. ⦃G1, L, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄ →
+                       L1 ≡[T1, 0] L → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
 /3 width=5 by fpbs_strap2, fpbq_lleq/ qed-.
 
-lemma cpxs_fqus_fpbs: ∀h,g,G1,G2,L1,L2,T1,T,T2. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] T →
-                      ⦃G1, L1, T⦄ ⊐* ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
+lemma cpxs_fqus_fpbs: ∀h,o,G1,G2,L1,L2,T1,T,T2. ⦃G1, L1⦄ ⊢ T1 ➡*[h, o] T →
+                      ⦃G1, L1, T⦄ ⊐* ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
 /3 width=5 by fpbs_fqus_trans, cpxs_fpbs/ qed.
 
-lemma cpxs_fqup_fpbs: ∀h,g,G1,G2,L1,L2,T1,T,T2. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] T →
-                      ⦃G1, L1, T⦄ ⊐+ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
+lemma cpxs_fqup_fpbs: ∀h,o,G1,G2,L1,L2,T1,T,T2. ⦃G1, L1⦄ ⊢ T1 ➡*[h, o] T →
+                      ⦃G1, L1, T⦄ ⊐+ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
 /3 width=5 by fpbs_fqup_trans, cpxs_fpbs/ qed.
 
-lemma fqus_lpxs_fpbs: ∀h,g,G1,G2,L1,L,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L, T2⦄ →
-                      ⦃G2, L⦄ ⊢ ➡*[h, g] L2 → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
+lemma fqus_lpxs_fpbs: ∀h,o,G1,G2,L1,L,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L, T2⦄ →
+                      ⦃G2, L⦄ ⊢ ➡*[h, o] L2 → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
 /3 width=3 by fpbs_lpxs_trans, fqus_fpbs/ qed.
 
-lemma cpxs_fqus_lpxs_fpbs: ∀h,g,G1,G2,L1,L,L2,T1,T,T2. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] T →
-                           ⦃G1, L1, T⦄ ⊐* ⦃G2, L, T2⦄ → ⦃G2, L⦄ ⊢ ➡*[h, g] L2 → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
+lemma cpxs_fqus_lpxs_fpbs: ∀h,o,G1,G2,L1,L,L2,T1,T,T2. ⦃G1, L1⦄ ⊢ T1 ➡*[h, o] T →
+                           ⦃G1, L1, T⦄ ⊐* ⦃G2, L, T2⦄ → ⦃G2, L⦄ ⊢ ➡*[h, o] L2 → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
 /3 width=5 by cpxs_fqus_fpbs, fpbs_lpxs_trans/ qed.
 
-lemma lpxs_lleq_fpbs: ∀h,g,G,L1,L,L2,T. ⦃G, L1⦄ ⊢ ➡*[h, g] L →
-                      L ≡[T, 0] L2 → ⦃G, L1, T⦄ ≥[h, g] ⦃G, L2, T⦄.
+lemma lpxs_lleq_fpbs: ∀h,o,G,L1,L,L2,T. ⦃G, L1⦄ ⊢ ➡*[h, o] L →
+                      L ≡[T, 0] L2 → ⦃G, L1, T⦄ ≥[h, o] ⦃G, L2, T⦄.
 /3 width=3 by lpxs_fpbs_trans, lleq_fpbs/ qed.
 
 (* Note: this is used in the closure proof *)
-lemma cpr_lpr_fpbs: ∀h,g,G,L1,L2,T1,T2. ⦃G, L1⦄ ⊢ T1 ➡ T2 → ⦃G, L1⦄ ⊢ ➡ L2 →
-                    ⦃G, L1, T1⦄ ≥[h, g] ⦃G, L2, T2⦄.
+lemma cpr_lpr_fpbs: ∀h,o,G,L1,L2,T1,T2. ⦃G, L1⦄ ⊢ T1 ➡ T2 → ⦃G, L1⦄ ⊢ ➡ L2 →
+                    ⦃G, L1, T1⦄ ≥[h, o] ⦃G, L2, T2⦄.
 /4 width=5 by fpbs_strap1, fpbq_fpbs, lpr_fpbq, cpr_fpbq/
 qed.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/fpbs_aaa.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/fpbs_aaa.ma
index 372b571b6..5432caf82 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/fpbs_aaa.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/fpbs_aaa.ma
@@ -19,9 +19,9 @@ include "basic_2/computation/fpbs.ma".
 
 (* Properties on atomic arity assignment for terms **************************)
 
-lemma fpbs_aaa_conf: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ →
+lemma fpbs_aaa_conf: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄ →
                      ∀A1. ⦃G1, L1⦄ ⊢ T1 ⁝ A1 → ∃A2. ⦃G2, L2⦄ ⊢ T2 ⁝ A2.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpbs_ind … H) -G2 -L2 -T2 /2 width=2 by ex_intro/
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpbs_ind … H) -G2 -L2 -T2 /2 width=2 by ex_intro/
 #G #G2 #L #L2 #T #T2 #_ #H2 #IH1 #A #HA elim (IH1 … HA) -IH1 -A
 /2 width=8 by fpbq_aaa_conf/
 qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/fpbs_alt.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/fpbs_alt.ma
index cc6ae1362..2f6753934 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/fpbs_alt.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/fpbs_alt.ma
@@ -22,19 +22,19 @@ include "basic_2/computation/fpbs.ma".
 
 (* Note: alternative definition of fpbs *)
 definition fpbsa: ∀h. sd h → tri_relation genv lenv term ≝
-                  λh,g,G1,L1,T1,G2,L2,T2.
-                  ∃∃L0,L,T. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] T &
+                  λh,o,G1,L1,T1,G2,L2,T2.
+                  ∃∃L0,L,T. ⦃G1, L1⦄ ⊢ T1 ➡*[h, o] T &
                          ⦃G1, L1, T⦄ ⊐* ⦃G2, L0, T2⦄ &
-                         ⦃G2, L0⦄ ⊢ ➡*[h, g] L & L ≡[T2, 0] L2.
+                         ⦃G2, L0⦄ ⊢ ➡*[h, o] L & L ≡[T2, 0] L2.
 
 interpretation "'big tree' parallel computation (closure) alternative"
-   'BTPRedStarAlt h g G1 L1 T1 G2 L2 T2 = (fpbsa h g G1 L1 T1 G2 L2 T2).
+   'BTPRedStarAlt h o G1 L1 T1 G2 L2 T2 = (fpbsa h o G1 L1 T1 G2 L2 T2).
 
 (* Basic properties *********************************************************)
 
-lemma fpb_fpbsa_trans: ∀h,g,G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ≽[h, g] ⦃G, L, T⦄ →
-                       ∀G2,L2,T2. ⦃G, L, T⦄ ≥≥[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥≥[h, g] ⦃G2, L2, T2⦄.
-#h #g #G1 #G #L1 #L #T1 #T * -G -L -T [ #G #L #T #HG1 | #T #HT1 | #L #HL1 | #L #HL1 ]
+lemma fpb_fpbsa_trans: ∀h,o,G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ≽[h, o] ⦃G, L, T⦄ →
+                       ∀G2,L2,T2. ⦃G, L, T⦄ ≥≥[h, o] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥≥[h, o] ⦃G2, L2, T2⦄.
+#h #o #G1 #G #L1 #L #T1 #T * -G -L -T [ #G #L #T #HG1 | #T #HT1 | #L #HL1 | #L #HL1 ]
 #G2 #L2 #T2 * #L00 #L0 #T0 #HT0 #HG2 #HL00 #HL02
 [ elim (fquq_cpxs_trans … HT0 … HG1) -T
   /3 width=7 by fqus_strap2, ex4_3_intro/
@@ -52,31 +52,31 @@ qed-.
 
 (* Main properties **********************************************************)
 
-theorem fpbs_fpbsa: ∀h,g,G1,G2,L1,L2,T1,T2.
-                    ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥≥[h, g] ⦃G2, L2, T2⦄.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpbs_ind_dx … H) -G1 -L1 -T1
+theorem fpbs_fpbsa: ∀h,o,G1,G2,L1,L2,T1,T2.
+                    ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥≥[h, o] ⦃G2, L2, T2⦄.
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H @(fpbs_ind_dx … H) -G1 -L1 -T1
 /2 width=7 by fpb_fpbsa_trans, ex4_3_intro/
 qed.
 
 (* Main inversion lemmas ****************************************************)
 
-theorem fpbsa_inv_fpbs: ∀h,g,G1,G2,L1,L2,T1,T2.
-                        ⦃G1, L1, T1⦄ ≥≥[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 *
+theorem fpbsa_inv_fpbs: ∀h,o,G1,G2,L1,L2,T1,T2.
+                        ⦃G1, L1, T1⦄ ≥≥[h, o] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 *
 /3 width=5 by cpxs_fqus_lpxs_fpbs, fpbs_strap1, fpbq_lleq/
 qed-.
 
 (* Advanced properties ******************************************************)
 
-lemma fpbs_intro_alt: ∀h,g,G1,G2,L1,L0,L,L2,T1,T,T2.
-                      ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] T → ⦃G1, L1, T⦄ ⊐* ⦃G2, L0, T2⦄ →
-                      ⦃G2, L0⦄ ⊢ ➡*[h, g] L → L ≡[T2, 0] L2 →  ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ .
+lemma fpbs_intro_alt: ∀h,o,G1,G2,L1,L0,L,L2,T1,T,T2.
+                      ⦃G1, L1⦄ ⊢ T1 ➡*[h, o] T → ⦃G1, L1, T⦄ ⊐* ⦃G2, L0, T2⦄ →
+                      ⦃G2, L0⦄ ⊢ ➡*[h, o] L → L ≡[T2, 0] L2 →  ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄ .
 /3 width=7 by fpbsa_inv_fpbs, ex4_3_intro/ qed.
 
 (* Advanced inversion lemmas *************************************************)
 
-lemma fpbs_inv_alt: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ →
-                    ∃∃L0,L,T. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] T &
+lemma fpbs_inv_alt: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄ →
+                    ∃∃L0,L,T. ⦃G1, L1⦄ ⊢ T1 ➡*[h, o] T &
                               ⦃G1, L1, T⦄ ⊐* ⦃G2, L0, T2⦄ &
-                              ⦃G2, L0⦄ ⊢ ➡*[h, g] L & L ≡[T2, 0] L2.
+                              ⦃G2, L0⦄ ⊢ ➡*[h, o] L & L ≡[T2, 0] L2.
 /2 width=1 by  fpbs_fpbsa/ qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/fpbs_fpb.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/fpbs_fpb.ma
index 00d015628..b9c657735 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/fpbs_fpb.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/fpbs_fpb.ma
@@ -19,10 +19,10 @@ include "basic_2/computation/fpbs_alt.ma".
 
 (* Properties on extended context-sensitive parallel computation for terms **)
 
-lemma fpbs_cpx_trans_neq: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ →
-                          ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h, g] U2 → (T2 = U2 → ⊥) →
-                          ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡[h, g] U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ≥[h, g] ⦃G2, L2, U2⦄.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H #U2 #HTU2 #HnTU2 elim (fpbs_inv_alt … H) -H
+lemma fpbs_cpx_trans_neq: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄ →
+                          ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h, o] U2 → (T2 = U2 → ⊥) →
+                          ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡[h, o] U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ≥[h, o] ⦃G2, L2, U2⦄.
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H #U2 #HTU2 #HnTU2 elim (fpbs_inv_alt … H) -H
 #L00 #L0 #T0 #HT10 #H10 #HL00 #HL02 lapply (lleq_cpx_trans … HTU2 … HL02) -HTU2
 #HTU2 lapply (cpx_lleq_conf_sn … HTU2 … HL02) -HL02
 #HL02 lapply (lpxs_cpx_trans … HTU2 … HL00) -HTU2
@@ -36,6 +36,6 @@ qed-.
 
 (* Properties on "rst" proper parallel reduction on closures ****************)
 
-lemma fpb_fpbs: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G2, L2, T2⦄ →
-                ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
+lemma fpb_fpbs: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻[h, o] ⦃G2, L2, T2⦄ →
+                ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
 /3 width=1 by fpbq_fpbs, fpb_fpbq/ qed.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/fpbs_fpbs.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/fpbs_fpbs.ma
index 68b07b4d3..d3d14d184 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/fpbs_fpbs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/fpbs_fpbs.ma
@@ -18,5 +18,5 @@ include "basic_2/computation/fpbs.ma".
 
 (* Main properties **********************************************************)
 
-theorem fpbs_trans: ∀h,g. tri_transitive … (fpbs h g).
+theorem fpbs_trans: ∀h,o. tri_transitive … (fpbs h o).
 /2 width=5 by tri_TC_transitive/ qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/fpbs_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/fpbs_lift.ma
index a1edecf0c..69f1a77fa 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/fpbs_lift.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/fpbs_lift.ma
@@ -19,18 +19,18 @@ include "basic_2/computation/fpbs.ma".
 
 (* Advanced properties ******************************************************)
 
-lemma lstas_fpbs: ∀h,g,G,L,T1,T2,d2. ⦃G, L⦄ ⊢ T1 •*[h, d2] T2 →
-                  ∀d1. d2 ≤ d1 → ⦃G, L⦄ ⊢ T1 ▪[h, g] d1 → ⦃G, L, T1⦄ ≥[h, g] ⦃G, L, T2⦄.
+lemma lstas_fpbs: ∀h,o,G,L,T1,T2,d2. ⦃G, L⦄ ⊢ T1 •*[h, d2] T2 →
+                  ∀d1. d2 ≤ d1 → ⦃G, L⦄ ⊢ T1 ▪[h, o] d1 → ⦃G, L, T1⦄ ≥[h, o] ⦃G, L, T2⦄.
 /3 width=5 by cpxs_fpbs, lstas_cpxs/ qed.
 
-lemma sta_fpbs: ∀h,g,G,L,T,U,d.
-                ⦃G, L⦄ ⊢ T ▪[h, g] d+1 → ⦃G, L⦄ ⊢ T •*[h, 1] U →
-                ⦃G, L, T⦄ ≥[h, g] ⦃G, L, U⦄.
+lemma sta_fpbs: ∀h,o,G,L,T,U,d.
+                ⦃G, L⦄ ⊢ T ▪[h, o] d+1 → ⦃G, L⦄ ⊢ T •*[h, 1] U →
+                ⦃G, L, T⦄ ≥[h, o] ⦃G, L, U⦄.
 /2 width=5 by lstas_fpbs/ qed.
 
 (* Note: this is used in the closure proof *)
-lemma cpr_lpr_sta_fpbs: ∀h,g,G,L1,L2,T1,T2,U2,d2.
+lemma cpr_lpr_sta_fpbs: ∀h,o,G,L1,L2,T1,T2,U2,d2.
                         ⦃G, L1⦄ ⊢ T1 ➡ T2 → ⦃G, L1⦄ ⊢ ➡ L2 →
-                        ⦃G, L2⦄ ⊢ T2 ▪[h, g] d2+1 → ⦃G, L2⦄ ⊢ T2 •*[h, 1] U2 →
-                        ⦃G, L1, T1⦄ ≥[h, g] ⦃G, L2, U2⦄.
+                        ⦃G, L2⦄ ⊢ T2 ▪[h, o] d2+1 → ⦃G, L2⦄ ⊢ T2 •*[h, 1] U2 →
+                        ⦃G, L1, T1⦄ ≥[h, o] ⦃G, L2, U2⦄.
 /4 width=5 by fpbs_strap1, cpr_lpr_fpbs, sta_cpx, fpbq_cpx/ qed.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/fsb.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/fsb.ma
index c87433c2d..c27ffaba5 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/fsb.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/fsb.ma
@@ -18,30 +18,30 @@ include "basic_2/computation/csx.ma".
 
 (* "QRST" STRONGLY NORMALIZING CLOSURES *************************************)
 
-inductive fsb (h) (g): relation3 genv lenv term ≝
+inductive fsb (h) (o): relation3 genv lenv term ≝
 | fsb_intro: ∀G1,L1,T1. (
-                ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G2, L2, T2⦄ → fsb h g G2 L2 T2
-             ) → fsb h g G1 L1 T1
+                ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≻[h, o] ⦃G2, L2, T2⦄ → fsb h o G2 L2 T2
+             ) → fsb h o G1 L1 T1
 .
 
 interpretation
    "'qrst' strong normalization (closure)"
-   'BTSN h g G L T = (fsb h g G L T).
+   'BTSN h o G L T = (fsb h o G L T).
 
 (* Basic eliminators ********************************************************)
 
-lemma fsb_ind_alt: ∀h,g. ∀R: relation3 …. (
-                      ∀G1,L1,T1. ⦥[h,g] ⦃G1, L1, T1⦄ → (
-                         ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G2, L2, T2⦄ → R G2 L2 T2
+lemma fsb_ind_alt: ∀h,o. ∀R: relation3 …. (
+                      ∀G1,L1,T1. ⦥[h,o] ⦃G1, L1, T1⦄ → (
+                         ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≻[h, o] ⦃G2, L2, T2⦄ → R G2 L2 T2
                       ) → R G1 L1 T1
                    ) →
-                   ∀G,L,T. ⦥[h, g] ⦃G, L, T⦄ → R G L T.
-#h #g #R #IH #G #L #T #H elim H -G -L -T
+                   ∀G,L,T. ⦥[h, o] ⦃G, L, T⦄ → R G L T.
+#h #o #R #IH #G #L #T #H elim H -G -L -T
 /4 width=1 by fsb_intro/
 qed-.
 
 (* Basic inversion lemmas ***************************************************)
 
-lemma fsb_inv_csx: ∀h,g,G,L,T. ⦥[h, g] ⦃G, L, T⦄ → ⦃G, L⦄ ⊢ ⬊*[h, g] T.
-#h #g #G #L #T #H elim H -G -L -T /5 width=1 by csx_intro, fpb_cpx/
+lemma fsb_inv_csx: ∀h,o,G,L,T. ⦥[h, o] ⦃G, L, T⦄ → ⦃G, L⦄ ⊢ ⬊*[h, o] T.
+#h #o #G #L #T #H elim H -G -L -T /5 width=1 by csx_intro, fpb_cpx/
 qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/fsb_aaa.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/fsb_aaa.ma
index 5a8d5d240..05bb29c01 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/fsb_aaa.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/fsb_aaa.ma
@@ -21,50 +21,50 @@ include "basic_2/computation/fsb_csx.ma".
 (* Main properties **********************************************************)
 
 (* Note: this is the "big tree" theorem ("RST" version) *)
-theorem aaa_fsb: ∀h,g,G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ⦥[h, g] ⦃G, L, T⦄.
+theorem aaa_fsb: ∀h,o,G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ⦥[h, o] ⦃G, L, T⦄.
 /3 width=2 by aaa_csx, csx_fsb/ qed.
 
 (* Note: this is the "big tree" theorem ("QRST" version) *)
-theorem aaa_fsba: ∀h,g,G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ⦥⦥[h, g] ⦃G, L, T⦄.
+theorem aaa_fsba: ∀h,o,G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ⦥⦥[h, o] ⦃G, L, T⦄.
 /3 width=2 by fsb_fsba, aaa_fsb/ qed.
 
 (* Advanced eliminators on atomica arity assignment for terms ***************)
 
-fact aaa_ind_fpb_aux: ∀h,g. ∀R:relation3 genv lenv term.
+fact aaa_ind_fpb_aux: ∀h,o. ∀R:relation3 genv lenv term.
                       (∀G1,L1,T1,A. ⦃G1, L1⦄ ⊢ T1 ⁝ A →
-                                    (∀G2,L2,T2. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G2, L2, T2⦄ → R G2 L2 T2) →
+                                    (∀G2,L2,T2. ⦃G1, L1, T1⦄ ≻[h, o] ⦃G2, L2, T2⦄ → R G2 L2 T2) →
                                     R G1 L1 T1
                       ) →
-                      ∀G,L,T. ⦃G, L⦄ ⊢ ⬊*[h, g] T → ∀A. ⦃G, L⦄ ⊢ T ⁝ A → R G L T.
-#h #g #R #IH #G #L #T #H @(csx_ind_fpb … H) -G -L -T
+                      ∀G,L,T. ⦃G, L⦄ ⊢ ⬊*[h, o] T → ∀A. ⦃G, L⦄ ⊢ T ⁝ A → R G L T.
+#h #o #R #IH #G #L #T #H @(csx_ind_fpb … H) -G -L -T
 #G1 #L1 #T1 #H1 #IH1 #A1 #HTA1 @IH -IH //
-#G2 #L2 #T2 #H12 elim (fpbs_aaa_conf h g … G2 … L2 … T2 … HTA1) -A1
+#G2 #L2 #T2 #H12 elim (fpbs_aaa_conf h o … G2 … L2 … T2 … HTA1) -A1
 /2 width=2 by fpb_fpbs/
 qed-.
 
-lemma aaa_ind_fpb: ∀h,g. ∀R:relation3 genv lenv term.
+lemma aaa_ind_fpb: ∀h,o. ∀R:relation3 genv lenv term.
                    (∀G1,L1,T1,A. ⦃G1, L1⦄ ⊢ T1 ⁝ A →
-                                 (∀G2,L2,T2. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G2, L2, T2⦄ → R G2 L2 T2) →
+                                 (∀G2,L2,T2. ⦃G1, L1, T1⦄ ≻[h, o] ⦃G2, L2, T2⦄ → R G2 L2 T2) →
                                  R G1 L1 T1
                    ) →
                    ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → R G L T.
 /4 width=4 by aaa_ind_fpb_aux, aaa_csx/ qed-.
 
-fact aaa_ind_fpbg_aux: ∀h,g. ∀R:relation3 genv lenv term.
+fact aaa_ind_fpbg_aux: ∀h,o. ∀R:relation3 genv lenv term.
                        (∀G1,L1,T1,A. ⦃G1, L1⦄ ⊢ T1 ⁝ A →
-                                     (∀G2,L2,T2. ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄ → R G2 L2 T2) →
+                                     (∀G2,L2,T2. ⦃G1, L1, T1⦄ >≡[h, o] ⦃G2, L2, T2⦄ → R G2 L2 T2) →
                                      R G1 L1 T1
                        ) →
-                       ∀G,L,T. ⦃G, L⦄ ⊢ ⬊*[h, g] T → ∀A. ⦃G, L⦄ ⊢ T ⁝ A → R G L T.
-#h #g #R #IH #G #L #T #H @(csx_ind_fpbg … H) -G -L -T
+                       ∀G,L,T. ⦃G, L⦄ ⊢ ⬊*[h, o] T → ∀A. ⦃G, L⦄ ⊢ T ⁝ A → R G L T.
+#h #o #R #IH #G #L #T #H @(csx_ind_fpbg … H) -G -L -T
 #G1 #L1 #T1 #H1 #IH1 #A1 #HTA1 @IH -IH //
-#G2 #L2 #T2 #H12 elim (fpbs_aaa_conf h g … G2 … L2 … T2 … HTA1) -A1
+#G2 #L2 #T2 #H12 elim (fpbs_aaa_conf h o … G2 … L2 … T2 … HTA1) -A1
 /2 width=2 by fpbg_fwd_fpbs/
 qed-.
 
-lemma aaa_ind_fpbg: ∀h,g. ∀R:relation3 genv lenv term.
+lemma aaa_ind_fpbg: ∀h,o. ∀R:relation3 genv lenv term.
                     (∀G1,L1,T1,A. ⦃G1, L1⦄ ⊢ T1 ⁝ A →
-                                  (∀G2,L2,T2. ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄ → R G2 L2 T2) →
+                                  (∀G2,L2,T2. ⦃G1, L1, T1⦄ >≡[h, o] ⦃G2, L2, T2⦄ → R G2 L2 T2) →
                                   R G1 L1 T1
                     ) →
                     ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → R G L T.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/fsb_alt.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/fsb_alt.ma
index c132dae87..80c7a6449 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/fsb_alt.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/fsb_alt.ma
@@ -19,64 +19,64 @@ include "basic_2/computation/fsb.ma".
 (* "QRST" STRONGLY NORMALIZING CLOSURES *************************************)
 
 (* Note: alternative definition of fsb *)
-inductive fsba (h) (g): relation3 genv lenv term ≝
+inductive fsba (h) (o): relation3 genv lenv term ≝
 | fsba_intro: ∀G1,L1,T1. (
-                 ∀G2,L2,T2. ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄ → fsba h g G2 L2 T2
-              ) → fsba h g G1 L1 T1.
+                 ∀G2,L2,T2. ⦃G1, L1, T1⦄ >≡[h, o] ⦃G2, L2, T2⦄ → fsba h o G2 L2 T2
+              ) → fsba h o G1 L1 T1.
 
 interpretation
    "'big tree' strong normalization (closure) alternative"
-   'BTSNAlt h g G L T = (fsba h g G L T).
+   'BTSNAlt h o G L T = (fsba h o G L T).
 
 (* Basic eliminators ********************************************************)
 
-lemma fsba_ind_alt: ∀h,g. ∀R: relation3 …. (
-                       ∀G1,L1,T1. ⦥⦥[h,g] ⦃G1, L1, T1⦄ → (
-                          ∀G2,L2,T2. ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄ → R G2 L2 T2
+lemma fsba_ind_alt: ∀h,o. ∀R: relation3 …. (
+                       ∀G1,L1,T1. ⦥⦥[h,o] ⦃G1, L1, T1⦄ → (
+                          ∀G2,L2,T2. ⦃G1, L1, T1⦄ >≡[h, o] ⦃G2, L2, T2⦄ → R G2 L2 T2
                        ) → R G1 L1 T1
                     ) →
-                    ∀G,L,T. ⦥⦥[h, g] ⦃G, L, T⦄ → R G L T.
-#h #g #R #IH #G #L #T #H elim H -G -L -T
+                    ∀G,L,T. ⦥⦥[h, o] ⦃G, L, T⦄ → R G L T.
+#h #o #R #IH #G #L #T #H elim H -G -L -T
 /4 width=1 by fsba_intro/
 qed-.
 
 (* Basic properties *********************************************************)
 
-lemma fsba_fpbs_trans: ∀h,g,G1,L1,T1. ⦥⦥[h, g] ⦃G1, L1, T1⦄ →
-                       ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ → ⦥⦥[h, g] ⦃G2, L2, T2⦄.
-#h #g #G1 #L1 #T1 #H @(fsba_ind_alt … H) -G1 -L1 -T1
+lemma fsba_fpbs_trans: ∀h,o,G1,L1,T1. ⦥⦥[h, o] ⦃G1, L1, T1⦄ →
+                       ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄ → ⦥⦥[h, o] ⦃G2, L2, T2⦄.
+#h #o #G1 #L1 #T1 #H @(fsba_ind_alt … H) -G1 -L1 -T1
 /4 width=5 by fsba_intro, fpbs_fpbg_trans/
 qed-.
 
 (* Main properties **********************************************************)
 
-theorem fsb_fsba: ∀h,g,G,L,T. ⦥[h, g] ⦃G, L, T⦄  → ⦥⦥[h, g] ⦃G, L, T⦄.
-#h #g #G #L #T #H @(fsb_ind_alt … H) -G -L -T
+theorem fsb_fsba: ∀h,o,G,L,T. ⦥[h, o] ⦃G, L, T⦄  → ⦥⦥[h, o] ⦃G, L, T⦄.
+#h #o #G #L #T #H @(fsb_ind_alt … H) -G -L -T
 #G1 #L1 #T1 #_ #IH @fsba_intro
 #G2 #L2 #T2 * /3 width=5 by fsba_fpbs_trans/
 qed.
 
 (* Main inversion lemmas ****************************************************)
 
-theorem fsba_inv_fsb: ∀h,g,G,L,T. ⦥⦥[h, g] ⦃G, L, T⦄ → ⦥[h, g] ⦃G, L, T⦄.
-#h #g #G #L #T #H @(fsba_ind_alt … H) -G -L -T
+theorem fsba_inv_fsb: ∀h,o,G,L,T. ⦥⦥[h, o] ⦃G, L, T⦄ → ⦥[h, o] ⦃G, L, T⦄.
+#h #o #G #L #T #H @(fsba_ind_alt … H) -G -L -T
 /4 width=1 by fsb_intro, fpb_fpbg/
 qed-.
 
 (* Advanced properties ******************************************************)
 
-lemma fsb_fpbs_trans: ∀h,g,G1,L1,T1. ⦥[h, g] ⦃G1, L1, T1⦄ →
-                      ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ → ⦥[h, g] ⦃G2, L2, T2⦄.
+lemma fsb_fpbs_trans: ∀h,o,G1,L1,T1. ⦥[h, o] ⦃G1, L1, T1⦄ →
+                      ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄ → ⦥[h, o] ⦃G2, L2, T2⦄.
 /4 width=5 by fsba_inv_fsb, fsb_fsba, fsba_fpbs_trans/ qed-.
 
 (* Advanced eliminators *****************************************************)
 
-lemma fsb_ind_fpbg: ∀h,g. ∀R:relation3 genv lenv term.
-                    (∀G1,L1,T1. ⦥[h, g] ⦃G1, L1, T1⦄ →
-                                (∀G2,L2,T2. ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄ → R G2 L2 T2) →
+lemma fsb_ind_fpbg: ∀h,o. ∀R:relation3 genv lenv term.
+                    (∀G1,L1,T1. ⦥[h, o] ⦃G1, L1, T1⦄ →
+                                (∀G2,L2,T2. ⦃G1, L1, T1⦄ >≡[h, o] ⦃G2, L2, T2⦄ → R G2 L2 T2) →
                                 R G1 L1 T1
                     ) →
-                    ∀G1,L1,T1. ⦥[h, g] ⦃G1, L1, T1⦄ → R G1 L1 T1.
-#h #g #R #IH #G1 #L1 #T1 #H @(fsba_ind_alt h g … G1 L1 T1)
+                    ∀G1,L1,T1. ⦥[h, o] ⦃G1, L1, T1⦄ → R G1 L1 T1.
+#h #o #R #IH #G1 #L1 #T1 #H @(fsba_ind_alt h o … G1 L1 T1)
 /3 width=1 by fsba_inv_fsb, fsb_fsba/
 qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/fsb_csx.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/fsb_csx.ma
index 403226304..e86c302b8 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/fsb_csx.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/fsb_csx.ma
@@ -22,9 +22,9 @@ include "basic_2/computation/fsb_alt.ma".
 
 (* Advanced propreties on context-sensitive extended normalizing terms ******)
 
-lemma csx_fsb_fpbs: ∀h,g,G1,L1,T1. ⦃G1, L1⦄ ⊢ ⬊*[h, g] T1 →
-                    ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ → ⦥[h, g] ⦃G2, L2, T2⦄.
-#h #g #G1 #L1 #T1 #H @(csx_ind … H) -T1
+lemma csx_fsb_fpbs: ∀h,o,G1,L1,T1. ⦃G1, L1⦄ ⊢ ⬊*[h, o] T1 →
+                    ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄ → ⦥[h, o] ⦃G2, L2, T2⦄.
+#h #o #G1 #L1 #T1 #H @(csx_ind … H) -T1
 #T1 #HT1 #IHc #G2 #L2 #T2 @(fqup_wf_ind … G2 L2 T2) -G2 -L2 -T2
 #G0 #L0 #T0 #IHu #H10 lapply (csx_fpbs_conf … H10) // -HT1
 #HT0 generalize in match IHu; -IHu generalize in match H10; -H10
@@ -48,23 +48,23 @@ lemma csx_fsb_fpbs: ∀h,g,G1,L1,T1. ⦃G1, L1⦄ ⊢ ⬊*[h, g] T1 →
 ]
 qed.
 
-lemma csx_fsb: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ ⬊*[h, g] T → ⦥[h, g] ⦃G, L, T⦄.
+lemma csx_fsb: ∀h,o,G,L,T. ⦃G, L⦄ ⊢ ⬊*[h, o] T → ⦥[h, o] ⦃G, L, T⦄.
 /2 width=5 by csx_fsb_fpbs/ qed.
 
 (* Advanced eliminators *****************************************************)
 
-lemma csx_ind_fpb: ∀h,g. ∀R:relation3 genv lenv term.
-                   (∀G1,L1,T1. ⦃G1, L1⦄ ⊢ ⬊*[h, g] T1 →
-                               (∀G2,L2,T2. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G2, L2, T2⦄ → R G2 L2 T2) →
+lemma csx_ind_fpb: ∀h,o. ∀R:relation3 genv lenv term.
+                   (∀G1,L1,T1. ⦃G1, L1⦄ ⊢ ⬊*[h, o] T1 →
+                               (∀G2,L2,T2. ⦃G1, L1, T1⦄ ≻[h, o] ⦃G2, L2, T2⦄ → R G2 L2 T2) →
                                R G1 L1 T1
                    ) →
-                   ∀G,L,T. ⦃G, L⦄ ⊢ ⬊*[h, g] T → R G L T.
+                   ∀G,L,T. ⦃G, L⦄ ⊢ ⬊*[h, o] T → R G L T.
 /4 width=4 by fsb_inv_csx, csx_fsb, fsb_ind_alt/ qed-.
 
-lemma csx_ind_fpbg: ∀h,g. ∀R:relation3 genv lenv term.
-                    (∀G1,L1,T1. ⦃G1, L1⦄ ⊢ ⬊*[h, g] T1 →
-                                (∀G2,L2,T2. ⦃G1, L1, T1⦄ >≡[h, g] ⦃G2, L2, T2⦄ → R G2 L2 T2) →
+lemma csx_ind_fpbg: ∀h,o. ∀R:relation3 genv lenv term.
+                    (∀G1,L1,T1. ⦃G1, L1⦄ ⊢ ⬊*[h, o] T1 →
+                                (∀G2,L2,T2. ⦃G1, L1, T1⦄ >≡[h, o] ⦃G2, L2, T2⦄ → R G2 L2 T2) →
                                 R G1 L1 T1
                     ) →
-                    ∀G,L,T. ⦃G, L⦄ ⊢ ⬊*[h, g] T → R G L T.
+                    ∀G,L,T. ⦃G, L⦄ ⊢ ⬊*[h, o] T → R G L T.
 /4 width=4 by fsb_inv_csx, csx_fsb, fsb_ind_fpbg/ qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/gcp.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/gcp.ma
index 4266d9275..61e9182cd 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/gcp.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/gcp.ma
@@ -26,12 +26,12 @@ definition CP0 ≝ λRR:relation4 genv lenv term term. λRS:relation term.
                  ∀G. d_liftable1 (nf RR RS G) (Ⓕ).
 
 definition CP1 ≝ λRR:relation4 genv lenv term term. λRS:relation term.
-                 ∀G,L. ∃k. NF … (RR G L) RS (⋆k).
+                 ∀G,L. ∃s. NF … (RR G L) RS (⋆s).
 
 definition CP2 ≝ λRP:candidate. ∀G. d_liftable1 (RP G) (Ⓕ).
 
 definition CP3 ≝ λRP:candidate.
-                 ∀G,L,T,k. RP G L (ⓐ⋆k.T) → RP G L T.
+                 ∀G,L,T,s. RP G L (ⓐ⋆s.T) → RP G L T.
 
 (* requirements for generic computation properties *)
 record gcp (RR:relation4 genv lenv term term) (RS:relation term) (RP:candidate) : Prop ≝
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/gcp_aaa.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/gcp_aaa.ma
index 3428478ea..37dbb569e 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/gcp_aaa.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/gcp_aaa.ma
@@ -29,10 +29,10 @@ theorem acr_aaa_csubc_lifts: ∀RR,RS,RP.
                              ∀T0. ⬆*[cs] T ≡ T0 → ∀L2. G ⊢ L2 ⫃[RP] L0 →
                              ⦃G, L2, T0⦄ ϵ[RP] 〚A〛.
 #RR #RS #RP #H1RP #H2RP #G #L1 #T #A #H elim H -G -L1 -T -A
-[ #G #L #k #L0 #cs #HL0 #X #H #L2 #HL20
+[ #G #L #s #L0 #cs #HL0 #X #H #L2 #HL20
   >(lifts_inv_sort1 … H) -H
   lapply (acr_gcr … H1RP H2RP (⓪)) #HAtom
-  lapply (s4 … HAtom G L2 (◊)) /2 width=1 by/
+  lapply (c4 … HAtom G L2 (◊)) /2 width=1 by/
 | #I #G #L1 #K1 #V1 #B #i #HLK1 #HKV1B #IHB #L0 #cs #HL01 #X #H #L2 #HL20
   lapply (acr_gcr … H1RP H2RP B) #HB
   elim (lifts_inv_lref1 … H) -H #i1 #Hi1 #H destruct
@@ -45,7 +45,7 @@ theorem acr_aaa_csubc_lifts: ∀RR,RS,RP.
   [ #K2 #HK20 #H destruct
     elim (lift_total V0 0 (i0 +1)) #V #HV0
     elim (lifts_lift_trans  … Hi0 … Hcs0 … HV10 … HV0) -HV10 #V2 #HV12 #HV2
-    lapply (s5 … HB ? G ? ? (◊) … HV0 HLK2) /3 width=7 by drops_cons, lifts_cons/ (* Note: uses IHB HL20 V2 HV0 *)
+    lapply (c5 … HB ? G ? ? (◊) … HV0 HLK2) /3 width=7 by drops_cons, lifts_cons/ (* Note: uses IHB HL20 V2 HV0 *)
   | -HLK1 -IHB -HL01 -HL20 -HK1b -Hi0 -Hcs0
     #K2 #V2 #A2 #HKV2A #H1KV0A #H2KV0A #_ #H1 #H2 destruct
     lapply (drop_fwd_drop2 … HLK2) #HLK2b
@@ -53,15 +53,15 @@ theorem acr_aaa_csubc_lifts: ∀RR,RS,RP.
     lapply (aaa_mono … H2KV0A … HKV0B) #H destruct -H2KV0A -HKV0B
     elim (lift_total V0 0 (i0 +1)) #V3 #HV03
     elim (lift_total V2 0 (i0 +1)) #V #HV2
-    lapply (s5 … HB ? G ? ? (◊) … (ⓝV3.V) … HLK2) /2 width=1 by lift_flat/
-    lapply (s7 … HB G L2 (◊)) /3 width=7 by gcr_lift/
+    lapply (c5 … HB ? G ? ? (◊) … (ⓝV3.V) … HLK2) /2 width=1 by lift_flat/
+    lapply (c7 … HB G L2 (◊)) /3 width=7 by gcr_lift/
   ]
 | #a #G #L #V #T #B #A #_ #_ #IHB #IHA #L0 #cs #HL0 #X #H #L2 #HL20
   elim (lifts_inv_bind1 … H) -H #V0 #T0 #HV0 #HT0 #H destruct
   lapply (acr_gcr … H1RP H2RP A) #HA
   lapply (acr_gcr … H1RP H2RP B) #HB
-  lapply (s1 … HB) -HB #HB
-  lapply (s6 … HA G L2 (◊) (◊)) /4 width=5 by lsubc_pair, drops_skip, liftv_nil/
+  lapply (c1 … HB) -HB #HB
+  lapply (c6 … HA G L2 (◊) (◊)) /4 width=5 by lsubc_pair, drops_skip, liftv_nil/
 | #a #G #L #W #T #B #A #HLWB #_ #IHB #IHA #L0 #cs #HL0 #X #H #L2 #HL02
   elim (lifts_inv_bind1 … H) -H #W0 #T0 #HW0 #HT0 #H destruct
   @(acr_abst  … H1RP H2RP) /2 width=5 by/
@@ -76,7 +76,7 @@ theorem acr_aaa_csubc_lifts: ∀RR,RS,RP.
 | #G #L #V #T #A #_ #_ #IH1A #IH2A #L0 #cs #HL0 #X #H #L2 #HL20
   elim (lifts_inv_flat1 … H) -H #V0 #T0 #HV0 #HT0 #H destruct
   lapply (acr_gcr … H1RP H2RP A) #HA
-  lapply (s7 … HA G L2 (◊)) /3 width=5 by/
+  lapply (c7 … HA G L2 (◊)) /3 width=5 by/
 ]
 qed.
 
@@ -89,5 +89,5 @@ lemma gcr_aaa: ∀RR,RS,RP. gcp RR RS RP → gcr RR RS RP RP →
                ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → RP G L T.
 #RR #RS #RP #H1RP #H2RP #G #L #T #A #HT
 lapply (acr_gcr … H1RP H2RP A) #HA
-@(s1 … HA) /2 width=4 by acr_aaa/
+@(c1 … HA) /2 width=4 by acr_aaa/
 qed.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/gcp_cr.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/gcp_cr.ma
index b60a87aef..dee244aac 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/gcp_cr.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/gcp_cr.ma
@@ -36,28 +36,28 @@ definition S3 ≝ λC:candidate.
                 C G L (ⒶVs.ⓓ{a}ⓝW.V.T) → C G L (ⒶVs.ⓐV.ⓛ{a}W.T).
 
 definition S4 ≝ λRP,C:candidate.
-                ∀G,L,Vs. all … (RP G L) Vs → ∀k. C G L (ⒶVs.⋆k).
+                ∀G,L,Vs. all … (RP G L) Vs → ∀s. C G L (ⒶVs.⋆s).
 
 definition S5 ≝ λC:candidate. ∀I,G,L,K,Vs,V1,V2,i.
                 C G L (ⒶVs.V2) → ⬆[0, i+1] V1 ≡ V2 →
                 ⬇[i] L ≡ K.ⓑ{I}V1 → C G L (ⒶVs.#i).
 
 definition S6 ≝ λRP,C:candidate.
-                ∀G,L,V1s,V2s. ⬆[0, 1] V1s ≡ V2s →
-                ∀a,V,T. C G (L.ⓓV) (ⒶV2s.T) → RP G L V → C G L (ⒶV1s.ⓓ{a}V.T).
+                ∀G,L,V1c,V2c. ⬆[0, 1] V1c ≡ V2c →
+                ∀a,V,T. C G (L.ⓓV) (ⒶV2c.T) → RP G L V → C G L (ⒶV1c.ⓓ{a}V.T).
 
 definition S7 ≝ λC:candidate.
                 ∀G,L,Vs,T,W. C G L (ⒶVs.T) → C G L (ⒶVs.W) → C G L (ⒶVs.ⓝW.T).
 
 (* requirements for the generic reducibility candidate *)
 record gcr (RR:relation4 genv lenv term term) (RS:relation term) (RP,C:candidate) : Prop ≝
-{ s1: S1 RP C;
-  s2: S2 RR RS RP C;
-  s3: S3 C;
-  s4: S4 RP C;
-  s5: S5 C;
-  s6: S6 RP C;
-  s7: S7 C
+{ c1: S1 RP C;
+  c2: S2 RR RS RP C;
+  c3: S3 C;
+  c4: S4 RP C;
+  c5: S5 C;
+  c6: S6 RP C;
+  c7: S7 C
 }.
 
 (* the functional construction for candidates *)
@@ -97,28 +97,28 @@ lemma acr_gcr: ∀RR,RS,RP. gcp RR RS RP → gcr RR RS RP RP →
 #RR #RS #RP #H1RP #H2RP #A elim A -A //
 #B #A #IHB #IHA @mk_gcr
 [ #G #L #T #H
-  elim (cp1 … H1RP G L) #k #HK
-  lapply (H L (⋆k) T (◊) ? ? ?) -H //
-  [ lapply (s2 … IHB G L (◊) … HK) //
-  | /3 width=6 by s1, cp3/
+  elim (cp1 … H1RP G L) #s #HK
+  lapply (H L (⋆s) T (◊) ? ? ?) -H //
+  [ lapply (c2 … IHB G L (◊) … HK) //
+  | /3 width=6 by c1, cp3/
   ]
 | #G #L #Vs #HVs #T #H1T #H2T #L0 #V0 #X #cs #HL0 #H #HB
-  elim (lifts_inv_applv1 … H) -H #V0s #T0 #HV0s #HT0 #H destruct
-  lapply (s1 … IHB … HB) #HV0
-  @(s2 … IHA … (V0 @ V0s))
+  elim (lifts_inv_applv1 … H) -H #V0c #T0 #HV0c #HT0 #H destruct
+  lapply (c1 … IHB … HB) #HV0
+  @(c2 … IHA … (V0 @ V0c))
   /3 width=14 by gcp2_lifts_all, gcp2_lifts, gcp0_lifts, lifts_simple_dx, conj/
 | #a #G #L #Vs #U #T #W #HA #L0 #V0 #X #cs #HL0 #H #HB
-  elim (lifts_inv_applv1 … H) -H #V0s #Y #HV0s #HY #H destruct
+  elim (lifts_inv_applv1 … H) -H #V0c #Y #HV0c #HY #H destruct
   elim (lifts_inv_flat1 … HY) -HY #U0 #X #HU0 #HX #H destruct
   elim (lifts_inv_bind1 … HX) -HX #W0 #T0 #HW0 #HT0 #H destruct
-  @(s3 … IHA … (V0 @ V0s)) /5 width=6 by lifts_applv, lifts_flat, lifts_bind/
-| #G #L #Vs #HVs #k #L0 #V0 #X #cs #HL0 #H #HB
-  elim (lifts_inv_applv1 … H) -H #V0s #Y #HV0s #HY #H destruct
+  @(c3 … IHA … (V0 @ V0c)) /5 width=6 by lifts_applv, lifts_flat, lifts_bind/
+| #G #L #Vs #HVs #s #L0 #V0 #X #cs #HL0 #H #HB
+  elim (lifts_inv_applv1 … H) -H #V0c #Y #HV0c #HY #H destruct
   >(lifts_inv_sort1 … HY) -Y
-  lapply (s1 … IHB … HB) #HV0
-  @(s4 … IHA … (V0 @ V0s)) /3 width=7 by gcp2_lifts_all, conj/
+  lapply (c1 … IHB … HB) #HV0
+  @(c4 … IHA … (V0 @ V0c)) /3 width=7 by gcp2_lifts_all, conj/
 | #I #G #L #K #Vs #V1 #V2 #i #HA #HV12 #HLK #L0 #V0 #X #cs #HL0 #H #HB
-  elim (lifts_inv_applv1 … H) -H #V0s #Y #HV0s #HY #H destruct
+  elim (lifts_inv_applv1 … H) -H #V0c #Y #HV0c #HY #H destruct
   elim (lifts_inv_lref1 … HY) -HY #i0 #Hi0 #H destruct
   elim (drops_drop_trans … HL0 … HLK) #X #cs0 #i1 #HL02 #H #Hi1 #Hcs0
   >(at_mono … Hi1 … Hi0) in HL02; -i1 #HL02
@@ -126,23 +126,23 @@ lemma acr_gcr: ∀RR,RS,RP. gcp RR RS RP → gcr RR RS RP RP →
   elim (lift_total W1 0 (i0 + 1)) #W2 #HW12
   elim (lifts_lift_trans  … Hcs0 … HVW1 … HW12) // -Hcs0 -Hi0 #V3 #HV13 #HVW2
   >(lift_mono … HV13 … HV12) in HVW2; -V3 #HVW2
-  @(s5 … IHA … (V0 @ V0s) … HW12 HL02) /3 width=5 by lifts_applv/
-| #G #L #V1s #V2s #HV12s #a #V #T #HA #HV #L0 #V10 #X #cs #HL0 #H #HB
-  elim (lifts_inv_applv1 … H) -H #V10s #Y #HV10s #HY #H destruct
+  @(c5 … IHA … (V0 @ V0c) … HW12 HL02) /3 width=5 by lifts_applv/
+| #G #L #V1c #V2c #HV12c #a #V #T #HA #HV #L0 #V10 #X #cs #HL0 #H #HB
+  elim (lifts_inv_applv1 … H) -H #V10c #Y #HV10c #HY #H destruct
   elim (lifts_inv_bind1 … HY) -HY #V0 #T0 #HV0 #HT0 #H destruct
   elim (lift_total V10 0 1) #V20 #HV120
-  elim (liftv_total 0 1 V10s) #V20s #HV120s
-  @(s6 … IHA … (V10 @ V10s) (V20 @ V20s)) /3 width=7 by gcp2_lifts, liftv_cons/
+  elim (liftv_total 0 1 V10c) #V20c #HV120c
+  @(c6 … IHA … (V10 @ V10c) (V20 @ V20c)) /3 width=7 by gcp2_lifts, liftv_cons/
   @(HA … (cs + 1)) /2 width=2 by drops_skip/
   [ @lifts_applv //
-    elim (liftsv_liftv_trans_le … HV10s … HV120s) -V10s #V10s #HV10s #HV120s
-    >(liftv_mono … HV12s … HV10s) -V1s //
+    elim (liftsv_liftv_trans_le … HV10c … HV120c) -V10c #V10c #HV10c #HV120c
+    >(liftv_mono … HV12c … HV10c) -V1c //
   | @(gcr_lift … H1RP … HB … HV120) /2 width=2 by drop_drop/
   ]
 | #G #L #Vs #T #W #HA #HW #L0 #V0 #X #cs #HL0 #H #HB
-  elim (lifts_inv_applv1 … H) -H #V0s #Y #HV0s #HY #H destruct
+  elim (lifts_inv_applv1 … H) -H #V0c #Y #HV0c #HY #H destruct
   elim (lifts_inv_flat1 … HY) -HY #W0 #T0 #HW0 #HT0 #H destruct
-  @(s7 … IHA … (V0 @ V0s)) /3 width=5 by lifts_applv/
+  @(c7 … IHA … (V0 @ V0c)) /3 width=5 by lifts_applv/
 ]
 qed.
 
@@ -157,11 +157,11 @@ lapply (acr_gcr … H1RP H2RP A) #HCA
 lapply (acr_gcr … H1RP H2RP B) #HCB
 elim (lifts_inv_bind1 … H) -H #W0 #T0 #HW0 #HT0 #H destruct
 lapply (gcr_lifts … H1RP … HL0 … HW0 HW) -HW #HW0
-lapply (s3 … HCA … a G L0 (◊)) #H @H -H
-lapply (s6 … HCA G L0 (◊) (◊) ?) // #H @H -H
+lapply (c3 … HCA … a G L0 (◊)) #H @H -H
+lapply (c6 … HCA G L0 (◊) (◊) ?) // #H @H -H
 [ @(HA … HL0) //
-| lapply (s1 … HCB) -HCB #HCB
-  lapply (s7 … H2RP G L0 (◊)) /3 width=1 by/
+| lapply (c1 … HCB) -HCB #HCB
+  lapply (c7 … H2RP G L0 (◊)) /3 width=1 by/
 ]
 qed.
 
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/lcosx.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/lcosx.ma
index b19af4067..8f29e1aaf 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/lcosx.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/lcosx.ma
@@ -17,27 +17,27 @@ include "basic_2/computation/lsx.ma".
 
 (* SN EXTENDED STRONGLY CONORMALIZING LOCAL ENVIRONMENTS ********************)
 
-inductive lcosx (h) (g) (G): relation2 ynat lenv ≝
-| lcosx_sort: ∀l. lcosx h g G l (⋆)
-| lcosx_skip: ∀I,L,T. lcosx h g G 0 L → lcosx h g G 0 (L.ⓑ{I}T)
-| lcosx_pair: ∀I,L,T,l. G ⊢ ⬊*[h, g, T, l] L →
-              lcosx h g G l L → lcosx h g G (⫯l) (L.ⓑ{I}T)
+inductive lcosx (h) (o) (G): relation2 ynat lenv ≝
+| lcosx_sort: ∀l. lcosx h o G l (⋆)
+| lcosx_skip: ∀I,L,T. lcosx h o G 0 L → lcosx h o G 0 (L.ⓑ{I}T)
+| lcosx_pair: ∀I,L,T,l. G ⊢ ⬊*[h, o, T, l] L →
+              lcosx h o G l L → lcosx h o G (⫯l) (L.ⓑ{I}T)
 .
 
 interpretation
    "sn extended strong conormalization (local environment)"
-   'CoSN h g l G L = (lcosx h g G l L).
+   'CoSN h o l G L = (lcosx h o G l L).
 
 (* Basic properties *********************************************************)
 
-lemma lcosx_O: ∀h,g,G,L. G ⊢ ~⬊*[h, g, 0] L.
-#h #g #G #L elim L /2 width=1 by lcosx_skip/
+lemma lcosx_O: ∀h,o,G,L. G ⊢ ~⬊*[h, o, 0] L.
+#h #o #G #L elim L /2 width=1 by lcosx_skip/
 qed.
 
-lemma lcosx_drop_trans_lt: ∀h,g,G,L,l. G ⊢ ~⬊*[h, g, l] L →
+lemma lcosx_drop_trans_lt: ∀h,o,G,L,l. G ⊢ ~⬊*[h, o, l] L →
                             ∀I,K,V,i. ⬇[i] L ≡ K.ⓑ{I}V → i < l →
-                            G ⊢ ~⬊*[h, g, ⫰(l-i)] K ∧ G ⊢ ⬊*[h, g, V, ⫰(l-i)] K.
-#h #g #G #L #l #H elim H -L -l
+                            G ⊢ ~⬊*[h, o, ⫰(l-i)] K ∧ G ⊢ ⬊*[h, o, V, ⫰(l-i)] K.
+#h #o #G #L #l #H elim H -L -l
 [ #l #J #K #V #i #H elim (drop_inv_atom1 … H) -H #H destruct
 | #I #L #T #_ #_ #J #K #V #i #_ #H elim (ylt_yle_false … H) -H //
 | #I #L #T #l #HT #HL #IHL #J #K #V #i #H #Hil
@@ -52,25 +52,25 @@ qed-.
 
 (* Basic inversion lemmas ***************************************************)
 
-fact lcosx_inv_succ_aux: ∀h,g,G,L,x. G ⊢ ~⬊*[h, g, x] L → ∀l. x = ⫯l →
+fact lcosx_inv_succ_aux: ∀h,o,G,L,x. G ⊢ ~⬊*[h, o, x] L → ∀l. x = ⫯l →
                          L = ⋆ ∨
-                         ∃∃I,K,V. L = K.ⓑ{I}V & G ⊢ ~⬊*[h, g, l] K &
-                                  G ⊢ ⬊*[h, g, V, l] K.
-#h #g #G #L #l * -L -l /2 width=1 by or_introl/
+                         ∃∃I,K,V. L = K.ⓑ{I}V & G ⊢ ~⬊*[h, o, l] K &
+                                  G ⊢ ⬊*[h, o, V, l] K.
+#h #o #G #L #l * -L -l /2 width=1 by or_introl/
 [ #I #L #T #_ #x #H elim (ysucc_inv_O_sn … H)
 | #I #L #T #l #HT #HL #x #H <(ysucc_inv_inj … H) -x
   /3 width=6 by ex3_3_intro, or_intror/
 ]
 qed-.
 
-lemma lcosx_inv_succ: ∀h,g,G,L,l. G ⊢ ~⬊*[h, g, ⫯l] L → L = ⋆ ∨
-                      ∃∃I,K,V. L = K.ⓑ{I}V & G ⊢ ~⬊*[h, g, l] K &
-                               G ⊢ ⬊*[h, g, V, l] K.
+lemma lcosx_inv_succ: ∀h,o,G,L,l. G ⊢ ~⬊*[h, o, ⫯l] L → L = ⋆ ∨
+                      ∃∃I,K,V. L = K.ⓑ{I}V & G ⊢ ~⬊*[h, o, l] K &
+                               G ⊢ ⬊*[h, o, V, l] K.
 /2 width=3 by lcosx_inv_succ_aux/ qed-.
 
-lemma lcosx_inv_pair: ∀h,g,I,G,L,T,l. G ⊢ ~⬊*[h, g, ⫯l] L.ⓑ{I}T →
-                      G ⊢ ~⬊*[h, g, l] L ∧ G ⊢ ⬊*[h, g, T, l] L.
-#h #g #I #G #L #T #l #H elim (lcosx_inv_succ … H) -H
+lemma lcosx_inv_pair: ∀h,o,I,G,L,T,l. G ⊢ ~⬊*[h, o, ⫯l] L.ⓑ{I}T →
+                      G ⊢ ~⬊*[h, o, l] L ∧ G ⊢ ⬊*[h, o, T, l] L.
+#h #o #I #G #L #T #l #H elim (lcosx_inv_succ … H) -H
 [ #H destruct
 | * #Z #Y #X #H destruct /2 width=1 by conj/
 ]
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/lcosx_cpx.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/lcosx_cpx.ma
index eaa8649d4..756b8a7a1 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/lcosx_cpx.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/lcosx_cpx.ma
@@ -12,7 +12,6 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "ground_2/ynat/ynat_max.ma".
 include "basic_2/computation/lsx_drop.ma".
 include "basic_2/computation/lsx_lpx.ma".
 include "basic_2/computation/lsx_lpxs.ma".
@@ -22,10 +21,10 @@ include "basic_2/computation/lcosx.ma".
 
 (* Properties on extended context-sensitive parallel reduction for term *****)
 
-lemma lsx_cpx_trans_lcosx: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 →
-                           ∀l. G ⊢ ~⬊*[h, g, l] L →
-                           G ⊢ ⬊*[h, g, T1, l] L → G ⊢ ⬊*[h, g, T2, l] L.
-#h #g #G #L #T1 #T2 #H elim H -G -L -T1 -T2 //
+lemma lsx_cpx_trans_lcosx: ∀h,o,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h, o] T2 →
+                           ∀l. G ⊢ ~⬊*[h, o, l] L →
+                           G ⊢ ⬊*[h, o, T1, l] L → G ⊢ ⬊*[h, o, T2, l] L.
+#h #o #G #L #T1 #T2 #H elim H -G -L -T1 -T2 //
 [ #I #G #L #K #V1 #V2 #W2 #i #HLK #_ #HVW2 #IHV12 #l #HL #H
   elim (ylt_split i l) #Hli [ -H | -HL ]
   [ <(ymax_pre_sn l (⫯i)) /2 width=1 by ylt_fwd_le_succ1/
@@ -63,6 +62,6 @@ lemma lsx_cpx_trans_lcosx: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 →
 ]
 qed-.
 
-lemma lsx_cpx_trans_O: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 →
-                       G ⊢ ⬊*[h, g, T1, 0] L → G ⊢ ⬊*[h, g, T2, 0] L.
+lemma lsx_cpx_trans_O: ∀h,o,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h, o] T2 →
+                       G ⊢ ⬊*[h, o, T1, 0] L → G ⊢ ⬊*[h, o, T2, 0] L.
 /2 width=3 by lsx_cpx_trans_lcosx/ qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/lprs_cprs.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/lprs_cprs.ma
index 05138eb57..ff82ee0a5 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/lprs_cprs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/lprs_cprs.ma
@@ -48,14 +48,14 @@ lemma lprs_ind_alt: ∀G. ∀R:relation lenv.
 
 (* Properties on context-sensitive parallel computation for terms ***********)
 
-lemma lprs_cpr_trans: ∀G. s_r_transitive … (cpr G) (λ_. lprs G).
-/3 width=5 by s_r_trans_LTC2, lpr_cprs_trans/ qed-.
+lemma lprs_cpr_trans: ∀G. c_r_transitive … (cpr G) (λ_. lprs G).
+/3 width=5 by c_r_trans_LTC2, lpr_cprs_trans/ qed-.
 
 (* Basic_1: was just: pr3_pr3_pr3_t *)
 (* Note: alternative proof /3 width=5 by s_r_trans_LTC1, lprs_cpr_trans/ *)
-lemma lprs_cprs_trans: ∀G. s_rs_transitive … (cpr G) (λ_. lprs G).
-#G @s_r_to_s_rs_trans @s_r_trans_LTC2
-@s_rs_trans_TC1 /2 width=3 by lpr_cprs_trans/ (**) (* full auto too slow *)
+lemma lprs_cprs_trans: ∀G. c_rs_transitive … (cpr G) (λ_. lprs G).
+#G @c_r_to_c_rs_trans @c_r_trans_LTC2
+@c_rs_trans_TC1 /2 width=3 by lpr_cprs_trans/ (**) (* full auto too slow *)
 qed-.
 
 lemma lprs_cprs_conf_dx: ∀G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡* T1 →
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/lpxs.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/lpxs.ma
index ee557e438..b6c91c8b1 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/lpxs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/lpxs.ma
@@ -19,56 +19,56 @@ include "basic_2/computation/lprs.ma".
 (* SN EXTENDED PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS *******************)
 
 definition lpxs: ∀h. sd h → relation3 genv lenv lenv ≝
-                 λh,g,G. TC … (lpx h g G).
+                 λh,o,G. TC … (lpx h o G).
 
 interpretation "extended parallel computation (local environment, sn variant)"
-   'PRedSnStar h g G L1 L2 = (lpxs h g G L1 L2).
+   'PRedSnStar h o G L1 L2 = (lpxs h o G L1 L2).
 
 (* Basic eliminators ********************************************************)
 
-lemma lpxs_ind: ∀h,g,G,L1. ∀R:predicate lenv. R L1 →
-                (∀L,L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L → ⦃G, L⦄ ⊢ ➡[h, g] L2 → R L → R L2) →
-                ∀L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 → R L2.
-#h #g #G #L1 #R #HL1 #IHL1 #L2 #HL12
+lemma lpxs_ind: ∀h,o,G,L1. ∀R:predicate lenv. R L1 →
+                (∀L,L2. ⦃G, L1⦄ ⊢ ➡*[h, o] L → ⦃G, L⦄ ⊢ ➡[h, o] L2 → R L → R L2) →
+                ∀L2. ⦃G, L1⦄ ⊢ ➡*[h, o] L2 → R L2.
+#h #o #G #L1 #R #HL1 #IHL1 #L2 #HL12
 @(TC_star_ind … HL1 IHL1 … HL12) //
 qed-.
 
-lemma lpxs_ind_dx: ∀h,g,G,L2. ∀R:predicate lenv. R L2 →
-                   (∀L1,L. ⦃G, L1⦄ ⊢ ➡[h, g] L → ⦃G, L⦄ ⊢ ➡*[h, g] L2 → R L → R L1) →
-                   ∀L1. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 → R L1.
-#h #g #G #L2 #R #HL2 #IHL2 #L1 #HL12
+lemma lpxs_ind_dx: ∀h,o,G,L2. ∀R:predicate lenv. R L2 →
+                   (∀L1,L. ⦃G, L1⦄ ⊢ ➡[h, o] L → ⦃G, L⦄ ⊢ ➡*[h, o] L2 → R L → R L1) →
+                   ∀L1. ⦃G, L1⦄ ⊢ ➡*[h, o] L2 → R L1.
+#h #o #G #L2 #R #HL2 #IHL2 #L1 #HL12
 @(TC_star_ind_dx … HL2 IHL2 … HL12) //
 qed-.
 
 (* Basic properties *********************************************************)
 
-lemma lprs_lpxs: ∀h,g,G,L1,L2. ⦃G, L1⦄ ⊢ ➡* L2 → ⦃G, L1⦄ ⊢ ➡*[h, g] L2.
+lemma lprs_lpxs: ∀h,o,G,L1,L2. ⦃G, L1⦄ ⊢ ➡* L2 → ⦃G, L1⦄ ⊢ ➡*[h, o] L2.
 /3 width=3 by lpr_lpx, monotonic_TC/ qed.
 
-lemma lpx_lpxs: ∀h,g,G,L1,L2. ⦃G, L1⦄ ⊢ ➡[h, g] L2 → ⦃G, L1⦄ ⊢ ➡*[h, g] L2.
+lemma lpx_lpxs: ∀h,o,G,L1,L2. ⦃G, L1⦄ ⊢ ➡[h, o] L2 → ⦃G, L1⦄ ⊢ ➡*[h, o] L2.
 /2 width=1 by inj/ qed.
 
-lemma lpxs_refl: ∀h,g,G,L. ⦃G, L⦄ ⊢ ➡*[h, g] L.
+lemma lpxs_refl: ∀h,o,G,L. ⦃G, L⦄ ⊢ ➡*[h, o] L.
 /2 width=1 by lprs_lpxs/ qed.
 
-lemma lpxs_strap1: ∀h,g,G,L1,L,L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L → ⦃G, L⦄ ⊢ ➡[h, g] L2 → ⦃G, L1⦄ ⊢ ➡*[h, g] L2.
+lemma lpxs_strap1: ∀h,o,G,L1,L,L2. ⦃G, L1⦄ ⊢ ➡*[h, o] L → ⦃G, L⦄ ⊢ ➡[h, o] L2 → ⦃G, L1⦄ ⊢ ➡*[h, o] L2.
 /2 width=3 by step/ qed.
 
-lemma lpxs_strap2: ∀h,g,G,L1,L,L2. ⦃G, L1⦄ ⊢ ➡[h, g] L → ⦃G, L⦄ ⊢ ➡*[h, g] L2 → ⦃G, L1⦄ ⊢ ➡*[h, g] L2.
+lemma lpxs_strap2: ∀h,o,G,L1,L,L2. ⦃G, L1⦄ ⊢ ➡[h, o] L → ⦃G, L⦄ ⊢ ➡*[h, o] L2 → ⦃G, L1⦄ ⊢ ➡*[h, o] L2.
 /2 width=3 by TC_strap/ qed.
 
-lemma lpxs_pair_refl: ∀h,g,G,L1,L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 → ∀I,V. ⦃G, L1.ⓑ{I}V⦄ ⊢ ➡*[h, g] L2.ⓑ{I}V.
+lemma lpxs_pair_refl: ∀h,o,G,L1,L2. ⦃G, L1⦄ ⊢ ➡*[h, o] L2 → ∀I,V. ⦃G, L1.ⓑ{I}V⦄ ⊢ ➡*[h, o] L2.ⓑ{I}V.
 /2 width=1 by TC_lpx_sn_pair_refl/ qed.
 
 (* Basic inversion lemmas ***************************************************)
 
-lemma lpxs_inv_atom1: ∀h,g,G,L2. ⦃G, ⋆⦄ ⊢ ➡*[h, g] L2 → L2 = ⋆.
+lemma lpxs_inv_atom1: ∀h,o,G,L2. ⦃G, ⋆⦄ ⊢ ➡*[h, o] L2 → L2 = ⋆.
 /2 width=2 by TC_lpx_sn_inv_atom1/ qed-.
 
-lemma lpxs_inv_atom2: ∀h,g,G,L1. ⦃G, L1⦄ ⊢ ➡*[h, g] ⋆ → L1 = ⋆.
+lemma lpxs_inv_atom2: ∀h,o,G,L1. ⦃G, L1⦄ ⊢ ➡*[h, o] ⋆ → L1 = ⋆.
 /2 width=2 by TC_lpx_sn_inv_atom2/ qed-.
 
 (* Basic forward lemmas *****************************************************)
 
-lemma lpxs_fwd_length: ∀h,g,G,L1,L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 → |L1| = |L2|.
+lemma lpxs_fwd_length: ∀h,o,G,L1,L2. ⦃G, L1⦄ ⊢ ➡*[h, o] L2 → |L1| = |L2|.
 /2 width=2 by TC_lpx_sn_fwd_length/ qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/lpxs_aaa.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/lpxs_aaa.ma
index d335f57da..0dfdd95e2 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/lpxs_aaa.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/lpxs_aaa.ma
@@ -19,9 +19,9 @@ include "basic_2/computation/lpxs.ma".
 
 (* Properties about atomic arity assignment on terms ************************)
 
-lemma lpxs_aaa_conf: ∀h,g,G,L1,T,A. ⦃G, L1⦄ ⊢ T ⁝ A →
-                     ∀L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 → ⦃G, L2⦄ ⊢ T ⁝ A.
-#h #g #G #L1 #T #A #HT #L2 #HL12
+lemma lpxs_aaa_conf: ∀h,o,G,L1,T,A. ⦃G, L1⦄ ⊢ T ⁝ A →
+                     ∀L2. ⦃G, L1⦄ ⊢ ➡*[h, o] L2 → ⦃G, L2⦄ ⊢ T ⁝ A.
+#h #o #G #L1 #T #A #HT #L2 #HL12
 @(TC_Conf3 … (λL,A. ⦃G, L⦄ ⊢ T ⁝ A) … HT ? HL12) /2 width=5 by lpx_aaa_conf/
 qed-.
 
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/lpxs_cpxs.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/lpxs_cpxs.ma
index c1928ff10..41316888e 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/lpxs_cpxs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/lpxs_cpxs.ma
@@ -19,66 +19,66 @@ include "basic_2/computation/lpxs.ma".
 
 (* Advanced properties ******************************************************)
 
-lemma lpxs_pair: ∀h,g,I,G,L1,L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 →
-                 ∀V1,V2. ⦃G, L1⦄ ⊢ V1 ➡*[h, g] V2 →
-                 ⦃G, L1.ⓑ{I}V1⦄ ⊢ ➡*[h, g] L2.ⓑ{I}V2.
+lemma lpxs_pair: ∀h,o,I,G,L1,L2. ⦃G, L1⦄ ⊢ ➡*[h, o] L2 →
+                 ∀V1,V2. ⦃G, L1⦄ ⊢ V1 ➡*[h, o] V2 →
+                 ⦃G, L1.ⓑ{I}V1⦄ ⊢ ➡*[h, o] L2.ⓑ{I}V2.
 /2 width=1 by TC_lpx_sn_pair/ qed.
 
 (* Advanced inversion lemmas ************************************************)
 
-lemma lpxs_inv_pair1: ∀h,g,I,G,K1,L2,V1. ⦃G, K1.ⓑ{I}V1⦄ ⊢ ➡*[h, g] L2 →
-                      ∃∃K2,V2. ⦃G, K1⦄ ⊢ ➡*[h, g] K2 & ⦃G, K1⦄ ⊢ V1 ➡*[h, g] V2 & L2 = K2.ⓑ{I}V2.
+lemma lpxs_inv_pair1: ∀h,o,I,G,K1,L2,V1. ⦃G, K1.ⓑ{I}V1⦄ ⊢ ➡*[h, o] L2 →
+                      ∃∃K2,V2. ⦃G, K1⦄ ⊢ ➡*[h, o] K2 & ⦃G, K1⦄ ⊢ V1 ➡*[h, o] V2 & L2 = K2.ⓑ{I}V2.
 /3 width=3 by TC_lpx_sn_inv_pair1, lpx_cpxs_trans/ qed-.
 
-lemma lpxs_inv_pair2: ∀h,g,I,G,L1,K2,V2. ⦃G, L1⦄ ⊢ ➡*[h, g] K2.ⓑ{I}V2 →
-                      ∃∃K1,V1. ⦃G, K1⦄ ⊢ ➡*[h, g] K2 & ⦃G, K1⦄ ⊢ V1 ➡*[h, g] V2 & L1 = K1.ⓑ{I}V1.
+lemma lpxs_inv_pair2: ∀h,o,I,G,L1,K2,V2. ⦃G, L1⦄ ⊢ ➡*[h, o] K2.ⓑ{I}V2 →
+                      ∃∃K1,V1. ⦃G, K1⦄ ⊢ ➡*[h, o] K2 & ⦃G, K1⦄ ⊢ V1 ➡*[h, o] V2 & L1 = K1.ⓑ{I}V1.
 /3 width=3 by TC_lpx_sn_inv_pair2, lpx_cpxs_trans/ qed-.
 
 (* Advanced eliminators *****************************************************)
 
-lemma lpxs_ind_alt: ∀h,g,G. ∀R:relation lenv.
+lemma lpxs_ind_alt: ∀h,o,G. ∀R:relation lenv.
                     R (⋆) (⋆) → (
                        ∀I,K1,K2,V1,V2.
-                       ⦃G, K1⦄ ⊢ ➡*[h, g] K2 → ⦃G, K1⦄ ⊢ V1 ➡*[h, g] V2 →
+                       ⦃G, K1⦄ ⊢ ➡*[h, o] K2 → ⦃G, K1⦄ ⊢ V1 ➡*[h, o] V2 →
                        R K1 K2 → R (K1.ⓑ{I}V1) (K2.ⓑ{I}V2)
                     ) →
-                    ∀L1,L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 → R L1 L2.
+                    ∀L1,L2. ⦃G, L1⦄ ⊢ ➡*[h, o] L2 → R L1 L2.
 /3 width=4 by TC_lpx_sn_ind, lpx_cpxs_trans/ qed-.
 
 (* Properties on context-sensitive extended parallel computation for terms **)
 
-lemma lpxs_cpx_trans: ∀h,g,G. s_r_transitive … (cpx h g G) (λ_.lpxs h g G).
-/3 width=5 by s_r_trans_LTC2, lpx_cpxs_trans/ qed-.
+lemma lpxs_cpx_trans: ∀h,o,G. c_r_transitive … (cpx h o G) (λ_.lpxs h o G).
+/3 width=5 by c_r_trans_LTC2, lpx_cpxs_trans/ qed-.
 
 (* Note: alternative proof: /3 width=5 by s_r_trans_TC1, lpxs_cpx_trans/ *)
-lemma lpxs_cpxs_trans: ∀h,g,G. s_rs_transitive … (cpx h g G) (λ_.lpxs h g G).
-#h #g #G @s_r_to_s_rs_trans @s_r_trans_LTC2
-@s_rs_trans_TC1 /2 width=3 by lpx_cpxs_trans/ (**) (* full auto too slow *)
+lemma lpxs_cpxs_trans: ∀h,o,G. c_rs_transitive … (cpx h o G) (λ_.lpxs h o G).
+#h #o #G @c_r_to_c_rs_trans @c_r_trans_LTC2
+@c_rs_trans_TC1 /2 width=3 by lpx_cpxs_trans/ (**) (* full auto too slow *)
 qed-.
 
-lemma cpxs_bind2: ∀h,g,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡*[h, g] V2 →
-                  ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ➡*[h, g] T2 →
-                  ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡*[h, g] ⓑ{a,I}V2.T2.
+lemma cpxs_bind2: ∀h,o,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡*[h, o] V2 →
+                  ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ➡*[h, o] T2 →
+                  ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡*[h, o] ⓑ{a,I}V2.T2.
 /4 width=5 by lpxs_cpxs_trans, lpxs_pair, cpxs_bind/ qed.
 
 (* Inversion lemmas on context-sensitive ext parallel computation for terms *)
 
-lemma cpxs_inv_abst1: ∀h,g,a,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓛ{a}V1.T1 ➡*[h, g] U2 →
-                      ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡*[h, g] V2 & ⦃G, L.ⓛV1⦄ ⊢ T1 ➡*[h, g] T2 &
+lemma cpxs_inv_abst1: ∀h,o,a,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓛ{a}V1.T1 ➡*[h, o] U2 →
+                      ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡*[h, o] V2 & ⦃G, L.ⓛV1⦄ ⊢ T1 ➡*[h, o] T2 &
                                U2 = ⓛ{a}V2.T2.
-#h #g #a #G #L #V1 #T1 #U2 #H @(cpxs_ind … H) -U2 /2 width=5 by ex3_2_intro/
+#h #o #a #G #L #V1 #T1 #U2 #H @(cpxs_ind … H) -U2 /2 width=5 by ex3_2_intro/
 #U0 #U2 #_ #HU02 * #V0 #T0 #HV10 #HT10 #H destruct
 elim (cpx_inv_abst1 … HU02) -HU02 #V2 #T2 #HV02 #HT02 #H destruct
 lapply (lpxs_cpx_trans … HT02 (L.ⓛV1) ?)
 /3 width=5 by lpxs_pair, cpxs_trans, cpxs_strap1, ex3_2_intro/
 qed-.
 
-lemma cpxs_inv_abbr1: ∀h,g,a,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓓ{a}V1.T1 ➡*[h, g] U2 → (
-                      ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡*[h, g] V2 & ⦃G, L.ⓓV1⦄ ⊢ T1 ➡*[h, g] T2 &
+lemma cpxs_inv_abbr1: ∀h,o,a,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓓ{a}V1.T1 ➡*[h, o] U2 → (
+                      ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡*[h, o] V2 & ⦃G, L.ⓓV1⦄ ⊢ T1 ➡*[h, o] T2 &
                                U2 = ⓓ{a}V2.T2
                       ) ∨
-                      ∃∃T2. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡*[h, g] T2 & ⬆[0, 1] U2 ≡ T2 & a = true.
-#h #g #a #G #L #V1 #T1 #U2 #H @(cpxs_ind … H) -U2 /3 width=5 by ex3_2_intro, or_introl/
+                      ∃∃T2. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡*[h, o] T2 & ⬆[0, 1] U2 ≡ T2 & a = true.
+#h #o #a #G #L #V1 #T1 #U2 #H @(cpxs_ind … H) -U2 /3 width=5 by ex3_2_intro, or_introl/
 #U0 #U2 #_ #HU02 * *
 [ #V0 #T0 #HV10 #HT10 #H destruct
   elim (cpx_inv_abbr1 … HU02) -HU02 *
@@ -97,16 +97,16 @@ qed-.
 
 (* More advanced properties *************************************************)
 
-lemma lpxs_pair2: ∀h,g,I,G,L1,L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 →
-                  ∀V1,V2. ⦃G, L2⦄ ⊢ V1 ➡*[h, g] V2 → ⦃G, L1.ⓑ{I}V1⦄ ⊢ ➡*[h, g] L2.ⓑ{I}V2.
+lemma lpxs_pair2: ∀h,o,I,G,L1,L2. ⦃G, L1⦄ ⊢ ➡*[h, o] L2 →
+                  ∀V1,V2. ⦃G, L2⦄ ⊢ V1 ➡*[h, o] V2 → ⦃G, L1.ⓑ{I}V1⦄ ⊢ ➡*[h, o] L2.ⓑ{I}V2.
 /3 width=3 by lpxs_pair, lpxs_cpxs_trans/ qed.
 
 (* Properties on supclosure *************************************************)
 
-lemma lpx_fqup_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
-                      ∀K1. ⦃G1, K1⦄ ⊢ ➡[h, g] L1 →
-                      ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ➡*[h, g] T & ⦃G1, K1, T⦄ ⊐+ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h, g] L2.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
+lemma lpx_fqup_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
+                      ∀K1. ⦃G1, K1⦄ ⊢ ➡[h, o] L1 →
+                      ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ➡*[h, o] T & ⦃G1, K1, T⦄ ⊐+ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h, o] L2.
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
 [ #G2 #L2 #T2 #H12 #K1 #HKL1 elim (lpx_fqu_trans … H12 … HKL1) -L1
   /3 width=5 by cpx_cpxs, fqu_fqup, ex3_2_intro/
 | #G #G2 #L #L2 #T #T2 #_ #H2 #IH1 #K1 #HLK1 elim (IH1 … HLK1) -L1
@@ -116,20 +116,20 @@ lemma lpx_fqup_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2,
 ]
 qed-.
 
-lemma lpx_fqus_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
-                      ∀K1. ⦃G1, K1⦄ ⊢ ➡[h, g] L1 →
-                      ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ➡*[h, g] T & ⦃G1, K1, T⦄ ⊐* ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h, g] L2.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqus_ind … H) -G2 -L2 -T2 [ /2 width=5 by ex3_2_intro/ ]
+lemma lpx_fqus_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
+                      ∀K1. ⦃G1, K1⦄ ⊢ ➡[h, o] L1 →
+                      ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ➡*[h, o] T & ⦃G1, K1, T⦄ ⊐* ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h, o] L2.
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqus_ind … H) -G2 -L2 -T2 [ /2 width=5 by ex3_2_intro/ ]
 #G #G2 #L #L2 #T #T2 #_ #H2 #IH1 #K1 #HLK1 elim (IH1 … HLK1) -L1
 #L0 #T0 #HT10 #HT0 #HL0 elim (lpx_fquq_trans … H2 … HL0) -L
 #L #T3 #HT3 #HT32 #HL2 elim (fqus_cpx_trans … HT0 … HT3) -T
 /3 width=7 by cpxs_strap1, fqus_strap1, ex3_2_intro/
 qed-.
 
-lemma lpxs_fquq_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
-                       ∀K1. ⦃G1, K1⦄ ⊢ ➡*[h, g] L1 →
-                       ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ➡*[h, g] T & ⦃G1, K1, T⦄ ⊐⸮ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, g] L2.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 #HT12 #K1 #H @(lpxs_ind_dx … H) -K1
+lemma lpxs_fquq_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
+                       ∀K1. ⦃G1, K1⦄ ⊢ ➡*[h, o] L1 →
+                       ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ➡*[h, o] T & ⦃G1, K1, T⦄ ⊐⸮ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, o] L2.
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 #HT12 #K1 #H @(lpxs_ind_dx … H) -K1
 [ /2 width=5 by ex3_2_intro/
 | #K1 #K #HK1 #_ * #L #T #HT1 #HT2 #HL2 -HT12
   lapply (lpx_cpxs_trans … HT1 … HK1) -HT1
@@ -138,10 +138,10 @@ lemma lpxs_fquq_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2,
 ]
 qed-.
 
-lemma lpxs_fqup_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
-                       ∀K1. ⦃G1, K1⦄ ⊢ ➡*[h, g] L1 →
-                       ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ➡*[h, g] T & ⦃G1, K1, T⦄ ⊐+ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, g] L2.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 #HT12 #K1 #H @(lpxs_ind_dx … H) -K1
+lemma lpxs_fqup_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
+                       ∀K1. ⦃G1, K1⦄ ⊢ ➡*[h, o] L1 →
+                       ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ➡*[h, o] T & ⦃G1, K1, T⦄ ⊐+ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, o] L2.
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 #HT12 #K1 #H @(lpxs_ind_dx … H) -K1
 [ /2 width=5 by ex3_2_intro/
 | #K1 #K #HK1 #_ * #L #T #HT1 #HT2 #HL2 -HT12
   lapply (lpx_cpxs_trans … HT1 … HK1) -HT1
@@ -150,10 +150,10 @@ lemma lpxs_fqup_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2
 ]
 qed-.
 
-lemma lpxs_fqus_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
-                       ∀K1. ⦃G1, K1⦄ ⊢ ➡*[h, g] L1 →
-                       ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ➡*[h, g] T & ⦃G1, K1, T⦄ ⊐* ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, g] L2.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqus_ind … H) -G2 -L2 -T2 /2 width=5 by ex3_2_intro/
+lemma lpxs_fqus_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
+                       ∀K1. ⦃G1, K1⦄ ⊢ ➡*[h, o] L1 →
+                       ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ➡*[h, o] T & ⦃G1, K1, T⦄ ⊐* ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, o] L2.
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqus_ind … H) -G2 -L2 -T2 /2 width=5 by ex3_2_intro/
 #G #G2 #L #L2 #T #T2 #_ #H2 #IH1 #K1 #HLK1 elim (IH1 … HLK1) -L1
 #L0 #T0 #HT10 #HT0 #HL0 elim (lpxs_fquq_trans … H2 … HL0) -L
 #L #T3 #HT3 #HT32 #HL2 elim (fqus_cpxs_trans … HT3 … HT0) -T
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/lpxs_drop.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/lpxs_drop.ma
index dba41c9dd..c75d754f9 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/lpxs_drop.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/lpxs_drop.ma
@@ -19,11 +19,11 @@ include "basic_2/computation/lpxs.ma".
 
 (* Properties on local environment slicing ***********************************)
 
-lemma lpxs_drop_conf: ∀h,g,G. dropable_sn (lpxs h g G).
+lemma lpxs_drop_conf: ∀h,o,G. dropable_sn (lpxs h o G).
 /3 width=3 by dropable_sn_TC, lpx_drop_conf/ qed-.
 
-lemma drop_lpxs_trans: ∀h,g,G. dedropable_sn (lpxs h g G).
+lemma drop_lpxs_trans: ∀h,o,G. dedropable_sn (lpxs h o G).
 /3 width=3 by dedropable_sn_TC, drop_lpx_trans/ qed-.
 
-lemma lpxs_drop_trans_O1: ∀h,g,G. dropable_dx (lpxs h g G).
+lemma lpxs_drop_trans_O1: ∀h,o,G. dropable_dx (lpxs h o G).
 /3 width=3 by dropable_dx_TC, lpx_drop_trans_O1/ qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/lpxs_lleq.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/lpxs_lleq.ma
index 6f6045dab..159c84f81 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/lpxs_lleq.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/lpxs_lleq.ma
@@ -22,18 +22,18 @@ include "basic_2/computation/lpxs_cpxs.ma".
 
 (* Properties on lazy equivalence for local environments ********************)
 
-lemma lleq_lpxs_trans: ∀h,g,G,L2,K2. ⦃G, L2⦄ ⊢ ➡*[h, g] K2 →
+lemma lleq_lpxs_trans: ∀h,o,G,L2,K2. ⦃G, L2⦄ ⊢ ➡*[h, o] K2 →
                        ∀L1,T,l. L1 ≡[T, l] L2 →
-                       ∃∃K1. ⦃G, L1⦄ ⊢ ➡*[h, g] K1 & K1 ≡[T, l] K2.
-#h #g #G #L2 #K2 #H @(lpxs_ind … H) -K2 /2 width=3 by ex2_intro/
+                       ∃∃K1. ⦃G, L1⦄ ⊢ ➡*[h, o] K1 & K1 ≡[T, l] K2.
+#h #o #G #L2 #K2 #H @(lpxs_ind … H) -K2 /2 width=3 by ex2_intro/
 #K #K2 #_ #HK2 #IH #L1 #T #l #HT elim (IH … HT) -L2
 #L #HL1 #HT elim (lleq_lpx_trans … HK2 … HT) -K
 /3 width=3 by lpxs_strap1, ex2_intro/
 qed-.
 
-lemma lpxs_nlleq_inv_step_sn: ∀h,g,G,L1,L2,T,l. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 → (L1 ≡[T, l] L2 → ⊥) →
-                              ∃∃L,L0. ⦃G, L1⦄ ⊢ ➡[h, g] L & L1 ≡[T, l] L → ⊥ & ⦃G, L⦄ ⊢ ➡*[h, g] L0 & L0 ≡[T, l] L2.
-#h #g #G #L1 #L2 #T #l #H @(lpxs_ind_dx … H) -L1
+lemma lpxs_nlleq_inv_step_sn: ∀h,o,G,L1,L2,T,l. ⦃G, L1⦄ ⊢ ➡*[h, o] L2 → (L1 ≡[T, l] L2 → ⊥) →
+                              ∃∃L,L0. ⦃G, L1⦄ ⊢ ➡[h, o] L & L1 ≡[T, l] L → ⊥ & ⦃G, L⦄ ⊢ ➡*[h, o] L0 & L0 ≡[T, l] L2.
+#h #o #G #L1 #L2 #T #l #H @(lpxs_ind_dx … H) -L1
 [ #H elim H -H //
 | #L1 #L #H1 #H2 #IH2 #H12 elim (lleq_dec T L1 L l) #H
   [ -H1 -H2 elim IH2 -IH2 /3 width=3 by lleq_trans/ -H12
@@ -47,10 +47,10 @@ lemma lpxs_nlleq_inv_step_sn: ∀h,g,G,L1,L2,T,l. ⦃G, L1⦄ ⊢ ➡*[h, g] L2
 ]
 qed-.
 
-lemma lpxs_lleq_fqu_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
-                           ∀K1. ⦃G1, K1⦄ ⊢ ➡*[h, g] L1 → K1 ≡[T1, 0] L1 →
-                           ∃∃K2. ⦃G1, K1, T1⦄ ⊐ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, g] L2 & K2 ≡[T2, 0] L2.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
+lemma lpxs_lleq_fqu_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
+                           ∀K1. ⦃G1, K1⦄ ⊢ ➡*[h, o] L1 → K1 ≡[T1, 0] L1 →
+                           ∃∃K2. ⦃G1, K1, T1⦄ ⊐ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, o] L2 & K2 ≡[T2, 0] L2.
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
 [ #I #G1 #L1 #V1 #X #H1 #H2 elim (lpxs_inv_pair2 … H1) -H1
   #K0 #V0 #H1KL1 #_ #H destruct
   elim (lleq_inv_lref_ge_dx … H2 ? I L1 V1) -H2 //
@@ -64,22 +64,22 @@ lemma lpxs_lleq_fqu_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2,
   /3 width=4 by lpxs_pair, fqu_bind_dx, ex3_intro/
 | #I #G1 #L1 #V1 #T1 #K1 #HLK1 #H elim (lleq_inv_flat … H) -H
   /2 width=4 by fqu_flat_dx, ex3_intro/
-| #G1 #L1 #L #T1 #U1 #m #HL1 #HTU1 #K1 #H1KL1 #H2KL1
-  elim (drop_O1_le (Ⓕ) (m+1) K1)
+| #G1 #L1 #L #T1 #U1 #k #HL1 #HTU1 #K1 #H1KL1 #H2KL1
+  elim (drop_O1_le (Ⓕ) (k+1) K1)
   [ #K #HK1 lapply (lleq_inv_lift_le … H2KL1 … HK1 HL1 … HTU1 ?) -H2KL1 //
     #H2KL elim (lpxs_drop_trans_O1 … H1KL1 … HL1) -L1
     #K0 #HK10 #H1KL lapply (drop_mono … HK10 … HK1) -HK10 #H destruct
     /3 width=4 by fqu_drop, ex3_intro/
-  | lapply (drop_fwd_length_le2 … HL1) -L -T1 -g
+  | lapply (drop_fwd_length_le2 … HL1) -L -T1 -o
     lapply (lleq_fwd_length … H2KL1) //
   ]
 ]
 qed-.
 
-lemma lpxs_lleq_fquq_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
-                            ∀K1. ⦃G1, K1⦄ ⊢ ➡*[h, g] L1 → K1 ≡[T1, 0] L1 →
-                            ∃∃K2. ⦃G1, K1, T1⦄ ⊐⸮ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, g] L2 & K2 ≡[T2, 0] L2.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H #K1 #H1KL1 #H2KL1
+lemma lpxs_lleq_fquq_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
+                            ∀K1. ⦃G1, K1⦄ ⊢ ➡*[h, o] L1 → K1 ≡[T1, 0] L1 →
+                            ∃∃K2. ⦃G1, K1, T1⦄ ⊐⸮ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, o] L2 & K2 ≡[T2, 0] L2.
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H #K1 #H1KL1 #H2KL1
 elim (fquq_inv_gen … H) -H
 [ #H elim (lpxs_lleq_fqu_trans … H … H1KL1 H2KL1) -L1
   /3 width=4 by fqu_fquq, ex3_intro/
@@ -87,10 +87,10 @@ elim (fquq_inv_gen … H) -H
 ]
 qed-.
 
-lemma lpxs_lleq_fqup_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
-                            ∀K1. ⦃G1, K1⦄ ⊢ ➡*[h, g] L1 → K1 ≡[T1, 0] L1 →
-                            ∃∃K2. ⦃G1, K1, T1⦄ ⊐+ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, g] L2 & K2 ≡[T2, 0] L2.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
+lemma lpxs_lleq_fqup_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
+                            ∀K1. ⦃G1, K1⦄ ⊢ ➡*[h, o] L1 → K1 ≡[T1, 0] L1 →
+                            ∃∃K2. ⦃G1, K1, T1⦄ ⊐+ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, o] L2 & K2 ≡[T2, 0] L2.
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
 [ #G2 #L2 #T2 #H #K1 #H1KL1 #H2KL1 elim (lpxs_lleq_fqu_trans … H … H1KL1 H2KL1) -L1
   /3 width=4 by fqu_fqup, ex3_intro/
 | #G #G2 #L #L2 #T #T2 #_ #HT2 #IHT1 #K1 #H1KL1 #H2KL1 elim (IHT1 … H2KL1) // -L1
@@ -99,10 +99,10 @@ lemma lpxs_lleq_fqup_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G
 ]
 qed-.
 
-lemma lpxs_lleq_fqus_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
-                            ∀K1. ⦃G1, K1⦄ ⊢ ➡*[h, g] L1 → K1 ≡[T1, 0] L1 →
-                            ∃∃K2. ⦃G1, K1, T1⦄ ⊐* ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, g] L2 & K2 ≡[T2, 0] L2.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H #K1 #H1KL1 #H2KL1
+lemma lpxs_lleq_fqus_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
+                            ∀K1. ⦃G1, K1⦄ ⊢ ➡*[h, o] L1 → K1 ≡[T1, 0] L1 →
+                            ∃∃K2. ⦃G1, K1, T1⦄ ⊐* ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, o] L2 & K2 ≡[T2, 0] L2.
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H #K1 #H1KL1 #H2KL1
 elim (fqus_inv_gen … H) -H
 [ #H elim (lpxs_lleq_fqup_trans … H … H1KL1 H2KL1) -L1
   /3 width=4 by fqup_fqus, ex3_intro/
@@ -110,22 +110,22 @@ elim (fqus_inv_gen … H) -H
 ]
 qed-.
 
-fact lreq_lpxs_trans_lleq_aux: ∀h,g,G,L1,L0,l,m. L1 ⩬[l, m] L0 → m = ∞ →
-                               ∀L2. ⦃G, L0⦄ ⊢ ➡*[h, g] L2 →
-                               ∃∃L. L ⩬[l, m] L2 & ⦃G, L1⦄ ⊢ ➡*[h, g] L &
+fact lreq_lpxs_trans_lleq_aux: ∀h,o,G,L1,L0,l,k. L1 ⩬[l, k] L0 → k = ∞ →
+                               ∀L2. ⦃G, L0⦄ ⊢ ➡*[h, o] L2 →
+                               ∃∃L. L ⩬[l, k] L2 & ⦃G, L1⦄ ⊢ ➡*[h, o] L &
                                     (∀T. L0 ≡[T, l] L2 ↔ L1 ≡[T, l] L).
-#h #g #G #L1 #L0 #l #m #H elim H -L1 -L0 -l -m
-[ #l #m #_ #L2 #H >(lpxs_inv_atom1 … H) -H
+#h #o #G #L1 #L0 #l #k #H elim H -L1 -L0 -l -k
+[ #l #k #_ #L2 #H >(lpxs_inv_atom1 … H) -H
   /3 width=5 by ex3_intro, conj/
 | #I1 #I0 #L1 #L0 #V1 #V0 #_ #_ #Hm destruct
-| #I #L1 #L0 #V1 #m #HL10 #IHL10 #Hm #Y #H
+| #I #L1 #L0 #V1 #k #HL10 #IHL10 #Hm #Y #H
   elim (lpxs_inv_pair1 … H) -H #L2 #V2 #HL02 #HV02 #H destruct
   lapply (ysucc_inv_Y_dx … Hm) -Hm #Hm
   elim (IHL10 … HL02) // -IHL10 -HL02 #L #HL2 #HL1 #IH
   @(ex3_intro … (L.ⓑ{I}V2)) /3 width=3 by lpxs_pair, lreq_cpxs_trans, lreq_pair/
   #T elim (IH T) #HL0dx #HL0sn
   @conj #H @(lleq_lreq_repl … H) -H /3 width=1 by lreq_sym, lreq_pair_O_Y/
-| #I1 #I0 #L1 #L0 #V1 #V0 #l #m #HL10 #IHL10 #Hm #Y #H
+| #I1 #I0 #L1 #L0 #V1 #V0 #l #k #HL10 #IHL10 #Hm #Y #H
   elim (lpxs_inv_pair1 … H) -H #L2 #V2 #HL02 #HV02 #H destruct
   elim (IHL10 … HL02) // -IHL10 -HL02 #L #HL2 #HL1 #IH
   @(ex3_intro … (L.ⓑ{I1}V1)) /3 width=1 by lpxs_pair, lreq_succ/
@@ -134,8 +134,8 @@ fact lreq_lpxs_trans_lleq_aux: ∀h,g,G,L1,L0,l,m. L1 ⩬[l, m] L0 → m = ∞ 
 ]
 qed-.
 
-lemma lreq_lpxs_trans_lleq: ∀h,g,G,L1,L0,l. L1 ⩬[l, ∞] L0 →
-                            ∀L2. ⦃G, L0⦄ ⊢ ➡*[h, g] L2 →
-                            ∃∃L. L ⩬[l, ∞] L2 & ⦃G, L1⦄ ⊢ ➡*[h, g] L &
+lemma lreq_lpxs_trans_lleq: ∀h,o,G,L1,L0,l. L1 ⩬[l, ∞] L0 →
+                            ∀L2. ⦃G, L0⦄ ⊢ ➡*[h, o] L2 →
+                            ∃∃L. L ⩬[l, ∞] L2 & ⦃G, L1⦄ ⊢ ➡*[h, o] L &
                                  (∀T. L0 ≡[T, l] L2 ↔ L1 ≡[T, l] L).
 /2 width=1 by lreq_lpxs_trans_lleq_aux/ qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/lpxs_lpxs.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/lpxs_lpxs.ma
index 1d287d368..2e0c14570 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/lpxs_lpxs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/lpxs_lpxs.ma
@@ -18,5 +18,5 @@ include "basic_2/computation/lpxs.ma".
 
 (* Main properties **********************************************************)
 
-theorem lpxs_trans: ∀h,g,G. Transitive … (lpxs h g G).
+theorem lpxs_trans: ∀h,o,G. Transitive … (lpxs h o G).
 /2 width=3 by trans_TC/ qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/lsubc_drop.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/lsubc_drop.ma
index 64b236cb0..678604fdd 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/lsubc_drop.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/lsubc_drop.ma
@@ -21,19 +21,19 @@ include "basic_2/computation/lsubc.ma".
 
 (* Basic_1: was: csubc_drop_conf_O *)
 (* Note: the constant 0 can not be generalized *)
-lemma lsubc_drop_O1_trans: ∀RP,G,L1,L2. G ⊢ L1 ⫃[RP] L2 → ∀K2,s,m. ⬇[s, 0, m] L2 ≡ K2 →
-                           ∃∃K1. ⬇[s, 0, m] L1 ≡ K1 & G ⊢ K1 ⫃[RP] K2.
+lemma lsubc_drop_O1_trans: ∀RP,G,L1,L2. G ⊢ L1 ⫃[RP] L2 → ∀K2,c,k. ⬇[c, 0, k] L2 ≡ K2 →
+                           ∃∃K1. ⬇[c, 0, k] L1 ≡ K1 & G ⊢ K1 ⫃[RP] K2.
 #RP #G #L1 #L2 #H elim H -L1 -L2
-[ #X #s #m #H elim (drop_inv_atom1 … H) -H /4 width=3 by drop_atom, ex2_intro/
-| #I #L1 #L2 #V #_ #IHL12 #X #s #m #H
+[ #X #c #k #H elim (drop_inv_atom1 … H) -H /4 width=3 by drop_atom, ex2_intro/
+| #I #L1 #L2 #V #_ #IHL12 #X #c #k #H
   elim (drop_inv_O1_pair1 … H) -H * #Hm #H destruct
-  [ elim (IHL12 L2 s 0) -IHL12 // #X #H <(drop_inv_O2 … H) -H
+  [ elim (IHL12 L2 c 0) -IHL12 // #X #H <(drop_inv_O2 … H) -H
     /3 width=3 by lsubc_pair, drop_pair, ex2_intro/
   | elim (IHL12 … H) -L2 /3 width=3 by drop_drop_lt, ex2_intro/
   ]
-| #L1 #L2 #V #W #A #HV #H1W #H2W #_ #IHL12 #X #s #m #H
+| #L1 #L2 #V #W #A #HV #H1W #H2W #_ #IHL12 #X #c #k #H
   elim (drop_inv_O1_pair1 … H) -H * #Hm #H destruct
-  [ elim (IHL12 L2 s 0) -IHL12 // #X #H <(drop_inv_O2 … H) -H
+  [ elim (IHL12 L2 c 0) -IHL12 // #X #H <(drop_inv_O2 … H) -H
     /3 width=8 by lsubc_beta, drop_pair, ex2_intro/
   | elim (IHL12 … H) -L2 /3 width=3 by drop_drop_lt, ex2_intro/
   ]
@@ -42,10 +42,10 @@ qed-.
 
 (* Basic_1: was: csubc_drop_conf_rev *)
 lemma drop_lsubc_trans: ∀RR,RS,RP. gcp RR RS RP →
-                        ∀G,L1,K1,l,m. ⬇[Ⓕ, l, m] L1 ≡ K1 → ∀K2. G ⊢ K1 ⫃[RP] K2 →
-                        ∃∃L2. G ⊢ L1 ⫃[RP] L2 & ⬇[Ⓕ, l, m] L2 ≡ K2.
-#RR #RS #RP #Hgcp #G #L1 #K1 #l #m #H elim H -L1 -K1 -l -m
-[ #l #m #Hm #X #H elim (lsubc_inv_atom1 … H) -H
+                        ∀G,L1,K1,l,k. ⬇[Ⓕ, l, k] L1 ≡ K1 → ∀K2. G ⊢ K1 ⫃[RP] K2 →
+                        ∃∃L2. G ⊢ L1 ⫃[RP] L2 & ⬇[Ⓕ, l, k] L2 ≡ K2.
+#RR #RS #RP #Hgcp #G #L1 #K1 #l #k #H elim H -L1 -K1 -l -k
+[ #l #k #Hm #X #H elim (lsubc_inv_atom1 … H) -H
   >Hm /2 width=3 by ex2_intro/
 | #L1 #I #V1 #X #H
   elim (lsubc_inv_pair1 … H) -H *
@@ -53,9 +53,9 @@ lemma drop_lsubc_trans: ∀RR,RS,RP. gcp RR RS RP →
   | #K1 #V #W1 #A #HV1 #H1W1 #H2W1 #HLK1 #H1 #H2 #H3 destruct
     /3 width=4 by lsubc_beta, drop_pair, ex2_intro/
   ]
-| #I #L1 #K1 #V1 #m #_ #IHLK1 #K2 #HK12
+| #I #L1 #K1 #V1 #k #_ #IHLK1 #K2 #HK12
   elim (IHLK1 … HK12) -K1 /3 width=5 by lsubc_pair, drop_drop, ex2_intro/
-| #I #L1 #K1 #V1 #V2 #l #m #HLK1 #HV21 #IHLK1 #X #H
+| #I #L1 #K1 #V1 #V2 #l #k #HLK1 #HV21 #IHLK1 #X #H
   elim (lsubc_inv_pair1 … H) -H *
   [ #K2 #HK12 #H destruct
     elim (IHLK1 … HK12) -K1 /3 width=5 by lsubc_pair, drop_skip, ex2_intro/
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/lsubc_drops.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/lsubc_drops.ma
index df89d984f..8fa9adfef 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/lsubc_drops.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/lsubc_drops.ma
@@ -24,7 +24,7 @@ lemma drops_lsubc_trans: ∀RR,RS,RP. gcp RR RS RP →
                          ∃∃L2. G ⊢ L1 ⫃[RP] L2 & ⬇*[Ⓕ, cs] L2 ≡ K2.
 #RR #RS #RP #Hgcp #G #L1 #K1 #cs #H elim H -L1 -K1 -cs
 [ /2 width=3 by drops_nil, ex2_intro/
-| #L1 #L #K1 #cs #l #m #_ #HLK1 #IHL #K2 #HK12
+| #L1 #L #K1 #cs #l #k #_ #HLK1 #IHL #K2 #HK12
   elim (drop_lsubc_trans … Hgcp … HLK1 … HK12) -Hgcp -K1 #K #HLK #HK2
   elim (IHL … HLK) -IHL -HLK /3 width=5 by drops_cons, ex2_intro/
 ]
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/lsx.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/lsx.ma
index 1cc24d432..7bb8aec4a 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/lsx.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/lsx.ma
@@ -19,91 +19,91 @@ include "basic_2/reduction/lpx.ma".
 (* SN EXTENDED STRONGLY NORMALIZING LOCAL ENVIRONMENTS **********************)
 
 definition lsx: ∀h. sd h → relation4 ynat term genv lenv ≝
-                λh,g,l,T,G. SN … (lpx h g G) (lleq l T).
+                λh,o,l,T,G. SN … (lpx h o G) (lleq l T).
 
 interpretation
    "extended strong normalization (local environment)"
-   'SN h g l T G L = (lsx h g T l G L).
+   'SN h o l T G L = (lsx h o T l G L).
 
 (* Basic eliminators ********************************************************)
 
-lemma lsx_ind: ∀h,g,G,T,l. ∀R:predicate lenv.
-               (∀L1. G ⊢ ⬊*[h, g, T, l] L1 →
-                     (∀L2. ⦃G, L1⦄ ⊢ ➡[h, g] L2 → (L1 ≡[T, l] L2 → ⊥) → R L2) →
+lemma lsx_ind: ∀h,o,G,T,l. ∀R:predicate lenv.
+               (∀L1. G ⊢ ⬊*[h, o, T, l] L1 →
+                     (∀L2. ⦃G, L1⦄ ⊢ ➡[h, o] L2 → (L1 ≡[T, l] L2 → ⊥) → R L2) →
                      R L1
                ) →
-               ∀L. G ⊢ ⬊*[h, g, T, l] L → R L.
-#h #g #G #T #l #R #H0 #L1 #H elim H -L1
+               ∀L. G ⊢ ⬊*[h, o, T, l] L → R L.
+#h #o #G #T #l #R #H0 #L1 #H elim H -L1
 /5 width=1 by lleq_sym, SN_intro/
 qed-.
 
 (* Basic properties *********************************************************)
 
-lemma lsx_intro: ∀h,g,G,L1,T,l.
-                 (∀L2. ⦃G, L1⦄ ⊢ ➡[h, g] L2 → (L1 ≡[T, l] L2 → ⊥) → G ⊢ ⬊*[h, g, T, l] L2) →
-                 G ⊢ ⬊*[h, g, T, l] L1.
+lemma lsx_intro: ∀h,o,G,L1,T,l.
+                 (∀L2. ⦃G, L1⦄ ⊢ ➡[h, o] L2 → (L1 ≡[T, l] L2 → ⊥) → G ⊢ ⬊*[h, o, T, l] L2) →
+                 G ⊢ ⬊*[h, o, T, l] L1.
 /5 width=1 by lleq_sym, SN_intro/ qed.
 
-lemma lsx_atom: ∀h,g,G,T,l. G ⊢ ⬊*[h, g, T, l] ⋆.
-#h #g #G #T #l @lsx_intro
+lemma lsx_atom: ∀h,o,G,T,l. G ⊢ ⬊*[h, o, T, l] ⋆.
+#h #o #G #T #l @lsx_intro
 #X #H #HT lapply (lpx_inv_atom1 … H) -H
 #H destruct elim HT -HT //
 qed.
 
-lemma lsx_sort: ∀h,g,G,L,l,k. G ⊢ ⬊*[h, g, ⋆k, l] L.
-#h #g #G #L1 #l #k @lsx_intro
+lemma lsx_sort: ∀h,o,G,L,l,s. G ⊢ ⬊*[h, o, ⋆s, l] L.
+#h #o #G #L1 #l #s @lsx_intro
 #L2 #HL12 #H elim H -H
 /3 width=4 by lpx_fwd_length, lleq_sort/
 qed.
 
-lemma lsx_gref: ∀h,g,G,L,l,p. G ⊢ ⬊*[h, g, §p, l] L.
-#h #g #G #L1 #l #p @lsx_intro
+lemma lsx_gref: ∀h,o,G,L,l,p. G ⊢ ⬊*[h, o, §p, l] L.
+#h #o #G #L1 #l #p @lsx_intro
 #L2 #HL12 #H elim H -H
 /3 width=4 by lpx_fwd_length, lleq_gref/
 qed.
 
-lemma lsx_ge_up: ∀h,g,G,L,T,U,lt,l,m. lt ≤ yinj l + yinj m →
-                 ⬆[l, m] T ≡ U → G ⊢ ⬊*[h, g, U, lt] L → G ⊢ ⬊*[h, g, U, l] L.
-#h #g #G #L #T #U #lt #l #m #Hltlm #HTU #H @(lsx_ind … H) -L
+lemma lsx_ge_up: ∀h,o,G,L,T,U,lt,l,k. lt ≤ yinj l + yinj k →
+                 ⬆[l, k] T ≡ U → G ⊢ ⬊*[h, o, U, lt] L → G ⊢ ⬊*[h, o, U, l] L.
+#h #o #G #L #T #U #lt #l #k #Hltlm #HTU #H @(lsx_ind … H) -L
 /5 width=7 by lsx_intro, lleq_ge_up/
 qed-.
 
-lemma lsx_ge: ∀h,g,G,L,T,l1,l2. l1 ≤ l2 →
-              G ⊢ ⬊*[h, g, T, l1] L → G ⊢ ⬊*[h, g, T, l2] L.
-#h #g #G #L #T #l1 #l2 #Hl12 #H @(lsx_ind … H) -L
+lemma lsx_ge: ∀h,o,G,L,T,l1,l2. l1 ≤ l2 →
+              G ⊢ ⬊*[h, o, T, l1] L → G ⊢ ⬊*[h, o, T, l2] L.
+#h #o #G #L #T #l1 #l2 #Hl12 #H @(lsx_ind … H) -L
 /5 width=7 by lsx_intro, lleq_ge/
 qed-.
 
 (* Basic forward lemmas *****************************************************)
 
-lemma lsx_fwd_bind_sn: ∀h,g,a,I,G,L,V,T,l. G ⊢ ⬊*[h, g, ⓑ{a,I}V.T, l] L →
-                       G ⊢ ⬊*[h, g, V, l] L.
-#h #g #a #I #G #L #V #T #l #H @(lsx_ind … H) -L
+lemma lsx_fwd_bind_sn: ∀h,o,a,I,G,L,V,T,l. G ⊢ ⬊*[h, o, ⓑ{a,I}V.T, l] L →
+                       G ⊢ ⬊*[h, o, V, l] L.
+#h #o #a #I #G #L #V #T #l #H @(lsx_ind … H) -L
 #L1 #_ #IHL1 @lsx_intro
 #L2 #HL12 #HV @IHL1 /3 width=4 by lleq_fwd_bind_sn/
 qed-.
 
-lemma lsx_fwd_flat_sn: ∀h,g,I,G,L,V,T,l. G ⊢ ⬊*[h, g, ⓕ{I}V.T, l] L →
-                       G ⊢ ⬊*[h, g, V, l] L.
-#h #g #I #G #L #V #T #l #H @(lsx_ind … H) -L
+lemma lsx_fwd_flat_sn: ∀h,o,I,G,L,V,T,l. G ⊢ ⬊*[h, o, ⓕ{I}V.T, l] L →
+                       G ⊢ ⬊*[h, o, V, l] L.
+#h #o #I #G #L #V #T #l #H @(lsx_ind … H) -L
 #L1 #_ #IHL1 @lsx_intro
 #L2 #HL12 #HV @IHL1 /3 width=3 by lleq_fwd_flat_sn/
 qed-.
 
-lemma lsx_fwd_flat_dx: ∀h,g,I,G,L,V,T,l. G ⊢ ⬊*[h, g, ⓕ{I}V.T, l] L →
-                       G ⊢ ⬊*[h, g, T, l] L.
-#h #g #I #G #L #V #T #l #H @(lsx_ind … H) -L
+lemma lsx_fwd_flat_dx: ∀h,o,I,G,L,V,T,l. G ⊢ ⬊*[h, o, ⓕ{I}V.T, l] L →
+                       G ⊢ ⬊*[h, o, T, l] L.
+#h #o #I #G #L #V #T #l #H @(lsx_ind … H) -L
 #L1 #_ #IHL1 @lsx_intro
 #L2 #HL12 #HV @IHL1 /3 width=3 by lleq_fwd_flat_dx/
 qed-.
 
-lemma lsx_fwd_pair_sn: ∀h,g,I,G,L,V,T,l. G ⊢ ⬊*[h, g, ②{I}V.T, l] L →
-                       G ⊢ ⬊*[h, g, V, l] L.
-#h #g * /2 width=4 by lsx_fwd_bind_sn, lsx_fwd_flat_sn/
+lemma lsx_fwd_pair_sn: ∀h,o,I,G,L,V,T,l. G ⊢ ⬊*[h, o, ②{I}V.T, l] L →
+                       G ⊢ ⬊*[h, o, V, l] L.
+#h #o * /2 width=4 by lsx_fwd_bind_sn, lsx_fwd_flat_sn/
 qed-.
 
 (* Basic inversion lemmas ***************************************************)
 
-lemma lsx_inv_flat: ∀h,g,I,G,L,V,T,l. G ⊢ ⬊*[h, g, ⓕ{I}V.T, l] L →
-                    G ⊢ ⬊*[h, g, V, l] L ∧ G ⊢ ⬊*[h, g, T, l] L.
+lemma lsx_inv_flat: ∀h,o,I,G,L,V,T,l. G ⊢ ⬊*[h, o, ⓕ{I}V.T, l] L →
+                    G ⊢ ⬊*[h, o, V, l] L ∧ G ⊢ ⬊*[h, o, T, l] L.
 /3 width=3 by lsx_fwd_flat_sn, lsx_fwd_flat_dx, conj/ qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/lsx_alt.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/lsx_alt.ma
index 05637c0e7..07f3d94b2 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/lsx_alt.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/lsx_alt.ma
@@ -20,54 +20,54 @@ include "basic_2/computation/lsx.ma".
 
 (* alternative definition of lsx *)
 definition lsxa: ∀h. sd h → relation4 ynat term genv lenv ≝
-                 λh,g,l,T,G. SN … (lpxs h g G) (lleq l T).
+                 λh,o,l,T,G. SN … (lpxs h o G) (lleq l T).
 
 interpretation
    "extended strong normalization (local environment) alternative"
-   'SNAlt h g l T G L = (lsxa h g T l G L).
+   'SNAlt h o l T G L = (lsxa h o T l G L).
 
 (* Basic eliminators ********************************************************)
 
-lemma lsxa_ind: ∀h,g,G,T,l. ∀R:predicate lenv.
-                (∀L1. G ⊢ ⬊⬊*[h, g, T, l] L1 →
-                      (∀L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 → (L1 ≡[T, l] L2 → ⊥) → R L2) →
+lemma lsxa_ind: ∀h,o,G,T,l. ∀R:predicate lenv.
+                (∀L1. G ⊢ ⬊⬊*[h, o, T, l] L1 →
+                      (∀L2. ⦃G, L1⦄ ⊢ ➡*[h, o] L2 → (L1 ≡[T, l] L2 → ⊥) → R L2) →
                       R L1
                 ) →
-                ∀L. G ⊢ ⬊⬊*[h, g, T, l] L → R L.
-#h #g #G #T #l #R #H0 #L1 #H elim H -L1
+                ∀L. G ⊢ ⬊⬊*[h, o, T, l] L → R L.
+#h #o #G #T #l #R #H0 #L1 #H elim H -L1
 /5 width=1 by lleq_sym, SN_intro/
 qed-.
 
 (* Basic properties *********************************************************)
 
-lemma lsxa_intro: ∀h,g,G,L1,T,l.
-                  (∀L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 → (L1 ≡[T, l] L2 → ⊥) → G ⊢ ⬊⬊*[h, g, T, l] L2) →
-                  G ⊢ ⬊⬊*[h, g, T, l] L1.
+lemma lsxa_intro: ∀h,o,G,L1,T,l.
+                  (∀L2. ⦃G, L1⦄ ⊢ ➡*[h, o] L2 → (L1 ≡[T, l] L2 → ⊥) → G ⊢ ⬊⬊*[h, o, T, l] L2) →
+                  G ⊢ ⬊⬊*[h, o, T, l] L1.
 /5 width=1 by lleq_sym, SN_intro/ qed.
 
-fact lsxa_intro_aux: ∀h,g,G,L1,T,l.
-                     (∀L,L2. ⦃G, L⦄ ⊢ ➡*[h, g] L2 → L1 ≡[T, l] L → (L1 ≡[T, l] L2 → ⊥) → G ⊢ ⬊⬊*[h, g, T, l] L2) →
-                     G ⊢ ⬊⬊*[h, g, T, l] L1.
+fact lsxa_intro_aux: ∀h,o,G,L1,T,l.
+                     (∀L,L2. ⦃G, L⦄ ⊢ ➡*[h, o] L2 → L1 ≡[T, l] L → (L1 ≡[T, l] L2 → ⊥) → G ⊢ ⬊⬊*[h, o, T, l] L2) →
+                     G ⊢ ⬊⬊*[h, o, T, l] L1.
 /4 width=3 by lsxa_intro/ qed-.
 
-lemma lsxa_lleq_trans: ∀h,g,T,G,L1,l. G ⊢ ⬊⬊*[h, g, T, l] L1 →
-                       ∀L2. L1 ≡[T, l] L2 → G ⊢ ⬊⬊*[h, g, T, l] L2.
-#h #g #T #G #L1 #l #H @(lsxa_ind … H) -L1
+lemma lsxa_lleq_trans: ∀h,o,T,G,L1,l. G ⊢ ⬊⬊*[h, o, T, l] L1 →
+                       ∀L2. L1 ≡[T, l] L2 → G ⊢ ⬊⬊*[h, o, T, l] L2.
+#h #o #T #G #L1 #l #H @(lsxa_ind … H) -L1
 #L1 #_ #IHL1 #L2 #HL12 @lsxa_intro
 #K2 #HLK2 #HnLK2 elim (lleq_lpxs_trans … HLK2 … HL12) -HLK2
 /5 width=4 by lleq_canc_sn, lleq_trans/
 qed-.
 
-lemma lsxa_lpxs_trans: ∀h,g,T,G,L1,l. G ⊢ ⬊⬊*[h, g, T, l] L1 →
-                       ∀L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 → G ⊢ ⬊⬊*[h, g, T, l] L2.
-#h #g #T #G #L1 #l #H @(lsxa_ind … H) -L1 #L1 #HL1 #IHL1 #L2 #HL12
+lemma lsxa_lpxs_trans: ∀h,o,T,G,L1,l. G ⊢ ⬊⬊*[h, o, T, l] L1 →
+                       ∀L2. ⦃G, L1⦄ ⊢ ➡*[h, o] L2 → G ⊢ ⬊⬊*[h, o, T, l] L2.
+#h #o #T #G #L1 #l #H @(lsxa_ind … H) -L1 #L1 #HL1 #IHL1 #L2 #HL12
 elim (lleq_dec T L1 L2 l) /3 width=4 by lsxa_lleq_trans/
 qed-.
 
-lemma lsxa_intro_lpx: ∀h,g,G,L1,T,l.
-                      (∀L2. ⦃G, L1⦄ ⊢ ➡[h, g] L2 → (L1 ≡[T, l] L2 → ⊥) → G ⊢ ⬊⬊*[h, g, T, l] L2) →
-                      G ⊢ ⬊⬊*[h, g, T, l] L1.
-#h #g #G #L1 #T #l #IH @lsxa_intro_aux
+lemma lsxa_intro_lpx: ∀h,o,G,L1,T,l.
+                      (∀L2. ⦃G, L1⦄ ⊢ ➡[h, o] L2 → (L1 ≡[T, l] L2 → ⊥) → G ⊢ ⬊⬊*[h, o, T, l] L2) →
+                      G ⊢ ⬊⬊*[h, o, T, l] L1.
+#h #o #G #L1 #T #l #IH @lsxa_intro_aux
 #L #L2 #H @(lpxs_ind_dx … H) -L
 [ #H destruct #H elim H //
 | #L0 #L elim (lleq_dec T L1 L l) /3 width=1 by/
@@ -79,37 +79,37 @@ qed-.
 
 (* Main properties **********************************************************)
 
-theorem lsx_lsxa: ∀h,g,G,L,T,l. G ⊢ ⬊*[h, g, T, l] L → G ⊢ ⬊⬊*[h, g, T, l] L.
-#h #g #G #L #T #l #H @(lsx_ind … H) -L
+theorem lsx_lsxa: ∀h,o,G,L,T,l. G ⊢ ⬊*[h, o, T, l] L → G ⊢ ⬊⬊*[h, o, T, l] L.
+#h #o #G #L #T #l #H @(lsx_ind … H) -L
 /4 width=1 by lsxa_intro_lpx/
 qed.
 
 (* Main inversion lemmas ****************************************************)
 
-theorem lsxa_inv_lsx: ∀h,g,G,L,T,l. G ⊢ ⬊⬊*[h, g, T, l] L → G ⊢ ⬊*[h, g, T, l] L.
-#h #g #G #L #T #l #H @(lsxa_ind … H) -L
+theorem lsxa_inv_lsx: ∀h,o,G,L,T,l. G ⊢ ⬊⬊*[h, o, T, l] L → G ⊢ ⬊*[h, o, T, l] L.
+#h #o #G #L #T #l #H @(lsxa_ind … H) -L
 /4 width=1 by lsx_intro, lpx_lpxs/
 qed-.
 
 (* Advanced properties ******************************************************)
 
-lemma lsx_intro_alt: ∀h,g,G,L1,T,l.
-                     (∀L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 → (L1 ≡[T, l] L2 → ⊥) → G ⊢ ⬊*[h, g, T, l] L2) →
-                     G ⊢ ⬊*[h, g, T, l] L1.
+lemma lsx_intro_alt: ∀h,o,G,L1,T,l.
+                     (∀L2. ⦃G, L1⦄ ⊢ ➡*[h, o] L2 → (L1 ≡[T, l] L2 → ⊥) → G ⊢ ⬊*[h, o, T, l] L2) →
+                     G ⊢ ⬊*[h, o, T, l] L1.
 /6 width=1 by lsxa_inv_lsx, lsx_lsxa, lsxa_intro/ qed.
 
-lemma lsx_lpxs_trans: ∀h,g,G,L1,T,l. G ⊢ ⬊*[h, g, T, l] L1 →
-                      ∀L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 → G ⊢ ⬊*[h, g, T, l] L2.
+lemma lsx_lpxs_trans: ∀h,o,G,L1,T,l. G ⊢ ⬊*[h, o, T, l] L1 →
+                      ∀L2. ⦃G, L1⦄ ⊢ ➡*[h, o] L2 → G ⊢ ⬊*[h, o, T, l] L2.
 /4 width=3 by lsxa_inv_lsx, lsx_lsxa, lsxa_lpxs_trans/ qed-.
 
 (* Advanced eliminators *****************************************************)
 
-lemma lsx_ind_alt: ∀h,g,G,T,l. ∀R:predicate lenv.
-                   (∀L1. G ⊢ ⬊*[h, g, T, l] L1 →
-                         (∀L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 → (L1 ≡[T, l] L2 → ⊥) → R L2) →
+lemma lsx_ind_alt: ∀h,o,G,T,l. ∀R:predicate lenv.
+                   (∀L1. G ⊢ ⬊*[h, o, T, l] L1 →
+                         (∀L2. ⦃G, L1⦄ ⊢ ➡*[h, o] L2 → (L1 ≡[T, l] L2 → ⊥) → R L2) →
                          R L1
                    ) →
-                   ∀L. G ⊢ ⬊*[h, g, T, l] L → R L.
-#h #g #G #T #l #R #IH #L #H @(lsxa_ind h g G T l … L)
+                   ∀L. G ⊢ ⬊*[h, o, T, l] L → R L.
+#h #o #G #T #l #R #IH #L #H @(lsxa_ind h o G T l … L)
 /4 width=1 by lsxa_inv_lsx, lsx_lsxa/
 qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/lsx_csx.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/lsx_csx.ma
index 67f754b7b..fa00a907d 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/lsx_csx.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/lsx_csx.ma
@@ -19,10 +19,10 @@ include "basic_2/computation/lcosx_cpx.ma".
 
 (* Advanced properties ******************************************************)
 
-lemma lsx_lref_be_lpxs: ∀h,g,I,G,K1,V,i,l. l ≤ yinj i → ⦃G, K1⦄ ⊢ ⬊*[h, g] V →
-                        ∀K2. G ⊢ ⬊*[h, g, V, 0] K2 → ⦃G, K1⦄ ⊢ ➡*[h, g] K2 →
-                        ∀L2. ⬇[i] L2 ≡ K2.ⓑ{I}V → G ⊢ ⬊*[h, g, #i, l] L2.
-#h #g #I #G #K1 #V #i #l #Hli #H @(csx_ind_alt … H) -V
+lemma lsx_lref_be_lpxs: ∀h,o,I,G,K1,V,i,l. l ≤ yinj i → ⦃G, K1⦄ ⊢ ⬊*[h, o] V →
+                        ∀K2. G ⊢ ⬊*[h, o, V, 0] K2 → ⦃G, K1⦄ ⊢ ➡*[h, o] K2 →
+                        ∀L2. ⬇[i] L2 ≡ K2.ⓑ{I}V → G ⊢ ⬊*[h, o, #i, l] L2.
+#h #o #I #G #K1 #V #i #l #Hli #H @(csx_ind_alt … H) -V
 #V0 #_ #IHV0 #K2 #H @(lsx_ind … H) -K2
 #K0 #HK0 #IHK0 #HK10 #L0 #HLK0 @lsx_intro
 #L2 #HL02 #HnL02 elim (lpx_drop_conf … HLK0 … HL02) -HL02
@@ -35,15 +35,15 @@ elim (eq_term_dec V0 V2) #HnV02 destruct [ -IHV0 -HV02 -HK0 | -IHK0 -HnL02 -HLK0
 ]
 qed.
 
-lemma lsx_lref_be: ∀h,g,I,G,K,V,i,l. l ≤ yinj i → ⦃G, K⦄ ⊢ ⬊*[h, g] V →
-                   G ⊢ ⬊*[h, g, V, 0] K →
-                   ∀L. ⬇[i] L ≡ K.ⓑ{I}V → G ⊢ ⬊*[h, g, #i, l] L.
+lemma lsx_lref_be: ∀h,o,I,G,K,V,i,l. l ≤ yinj i → ⦃G, K⦄ ⊢ ⬊*[h, o] V →
+                   G ⊢ ⬊*[h, o, V, 0] K →
+                   ∀L. ⬇[i] L ≡ K.ⓑ{I}V → G ⊢ ⬊*[h, o, #i, l] L.
 /2 width=8 by lsx_lref_be_lpxs/ qed.
 
 (* Main properties **********************************************************)
 
-theorem csx_lsx: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ ⬊*[h, g] T → ∀l. G ⊢ ⬊*[h, g, T, l] L.
-#h #g #G #L #T @(fqup_wf_ind_eq … G L T) -G -L -T
+theorem csx_lsx: ∀h,o,G,L,T. ⦃G, L⦄ ⊢ ⬊*[h, o] T → ∀l. G ⊢ ⬊*[h, o, T, l] L.
+#h #o #G #L #T @(fqup_wf_ind_eq … G L T) -G -L -T
 #Z #Y #X #IH #G #L * * //
 [ #i #HG #HL #HT #H #l destruct
   elim (lt_or_ge i (|L|)) /2 width=1 by lsx_lref_free/
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/lsx_drop.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/lsx_drop.ma
index f1a4c7b64..c70d6fbc5 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/lsx_drop.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/lsx_drop.ma
@@ -20,23 +20,23 @@ include "basic_2/computation/lsx.ma".
 
 (* Advanced properties ******************************************************)
 
-lemma lsx_lref_free: ∀h,g,G,L,l,i. |L| ≤ i → G ⊢ ⬊*[h, g, #i, l] L.
-#h #g #G #L1 #l #i #HL1 @lsx_intro
+lemma lsx_lref_free: ∀h,o,G,L,l,i. |L| ≤ i → G ⊢ ⬊*[h, o, #i, l] L.
+#h #o #G #L1 #l #i #HL1 @lsx_intro
 #L2 #HL12 #H elim H -H
 /4 width=6 by lpx_fwd_length, lleq_free, le_repl_sn_conf_aux/
 qed.
 
-lemma lsx_lref_skip: ∀h,g,G,L,l,i. yinj i < l → G ⊢ ⬊*[h, g, #i, l] L.
-#h #g #G #L1 #l #i #HL1 @lsx_intro
+lemma lsx_lref_skip: ∀h,o,G,L,l,i. yinj i < l → G ⊢ ⬊*[h, o, #i, l] L.
+#h #o #G #L1 #l #i #HL1 @lsx_intro
 #L2 #HL12 #H elim H -H
 /3 width=4 by lpx_fwd_length, lleq_skip/
 qed.
 
 (* Advanced forward lemmas **************************************************)
 
-lemma lsx_fwd_lref_be: ∀h,g,I,G,L,l,i. l ≤ yinj i → G ⊢ ⬊*[h, g, #i, l] L →
-                       ∀K,V. ⬇[i] L ≡ K.ⓑ{I}V → G ⊢ ⬊*[h, g, V, 0] K.
-#h #g #I #G #L #l #i #Hli #H @(lsx_ind … H) -L
+lemma lsx_fwd_lref_be: ∀h,o,I,G,L,l,i. l ≤ yinj i → G ⊢ ⬊*[h, o, #i, l] L →
+                       ∀K,V. ⬇[i] L ≡ K.ⓑ{I}V → G ⊢ ⬊*[h, o, V, 0] K.
+#h #o #I #G #L #l #i #Hli #H @(lsx_ind … H) -L
 #L1 #_ #IHL1 #K1 #V #HLK1 @lsx_intro
 #K2 #HK12 #HnK12 lapply (drop_fwd_drop2 … HLK1)
 #H2LK1 elim (drop_lpx_trans … H2LK1 … HK12) -H2LK1 -HK12
@@ -48,19 +48,19 @@ qed-.
 
 (* Properties on relocation *************************************************)
 
-lemma lsx_lift_le: ∀h,g,G,K,T,U,lt,l,m. lt ≤ yinj l →
-                   ⬆[l, m] T ≡ U → G ⊢ ⬊*[h, g, T, lt] K →
-                   ∀L. ⬇[Ⓕ, l, m] L ≡ K → G ⊢ ⬊*[h, g, U, lt] L.
-#h #g #G #K #T #U #lt #l #m #Hltl #HTU #H @(lsx_ind … H) -K
+lemma lsx_lift_le: ∀h,o,G,K,T,U,lt,l,k. lt ≤ yinj l →
+                   ⬆[l, k] T ≡ U → G ⊢ ⬊*[h, o, T, lt] K →
+                   ∀L. ⬇[Ⓕ, l, k] L ≡ K → G ⊢ ⬊*[h, o, U, lt] L.
+#h #o #G #K #T #U #lt #l #k #Hltl #HTU #H @(lsx_ind … H) -K
 #K1 #_ #IHK1 #L1 #HLK1 @lsx_intro
 #L2 #HL12 #HnU elim (lpx_drop_conf … HLK1 … HL12) -HL12
 /4 width=10 by lleq_lift_le/
 qed-.
 
-lemma lsx_lift_ge: ∀h,g,G,K,T,U,lt,l,m. yinj l ≤ lt →
-                   ⬆[l, m] T ≡ U → G ⊢ ⬊*[h, g, T, lt] K →
-                   ∀L. ⬇[Ⓕ, l, m] L ≡ K → G ⊢ ⬊*[h, g, U, lt + m] L.
-#h #g #G #K #T #U #lt #l #m #Hllt #HTU #H @(lsx_ind … H) -K
+lemma lsx_lift_ge: ∀h,o,G,K,T,U,lt,l,k. yinj l ≤ lt →
+                   ⬆[l, k] T ≡ U → G ⊢ ⬊*[h, o, T, lt] K →
+                   ∀L. ⬇[Ⓕ, l, k] L ≡ K → G ⊢ ⬊*[h, o, U, lt + k] L.
+#h #o #G #K #T #U #lt #l #k #Hllt #HTU #H @(lsx_ind … H) -K
 #K1 #_ #IHK1 #L1 #HLK1 @lsx_intro
 #L2 #HL12 #HnU elim (lpx_drop_conf … HLK1 … HL12) -HL12
 /4 width=9 by lleq_lift_ge/
@@ -68,28 +68,28 @@ qed-.
 
 (* Inversion lemmas on relocation *******************************************)
 
-lemma lsx_inv_lift_le: ∀h,g,G,L,T,U,lt,l,m. lt ≤ yinj l →
-                       ⬆[l, m] T ≡ U → G ⊢ ⬊*[h, g, U, lt] L →
-                       ∀K. ⬇[Ⓕ, l, m] L ≡ K → G ⊢ ⬊*[h, g, T, lt] K.
-#h #g #G #L #T #U #lt #l #m #Hltl #HTU #H @(lsx_ind … H) -L
+lemma lsx_inv_lift_le: ∀h,o,G,L,T,U,lt,l,k. lt ≤ yinj l →
+                       ⬆[l, k] T ≡ U → G ⊢ ⬊*[h, o, U, lt] L →
+                       ∀K. ⬇[Ⓕ, l, k] L ≡ K → G ⊢ ⬊*[h, o, T, lt] K.
+#h #o #G #L #T #U #lt #l #k #Hltl #HTU #H @(lsx_ind … H) -L
 #L1 #_ #IHL1 #K1 #HLK1 @lsx_intro
 #K2 #HK12 #HnT elim (drop_lpx_trans … HLK1 … HK12) -HK12
 /4 width=10 by lleq_inv_lift_le/
 qed-.
 
-lemma lsx_inv_lift_be: ∀h,g,G,L,T,U,lt,l,m. yinj l ≤ lt → lt ≤ l + m →
-                       ⬆[l, m] T ≡ U → G ⊢ ⬊*[h, g, U, lt] L →
-                       ∀K. ⬇[Ⓕ, l, m] L ≡ K → G ⊢ ⬊*[h, g, T, l] K.
-#h #g #G #L #T #U #lt #l #m #Hllt #Hltlm #HTU #H @(lsx_ind … H) -L
+lemma lsx_inv_lift_be: ∀h,o,G,L,T,U,lt,l,k. yinj l ≤ lt → lt ≤ l + k →
+                       ⬆[l, k] T ≡ U → G ⊢ ⬊*[h, o, U, lt] L →
+                       ∀K. ⬇[Ⓕ, l, k] L ≡ K → G ⊢ ⬊*[h, o, T, l] K.
+#h #o #G #L #T #U #lt #l #k #Hllt #Hltlm #HTU #H @(lsx_ind … H) -L
 #L1 #_ #IHL1 #K1 #HLK1 @lsx_intro
 #K2 #HK12 #HnT elim (drop_lpx_trans … HLK1 … HK12) -HK12
 /4 width=11 by lleq_inv_lift_be/
 qed-.
 
-lemma lsx_inv_lift_ge: ∀h,g,G,L,T,U,lt,l,m. yinj l + yinj m ≤ lt →
-                       ⬆[l, m] T ≡ U → G ⊢ ⬊*[h, g, U, lt] L →
-                       ∀K. ⬇[Ⓕ, l, m] L ≡ K → G ⊢ ⬊*[h, g, T, lt-m] K.
-#h #g #G #L #T #U #lt #l #m #Hlmlt #HTU #H @(lsx_ind … H) -L
+lemma lsx_inv_lift_ge: ∀h,o,G,L,T,U,lt,l,k. yinj l + yinj k ≤ lt →
+                       ⬆[l, k] T ≡ U → G ⊢ ⬊*[h, o, U, lt] L →
+                       ∀K. ⬇[Ⓕ, l, k] L ≡ K → G ⊢ ⬊*[h, o, T, lt-k] K.
+#h #o #G #L #T #U #lt #l #k #Hlmlt #HTU #H @(lsx_ind … H) -L
 #L1 #_ #IHL1 #K1 #HLK1 @lsx_intro
 #K2 #HK12 #HnT elim (drop_lpx_trans … HLK1 … HK12) -HK12
 /4 width=9 by lleq_inv_lift_ge/
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/lsx_lpx.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/lsx_lpx.ma
index 05fc77f90..81dc6e73d 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/lsx_lpx.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/lsx_lpx.ma
@@ -20,23 +20,23 @@ include "basic_2/computation/lsx.ma".
 
 (* Advanced properties ******************************************************)
 
-lemma lsx_lleq_trans: ∀h,g,T,G,L1,l. G ⊢ ⬊*[h, g, T, l] L1 →
-                      ∀L2. L1 ≡[T, l] L2 → G ⊢ ⬊*[h, g, T, l] L2.
-#h #g #T #G #L1 #l #H @(lsx_ind … H) -L1
+lemma lsx_lleq_trans: ∀h,o,T,G,L1,l. G ⊢ ⬊*[h, o, T, l] L1 →
+                      ∀L2. L1 ≡[T, l] L2 → G ⊢ ⬊*[h, o, T, l] L2.
+#h #o #T #G #L1 #l #H @(lsx_ind … H) -L1
 #L1 #_ #IHL1 #L2 #HL12 @lsx_intro
 #K2 #HLK2 #HnLK2 elim (lleq_lpx_trans … HLK2 … HL12) -HLK2
 /5 width=4 by lleq_canc_sn, lleq_trans/
 qed-.
 
-lemma lsx_lpx_trans: ∀h,g,T,G,L1,l. G ⊢ ⬊*[h, g, T, l] L1 →
-                     ∀L2. ⦃G, L1⦄ ⊢ ➡[h, g] L2 → G ⊢ ⬊*[h, g, T, l] L2.
-#h #g #T #G #L1 #l #H @(lsx_ind … H) -L1 #L1 #HL1 #IHL1 #L2 #HL12
+lemma lsx_lpx_trans: ∀h,o,T,G,L1,l. G ⊢ ⬊*[h, o, T, l] L1 →
+                     ∀L2. ⦃G, L1⦄ ⊢ ➡[h, o] L2 → G ⊢ ⬊*[h, o, T, l] L2.
+#h #o #T #G #L1 #l #H @(lsx_ind … H) -L1 #L1 #HL1 #IHL1 #L2 #HL12
 elim (lleq_dec T L1 L2 l) /3 width=4 by lsx_lleq_trans/
 qed-.
 
-lemma lsx_lreq_conf: ∀h,g,G,L1,T,l. G ⊢ ⬊*[h, g, T, l] L1 →
-                    ∀L2. L1 ⩬[l, ∞] L2 → G ⊢ ⬊*[h, g, T, l] L2.
-#h #g #G #L1 #T #l #H @(lsx_ind … H) -L1
+lemma lsx_lreq_conf: ∀h,o,G,L1,T,l. G ⊢ ⬊*[h, o, T, l] L1 →
+                    ∀L2. L1 ⩬[l, ∞] L2 → G ⊢ ⬊*[h, o, T, l] L2.
+#h #o #G #L1 #T #l #H @(lsx_ind … H) -L1
 #L1 #_ #IHL1 #L2 #HL12 @lsx_intro
 #L3 #HL23 #HnL23 elim (lreq_lpx_trans_lleq … HL12 … HL23) -HL12 -HL23
 #L0 #HL03 #HL10 #H elim (H T) -H /4 width=4 by/
@@ -44,9 +44,9 @@ qed-.
 
 (* Advanced forward lemmas **************************************************)
 
-lemma lsx_fwd_bind_dx: ∀h,g,a,I,G,L,V,T,l. G ⊢ ⬊*[h, g, ⓑ{a,I}V.T, l] L →
-                       G ⊢ ⬊*[h, g, T, ⫯l] L.ⓑ{I}V.
-#h #g #a #I #G #L #V1 #T #l #H @(lsx_ind … H) -L
+lemma lsx_fwd_bind_dx: ∀h,o,a,I,G,L,V,T,l. G ⊢ ⬊*[h, o, ⓑ{a,I}V.T, l] L →
+                       G ⊢ ⬊*[h, o, T, ⫯l] L.ⓑ{I}V.
+#h #o #a #I #G #L #V1 #T #l #H @(lsx_ind … H) -L
 #L1 #_ #IHL1 @lsx_intro
 #Y #H #HT elim (lpx_inv_pair1 … H) -H
 #L2 #V2 #HL12 #_ #H destruct
@@ -58,6 +58,6 @@ qed-.
 
 (* Advanced inversion lemmas ************************************************)
 
-lemma lsx_inv_bind: ∀h,g,a,I,G,L,V,T,l. G ⊢ ⬊*[h, g, ⓑ{a, I}V.T, l] L →
-                    G ⊢ ⬊*[h, g, V, l] L ∧ G ⊢ ⬊*[h, g, T, ⫯l] L.ⓑ{I}V.
+lemma lsx_inv_bind: ∀h,o,a,I,G,L,V,T,l. G ⊢ ⬊*[h, o, ⓑ{a, I}V.T, l] L →
+                    G ⊢ ⬊*[h, o, V, l] L ∧ G ⊢ ⬊*[h, o, T, ⫯l] L.ⓑ{I}V.
 /3 width=4 by lsx_fwd_bind_sn, lsx_fwd_bind_dx, conj/ qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/lsx_lpxs.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/lsx_lpxs.ma
index d84202bbc..2e6da2ed4 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/lsx_lpxs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/lsx_lpxs.ma
@@ -19,11 +19,11 @@ include "basic_2/computation/lsx_alt.ma".
 
 (* Advanced properties ******************************************************)
 
-fact lsx_bind_lpxs_aux: ∀h,g,a,I,G,L1,V,l. G ⊢ ⬊*[h, g, V, l] L1 →
-                        ∀Y,T. G ⊢ ⬊*[h, g, T, ⫯l] Y →
-                        ∀L2. Y = L2.ⓑ{I}V → ⦃G, L1⦄ ⊢ ➡*[h, g] L2 →
-                        G ⊢ ⬊*[h, g, ⓑ{a,I}V.T, l] L2.
-#h #g #a #I #G #L1 #V #l #H @(lsx_ind_alt … H) -L1
+fact lsx_bind_lpxs_aux: ∀h,o,a,I,G,L1,V,l. G ⊢ ⬊*[h, o, V, l] L1 →
+                        ∀Y,T. G ⊢ ⬊*[h, o, T, ⫯l] Y →
+                        ∀L2. Y = L2.ⓑ{I}V → ⦃G, L1⦄ ⊢ ➡*[h, o] L2 →
+                        G ⊢ ⬊*[h, o, ⓑ{a,I}V.T, l] L2.
+#h #o #a #I #G #L1 #V #l #H @(lsx_ind_alt … H) -L1
 #L1 #HL1 #IHL1 #Y #T #H @(lsx_ind_alt … H) -Y
 #Y #HY #IHY #L2 #H #HL12 destruct @lsx_intro_alt
 #L0 #HL20 lapply (lpxs_trans … HL12 … HL20)
@@ -36,15 +36,15 @@ fact lsx_bind_lpxs_aux: ∀h,g,a,I,G,L1,V,l. G ⊢ ⬊*[h, g, V, l] L1 →
 ]
 qed-.
 
-lemma lsx_bind: ∀h,g,a,I,G,L,V,l. G ⊢ ⬊*[h, g, V, l] L →
-                ∀T. G ⊢ ⬊*[h, g, T, ⫯l] L.ⓑ{I}V →
-                G ⊢ ⬊*[h, g, ⓑ{a,I}V.T, l] L.
+lemma lsx_bind: ∀h,o,a,I,G,L,V,l. G ⊢ ⬊*[h, o, V, l] L →
+                ∀T. G ⊢ ⬊*[h, o, T, ⫯l] L.ⓑ{I}V →
+                G ⊢ ⬊*[h, o, ⓑ{a,I}V.T, l] L.
 /2 width=3 by lsx_bind_lpxs_aux/ qed.
 
-lemma lsx_flat_lpxs: ∀h,g,I,G,L1,V,l. G ⊢ ⬊*[h, g, V, l] L1 →
-                     ∀L2,T. G ⊢ ⬊*[h, g, T, l] L2 → ⦃G, L1⦄ ⊢ ➡*[h, g] L2 →
-                     G ⊢ ⬊*[h, g, ⓕ{I}V.T, l] L2.
-#h #g #I #G #L1 #V #l #H @(lsx_ind_alt … H) -L1
+lemma lsx_flat_lpxs: ∀h,o,I,G,L1,V,l. G ⊢ ⬊*[h, o, V, l] L1 →
+                     ∀L2,T. G ⊢ ⬊*[h, o, T, l] L2 → ⦃G, L1⦄ ⊢ ➡*[h, o] L2 →
+                     G ⊢ ⬊*[h, o, ⓕ{I}V.T, l] L2.
+#h #o #I #G #L1 #V #l #H @(lsx_ind_alt … H) -L1
 #L1 #HL1 #IHL1 #L2 #T #H @(lsx_ind_alt … H) -L2
 #L2 #HL2 #IHL2 #HL12 @lsx_intro_alt
 #L0 #HL20 lapply (lpxs_trans … HL12 … HL20)
@@ -57,6 +57,6 @@ lemma lsx_flat_lpxs: ∀h,g,I,G,L1,V,l. G ⊢ ⬊*[h, g, V, l] L1 →
 ]
 qed-.
 
-lemma lsx_flat: ∀h,g,I,G,L,V,l. G ⊢ ⬊*[h, g, V, l] L →
-                ∀T. G ⊢ ⬊*[h, g, T, l] L → G ⊢ ⬊*[h, g, ⓕ{I}V.T, l] L.
+lemma lsx_flat: ∀h,o,I,G,L,V,l. G ⊢ ⬊*[h, o, V, l] L →
+                ∀T. G ⊢ ⬊*[h, o, T, l] L → G ⊢ ⬊*[h, o, ⓕ{I}V.T, l] L.
 /2 width=3 by lsx_flat_lpxs/ qed.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/scpds.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/scpds.ma
index abc22cc37..1f189c15e 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/scpds.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/scpds.ma
@@ -19,30 +19,30 @@ include "basic_2/computation/cprs.ma".
 (* STRATIFIED DECOMPOSED PARALLEL COMPUTATION ON TERMS **********************)
 
 definition scpds: ∀h. sd h → nat → relation4 genv lenv term term ≝
-                  λh,g,d2,G,L,T1,T2.
-                  ∃∃T,d1. d2 ≤ d1 & ⦃G, L⦄ ⊢ T1 ▪[h, g] d1 & ⦃G, L⦄ ⊢ T1 •*[h, d2] T & ⦃G, L⦄ ⊢ T ➡* T2.
+                  λh,o,d2,G,L,T1,T2.
+                  ∃∃T,d1. d2 ≤ d1 & ⦃G, L⦄ ⊢ T1 ▪[h, o] d1 & ⦃G, L⦄ ⊢ T1 •*[h, d2] T & ⦃G, L⦄ ⊢ T ➡* T2.
 
 interpretation "stratified decomposed parallel computation (term)"
-   'DPRedStar h g d G L T1 T2 = (scpds h g d G L T1 T2).
+   'DPRedStar h o d G L T1 T2 = (scpds h o d G L T1 T2).
 
 (* Basic properties *********************************************************)
 
-lemma sta_cprs_scpds: ∀h,g,G,L,T1,T,T2,d. ⦃G, L⦄ ⊢ T1 ▪[h, g] d+1 → ⦃G, L⦄ ⊢ T1 •*[h, 1] T →
-                      ⦃G, L⦄ ⊢ T ➡* T2 → ⦃G, L⦄ ⊢ T1 •*➡*[h, g, 1] T2.
+lemma sta_cprs_scpds: ∀h,o,G,L,T1,T,T2,d. ⦃G, L⦄ ⊢ T1 ▪[h, o] d+1 → ⦃G, L⦄ ⊢ T1 •*[h, 1] T →
+                      ⦃G, L⦄ ⊢ T ➡* T2 → ⦃G, L⦄ ⊢ T1 •*➡*[h, o, 1] T2.
 /2 width=6 by ex4_2_intro/ qed.
 
-lemma lstas_scpds: ∀h,g,G,L,T1,T2,d1. ⦃G, L⦄ ⊢ T1 ▪[h, g] d1 →
-                   ∀d2. d2 ≤ d1 → ⦃G, L⦄ ⊢ T1 •*[h, d2] T2 → ⦃G, L⦄ ⊢ T1 •*➡*[h, g, d2] T2.
+lemma lstas_scpds: ∀h,o,G,L,T1,T2,d1. ⦃G, L⦄ ⊢ T1 ▪[h, o] d1 →
+                   ∀d2. d2 ≤ d1 → ⦃G, L⦄ ⊢ T1 •*[h, d2] T2 → ⦃G, L⦄ ⊢ T1 •*➡*[h, o, d2] T2.
 /2 width=6 by ex4_2_intro/ qed.
 
-lemma scpds_strap1: ∀h,g,G,L,T1,T,T2,d.
-                    ⦃G, L⦄ ⊢ T1 •*➡*[h, g, d] T → ⦃G, L⦄ ⊢ T ➡ T2 → ⦃G, L⦄ ⊢ T1 •*➡*[h, g, d] T2.
-#h #g #G #L #T1 #T #T2 #d * /3 width=8 by cprs_strap1, ex4_2_intro/
+lemma scpds_strap1: ∀h,o,G,L,T1,T,T2,d.
+                    ⦃G, L⦄ ⊢ T1 •*➡*[h, o, d] T → ⦃G, L⦄ ⊢ T ➡ T2 → ⦃G, L⦄ ⊢ T1 •*➡*[h, o, d] T2.
+#h #o #G #L #T1 #T #T2 #d * /3 width=8 by cprs_strap1, ex4_2_intro/
 qed.
 
 (* Basic forward lemmas *****************************************************)
 
-lemma scpds_fwd_cprs: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 •*➡*[h, g, 0] T2 →
+lemma scpds_fwd_cprs: ∀h,o,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 •*➡*[h, o, 0] T2 →
                       ⦃G, L⦄ ⊢ T1 ➡* T2.
-#h #g #G #L #T1 #T2 * /3 width=3 by cprs_strap2, lstas_cpr/
+#h #o #G #L #T1 #T2 * /3 width=3 by cprs_strap2, lstas_cpr/
 qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/scpds_aaa.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/scpds_aaa.ma
index f14723fdc..eef78441b 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/scpds_aaa.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/scpds_aaa.ma
@@ -20,6 +20,6 @@ include "basic_2/computation/scpds.ma".
 
 (* Properties on atomic arity assignment for terms **************************)
 
-lemma scpds_aaa_conf: ∀h,g,G,L,d. Conf3 … (aaa G L) (scpds h g d G L).
-#h #g #G #L #d #A #T #HT #U * /3 width=6 by lstas_aaa_conf, cprs_aaa_conf/
+lemma scpds_aaa_conf: ∀h,o,G,L,d. Conf3 … (aaa G L) (scpds h o d G L).
+#h #o #G #L #d #A #T #HT #U * /3 width=6 by lstas_aaa_conf, cprs_aaa_conf/
 qed.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/scpds_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/scpds_lift.ma
index ee212e99c..f63fd3ad6 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/scpds_lift.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/scpds_lift.ma
@@ -21,14 +21,14 @@ include "basic_2/computation/scpds.ma".
 
 (* Relocation properties ****************************************************)
 
-lemma scpds_lift: ∀h,g,G,d. d_liftable (scpds h g d G).
-#h #g #G #d2 #K #T1 #T2 * #T #d1 #Hd21 #Hd1 #HT1 #HT2 #L #s #l #m
-elim (lift_total T l m)
+lemma scpds_lift: ∀h,o,G,d. d_liftable (scpds h o d G).
+#h #o #G #d2 #K #T1 #T2 * #T #d1 #Hd21 #Hd1 #HT1 #HT2 #L #c #l #k
+elim (lift_total T l k)
 /3 width=15 by cprs_lift, da_lift, lstas_lift, ex4_2_intro/
 qed.
 
-lemma scpds_inv_lift1: ∀h,g,G,d. d_deliftable_sn (scpds h g d G).
-#h #g #G #d2 #L #U1 #U2 * #U #d1 #Hd21 #Hd1 #HU1 #HU2 #K #s #l #m #HLK #T1 #HTU1
+lemma scpds_inv_lift1: ∀h,o,G,d. d_deliftable_sn (scpds h o d G).
+#h #o #G #d2 #L #U1 #U2 * #U #d1 #Hd21 #Hd1 #HU1 #HU2 #K #c #l #k #HLK #T1 #HTU1
 lapply (da_inv_lift … Hd1 … HLK … HTU1) -Hd1 #Hd1
 elim (lstas_inv_lift1 … HU1 … HLK … HTU1) -U1 #T #HTU #HT1
 elim (cprs_inv_lift1 … HU2 … HLK … HTU) -U -L
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/scpds_scpds.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/scpds_scpds.ma
index e9afcf4ef..6c5de0687 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/scpds_scpds.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/scpds_scpds.ma
@@ -21,26 +21,26 @@ include "basic_2/computation/scpds.ma".
 
 (* Advanced properties ******************************************************)
 
-lemma scpds_strap2: ∀h,g,G,L,T1,T,T2,d1,d. ⦃G, L⦄ ⊢ T1 ▪[h, g] d1+1 →
-                    ⦃G, L⦄ ⊢ T1 •*[h, 1] T → ⦃G, L⦄ ⊢ T •*➡*[h, g, d] T2 →
-                    ⦃G, L⦄ ⊢ T1 •*➡*[h, g, d+1] T2.
-#h #g #G #L #T1 #T #T2 #d1 #d #Hd1 #HT1 *
+lemma scpds_strap2: ∀h,o,G,L,T1,T,T2,d1,d. ⦃G, L⦄ ⊢ T1 ▪[h, o] d1+1 →
+                    ⦃G, L⦄ ⊢ T1 •*[h, 1] T → ⦃G, L⦄ ⊢ T •*➡*[h, o, d] T2 →
+                    ⦃G, L⦄ ⊢ T1 •*➡*[h, o, d+1] T2.
+#h #o #G #L #T1 #T #T2 #d1 #d #Hd1 #HT1 *
 #T0 #d0 #Hd0 #HTd0 #HT0 #HT02
-lapply (lstas_da_conf … HT1 … Hd1) <minus_plus_m_m #HTd1
+lapply (lstas_da_conf … HT1 … Hd1) <minus_plus_k_k #HTd1
 lapply (da_mono … HTd0 … HTd1) -HTd0 -HTd1 #H destruct
 lapply (lstas_trans … HT1 … HT0) -T >commutative_plus
 /3 width=6 by le_S_S, ex4_2_intro/
 qed.
 
-lemma scpds_cprs_trans: ∀h,g,G,L,T1,T,T2,d.
-                        ⦃G, L⦄ ⊢ T1 •*➡*[h, g, d] T → ⦃G, L⦄ ⊢ T ➡* T2 → ⦃G, L⦄ ⊢ T1 •*➡*[h, g, d] T2.
-#h #g #G #L #T1 #T #T2 #d * /3 width=8 by cprs_trans, ex4_2_intro/
+lemma scpds_cprs_trans: ∀h,o,G,L,T1,T,T2,d.
+                        ⦃G, L⦄ ⊢ T1 •*➡*[h, o, d] T → ⦃G, L⦄ ⊢ T ➡* T2 → ⦃G, L⦄ ⊢ T1 •*➡*[h, o, d] T2.
+#h #o #G #L #T1 #T #T2 #d * /3 width=8 by cprs_trans, ex4_2_intro/
 qed-.
 
-lemma lstas_scpds_trans: ∀h,g,G,L,T1,T,T2,d1,d2,d.
-                         d2 ≤ d1 → ⦃G, L⦄ ⊢ T1 ▪[h, g] d1 →
-                         ⦃G, L⦄ ⊢ T1 •*[h, d2] T → ⦃G, L⦄ ⊢ T •*➡*[h, g, d] T2 → ⦃G, L⦄ ⊢ T1 •*➡*[h, g, d2+d] T2.
-#h #g #G #L #T1 #T #T2 #d1 #d2 #d #Hd21 #HTd1 #HT1 * #T0 #d0 #Hd0 #HTd0 #HT0 #HT02
+lemma lstas_scpds_trans: ∀h,o,G,L,T1,T,T2,d1,d2,d.
+                         d2 ≤ d1 → ⦃G, L⦄ ⊢ T1 ▪[h, o] d1 →
+                         ⦃G, L⦄ ⊢ T1 •*[h, d2] T → ⦃G, L⦄ ⊢ T •*➡*[h, o, d] T2 → ⦃G, L⦄ ⊢ T1 •*➡*[h, o, d2+d] T2.
+#h #o #G #L #T1 #T #T2 #d1 #d2 #d #Hd21 #HTd1 #HT1 * #T0 #d0 #Hd0 #HTd0 #HT0 #HT02
 lapply (lstas_da_conf … HT1 … HTd1) #HTd12
 lapply (da_mono … HTd12 … HTd0) -HTd12 -HTd0 #H destruct
 lapply (le_minus_to_plus_r … Hd21 Hd0) -Hd21 -Hd0
@@ -49,19 +49,19 @@ qed-.
 
 (* Advanced inversion lemmas ************************************************)
 
-lemma scpds_inv_abst1: ∀h,g,a,G,L,V1,T1,U2,d. ⦃G, L⦄ ⊢ ⓛ{a}V1.T1 •*➡*[h, g, d] U2 →
-                       ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡* V2 & ⦃G, L.ⓛV1⦄ ⊢ T1 •*➡*[h, g, d] T2 &
+lemma scpds_inv_abst1: ∀h,o,a,G,L,V1,T1,U2,d. ⦃G, L⦄ ⊢ ⓛ{a}V1.T1 •*➡*[h, o, d] U2 →
+                       ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡* V2 & ⦃G, L.ⓛV1⦄ ⊢ T1 •*➡*[h, o, d] T2 &
                                 U2 = ⓛ{a}V2.T2.
-#h #g #a #G #L #V1 #T1 #U2 #d2 * #X #d1 #Hd21 #Hd1 #H1 #H2
+#h #o #a #G #L #V1 #T1 #U2 #d2 * #X #d1 #Hd21 #Hd1 #H1 #H2
 lapply (da_inv_bind … Hd1) -Hd1 #Hd1
 elim (lstas_inv_bind1 … H1) -H1 #U #HTU1 #H destruct
 elim (cprs_inv_abst1 … H2) -H2 #V2 #T2 #HV12 #HUT2 #H destruct
 /3 width=6 by ex4_2_intro, ex3_2_intro/
 qed-.
 
-lemma scpds_inv_abbr_abst: ∀h,g,a1,a2,G,L,V1,W2,T1,T2,d. ⦃G, L⦄ ⊢ ⓓ{a1}V1.T1 •*➡*[h, g, d] ⓛ{a2}W2.T2 →
-                           ∃∃T. ⦃G, L.ⓓV1⦄ ⊢ T1 •*➡*[h, g, d] T & ⬆[0, 1] ⓛ{a2}W2.T2 ≡ T & a1 = true.
-#h #g #a1 #a2 #G #L #V1 #W2 #T1 #T2 #d2 * #X #d1 #Hd21 #Hd1 #H1 #H2
+lemma scpds_inv_abbr_abst: ∀h,o,a1,a2,G,L,V1,W2,T1,T2,d. ⦃G, L⦄ ⊢ ⓓ{a1}V1.T1 •*➡*[h, o, d] ⓛ{a2}W2.T2 →
+                           ∃∃T. ⦃G, L.ⓓV1⦄ ⊢ T1 •*➡*[h, o, d] T & ⬆[0, 1] ⓛ{a2}W2.T2 ≡ T & a1 = true.
+#h #o #a1 #a2 #G #L #V1 #W2 #T1 #T2 #d2 * #X #d1 #Hd21 #Hd1 #H1 #H2
 lapply (da_inv_bind … Hd1) -Hd1 #Hd1
 elim (lstas_inv_bind1 … H1) -H1 #U1 #HTU1 #H destruct
 elim (cprs_inv_abbr1 … H2) -H2 *
@@ -70,24 +70,24 @@ elim (cprs_inv_abbr1 … H2) -H2 *
 ]
 qed-.
 
-lemma scpds_inv_lstas_eq: ∀h,g,G,L,T1,T2,d. ⦃G, L⦄ ⊢ T1 •*➡*[h, g, d] T2 →
+lemma scpds_inv_lstas_eq: ∀h,o,G,L,T1,T2,d. ⦃G, L⦄ ⊢ T1 •*➡*[h, o, d] T2 →
                           ∀T. ⦃G, L⦄ ⊢ T1 •*[h, d] T → ⦃G, L⦄ ⊢ T ➡* T2.
-#h #g #G #L #T1 #T2 #d2 *
+#h #o #G #L #T1 #T2 #d2 *
 #T0 #d1 #_ #_ #HT10 #HT02 #T #HT1
 lapply (lstas_mono … HT10 … HT1) #H destruct //
 qed-.
 
 (* Advanced forward lemmas **************************************************)
 
-lemma scpds_fwd_cpxs: ∀h,g,G,L,T1,T2,d. ⦃G, L⦄ ⊢ T1 •*➡*[h, g, d] T2 → ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2.
-#h #g #G #L #T1 #T2 #d * /3 width=5 by cpxs_trans, lstas_cpxs, cprs_cpxs/
+lemma scpds_fwd_cpxs: ∀h,o,G,L,T1,T2,d. ⦃G, L⦄ ⊢ T1 •*➡*[h, o, d] T2 → ⦃G, L⦄ ⊢ T1 ➡*[h, o] T2.
+#h #o #G #L #T1 #T2 #d * /3 width=5 by cpxs_trans, lstas_cpxs, cprs_cpxs/
 qed-.
 
 (* Main properties **********************************************************)
 
-theorem scpds_conf_eq: ∀h,g,G,L,T0,T1,d. ⦃G, L⦄ ⊢ T0 •*➡*[h, g, d] T1 →
-                       ∀T2. ⦃G, L⦄ ⊢ T0 •*➡*[h, g, d] T2 →
+theorem scpds_conf_eq: ∀h,o,G,L,T0,T1,d. ⦃G, L⦄ ⊢ T0 •*➡*[h, o, d] T1 →
+                       ∀T2. ⦃G, L⦄ ⊢ T0 •*➡*[h, o, d] T2 →
                        ∃∃T. ⦃G, L⦄ ⊢ T1 ➡* T & ⦃G, L⦄ ⊢ T2 ➡* T.
-#h #g #G #L #T0 #T1 #d0 * #U1 #d1 #_ #_ #H1 #HUT1 #T2 * #U2 #d2 #_ #_ #H2 #HUT2 -d1 -d2
+#h #o #G #L #T0 #T1 #d0 * #U1 #d1 #_ #_ #H1 #HUT1 #T2 * #U2 #d2 #_ #_ #H2 #HUT2 -d1 -d2
 lapply (lstas_mono … H1 … H2) #H destruct -h -d0 /2 width=3 by cprs_conf/
 qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_new/drops/drops.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_new/drops/drops.etc
index 86ca4c76a..6c07f6f34 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/etc_new/drops/drops.etc
+++ b/matita/matita/contribs/lambdadelta/basic_2/etc_new/drops/drops.etc
@@ -1,27 +1,3 @@
-lemma drop_split: ∀L1,L2,l,m2,s. ⬇[s, l, m2] L1 ≡ L2 → ∀m1. m1 ≤ m2 →
-                  ∃∃L. ⬇[s, l, m2 - m1] L1 ≡ L & ⬇[s, l, m1] L ≡ L2.
-#L1 #L2 #l #m2 #s #H elim H -L1 -L2 -l -m2
-[ #l #m2 #Hs #m1 #Hm12 @(ex2_intro … (⋆))
-  @drop_atom #H lapply (Hs H) -s #H destruct /2 width=1 by le_n_O_to_eq/
-| #I #L1 #V #m1 #Hm1 lapply (le_n_O_to_eq … Hm1) -Hm1
-  #H destruct /2 width=3 by ex2_intro/
-| #I #L1 #L2 #V #m2 #HL12 #IHL12 #m1 @(nat_ind_plus … m1) -m1
-  [ /3 width=3 by drop_drop, ex2_intro/
-  | -HL12 #m1 #_ #Hm12 lapply (le_plus_to_le_r … Hm12) -Hm12
-    #Hm12 elim (IHL12 … Hm12) -IHL12 >minus_plus_plus_l
-    #L #HL1 #HL2 elim (lt_or_ge (|L1|) (m2-m1)) #H0
-    [ elim (drop_inv_O1_gt … HL1 H0) -HL1 #H1 #H2 destruct
-      elim (drop_inv_atom1 … HL2) -HL2 #H #_ destruct
-      @(ex2_intro … (⋆)) [ @drop_O1_ge normalize // ]
-      @drop_atom #H destruct
-    | elim (drop_O1_pair … HL1 H0 I V) -HL1 -H0 /3 width=5 by drop_drop, ex2_intro/
-    ]
-  ]
-| #I #L1 #L2 #V1 #V2 #l #m2 #_ #HV21 #IHL12 #m1 #Hm12 elim (IHL12 … Hm12) -IHL12
-  #L #HL1 #HL2 elim (lift_split … HV21 l m1) -HV21 /3 width=5 by drop_skip, ex2_intro/
-]
-qed-.
-
 lemma drop_inv_refl: ∀L,l,m. ⬇[Ⓕ, l, m] L ≡ L → m = 0.
 /2 width=5 by drop_inv_length_eq/ qed-.
 
diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/drops.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/drops.ma
index 2e21e55ef..fd47afae2 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/relocation/drops.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/drops.ma
@@ -238,6 +238,16 @@ lemma drops_inv_skip2: ∀I,X,K2,V2,c,f. ⬇*[c, ↑f] X ≡ K2.ⓑ{I}V2 →
 
 (* Inversion lemmas with test for uniformity ********************************)
 
+lemma drops_inv_isuni: ∀L1,L2,f. ⬇*[Ⓣ, f] L1 ≡ L2 → 𝐔⦃f⦄ →
+                       (𝐈⦃f⦄ ∧ L2 = L1) ∨
+                       ∃∃I,K,V,g. ⬇*[Ⓣ, g] K ≡ L2 & L1 = K.ⓑ{I}V & f = ⫯g.
+#L1 #L2 #f * -L1 -L2 -f
+[ /4 width=1 by or_introl, conj/
+| /4 width=7 by ex3_4_intro, or_intror/
+| /7 width=6 by drops_fwd_isid, lifts_fwd_isid, isuni_inv_push, isid_push, or_introl, conj, eq_f3, sym_eq/
+]
+qed-.
+
 (* Basic_2A1: was: drop_inv_O1_pair1 *)
 lemma drops_inv_pair1_isuni: ∀I,K,L2,V,c,f. 𝐔⦃f⦄ → ⬇*[c, f] K.ⓑ{I}V ≡ L2 →
                              (𝐈⦃f⦄ ∧ L2 = K.ⓑ{I}V) ∨
@@ -271,6 +281,16 @@ lemma drops_inv_pair2_isuni_next: ∀I,K,V,c,f,L1. 𝐔⦃f⦄ → ⬇*[c, ⫯f]
 ]
 qed-. 
 
+(* Inversion lemmas with uniform relocations ********************************)
+
+lemma drops_inv_succ: ∀L1,L2,l. ⬇*[⫯l] L1 ≡ L2 →
+                      ∃∃I,K,V. ⬇*[l] K ≡ L2 & L1 = K.ⓑ{I}V.
+#L1 #L2 #l #H elim (drops_inv_isuni … H) -H // *
+[ #H elim (isid_inv_next … H) -H //
+| /2 width=5 by ex2_3_intro/
+]
+qed-. 
+
 (* Basic_2A1: removed theorems 12:
               drops_inv_nil drops_inv_cons d1_liftable_liftables
               drop_refl_atom_O2 drop_inv_pair1
diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/lifts.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/lifts.ma
index a72f58604..97db4d98e 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/relocation/lifts.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/lifts.ma
@@ -1,3 +1,4 @@
+
 (**************************************************************************)
 (*       ___                                                              *)
 (*      ||M||                                                             *)
@@ -295,8 +296,9 @@ elim (IHV1 f) -IHV1 #V2 #HV12
 ]
 qed-.
 
-lemma lift_SO: ∀i. ⬆*[1] #i ≡ #(⫯i).
-/2 width=1 by lifts_lref/ qed.
+lemma lift_lref_uni: ∀l,i. ⬆*[l] #i ≡ #(l+i).
+#l elim l -l /2 width=1 by lifts_lref/
+qed.
 
 (* Basic_1: includes: lift_free (right to left) *)
 (* Basic_2A1: includes: lift_split *)
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/fqup.ma b/matita/matita/contribs/lambdadelta/basic_2/s_computation/fqup.ma
similarity index 69%
rename from matita/matita/contribs/lambdadelta/basic_2/computation/fqup.ma
rename to matita/matita/contribs/lambdadelta/basic_2/s_computation/fqup.ma
index 3ac1a812d..808164b69 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/fqup.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/s_computation/fqup.ma
@@ -13,7 +13,7 @@
 (**************************************************************************)
 
 include "basic_2/notation/relations/suptermplus_6.ma".
-include "basic_2/substitution/fqu.ma".
+include "basic_2/s_transition/fqu.ma".
 
 (* PLUS-ITERATED SUPCLOSURE *************************************************)
 
@@ -37,17 +37,6 @@ lemma fqup_strap2: ∀G1,G,G2,L1,L,L2,T1,T,T2.
                    ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄.
 /2 width=5 by tri_TC_strap/ qed.
 
-lemma fqup_drop: ∀G1,G2,L1,K1,K2,T1,T2,U1,k. ⬇[k] L1 ≡ K1 → ⬆[0, k] T1 ≡ U1 →
-                  ⦃G1, K1, T1⦄ ⊐+ ⦃G2, K2, T2⦄ → ⦃G1, L1, U1⦄ ⊐+ ⦃G2, K2, T2⦄.
-#G1 #G2 #L1 #K1 #K2 #T1 #T2 #U1 #k #HLK1 #HTU1 #HT12 elim (eq_or_gt … k) #H destruct
-[ >(drop_inv_O2 … HLK1) -L1 <(lift_inv_O2 … HTU1) -U1 //
-| /3 width=5 by fqup_strap2, fqu_drop_lt/
-]
-qed-.
-
-lemma fqup_lref: ∀I,G,L,K,V,i. ⬇[i] L ≡ K.ⓑ{I}V → ⦃G, L, #i⦄ ⊐+ ⦃G, K, V⦄.
-/3 width=6 by fqu_lref_O, fqu_fqup, lift_lref_ge, fqup_drop/ qed.
-
 lemma fqup_pair_sn: ∀I,G,L,V,T. ⦃G, L, ②{I}V.T⦄ ⊐+ ⦃G, L, V⦄.
 /2 width=1 by fqu_pair_sn, fqu_fqup/ qed.
 
@@ -84,26 +73,4 @@ lemma fqup_ind_dx: ∀G2,L2,T2. ∀R:relation3 ….
 @(tri_TC_ind_dx … IH1 IH2 G1 L1 T1 H)
 qed-.
 
-(* Basic forward lemmas *****************************************************)
-
-lemma fqup_fwd_fw: ∀G1,G2,L1,L2,T1,T2.
-                   ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ → ♯{G2, L2, T2} < ♯{G1, L1, T1}.
-#G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
-/3 width=3 by fqu_fwd_fw, transitive_lt/
-qed-.
-
-(* Advanced eliminators *****************************************************)
-
-lemma fqup_wf_ind: ∀R:relation3 …. (
-                      ∀G1,L1,T1. (∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ → R G2 L2 T2) →
-                      R G1 L1 T1
-                   ) → ∀G1,L1,T1. R G1 L1 T1.
-#R #HR @(f3_ind … fw) #x #IHx #G1 #L1 #T1 #H destruct /4 width=1 by fqup_fwd_fw/
-qed-.
-
-lemma fqup_wf_ind_eq: ∀R:relation3 …. (
-                         ∀G1,L1,T1. (∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ → R G2 L2 T2) →
-                         ∀G2,L2,T2. G1 = G2 → L1 = L2 → T1 = T2 → R G2 L2 T2
-                      ) → ∀G1,L1,T1. R G1 L1 T1.
-#R #HR @(f3_ind … fw) #x #IHx #G1 #L1 #T1 #H destruct /4 width=7 by fqup_fwd_fw/
-qed-.
+(* Basic_2A1: removed theorems 1: fqup_drop *)
diff --git a/matita/matita/contribs/lambdadelta/basic_2/s_computation/fqup_drops.ma b/matita/matita/contribs/lambdadelta/basic_2/s_computation/fqup_drops.ma
new file mode 100644
index 000000000..c706f6b8d
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/s_computation/fqup_drops.ma
@@ -0,0 +1,46 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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 "basic_2/relocation/drops.ma".
+include "basic_2/s_computation/fqup.ma".
+
+(* PLUS-ITERATED SUPCLOSURE *************************************************)
+
+(* Properties with generic slicing for local environments *******************)
+
+lemma fqup_drops_succ: ∀G,K,T,l,L,U. ⬇*[⫯l] L ≡ K → ⬆*[⫯l] T ≡ U →
+                       ⦃G, L, U⦄ ⊐+ ⦃G, K, T⦄.
+#G #K #T #l elim l -l
+[ #L #U #HLK #HTU elim (drops_inv_succ … HLK) -HLK
+  #I #Y #V #HY #H destruct <(drops_fwd_isid … HY) -K
+  /3 width=1 by fqu_fqup, fqu_drop/
+| #l #IH #L #U #HLK #HTU elim (drops_inv_succ … HLK) -HLK
+  #I #Y #V #HY #H destruct
+  elim (lifts_split_trans … HTU … (𝐔❴⫯l❵) (𝐔❴1❵)) -HTU
+  /4 width=5 by fqup_strap2, fqu_drop/
+]
+qed.
+
+lemma fqup_drops_strap1: ∀G1,G2,L1,K1,K2,T1,T2,U1,l. ⬇*[l] L1 ≡ K1 → ⬆*[l] T1 ≡ U1 →
+                         ⦃G1, K1, T1⦄ ⊐ ⦃G2, K2, T2⦄ → ⦃G1, L1, U1⦄ ⊐+ ⦃G2, K2, T2⦄.
+#G1 #G2 #L1 #K1 #K2 #T1 #T2 #U1 *
+[ #HLK1 #HTU1 #HT12
+  >(drops_fwd_isid … HLK1) -L1 //
+  <(lifts_fwd_isid … HTU1) -U1 /2 width=1 by fqu_fqup/
+| /3 width=5 by fqup_strap1, fqup_drops_succ/
+]
+qed-.
+
+lemma fqup_lref: ∀I,G,L,K,V,i. ⬇*[i] L ≡ K.ⓑ{I}V → ⦃G, L, #i⦄ ⊐+ ⦃G, K, V⦄.
+/2 width=6 by fqup_drops_strap1/ qed. 
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/fqup_fqup.ma b/matita/matita/contribs/lambdadelta/basic_2/s_computation/fqup_fqup.ma
similarity index 96%
rename from matita/matita/contribs/lambdadelta/basic_2/computation/fqup_fqup.ma
rename to matita/matita/contribs/lambdadelta/basic_2/s_computation/fqup_fqup.ma
index 48420a7cf..4524e8a77 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/fqup_fqup.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/s_computation/fqup_fqup.ma
@@ -12,7 +12,7 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "basic_2/multiple/fqup.ma".
+include "basic_2/s_computation/fqup.ma".
 
 (* PLUS-ITERATED SUPCLOSURE *************************************************)
 
diff --git a/matita/matita/contribs/lambdadelta/basic_2/s_computation/fqup_weight.ma b/matita/matita/contribs/lambdadelta/basic_2/s_computation/fqup_weight.ma
new file mode 100644
index 000000000..ec05d1e98
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/s_computation/fqup_weight.ma
@@ -0,0 +1,42 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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 "basic_2/s_transition/fqu_weight.ma".
+include "basic_2/s_computation/fqup.ma".
+
+(* PLUS-ITERATED SUPCLOSURE *************************************************)
+
+(* Forward lemmas with weight for closures **********************************)
+
+lemma fqup_fwd_fw: ∀G1,G2,L1,L2,T1,T2.
+                   ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ → ♯{G2, L2, T2} < ♯{G1, L1, T1}.
+#G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
+/3 width=3 by fqu_fwd_fw, transitive_lt/
+qed-.
+
+(* Advanced eliminators *****************************************************)
+
+lemma fqup_wf_ind: ∀R:relation3 …. (
+                      ∀G1,L1,T1. (∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ → R G2 L2 T2) →
+                      R G1 L1 T1
+                   ) → ∀G1,L1,T1. R G1 L1 T1.
+#R #HR @(f3_ind … fw) #x #IHx #G1 #L1 #T1 #H destruct /4 width=1 by fqup_fwd_fw/
+qed-.
+
+lemma fqup_wf_ind_eq: ∀R:relation3 …. (
+                         ∀G1,L1,T1. (∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ → R G2 L2 T2) →
+                         ∀G2,L2,T2. G1 = G2 → L1 = L2 → T1 = T2 → R G2 L2 T2
+                      ) → ∀G1,L1,T1. R G1 L1 T1.
+#R #HR @(f3_ind … fw) #x #IHx #G1 #L1 #T1 #H destruct /4 width=7 by fqup_fwd_fw/
+qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/fqus.ma b/matita/matita/contribs/lambdadelta/basic_2/s_computation/fqus.ma
similarity index 67%
rename from matita/matita/contribs/lambdadelta/basic_2/computation/fqus.ma
rename to matita/matita/contribs/lambdadelta/basic_2/s_computation/fqus.ma
index 0a2061220..851ac5108 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/fqus.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/s_computation/fqus.ma
@@ -13,8 +13,7 @@
 (**************************************************************************)
 
 include "basic_2/notation/relations/suptermstar_6.ma".
-include "basic_2/substitution/fquq.ma".
-include "basic_2/multiple/fqup.ma".
+include "basic_2/s_transition/fquq.ma".
 
 (* STAR-ITERATED SUPCLOSURE *************************************************)
 
@@ -55,29 +54,4 @@ lemma fqus_strap2: ∀G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G, L,
                    ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄.
 /2 width=5 by tri_TC_strap/ qed-.
 
-lemma fqus_drop: ∀G1,G2,K1,K2,T1,T2. ⦃G1, K1, T1⦄ ⊐* ⦃G2, K2, T2⦄ →
-                  ∀L1,U1,k. ⬇[k] L1 ≡ K1 → ⬆[0, k] T1 ≡ U1 →
-                  ⦃G1, L1, U1⦄ ⊐* ⦃G2, K2, T2⦄.
-#G1 #G2 #K1 #K2 #T1 #T2 #H @(fqus_ind … H) -G2 -K2 -T2
-/3 width=5 by fqus_strap1, fquq_fqus, fquq_drop/
-qed-.
-
-lemma fqup_fqus: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄.
-#G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
-/3 width=5 by fqus_strap1, fquq_fqus, fqu_fquq/
-qed.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma fqus_fwd_fw: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ → ♯{G2, L2, T2} ≤ ♯{G1, L1, T1}.
-#G1 #G2 #L1 #L2 #T1 #T2 #H @(fqus_ind … H) -L2 -T2
-/3 width=3 by fquq_fwd_fw, transitive_le/
-qed-.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma fqup_inv_step_sn: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
-                        ∃∃G,L,T. ⦃G1, L1, T1⦄ ⊐ ⦃G, L, T⦄ & ⦃G, L, T⦄ ⊐* ⦃G2, L2, T2⦄.
-#G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind_dx … H) -G1 -L1 -T1 /2 width=5 by ex2_3_intro/
-#G1 #G #L1 #L #T1 #T #H1 #_ * /4 width=9 by fqus_strap2, fqu_fquq, ex2_3_intro/
-qed-.
+(* Basic_2A1: removed theorems 1: fqus_drop *)
diff --git a/matita/matita/contribs/lambdadelta/basic_2/s_computation/fqus_drops.ma b/matita/matita/contribs/lambdadelta/basic_2/s_computation/fqus_drops.ma
new file mode 100644
index 000000000..8850c1f54
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/s_computation/fqus_drops.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 "basic_2/s_computation/fqup_drops.ma".
+include "basic_2/s_computation/fqus_fqup.ma".
+
+(* STAR-ITERATED SUPCLOSURE *************************************************)
+
+(* Properties with generic slicing for local environments *******************)
+
+lemma fqus_drops: ∀G,L,K,T,U,l. ⬇*[l] L ≡ K → ⬆*[l] T ≡ U →
+                  ⦃G, L, U⦄ ⊐* ⦃G, K, T⦄.
+#G #L #K #T #U * /3 width=3 by fqup_drops_succ, fqup_fqus/
+#HLK #HTU <(lifts_fwd_isid … HTU) -U // <(drops_fwd_isid … HLK) -K //
+qed.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/fqus_alt.ma b/matita/matita/contribs/lambdadelta/basic_2/s_computation/fqus_fqup.ma
similarity index 61%
rename from matita/matita/contribs/lambdadelta/basic_2/computation/fqus_alt.ma
rename to matita/matita/contribs/lambdadelta/basic_2/s_computation/fqus_fqup.ma
index c9ec457b6..6c120326e 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/fqus_alt.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/s_computation/fqus_fqup.ma
@@ -12,21 +12,27 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "basic_2/substitution/fquq_alt.ma".
-include "basic_2/multiple/fqus.ma".
+include "basic_2/s_computation/fqup.ma".
+include "basic_2/s_computation/fqus.ma".
 
 (* STAR-ITERATED SUPCLOSURE *************************************************)
 
-(* Advanced inversion lemmas ************************************************)
+(* Alternative definition with plus-iterated supclosure *********************)
 
-lemma fqus_inv_gen: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
-                    ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ ∨ (∧∧ G1 = G2 & L1 = L2 & T1 = T2).
+lemma fqup_fqus: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄.
+#G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
+/3 width=5 by fqus_strap1, fquq_fqus, fqu_fquq/
+qed.
+
+(* Basic_2A1: was: fqus_inv_gen *)
+lemma fqus_inv_fqup: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
+                     ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ ∨ (∧∧ G1 = G2 & L1 = L2 & T1 = T2).
 #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqus_ind … H) -G2 -L2 -T2 //
-#G #G2 #L #L2 #T #T2 #_ #H2 * elim (fquq_inv_gen … H2) -H2
-[ /3 width=5 by fqup_strap1, or_introl/
-| * #HG #HL #HT destruct /2 width=1 by or_introl/
-| #H2 * #HG #HL #HT destruct /3 width=1 by fqu_fqup, or_introl/
-| * #H1G #H1L #H1T * #H2G #H2L #H2T destruct /2 width=1 by or_intror/
+#G #G2 #L #L2 #T #T2 #_ *
+[ #H2 * /3 width=5 by fqup_strap1, or_introl/
+  * /3 width=1 by fqu_fqup, or_introl/
+| * #HG #HL #HT destruct * /2 width=1 by or_introl/
+  * /2 width=1 by or_intror/
 ]
 qed-.
 
@@ -34,17 +40,17 @@ qed-.
 
 lemma fqus_strap1_fqu: ∀G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐ ⦃G2, L2, T2⦄ →
                        ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄.
-#G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2 elim (fqus_inv_gen … H1) -H1
+#G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2 elim (fqus_inv_fqup … H1) -H1
 [ /2 width=5 by fqup_strap1/
-| * #HG #HL #HT destruct /2 width=1 by fqu_fqup/
+| * /2 width=1 by fqu_fqup/
 ]
 qed-.
 
 lemma fqus_strap2_fqu: ∀G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐* ⦃G2, L2, T2⦄ →
                        ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄.
-#G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2 elim (fqus_inv_gen … H2) -H2
+#G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2 elim (fqus_inv_fqup … H2) -H2
 [ /2 width=5 by fqup_strap2/
-| * #HG #HL #HT destruct /2 width=1 by fqu_fqup/
+| * /2 width=1 by fqu_fqup/
 ]
 qed-.
 
@@ -59,3 +65,11 @@ lemma fqup_fqus_trans: ∀G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G, L
 #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 @(fqup_ind_dx … H1) -H1 -G1 -L1 -T1
 /3 width=5 by fqus_strap2_fqu, fqup_strap2/
 qed-.
+
+(* Advanced inversion lemmas for plus-iterated supclosure *******************)
+
+lemma fqup_inv_step_sn: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
+                        ∃∃G,L,T. ⦃G1, L1, T1⦄ ⊐ ⦃G, L, T⦄ & ⦃G, L, T⦄ ⊐* ⦃G2, L2, T2⦄.
+#G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind_dx … H) -G1 -L1 -T1 /2 width=5 by ex2_3_intro/
+#G1 #G #L1 #L #T1 #T #H1 #_ * /4 width=9 by fqus_strap2, fqu_fquq, ex2_3_intro/
+qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/fqus_fqus.ma b/matita/matita/contribs/lambdadelta/basic_2/s_computation/fqus_fqus.ma
similarity index 96%
rename from matita/matita/contribs/lambdadelta/basic_2/computation/fqus_fqus.ma
rename to matita/matita/contribs/lambdadelta/basic_2/s_computation/fqus_fqus.ma
index ef9902931..26da77526 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/fqus_fqus.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/s_computation/fqus_fqus.ma
@@ -12,7 +12,7 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "basic_2/multiple/fqus.ma".
+include "basic_2/s_computation/fqus.ma".
 
 (* STAR-ITERATED SUPCLOSURE *************************************************)
 
diff --git a/matita/matita/contribs/lambdadelta/basic_2/s_computation/fqus_weight.ma b/matita/matita/contribs/lambdadelta/basic_2/s_computation/fqus_weight.ma
new file mode 100644
index 000000000..5a29848cc
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/s_computation/fqus_weight.ma
@@ -0,0 +1,25 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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 "basic_2/s_transition/fquq_weight.ma".
+include "basic_2/s_computation/fqus.ma".
+
+(* STAR-ITERATED SUPCLOSURE *************************************************)
+
+(* Forward lemmas with weight for closures **********************************)
+
+lemma fqus_fwd_fw: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ → ♯{G2, L2, T2} ≤ ♯{G1, L1, T1}.
+#G1 #G2 #L1 #L2 #T1 #T2 #H @(fqus_ind … H) -L2 -T2
+/3 width=3 by fquq_fwd_fw, transitive_le/
+qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/frees.ma b/matita/matita/contribs/lambdadelta/basic_2/static/frees.ma
similarity index 100%
rename from matita/matita/contribs/lambdadelta/basic_2/relocation/frees.ma
rename to matita/matita/contribs/lambdadelta/basic_2/static/frees.ma
diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/frees_frees.ma b/matita/matita/contribs/lambdadelta/basic_2/static/frees_frees.ma
similarity index 100%
rename from matita/matita/contribs/lambdadelta/basic_2/relocation/frees_frees.ma
rename to matita/matita/contribs/lambdadelta/basic_2/static/frees_frees.ma
diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/frees_lreq.ma b/matita/matita/contribs/lambdadelta/basic_2/static/frees_lreq.ma
similarity index 100%
rename from matita/matita/contribs/lambdadelta/basic_2/relocation/frees_lreq.ma
rename to matita/matita/contribs/lambdadelta/basic_2/static/frees_lreq.ma
diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/frees_weight.ma b/matita/matita/contribs/lambdadelta/basic_2/static/frees_weight.ma
similarity index 100%
rename from matita/matita/contribs/lambdadelta/basic_2/relocation/frees_weight.ma
rename to matita/matita/contribs/lambdadelta/basic_2/static/frees_weight.ma
diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/freq.ma b/matita/matita/contribs/lambdadelta/basic_2/static/freq.ma
similarity index 100%
rename from matita/matita/contribs/lambdadelta/basic_2/relocation/freq.ma
rename to matita/matita/contribs/lambdadelta/basic_2/static/freq.ma
diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/freq_freq.ma b/matita/matita/contribs/lambdadelta/basic_2/static/freq_freq.ma
similarity index 100%
rename from matita/matita/contribs/lambdadelta/basic_2/relocation/freq_freq.ma
rename to matita/matita/contribs/lambdadelta/basic_2/static/freq_freq.ma
diff --git a/matita/matita/contribs/lambdadelta/basic_2/web/basic_2.ldw.xml b/matita/matita/contribs/lambdadelta/basic_2/web/basic_2.ldw.xml
index 35da29879..a50dd0d88 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/web/basic_2.ldw.xml
+++ b/matita/matita/contribs/lambdadelta/basic_2/web/basic_2.ldw.xml
@@ -3,6 +3,7 @@
 <page xmlns="http://lambdadelta.info/"
       description = "\lambda\delta home page"
       title = "\lambda\delta home page"
+      logo = "crux"
       head = "cic:/matita/lambdadelta/basic_2/ (core λδ version 2)"
 >
    <sitemap name="sitemap"/>
@@ -32,6 +33,11 @@
    </news>
 
    <subsection name="A2">Stage "A2": "Extending the Applicability Condition"</subsection>
+
+   <news class="alpha" date="2016 March 25.">
+         Relocation with reference transforming maps (rtmap).
+   </news>
+
    <news class="alpha" date="2015 October 9.">
          λδ version 2A2 is started.
    </news>
diff --git a/matita/matita/contribs/lambdadelta/basic_2/web/basic_2_src.tbl b/matita/matita/contribs/lambdadelta/basic_2/web/basic_2_src.tbl
index 0d288fdc2..73a5a3977 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/web/basic_2_src.tbl
+++ b/matita/matita/contribs/lambdadelta/basic_2/web/basic_2_src.tbl
@@ -216,8 +216,9 @@ table {
    ]
    class "grass"
    [ { "s-computation" * } {
-        [ { "" * } {
-             [ * ]
+        [ { "iterated structural successor for closures" * } {
+             [ "fqus ( ⦃?,?,?⦄ ⊐* ⦃?,?,?⦄ )" "fqus_weight" + "fqus_drops" + "fqus_fqup" + "fqus_fqus" * ]
+             [ "fqup ( ⦃?,?,?⦄ ⊐+ ⦃?,?,?⦄ )" "fqup_weight" + "fqup_drops" + "fqup_fqup" * ]
           }
         ]
      }
@@ -304,11 +305,6 @@ class "italic"            { 1 }
              [ "cpys ( ⦃?,?⦄ ⊢ ? ▶*[?,?] ? )" "cpys_alt ( ⦃?,?⦄ ⊢ ? ▶▶*[?,?] ? )" "cpys_lift" + "cpys_cpys" * ]
           }
         ]
-        [ { "iterated structural successor for closures" * } {
-             [ "fqus ( ⦃?,?,?⦄ ⊐* ⦃?,?,?⦄ )" "fqus_alt" + "fqus_fqus" * ]
-             [ "fqup ( ⦃?,?,?⦄ ⊐+ ⦃?,?,?⦄ )" "fqup_fqup" * ]
-          }
-        ]
         [ { "pointwise union for local environments" * } {
              [ "llor ( ? ⋓[?,?] ? ≡ ? )" "llor_alt" + "llor_drop" * ]
           }
diff --git a/matita/matita/contribs/lambdadelta/ground_2/relocation/rtmap_after.ma b/matita/matita/contribs/lambdadelta/ground_2/relocation/rtmap_after.ma
index 222bc07f7..09bab5e10 100644
--- a/matita/matita/contribs/lambdadelta/ground_2/relocation/rtmap_after.ma
+++ b/matita/matita/contribs/lambdadelta/ground_2/relocation/rtmap_after.ma
@@ -320,6 +320,24 @@ lemma after_isid_isuni: ∀f1,f2. 𝐈⦃f2⦄ → 𝐔⦃f1⦄ → f1 ⊚ ⫯f2
 /5 width=7 by isid_after_dx, after_eq_repl_back_2, after_next, after_push, eq_push_inv_isid/
 qed.
 
+lemma after_uni_next2: ∀f2. 𝐔⦃f2⦄ → ∀f1,f. ⫯f2 ⊚ f1 ≡ f → f2 ⊚ ⫯f1 ≡ f.
+#f2 #H elim H -f2
+[ #f2 #Hf2 #f1 #f #Hf
+  elim (after_inv_nxx … Hf) -Hf [2,3: // ] #g #Hg #H0 destruct
+  /4 width=7 by after_isid_inv_sn, isid_after_sn, after_eq_repl_back_0, eq_next/
+| #f2 #_ #g2 #H2 #IH #f1 #f #Hf
+  elim (after_inv_nxx … Hf) -Hf [2,3: // ] #g #Hg #H0 destruct
+  /3 width=5 by after_next/
+]
+qed.
+
+(* Properties on uni ********************************************************)
+
+lemma after_uni: ∀n1,n2. 𝐔❴n1❵ ⊚ 𝐔❴n2❵ ≡ 𝐔❴n1+n2❵.
+@nat_elim2
+/4 width=5 by after_uni_next2, isid_after_sn, isid_after_dx, after_next/
+qed.
+
 (* Forward lemmas on at *****************************************************)
 
 lemma after_at_fwd: ∀i,i1,f. @⦃i1, f⦄ ≡ i → ∀f2,f1. f2 ⊚ f1 ≡ f →
diff --git a/matita/matita/contribs/lambdadelta/ground_2/web/ground_2.ldw.xml b/matita/matita/contribs/lambdadelta/ground_2/web/ground_2.ldw.xml
index 2dbdd3444..5b071cd95 100644
--- a/matita/matita/contribs/lambdadelta/ground_2/web/ground_2.ldw.xml
+++ b/matita/matita/contribs/lambdadelta/ground_2/web/ground_2.ldw.xml
@@ -3,6 +3,7 @@
 <page xmlns="http://lambdadelta.info/"
       description = "\lambda\delta home page"
       title = "\lambda\delta home page"
+      logo = "crux"
       head = "cic:/matita/lambdadelta/ground_2/ (background for λδ version 2)"
 >
    <sitemap name="sitemap"/>
-- 
2.39.2