From 29973426e0227ee48368d1c24dc0c17bf2baef77 Mon Sep 17 00:00:00 2001
From: Ferruccio Guidi <ferruccio.guidi@unibo.it>
Date: Thu, 1 Aug 2013 12:32:09 +0000
Subject: [PATCH] partial commit: just the components before "static" ...

---
 .../basic_2/computation/acp_aaa.ma            |   4 +-
 .../lambdadelta/basic_2/computation/cpds.ma   |  14 +-
 .../basic_2/computation/cpds_aaa.ma           |   2 +-
 .../basic_2/computation/cpds_cpds.ma          |  14 +-
 .../basic_2/computation/cpds_lift.ma          |   8 +-
 .../lambdadelta/basic_2/computation/cpre.ma   |   4 +-
 .../basic_2/computation/cpre_cpre.ma          |   2 +-
 .../lambdadelta/basic_2/computation/cprs.ma   |  50 ++---
 .../basic_2/computation/cprs_cprs.ma          |  46 ++---
 .../basic_2/computation/cprs_lift.ma          |   4 +-
 .../lambdadelta/basic_2/computation/cpxe.ma   |   4 +-
 .../lambdadelta/basic_2/computation/cpxs.ma   |  68 +++----
 .../basic_2/computation/cpxs_aaa.ma           |   6 +-
 .../basic_2/computation/cpxs_cpxs.ma          |  54 +++---
 .../basic_2/computation/cpxs_lift.ma          |  20 +-
 .../basic_2/computation/cpxs_tstc.ma          |  22 +--
 .../basic_2/computation/cpxs_tstc_vector.ma   |  26 +--
 .../lambdadelta/basic_2/computation/csn.ma    |  40 ++--
 .../basic_2/computation/csn_aaa.ma            |   2 +-
 .../basic_2/computation/csn_alt.ma            |  36 ++--
 .../basic_2/computation/csn_lift.ma           |  18 +-
 .../basic_2/computation/csn_lpx.ma            |  30 +--
 .../basic_2/computation/csn_tstc_vector.ma    |  22 +--
 .../basic_2/computation/csn_vector.ma         |   8 +-
 .../lambdadelta/basic_2/computation/lprs.ma   |  10 +-
 .../basic_2/computation/lprs_cprs.ma          |  16 +-
 .../lambdadelta/basic_2/computation/lpxs.ma   |  26 +--
 .../basic_2/computation/lpxs_aaa.ma           |   4 +-
 .../basic_2/computation/lpxs_alt.ma           |   8 +-
 .../basic_2/computation/lpxs_cpxs.ma          |  32 +--
 .../lambdadelta/basic_2/conversion/cpc.ma     |   4 +-
 .../lambdadelta/basic_2/conversion/cpc_cpc.ma |   4 +-
 .../lambdadelta/basic_2/dynamic/lsubsv.ma     |  58 +++---
 .../basic_2/dynamic/lsubsv_cpcs.ma            |   2 +-
 .../basic_2/dynamic/lsubsv_cpds.ma            |  28 +--
 .../basic_2/dynamic/lsubsv_ldrop.ma           |   8 +-
 .../basic_2/dynamic/lsubsv_lsuba.ma           |   2 +-
 .../lambdadelta/basic_2/dynamic/lsubsv_snv.ma |   8 +-
 .../basic_2/dynamic/lsubsv_ssta.ma            |   6 +-
 .../lambdadelta/basic_2/dynamic/snv.ma        |  56 +++---
 .../lambdadelta/basic_2/dynamic/snv_aaa.ma    |   4 +-
 .../lambdadelta/basic_2/dynamic/snv_cpcs.ma   |  94 ++++-----
 .../lambdadelta/basic_2/dynamic/snv_lift.ma   |  10 +-
 .../lambdadelta/basic_2/dynamic/snv_lpr.ma    |   8 +-
 .../lambdadelta/basic_2/dynamic/snv_ssta.ma   |   6 +-
 .../basic_2/dynamic/snv_ssta_lpr.ma           |   6 +-
 .../lambdadelta/basic_2/dynamic/snv_sstas.ma  |   4 +-
 .../lambdadelta/basic_2/dynamic/ygt.ma        |  42 ++--
 .../lambdadelta/basic_2/dynamic/ypr.ma        |   4 +-
 .../lambdadelta/basic_2/dynamic/yprs.ma       |  34 ++--
 .../lambdadelta/basic_2/dynamic/ysc.ma        |  12 +-
 .../lambdadelta/basic_2/equivalence/cpcs.ma   |  28 +--
 .../basic_2/equivalence/cpcs_aaa.ma           |   4 +-
 .../basic_2/equivalence/cpcs_cpcs.ma          |  64 +++---
 .../basic_2/equivalence/cpcs_cprs.ma          |  18 +-
 .../lambdadelta/basic_2/grammar/cl_weight.ma  |  20 +-
 .../lambdadelta/basic_2/grammar/genv.ma       |   2 +-
 .../functions/{weight_2.ma => weight_3.ma}    |   4 +-
 .../lambdadelta/basic_2/notation/notation.ma  |   2 +-
 .../notation/relations/atomicarity_3.ma       |   2 +-
 .../basic_2/notation/relations/normal_2.ma    |   2 +-
 .../notation/relations/notreducible_2.ma      |   2 +-
 .../basic_2/notation/relations/pconv_3.ma     |   2 +-
 .../basic_2/notation/relations/pconvstar_3.ma |   2 +-
 .../basic_2/notation/relations/peval_3.ma     |   2 +-
 .../basic_2/notation/relations/pred_3.ma      |   2 +-
 .../basic_2/notation/relations/predstar_3.ma  |   2 +-
 .../basic_2/notation/relations/reducible_2.ma |   2 +-
 .../{suptermopt_4.ma => supterm_6.ma}         |   4 +-
 .../{suptermoptalt_4.ma => suptermopt_6.ma}   |   4 +-
 .../notation/relations/suptermoptalt_6.ma     |  19 ++
 .../{suptermplus_4.ma => suptermplus_6.ma}    |   4 +-
 .../notation/relations/suptermstar_4.ma       |  19 --
 .../{supterm_4.ma => suptermstar_6.ma}        |   4 +-
 .../lambdadelta/basic_2/reduction/cir.ma      |  28 +--
 .../basic_2/reduction/cir_append.ma           |   4 +-
 .../lambdadelta/basic_2/reduction/cir_lift.ma |   4 +-
 .../lambdadelta/basic_2/reduction/cix.ma      |  38 ++--
 .../basic_2/reduction/cix_append.ma           |   4 +-
 .../lambdadelta/basic_2/reduction/cix_lift.ma |  10 +-
 .../lambdadelta/basic_2/reduction/cnr.ma      |  22 +--
 .../lambdadelta/basic_2/reduction/cnr_cir.ma  |   4 +-
 .../lambdadelta/basic_2/reduction/cnr_crr.ma  |   2 +-
 .../lambdadelta/basic_2/reduction/cnr_lift.ma |   8 +-
 .../lambdadelta/basic_2/reduction/cnx.ma      |  38 ++--
 .../lambdadelta/basic_2/reduction/cnx_cix.ma  |   4 +-
 .../lambdadelta/basic_2/reduction/cnx_crx.ma  |   2 +-
 .../lambdadelta/basic_2/reduction/cnx_lift.ma |  10 +-
 .../lambdadelta/basic_2/reduction/cpr.ma      |  86 ++++-----
 .../lambdadelta/basic_2/reduction/cpr_cir.ma  |   2 +-
 .../lambdadelta/basic_2/reduction/cpx.ma      | 116 +++++------
 .../lambdadelta/basic_2/reduction/cpx_cix.ma  |   2 +-
 .../lambdadelta/basic_2/reduction/cpx_lift.ma |  16 +-
 .../lambdadelta/basic_2/reduction/crr.ma      |  26 +--
 .../basic_2/reduction/crr_append.ma           |   6 +-
 .../lambdadelta/basic_2/reduction/crr_lift.ma |   4 +-
 .../lambdadelta/basic_2/reduction/crx.ma      |  32 +--
 .../basic_2/reduction/crx_append.ma           |   4 +-
 .../lambdadelta/basic_2/reduction/crx_lift.ma |   8 +-
 .../lambdadelta/basic_2/reduction/lpr.ma      |   4 +-
 .../basic_2/reduction/lpr_ldrop.ma            |   2 +-
 .../lambdadelta/basic_2/reduction/lpr_lpr.ma  |  48 ++---
 .../lambdadelta/basic_2/reduction/lpx.ma      |  34 ++--
 .../lambdadelta/basic_2/reduction/lpx_aaa.ma  |  10 +-
 .../lambdadelta/basic_2/relocation/fsup.ma    |  51 ++---
 .../lambdadelta/basic_2/relocation/fsupq.ma   |  50 ++---
 .../basic_2/relocation/fsupq_alt.ma           |  24 +--
 .../basic_2/relocation/{gdrop.ma => gget.ma}  |  34 ++--
 .../{gdrop_gdrop.ma => gget_gget.ma}          |  18 +-
 .../lambdadelta/basic_2/static/aaa.ma         |  44 ++---
 .../lambdadelta/basic_2/static/aaa_aaa.ma     |   2 +-
 .../lambdadelta/basic_2/static/ssta.ma        |  46 ++---
 .../lambdadelta/basic_2/static/ssta_aaa.ma    |   2 +-
 .../lambdadelta/basic_2/static/ssta_lift.ma   |  12 +-
 .../lambdadelta/basic_2/static/ssta_ssta.ma   |   6 +-
 .../lambdadelta/basic_2/substitution/fsupp.ma |  81 ++++----
 .../basic_2/substitution/fsupp_fsupp.ma       |   4 +-
 .../lambdadelta/basic_2/substitution/fsups.ma |  68 +++----
 .../basic_2/substitution/fsups_fsups.ma       |   4 +-
 .../lambdadelta/basic_2/unfold/sstas.ma       |  26 +--
 .../lambdadelta/basic_2/unfold/sstas_aaa.ma   |   4 +-
 .../lambdadelta/basic_2/unfold/sstas_lift.ma  |  14 +-
 .../lambdadelta/basic_2/unfold/sstas_sstas.ma |  20 +-
 .../lambdadelta/basic_2/web/basic_2_src.tbl   |  16 +-
 .../contribs/lambdadelta/ground_2/star.ma     | 111 +++++++++--
 matita/matita/lib/arithmetics/nat.ma          |  13 ++
 matita/matita/lib/basics/relations.ma         |  23 ++-
 matita/matita/lib/basics/star.ma              | 182 ++++++++++++++----
 128 files changed, 1439 insertions(+), 1208 deletions(-)
 rename matita/matita/contribs/lambdadelta/basic_2/notation/functions/{weight_2.ma => weight_3.ma} (91%)
 rename matita/matita/contribs/lambdadelta/basic_2/notation/relations/{suptermopt_4.ma => supterm_6.ma} (85%)
 rename matita/matita/contribs/lambdadelta/basic_2/notation/relations/{suptermoptalt_4.ma => suptermopt_6.ma} (85%)
 create mode 100644 matita/matita/contribs/lambdadelta/basic_2/notation/relations/suptermoptalt_6.ma
 rename matita/matita/contribs/lambdadelta/basic_2/notation/relations/{suptermplus_4.ma => suptermplus_6.ma} (86%)
 delete mode 100644 matita/matita/contribs/lambdadelta/basic_2/notation/relations/suptermstar_4.ma
 rename matita/matita/contribs/lambdadelta/basic_2/notation/relations/{supterm_4.ma => suptermstar_6.ma} (86%)
 rename matita/matita/contribs/lambdadelta/basic_2/relocation/{gdrop.ma => gget.ma} (74%)
 rename matita/matita/contribs/lambdadelta/basic_2/relocation/{gdrop_gdrop.ma => gget_gget.ma} (73%)

diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/acp_aaa.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/acp_aaa.ma
index 77b705f97..168d5dfcc 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/acp_aaa.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/acp_aaa.ma
@@ -81,11 +81,11 @@ qed.
 
 (* Basic_1: was: sc3_arity *)
 lemma aacr_aaa: ∀RR,RS,RP. acp RR RS RP → acr RR RS RP (λL,T. RP L T) →
-                ∀L,T,A. L ⊢ T ⁝ A → ⦃L, T⦄ ϵ[RP] 〚A〛.
+                ∀L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ⦃L, T⦄ ϵ[RP] 〚A〛.
 /2 width=8/ qed.
 
 lemma acp_aaa: ∀RR,RS,RP. acp RR RS RP → acr RR RS RP (λL,T. RP L T) →
-               ∀L,T,A. L ⊢ T ⁝ A → RP L T.
+               ∀L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → RP L T.
 #RR #RS #RP #H1RP #H2RP #L #T #A #HT
 lapply (aacr_acr … H1RP H2RP A) #HA
 @(s1 … HA) /2 width=4/
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/cpds.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/cpds.ma
index 981809df8..b725a809a 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/cpds.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/cpds.ma
@@ -19,7 +19,7 @@ include "basic_2/computation/cprs.ma".
 (* DECOMPOSED EXTENDED PARALLEL COMPUTATION ON TERMS ************************)
 
 definition cpds: ∀h. sd h → lenv → relation term ≝ λh,g,L,T1,T2.
-                 ∃∃T. ⦃h, L⦄ ⊢ T1 •*[g] T & L ⊢ T ➡* T2.
+                 ∃∃T. ⦃G, L⦄ ⊢ T1 •*[h, g] T & ⦃G, L⦄ ⊢ T ➡* T2.
 
 interpretation "decomposed extended parallel computation (term)"
    'DPRedStar h g L T1 T2 = (cpds h g L T1 T2).
@@ -29,22 +29,22 @@ interpretation "decomposed extended parallel computation (term)"
 lemma cpds_refl: ∀h,g,L. reflexive … (cpds h g L).
 /2 width=3/ qed.
 
-lemma sstas_cpds: ∀h,g,L,T1,T2. ⦃h, L⦄ ⊢ T1 •*[g] T2 → ⦃h, L⦄ ⊢ T1 •*➡*[g] T2.
+lemma sstas_cpds: ∀h,g,L,T1,T2. ⦃G, L⦄ ⊢ T1 •*[h, g] T2 → ⦃G, L⦄ ⊢ T1 •*➡*[h, g] T2.
 /2 width=3/ qed.
 
-lemma cprs_cpds: ∀h,g,L,T1,T2.  L ⊢ T1 ➡* T2 → ⦃h, L⦄ ⊢ T1 •*➡*[g] T2.
+lemma cprs_cpds: ∀h,g,L,T1,T2.  ⦃G, L⦄ ⊢ T1 ➡* T2 → ⦃G, L⦄ ⊢ T1 •*➡*[h, g] T2.
 /2 width=3/ qed.
 
 lemma cpds_strap1: ∀h,g,L,T1,T,T2.
-                   ⦃h, L⦄ ⊢ T1 •*➡*[g] T → L ⊢ T ➡ T2 → ⦃h, L⦄ ⊢ T1 •*➡*[g] T2.
+                   ⦃G, L⦄ ⊢ T1 •*➡*[h, g] T → ⦃G, L⦄ ⊢ T ➡ T2 → ⦃G, L⦄ ⊢ T1 •*➡*[h, g] T2.
 #h #g #L #T1 #T #T2 * /3 width=5/
 qed.
 
 lemma cpds_strap2: ∀h,g,L,T1,T,T2,l.
-                   ⦃h, L⦄ ⊢ T1 •[g] ⦃l+1, T⦄ → ⦃h, L⦄ ⊢ T •*➡*[g] T2 → ⦃h, L⦄ ⊢ T1 •*➡*[g] T2.
+                   ⦃G, L⦄ ⊢ T1 •[h, g] ⦃l+1, T⦄ → ⦃G, L⦄ ⊢ T •*➡*[h, g] T2 → ⦃G, L⦄ ⊢ T1 •*➡*[h, g] T2.
 #h #g #L #T1 #T #T2 #l #HT1 * /3 width=4/
 qed.
 
-lemma ssta_cprs_cpds: ∀h,g,L,T1,T,T2,l. ⦃h, L⦄ ⊢ T1 •[g] ⦃l+1, T⦄ →
-                      L ⊢ T ➡* T2 → ⦃h, L⦄ ⊢ T1 •*➡*[g] T2.
+lemma ssta_cprs_cpds: ∀h,g,L,T1,T,T2,l. ⦃G, L⦄ ⊢ T1 •[h, g] ⦃l+1, T⦄ →
+                      ⦃G, L⦄ ⊢ T ➡* T2 → ⦃G, L⦄ ⊢ T1 •*➡*[h, g] T2.
 /3 width=3/ qed.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/cpds_aaa.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/cpds_aaa.ma
index ed3b092fc..7ab1b9b0b 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/cpds_aaa.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/cpds_aaa.ma
@@ -20,6 +20,6 @@ include "basic_2/computation/cpds.ma".
 
 (* Properties on atomic arity assignment for terms **************************)
 
-lemma cpds_aaa: ∀h,g,L,T,A. L ⊢ T ⁝ A → ∀U. ⦃h, L⦄ ⊢ T •*➡*[g] U → L ⊢ U ⁝ A.
+lemma cpds_aaa: ∀h,g,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀U. ⦃G, L⦄ ⊢ T •*➡*[h, g] U → ⦃G, L⦄ ⊢ U ⁝ A.
 #h #g #L #T #A #HT #U * /3 width=5 by sstas_aaa, aaa_cprs_conf/
 qed.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/cpds_cpds.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/cpds_cpds.ma
index ec3d035f0..f5b7792cd 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/cpds_cpds.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/cpds_cpds.ma
@@ -22,29 +22,29 @@ include "basic_2/computation/cpds.ma".
 (* Advanced properties ******************************************************)
 
 lemma cpds_cprs_trans: ∀h,g,L,T1,T,T2.
-                       ⦃h, L⦄ ⊢ T1 •*➡*[g] T → L ⊢ T ➡* T2 → ⦃h, L⦄ ⊢ T1 •*➡*[g] T2.
+                       ⦃G, L⦄ ⊢ T1 •*➡*[h, g] T → ⦃G, L⦄ ⊢ T ➡* T2 → ⦃G, L⦄ ⊢ T1 •*➡*[h, g] T2.
 #h #g #L #T1 #T #T2 * #T0 #HT10 #HT0 #HT2
 lapply (cprs_trans … HT0 … HT2) -T /2 width=3/
 qed-.
 
 lemma sstas_cpds_trans: ∀h,g,L,T1,T,T2.
-                        ⦃h, L⦄ ⊢ T1 •*[g] T → ⦃h, L⦄ ⊢ T •*➡*[g] T2 → ⦃h, L⦄ ⊢ T1 •*➡*[g] T2.
+                        ⦃G, L⦄ ⊢ T1 •*[h, g] T → ⦃G, L⦄ ⊢ T •*➡*[h, g] T2 → ⦃G, L⦄ ⊢ T1 •*➡*[h, g] T2.
 #h #g #L #T1 #T #T2 #HT1 * #T0 #HT0 #HT02
 lapply (sstas_trans … HT1 … HT0) -T /2 width=3/
 qed-.
 
 (* Advanced inversion lemmas ************************************************)
 
-lemma cpds_inv_abst1: ∀h,g,a,L,V1,T1,U2. ⦃h, L⦄ ⊢ ⓛ{a}V1. T1 •*➡*[g] U2 →
-                      ∃∃V2,T2. L ⊢ V1 ➡* V2 & ⦃h, L.ⓛV1⦄ ⊢ T1 •*➡*[g] T2 &
+lemma cpds_inv_abst1: ∀h,g,a,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓛ{a}V1. T1 •*➡*[h, g] U2 →
+                      ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡* V2 & ⦃h, L.ⓛV1⦄ ⊢ T1 •*➡*[h, g] T2 &
                                U2 = ⓛ{a}V2. T2.
 #h #g #a #L #V1 #T1 #U2 * #X #H1 #H2
 elim (sstas_inv_bind1 … H1) -H1 #U #HTU1 #H destruct
 elim (cprs_inv_abst1 … H2) -H2 #V2 #T2 #HV12 #HUT2 #H destruct /3 width=5/
 qed-.
 
-lemma cpds_inv_abbr_abst: ∀h,g,a1,a2,L,V1,W2,T1,T2. ⦃h, L⦄ ⊢ ⓓ{a1}V1.T1 •*➡*[g] ⓛ{a2}W2.T2 →
-                          ∃∃T. ⦃h, L.ⓓV1⦄ ⊢ T1 •*➡*[g] T & ⇧[0, 1] ⓛ{a2}W2.T2 ≡ T & a1 = true.
+lemma cpds_inv_abbr_abst: ∀h,g,a1,a2,L,V1,W2,T1,T2. ⦃G, L⦄ ⊢ ⓓ{a1}V1.T1 •*➡*[h, g] ⓛ{a2}W2.T2 →
+                          ∃∃T. ⦃h, L.ⓓV1⦄ ⊢ T1 •*➡*[h, g] T & ⇧[0, 1] ⓛ{a2}W2.T2 ≡ T & a1 = true.
 #h #g #a1 #a2 #L #V1 #W2 #T1 #T2 * #X #H1 #H2
 elim (sstas_inv_bind1 … H1) -H1 #U1 #HTU1 #H destruct
 elim (cprs_inv_abbr1 … H2) -H2 *
@@ -55,6 +55,6 @@ qed-.
 
 (* Advanced forward lemmas **************************************************)
 
-lemma cpds_fwd_cpxs: ∀h,g,L,T1,T2. ⦃h, L⦄ ⊢ T1 •*➡*[g] T2 → ⦃h, L⦄ ⊢ T1 ➡*[g] T2.
+lemma cpds_fwd_cpxs: ∀h,g,L,T1,T2. ⦃G, L⦄ ⊢ T1 •*➡*[h, g] T2 → ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2.
 #h #g #L #T1 #T2 * /3 width=3 by cpxs_trans, sstas_cpxs, cprs_cpxs/
 qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/cpds_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/cpds_lift.ma
index 1dd32f7a0..40edbb07f 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/cpds_lift.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/cpds_lift.ma
@@ -21,15 +21,15 @@ include "basic_2/computation/cpds.ma".
 (* Relocation properties ****************************************************)
 
 lemma cpds_lift: ∀L,K,d,e. ⇩[d, e] L ≡ K → ∀T1,U1. ⇧[d, e] T1 ≡ U1 →
-                 ∀h,g,T2. ⦃h, K⦄ ⊢ T1 •*➡*[g] T2 → ∀U2. ⇧[d, e] T2 ≡ U2 →
-                 ⦃h, L⦄ ⊢ U1 •*➡*[g] U2.
+                 ∀h,g,T2. ⦃h, K⦄ ⊢ T1 •*➡*[h, g] T2 → ∀U2. ⇧[d, e] T2 ≡ U2 →
+                 ⦃G, L⦄ ⊢ U1 •*➡*[h, g] U2.
 #L #K #d #e #HLK #T1 #U1 #HTU1 #h #g #T2 * #T
 elim (lift_total T d e) /3 width=11/
 qed.
 
 lemma cpds_inv_lift1: ∀L,K,d,e. ⇩[d, e] L ≡ K →
-                      ∀T1,U1. ⇧[d, e] T1 ≡ U1 → ∀h,g,U2. ⦃h, L⦄ ⊢ U1 •*➡*[g] U2 →
-                      ∃∃T2. ⇧[d, e] T2 ≡ U2 & ⦃h, K⦄ ⊢ T1 •*➡*[g] T2.
+                      ∀T1,U1. ⇧[d, e] T1 ≡ U1 → ∀h,g,U2. ⦃G, L⦄ ⊢ U1 •*➡*[h, g] U2 →
+                      ∃∃T2. ⇧[d, e] T2 ≡ U2 & ⦃h, K⦄ ⊢ T1 •*➡*[h, g] T2.
 #L #K #d #e #HLK #T1 #U1 #HTU1 #h #g #U2 * #U #HU1 #HU2
 elim (sstas_inv_lift1 … HU1 … HLK … HTU1) -U1 #T #HT1 #HTU
 elim (cprs_inv_lift1 … HU2 … HLK … HTU) -U -L /3 width=5/
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/cpre.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/cpre.ma
index a31e4023d..d6f6e3dce 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/cpre.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/cpre.ma
@@ -19,7 +19,7 @@ include "basic_2/computation/csn.ma".
 (* CONTEXT-SENSITIVE PARALLEL EVALUATION ON TERMS **************************)
 
 definition cpre: lenv → relation term ≝
-                 λL,T1,T2. L ⊢ T1 ➡* T2 ∧ L ⊢ 𝐍⦃T2⦄.
+                 λL,T1,T2. ⦃G, L⦄ ⊢ T1 ➡* T2 ∧ ⦃G, L⦄ ⊢ 𝐍⦃T2⦄.
 
 interpretation "context-sensitive parallel evaluation (term)"
    'PEval L T1 T2 = (cpre L T1 T2).
@@ -27,7 +27,7 @@ interpretation "context-sensitive parallel evaluation (term)"
 (* Basic_properties *********************************************************)
 
 (* Basic_1: was just: nf2_sn3 *)
-lemma csn_cpre: ∀h,g,L,T1. ⦃h, L⦄ ⊢ ⬊*[g] T1 → ∃T2. L ⊢ T1 ➡* 𝐍⦃T2⦄.
+lemma csn_cpre: ∀h,g,L,T1. ⦃G, L⦄ ⊢ ⬊*[h, g] T1 → ∃T2. ⦃G, L⦄ ⊢ T1 ➡* 𝐍⦃T2⦄.
 #h #g #L #T1 #H @(csn_ind … H) -T1
 #T1 #_ #IHT1
 elim (cnr_dec L T1) /3 width=3/
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/cpre_cpre.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/cpre_cpre.ma
index cac970e9a..353350192 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/cpre_cpre.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/cpre_cpre.ma
@@ -20,7 +20,7 @@ include "basic_2/computation/cpre.ma".
 (* Main properties *********************************************************)
 
 (* Basic_1: was: nf2_pr3_confluence *)
-theorem cpre_mono: ∀L,T,T1. L ⊢ T ➡* 𝐍⦃T1⦄ → ∀T2. L ⊢ T ➡* 𝐍⦃T2⦄ → T1 = T2.
+theorem cpre_mono: ∀L,T,T1. ⦃G, L⦄ ⊢ T ➡* 𝐍⦃T1⦄ → ∀T2. ⦃G, L⦄ ⊢ T ➡* 𝐍⦃T2⦄ → T1 = T2.
 #L #T #T1 * #H1T1 #H2T1 #T2 * #H1T2 #H2T2
 elim (cprs_conf … H1T1 … H1T2) -T #T #HT1
 >(cprs_inv_cnr1 … HT1 H2T1) -T1 #HT2
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/cprs.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/cprs.ma
index 1bcc38d80..e4af5fdfb 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/cprs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/cprs.ma
@@ -26,15 +26,15 @@ interpretation "context-sensitive parallel computation (term)"
 (* Basic eliminators ********************************************************)
 
 lemma cprs_ind: ∀L,T1. ∀R:predicate term. R T1 →
-                (∀T,T2. L ⊢ T1 ➡* T → L ⊢ T ➡ T2 → R T → R T2) →
-                ∀T2. L ⊢ T1 ➡* T2 → R T2.
+                (∀T,T2. ⦃G, L⦄ ⊢ T1 ➡* T → ⦃G, L⦄ ⊢ T ➡ T2 → R T → R T2) →
+                ∀T2. ⦃G, L⦄ ⊢ T1 ➡* T2 → R T2.
 #L #T1 #R #HT1 #IHT1 #T2 #HT12
 @(TC_star_ind … HT1 IHT1 … HT12) //
 qed-.
 
 lemma cprs_ind_dx: ∀L,T2. ∀R:predicate term. R T2 →
-                   (∀T1,T. L ⊢ T1 ➡ T → L ⊢ T ➡* T2 → R T → R T1) →
-                   ∀T1. L ⊢ T1 ➡* T2 → R T1.
+                   (∀T1,T. ⦃G, L⦄ ⊢ T1 ➡ T → ⦃G, L⦄ ⊢ T ➡* T2 → R T → R T1) →
+                   ∀T1. ⦃G, L⦄ ⊢ T1 ➡* T2 → R T1.
 #L #T2 #R #HT2 #IHT2 #T1 #HT12
 @(TC_star_ind_dx … HT2 IHT2 … HT12) //
 qed-.
@@ -42,20 +42,20 @@ qed-.
 (* Basic properties *********************************************************)
 
 (* Basic_1: was: pr3_pr2 *)
-lemma cpr_cprs: ∀L,T1,T2. L ⊢ T1 ➡ T2 → L ⊢ T1 ➡* T2.
+lemma cpr_cprs: ∀L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡ T2 → ⦃G, L⦄ ⊢ T1 ➡* T2.
 /2 width=1/ qed.
 
 (* Basic_1: was: pr3_refl *)
-lemma cprs_refl: ∀L,T. L ⊢ T ➡* T.
+lemma cprs_refl: ∀L,T. ⦃G, L⦄ ⊢ T ➡* T.
 /2 width=1/ qed.
 
 lemma cprs_strap1: ∀L,T1,T,T2.
-                   L ⊢ T1 ➡* T → L ⊢ T ➡ T2 → L ⊢ T1 ➡* T2.
+                   ⦃G, L⦄ ⊢ T1 ➡* T → ⦃G, L⦄ ⊢ T ➡ T2 → ⦃G, L⦄ ⊢ T1 ➡* T2.
 normalize /2 width=3/ qed.
 
 (* Basic_1: was: pr3_step *)
 lemma cprs_strap2: ∀L,T1,T,T2.
-                   L ⊢ T1 ➡ T → L ⊢ T ➡* T2 → L ⊢ T1 ➡* T2.
+                   ⦃G, L⦄ ⊢ T1 ➡ T → ⦃G, L⦄ ⊢ T ➡* T2 → ⦃G, L⦄ ⊢ T1 ➡* T2.
 normalize /2 width=3/ qed.
 
 lemma lsubr_cprs_trans: lsub_trans … cprs lsubr.
@@ -63,50 +63,50 @@ lemma lsubr_cprs_trans: lsub_trans … cprs lsubr.
 qed-.
 
 (* Basic_1: was: pr3_pr1 *)
-lemma tprs_cprs: ∀L,T1,T2. ⋆ ⊢ T1 ➡* T2 → L ⊢ T1 ➡* T2.
+lemma tprs_cprs: ∀L,T1,T2. ⋆ ⊢ T1 ➡* T2 → ⦃G, L⦄ ⊢ T1 ➡* T2.
 #L #T1 #T2 #H @(lsubr_cprs_trans … H) -H //
 qed.
 
-lemma cprs_bind_dx: ∀L,V1,V2. L ⊢ V1 ➡ V2 → ∀I,T1,T2. L. ⓑ{I}V1 ⊢ T1 ➡* T2 →
-                    ∀a. L ⊢ ⓑ{a,I}V1. T1 ➡* ⓑ{a,I}V2. T2.
+lemma cprs_bind_dx: ∀L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡ V2 → ∀I,T1,T2. L. ⓑ{I}V1 ⊢ T1 ➡* T2 →
+                    ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1. T1 ➡* ⓑ{a,I}V2. T2.
 #L #V1 #V2 #HV12 #I #T1 #T2 #HT12 #a @(cprs_ind_dx … HT12) -T1
 /3 width=1/ /3 width=3/
 qed.
 
 (* Basic_1: was only: pr3_thin_dx *)
-lemma cprs_flat_dx: ∀I,L,V1,V2. L ⊢ V1 ➡ V2 → ∀T1,T2. L ⊢ T1 ➡* T2 →
-                    L ⊢ ⓕ{I} V1. T1 ➡* ⓕ{I} V2. T2.
+lemma cprs_flat_dx: ∀I,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡ V2 → ∀T1,T2. ⦃G, L⦄ ⊢ T1 ➡* T2 →
+                    ⦃G, L⦄ ⊢ ⓕ{I} V1. T1 ➡* ⓕ{I} V2. T2.
 #I #L #V1 #V2 #HV12 #T1 #T2 #HT12 @(cprs_ind … HT12) -T2 /3 width=1/
 #T #T2 #_ #HT2 #IHT1
 @(cprs_strap1 … IHT1) -V1 -T1 /2 width=1/
 qed.
 
-lemma cprs_flat_sn: ∀I,L,T1,T2. L ⊢ T1 ➡ T2 → ∀V1,V2. L ⊢ V1 ➡* V2 →
-                    L ⊢ ⓕ{I} V1. T1 ➡* ⓕ{I} V2. T2.
+lemma cprs_flat_sn: ∀I,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡ T2 → ∀V1,V2. ⦃G, L⦄ ⊢ V1 ➡* V2 →
+                    ⦃G, L⦄ ⊢ ⓕ{I} V1. T1 ➡* ⓕ{I} V2. T2.
 #I #L #T1 #T2 #HT12 #V1 #V2 #H @(cprs_ind … H) -V2 /3 width=1/
 #V #V2 #_ #HV2 #IHV1
 @(cprs_strap1 … IHV1) -V1 -T1 /2 width=1/
 qed.
 
 lemma cprs_zeta: ∀L,V,T1,T,T2. ⇧[0, 1] T2 ≡ T →
-                 L.ⓓV ⊢ T1 ➡* T → L ⊢ +ⓓV.T1 ➡* T2.
+                 L.ⓓV ⊢ T1 ➡* T → ⦃G, L⦄ ⊢ +ⓓV.T1 ➡* T2.
 #L #V #T1 #T #T2 #HT2 #H @(TC_ind_dx … T1 H) -T1 /3 width=3/
 qed.
 
-lemma cprs_tau: ∀L,T1,T2. L ⊢ T1 ➡* T2 → ∀V. L ⊢ ⓝV.T1 ➡* T2.
+lemma cprs_tau: ∀L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡* T2 → ∀V. ⦃G, L⦄ ⊢ ⓝV.T1 ➡* T2.
 #L #T1 #T2 #H elim H -T2 /2 width=3/ /3 width=1/
 qed.
 
 lemma cprs_beta_dx: ∀a,L,V1,V2,W1,W2,T1,T2.
-                    L ⊢ V1 ➡ V2 → L ⊢ W1 ➡ W2 → L.ⓛW1 ⊢ T1 ➡* T2 →
-                    L ⊢ ⓐV1.ⓛ{a}W1.T1 ➡* ⓓ{a}ⓝW2.V2.T2.
+                    ⦃G, L⦄ ⊢ V1 ➡ V2 → ⦃G, L⦄ ⊢ W1 ➡ W2 → L.ⓛW1 ⊢ T1 ➡* T2 →
+                    ⦃G, L⦄ ⊢ ⓐV1.ⓛ{a}W1.T1 ➡* ⓓ{a}ⓝW2.V2.T2.
 #a #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 #HW12 * -T2 /3 width=1/
 /4 width=7 by cprs_strap1, cprs_bind_dx, cprs_flat_dx, cpr_beta/ (**) (* auto too slow without trace *)
 qed.
 
 lemma cprs_theta_dx: ∀a,L,V1,V,V2,W1,W2,T1,T2.
-                     L ⊢ V1 ➡ V → ⇧[0, 1] V ≡ V2 → L.ⓓW1 ⊢ T1 ➡* T2 →
-                     L ⊢ W1 ➡ W2 → L ⊢ ⓐV1.ⓓ{a}W1.T1 ➡* ⓓ{a}W2.ⓐV2.T2.
+                     ⦃G, L⦄ ⊢ V1 ➡ V → ⇧[0, 1] V ≡ V2 → L.ⓓW1 ⊢ T1 ➡* T2 →
+                     ⦃G, L⦄ ⊢ W1 ➡ W2 → ⦃G, L⦄ ⊢ ⓐV1.ⓓ{a}W1.T1 ➡* ⓓ{a}W2.ⓐV2.T2.
 #a #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 * -T2 [ /3 width=3/ ]
 /4 width=9 by cprs_strap1, cprs_bind_dx, cprs_flat_dx, cpr_theta/ (**) (* auto too slow without trace *)
 qed.
@@ -114,15 +114,15 @@ qed.
 (* Basic inversion lemmas ***************************************************)
 
 (* Basic_1: was: pr3_gen_sort *)
-lemma cprs_inv_sort1: ∀L,U2,k. L ⊢ ⋆k ➡* U2 → U2 = ⋆k.
+lemma cprs_inv_sort1: ∀L,U2,k. ⦃G, L⦄ ⊢ ⋆k ➡* U2 → U2 = ⋆k.
 #L #U2 #k #H @(cprs_ind … H) -U2 //
 #U2 #U #_ #HU2 #IHU2 destruct
 >(cpr_inv_sort1 … HU2) -HU2 //
 qed-.
 
 (* Basic_1: was: pr3_gen_cast *)
-lemma cprs_inv_cast1: ∀L,W1,T1,U2. L ⊢ ⓝW1.T1 ➡* U2 → L ⊢ T1 ➡* U2 ∨
-                      ∃∃W2,T2. L ⊢ W1 ➡* W2 & L ⊢ T1 ➡* T2 & U2 = ⓝW2.T2.
+lemma cprs_inv_cast1: ∀L,W1,T1,U2. ⦃G, L⦄ ⊢ ⓝW1.T1 ➡* U2 → ⦃G, L⦄ ⊢ T1 ➡* U2 ∨
+                      ∃∃W2,T2. ⦃G, L⦄ ⊢ W1 ➡* W2 & ⦃G, L⦄ ⊢ T1 ➡* T2 & U2 = ⓝW2.T2.
 #L #W1 #T1 #U2 #H @(cprs_ind … H) -U2 /3 width=5/
 #U2 #U #_ #HU2 * /3 width=3/ *
 #W #T #HW1 #HT1 #H destruct
@@ -131,7 +131,7 @@ elim (cpr_inv_cast1 … HU2) -HU2 /3 width=3/ *
 qed-.
 
 (* Basic_1: was: nf2_pr3_unfold *)
-lemma cprs_inv_cnr1: ∀L,T,U. L ⊢ T ➡* U → L ⊢ 𝐍⦃T⦄ → T = U.
+lemma cprs_inv_cnr1: ∀L,T,U. ⦃G, L⦄ ⊢ T ➡* U → ⦃G, L⦄ ⊢ 𝐍⦃T⦄ → T = U.
 #L #T #U #H @(cprs_ind_dx … H) -T //
 #T0 #T #H1T0 #_ #IHT #H2T0
 lapply (H2T0 … H1T0) -H1T0 #H destruct /2 width=1/
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 7e8723fb1..1941a33bb 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/cprs_cprs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/cprs_cprs.ma
@@ -29,48 +29,48 @@ theorem cprs_trans: ∀L. Transitive … (cprs L).
 theorem cprs_conf: ∀L. confluent2 … (cprs L) (cprs L).
 #L @TC_confluent2 /2 width=3 by cpr_conf/ qed-. (**) (* auto /3 width=3/ does not work because a δ-expansion gets in the way *)
 
-theorem cprs_bind: ∀a,I,L,V1,V2,T1,T2. L. ⓑ{I}V1 ⊢ T1 ➡* T2 → L ⊢ V1 ➡* V2 →
-                   L ⊢ ⓑ{a,I}V1. T1 ➡* ⓑ{a,I}V2. T2.
+theorem cprs_bind: ∀a,I,L,V1,V2,T1,T2. L. ⓑ{I}V1 ⊢ T1 ➡* T2 → ⦃G, L⦄ ⊢ V1 ➡* V2 →
+                   ⦃G, L⦄ ⊢ ⓑ{a,I}V1. T1 ➡* ⓑ{a,I}V2. T2.
 #a #I #L #V1 #V2 #T1 #T2 #HT12 #H @(cprs_ind … H) -V2 /2 width=1/
 #V #V2 #_ #HV2 #IHV1
 @(cprs_trans … IHV1) -V1 /2 width=1/
 qed.
 
 (* Basic_1: was: pr3_flat *)
-theorem cprs_flat: ∀I,L,V1,V2,T1,T2. L ⊢ T1 ➡* T2 → L ⊢ V1 ➡* V2 →
-                   L ⊢ ⓕ{I} V1. T1 ➡* ⓕ{I} V2. T2.
+theorem cprs_flat: ∀I,L,V1,V2,T1,T2. ⦃G, L⦄ ⊢ T1 ➡* T2 → ⦃G, L⦄ ⊢ V1 ➡* V2 →
+                   ⦃G, L⦄ ⊢ ⓕ{I} V1. T1 ➡* ⓕ{I} V2. T2.
 #I #L #V1 #V2 #T1 #T2 #HT12 #H @(cprs_ind … H) -V2 /2 width=1/
 #V #V2 #_ #HV2 #IHV1
 @(cprs_trans … IHV1) -IHV1 /2 width=1/
 qed.
 
 theorem cprs_beta_rc: ∀a,L,V1,V2,W1,W2,T1,T2.
-                      L ⊢ V1 ➡ V2 → L.ⓛW1 ⊢ T1 ➡* T2 → L ⊢ W1 ➡* W2 →
-                      L ⊢ ⓐV1.ⓛ{a}W1.T1 ➡* ⓓ{a}ⓝW2.V2.T2.
+                      ⦃G, L⦄ ⊢ V1 ➡ V2 → L.ⓛW1 ⊢ T1 ➡* T2 → ⦃G, L⦄ ⊢ W1 ➡* W2 →
+                      ⦃G, L⦄ ⊢ ⓐV1.ⓛ{a}W1.T1 ➡* ⓓ{a}ⓝW2.V2.T2.
 #a #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 #HT12 #H @(cprs_ind … H) -W2 /2 width=1/
 #W #W2 #_ #HW2 #IHW1
 @(cprs_trans … IHW1) -IHW1 /3 width=1/
 qed.
 
 theorem cprs_beta: ∀a,L,V1,V2,W1,W2,T1,T2.
-                   L.ⓛW1 ⊢ T1 ➡* T2 → L ⊢ W1 ➡* W2 → L ⊢ V1 ➡* V2 →
-                   L ⊢ ⓐV1.ⓛ{a}W1.T1 ➡* ⓓ{a}ⓝW2.V2.T2.
+                   L.ⓛW1 ⊢ T1 ➡* T2 → ⦃G, L⦄ ⊢ W1 ➡* W2 → ⦃G, L⦄ ⊢ V1 ➡* V2 →
+                   ⦃G, L⦄ ⊢ ⓐV1.ⓛ{a}W1.T1 ➡* ⓓ{a}ⓝW2.V2.T2.
 #a #L #V1 #V2 #W1 #W2 #T1 #T2 #HT12 #HW12 #H @(cprs_ind … H) -V2 /2 width=1/
 #V #V2 #_ #HV2 #IHV1
 @(cprs_trans … IHV1) -IHV1 /3 width=1/
 qed.
 
 theorem cprs_theta_rc: ∀a,L,V1,V,V2,W1,W2,T1,T2.
-                       L ⊢ V1 ➡ V → ⇧[0, 1] V ≡ V2 → L.ⓓW1 ⊢ T1 ➡* T2 →
-                       L ⊢ W1 ➡* W2 → L ⊢ ⓐV1.ⓓ{a}W1.T1 ➡* ⓓ{a}W2.ⓐV2.T2.
+                       ⦃G, L⦄ ⊢ V1 ➡ V → ⇧[0, 1] V ≡ V2 → L.ⓓW1 ⊢ T1 ➡* T2 →
+                       ⦃G, L⦄ ⊢ W1 ➡* W2 → ⦃G, L⦄ ⊢ ⓐV1.ⓓ{a}W1.T1 ➡* ⓓ{a}W2.ⓐV2.T2.
 #a #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 #HT12 #H elim H -W2 /2 width=3/
 #W #W2 #_ #HW2 #IHW1
 @(cprs_trans … IHW1) /2 width=1/
 qed.
 
 theorem cprs_theta: ∀a,L,V1,V,V2,W1,W2,T1,T2.
-                    ⇧[0, 1] V ≡ V2 → L ⊢ W1 ➡* W2 → L.ⓓW1 ⊢ T1 ➡* T2 →
-                    L ⊢ V1 ➡* V → L ⊢ ⓐV1.ⓓ{a}W1.T1 ➡* ⓓ{a}W2.ⓐV2.T2.
+                    ⇧[0, 1] V ≡ V2 → ⦃G, L⦄ ⊢ W1 ➡* W2 → L.ⓓW1 ⊢ T1 ➡* T2 →
+                    ⦃G, L⦄ ⊢ V1 ➡* V → ⦃G, L⦄ ⊢ ⓐV1.ⓓ{a}W1.T1 ➡* ⓓ{a}W2.ⓐV2.T2.
 #a #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV2 #HW12 #HT12 #H @(TC_ind_dx … V1 H) -V1 /2 width=3/
 #V1 #V0 #HV10 #_ #IHV0
 @(cprs_trans … IHV0) /2 width=1/
@@ -79,14 +79,14 @@ qed.
 (* Advanced inversion lemmas ************************************************)
 
 (* Basic_1: was pr3_gen_appl *)
-lemma cprs_inv_appl1: ∀L,V1,T1,U2. L ⊢ ⓐV1.T1 ➡* U2 →
-                      ∨∨ ∃∃V2,T2.       L ⊢ V1 ➡* V2 & L ⊢ T1 ➡* T2 &
+lemma cprs_inv_appl1: ∀L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓐV1.T1 ➡* U2 →
+                      ∨∨ ∃∃V2,T2.       ⦃G, L⦄ ⊢ V1 ➡* V2 & ⦃G, L⦄ ⊢ T1 ➡* T2 &
                                         U2 = ⓐV2. T2
-                       | ∃∃a,W,T.       L ⊢ T1 ➡* ⓛ{a}W.T &
-                                        L ⊢ ⓓ{a}ⓝW.V1.T ➡* U2
-                       | ∃∃a,V0,V2,V,T. L ⊢ V1 ➡* V0 & ⇧[0,1] V0 ≡ V2 &
-                                        L ⊢ T1 ➡* ⓓ{a}V.T &
-                                        L ⊢ ⓓ{a}V.ⓐV2.T ➡* U2.
+                       | ∃∃a,W,T.       ⦃G, L⦄ ⊢ T1 ➡* ⓛ{a}W.T &
+                                        ⦃G, L⦄ ⊢ ⓓ{a}ⓝW.V1.T ➡* U2
+                       | ∃∃a,V0,V2,V,T. ⦃G, L⦄ ⊢ V1 ➡* V0 & ⇧[0,1] V0 ≡ V2 &
+                                        ⦃G, L⦄ ⊢ T1 ➡* ⓓ{a}V.T &
+                                        ⦃G, L⦄ ⊢ ⓓ{a}V.ⓐV2.T ➡* U2.
 #L #V1 #T1 #U2 #H @(cprs_ind … H) -U2 [ /3 width=5/ ]
 #U #U2 #_ #HU2 * *
 [ #V0 #T0 #HV10 #HT10 #H destruct
@@ -128,8 +128,8 @@ lemma lpr_cpr_trans: s_r_trans … cpr lpr.
 ]
 qed-.
 
-lemma cpr_bind2: ∀L,V1,V2. L ⊢ V1 ➡ V2 → ∀I,T1,T2. L. ⓑ{I}V2 ⊢ T1 ➡ T2 →
-                 ∀a. L ⊢ ⓑ{a,I}V1. T1 ➡* ⓑ{a,I}V2. T2.
+lemma cpr_bind2: ∀L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡ V2 → ∀I,T1,T2. L. ⓑ{I}V2 ⊢ T1 ➡ T2 →
+                 ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1. T1 ➡* ⓑ{a,I}V2. T2.
 #L #V1 #V2 #HV12 #I #T1 #T2 #HT12
 lapply (lpr_cpr_trans … HT12 (L.ⓑ{I}V1) ?) /2 width=1/
 qed.
@@ -166,8 +166,8 @@ elim (cprs_lpr_conf_dx … HT01 … HL01) -HT01 #T #HT1
 lapply (lpr_cprs_trans … HT1 … HL01) -HT1 /2 width=3/
 qed-.
 
-lemma cprs_bind2_dx: ∀L,V1,V2. L ⊢ V1 ➡ V2 → ∀I,T1,T2. L. ⓑ{I}V2 ⊢ T1 ➡* T2 →
-                     ∀a. L ⊢ ⓑ{a,I}V1. T1 ➡* ⓑ{a,I}V2. T2.
+lemma cprs_bind2_dx: ∀L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡ V2 → ∀I,T1,T2. L. ⓑ{I}V2 ⊢ T1 ➡* T2 →
+                     ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1. T1 ➡* ⓑ{a,I}V2. T2.
 #L #V1 #V2 #HV12 #I #T1 #T2 #HT12
 lapply (lpr_cprs_trans … HT12 (L.ⓑ{I}V1) ?) /2 width=1/
 qed.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/cprs_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/cprs_lift.ma
index 7e75767b1..7e5961604 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/cprs_lift.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/cprs_lift.ma
@@ -22,7 +22,7 @@ include "basic_2/computation/cprs.ma".
 (* Note: apparently this was missing in basic_1 *)
 lemma cprs_delta: ∀L,K,V,V2,i.
                   ⇩[0, i] L ≡ K. ⓓV → K ⊢ V ➡* V2 →
-                  ∀W2. ⇧[0, i + 1] V2 ≡ W2 → L ⊢ #i ➡* W2.
+                  ∀W2. ⇧[0, i + 1] V2 ≡ W2 → ⦃G, L⦄ ⊢ #i ➡* W2.
 #L #K #V #V2 #i #HLK #H elim H -V2 [ /3 width=6/ ]
 #V1 #V2 #_ #HV12 #IHV1 #W2 #HVW2
 lapply (ldrop_fwd_ldrop2 … HLK) -HLK #HLK
@@ -32,7 +32,7 @@ qed.
 (* Advanced inversion lemmas ************************************************)
 
 (* Basic_1: was: pr3_gen_lref *)
-lemma cprs_inv_lref1: ∀L,T2,i. L ⊢ #i ➡* T2 →
+lemma cprs_inv_lref1: ∀L,T2,i. ⦃G, L⦄ ⊢ #i ➡* T2 →
                       T2 = #i ∨
                       ∃∃K,V1,T1. ⇩[0, i] L ≡ K. ⓓV1 &
                                  K ⊢ V1 ➡* T1 &
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/cpxe.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/cpxe.ma
index 0a150d198..fd875138c 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/cpxe.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/cpxe.ma
@@ -19,14 +19,14 @@ include "basic_2/computation/csn.ma".
 (* CONTEXT-SENSITIVE EXTENDED PARALLEL EVALUATION ON TERMS ******************)
 
 definition cpxe: ∀h. sd h → lenv → relation term ≝
-                 λh,g,L,T1,T2. ⦃h, L⦄ ⊢ T1 ➡*[g] T2 ∧ ⦃h, L⦄ ⊢ 𝐍[g]⦃T2⦄.
+                 λh,g,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 ∧ ⦃G, L⦄ ⊢ 𝐍[h, g]⦃T2⦄.
 
 interpretation "context-sensitive extended parallel evaluation (term)"
    'PEval h g L T1 T2 = (cpxe h g L T1 T2).
 
 (* Basic_properties *********************************************************)
 
-lemma csn_cpxe: ∀h,g,L,T1. ⦃h, L⦄ ⊢ ⬊*[g] T1 → ∃T2. ⦃h, L⦄ ⊢ T1 ➡*[g] 𝐍⦃T2⦄.
+lemma csn_cpxe: ∀h,g,L,T1. ⦃G, L⦄ ⊢ ⬊*[h, g] T1 → ∃T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] 𝐍⦃T2⦄.
 #h #g #L #T1 #H @(csn_ind … H) -T1
 #T1 #_ #IHT1
 elim (cnx_dec h g L T1) /3 width=3/
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs.ma
index 635aff910..ba5fd7c5a 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs.ma
@@ -28,97 +28,97 @@ interpretation "extended context-sensitive parallel computation (term)"
 (* Basic eliminators ********************************************************)
 
 lemma cpxs_ind: ∀h,g,L,T1. ∀R:predicate term. R T1 →
-                (∀T,T2. ⦃h, L⦄ ⊢ T1 ➡*[g] T → ⦃h, L⦄ ⊢ T ➡[g] T2 → R T → R T2) →
-                ∀T2. ⦃h, L⦄ ⊢ T1 ➡*[g] T2 → R T2.
+                (∀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 #T1 #R #HT1 #IHT1 #T2 #HT12
 @(TC_star_ind … HT1 IHT1 … HT12) //
 qed-.
 
 lemma cpxs_ind_dx: ∀h,g,L,T2. ∀R:predicate term. R T2 →
-                   (∀T1,T. ⦃h, L⦄ ⊢ T1 ➡[g] T → ⦃h, L⦄ ⊢ T ➡*[g] T2 → R T → R T1) →
-                   ∀T1. ⦃h, L⦄ ⊢ T1 ➡*[g] T2 → R T1.
+                   (∀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 #L #T2 #R #HT2 #IHT2 #T1 #HT12
 @(TC_star_ind_dx … HT2 IHT2 … HT12) //
 qed-.
 
 (* Basic properties *********************************************************)
 
-lemma cpxs_refl: ∀h,g,L,T. ⦃h, L⦄ ⊢ T ➡*[g] T.
+lemma cpxs_refl: ∀h,g,L,T. ⦃G, L⦄ ⊢ T ➡*[h, g] T.
 /2 width=1/ qed.
 
-lemma cpx_cpxs: ∀h,g,L,T1,T2. ⦃h, L⦄ ⊢ T1 ➡[g] T2 → ⦃h, L⦄ ⊢ T1 ➡*[g] T2.
+lemma cpx_cpxs: ∀h,g,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 → ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2.
 /2 width=1/ qed.
 
-lemma cpxs_strap1: ∀h,g,L,T1,T. ⦃h, L⦄ ⊢ T1 ➡*[g] T →
-                   ∀T2. ⦃h, L⦄ ⊢ T ➡[g] T2 → ⦃h, L⦄ ⊢ T1 ➡*[g] T2.
+lemma cpxs_strap1: ∀h,g,L,T1,T. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T →
+                   ∀T2. ⦃G, L⦄ ⊢ T ➡[h, g] T2 → ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2.
 normalize /2 width=3/ qed.
 
-lemma cpxs_strap2: ∀h,g,L,T1,T. ⦃h, L⦄ ⊢ T1 ➡[g] T →
-                   ∀T2. ⦃h, L⦄ ⊢ T ➡*[g] T2 → ⦃h, L⦄ ⊢ T1 ➡*[g] T2.
+lemma cpxs_strap2: ∀h,g,L,T1,T. ⦃G, L⦄ ⊢ T1 ➡[h, g] T →
+                   ∀T2. ⦃G, L⦄ ⊢ T ➡*[h, g] T2 → ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2.
 normalize /2 width=3/ qed.
 
 lemma lsubr_cpxs_trans: ∀h,g. lsub_trans … (cpxs h g) lsubr.
 /3 width=5 by lsubr_cpx_trans, TC_lsub_trans/
 qed-.
 
-lemma sstas_cpxs: ∀h,g,L,T1,T2. ⦃h, L⦄ ⊢ T1 •* [g] T2 → ⦃h, L⦄ ⊢ T1 ➡*[g] T2.
+lemma sstas_cpxs: ∀h,g,L,T1,T2. ⦃G, L⦄ ⊢ T1 •* [h, g] T2 → ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2.
 #h #g #L #T1 #T2 #H @(sstas_ind … H) -T2 //
 /3 width=4 by cpxs_strap1, ssta_cpx/
 qed.
 
-lemma cprs_cpxs: ∀h,g,L,T1,T2. L ⊢ T1 ➡* T2 → ⦃h, L⦄ ⊢ T1 ➡*[g] T2.
+lemma cprs_cpxs: ∀h,g,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡* T2 → ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2.
 #h #g #L #T1 #T2 #H @(cprs_ind … H) -T2 // /3 width=3/
 qed.
 
-lemma cpxs_bind_dx: ∀h,g,L,V1,V2. ⦃h, L⦄ ⊢ V1 ➡[g] V2 →
-                    ∀I,T1,T2. ⦃h, L. ⓑ{I}V1⦄ ⊢ T1 ➡*[g] T2 →
-                    ∀a. ⦃h, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡*[g] ⓑ{a,I}V2.T2.
+lemma cpxs_bind_dx: ∀h,g,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 →
+                    ∀I,T1,T2. ⦃h, L. ⓑ{I}V1⦄ ⊢ T1 ➡*[h, g] T2 →
+                    ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡*[h, g] ⓑ{a,I}V2.T2.
 #h #g #L #V1 #V2 #HV12 #I #T1 #T2 #HT12 #a @(cpxs_ind_dx … HT12) -T1
 /3 width=1/ /3 width=3/
 qed.
 
-lemma cpxs_flat_dx: ∀h,g,L,V1,V2. ⦃h, L⦄ ⊢ V1 ➡[g] V2 →
-                    ∀T1,T2. ⦃h, L⦄ ⊢ T1 ➡*[g] T2 →
-                    ∀I. ⦃h, L⦄ ⊢ ⓕ{I} V1. T1 ➡*[g] ⓕ{I} V2. T2.
+lemma cpxs_flat_dx: ∀h,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 #L #V1 #V2 #HV12 #T1 #T2 #HT12 @(cpxs_ind … HT12) -T2 /3 width=1/ /3 width=5/
 qed.
 
-lemma cpxs_flat_sn: ∀h,g,L,T1,T2. ⦃h, L⦄ ⊢ T1 ➡[g] T2 →
-                    ∀V1,V2. ⦃h, L⦄ ⊢ V1 ➡*[g] V2 →
-                    ∀I. ⦃h, L⦄ ⊢ ⓕ{I} V1. T1 ➡*[g] ⓕ{I} V2. T2.
+lemma cpxs_flat_sn: ∀h,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 #L #T1 #T2 #HT12 #V1 #V2 #H @(cpxs_ind … H) -V2 /3 width=1/ /3 width=5/
 qed.
 
 lemma cpxs_zeta: ∀h,g,L,V,T1,T,T2. ⇧[0, 1] T2 ≡ T →
-                 ⦃h, L.ⓓV⦄ ⊢ T1 ➡*[g] T → ⦃h, L⦄ ⊢ +ⓓV.T1 ➡*[g] T2.
+                 ⦃h, L.ⓓV⦄ ⊢ T1 ➡*[h, g] T → ⦃G, L⦄ ⊢ +ⓓV.T1 ➡*[h, g] T2.
 #h #g #L #V #T1 #T #T2 #HT2 #H @(TC_ind_dx … T1 H) -T1 /3 width=3/
 qed.
 
-lemma cpxs_tau: ∀h,g,L,T1,T2. ⦃h, L⦄ ⊢ T1 ➡*[g] T2 → ∀V. ⦃h, L⦄ ⊢ ⓝV.T1 ➡*[g] T2.
+lemma cpxs_tau: ∀h,g,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 → ∀V. ⦃G, L⦄ ⊢ ⓝV.T1 ➡*[h, g] T2.
 #h #g #L #T1 #T2 #H elim H -T2 /2 width=3/ /3 width=1/
 qed.
 
-lemma cpxs_ti: ∀h,g,L,V1,V2. ⦃h, L⦄ ⊢ V1 ➡*[g] V2 → ∀T. ⦃h, L⦄ ⊢ ⓝV1.T ➡*[g] V2.
+lemma cpxs_ti: ∀h,g,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡*[h, g] V2 → ∀T. ⦃G, L⦄ ⊢ ⓝV1.T ➡*[h, g] V2.
 #h #g #L #V1 #V2 #H elim H -V2 /2 width=3/ /3 width=1/
 qed.
 
 lemma cpxs_beta_dx: ∀h,g,a,L,V1,V2,W1,W2,T1,T2.
-                    ⦃h, L⦄ ⊢ V1 ➡[g] V2 → ⦃h, L.ⓛW1⦄ ⊢ T1 ➡*[g] T2 → ⦃h, L⦄ ⊢ W1 ➡[g] W2 →
-                    ⦃h, L⦄ ⊢ ⓐV1.ⓛ{a}W1.T1 ➡*[g] ⓓ{a}ⓝW2.V2.T2.
+                    ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 → ⦃h, 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 #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 * -T2 /3 width=1/
 /4 width=7 by cpxs_strap1, cpxs_bind_dx, cpxs_flat_dx, cpx_beta/ (**) (* auto too slow without trace *)
 qed.
 
 lemma cpxs_theta_dx: ∀h,g,a,L,V1,V,V2,W1,W2,T1,T2.
-                     ⦃h, L⦄ ⊢ V1 ➡[g] V → ⇧[0, 1] V ≡ V2 → ⦃h, L.ⓓW1⦄ ⊢ T1 ➡*[g] T2 →
-                     ⦃h, L⦄ ⊢ W1 ➡[g] W2 → ⦃h, L⦄ ⊢ ⓐV1.ⓓ{a}W1.T1 ➡*[g] ⓓ{a}W2.ⓐV2.T2.
+                     ⦃G, L⦄ ⊢ V1 ➡[h, g] V → ⇧[0, 1] V ≡ V2 → ⦃h, 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 #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 * -T2 [ /3 width=3/ ]
 /4 width=9 by cpxs_strap1, cpxs_bind_dx, cpxs_flat_dx, cpx_theta/ (**) (* auto too slow without trace *)
 qed.
 
 (* Basic inversion lemmas ***************************************************)
 
-lemma cpxs_inv_sort1: ∀h,g,L,U2,k. ⦃h, L⦄ ⊢ ⋆k ➡*[g] U2 →
+lemma cpxs_inv_sort1: ∀h,g,L,U2,k. ⦃G, L⦄ ⊢ ⋆k ➡*[h, g] U2 →
                       ∃∃n,l. deg h g k (n+l) & U2 = ⋆((next h)^n k).
 #h #g #L #U2 #k #H @(cpxs_ind … H) -U2
 [ elim (deg_total h g k) #l #Hkl
@@ -132,10 +132,10 @@ lemma cpxs_inv_sort1: ∀h,g,L,U2,k. ⦃h, L⦄ ⊢ ⋆k ➡*[g] U2 →
 ]
 qed-.
 
-lemma cpxs_inv_cast1: ∀h,g,L,W1,T1,U2. ⦃h, L⦄ ⊢ ⓝW1.T1 ➡*[g] U2 →
-                      ∨∨ ∃∃W2,T2. ⦃h, L⦄ ⊢ W1 ➡*[g] W2 & ⦃h, L⦄ ⊢ T1 ➡*[g] T2 & U2 = ⓝW2.T2
-                       | ⦃h, L⦄ ⊢ T1 ➡*[g] U2
-                       | ⦃h, L⦄ ⊢ W1 ➡*[g] U2.
+lemma cpxs_inv_cast1: ∀h,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 #L #W1 #T1 #U2 #H @(cpxs_ind … H) -U2 /3 width=5/
 #U2 #U #_ #HU2 * /3 width=3/ *
 #W #T #HW1 #HT1 #H destruct
@@ -145,7 +145,7 @@ lapply (cpxs_strap1 … HW1 … HW2) -W
 lapply (cpxs_strap1 … HT1 … HT2) -T /3 width=5/
 qed-.
 
-lemma cpxs_inv_cnx1: ∀h,g,L,T,U. ⦃h, L⦄ ⊢ T ➡*[g] U → ⦃h, L⦄ ⊢ 𝐍[g]⦃T⦄ → T = U.
+lemma cpxs_inv_cnx1: ∀h,g,L,T,U. ⦃G, L⦄ ⊢ T ➡*[h, g] U → ⦃G, L⦄ ⊢ 𝐍[h, g]⦃T⦄ → T = U.
 #h #g #L #T #U #H @(cpxs_ind_dx … H) -T //
 #T0 #T #H1T0 #_ #IHT #H2T0
 lapply (H2T0 … H1T0) -H1T0 #H destruct /2 width=1/
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 ed2329c54..5f377731a 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs_aaa.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs_aaa.ma
@@ -19,11 +19,11 @@ include "basic_2/computation/cpxs.ma".
 
 (* Properties about atomic arity assignment on terms ************************)
 
-lemma aaa_cpxs_conf: ∀h,g,L,T1,A. L ⊢ T1 ⁝ A →
-                     ∀T2. ⦃h, L⦄ ⊢ T1 ➡*[g] T2 → L ⊢ T2 ⁝ A.
+lemma aaa_cpxs_conf: ∀h,g,L,T1,A. ⦃G, L⦄ ⊢ T1 ⁝ A →
+                     ∀T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 → ⦃G, L⦄ ⊢ T2 ⁝ A.
 #h #g #L #T1 #A #HT1 #T2 #HT12
 @(TC_Conf3 … HT1 ? HT12) -A -T1 -T2 /2 width=5 by aaa_cpx_conf/
 qed-.
 
-lemma aaa_cprs_conf: ∀L,T1,A. L ⊢ T1 ⁝ A → ∀T2. L ⊢ T1 ➡* T2 → L ⊢ T2 ⁝ A.
+lemma aaa_cprs_conf: ∀L,T1,A. ⦃G, L⦄ ⊢ T1 ⁝ A → ∀T2. ⦃G, L⦄ ⊢ T1 ➡* T2 → ⦃G, L⦄ ⊢ T2 ⁝ A.
 /3 width=5 by aaa_cpxs_conf, cprs_cpxs/ 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 2399ca665..895832329 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs_cpxs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs_cpxs.ma
@@ -22,51 +22,51 @@ include "basic_2/computation/cpxs_lift.ma".
 theorem cpxs_trans: ∀h,g,L. Transitive … (cpxs h g L).
 #h #g #L #T1 #T #HT1 #T2 @trans_TC @HT1 qed-. (**) (* auto /3 width=3/ does not work because a δ-expansion gets in the way *)
 
-theorem cpxs_bind: ∀h,g,a,I,L,V1,V2,T1,T2. ⦃h, L.ⓑ{I}V1⦄ ⊢ T1 ➡*[g] T2 →
-                   ⦃h, L⦄ ⊢ V1 ➡*[g] V2 →
-                   ⦃h, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡*[g] ⓑ{a,I}V2.T2.
+theorem cpxs_bind: ∀h,g,a,I,L,V1,V2,T1,T2. ⦃h, 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 #L #V1 #V2 #T1 #T2 #HT12 #H @(cpxs_ind … H) -V2 /2 width=1/
 #V #V2 #_ #HV2 #IHV1
 @(cpxs_trans … IHV1) -V1 /2 width=1/
 qed.
 
-theorem cpxs_flat: ∀h,g,I,L,V1,V2,T1,T2. ⦃h, L⦄ ⊢ T1 ➡*[g] T2 →
-                   ⦃h, L⦄ ⊢ V1 ➡*[g] V2 →
-                   ⦃h, L⦄ ⊢ ⓕ{I} V1.T1 ➡*[g] ⓕ{I} V2.T2.
+theorem cpxs_flat: ∀h,g,I,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 #L #V1 #V2 #T1 #T2 #HT12 #H @(cpxs_ind … H) -V2 /2 width=1/
 #V #V2 #_ #HV2 #IHV1
 @(cpxs_trans … IHV1) -IHV1 /2 width=1/
 qed.
 
 theorem cpxs_beta_rc: ∀h,g,a,L,V1,V2,W1,W2,T1,T2.
-                      ⦃h, L⦄ ⊢ V1 ➡[g] V2 → ⦃h, L.ⓛW1⦄ ⊢ T1 ➡*[g] T2 → ⦃h, L⦄ ⊢ W1 ➡*[g] W2 →
-                      ⦃h, L⦄ ⊢ ⓐV1.ⓛ{a}W1.T1 ➡*[g] ⓓ{a}ⓝW2.V2.T2.
+                      ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 → ⦃h, 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 #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 #HT12 #H @(cpxs_ind … H) -W2 /2 width=1/
 #W #W2 #_ #HW2 #IHW1
 @(cpxs_trans … IHW1) -IHW1 /3 width=1/
 qed.
 
 theorem cpxs_beta: ∀h,g,a,L,V1,V2,W1,W2,T1,T2.
-                   ⦃h, L.ⓛW1⦄ ⊢ T1 ➡*[g] T2 → ⦃h, L⦄ ⊢ W1 ➡*[g] W2 → ⦃h, L⦄ ⊢ V1 ➡*[g] V2 →
-                   ⦃h, L⦄ ⊢ ⓐV1.ⓛ{a}W1.T1 ➡*[g] ⓓ{a}ⓝW2.V2.T2.
+                   ⦃h, 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 #L #V1 #V2 #W1 #W2 #T1 #T2 #HT12 #HW12 #H @(cpxs_ind … H) -V2 /2 width=1/
 #V #V2 #_ #HV2 #IHV1
 @(cpxs_trans … IHV1) -IHV1 /3 width=1/
 qed.
 
 theorem cpxs_theta_rc: ∀h,g,a,L,V1,V,V2,W1,W2,T1,T2.
-                       ⦃h, L⦄ ⊢ V1 ➡[g] V → ⇧[0, 1] V ≡ V2 →
-                       ⦃h, L.ⓓW1⦄ ⊢ T1 ➡*[g] T2 → ⦃h, L⦄ ⊢ W1 ➡*[g] W2 →
-                       ⦃h, L⦄ ⊢ ⓐV1.ⓓ{a}W1.T1 ➡*[g] ⓓ{a}W2.ⓐV2.T2.
+                       ⦃G, L⦄ ⊢ V1 ➡[h, g] V → ⇧[0, 1] V ≡ V2 →
+                       ⦃h, 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 #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 #HT12 #H elim H -W2 /2 width=3/
 #W #W2 #_ #HW2 #IHW1
 @(cpxs_trans … IHW1) -IHW1 /2 width=1/
 qed.
 
 theorem cpxs_theta: ∀h,g,a,L,V1,V,V2,W1,W2,T1,T2.
-                    ⇧[0, 1] V ≡ V2 → ⦃h, L⦄ ⊢ W1 ➡*[g] W2 →
-                    ⦃h, L.ⓓW1⦄ ⊢ T1 ➡*[g] T2 → ⦃h, L⦄ ⊢ V1 ➡*[g] V →
-                    ⦃h, L⦄ ⊢ ⓐV1.ⓓ{a}W1.T1 ➡*[g] ⓓ{a}W2.ⓐV2.T2.
+                    ⇧[0, 1] V ≡ V2 → ⦃G, L⦄ ⊢ W1 ➡*[h, g] W2 →
+                    ⦃h, 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 #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV2 #HW12 #HT12 #H @(TC_ind_dx … V1 H) -V1 /2 width=3/
 #V1 #V0 #HV10 #_ #IHV0
 @(cpxs_trans … IHV0) -IHV0 /2 width=1/
@@ -74,12 +74,12 @@ qed.
 
 (* Advanced inversion lemmas ************************************************)
 
-lemma cpxs_inv_appl1: ∀h,g,L,V1,T1,U2. ⦃h, L⦄ ⊢ ⓐV1.T1 ➡*[g] U2 →
-                      ∨∨ ∃∃V2,T2.       ⦃h, L⦄ ⊢ V1 ➡*[g] V2 & ⦃h, L⦄ ⊢ T1 ➡*[g] T2 &
+lemma cpxs_inv_appl1: ∀h,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 &
                                         U2 = ⓐV2. T2
-                       | ∃∃a,W,T.       ⦃h, L⦄ ⊢ T1 ➡*[g] ⓛ{a}W.T & ⦃h, L⦄ ⊢ ⓓ{a}ⓝW.V1.T ➡*[g] U2
-                       | ∃∃a,V0,V2,V,T. ⦃h, L⦄ ⊢ V1 ➡*[g] V0 & ⇧[0,1] V0 ≡ V2 &
-                                        ⦃h, L⦄ ⊢ T1 ➡*[g] ⓓ{a}V.T & ⦃h, L⦄ ⊢ ⓓ{a}V.ⓐV2.T ➡*[g] U2.
+                       | ∃∃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 #L #V1 #T1 #U2 #H @(cpxs_ind … H) -U2 [ /3 width=5/ ]
 #U #U2 #_ #HU2 * *
 [ #V0 #T0 #HV10 #HT10 #H destruct
@@ -120,9 +120,9 @@ lemma lpx_cpx_trans: ∀h,g. s_r_trans … (cpx h g) (lpx h g).
 ]
 qed-.
 
-lemma cpx_bind2: ∀h,g,L,V1,V2. ⦃h, L⦄ ⊢ V1 ➡[g] V2 →
-                 ∀I,T1,T2. ⦃h, L.ⓑ{I}V2⦄ ⊢ T1 ➡[g] T2 →
-                 ∀a. ⦃h, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡*[g] ⓑ{a,I}V2.T2.
+lemma cpx_bind2: ∀h,g,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 →
+                 ∀I,T1,T2. ⦃h, L.ⓑ{I}V2⦄ ⊢ T1 ➡[h, g] T2 →
+                 ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡*[h, g] ⓑ{a,I}V2.T2.
 #h #g #L #V1 #V2 #HV12 #I #T1 #T2 #HT12
 lapply (lpx_cpx_trans … HT12 (L.ⓑ{I}V1) ?) /2 width=1/
 qed.
@@ -132,9 +132,9 @@ qed.
 lemma lpx_cpxs_trans: ∀h,g. s_rs_trans … (cpx h g) (lpx h g).
 /3 width=5 by s_r_trans_TC1, lpx_cpx_trans/ qed-.
 
-lemma cpxs_bind2_dx: ∀h,g,L,V1,V2. ⦃h, L⦄ ⊢ V1 ➡[g] V2 →
-                     ∀I,T1,T2. ⦃h, L.ⓑ{I}V2⦄ ⊢ T1 ➡*[g] T2 →
-                     ∀a. ⦃h, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡*[g] ⓑ{a,I}V2.T2.
+lemma cpxs_bind2_dx: ∀h,g,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 →
+                     ∀I,T1,T2. ⦃h, L.ⓑ{I}V2⦄ ⊢ T1 ➡*[h, g] T2 →
+                     ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡*[h, g] ⓑ{a,I}V2.T2.
 #h #g #L #V1 #V2 #HV12 #I #T1 #T2 #HT12
 lapply (lpx_cpxs_trans … HT12 (L.ⓑ{I}V1) ?) /2 width=1/
 qed.
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 2e86e82f4..7bba480e4 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs_lift.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs_lift.ma
@@ -21,8 +21,8 @@ include "basic_2/computation/cpxs.ma".
 (* Advanced properties ******************************************************)
 
 lemma cpxs_delta: ∀h,g,I,L,K,V,V2,i.
-                  ⇩[0, i] L ≡ K. ⓑ{I}V → ⦃h, K⦄ ⊢ V ➡*[g] V2 →
-                  ∀W2. ⇧[0, i + 1] V2 ≡ W2 → ⦃h, L⦄ ⊢ #i ➡*[g] W2.
+                  ⇩[0, i] L ≡ K. ⓑ{I}V → ⦃h, K⦄ ⊢ V ➡*[h, g] V2 →
+                  ∀W2. ⇧[0, i + 1] V2 ≡ W2 → ⦃G, L⦄ ⊢ #i ➡*[h, g] W2.
 #h #g #I #L #K #V #V2 #i #HLK #H elim H -V2 [ /3 width=9/ ]
 #V1 #V2 #_ #HV12 #IHV1 #W2 #HVW2
 lapply (ldrop_fwd_ldrop2 … HLK) -HLK #HLK
@@ -31,9 +31,9 @@ qed.
 
 (* Advanced inversion lemmas ************************************************)
 
-lemma cpxs_inv_lref1: ∀h,g,L,T2,i. ⦃h, L⦄ ⊢ #i ➡*[g] T2 →
+lemma cpxs_inv_lref1: ∀h,g,L,T2,i. ⦃G, L⦄ ⊢ #i ➡*[h, g] T2 →
                       T2 = #i ∨
-                      ∃∃I,K,V1,T1. ⇩[0, i] L ≡ K.ⓑ{I}V1 & ⦃h, K⦄ ⊢ V1 ➡*[g] T1 &
+                      ∃∃I,K,V1,T1. ⇩[0, i] L ≡ K.ⓑ{I}V1 & ⦃h, K⦄ ⊢ V1 ➡*[h, g] T1 &
                                    ⇧[0, i + 1] T1 ≡ T2.
 #h #g #L #T2 #i #H @(cpxs_ind … H) -T2 /2 width=1/
 #T #T2 #_ #HT2 *
@@ -57,9 +57,9 @@ qed-.
 
 (* Properties on supclosure *************************************************)
 
-lemma fsupq_cpxs_trans: ∀h,g,L1,L2,T2,U2. ⦃h, L2⦄ ⊢ T2 ➡*[g] U2 →
+lemma fsupq_cpxs_trans: ∀h,g,L1,L2,T2,U2. ⦃h, L2⦄ ⊢ T2 ➡*[h, g] U2 →
                         ∀T1. ⦃L1, T1⦄ ⊃⸮ ⦃L2, T2⦄ →
-                        ∃∃U1. ⦃h, L1⦄ ⊢ T1 ➡*[g] U1 & ⦃L1, U1⦄ ⊃* ⦃L2, U2⦄.
+                        ∃∃U1. ⦃h, L1⦄ ⊢ T1 ➡*[h, g] U1 & ⦃L1, U1⦄ ⊃* ⦃L2, U2⦄.
 #h #g #L1 #L2 #T2 #U2 #H @(cpxs_ind_dx … H) -T2 [ /3 width=3/ ]
 #T #T2 #HT2 #_ #IHTU2 #T1 #HT1
 elim (fsupq_cpx_trans … HT1 … HT2) -T #T #HT1 #HT2
@@ -67,8 +67,8 @@ elim (IHTU2 … HT2) -T2 /3 width=3/
 qed-.
 
 lemma fsups_cpxs_trans: ∀h,g,L1,L2,T1,T2. ⦃L1, T1⦄ ⊃* ⦃L2, T2⦄ →
-                        ∀U2. ⦃h, L2⦄ ⊢ T2 ➡*[g] U2 →
-                        ∃∃U1. ⦃h, L1⦄ ⊢ T1 ➡*[g] U1 & ⦃L1, U1⦄ ⊃* ⦃L2, U2⦄.
+                        ∀U2. ⦃h, L2⦄ ⊢ T2 ➡*[h, g] U2 →
+                        ∃∃U1. ⦃h, L1⦄ ⊢ T1 ➡*[h, g] U1 & ⦃L1, U1⦄ ⊃* ⦃L2, U2⦄.
 #h #g #L1 #L2 #T1 #T2 #H @(fsups_ind … H) -L2 -T2 [ /2 width=3/ ]
 #L #L2 #T #T2 #_ #HT2 #IHT1 #U2 #HTU2
 elim (fsupq_cpxs_trans … HTU2 … HT2) -T2 #T2 #HT2 #HTU2
@@ -77,6 +77,6 @@ lapply (fsups_trans … HT2 … HTU2) -L -T2 /2 width=3/
 qed-.
 
 lemma fsup_ssta_trans: ∀h,g,L1,L2,T1,T2. ⦃L1, T1⦄ ⊃ ⦃L2, T2⦄ →
-                       ∀U2,l. ⦃h, L2⦄ ⊢ T2 •[g] ⦃l+1, U2⦄ →
-                       ∃∃U1. ⦃h, L1⦄ ⊢ T1 ➡[g] U1 & ⦃L1, U1⦄ ⊃⸮ ⦃L2, U2⦄.
+                       ∀U2,l. ⦃h, L2⦄ ⊢ T2 •[h, g] ⦃l+1, U2⦄ →
+                       ∃∃U1. ⦃h, L1⦄ ⊢ T1 ➡[h, g] U1 & ⦃L1, U1⦄ ⊃⸮ ⦃L2, U2⦄.
 /3 width=4 by fsup_cpx_trans, ssta_cpx/ qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs_tstc.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs_tstc.ma
index 87ae419aa..03566d556 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs_tstc.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs_tstc.ma
@@ -19,13 +19,13 @@ include "basic_2/computation/lpxs_cpxs.ma".
 
 (* Forward lemmas involving same top term constructor ***********************)
 
-lemma cpxs_fwd_cnx: ∀h,g,L,T. ⦃h, L⦄ ⊢ 𝐍[g]⦃T⦄ → ∀U. ⦃h, L⦄ ⊢ T ➡*[g] U → T ≃ U.
+lemma cpxs_fwd_cnx: ∀h,g,L,T. ⦃G, L⦄ ⊢ 𝐍[h, g]⦃T⦄ → ∀U. ⦃G, L⦄ ⊢ T ➡*[h, g] U → T ≃ U.
 #h #g #L #T #HT #U #H
 >(cpxs_inv_cnx1 … H HT) -L -T //
 qed-.
 
-lemma cpxs_fwd_sort: ∀h,g,L,U,k. ⦃h, L⦄ ⊢ ⋆k ➡*[g] U →
-                     ⋆k ≃ U ∨ ⦃h, L⦄ ⊢ ⋆(next h k) ➡*[g] U.
+lemma cpxs_fwd_sort: ∀h,g,L,U,k. ⦃G, L⦄ ⊢ ⋆k ➡*[h, g] U →
+                     ⋆k ≃ U ∨ ⦃G, L⦄ ⊢ ⋆(next h k) ➡*[h, g] U.
 #h #g #L #U #k #H
 elim (cpxs_inv_sort1 … H) -H #n #l generalize in match k; -k @(nat_ind_plus … n) -n
 [ #k #_ #H -l destruct /2 width=1/
@@ -41,8 +41,8 @@ elim (cpxs_inv_sort1 … H) -H #n #l generalize in match k; -k @(nat_ind_plus 
 qed-.
 
 (* Basic_1: was just: pr3_iso_beta *)
-lemma cpxs_fwd_beta: ∀h,g,a,L,V,W,T,U. ⦃h, L⦄ ⊢ ⓐV.ⓛ{a}W.T ➡*[g] U →
-                     ⓐV.ⓛ{a}W.T ≃ U ∨ ⦃h, L⦄ ⊢ ⓓ{a}ⓝW.V.T ➡*[g] U.
+lemma cpxs_fwd_beta: ∀h,g,a,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 #L #V #W #T #U #H
 elim (cpxs_inv_appl1 … H) -H *
 [ #V0 #T0 #_ #_ #H destruct /2 width=1/
@@ -58,8 +58,8 @@ qed-.
 (* Note: probably this is an inversion lemma *)
 lemma cpxs_fwd_delta: ∀h,g,I,L,K,V1,i. ⇩[0, i] L ≡ K.ⓑ{I}V1 →
                       ∀V2. ⇧[0, i + 1] V1 ≡ V2 →
-                      ∀U. ⦃h, L⦄ ⊢ #i ➡*[g] U →
-                      #i ≃ U ∨ ⦃h, L⦄ ⊢ V2 ➡*[g] U.
+                      ∀U. ⦃G, L⦄ ⊢ #i ➡*[h, g] U →
+                      #i ≃ U ∨ ⦃G, L⦄ ⊢ V2 ➡*[h, g] U.
 #h #g #I #L #K #V1 #i #HLK #V2 #HV12 #U #H
 elim (cpxs_inv_lref1 … H) -H /2 width=1/
 * #I0 #K0 #V0 #U0 #HLK0 #HVU0 #HU0
@@ -67,9 +67,9 @@ lapply (ldrop_mono … HLK0 … HLK) -HLK0 #H destruct
 lapply (ldrop_fwd_ldrop2 … HLK) -HLK /3 width=9/
 qed-.
 
-lemma cpxs_fwd_theta: ∀h,g,a,L,V1,V,T,U. ⦃h, L⦄ ⊢ ⓐV1.ⓓ{a}V.T ➡*[g] U →
+lemma cpxs_fwd_theta: ∀h,g,a,L,V1,V,T,U. ⦃G, L⦄ ⊢ ⓐV1.ⓓ{a}V.T ➡*[h, g] U →
                       ∀V2. ⇧[0, 1] V1 ≡ V2 → ⓐV1.ⓓ{a}V.T ≃ U ∨
-                      ⦃h, L⦄ ⊢ ⓓ{a}V.ⓐV2.T ➡*[g] U.
+                      ⦃G, L⦄ ⊢ ⓓ{a}V.ⓐV2.T ➡*[h, g] U.
 #h #g #a #L #V1 #V #T #U #H #V2 #HV12
 elim (cpxs_inv_appl1 … H) -H *
 [ -HV12 #V0 #T0 #_ #_ #H destruct /2 width=1/
@@ -99,8 +99,8 @@ elim (cpxs_inv_appl1 … H) -H *
 ]
 qed-.
 
-lemma cpxs_fwd_cast: ∀h,g,L,W,T,U. ⦃h, L⦄ ⊢ ⓝW.T ➡*[g] U →
-                     ∨∨ ⓝW. T ≃ U | ⦃h, L⦄ ⊢ T ➡*[g] U | ⦃h, L⦄ ⊢ W ➡*[g] U.
+lemma cpxs_fwd_cast: ∀h,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 #L #W #T #U #H
 elim (cpxs_inv_cast1 … H) -H /2 width=1/ *
 #W0 #T0 #_ #_ #H destruct /2 width=1/
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs_tstc_vector.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs_tstc_vector.ma
index 61f43dda6..a166abc43 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs_tstc_vector.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs_tstc_vector.ma
@@ -21,8 +21,8 @@ include "basic_2/computation/cpxs_tstc.ma".
 (* Vector form of forward lemmas involving same top term constructor ********)
 
 (* Basic_1: was just: nf2_iso_appls_lref *)
-lemma cpxs_fwd_cnx_vector: ∀h,g,L,T.  𝐒⦃T⦄ → ⦃h, L⦄ ⊢ 𝐍[g]⦃T⦄ →
-                           ∀Vs,U. ⦃h, L⦄ ⊢ ⒶVs.T ➡*[g] U → ⒶVs.T ≃ U.
+lemma cpxs_fwd_cnx_vector: ∀h,g,L,T.  𝐒⦃T⦄ → ⦃G, L⦄ ⊢ 𝐍[h, g]⦃T⦄ →
+                           ∀Vs,U. ⦃G, L⦄ ⊢ ⒶVs.T ➡*[h, g] U → ⒶVs.T ≃ U.
 #h #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 *
@@ -36,8 +36,8 @@ elim (cpxs_inv_appl1 … H) -H *
 ]
 qed-.
 
-lemma cpxs_fwd_sort_vector: ∀h,g,L,k,Vs,U. ⦃h, L⦄ ⊢ ⒶVs.⋆k ➡*[g] U →
-                            ⒶVs.⋆k ≃ U ∨ ⦃h, L⦄ ⊢ ⒶVs.⋆(next h k) ➡*[g] U.
+lemma cpxs_fwd_sort_vector: ∀h,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 #L #k #Vs elim Vs -Vs /2 width=1 by cpxs_fwd_sort/
 #V #Vs #IHVs #U #H
 elim (cpxs_inv_appl1 … H) -H *
@@ -61,8 +61,8 @@ qed-.
 
 
 (* Basic_1: was just: pr3_iso_appls_beta *)
-lemma cpxs_fwd_beta_vector: ∀h,g,a,L,Vs,V,W,T,U. ⦃h, L⦄ ⊢ ⒶVs.ⓐV.ⓛ{a}W.T ➡*[g] U →
-                            ⒶVs. ⓐV. ⓛ{a}W. T ≃ U ∨ ⦃h, L⦄ ⊢ ⒶVs.ⓓ{a}ⓝW.V.T ➡*[g] U.
+lemma cpxs_fwd_beta_vector: ∀h,g,a,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 #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 *
@@ -86,8 +86,8 @@ qed-.
 
 lemma cpxs_fwd_delta_vector: ∀h,g,I,L,K,V1,i. ⇩[0, i] L ≡ K.ⓑ{I}V1 →
                              ∀V2. ⇧[0, i + 1] V1 ≡ V2 →
-                             ∀Vs,U. ⦃h, L⦄ ⊢ ⒶVs.#i ➡*[g] U →
-                             ⒶVs.#i ≃ U ∨ ⦃h, L⦄ ⊢ ⒶVs.V2 ➡*[g] U.
+                             ∀Vs,U. ⦃G, L⦄ ⊢ ⒶVs.#i ➡*[h, g] U →
+                             ⒶVs.#i ≃ U ∨ ⦃G, L⦄ ⊢ ⒶVs.V2 ➡*[h, g] U.
 #h #g #I #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 *
@@ -111,8 +111,8 @@ qed-.
 
 (* Basic_1: was just: pr3_iso_appls_abbr *)
 lemma cpxs_fwd_theta_vector: ∀h,g,L,V1s,V2s. ⇧[0, 1] V1s ≡ V2s →
-                             ∀a,V,T,U. ⦃h, L⦄ ⊢ ⒶV1s.ⓓ{a}V.T ➡*[g] U →
-                             ⒶV1s. ⓓ{a}V. T ≃ U ∨ ⦃h, L⦄ ⊢ ⓓ{a}V.ⒶV2s.T ➡*[g] U.
+                             ∀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 #L #V1s #V2s * -V1s -V2s /3 width=1/
 #V1s #V2s #V1a #V2a #HV12a #HV12s #a
 generalize in match HV12a; -HV12a
@@ -159,10 +159,10 @@ elim (cpxs_inv_appl1 … H) -H *
 qed-.
 
 (* Basic_1: was just: pr3_iso_appls_cast *)
-lemma cpxs_fwd_cast_vector: ∀h,g,L,Vs,W,T,U. ⦃h, L⦄ ⊢ ⒶVs.ⓝW.T ➡*[g] U →
+lemma cpxs_fwd_cast_vector: ∀h,g,L,Vs,W,T,U. ⦃G, L⦄ ⊢ ⒶVs.ⓝW.T ➡*[h, g] U →
                             ∨∨ ⒶVs. ⓝW. T ≃ U
-                             | ⦃h, L⦄ ⊢ ⒶVs.T ➡*[g] U
-                             | ⦃h, L⦄ ⊢ ⒶVs.W ➡*[g] U.
+                             | ⦃G, L⦄ ⊢ ⒶVs.T ➡*[h, g] U
+                             | ⦃G, L⦄ ⊢ ⒶVs.W ➡*[h, g] U.
 #h #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 *
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/csn.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/csn.ma
index 9aead33d3..e9eca9679 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/csn.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/csn.ma
@@ -27,11 +27,11 @@ interpretation
 (* Basic eliminators ********************************************************)
 
 lemma csn_ind: ∀h,g,L. ∀R:predicate term.
-               (∀T1. ⦃h, L⦄ ⊢ ⬊*[g] T1 →
-                     (∀T2. ⦃h, L⦄ ⊢ T1 ➡[g] T2 → (T1 = T2 → ⊥) → R T2) →
+               (∀T1. ⦃G, L⦄ ⊢ ⬊*[h, g] T1 →
+                     (∀T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 → (T1 = T2 → ⊥) → R T2) →
                      R T1
                ) →
-               ∀T. ⦃h, L⦄ ⊢ ⬊*[g] T → R T.
+               ∀T. ⦃G, L⦄ ⊢ ⬊*[h, g] T → R T.
 #h #g #L #R #H0 #T1 #H elim H -T1 #T1 #HT1 #IHT1
 @H0 -H0 /3 width=1/ -IHT1 /4 width=1/
 qed-.
@@ -40,12 +40,12 @@ qed-.
 
 (* Basic_1: was just: sn3_pr2_intro *)
 lemma csn_intro: ∀h,g,L,T1.
-                 (∀T2. ⦃h, L⦄ ⊢ T1 ➡[g] T2 → (T1 = T2 → ⊥) → ⦃h, L⦄ ⊢ ⬊*[g] T2) →
-                 ⦃h, L⦄ ⊢ ⬊*[g] T1.
+                 (∀T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 → (T1 = T2 → ⊥) → ⦃G, L⦄ ⊢ ⬊*[h, g] T2) →
+                 ⦃G, L⦄ ⊢ ⬊*[h, g] T1.
 /4 width=1/ qed.
 
-lemma csn_cpx_trans: ∀h,g,L,T1. ⦃h, L⦄ ⊢ ⬊*[g] T1 →
-                     ∀T2. ⦃h, L⦄ ⊢ T1 ➡[g] T2 → ⦃h, L⦄ ⊢ ⬊*[g] T2.
+lemma csn_cpx_trans: ∀h,g,L,T1. ⦃G, L⦄ ⊢ ⬊*[h, g] T1 →
+                     ∀T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 → ⦃G, L⦄ ⊢ ⬊*[h, g] T2.
 #h #g #L #T1 #H elim H -T1 #T1 #HT1 #IHT1 #T2 #HLT12
 @csn_intro #T #HLT2 #HT2
 elim (term_eq_dec T1 T2) #HT12
@@ -54,10 +54,10 @@ elim (term_eq_dec T1 T2) #HT12
 qed-.
 
 (* Basic_1: was just: sn3_nf2 *)
-lemma cnx_csn: ∀h,g,L,T. ⦃h, L⦄ ⊢ 𝐍[g]⦃T⦄ → ⦃h, L⦄ ⊢ ⬊*[g] T.
+lemma cnx_csn: ∀h,g,L,T. ⦃G, L⦄ ⊢ 𝐍[h, g]⦃T⦄ → ⦃G, L⦄ ⊢ ⬊*[h, g] T.
 /2 width=1/ qed.
 
-lemma cnx_sort: ∀h,g,L,k. ⦃h, L⦄ ⊢ ⬊*[g] ⋆k.
+lemma cnx_sort: ∀h,g,L,k. ⦃G, L⦄ ⊢ ⬊*[h, g] ⋆k.
 #h #g #L #k elim (deg_total h g k)
 #l generalize in match k; -k @(nat_ind_plus … l) -l /3 width=1/
 #l #IHl #k #Hkl lapply (deg_next_SO … Hkl) -Hkl
@@ -68,8 +68,8 @@ lemma cnx_sort: ∀h,g,L,k. ⦃h, L⦄ ⊢ ⬊*[g] ⋆k.
 qed.
 
 (* Basic_1: was just: sn3_cast *)
-lemma csn_cast: ∀h,g,L,W. ⦃h, L⦄ ⊢ ⬊*[g] W →
-                ∀T. ⦃h, L⦄ ⊢ ⬊*[g] T → ⦃h, L⦄ ⊢ ⬊*[g] ⓝW.T.
+lemma csn_cast: ∀h,g,L,W. ⦃G, L⦄ ⊢ ⬊*[h, g] W →
+                ∀T. ⦃G, L⦄ ⊢ ⬊*[h, g] T → ⦃G, L⦄ ⊢ ⬊*[h, g] ⓝW.T.
 #h #g #L #W #HW @(csn_ind … HW) -W #W #HW #IHW #T #HT @(csn_ind … HT) -T #T #HT #IHT
 @csn_intro #X #H1 #H2
 elim (cpx_inv_cast1 … H1) -H1
@@ -84,37 +84,37 @@ qed.
 
 (* Basic forward lemmas *****************************************************)
 
-fact csn_fwd_pair_sn_aux: ∀h,g,L,U. ⦃h, L⦄ ⊢ ⬊*[g] U →
-                          ∀I,V,T. U = ②{I}V.T → ⦃h, L⦄ ⊢ ⬊*[g] V.
+fact csn_fwd_pair_sn_aux: ∀h,g,L,U. ⦃G, L⦄ ⊢ ⬊*[h, g] U →
+                          ∀I,V,T. U = ②{I}V.T → ⦃G, L⦄ ⊢ ⬊*[h, g] V.
 #h #g #L #U #H elim H -H #U0 #_ #IH #I #V #T #H destruct
 @csn_intro #V2 #HLV2 #HV2
 @(IH (②{I}V2.T)) -IH // /2 width=1/ -HLV2 #H destruct /2 width=1/
 qed-.
 
 (* Basic_1: was just: sn3_gen_head *)
-lemma csn_fwd_pair_sn: ∀h,g,I,L,V,T. ⦃h, L⦄ ⊢ ⬊*[g] ②{I}V.T → ⦃h, L⦄ ⊢ ⬊*[g] V.
+lemma csn_fwd_pair_sn: ∀h,g,I,L,V,T. ⦃G, L⦄ ⊢ ⬊*[h, g] ②{I}V.T → ⦃G, L⦄ ⊢ ⬊*[h, g] V.
 /2 width=5 by csn_fwd_pair_sn_aux/ qed-.
 
-fact csn_fwd_bind_dx_aux: ∀h,g,L,U. ⦃h, L⦄ ⊢ ⬊*[g] U →
-                          ∀a,I,V,T. U = ⓑ{a,I}V.T → ⦃h, L.ⓑ{I}V⦄ ⊢ ⬊*[g] T.
+fact csn_fwd_bind_dx_aux: ∀h,g,L,U. ⦃G, L⦄ ⊢ ⬊*[h, g] U →
+                          ∀a,I,V,T. U = ⓑ{a,I}V.T → ⦃h, L.ⓑ{I}V⦄ ⊢ ⬊*[h, g] T.
 #h #g #L #U #H elim H -H #U0 #_ #IH #a #I #V #T #H destruct
 @csn_intro #T2 #HLT2 #HT2
 @(IH (ⓑ{a,I} V. T2)) -IH // /2 width=1/ -HLT2 #H destruct /2 width=1/
 qed-.
 
 (* Basic_1: was just: sn3_gen_bind *)
-lemma csn_fwd_bind_dx: ∀h,g,a,I,L,V,T. ⦃h, L⦄ ⊢ ⬊*[g] ⓑ{a,I}V.T → ⦃h, L.ⓑ{I}V⦄ ⊢ ⬊*[g] T.
+lemma csn_fwd_bind_dx: ∀h,g,a,I,L,V,T. ⦃G, L⦄ ⊢ ⬊*[h, g] ⓑ{a,I}V.T → ⦃h, L.ⓑ{I}V⦄ ⊢ ⬊*[h, g] T.
 /2 width=4 by csn_fwd_bind_dx_aux/ qed-.
 
-fact csn_fwd_flat_dx_aux: ∀h,g,L,U. ⦃h, L⦄ ⊢ ⬊*[g] U →
-                          ∀I,V,T. U = ⓕ{I}V.T → ⦃h, L⦄ ⊢ ⬊*[g] T.
+fact csn_fwd_flat_dx_aux: ∀h,g,L,U. ⦃G, L⦄ ⊢ ⬊*[h, g] U →
+                          ∀I,V,T. U = ⓕ{I}V.T → ⦃G, L⦄ ⊢ ⬊*[h, g] T.
 #h #g #L #U #H elim H -H #U0 #_ #IH #I #V #T #H destruct
 @csn_intro #T2 #HLT2 #HT2
 @(IH (ⓕ{I}V.T2)) -IH // /2 width=1/ -HLT2 #H destruct /2 width=1/
 qed-.
 
 (* Basic_1: was just: sn3_gen_flat *)
-lemma csn_fwd_flat_dx: ∀h,g,I,L,V,T. ⦃h, L⦄ ⊢ ⬊*[g] ⓕ{I}V.T → ⦃h, L⦄ ⊢ ⬊*[g] T.
+lemma csn_fwd_flat_dx: ∀h,g,I,L,V,T. ⦃G, L⦄ ⊢ ⬊*[h, g] ⓕ{I}V.T → ⦃G, L⦄ ⊢ ⬊*[h, g] T.
 /2 width=5 by csn_fwd_flat_dx_aux/ qed-.
 
 (* Basic_1: removed theorems 14:
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/csn_aaa.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/csn_aaa.ma
index cd8d5a64b..f5229400f 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/csn_aaa.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/csn_aaa.ma
@@ -19,7 +19,7 @@ include "basic_2/computation/csn_tstc_vector.ma".
 
 (* Main properties concerning atomic arity assignment ***********************)
 
-theorem csn_aaa: ∀h,g,L,T,A. L ⊢ T ⁝ A → ⦃h, L⦄ ⊢ ⬊*[g] T.
+theorem csn_aaa: ∀h,g,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ⦃G, L⦄ ⊢ ⬊*[h, g] T.
 #h #g #L #T #A #H
 @(acp_aaa … (csn_acp h g) (csn_acr h g) … H)
 qed.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/csn_alt.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/csn_alt.ma
index 954cb9c01..07716de62 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/csn_alt.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/csn_alt.ma
@@ -29,10 +29,10 @@ interpretation
 (* Basic eliminators ********************************************************)
 
 lemma csna_ind: ∀h,g,L. ∀R:predicate term.
-                (∀T1. ⦃h, L⦄ ⊢ ⬊⬊*[g] T1 →
-                      (∀T2. ⦃h, L⦄ ⊢ T1 ➡*[g] T2 → (T1 = T2 → ⊥) → R T2) → R T1
+                (∀T1. ⦃G, L⦄ ⊢ ⬊⬊*[h, g] T1 →
+                      (∀T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 → (T1 = T2 → ⊥) → R T2) → R T1
                 ) →
-                ∀T. ⦃h, L⦄ ⊢ ⬊⬊*[g] T → R T.
+                ∀T. ⦃G, L⦄ ⊢ ⬊⬊*[h, g] T → R T.
 #h #g #L #R #H0 #T1 #H elim H -T1 #T1 #HT1 #IHT1
 @H0 -H0 /3 width=1/ -IHT1 /4 width=1/
 qed-.
@@ -41,18 +41,18 @@ qed-.
 
 (* Basic_1: was just: sn3_intro *)
 lemma csna_intro: ∀h,g,L,T1.
-                  (∀T2. ⦃h, L⦄ ⊢ T1 ➡*[g] T2 → (T1 = T2 → ⊥) → ⦃h, L⦄ ⊢ ⬊⬊*[g] T2) →
-                  ⦃h, L⦄ ⊢ ⬊⬊*[g] T1.
+                  (∀T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 → (T1 = T2 → ⊥) → ⦃G, L⦄ ⊢ ⬊⬊*[h, g] T2) →
+                  ⦃G, L⦄ ⊢ ⬊⬊*[h, g] T1.
 /4 width=1/ qed.
 
 fact csna_intro_aux: ∀h,g,L,T1. (
-                        ∀T,T2. ⦃h, L⦄ ⊢ T ➡*[g] T2 → T1 = T → (T1 = T2 → ⊥) → ⦃h, L⦄ ⊢ ⬊⬊*[g] T2
-                     ) → ⦃h, L⦄ ⊢ ⬊⬊*[g] T1.
+                        ∀T,T2. ⦃G, L⦄ ⊢ T ➡*[h, g] T2 → T1 = T → (T1 = T2 → ⊥) → ⦃G, L⦄ ⊢ ⬊⬊*[h, g] T2
+                     ) → ⦃G, L⦄ ⊢ ⬊⬊*[h, g] T1.
 /4 width=3/ qed-.
 
 (* Basic_1: was just: sn3_pr3_trans (old version) *)
-lemma csna_cpxs_trans: ∀h,g,L,T1. ⦃h, L⦄ ⊢ ⬊⬊*[g] T1 →
-                       ∀T2. ⦃h, L⦄ ⊢ T1 ➡*[g] T2 → ⦃h, L⦄ ⊢ ⬊⬊*[g] T2.
+lemma csna_cpxs_trans: ∀h,g,L,T1. ⦃G, L⦄ ⊢ ⬊⬊*[h, g] T1 →
+                       ∀T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 → ⦃G, L⦄ ⊢ ⬊⬊*[h, g] T2.
 #h #g #L #T1 #H elim H -T1 #T1 #HT1 #IHT1 #T2 #HLT12
 @csna_intro #T #HLT2 #HT2
 elim (term_eq_dec T1 T2) #HT12
@@ -62,8 +62,8 @@ qed.
 
 (* Basic_1: was just: sn3_pr2_intro (old version) *)
 lemma csna_intro_cpx: ∀h,g,L,T1. (
-                         ∀T2. ⦃h, L⦄ ⊢ T1 ➡[g] T2 → (T1 = T2 → ⊥) → ⦃h, L⦄ ⊢ ⬊⬊*[g] T2
-                      ) → ⦃h, L⦄ ⊢ ⬊⬊*[g] T1.
+                         ∀T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 → (T1 = T2 → ⊥) → ⦃G, L⦄ ⊢ ⬊⬊*[h, g] T2
+                      ) → ⦃G, L⦄ ⊢ ⬊⬊*[h, g] T1.
 #h #g #L #T1 #H
 @csna_intro_aux #T #T2 #H @(cpxs_ind_dx … H) -T
 [ -H #H destruct #H
@@ -78,27 +78,27 @@ qed.
 
 (* Main properties **********************************************************)
 
-theorem csn_csna: ∀h,g,L,T. ⦃h, L⦄ ⊢ ⬊*[g] T → ⦃h, L⦄ ⊢ ⬊⬊*[g] T.
+theorem csn_csna: ∀h,g,L,T. ⦃G, L⦄ ⊢ ⬊*[h, g] T → ⦃G, L⦄ ⊢ ⬊⬊*[h, g] T.
 #h #g #L #T #H @(csn_ind … H) -T /4 width=1/
 qed.
 
-theorem csna_csn: ∀h,g,L,T. ⦃h, L⦄ ⊢ ⬊⬊*[g] T → ⦃h, L⦄ ⊢ ⬊*[g] T.
+theorem csna_csn: ∀h,g,L,T. ⦃G, L⦄ ⊢ ⬊⬊*[h, g] T → ⦃G, L⦄ ⊢ ⬊*[h, g] T.
 #h #g #L #T #H @(csna_ind … H) -T /4 width=1/
 qed.
 
 (* Basic_1: was just: sn3_pr3_trans *)
-lemma csn_cpxs_trans: ∀h,g,L,T1. ⦃h, L⦄ ⊢ ⬊*[g] T1 →
-                      ∀T2. ⦃h, L⦄ ⊢ T1 ➡*[g] T2 → ⦃h, L⦄ ⊢ ⬊*[g] T2.
+lemma csn_cpxs_trans: ∀h,g,L,T1. ⦃G, L⦄ ⊢ ⬊*[h, g] T1 →
+                      ∀T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 → ⦃G, L⦄ ⊢ ⬊*[h, g] T2.
 #h #g #L #T1 #HT1 #T2 #H @(cpxs_ind … H) -T2 // /2 width=3 by csn_cpx_trans/
 qed-.
 
 (* Main eliminators *********************************************************)
 
 lemma csn_ind_alt: ∀h,g,L. ∀R:predicate term.
-                   (∀T1. ⦃h, L⦄ ⊢ ⬊*[g] T1 →
-                         (∀T2. ⦃h, L⦄ ⊢ T1 ➡*[g] T2 → (T1 = T2 → ⊥) → R T2) → R T1
+                   (∀T1. ⦃G, L⦄ ⊢ ⬊*[h, g] T1 →
+                         (∀T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 → (T1 = T2 → ⊥) → R T2) → R T1
                    ) →
-                   ∀T. ⦃h, L⦄ ⊢ ⬊*[g] T → R T.
+                   ∀T. ⦃G, L⦄ ⊢ ⬊*[h, g] T → R T.
 #h #g #L #R #H0 #T1 #H @(csna_ind … (csn_csna … H)) -T1 #T1 #HT1 #IHT1
 @H0 -H0 /2 width=1/ -HT1 /3 width=1/
 qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/csn_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/csn_lift.ma
index e57f37d9d..7a4513fb9 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/csn_lift.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/csn_lift.ma
@@ -21,8 +21,8 @@ include "basic_2/computation/csn.ma".
 (* Relocation properties ****************************************************)
 
 (* Basic_1: was just: sn3_lift *)
-lemma csn_lift: ∀h,g,L2,L1,T1,d,e. ⦃h, L1⦄ ⊢ ⬊*[g] T1 →
-                ∀T2. ⇩[d, e] L2 ≡ L1 → ⇧[d, e] T1 ≡ T2 → ⦃h, L2⦄ ⊢ ⬊*[g] T2.
+lemma csn_lift: ∀h,g,L2,L1,T1,d,e. ⦃h, L1⦄ ⊢ ⬊*[h, g] T1 →
+                ∀T2. ⇩[d, e] L2 ≡ L1 → ⇧[d, e] T1 ≡ T2 → ⦃h, L2⦄ ⊢ ⬊*[h, g] T2.
 #h #g #L2 #L1 #T1 #d #e #H elim H -T1 #T1 #_ #IHT1 #T2 #HL21 #HT12
 @csn_intro #T #HLT2 #HT2
 elim (cpx_inv_lift1 … HLT2 … HL21 … HT12) -HLT2 #T0 #HT0 #HLT10
@@ -31,8 +31,8 @@ elim (cpx_inv_lift1 … HLT2 … HL21 … HT12) -HLT2 #T0 #HT0 #HLT10
 qed.
 
 (* Basic_1: was just: sn3_gen_lift *)
-lemma csn_inv_lift: ∀h,g,L2,L1,T1,d,e. ⦃h, L1⦄ ⊢ ⬊*[g] T1 →
-                    ∀T2. ⇩[d, e] L1 ≡ L2 → ⇧[d, e] T2 ≡ T1 → ⦃h, L2⦄ ⊢ ⬊*[g] T2.
+lemma csn_inv_lift: ∀h,g,L2,L1,T1,d,e. ⦃h, L1⦄ ⊢ ⬊*[h, g] T1 →
+                    ∀T2. ⇩[d, e] L1 ≡ L2 → ⇧[d, e] T2 ≡ T1 → ⦃h, L2⦄ ⊢ ⬊*[h, g] T2.
 #h #g #L2 #L1 #T1 #d #e #H elim H -T1 #T1 #_ #IHT1 #T2 #HL12 #HT21
 @csn_intro #T #HLT2 #HT2
 elim (lift_total T d e) #T0 #HT0
@@ -44,7 +44,7 @@ qed.
 (* Advanced properties ******************************************************)
 
 (* Basic_1: was just: sn3_abbr *)
-lemma csn_lref_bind: ∀h,g,I,L,K,V,i. ⇩[0, i] L ≡ K.ⓑ{I}V → ⦃h, K⦄ ⊢ ⬊*[g] V → ⦃h, L⦄ ⊢ ⬊*[g] #i.
+lemma csn_lref_bind: ∀h,g,I,L,K,V,i. ⇩[0, i] L ≡ K.ⓑ{I}V → ⦃h, K⦄ ⊢ ⬊*[h, g] V → ⦃G, L⦄ ⊢ ⬊*[h, g] #i.
 #h #g #I #L #K #V #i #HLK #HV
 @csn_intro #X #H #Hi
 elim (cpx_inv_lref1 … H) -H
@@ -57,9 +57,9 @@ elim (cpx_inv_lref1 … H) -H
 ]
 qed.
 
-lemma csn_appl_simple: ∀h,g,L,V. ⦃h, L⦄ ⊢ ⬊*[g] V → ∀T1.
-                       (∀T2. ⦃h, L⦄ ⊢ T1 ➡[g] T2 → (T1 = T2 → ⊥) → ⦃h, L⦄ ⊢ ⬊*[g] ⓐV.T2) →
-                       𝐒⦃T1⦄ → ⦃h, L⦄ ⊢ ⬊*[g] ⓐV.T1.
+lemma csn_appl_simple: ∀h,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 #L #V #H @(csn_ind … H) -V #V #_ #IHV #T1 #IHT1 #HT1
 @csn_intro #X #H1 #H2
 elim (cpx_inv_appl1_simple … H1) // -H1
@@ -80,7 +80,7 @@ qed.
 
 (* Basic_1: was: sn3_gen_def *)
 lemma csn_inv_lref_bind: ∀h,g,I,L,K,V,i. ⇩[0, i] L ≡ K.ⓑ{I}V →
-                         ⦃h, L⦄ ⊢ ⬊*[g] #i → ⦃h, K⦄ ⊢ ⬊*[g] V.
+                         ⦃G, L⦄ ⊢ ⬊*[h, g] #i → ⦃h, K⦄ ⊢ ⬊*[h, g] V.
 #h #g #I #L #K #V #i #HLK #Hi
 elim (lift_total V 0 (i+1)) #V0 #HV0
 lapply (ldrop_fwd_ldrop2 … HLK) #H0LK
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/csn_lpx.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/csn_lpx.ma
index 3f97c882a..ca40d78e9 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/csn_lpx.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/csn_lpx.ma
@@ -21,15 +21,15 @@ include "basic_2/computation/csn_lift.ma".
 
 (* Advanced properties ******************************************************)
 
-lemma csn_lpx_conf: ∀h,g,L1,L2. ⦃h, L1⦄ ⊢ ➡[g] L2 →
-                    ∀T. ⦃h, L1⦄ ⊢ ⬊*[g] T → ⦃h, L2⦄ ⊢ ⬊*[g] T.
+lemma csn_lpx_conf: ∀h,g,L1,L2. ⦃h, L1⦄ ⊢ ➡[h, g] L2 →
+                    ∀T. ⦃h, L1⦄ ⊢ ⬊*[h, g] T → ⦃h, L2⦄ ⊢ ⬊*[h, g] T.
 #h #g #L1 #L2 #HL12 #T #H @(csn_ind_alt … H) -T #T #_ #IHT
 @csn_intro #T0 #HLT0 #HT0
 @IHT /2 width=2/ -IHT -HT0 /2 width=3 by lpx_cpx_trans/
 qed.
 
-lemma csn_abst: ∀h,g,a,L,W. ⦃h, L⦄ ⊢ ⬊*[g] W →
-                ∀T. ⦃h, L.ⓛW⦄ ⊢ ⬊*[g] T → ⦃h, L⦄ ⊢ ⬊*[g] ⓛ{a}W.T.
+lemma csn_abst: ∀h,g,a,L,W. ⦃G, L⦄ ⊢ ⬊*[h, g] W →
+                ∀T. ⦃h, L.ⓛW⦄ ⊢ ⬊*[h, g] T → ⦃G, L⦄ ⊢ ⬊*[h, g] ⓛ{a}W.T.
 #h #g #a #L #W #HW @(csn_ind … HW) -W #W #_ #IHW #T #HT @(csn_ind … HT) -T #T #HT #IHT
 @csn_intro #X #H1 #H2
 elim (cpx_inv_abst1 … H1) -H1
@@ -41,8 +41,8 @@ elim (eq_false_inv_tpair_sn … H2) -H2
 ]
 qed.
 
-lemma csn_abbr: ∀h,g,a,L,V. ⦃h, L⦄ ⊢ ⬊*[g] V →
-                ∀T. ⦃h, L.ⓓV⦄ ⊢ ⬊*[g] T → ⦃h, L⦄ ⊢ ⬊*[g] ⓓ{a}V. T.
+lemma csn_abbr: ∀h,g,a,L,V. ⦃G, L⦄ ⊢ ⬊*[h, g] V →
+                ∀T. ⦃h, L.ⓓV⦄ ⊢ ⬊*[h, g] T → ⦃G, L⦄ ⊢ ⬊*[h, g] ⓓ{a}V. T.
 #h #g #a #L #V #HV elim HV -V #V #_ #IHV #T #HT @(csn_ind_alt … HT) -T #T #HT #IHT
 @csn_intro #X #H1 #H2
 elim (cpx_inv_abbr1 … H1) -H1 *
@@ -58,8 +58,8 @@ elim (cpx_inv_abbr1 … H1) -H1 *
 ]
 qed.
 
-fact csn_appl_beta_aux: ∀h,g,a,L,U1. ⦃h, L⦄ ⊢ ⬊*[g] U1 →
-                        ∀V,W,T1. U1 = ⓓ{a}ⓝW.V.T1 → ⦃h, L⦄ ⊢ ⬊*[g] ⓐV.ⓛ{a}W.T1.
+fact csn_appl_beta_aux: ∀h,g,a,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 #L #X #H @(csn_ind … H) -X
 #X #HT1 #IHT1 #V #W #T1 #H1 destruct
 @csn_intro #X #H1 #H2
@@ -76,11 +76,11 @@ elim (cpx_inv_appl1 … H1) -H1 *
 qed-.
 
 (* Basic_1: was just: sn3_beta *)
-lemma csn_appl_beta: ∀h,g,a,L,V,W,T. ⦃h, L⦄ ⊢ ⬊*[g] ⓓ{a}ⓝW.V.T → ⦃h, L⦄ ⊢ ⬊*[g] ⓐV.ⓛ{a}W.T.
+lemma csn_appl_beta: ∀h,g,a,L,V,W,T. ⦃G, L⦄ ⊢ ⬊*[h, g] ⓓ{a}ⓝW.V.T → ⦃G, L⦄ ⊢ ⬊*[h, g] ⓐV.ⓛ{a}W.T.
 /2 width=3 by csn_appl_beta_aux/ qed.
 
-fact csn_appl_theta_aux: ∀h,g,a,L,U. ⦃h, L⦄ ⊢ ⬊*[g] U → ∀V1,V2. ⇧[0, 1] V1 ≡ V2 →
-                         ∀V,T. U = ⓓ{a}V.ⓐV2.T → ⦃h, L⦄ ⊢ ⬊*[g] ⓐV1.ⓓ{a}V.T.
+fact csn_appl_theta_aux: ∀h,g,a,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 #L #X #H @(csn_ind_alt … H) -X #X #HVT #IHVT #V1 #V2 #HV12 #V #T #H destruct
 lapply (csn_fwd_pair_sn … HVT) #HV
 lapply (csn_fwd_bind_dx … HVT) -HVT #HVT
@@ -117,13 +117,13 @@ elim (cpx_inv_appl1 … HL) -HL *
 qed-.
 
 lemma csn_appl_theta: ∀h,g,a,V1,V2. ⇧[0, 1] V1 ≡ V2 →
-                      ∀L,V,T. ⦃h, L⦄ ⊢ ⬊*[g] ⓓ{a}V.ⓐV2.T → ⦃h, L⦄ ⊢ ⬊*[g] ⓐV1.ⓓ{a}V.T.
+                      ∀L,V,T. ⦃G, L⦄ ⊢ ⬊*[h, g] ⓓ{a}V.ⓐV2.T → ⦃G, L⦄ ⊢ ⬊*[h, g] ⓐV1.ⓓ{a}V.T.
 /2 width=5 by csn_appl_theta_aux/ qed.
 
 (* Basic_1: was just: sn3_appl_appl *)
-lemma csn_appl_simple_tstc: ∀h,g,L,V. ⦃h, L⦄ ⊢ ⬊*[g] V → ∀T1. ⦃h, L⦄ ⊢ ⬊*[g] T1 →
-                            (∀T2. ⦃h, L⦄ ⊢ T1 ➡*[g] T2 → (T1 ≃ T2 → ⊥) → ⦃h, L⦄ ⊢ ⬊*[g] ⓐV.T2) →
-                            𝐒⦃T1⦄ → ⦃h, L⦄ ⊢ ⬊*[g] ⓐV.T1.
+lemma csn_appl_simple_tstc: ∀h,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 #L #V #H @(csn_ind … H) -V #V #_ #IHV #T1 #H @(csn_ind … H) -T1 #T1 #H1T1 #IHT1 #H2T1 #H3T1
 @csn_intro #X #HL #H
 elim (cpx_inv_appl1_simple … HL) -HL //
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/csn_tstc_vector.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/csn_tstc_vector.ma
index 123704979..4b19e3db2 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/csn_tstc_vector.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/csn_tstc_vector.ma
@@ -22,8 +22,8 @@ include "basic_2/computation/csn_vector.ma".
 (* Advanced properties ******************************************************)
 
 (* Basic_1: was just: sn3_appls_lref *)
-lemma csn_applv_cnx: ∀h,g,L,T. 𝐒⦃T⦄ → ⦃h, L⦄ ⊢ 𝐍[g]⦃T⦄ →
-                     ∀Vs. ⦃h, L⦄ ⊢ ⬊*[g] Vs → ⦃h, L⦄ ⊢ ⬊*[g] ⒶVs.T.
+lemma csn_applv_cnx: ∀h,g,L,T. 𝐒⦃T⦄ → ⦃G, L⦄ ⊢ 𝐍[h, g]⦃T⦄ →
+                     ∀Vs. ⦃G, L⦄ ⊢ ⬊*[h, g] Vs → ⦃G, L⦄ ⊢ ⬊*[h, g] ⒶVs.T.
 #h #g #L #T #H1T #H2T #Vs elim Vs -Vs [ #_ @(cnx_csn … H2T) ] (**) (* /2 width=1/ does not work *)
 #V #Vs #IHV #H
 elim (csnv_inv_cons … H) -H #HV #HVs
@@ -33,7 +33,7 @@ lapply (cpxs_fwd_cnx_vector … H) -H // -H1T -H2T #H
 elim (H0) -H0 //
 qed.
 
-lemma csn_applv_sort: ∀h,g,L,k,Vs. ⦃h, L⦄ ⊢ ⬊*[g] Vs → ⦃h, L⦄ ⊢ ⬊*[g] ⒶVs.⋆k.
+lemma csn_applv_sort: ∀h,g,L,k,Vs. ⦃G, L⦄ ⊢ ⬊*[h, g] Vs → ⦃G, L⦄ ⊢ ⬊*[h, g] ⒶVs.⋆k.
 #h #g #L #k elim (deg_total h g k)
 #l generalize in match k; -k @(nat_ind_plus … l) -l [ /3 width=1/ ]
 #l #IHl #k #Hkl lapply (deg_next_SO … Hkl) -Hkl
@@ -49,8 +49,8 @@ elim (cpxs_fwd_sort_vector … H) -H #H
 qed.
 
 (* Basic_1: was just: sn3_appls_beta *)
-lemma csn_applv_beta: ∀h,g,a,L,Vs,V,W,T. ⦃h, L⦄ ⊢ ⬊*[g] ⒶVs.ⓓ{a}ⓝW.V.T →
-                      ⦃h, L⦄ ⊢ ⬊*[g] ⒶVs. ⓐV.ⓛ{a}W.T.
+lemma csn_applv_beta: ∀h,g,a,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 #L #Vs elim Vs -Vs /2 width=1/
 #V0 #Vs #IHV #V #W #T #H1T
 lapply (csn_fwd_pair_sn … H1T) #HV0
@@ -65,7 +65,7 @@ qed.
 
 lemma csn_applv_delta: ∀h,g,I,L,K,V1,i. ⇩[0, i] L ≡ K.ⓑ{I}V1 →
                        ∀V2. ⇧[0, i + 1] V1 ≡ V2 →
-                       ∀Vs. ⦃h, L⦄ ⊢ ⬊*[g] (ⒶVs.V2) → ⦃h, L⦄ ⊢ ⬊*[g] (ⒶVs.#i).
+                       ∀Vs. ⦃G, L⦄ ⊢ ⬊*[h, g] (ⒶVs.V2) → ⦃G, L⦄ ⊢ ⬊*[h, g] (ⒶVs.#i).
 #h #g #I #L #K #V1 #i #HLK #V2 #HV12 #Vs elim Vs -Vs
 [ #H
   lapply (ldrop_fwd_ldrop2 … HLK) #HLK0
@@ -84,8 +84,8 @@ qed.
 
 (* Basic_1: was just: sn3_appls_abbr *)
 lemma csn_applv_theta: ∀h,g,a,L,V1s,V2s. ⇧[0, 1] V1s ≡ V2s →
-                       ∀V,T. ⦃h, L⦄ ⊢ ⬊*[g] ⓓ{a}V.ⒶV2s.T →
-                       ⦃h, L⦄ ⊢ ⬊*[g] ⒶV1s.ⓓ{a}V.T.
+                       ∀V,T. ⦃G, L⦄ ⊢ ⬊*[h, g] ⓓ{a}V.ⒶV2s.T →
+                       ⦃G, L⦄ ⊢ ⬊*[h, g] ⒶV1s.ⓓ{a}V.T.
 #h #g #a #L #V1s #V2s * -V1s -V2s /2 width=1/
 #V1s #V2s #V1 #V2 #HV12 #H
 generalize in match HV12; -HV12 generalize in match V2; -V2 generalize in match V1; -V1
@@ -102,8 +102,8 @@ elim (cpxs_fwd_theta_vector … (V2@V2s) … H1) -H1 /2 width=1/ -HV12s -HV12
 qed.
 
 (* Basic_1: was just: sn3_appls_cast *)
-lemma csn_applv_cast: ∀h,g,L,Vs,W,T. ⦃h, L⦄ ⊢ ⬊*[g] ⒶVs.W → ⦃h, L⦄ ⊢ ⬊*[g] ⒶVs.T →
-                      ⦃h, L⦄ ⊢ ⬊*[g] ⒶVs.ⓝW.T.
+lemma csn_applv_cast: ∀h,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 #L #Vs elim Vs -Vs /2 width=1/
 #V #Vs #IHV #W #T #H1W #H1T
 lapply (csn_fwd_pair_sn … H1W) #HV
@@ -118,7 +118,7 @@ elim (cpxs_fwd_cast_vector … H) -H #H
 ]
 qed.
 
-theorem csn_acr: ∀h,g. acr (cpx h g) (eq …) (csn h g) (λL,T. ⦃h, L⦄ ⊢ ⬊*[g] T).
+theorem csn_acr: ∀h,g. acr (cpx h g) (eq …) (csn h g) (λL,T. ⦃G, L⦄ ⊢ ⬊*[h, g] T).
 #h #g @mk_acr //
 [ /3 width=1/
 |2,3,6: /2 width=1/
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/csn_vector.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/csn_vector.ma
index c655be403..f29cedc04 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/csn_vector.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/csn_vector.ma
@@ -26,14 +26,14 @@ interpretation
 
 (* Basic inversion lemmas ***************************************************)
 
-lemma csnv_inv_cons: ∀h,g,L,T,Ts. ⦃h, L⦄ ⊢ ⬊*[g] T @ Ts →
-                     ⦃h, L⦄ ⊢ ⬊*[g] T ∧ ⦃h, L⦄ ⊢ ⬊*[g] Ts.
+lemma csnv_inv_cons: ∀h,g,L,T,Ts. ⦃G, L⦄ ⊢ ⬊*[h, g] T @ Ts →
+                     ⦃G, L⦄ ⊢ ⬊*[h, g] T ∧ ⦃G, L⦄ ⊢ ⬊*[h, g] Ts.
 normalize // qed-.
 
 (* Basic forward lemmas *****************************************************)
 
-lemma csn_fwd_applv: ∀h,g,L,T,Vs. ⦃h, L⦄ ⊢ ⬊*[g] Ⓐ Vs.T →
-                     ⦃h, L⦄ ⊢ ⬊*[g] Vs ∧ ⦃h, L⦄ ⊢ ⬊*[g] T.
+lemma csn_fwd_applv: ∀h,g,L,T,Vs. ⦃G, L⦄ ⊢ ⬊*[h, g] Ⓐ Vs.T →
+                     ⦃G, L⦄ ⊢ ⬊*[h, g] Vs ∧ ⦃G, L⦄ ⊢ ⬊*[h, g] T.
 #h #g #L #T #Vs elim Vs -Vs /2 width=1/
 #V #Vs #IHVs #HVs
 lapply (csn_fwd_pair_sn … HVs) #HV
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/lprs.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/lprs.ma
index 2916145ee..2e6a25167 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/lprs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/lprs.ma
@@ -26,14 +26,14 @@ interpretation "parallel computation (local environment, sn variant)"
 (* Basic eliminators ********************************************************)
 
 lemma lprs_ind: ∀L1. ∀R:predicate lenv. R L1 →
-                (∀L,L2. L1 ⊢ ➡* L → L ⊢ ➡ L2 → R L → R L2) →
+                (∀L,L2. L1 ⊢ ➡* L → ⦃G, L⦄ ⊢ ➡ L2 → R L → R L2) →
                 ∀L2. L1 ⊢ ➡* L2 → R L2.
 #L1 #R #HL1 #IHL1 #L2 #HL12
 @(TC_star_ind … HL1 IHL1 … HL12) //
 qed-.
 
 lemma lprs_ind_dx: ∀L2. ∀R:predicate lenv. R L2 →
-                   (∀L1,L. L1 ⊢ ➡ L → L ⊢ ➡* L2 → R L → R L1) →
+                   (∀L1,L. L1 ⊢ ➡ L → ⦃G, L⦄ ⊢ ➡* L2 → R L → R L1) →
                    ∀L1. L1 ⊢ ➡* L2 → R L1.
 #L2 #R #HL2 #IHL2 #L1 #HL12
 @(TC_star_ind_dx … HL2 IHL2 … HL12) //
@@ -44,13 +44,13 @@ qed-.
 lemma lpr_lprs: ∀L1,L2. L1 ⊢ ➡ L2 → L1 ⊢ ➡* L2.
 /2 width=1/ qed.
 
-lemma lprs_refl: ∀L. L ⊢ ➡* L.
+lemma lprs_refl: ∀L. ⦃G, L⦄ ⊢ ➡* L.
 /2 width=1/ qed.
 
-lemma lprs_strap1: ∀L1,L,L2. L1 ⊢ ➡* L → L ⊢ ➡ L2 → L1 ⊢ ➡* L2.
+lemma lprs_strap1: ∀L1,L,L2. L1 ⊢ ➡* L → ⦃G, L⦄ ⊢ ➡ L2 → L1 ⊢ ➡* L2.
 /2 width=3/ qed.
 
-lemma lprs_strap2: ∀L1,L,L2. L1 ⊢ ➡ L → L ⊢ ➡* L2 → L1 ⊢ ➡* L2.
+lemma lprs_strap2: ∀L1,L,L2. L1 ⊢ ➡ L → ⦃G, L⦄ ⊢ ➡* L2 → L1 ⊢ ➡* L2.
 /2 width=3/ qed.
 
 lemma lprs_pair_refl: ∀L1,L2. L1 ⊢ ➡* L2 → ∀I,V. L1. ⓑ{I} V ⊢ ➡* L2. ⓑ{I} V.
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 cc52394c4..c3ba54ef2 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/lprs_cprs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/lprs_cprs.ma
@@ -71,8 +71,8 @@ lemma lprs_cpr_conf_sn: ∀L0,T0,T1. L0 ⊢ T0 ➡ T1 → ∀L1. L0 ⊢ ➡* L1
                         ∃∃T. L0 ⊢ T1 ➡* T & L1 ⊢ T0 ➡* T.
 /3 width=3 by lprs_cprs_conf_sn, cpr_cprs/ qed-.
 
-lemma cprs_bind2: ∀L,V1,V2. L ⊢ V1 ➡* V2 → ∀I,T1,T2. L. ⓑ{I}V2 ⊢ T1 ➡* T2 →
-                  ∀a. L ⊢ ⓑ{a,I}V1. T1 ➡* ⓑ{a,I}V2. T2.
+lemma cprs_bind2: ∀L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡* V2 → ∀I,T1,T2. L. ⓑ{I}V2 ⊢ T1 ➡* T2 →
+                  ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1. T1 ➡* ⓑ{a,I}V2. T2.
 #L #V1 #V2 #HV12 #I #T1 #T2 #HT12
 lapply (lprs_cprs_trans … HT12 (L.ⓑ{I}V1) ?) /2 width=1/
 qed.
@@ -80,8 +80,8 @@ qed.
 (* Inversion lemmas on context-sensitive parallel computation for terms *****)
 
 (* Basic_1: was: pr3_gen_abst *)
-lemma cprs_inv_abst1: ∀a,L,W1,T1,U2. L ⊢ ⓛ{a}W1.T1 ➡* U2 →
-                      ∃∃W2,T2. L ⊢ W1 ➡* W2 & L.ⓛW1 ⊢ T1 ➡* T2 &
+lemma cprs_inv_abst1: ∀a,L,W1,T1,U2. ⦃G, L⦄ ⊢ ⓛ{a}W1.T1 ➡* U2 →
+                      ∃∃W2,T2. ⦃G, L⦄ ⊢ W1 ➡* W2 & L.ⓛW1 ⊢ T1 ➡* T2 &
                                U2 = ⓛ{a}W2.T2.
 #a #L #V1 #T1 #U2 #H @(cprs_ind … H) -U2 /2 width=5/
 #U0 #U2 #_ #HU02 * #V0 #T0 #HV10 #HT10 #H destruct
@@ -91,15 +91,15 @@ lapply (cprs_strap1 … HV10 … HV02) -V0
 lapply (cprs_trans … HT10 … HT02) -T0 /2 width=5/
 qed-.
 
-lemma cprs_inv_abst: ∀a,L,W1,W2,T1,T2. L ⊢ ⓛ{a}W1.T1 ➡* ⓛ{a}W2.T2 →
-                     L ⊢ W1 ➡* W2 ∧ L.ⓛW1 ⊢ T1 ➡* T2.
+lemma cprs_inv_abst: ∀a,L,W1,W2,T1,T2. ⦃G, L⦄ ⊢ ⓛ{a}W1.T1 ➡* ⓛ{a}W2.T2 →
+                     ⦃G, L⦄ ⊢ W1 ➡* W2 ∧ L.ⓛW1 ⊢ T1 ➡* T2.
 #a #L #W1 #W2 #T1 #T2 #H
 elim (cprs_inv_abst1 … H) -H #W #T #HW1 #HT1 #H destruct /2 width=1/
 qed-.
 
 (* Basic_1: was pr3_gen_abbr *)
-lemma cprs_inv_abbr1: ∀a,L,V1,T1,U2. L ⊢ ⓓ{a}V1.T1 ➡* U2 → (
-                      ∃∃V2,T2. L ⊢ V1 ➡* V2 & L. ⓓV1 ⊢ T1 ➡* T2 &
+lemma cprs_inv_abbr1: ∀a,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓓ{a}V1.T1 ➡* U2 → (
+                      ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡* V2 & L. ⓓV1 ⊢ T1 ➡* T2 &
                                U2 = ⓓ{a}V2.T2
                       ) ∨
                       ∃∃T2. L. ⓓV1 ⊢ T1 ➡* T2 & ⇧[0, 1] U2 ≡ T2 & a = true.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/lpxs.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/lpxs.ma
index 4cf6a28b4..c2d0eb2cf 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/lpxs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/lpxs.ma
@@ -26,48 +26,48 @@ interpretation "extended parallel computation (local environment, sn variant)"
 (* Basic eliminators ********************************************************)
 
 lemma lpxs_ind: ∀h,g,L1. ∀R:predicate lenv. R L1 →
-                (∀L,L2. ⦃h, L1⦄ ⊢ ➡*[g] L → ⦃h, L⦄ ⊢ ➡[g] L2 → R L → R L2) →
-                ∀L2. ⦃h, L1⦄ ⊢ ➡*[g] L2 → R L2.
+                (∀L,L2. ⦃h, L1⦄ ⊢ ➡*[h, g] L → ⦃G, L⦄ ⊢ ➡[h, g] L2 → R L → R L2) →
+                ∀L2. ⦃h, L1⦄ ⊢ ➡*[h, g] L2 → R L2.
 #h #g #L1 #R #HL1 #IHL1 #L2 #HL12
 @(TC_star_ind … HL1 IHL1 … HL12) //
 qed-.
 
 lemma lpxs_ind_dx: ∀h,g,L2. ∀R:predicate lenv. R L2 →
-                   (∀L1,L. ⦃h, L1⦄ ⊢ ➡[g] L → ⦃h, L⦄ ⊢ ➡*[g] L2 → R L → R L1) →
-                   ∀L1. ⦃h, L1⦄ ⊢ ➡*[g] L2 → R L1.
+                   (∀L1,L. ⦃h, L1⦄ ⊢ ➡[h, g] L → ⦃G, L⦄ ⊢ ➡*[h, g] L2 → R L → R L1) →
+                   ∀L1. ⦃h, L1⦄ ⊢ ➡*[h, g] L2 → R L1.
 #h #g #L2 #R #HL2 #IHL2 #L1 #HL12
 @(TC_star_ind_dx … HL2 IHL2 … HL12) //
 qed-.
 
 (* Basic properties *********************************************************)
 
-lemma lprs_lpxs: ∀h,g,L1,L2. L1 ⊢ ➡* L2 → ⦃h, L1⦄ ⊢ ➡*[g] L2.
+lemma lprs_lpxs: ∀h,g,L1,L2. L1 ⊢ ➡* L2 → ⦃h, L1⦄ ⊢ ➡*[h, g] L2.
 /3 width=3/ qed.
 
-lemma lpx_lpxs: ∀h,g,L1,L2. ⦃h, L1⦄ ⊢ ➡[g] L2 → ⦃h, L1⦄ ⊢ ➡*[g] L2.
+lemma lpx_lpxs: ∀h,g,L1,L2. ⦃h, L1⦄ ⊢ ➡[h, g] L2 → ⦃h, L1⦄ ⊢ ➡*[h, g] L2.
 /2 width=1/ qed.
 
-lemma lpxs_refl: ∀h,g,L. ⦃h, L⦄ ⊢ ➡*[g] L.
+lemma lpxs_refl: ∀h,g,L. ⦃G, L⦄ ⊢ ➡*[h, g] L.
 /2 width=1/ qed.
 
-lemma lpxs_strap1: ∀h,g,L1,L,L2. ⦃h, L1⦄ ⊢ ➡*[g] L → ⦃h, L⦄ ⊢ ➡[g] L2 → ⦃h, L1⦄ ⊢ ➡*[g] L2.
+lemma lpxs_strap1: ∀h,g,L1,L,L2. ⦃h, L1⦄ ⊢ ➡*[h, g] L → ⦃G, L⦄ ⊢ ➡[h, g] L2 → ⦃h, L1⦄ ⊢ ➡*[h, g] L2.
 /2 width=3/ qed.
 
-lemma lpxs_strap2: ∀h,g,L1,L,L2. ⦃h, L1⦄ ⊢ ➡[g] L → ⦃h, L⦄ ⊢ ➡*[g] L2 → ⦃h, L1⦄ ⊢ ➡*[g] L2.
+lemma lpxs_strap2: ∀h,g,L1,L,L2. ⦃h, L1⦄ ⊢ ➡[h, g] L → ⦃G, L⦄ ⊢ ➡*[h, g] L2 → ⦃h, L1⦄ ⊢ ➡*[h, g] L2.
 /2 width=3/ qed.
 
-lemma lpxs_pair_refl: ∀h,g,L1,L2. ⦃h, L1⦄ ⊢ ➡*[g] L2 → ∀I,V. ⦃h, L1. ⓑ{I}V⦄ ⊢ ➡*[g] L2. ⓑ{I}V.
+lemma lpxs_pair_refl: ∀h,g,L1,L2. ⦃h, L1⦄ ⊢ ➡*[h, g] L2 → ∀I,V. ⦃h, L1. ⓑ{I}V⦄ ⊢ ➡*[h, g] L2. ⓑ{I}V.
 /2 width=1 by TC_lpx_sn_pair_refl/ qed.
 
 (* Basic inversion lemmas ***************************************************)
 
-lemma lpxs_inv_atom1: ∀h,g,L2. ⦃h, ⋆⦄ ⊢ ➡*[g] L2 → L2 = ⋆.
+lemma lpxs_inv_atom1: ∀h,g,L2. ⦃h, ⋆⦄ ⊢ ➡*[h, g] L2 → L2 = ⋆.
 /2 width=2 by TC_lpx_sn_inv_atom1/ qed-.
 
-lemma lpxs_inv_atom2: ∀h,g,L1. ⦃h, L1⦄ ⊢ ➡*[g] ⋆ → L1 = ⋆.
+lemma lpxs_inv_atom2: ∀h,g,L1. ⦃h, L1⦄ ⊢ ➡*[h, g] ⋆ → L1 = ⋆.
 /2 width=2 by TC_lpx_sn_inv_atom2/ qed-.
 
 (* Basic forward lemmas *****************************************************)
 
-lemma lpxs_fwd_length: ∀h,g,L1,L2. ⦃h, L1⦄ ⊢ ➡*[g] L2 → |L1| = |L2|.
+lemma lpxs_fwd_length: ∀h,g,L1,L2. ⦃h, L1⦄ ⊢ ➡*[h, g] 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 b05385da5..e866102d4 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/lpxs_aaa.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/lpxs_aaa.ma
@@ -20,9 +20,9 @@ include "basic_2/computation/lpxs.ma".
 (* Properties about atomic arity assignment on terms ************************)
 
 lemma aaa_lpxs_conf: ∀h,g,L1,T,A.
-                     L1 ⊢ T ⁝ A → ∀L2. ⦃h, L1⦄ ⊢ ➡*[g] L2 → L2 ⊢ T ⁝ A.
+                     L1 ⊢ T ⁝ A → ∀L2. ⦃h, L1⦄ ⊢ ➡*[h, g] L2 → L2 ⊢ T ⁝ A.
 #h #g #L1 #T #A #HT #L2 #HL12
-@(TC_Conf3 … (λL,A. L ⊢ T ⁝ A) … HT ? HL12) /2 width=5 by aaa_lpx_conf/
+@(TC_Conf3 … (λL,A. ⦃G, L⦄ ⊢ T ⁝ A) … HT ? HL12) /2 width=5 by aaa_lpx_conf/
 qed-.
 
 lemma aaa_lprs_conf: ∀L1,T,A. L1 ⊢ T ⁝ A → ∀L2. L1 ⊢ ➡* L2 → L2 ⊢ T ⁝ A.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/lpxs_alt.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/lpxs_alt.ma
index 7a77b4b30..95a171de8 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/lpxs_alt.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/lpxs_alt.ma
@@ -26,12 +26,12 @@ interpretation "extended parallel computation (local environment, sn variant) al
 
 (* Main properties on the alternative definition ****************************)
 
-theorem lpxsa_lpxs: ∀h,g,L1,L2. ⦃h, L1⦄ ⊢ ➡➡*[g] L2 → ⦃h, L1⦄ ⊢ ➡*[g] L2.
+theorem lpxsa_lpxs: ∀h,g,L1,L2. ⦃h, L1⦄ ⊢ ➡➡*[h, g] L2 → ⦃h, L1⦄ ⊢ ➡*[h, g] L2.
 /2 width=1 by lpx_sn_LTC_TC_lpx_sn/ qed-.
 
 (* Main inversion lemmas on the alternative definition **********************)
 
-theorem lpxs_inv_lpxsa: ∀h,g,L1,L2. ⦃h, L1⦄ ⊢ ➡*[g] L2 → ⦃h, L1⦄ ⊢ ➡➡*[g] L2.
+theorem lpxs_inv_lpxsa: ∀h,g,L1,L2. ⦃h, L1⦄ ⊢ ➡*[h, g] L2 → ⦃h, L1⦄ ⊢ ➡➡*[h, g] L2.
 /3 width=3 by TC_lpx_sn_inv_lpx_sn_LTC, lpx_cpxs_trans/ qed-.
 
 (* Alternative eliminators **************************************************)
@@ -39,8 +39,8 @@ theorem lpxs_inv_lpxsa: ∀h,g,L1,L2. ⦃h, L1⦄ ⊢ ➡*[g] L2 → ⦃h, L1⦄
 lemma lpxs_ind_alt: ∀h,g. ∀R:relation lenv.
                     R (⋆) (⋆) → (
                        ∀I,K1,K2,V1,V2.
-                       ⦃h, K1⦄ ⊢ ➡*[g] K2 → ⦃h, K1⦄ ⊢ V1 ➡*[g] V2 →
+                       ⦃h, K1⦄ ⊢ ➡*[h, g] K2 → ⦃h, K1⦄ ⊢ V1 ➡*[h, g] V2 →
                        R K1 K2 → R (K1.ⓑ{I}V1) (K2.ⓑ{I}V2)
                     ) →
-                    ∀L1,L2. ⦃h, L1⦄ ⊢ ➡*[g] L2 → R L1 L2.
+                    ∀L1,L2. ⦃h, L1⦄ ⊢ ➡*[h, g] L2 → R L1 L2.
 /3 width=4 by TC_lpx_sn_ind, lpx_cpxs_trans/ 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 960d6778b..381e6e939 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/lpxs_cpxs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/lpxs_cpxs.ma
@@ -19,18 +19,18 @@ include "basic_2/computation/lpxs.ma".
 
 (* Advanced properties ******************************************************)
 
-lemma lpxs_pair: ∀h,g,I,L1,L2. ⦃h, L1⦄ ⊢ ➡*[g] L2 → ∀V1,V2. ⦃h, L1⦄ ⊢ V1 ➡*[g] V2 →
-                 ⦃h, L1.ⓑ{I}V1⦄ ⊢ ➡*[g] L2.ⓑ{I}V2.
+lemma lpxs_pair: ∀h,g,I,L1,L2. ⦃h, L1⦄ ⊢ ➡*[h, g] L2 → ∀V1,V2. ⦃h, L1⦄ ⊢ V1 ➡*[h, g] V2 →
+                 ⦃h, L1.ⓑ{I}V1⦄ ⊢ ➡*[h, g] L2.ⓑ{I}V2.
 /2 width=1 by TC_lpx_sn_pair/ qed.
 
 (* Advanced inversion lemmas ************************************************)
 
-lemma lpxs_inv_pair1: ∀h,g,I,K1,L2,V1. ⦃h, K1.ⓑ{I}V1⦄ ⊢ ➡*[g] L2 →
-                      ∃∃K2,V2. ⦃h, K1⦄ ⊢ ➡*[g] K2 & ⦃h, K1⦄ ⊢ V1 ➡*[g] V2 & L2 = K2.ⓑ{I}V2.
+lemma lpxs_inv_pair1: ∀h,g,I,K1,L2,V1. ⦃h, K1.ⓑ{I}V1⦄ ⊢ ➡*[h, g] L2 →
+                      ∃∃K2,V2. ⦃h, K1⦄ ⊢ ➡*[h, g] K2 & ⦃h, K1⦄ ⊢ V1 ➡*[h, g] 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,L1,K2,V2. ⦃h, L1⦄ ⊢ ➡*[g] K2.ⓑ{I}V2 →
-                      ∃∃K1,V1. ⦃h, K1⦄ ⊢ ➡*[g] K2 & ⦃h, K1⦄ ⊢ V1 ➡*[g] V2 & L1 = K1.ⓑ{I}V1.
+lemma lpxs_inv_pair2: ∀h,g,I,L1,K2,V2. ⦃h, L1⦄ ⊢ ➡*[h, g] K2.ⓑ{I}V2 →
+                      ∃∃K1,V1. ⦃h, K1⦄ ⊢ ➡*[h, g] K2 & ⦃h, K1⦄ ⊢ V1 ➡*[h, g] V2 & L1 = K1.ⓑ{I}V1.
 /3 width=3 by TC_lpx_sn_inv_pair2, lpx_cpxs_trans/ qed-.
 
 (* Properties on context-sensitive extended parallel computation for terms **)
@@ -41,17 +41,17 @@ lemma lpxs_cpx_trans: ∀h,g. s_r_trans … (cpx h g) (lpxs h g).
 lemma lpxs_cpxs_trans: ∀h,g. s_rs_trans … (cpx h g) (lpxs h g).
 /3 width=5 by s_r_trans_TC1, lpxs_cpx_trans/ qed-.
 
-lemma cpxs_bind2: ∀h,g,L,V1,V2. ⦃h, L⦄ ⊢ V1 ➡*[g] V2 →
-                  ∀I,T1,T2. ⦃h, L.ⓑ{I}V2⦄ ⊢ T1 ➡*[g] T2 →
-                  ∀a. ⦃h, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡*[g] ⓑ{a,I}V2.T2.
+lemma cpxs_bind2: ∀h,g,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡*[h, g] V2 →
+                  ∀I,T1,T2. ⦃h, L.ⓑ{I}V2⦄ ⊢ T1 ➡*[h, g] T2 →
+                  ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡*[h, g] ⓑ{a,I}V2.T2.
 #h #g #L #V1 #V2 #HV12 #I #T1 #T2 #HT12
 lapply (lpxs_cpxs_trans … HT12 (L.ⓑ{I}V1) ?) /2 width=1/
 qed.
 
 (* Inversion lemmas on context-sensitive ext parallel computation for terms *)
 
-lemma cpxs_inv_abst1: ∀h,g,a,L,V1,T1,U2. ⦃h, L⦄ ⊢ ⓛ{a}V1.T1 ➡*[g] U2 →
-                      ∃∃V2,T2. ⦃h, L⦄ ⊢ V1 ➡*[g] V2 & ⦃h, L.ⓛV1⦄ ⊢ T1 ➡*[g] T2 &
+lemma cpxs_inv_abst1: ∀h,g,a,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓛ{a}V1.T1 ➡*[h, g] U2 →
+                      ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡*[h, g] V2 & ⦃h, L.ⓛV1⦄ ⊢ T1 ➡*[h, g] T2 &
                                U2 = ⓛ{a}V2.T2.
 #h #g #a #L #V1 #T1 #U2 #H @(cpxs_ind … H) -U2 /2 width=5/
 #U0 #U2 #_ #HU02 * #V0 #T0 #HV10 #HT10 #H destruct
@@ -61,11 +61,11 @@ lapply (cpxs_strap1 … HV10 … HV02) -V0
 lapply (cpxs_trans … HT10 … HT02) -T0 /2 width=5/
 qed-.
 
-lemma cpxs_inv_abbr1: ∀h,g,a,L,V1,T1,U2. ⦃h, L⦄ ⊢ ⓓ{a}V1.T1 ➡*[g] U2 → (
-                      ∃∃V2,T2. ⦃h, L⦄ ⊢ V1 ➡*[g] V2 & ⦃h, L.ⓓV1⦄ ⊢ T1 ➡*[g] T2 &
+lemma cpxs_inv_abbr1: ∀h,g,a,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓓ{a}V1.T1 ➡*[h, g] U2 → (
+                      ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡*[h, g] V2 & ⦃h, L.ⓓV1⦄ ⊢ T1 ➡*[h, g] T2 &
                                U2 = ⓓ{a}V2.T2
                       ) ∨
-                      ∃∃T2. ⦃h, L.ⓓV1⦄ ⊢ T1 ➡*[g] T2 & ⇧[0, 1] U2 ≡ T2 & a = true.
+                      ∃∃T2. ⦃h, L.ⓓV1⦄ ⊢ T1 ➡*[h, g] T2 & ⇧[0, 1] U2 ≡ T2 & a = true.
 #h #g #a #L #V1 #T1 #U2 #H @(cpxs_ind … H) -U2 /3 width=5/
 #U0 #U2 #_ #HU02 * *
 [ #V0 #T0 #HV10 #HT10 #H destruct
@@ -86,6 +86,6 @@ qed-.
 
 (* More advanced properties *************************************************)
 
-lemma lpxs_pair2: ∀h,g,I,L1,L2. ⦃h, L1⦄ ⊢ ➡*[g] L2 →
-                  ∀V1,V2. ⦃h, L2⦄ ⊢ V1 ➡*[g] V2 → ⦃h, L1.ⓑ{I}V1⦄ ⊢ ➡*[g] L2.ⓑ{I}V2.
+lemma lpxs_pair2: ∀h,g,I,L1,L2. ⦃h, L1⦄ ⊢ ➡*[h, g] L2 →
+                  ∀V1,V2. ⦃h, L2⦄ ⊢ V1 ➡*[h, g] V2 → ⦃h, L1.ⓑ{I}V1⦄ ⊢ ➡*[h, g] L2.ⓑ{I}V2.
 /3 width=3 by lpxs_pair, lpxs_cpxs_trans/ qed.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/conversion/cpc.ma b/matita/matita/contribs/lambdadelta/basic_2/conversion/cpc.ma
index 9b78b55c9..5be5601ea 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/conversion/cpc.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/conversion/cpc.ma
@@ -18,7 +18,7 @@ include "basic_2/reduction/cpr.ma".
 (* CONTEXT-SENSITIVE PARALLEL CONVERSION ON TERMS ***************************)
 
 definition cpc: lenv → relation term ≝
-   λL,T1,T2. L ⊢ T1 ➡ T2 ∨ L ⊢ T2 ➡ T1.
+   λL,T1,T2. ⦃G, L⦄ ⊢ T1 ➡ T2 ∨ ⦃G, L⦄ ⊢ T2 ➡ T1.
 
 interpretation
    "context-sensitive parallel conversion (term)"
@@ -35,6 +35,6 @@ qed.
 
 (* Basic forward lemmas *****************************************************)
 
-lemma cpc_fwd_cpr: ∀L,T1,T2. L ⊢ T1 ⬌ T2 → ∃∃T. L ⊢ T1 ➡ T & L ⊢ T2 ➡ T.
+lemma cpc_fwd_cpr: ∀L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬌ T2 → ∃∃T. ⦃G, L⦄ ⊢ T1 ➡ T & ⦃G, L⦄ ⊢ T2 ➡ T.
 #L #T1 #T2 * /2 width=3/
 qed.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/conversion/cpc_cpc.ma b/matita/matita/contribs/lambdadelta/basic_2/conversion/cpc_cpc.ma
index dcea07a8b..092d2a9a7 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/conversion/cpc_cpc.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/conversion/cpc_cpc.ma
@@ -18,6 +18,6 @@ include "basic_2/conversion/cpc.ma".
 
 (* Main properties **********************************************************)
 
-theorem cpc_conf: ∀L,T0,T1,T2. L ⊢ T0 ⬌ T1 → L ⊢ T0 ⬌ T2 →
-                  ∃∃T. L ⊢ T1 ⬌ T & L ⊢ T2 ⬌ T.
+theorem cpc_conf: ∀L,T0,T1,T2. ⦃G, L⦄ ⊢ T0 ⬌ T1 → ⦃G, L⦄ ⊢ T0 ⬌ T2 →
+                  ∃∃T. ⦃G, L⦄ ⊢ T1 ⬌ T & ⦃G, L⦄ ⊢ T2 ⬌ T.
 /3 width=3/ qed.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/lsubsv.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/lsubsv.ma
index 8d4e9b67a..29d6151fe 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/dynamic/lsubsv.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/lsubsv.ma
@@ -22,8 +22,8 @@ inductive lsubsv (h:sh) (g:sd h): relation lenv ≝
 | lsubsv_atom: lsubsv h g (⋆) (⋆)
 | lsubsv_pair: ∀I,L1,L2,V. lsubsv h g L1 L2 →
                lsubsv h g (L1.ⓑ{I}V) (L2.ⓑ{I}V)
-| lsubsv_abbr: ∀L1,L2,W,V,W1,V2,l. ⦃h, L1⦄ ⊢ ⓝW.V ¡[g] → ⦃h, L2⦄ ⊢ W ¡[g] →
-               ⦃h, L1⦄ ⊢ V •[g] ⦃l+1, W1⦄ → ⦃h, L2⦄ ⊢ W •[g] ⦃l, V2⦄ →
+| lsubsv_abbr: ∀L1,L2,W,V,W1,V2,l. ⦃h, L1⦄ ⊢ ⓝW.V ¡[h, g] → ⦃h, L2⦄ ⊢ W ¡[h, g] →
+               ⦃h, L1⦄ ⊢ V •[h, g] ⦃l+1, W1⦄ → ⦃h, L2⦄ ⊢ W •[h, g] ⦃l, V2⦄ →
                lsubsv h g L1 L2 → lsubsv h g (L1.ⓓⓝW.V) (L2.ⓛW)
 .
 
@@ -33,7 +33,7 @@ interpretation
 
 (* Basic inversion lemmas ***************************************************)
 
-fact lsubsv_inv_atom1_aux: ∀h,g,L1,L2. h ⊢ L1 ¡⊑[g] L2 → L1 = ⋆ → L2 = ⋆.
+fact lsubsv_inv_atom1_aux: ∀h,g,L1,L2. h ⊢ L1 ¡⊑[h, g] L2 → L1 = ⋆ → L2 = ⋆.
 #h #g #L1 #L2 * -L1 -L2
 [ //
 | #I #L1 #L2 #V #_ #H destruct
@@ -41,15 +41,15 @@ fact lsubsv_inv_atom1_aux: ∀h,g,L1,L2. h ⊢ L1 ¡⊑[g] L2 → L1 = ⋆ → L
 ]
 qed-.
 
-lemma lsubsv_inv_atom1: ∀h,g,L2. h ⊢ ⋆ ¡⊑[g] L2 → L2 = ⋆.
+lemma lsubsv_inv_atom1: ∀h,g,L2. h ⊢ ⋆ ¡⊑[h, g] L2 → L2 = ⋆.
 /2 width=5 by lsubsv_inv_atom1_aux/ qed-.
 
-fact lsubsv_inv_pair1_aux: ∀h,g,L1,L2. h ⊢ L1 ¡⊑[g] L2 →
+fact lsubsv_inv_pair1_aux: ∀h,g,L1,L2. h ⊢ L1 ¡⊑[h, g] L2 →
                            ∀I,K1,X. L1 = K1.ⓑ{I}X →
-                           (∃∃K2. h ⊢ K1 ¡⊑[g] K2 & L2 = K2.ⓑ{I}X) ∨
-                           ∃∃K2,W,V,W1,V2,l. ⦃h, K1⦄ ⊢ X ¡[g] & ⦃h, K2⦄ ⊢ W ¡[g] &
-                                             ⦃h, K1⦄ ⊢ V •[g] ⦃l+1, W1⦄ & ⦃h, K2⦄ ⊢ W •[g] ⦃l, V2⦄ &
-                                             h ⊢ K1 ¡⊑[g] K2 &
+                           (∃∃K2. h ⊢ K1 ¡⊑[h, g] K2 & L2 = K2.ⓑ{I}X) ∨
+                           ∃∃K2,W,V,W1,V2,l. ⦃h, K1⦄ ⊢ X ¡[h, g] & ⦃h, K2⦄ ⊢ W ¡[h, g] &
+                                             ⦃h, K1⦄ ⊢ V •[h, g] ⦃l+1, W1⦄ & ⦃h, K2⦄ ⊢ W •[h, g] ⦃l, V2⦄ &
+                                             h ⊢ K1 ¡⊑[h, g] K2 &
                                              I = Abbr & L2 = K2.ⓛW & X = ⓝW.V.
 #h #g #L1 #L2 * -L1 -L2
 [ #J #K1 #X #H destruct
@@ -58,15 +58,15 @@ fact lsubsv_inv_pair1_aux: ∀h,g,L1,L2. h ⊢ L1 ¡⊑[g] L2 →
 ]
 qed-.
 
-lemma lsubsv_inv_pair1: ∀h,g,I,K1,L2,X. h ⊢ K1.ⓑ{I}X ¡⊑[g] L2 →
-                        (∃∃K2. h ⊢ K1 ¡⊑[g] K2 & L2 = K2.ⓑ{I}X) ∨
-                        ∃∃K2,W,V,W1,V2,l. ⦃h, K1⦄ ⊢ X ¡[g] & ⦃h, K2⦄ ⊢ W ¡[g] &
-                                          ⦃h, K1⦄ ⊢ V •[g] ⦃l+1, W1⦄ & ⦃h, K2⦄ ⊢ W •[g] ⦃l, V2⦄ &
-                                          h ⊢ K1 ¡⊑[g] K2 &
+lemma lsubsv_inv_pair1: ∀h,g,I,K1,L2,X. h ⊢ K1.ⓑ{I}X ¡⊑[h, g] L2 →
+                        (∃∃K2. h ⊢ K1 ¡⊑[h, g] K2 & L2 = K2.ⓑ{I}X) ∨
+                        ∃∃K2,W,V,W1,V2,l. ⦃h, K1⦄ ⊢ X ¡[h, g] & ⦃h, K2⦄ ⊢ W ¡[h, g] &
+                                          ⦃h, K1⦄ ⊢ V •[h, g] ⦃l+1, W1⦄ & ⦃h, K2⦄ ⊢ W •[h, g] ⦃l, V2⦄ &
+                                          h ⊢ K1 ¡⊑[h, g] K2 &
                                           I = Abbr & L2 = K2.ⓛW & X = ⓝW.V.
 /2 width=3 by lsubsv_inv_pair1_aux/ qed-.
 
-fact lsubsv_inv_atom2_aux: ∀h,g,L1,L2. h ⊢ L1 ¡⊑[g] L2 → L2 = ⋆ → L1 = ⋆.
+fact lsubsv_inv_atom2_aux: ∀h,g,L1,L2. h ⊢ L1 ¡⊑[h, g] L2 → L2 = ⋆ → L1 = ⋆.
 #h #g #L1 #L2 * -L1 -L2
 [ //
 | #I #L1 #L2 #V #_ #H destruct
@@ -74,15 +74,15 @@ fact lsubsv_inv_atom2_aux: ∀h,g,L1,L2. h ⊢ L1 ¡⊑[g] L2 → L2 = ⋆ → L
 ]
 qed-.
 
-lemma lsubsv_inv_atom2: ∀h,g,L1. h ⊢ L1 ¡⊑[g] ⋆ → L1 = ⋆.
+lemma lsubsv_inv_atom2: ∀h,g,L1. h ⊢ L1 ¡⊑[h, g] ⋆ → L1 = ⋆.
 /2 width=5 by lsubsv_inv_atom2_aux/ qed-.
 
-fact lsubsv_inv_pair2_aux: ∀h,g,L1,L2. h ⊢ L1 ¡⊑[g] L2 →
+fact lsubsv_inv_pair2_aux: ∀h,g,L1,L2. h ⊢ L1 ¡⊑[h, g] L2 →
                            ∀I,K2,W. L2 = K2.ⓑ{I}W →
-                           (∃∃K1. h ⊢ K1 ¡⊑[g] K2 & L1 = K1.ⓑ{I}W) ∨
-                           ∃∃K1,V,W1,V2,l. ⦃h, K1⦄ ⊢ ⓝW.V ¡[g] & ⦃h, K2⦄ ⊢ W ¡[g] &
-                                           ⦃h, K1⦄ ⊢ V •[g] ⦃l+1, W1⦄ & ⦃h, K2⦄ ⊢ W •[g] ⦃l, V2⦄ &
-                                           h ⊢ K1 ¡⊑[g] K2 & I = Abst & L1 = K1. ⓓⓝW.V.
+                           (∃∃K1. h ⊢ K1 ¡⊑[h, g] K2 & L1 = K1.ⓑ{I}W) ∨
+                           ∃∃K1,V,W1,V2,l. ⦃h, K1⦄ ⊢ ⓝW.V ¡[h, g] & ⦃h, K2⦄ ⊢ W ¡[h, g] &
+                                           ⦃h, K1⦄ ⊢ V •[h, g] ⦃l+1, W1⦄ & ⦃h, K2⦄ ⊢ W •[h, g] ⦃l, V2⦄ &
+                                           h ⊢ K1 ¡⊑[h, g] K2 & I = Abst & L1 = K1. ⓓⓝW.V.
 #h #g #L1 #L2 * -L1 -L2
 [ #J #K2 #U #H destruct
 | #I #L1 #L2 #V #HL12 #J #K2 #U #H destruct /3 width=3/
@@ -90,26 +90,26 @@ fact lsubsv_inv_pair2_aux: ∀h,g,L1,L2. h ⊢ L1 ¡⊑[g] L2 →
 ]
 qed-.
 
-lemma lsubsv_inv_pair2: ∀h,g,I,L1,K2,W. h ⊢ L1 ¡⊑[g] K2.ⓑ{I}W →
-                        (∃∃K1. h ⊢ K1 ¡⊑[g] K2 & L1 = K1.ⓑ{I}W) ∨
-                        ∃∃K1,V,W1,V2,l. ⦃h, K1⦄ ⊢ ⓝW.V ¡[g] & ⦃h, K2⦄ ⊢ W ¡[g] &
-                                        ⦃h, K1⦄ ⊢ V •[g] ⦃l+1, W1⦄ & ⦃h, K2⦄ ⊢ W •[g] ⦃l, V2⦄ &
-                                        h ⊢ K1 ¡⊑[g] K2 & I = Abst & L1 = K1. ⓓⓝW.V.
+lemma lsubsv_inv_pair2: ∀h,g,I,L1,K2,W. h ⊢ L1 ¡⊑[h, g] K2.ⓑ{I}W →
+                        (∃∃K1. h ⊢ K1 ¡⊑[h, g] K2 & L1 = K1.ⓑ{I}W) ∨
+                        ∃∃K1,V,W1,V2,l. ⦃h, K1⦄ ⊢ ⓝW.V ¡[h, g] & ⦃h, K2⦄ ⊢ W ¡[h, g] &
+                                        ⦃h, K1⦄ ⊢ V •[h, g] ⦃l+1, W1⦄ & ⦃h, K2⦄ ⊢ W •[h, g] ⦃l, V2⦄ &
+                                        h ⊢ K1 ¡⊑[h, g] K2 & I = Abst & L1 = K1. ⓓⓝW.V.
 /2 width=3 by lsubsv_inv_pair2_aux/ qed-.
 
 (* Basic_forward lemmas *****************************************************)
 
-lemma lsubsv_fwd_lsubr: ∀h,g,L1,L2. h ⊢ L1 ¡⊑[g] L2 → L1 ⊑ L2.
+lemma lsubsv_fwd_lsubr: ∀h,g,L1,L2. h ⊢ L1 ¡⊑[h, g] L2 → L1 ⊑ L2.
 #h #g #L1 #L2 #H elim H -L1 -L2 // /2 width=1/
 qed-.
 
 (* Basic properties *********************************************************)
 
-lemma lsubsv_refl: ∀h,g,L. h ⊢ L ¡⊑[g] L.
+lemma lsubsv_refl: ∀h,g,L. h ⊢ L ¡⊑[h, g] L.
 #h #g #L elim L -L // /2 width=1/
 qed.
 
-lemma lsubsv_cprs_trans: ∀h,g,L1,L2. h ⊢ L1 ¡⊑[g] L2 →
+lemma lsubsv_cprs_trans: ∀h,g,L1,L2. h ⊢ L1 ¡⊑[h, g] L2 →
                          ∀T1,T2. L2 ⊢ T1 ➡* T2 → L1 ⊢ T1 ➡* T2.
 /3 width=5 by lsubsv_fwd_lsubr, lsubr_cprs_trans/
 qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/lsubsv_cpcs.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/lsubsv_cpcs.ma
index a66322d66..30e42ea09 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/dynamic/lsubsv_cpcs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/lsubsv_cpcs.ma
@@ -19,7 +19,7 @@ include "basic_2/dynamic/lsubsv.ma".
 
 (* Properties on context-sensitive parallel equivalence for terms ***********)
 
-lemma lsubsv_cpcs_trans: ∀h,g,L1,L2. h ⊢ L1 ¡⊑[g] L2 →
+lemma lsubsv_cpcs_trans: ∀h,g,L1,L2. h ⊢ L1 ¡⊑[h, g] L2 →
                          ∀T1,T2. L2 ⊢ T1 ⬌* T2 → L1 ⊢ T1 ⬌* T2.
 /3 width=5 by lsubsv_fwd_lsubr, lsubr_cpcs_trans/
 qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/lsubsv_cpds.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/lsubsv_cpds.ma
index 4ffc91f95..55a7e9a13 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/dynamic/lsubsv_cpds.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/lsubsv_cpds.ma
@@ -20,13 +20,13 @@ include "basic_2/dynamic/lsubsv_ssta.ma".
 (* Properties for the preservation results **********************************)
 
 fact lsubsv_sstas_aux: ∀h,g,L0,T0.
-                       (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_cpr_lpr h g L1 T1) →
-                       (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_ssta_cpr_lpr h g L1 T1) →
-                       (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ssta h g L1 T1) →
-                       (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_lsubsv h g L1 T1) →
-                       ∀L2,T. h ⊢ ⦃L0, T0⦄ >[g] ⦃L2, T⦄ → ⦃h, L2⦄ ⊢ T ¡[g] →
-                       ∀L1. h ⊢ L1 ¡⊑[g] L2 → ∀U2. ⦃h, L2⦄ ⊢ T •*[g] U2 →
-                       ∃∃U1. ⦃h, L1⦄ ⊢ T •*[g] U1 & L1 ⊢ U1 ⬌* U2.
+                       (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_snv_cpr_lpr h g L1 T1) →
+                       (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_ssta_cpr_lpr h g L1 T1) →
+                       (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_snv_ssta h g L1 T1) →
+                       (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_snv_lsubsv h g L1 T1) →
+                       ∀L2,T. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L2, T⦄ → ⦃h, L2⦄ ⊢ T ¡[h, g] →
+                       ∀L1. h ⊢ L1 ¡⊑[h, g] L2 → ∀U2. ⦃h, L2⦄ ⊢ T •*[h, g] U2 →
+                       ∃∃U1. ⦃h, L1⦄ ⊢ T •*[h, g] U1 & L1 ⊢ U1 ⬌* U2.
 #h #g #L0 #T0 #IH4 #IH3 #IH2 #IH1 #L2 #T #HLT0 #HT #L1 #HL12 #U2 #H @(sstas_ind … H) -U2 [ /2 width=3/ ]
 #U2 #W #l #HTU2 #HU2W * #U1 #HTU1 #HU12
 lapply (IH1 … HT … HL12) // #H
@@ -41,13 +41,13 @@ lapply (cpcs_trans … HW12 … HW2) -W2 /3 width=4/
 qed-.
 
 fact lsubsv_cpds_aux: ∀h,g,L0,T0.
-                      (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_cpr_lpr h g L1 T1) →
-                      (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_ssta_cpr_lpr h g L1 T1) →
-                      (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ssta h g L1 T1) →
-                      (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_lsubsv h g L1 T1) →
-                      ∀L2,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L2, T1⦄ → ⦃h, L2⦄ ⊢ T1 ¡[g] →
-                      ∀L1. h ⊢ L1 ¡⊑[g] L2 → ∀T2. ⦃h, L2⦄ ⊢ T1 •*➡*[g] T2 →
-                      ∃∃T. ⦃h, L1⦄ ⊢ T1 •*➡*[g] T & L1 ⊢ T2 ➡* T.
+                      (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_snv_cpr_lpr h g L1 T1) →
+                      (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_ssta_cpr_lpr h g L1 T1) →
+                      (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_snv_ssta h g L1 T1) →
+                      (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_snv_lsubsv h g L1 T1) →
+                      ∀L2,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L2, T1⦄ → ⦃h, L2⦄ ⊢ T1 ¡[h, g] →
+                      ∀L1. h ⊢ L1 ¡⊑[h, g] L2 → ∀T2. ⦃h, L2⦄ ⊢ T1 •*➡*[h, g] T2 →
+                      ∃∃T. ⦃h, L1⦄ ⊢ T1 •*➡*[h, g] T & L1 ⊢ T2 ➡* T.
 #h #g #L0 #T0 #IH4 #IH3 #IH2 #IH1 #L2 #T1 #HLT0 #HT1 #L1 #HL12 #T2 * #T #HT1T #HTT2
 lapply (lsubsv_cprs_trans … HL12 … HTT2) -HTT2 #HTT2
 elim (lsubsv_sstas_aux … IH4 IH3 IH2 IH1 … HLT0 … HL12 … HT1T) // -L2 -L0 -T0 #T0 #HT10 #HT0
diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/lsubsv_ldrop.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/lsubsv_ldrop.ma
index f8caa3aee..31d47c8c5 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/dynamic/lsubsv_ldrop.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/lsubsv_ldrop.ma
@@ -19,9 +19,9 @@ include "basic_2/dynamic/lsubsv.ma".
 (* Properties concerning basic local environment slicing ********************)
 
 (* Note: the constant 0 cannot be generalized *)
-lemma lsubsv_ldrop_O1_conf: ∀h,g,L1,L2. h ⊢ L1 ¡⊑[g] L2 →
+lemma lsubsv_ldrop_O1_conf: ∀h,g,L1,L2. h ⊢ L1 ¡⊑[h, g] L2 →
                             ∀K1,e. ⇩[0, e] L1 ≡ K1 →
-                            ∃∃K2. h ⊢ K1 ¡⊑[g] K2 & ⇩[0, e] L2 ≡ K2.
+                            ∃∃K2. h ⊢ K1 ¡⊑[h, g] K2 & ⇩[0, e] L2 ≡ K2.
 #h #g #L1 #L2 #H elim H -L1 -L2
 [ /2 width=3/
 | #I #L1 #L2 #V #_ #IHL12 #K1 #e #H
@@ -42,9 +42,9 @@ lemma lsubsv_ldrop_O1_conf: ∀h,g,L1,L2. h ⊢ L1 ¡⊑[g] L2 →
 qed-.
 
 (* Note: the constant 0 cannot be generalized *)
-lemma lsubsv_ldrop_O1_trans: ∀h,g,L1,L2. h ⊢ L1 ¡⊑[g] L2 →
+lemma lsubsv_ldrop_O1_trans: ∀h,g,L1,L2. h ⊢ L1 ¡⊑[h, g] L2 →
                              ∀K2,e. ⇩[0, e] L2 ≡ K2 →
-                             ∃∃K1. h ⊢ K1 ¡⊑[g] K2 & ⇩[0, e] L1 ≡ K1.
+                             ∃∃K1. h ⊢ K1 ¡⊑[h, g] K2 & ⇩[0, e] L1 ≡ K1.
 #h #g #L1 #L2 #H elim H -L1 -L2
 [ /2 width=3/
 | #I #L1 #L2 #V #_ #IHL12 #K2 #e #H
diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/lsubsv_lsuba.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/lsubsv_lsuba.ma
index 100c34da2..7d40d4081 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/dynamic/lsubsv_lsuba.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/lsubsv_lsuba.ma
@@ -19,7 +19,7 @@ include "basic_2/dynamic/lsubsv.ma".
 
 (* Forward lemmas on lenv refinement for atomic arity assignment ************)
 
-lemma lsubsv_fwd_lsuba: ∀h,g,L1,L2. h ⊢ L1 ¡⊑[g] L2 → L1 ⁝⊑ L2.
+lemma lsubsv_fwd_lsuba: ∀h,g,L1,L2. h ⊢ L1 ¡⊑[h, g] L2 → L1 ⁝⊑ L2.
 #h #g #L1 #L2 #H elim H -L1 -L2 // /2 width=1/
 #L1 #L2 #W #V #W1 #V2 #l #HV #HW #_ #_ #_ #IHL12 -W1 -V2
 elim (snv_fwd_aaa … HV) -HV #A #HV
diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/lsubsv_snv.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/lsubsv_snv.ma
index 078b8fa89..67230d980 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/dynamic/lsubsv_snv.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/lsubsv_snv.ma
@@ -21,10 +21,10 @@ include "basic_2/dynamic/lsubsv_cpcs.ma".
 (* Properties concerning stratified native validity *************************)
 
 fact snv_lsubsv_aux: ∀h,g,L0,T0.
-                     (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_cpr_lpr h g L1 T1) →
-                     (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_ssta_cpr_lpr h g L1 T1) →
-                     (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ssta h g L1 T1) →
-                     (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_lsubsv h g L1 T1) →
+                     (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_snv_cpr_lpr h g L1 T1) →
+                     (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_ssta_cpr_lpr h g L1 T1) →
+                     (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_snv_ssta h g L1 T1) →
+                     (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_snv_lsubsv h g L1 T1) →
                      ∀L1,T1. L0 = L1 → T0 = T1 → IH_snv_lsubsv h g L1 T1.
 #h #g #L0 #T0 #IH4 #IH3 #IH2 #IH1 #L2 * * [||||*] //
 [ #i #HL0 #HT0 #H #L1 #HL12 destruct -IH4 -IH3 -IH2
diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/lsubsv_ssta.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/lsubsv_ssta.ma
index b372dca9c..15f2e14b7 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/dynamic/lsubsv_ssta.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/lsubsv_ssta.ma
@@ -20,9 +20,9 @@ include "basic_2/dynamic/lsubsv_ldrop.ma".
 
 (* Properties on stratified static type assignment **************************)
 
-lemma lsubsv_ssta_trans: ∀h,g,L2,T,U2,l. ⦃h, L2⦄ ⊢ T •[g] ⦃l, U2⦄ →
-                         ∀L1. h ⊢ L1 ¡⊑[g] L2 →
-                         ∃∃U1. ⦃h, L1⦄ ⊢ T •[g] ⦃l, U1⦄ & L1 ⊢ U1 ⬌* U2.
+lemma lsubsv_ssta_trans: ∀h,g,L2,T,U2,l. ⦃h, L2⦄ ⊢ T •[h, g] ⦃l, U2⦄ →
+                         ∀L1. h ⊢ L1 ¡⊑[h, g] L2 →
+                         ∃∃U1. ⦃h, L1⦄ ⊢ T •[h, g] ⦃l, U1⦄ & L1 ⊢ U1 ⬌* U2.
 #h #g #L2 #T #U #l #H elim H -L2 -T -U -l
 [ /3 width=3/
 | #L2 #K2 #X #Y #U #i #l #HLK2 #_ #HYU #IHXY #L1 #HL12
diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv.ma
index d4e3beda1..3f62cff77 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv.ma
@@ -23,10 +23,10 @@ inductive snv (h:sh) (g:sd h): lenv → predicate term ≝
 | snv_lref: ∀I,L,K,V,i. ⇩[0, i] L ≡ K.ⓑ{I}V → snv h g K V → snv h g L (#i)
 | snv_bind: ∀a,I,L,V,T. snv h g L V → snv h g (L.ⓑ{I}V) T → snv h g L (ⓑ{a,I}V.T)
 | snv_appl: ∀a,L,V,W,W0,T,U,l. snv h g L V → snv h g L T →
-            ⦃h, L⦄ ⊢ V •[g] ⦃l+1, W⦄ → L ⊢ W ➡* W0 →
-            ⦃h, L⦄ ⊢ T •*➡*[g] ⓛ{a}W0.U → snv h g L (ⓐV.T)
+            ⦃G, L⦄ ⊢ V •[h, g] ⦃l+1, W⦄ → ⦃G, L⦄ ⊢ W ➡* W0 →
+            ⦃G, L⦄ ⊢ T •*➡*[h, g] ⓛ{a}W0.U → snv h g L (ⓐV.T)
 | snv_cast: ∀L,W,T,U,l. snv h g L W → snv h g L T →
-            ⦃h, L⦄ ⊢ T •[g] ⦃l+1, U⦄ → L ⊢ U ⬌* W → snv h g L (ⓝW.T)
+            ⦃G, L⦄ ⊢ T •[h, g] ⦃l+1, U⦄ → ⦃G, L⦄ ⊢ U ⬌* W → snv h g L (ⓝW.T)
 .
 
 interpretation "stratified native validity (term)"
@@ -34,8 +34,8 @@ interpretation "stratified native validity (term)"
 
 (* Basic inversion lemmas ***************************************************)
 
-fact snv_inv_lref_aux: ∀h,g,L,X. ⦃h, L⦄ ⊢ X ¡[g] → ∀i. X = #i →
-                       ∃∃I,K,V. ⇩[0, i] L ≡ K.ⓑ{I}V & ⦃h, K⦄ ⊢ V ¡[g].
+fact snv_inv_lref_aux: ∀h,g,L,X. ⦃G, L⦄ ⊢ X ¡[h, g] → ∀i. X = #i →
+                       ∃∃I,K,V. ⇩[0, i] L ≡ K.ⓑ{I}V & ⦃h, K⦄ ⊢ V ¡[h, g].
 #h #g #L #X * -L -X
 [ #L #k #i #H destruct
 | #I #L #K #V #i0 #HLK #HV #i #H destruct /2 width=5/
@@ -45,11 +45,11 @@ fact snv_inv_lref_aux: ∀h,g,L,X. ⦃h, L⦄ ⊢ X ¡[g] → ∀i. X = #i →
 ]
 qed.
 
-lemma snv_inv_lref: ∀h,g,L,i. ⦃h, L⦄ ⊢ #i ¡[g] →
-                    ∃∃I,K,V. ⇩[0, i] L ≡ K.ⓑ{I}V & ⦃h, K⦄ ⊢ V ¡[g].
+lemma snv_inv_lref: ∀h,g,L,i. ⦃G, L⦄ ⊢ #i ¡[h, g] →
+                    ∃∃I,K,V. ⇩[0, i] L ≡ K.ⓑ{I}V & ⦃h, K⦄ ⊢ V ¡[h, g].
 /2 width=3/ qed-.
 
-fact snv_inv_gref_aux: ∀h,g,L,X. ⦃h, L⦄ ⊢ X ¡[g] → ∀p. X = §p → ⊥.
+fact snv_inv_gref_aux: ∀h,g,L,X. ⦃G, L⦄ ⊢ X ¡[h, g] → ∀p. X = §p → ⊥.
 #h #g #L #X * -L -X
 [ #L #k #p #H destruct
 | #I #L #K #V #i #_ #_ #p #H destruct
@@ -59,11 +59,11 @@ fact snv_inv_gref_aux: ∀h,g,L,X. ⦃h, L⦄ ⊢ X ¡[g] → ∀p. X = §p →
 ]
 qed.
 
-lemma snv_inv_gref: ∀h,g,L,p. ⦃h, L⦄ ⊢ §p ¡[g] → ⊥.
+lemma snv_inv_gref: ∀h,g,L,p. ⦃G, L⦄ ⊢ §p ¡[h, g] → ⊥.
 /2 width=7/ qed-.
 
-fact snv_inv_bind_aux: ∀h,g,L,X. ⦃h, L⦄ ⊢ X ¡[g] → ∀a,I,V,T. X = ⓑ{a,I}V.T →
-                       ⦃h, L⦄ ⊢ V ¡[g] ∧ ⦃h, L.ⓑ{I}V⦄ ⊢ T ¡[g].
+fact snv_inv_bind_aux: ∀h,g,L,X. ⦃G, L⦄ ⊢ X ¡[h, g] → ∀a,I,V,T. X = ⓑ{a,I}V.T →
+                       ⦃G, L⦄ ⊢ V ¡[h, g] ∧ ⦃h, L.ⓑ{I}V⦄ ⊢ T ¡[h, g].
 #h #g #L #X * -L -X
 [ #L #k #a #I #V #T #H destruct
 | #I0 #L #K #V0 #i #_ #_ #a #I #V #T #H destruct
@@ -73,14 +73,14 @@ fact snv_inv_bind_aux: ∀h,g,L,X. ⦃h, L⦄ ⊢ X ¡[g] → ∀a,I,V,T. X = 
 ]
 qed.
 
-lemma snv_inv_bind: ∀h,g,a,I,L,V,T. ⦃h, L⦄ ⊢ ⓑ{a,I}V.T ¡[g] →
-                        ⦃h, L⦄ ⊢ V ¡[g] ∧ ⦃h, L.ⓑ{I}V⦄ ⊢ T ¡[g].
+lemma snv_inv_bind: ∀h,g,a,I,L,V,T. ⦃G, L⦄ ⊢ ⓑ{a,I}V.T ¡[h, g] →
+                        ⦃G, L⦄ ⊢ V ¡[h, g] ∧ ⦃h, L.ⓑ{I}V⦄ ⊢ T ¡[h, g].
 /2 width=4/ qed-.
 
-fact snv_inv_appl_aux: ∀h,g,L,X. ⦃h, L⦄ ⊢ X ¡[g] → ∀V,T. X = ⓐV.T →
-                       ∃∃a,W,W0,U,l. ⦃h, L⦄ ⊢ V ¡[g] & ⦃h, L⦄ ⊢ T ¡[g] &
-                                   ⦃h, L⦄ ⊢ V •[g] ⦃l+1, W⦄ & L ⊢ W ➡* W0 &
-                                   ⦃h, L⦄ ⊢ T •*➡*[g] ⓛ{a}W0.U.
+fact snv_inv_appl_aux: ∀h,g,L,X. ⦃G, L⦄ ⊢ X ¡[h, g] → ∀V,T. X = ⓐV.T →
+                       ∃∃a,W,W0,U,l. ⦃G, L⦄ ⊢ V ¡[h, g] & ⦃G, L⦄ ⊢ T ¡[h, g] &
+                                   ⦃G, L⦄ ⊢ V •[h, g] ⦃l+1, W⦄ & ⦃G, L⦄ ⊢ W ➡* W0 &
+                                   ⦃G, L⦄ ⊢ T •*➡*[h, g] ⓛ{a}W0.U.
 #h #g #L #X * -L -X
 [ #L #k #V #T #H destruct
 | #I #L #K #V0 #i #_ #_ #V #T #H destruct
@@ -90,15 +90,15 @@ fact snv_inv_appl_aux: ∀h,g,L,X. ⦃h, L⦄ ⊢ X ¡[g] → ∀V,T. X = ⓐV.T
 ]
 qed.
 
-lemma snv_inv_appl: ∀h,g,L,V,T. ⦃h, L⦄ ⊢ ⓐV.T ¡[g] →
-                    ∃∃a,W,W0,U,l. ⦃h, L⦄ ⊢ V ¡[g] & ⦃h, L⦄ ⊢ T ¡[g] &
-                                ⦃h, L⦄ ⊢ V •[g] ⦃l+1, W⦄ & L ⊢ W ➡* W0 &
-                                ⦃h, L⦄ ⊢ T •*➡*[g] ⓛ{a}W0.U.
+lemma snv_inv_appl: ∀h,g,L,V,T. ⦃G, L⦄ ⊢ ⓐV.T ¡[h, g] →
+                    ∃∃a,W,W0,U,l. ⦃G, L⦄ ⊢ V ¡[h, g] & ⦃G, L⦄ ⊢ T ¡[h, g] &
+                                ⦃G, L⦄ ⊢ V •[h, g] ⦃l+1, W⦄ & ⦃G, L⦄ ⊢ W ➡* W0 &
+                                ⦃G, L⦄ ⊢ T •*➡*[h, g] ⓛ{a}W0.U.
 /2 width=3/ qed-.
 
-fact snv_inv_cast_aux: ∀h,g,L,X. ⦃h, L⦄ ⊢ X ¡[g] → ∀W,T. X = ⓝW.T →
-                       ∃∃U,l. ⦃h, L⦄ ⊢ W ¡[g] & ⦃h, L⦄ ⊢ T ¡[g] &
-                              ⦃h, L⦄ ⊢ T •[g] ⦃l+1, U⦄ & L ⊢ U ⬌* W.
+fact snv_inv_cast_aux: ∀h,g,L,X. ⦃G, L⦄ ⊢ X ¡[h, g] → ∀W,T. X = ⓝW.T →
+                       ∃∃U,l. ⦃G, L⦄ ⊢ W ¡[h, g] & ⦃G, L⦄ ⊢ T ¡[h, g] &
+                              ⦃G, L⦄ ⊢ T •[h, g] ⦃l+1, U⦄ & ⦃G, L⦄ ⊢ U ⬌* W.
 #h #g #L #X * -L -X
 [ #L #k #W #T #H destruct
 | #I #L #K #V #i #_ #_ #W #T #H destruct
@@ -108,14 +108,14 @@ fact snv_inv_cast_aux: ∀h,g,L,X. ⦃h, L⦄ ⊢ X ¡[g] → ∀W,T. X = ⓝW.T
 ]
 qed.
 
-lemma snv_inv_cast: ∀h,g,L,W,T. ⦃h, L⦄ ⊢ ⓝW.T ¡[g] →
-                    ∃∃U,l. ⦃h, L⦄ ⊢ W ¡[g] & ⦃h, L⦄ ⊢ T ¡[g] &
-                           ⦃h, L⦄ ⊢ T •[g] ⦃l+1, U⦄ & L ⊢ U ⬌* W.
+lemma snv_inv_cast: ∀h,g,L,W,T. ⦃G, L⦄ ⊢ ⓝW.T ¡[h, g] →
+                    ∃∃U,l. ⦃G, L⦄ ⊢ W ¡[h, g] & ⦃G, L⦄ ⊢ T ¡[h, g] &
+                           ⦃G, L⦄ ⊢ T •[h, g] ⦃l+1, U⦄ & ⦃G, L⦄ ⊢ U ⬌* W.
 /2 width=3/ qed-.
 
 (* Basic forward lemmas *****************************************************)
 
-lemma snv_fwd_ssta: ∀h,g,L,T. ⦃h, L⦄ ⊢ T ¡[g] → ∃∃l,U. ⦃h, L⦄ ⊢ T •[g] ⦃l, U⦄.
+lemma snv_fwd_ssta: ∀h,g,L,T. ⦃G, L⦄ ⊢ T ¡[h, g] → ∃∃l,U. ⦃G, L⦄ ⊢ T •[h, g] ⦃l, U⦄.
 #h #g #L #T #H elim H -L -T
 [ #L #k elim (deg_total h g k) /3 width=3/
 | * #L #K #V #i #HLK #_ * #l0 #W #HVW
diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_aaa.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_aaa.ma
index 32807a499..3a161e1f9 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_aaa.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_aaa.ma
@@ -21,7 +21,7 @@ include "basic_2/dynamic/snv.ma".
 
 (* Forward lemmas on atomic arity assignment for terms **********************)
 
-lemma snv_fwd_aaa: ∀h,g,L,T. ⦃h, L⦄ ⊢ T ¡[g] → ∃A. L ⊢ T ⁝ A.
+lemma snv_fwd_aaa: ∀h,g,L,T. ⦃G, L⦄ ⊢ T ¡[h, g] → ∃A. ⦃G, L⦄ ⊢ T ⁝ A.
 #h #g #L #T #H elim H -L -T
 [ /2 width=2/
 | #I #L #K #V #i #HLK #_ * /3 width=6/
@@ -36,6 +36,6 @@ lemma snv_fwd_aaa: ∀h,g,L,T. ⦃h, L⦄ ⊢ T ¡[g] → ∃A. L ⊢ T ⁝ A.
 ]
 qed-.
 
-lemma snv_fwd_csn: ∀h,g,L,T. ⦃h, L⦄ ⊢ T ¡[g] → ⦃h, L⦄ ⊢ ⬊*[g] T.
+lemma snv_fwd_csn: ∀h,g,L,T. ⦃G, L⦄ ⊢ T ¡[h, g] → ⦃G, L⦄ ⊢ ⬊*[h, g] T.
 #h #g #L #T #H elim (snv_fwd_aaa … H) -H /2 width=2/
 qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_cpcs.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_cpcs.ma
index 707994c62..8aaf788a2 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_cpcs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_cpcs.ma
@@ -22,41 +22,41 @@ include "basic_2/dynamic/ygt.ma".
 (* Inductive premises for the preservation results **************************)
 
 definition IH_snv_cpr_lpr: ∀h:sh. sd h → relation2 lenv term ≝
-                           λh,g,L1,T1. ⦃h, L1⦄ ⊢ T1 ¡[g] →
-                           ∀T2. L1 ⊢ T1 ➡ T2 → ∀L2. L1 ⊢ ➡ L2 → ⦃h, L2⦄ ⊢ T2 ¡[g].
+                           λh,g,L1,T1. ⦃h, L1⦄ ⊢ T1 ¡[h, g] →
+                           ∀T2. L1 ⊢ T1 ➡ T2 → ∀L2. L1 ⊢ ➡ L2 → ⦃h, L2⦄ ⊢ T2 ¡[h, g].
 
 definition IH_ssta_cpr_lpr: ∀h:sh. sd h → relation2 lenv term ≝
-                            λh,g,L1,T1. ⦃h, L1⦄ ⊢ T1 ¡[g] →
-                            ∀U1,l. ⦃h, L1⦄ ⊢ T1 •[g] ⦃l, U1⦄ →
+                            λh,g,L1,T1. ⦃h, L1⦄ ⊢ T1 ¡[h, g] →
+                            ∀U1,l. ⦃h, L1⦄ ⊢ T1 •[h, g] ⦃l, U1⦄ →
                             ∀T2. L1 ⊢ T1 ➡ T2 → ∀L2. L1 ⊢ ➡ L2 →
-                            ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g] ⦃l, U2⦄ & L2 ⊢ U1 ⬌* U2.
+                            ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[h, g] ⦃l, U2⦄ & L2 ⊢ U1 ⬌* U2.
 
 definition IH_snv_ssta: ∀h:sh. sd h → relation2 lenv term ≝
-                        λh,g,L,T. ⦃h, L⦄ ⊢ T ¡[g] →
-                        ∀U,l. ⦃h, L⦄ ⊢ T •[g] ⦃l+1, U⦄ → ⦃h, L⦄ ⊢ U ¡[g].
+                        λh,g,L,T. ⦃G, L⦄ ⊢ T ¡[h, g] →
+                        ∀U,l. ⦃G, L⦄ ⊢ T •[h, g] ⦃l+1, U⦄ → ⦃G, L⦄ ⊢ U ¡[h, g].
 
 definition IH_snv_lsubsv: ∀h:sh. sd h → relation2 lenv term ≝
-                          λh,g,L2,T. ⦃h, L2⦄ ⊢ T ¡[g] →
-                          ∀L1. h ⊢ L1 ¡⊑[g] L2 → ⦃h, L1⦄ ⊢ T ¡[g].
+                          λh,g,L2,T. ⦃h, L2⦄ ⊢ T ¡[h, g] →
+                          ∀L1. h ⊢ L1 ¡⊑[h, g] L2 → ⦃h, L1⦄ ⊢ T ¡[h, g].
 
 (* Properties for the preservation results **********************************)
 
 fact snv_cprs_lpr_aux: ∀h,g,L0,T0.
-                       (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_cpr_lpr h g L1 T1) →
-                       ∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → ⦃h, L1⦄ ⊢ T1 ¡[g] →
-                       ∀T2. L1 ⊢ T1 ➡* T2 → ∀L2. L1 ⊢ ➡ L2 → ⦃h, L2⦄ ⊢ T2 ¡[g].
+                       (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_snv_cpr_lpr h g L1 T1) →
+                       ∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → ⦃h, L1⦄ ⊢ T1 ¡[h, g] →
+                       ∀T2. L1 ⊢ T1 ➡* T2 → ∀L2. L1 ⊢ ➡ L2 → ⦃h, L2⦄ ⊢ T2 ¡[h, g].
 #h #g #L0 #T0 #IH #L1 #T1 #HLT0 #HT1 #T2 #H
 elim H -T2 [ /2 width=6/ ] -HT1
 /4 width=6 by ygt_yprs_trans, cprs_yprs/
 qed-.
 
 fact ssta_cprs_lpr_aux: ∀h,g,L0,T0.
-                        (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_cpr_lpr h g L1 T1) →
-                        (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_ssta_cpr_lpr h g L1 T1) →
-                        ∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → ⦃h, L1⦄ ⊢ T1 ¡[g] →
-                        ∀U1,l. ⦃h, L1⦄ ⊢ T1 •[g] ⦃l, U1⦄ →
+                        (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_snv_cpr_lpr h g L1 T1) →
+                        (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_ssta_cpr_lpr h g L1 T1) →
+                        ∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → ⦃h, L1⦄ ⊢ T1 ¡[h, g] →
+                        ∀U1,l. ⦃h, L1⦄ ⊢ T1 •[h, g] ⦃l, U1⦄ →
                         ∀T2. L1 ⊢ T1 ➡* T2 → ∀L2. L1 ⊢ ➡ L2 →
-                        ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g] ⦃l, U2⦄ & L2 ⊢ U1 ⬌* U2.
+                        ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[h, g] ⦃l, U2⦄ & L2 ⊢ U1 ⬌* U2.
 #h #g #L0 #T0 #IH2 #IH1 #L1 #T1 #H01 #HT1 #U1 #l #HTU1 #T2 #H
 elim H -T2 [ /2 width=7/ ]
 #T #T2 #HT1T #HTT2 #IHT1 #L2 #HL12
@@ -70,12 +70,12 @@ lapply (cpcs_trans … HU1 … HU2) -U /2 width=3/
 qed-.
 
 fact ssta_cpcs_lpr_aux: ∀h,g,L0,T0.
-                        (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_cpr_lpr h g L1 T1) →
-                        (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_ssta_cpr_lpr h g L1 T1) →
-                        ∀L1,T1,T2. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T2⦄ →
-                        ⦃h, L1⦄ ⊢ T1 ¡[g] → ⦃h, L1⦄ ⊢ T2 ¡[g] →
-                        ∀U1,l1. ⦃h, L1⦄ ⊢ T1 •[g] ⦃l1, U1⦄ →
-                        ∀U2,l2. ⦃h, L1⦄ ⊢ T2 •[g] ⦃l2, U2⦄ →
+                        (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_snv_cpr_lpr h g L1 T1) →
+                        (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_ssta_cpr_lpr h g L1 T1) →
+                        ∀L1,T1,T2. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T2⦄ →
+                        ⦃h, L1⦄ ⊢ T1 ¡[h, g] → ⦃h, L1⦄ ⊢ T2 ¡[h, g] →
+                        ∀U1,l1. ⦃h, L1⦄ ⊢ T1 •[h, g] ⦃l1, U1⦄ →
+                        ∀U2,l2. ⦃h, L1⦄ ⊢ T2 •[h, g] ⦃l2, U2⦄ →
                         L1 ⊢ T1 ⬌* T2 → ∀L2. L1 ⊢ ➡ L2 →
                         l1 = l2 ∧ L2 ⊢ U1 ⬌* U2.
 #h #g #L0 #T0 #IH2 #IH1 #L1 #T1 #T2 #H01 #H02 #HT1 #HT2 #U1 #l1 #HTU1 #U2 #l2 #HTU2 #H #L2 #HL12
@@ -87,28 +87,28 @@ lapply (cpcs_canc_dx … HUW1 … HUW2) -W2 /2 width=1/
 qed-.
 
 fact snv_sstas_aux: ∀h,g,L0,T0.
-                    (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ssta h g L1 T1) →
-                    ∀L,T. h ⊢ ⦃L0, T0⦄ >[g] ⦃L, T⦄ → ⦃h, L⦄ ⊢ T ¡[g] →
-                    ∀U. ⦃h, L⦄ ⊢ T •*[g] U → ⦃h, L⦄ ⊢ U ¡[g].
+                    (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_snv_ssta h g L1 T1) →
+                    ∀L,T. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L, T⦄ → ⦃G, L⦄ ⊢ T ¡[h, g] →
+                    ∀U. ⦃G, L⦄ ⊢ T •*[h, g] U → ⦃G, L⦄ ⊢ U ¡[h, g].
 #h #g #L0 #T0 #IH #L #T #H01 #HT #U #H
 @(sstas_ind … H) -U // -HT /4 width=5 by ygt_yprs_trans, sstas_yprs/
 qed-.
 
 fact snv_sstas_lpr_aux: ∀h,g,L0,T0.
-                        (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_cpr_lpr h g L1 T1) →
-                        (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ssta h g L1 T1) →
-                        ∀L1,T. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T⦄ → ⦃h, L1⦄ ⊢ T ¡[g] →
-                        ∀U. ⦃h, L1⦄ ⊢ T •*[g] U → ∀L2. L1 ⊢ ➡ L2 → ⦃h, L2⦄ ⊢ U ¡[g].
+                        (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_snv_cpr_lpr h g L1 T1) →
+                        (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_snv_ssta h g L1 T1) →
+                        ∀L1,T. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T⦄ → ⦃h, L1⦄ ⊢ T ¡[h, g] →
+                        ∀U. ⦃h, L1⦄ ⊢ T •*[h, g] U → ∀L2. L1 ⊢ ➡ L2 → ⦃h, L2⦄ ⊢ U ¡[h, g].
 /4 width=7 by snv_sstas_aux, ygt_yprs_trans, sstas_yprs/
 qed-.
 
 fact sstas_cprs_lpr_aux: ∀h,g,L0,T0.
-                         (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ssta h g L1 T1) →
-                         (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_cpr_lpr h g L1 T1) →
-                         (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_ssta_cpr_lpr h g L1 T1) →
-                         ∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → ⦃h, L1⦄ ⊢ T1 ¡[g] →
-                         ∀U1. ⦃h, L1⦄ ⊢ T1 •*[g] U1 → ∀T2. L1 ⊢ T1 ➡* T2 → ∀L2. L1 ⊢ ➡ L2 →
-                         ∃∃U2. ⦃h, L2⦄ ⊢ T2 •*[g] U2 & L2 ⊢ U1 ⬌* U2.
+                         (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_snv_ssta h g L1 T1) →
+                         (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_snv_cpr_lpr h g L1 T1) →
+                         (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_ssta_cpr_lpr h g L1 T1) →
+                         ∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → ⦃h, L1⦄ ⊢ T1 ¡[h, g] →
+                         ∀U1. ⦃h, L1⦄ ⊢ T1 •*[h, g] U1 → ∀T2. L1 ⊢ T1 ➡* T2 → ∀L2. L1 ⊢ ➡ L2 →
+                         ∃∃U2. ⦃h, L2⦄ ⊢ T2 •*[h, g] U2 & L2 ⊢ U1 ⬌* U2.
 #h #g #L0 #T0 #IH3 #IH2 #IH1 #L1 #T1 #H01 #HT1 #U1 #H
 @(sstas_ind … H) -U1 [ /3 width=5 by lpr_cprs_conf, ex2_intro/ ]
 #U1 #W1 #l1 #HTU1 #HUW1 #IHTU1 #T2 #HT12 #L2 #HL12
@@ -128,13 +128,13 @@ lapply (cpcs_trans … HW1 … HW12) -W /3 width=4/
 qed-.
 
 fact cpds_cprs_lpr_aux: ∀h,g,L0,T0.
-                        (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ssta h g L1 T1) →
-                        (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_cpr_lpr h g L1 T1) →
-                        (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_ssta_cpr_lpr h g L1 T1) →
-                        ∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → ⦃h, L1⦄ ⊢ T1 ¡[g] →
-                        ∀U1. ⦃h, L1⦄ ⊢ T1 •*➡*[g] U1 →
+                        (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_snv_ssta h g L1 T1) →
+                        (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_snv_cpr_lpr h g L1 T1) →
+                        (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_ssta_cpr_lpr h g L1 T1) →
+                        ∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → ⦃h, L1⦄ ⊢ T1 ¡[h, g] →
+                        ∀U1. ⦃h, L1⦄ ⊢ T1 •*➡*[h, g] U1 →
                         ∀T2. L1 ⊢ T1 ➡* T2 → ∀L2. L1 ⊢ ➡ L2 →
-                        ∃∃U2. ⦃h, L2⦄ ⊢ T2 •*➡*[g] U2 & L2 ⊢ U1 ➡* U2.
+                        ∃∃U2. ⦃h, L2⦄ ⊢ T2 •*➡*[h, g] U2 & L2 ⊢ U1 ➡* U2.
 #h #g #L0 #T0 #IH3 #IH2 #IH1 #L1 #T1 #H01 #HT1 #U1 * #W1 #HTW1 #HWU1 #T2 #HT12 #L2 #HL12
 elim (sstas_cprs_lpr_aux … IH3 IH2 IH1 … H01 … HTW1 … HT12 … HL12) // -L0 -T0 -T1 #W2 #HTW2 #HW12
 lapply (lpr_cprs_conf … HL12 … HWU1) -L1 #HWU1
@@ -143,11 +143,11 @@ elim (cpcs_inv_cprs … H) -H /3 width=3/
 qed-.
 
 fact ssta_cpds_aux: ∀h,g,L0,T0.
-                    (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_cpr_lpr h g L1 T1) →
-                    (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_ssta_cpr_lpr h g L1 T1) →
-                    ∀L,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L, T1⦄ → ⦃h, L⦄ ⊢ T1 ¡[g] →
-                    ∀l,U1. ⦃h, L⦄ ⊢ T1 •[g] ⦃l+1, U1⦄ → ∀T2. ⦃h, L⦄ ⊢ T1 •*➡*[g] T2 →
-                    ∃∃U,U2. ⦃h, L⦄ ⊢ U1 •*[g] U & ⦃h, L⦄ ⊢ T2 •*[g] U2 & L ⊢ U ⬌* U2.
+                    (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_snv_cpr_lpr h g L1 T1) →
+                    (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_ssta_cpr_lpr h g L1 T1) →
+                    ∀L,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L, T1⦄ → ⦃G, L⦄ ⊢ T1 ¡[h, g] →
+                    ∀l,U1. ⦃G, L⦄ ⊢ T1 •[h, g] ⦃l+1, U1⦄ → ∀T2. ⦃G, L⦄ ⊢ T1 •*➡*[h, g] T2 →
+                    ∃∃U,U2. ⦃G, L⦄ ⊢ U1 •*[h, g] U & ⦃G, L⦄ ⊢ T2 •*[h, g] U2 & ⦃G, L⦄ ⊢ U ⬌* U2.
 #h #g #L0 #T0 #IH2 #IH1 #L #T1 #H01 #HT1 #l #U1 #HTU1 #T2 * #T #HT1T #HTT2
 elim (sstas_strip … HT1T … HTU1) #HU1T destruct [ -HT1T | -L0 -T0 -T1 ]
 [ elim (ssta_cprs_lpr_aux … IH2 IH1 … HTU1 … HTT2 L) // -L0 -T0 -T /3 width=5/
diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_lift.ma
index eb698c599..9b9513f0a 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_lift.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_lift.ma
@@ -20,8 +20,8 @@ include "basic_2/dynamic/snv.ma".
 
 (* Relocation properties ****************************************************)
 
-lemma snv_lift: ∀h,g,K,T. ⦃h, K⦄ ⊢ T ¡[g] → ∀L,d,e. ⇩[d, e] L ≡ K →
-                ∀U. ⇧[d, e] T ≡ U → ⦃h, L⦄ ⊢ U ¡[g].
+lemma snv_lift: ∀h,g,K,T. ⦃h, K⦄ ⊢ T ¡[h, g] → ∀L,d,e. ⇩[d, e] L ≡ K →
+                ∀U. ⇧[d, e] T ≡ U → ⦃G, L⦄ ⊢ U ¡[h, g].
 #h #g #K #T #H elim H -K -T
 [ #K #k #L #d #e #_ #X #H
   >(lift_inv_sort1 … H) -X -K -d -e //
@@ -50,8 +50,8 @@ lemma snv_lift: ∀h,g,K,T. ⦃h, K⦄ ⊢ T ¡[g] → ∀L,d,e. ⇩[d, e] L ≡
 ]
 qed.
 
-lemma snv_inv_lift: ∀h,g,L,U. ⦃h, L⦄ ⊢ U ¡[g] → ∀K,d,e. ⇩[d, e] L ≡ K →
-                    ∀T. ⇧[d, e] T ≡ U → ⦃h, K⦄ ⊢ T ¡[g].
+lemma snv_inv_lift: ∀h,g,L,U. ⦃G, L⦄ ⊢ U ¡[h, g] → ∀K,d,e. ⇩[d, e] L ≡ K →
+                    ∀T. ⇧[d, e] T ≡ U → ⦃h, K⦄ ⊢ T ¡[h, g].
 #h #g #L #U #H elim H -L -U
 [ #L #k #K #d #e #_ #X #H
   >(lift_inv_sort2 … H) -X -L -d -e //
@@ -81,7 +81,7 @@ qed-.
 (* Advanced properties ******************************************************)
 
 lemma snv_fsup_conf: ∀h,g,L1,L2,T1,T2. ⦃L1, T1⦄ ⊃ ⦃L2, T2⦄ →
-                     ⦃h, L1⦄ ⊢ T1 ¡[g] → ⦃h, L2⦄ ⊢ T2 ¡[g].
+                     ⦃h, L1⦄ ⊢ T1 ¡[h, g] → ⦃h, L2⦄ ⊢ T2 ¡[h, g].
 #h #g #L1 #L2 #T1 #T2 #H elim H -L1 -L2 -T1 -T2
 [ #I1 #L1 #V1 #H
   elim (snv_inv_lref … H) -H #I2 #L2 #V2 #H #HV2
diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_lpr.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_lpr.ma
index 39a60a97a..3007d7d67 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_lpr.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_lpr.ma
@@ -21,10 +21,10 @@ include "basic_2/dynamic/snv_cpcs.ma".
 (* Properties on context-free parallel reduction for local environments *****)
 
 fact snv_cpr_lpr_aux: ∀h,g,L0,T0.
-                      (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_lsubsv h g L1 T1) →
-                      (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_ssta_cpr_lpr h g L1 T1) →
-                      (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ssta h g L1 T1) →
-                      (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_cpr_lpr h g L1 T1) →
+                      (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_snv_lsubsv h g L1 T1) →
+                      (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_ssta_cpr_lpr h g L1 T1) →
+                      (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_snv_ssta h g L1 T1) →
+                      (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_snv_cpr_lpr h g L1 T1) →
                       ∀L1,T1. L0 = L1 → T0 = T1 → IH_snv_cpr_lpr h g L1 T1.
 #h #g #L0 #T0 #IH4 #IH3 #IH2 #IH1 #L1 * * [||||*]
 [ #k #HL0 #HT0 #H1 #X #H2 #L2 #_ destruct -IH4 -IH3 -IH2 -IH1 -H1
diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_ssta.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_ssta.ma
index 766e88393..3ba30937e 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_ssta.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_ssta.ma
@@ -20,9 +20,9 @@ include "basic_2/dynamic/snv_cpcs.ma".
 (* Properties on stratified static type assignment for terms ****************)
 
 fact snv_ssta_aux: ∀h,g,L0,T0.
-                   (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_cpr_lpr h g L1 T1) →
-                   (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_ssta_cpr_lpr h g L1 T1) →
-                   (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ssta h g L1 T1) →
+                   (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_snv_cpr_lpr h g L1 T1) →
+                   (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_ssta_cpr_lpr h g L1 T1) →
+                   (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_snv_ssta h g L1 T1) →
                    ∀L1,T1. L0 = L1 → T0 = T1 → IH_snv_ssta h g L1 T1.
 #h #g #L0 #T0 #IH3 #IH2 #IH1 #L1 * * [||||*]
 [ #k #HL0 #HT0 #_ #X #l #H2 destruct -IH3 -IH2 -IH1
diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_ssta_lpr.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_ssta_lpr.ma
index 6c5e0f6a5..bb07981b0 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_ssta_lpr.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_ssta_lpr.ma
@@ -21,9 +21,9 @@ include "basic_2/dynamic/lsubsv_ssta.ma".
 (* Properties on sn parallel reduction for local environments ***************)
 
 fact ssta_cpr_lpr_aux: ∀h,g,L0,T0.
-                       (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ssta h g L1 T1) →
-                       (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_cpr_lpr h g L1 T1) →
-                       (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_ssta_cpr_lpr h g L1 T1) →
+                       (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_snv_ssta h g L1 T1) →
+                       (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_snv_cpr_lpr h g L1 T1) →
+                       (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_ssta_cpr_lpr h g L1 T1) →
                        ∀L1,T1. L0 = L1 → T0 = T1 → IH_ssta_cpr_lpr h g L1 T1.
 #h #g #L0 #T0 #IH3 #IH2 #IH1 #L1 * * [|||| *]
 [ #k #_ #_ #_ #X2 #l #H2 #X3 #H3 #L2 #HL12 -IH3 -IH2 -IH1
diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_sstas.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_sstas.ma
index a663fecfc..bc2d74497 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_sstas.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_sstas.ma
@@ -19,8 +19,8 @@ include "basic_2/dynamic/snv.ma".
 
 (* Forward_lemmas on iterated stratified static type assignment for terms ***)
 
-lemma snv_sstas_fwd_correct: ∀h,g,L,T1,T2. ⦃h, L⦄ ⊢ T1 ¡[g] → ⦃h, L⦄ ⊢ T1 •* [g] T2 →
-                             ∃∃U2,l. ⦃h, L⦄ ⊢ T2 •[g] ⦃l, U2⦄.
+lemma snv_sstas_fwd_correct: ∀h,g,L,T1,T2. ⦃G, L⦄ ⊢ T1 ¡[h, g] → ⦃G, L⦄ ⊢ T1 •* [h, g] T2 →
+                             ∃∃U2,l. ⦃G, L⦄ ⊢ T2 •[h, g] ⦃l, U2⦄.
 #h #g #L #T1 #T2 #HT1 #HT12
 elim (snv_fwd_ssta … HT1) -HT1 /2 width=5 by sstas_fwd_correct/
 qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/ygt.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/ygt.ma
index b593d637d..7b50bfdcf 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/dynamic/ygt.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/ygt.ma
@@ -19,9 +19,9 @@ include "basic_2/dynamic/yprs.ma".
 (* "BIG TREE" PROPER PARALLEL COMPUTATION FOR CLOSURES **********************)
 
 inductive ygt (h) (g) (L1) (T1): relation2 lenv term ≝
-| ygt_inj : ∀L,L2,T,T2. h ⊢ ⦃L1, T1⦄ ≥[g] ⦃L, T⦄ → h ⊢ ⦃L, T⦄ ≻[g] ⦃L2, T2⦄ →
+| ygt_inj : ∀L,L2,T,T2. h ⊢ ⦃L1, T1⦄ ≥[h, g] ⦃L, T⦄ → h ⊢ ⦃L, T⦄ ≻[h, g] ⦃L2, T2⦄ →
             ygt h g L1 T1 L2 T2
-| ygt_step: ∀L,L2,T. ygt h g L1 T1 L T → L ⊢ ➡ L2 → ygt h g L1 T1 L2 T
+| ygt_step: ∀L,L2,T. ygt h g L1 T1 L T → ⦃G, L⦄ ⊢ ➡ L2 → ygt h g L1 T1 L2 T
 .
 
 interpretation "'big tree' proper parallel computation (closure)"
@@ -29,49 +29,49 @@ interpretation "'big tree' proper parallel computation (closure)"
 
 (* Basic forvard lemmas *****************************************************)
 
-lemma ygt_fwd_yprs: ∀h,g,L1,L2,T1,T2. h ⊢ ⦃L1, T1⦄ >[g] ⦃L2, T2⦄ →
-                    h ⊢ ⦃L1, T1⦄ ≥[g] ⦃L2, T2⦄.
+lemma ygt_fwd_yprs: ∀h,g,L1,L2,T1,T2. h ⊢ ⦃L1, T1⦄ >[h, g] ⦃L2, T2⦄ →
+                    h ⊢ ⦃L1, T1⦄ ≥[h, g] ⦃L2, T2⦄.
 #h #g #L1 #L2 #T1 #T2 #H elim H -L2 -T2
 /3 width=4 by yprs_strap1, ysc_ypr, ypr_lpr/
 qed-.
 
 (* Basic properties *********************************************************)
 
-lemma ysc_ygt: ∀h,g,L1,L2,T1,T2. h ⊢ ⦃L1, T1⦄ ≻[g] ⦃L2, T2⦄ →
-               h ⊢ ⦃L1, T1⦄ >[g] ⦃L2, T2⦄.
+lemma ysc_ygt: ∀h,g,L1,L2,T1,T2. h ⊢ ⦃L1, T1⦄ ≻[h, g] ⦃L2, T2⦄ →
+               h ⊢ ⦃L1, T1⦄ >[h, g] ⦃L2, T2⦄.
 /3 width=4/ qed.
 
-lemma ygt_strap1: ∀h,g,L1,L,L2,T1,T,T2. h ⊢ ⦃L1, T1⦄ >[g] ⦃L, T⦄ →
-                  h ⊢ ⦃L, T⦄ ≽[g] ⦃L2, T2⦄ → h ⊢ ⦃L1, T1⦄ >[g] ⦃L2, T2⦄.
+lemma ygt_strap1: ∀h,g,L1,L,L2,T1,T,T2. h ⊢ ⦃L1, T1⦄ >[h, g] ⦃L, T⦄ →
+                  h ⊢ ⦃L, T⦄ ≽[h, g] ⦃L2, T2⦄ → h ⊢ ⦃L1, T1⦄ >[h, g] ⦃L2, T2⦄.
 #h #g #L1 #L #L2 #T1 #T #T2 #H1 #H2
 lapply (ygt_fwd_yprs … H1) #H0
 elim (ypr_inv_ysc … H2) -H2 [| * #HL2 #H destruct ] /2 width=4/
 qed-.
 
-lemma ygt_strap2: ∀h,g,L1,L,L2,T1,T,T2. h ⊢ ⦃L1, T1⦄ ≽[g] ⦃L, T⦄ →
-                  h ⊢ ⦃L, T⦄ >[g] ⦃L2, T2⦄ → h ⊢ ⦃L1, T1⦄ >[g] ⦃L2, T2⦄.
+lemma ygt_strap2: ∀h,g,L1,L,L2,T1,T,T2. h ⊢ ⦃L1, T1⦄ ≽[h, g] ⦃L, T⦄ →
+                  h ⊢ ⦃L, T⦄ >[h, g] ⦃L2, T2⦄ → h ⊢ ⦃L1, T1⦄ >[h, g] ⦃L2, T2⦄.
 #h #g #L1 #L #L2 #T1 #T #T2 #H1 #H2 elim H2 -L2 -T2
 [ /3 width=4 by ygt_inj, yprs_strap2/ | /2 width=3/ ]
 qed-.
 
-lemma ygt_yprs_trans: ∀h,g,L1,L,L2,T1,T,T2. h ⊢ ⦃L1, T1⦄ >[g] ⦃L, T⦄ →
-                      h ⊢ ⦃L, T⦄ ≥[g] ⦃L2, T2⦄ → h ⊢ ⦃L1, T1⦄ >[g] ⦃L2, T2⦄.
+lemma ygt_yprs_trans: ∀h,g,L1,L,L2,T1,T,T2. h ⊢ ⦃L1, T1⦄ >[h, g] ⦃L, T⦄ →
+                      h ⊢ ⦃L, T⦄ ≥[h, g] ⦃L2, T2⦄ → h ⊢ ⦃L1, T1⦄ >[h, g] ⦃L2, T2⦄.
 #h #g #L1 #L #L2 #T1 #T #T2 #HT1 #HT2 @(yprs_ind … HT2) -L2 -T2 //
 /2 width=4 by ygt_strap1/
 qed-.
 
-lemma yprs_ygt_trans: ∀h,g,L1,L,T1,T. h ⊢ ⦃L1, T1⦄ ≥[g] ⦃L, T⦄ →
-                      ∀L2,T2. h ⊢ ⦃L, T⦄ >[g] ⦃L2, T2⦄ → h ⊢ ⦃L1, T1⦄ >[g] ⦃L2, T2⦄.
+lemma yprs_ygt_trans: ∀h,g,L1,L,T1,T. h ⊢ ⦃L1, T1⦄ ≥[h, g] ⦃L, T⦄ →
+                      ∀L2,T2. h ⊢ ⦃L, T⦄ >[h, g] ⦃L2, T2⦄ → h ⊢ ⦃L1, T1⦄ >[h, g] ⦃L2, T2⦄.
 #h #g #L1 #L #T1 #T #HT1 @(yprs_ind … HT1) -L -T //
 /3 width=4 by ygt_strap2/
 qed-.
 
-lemma fsupp_ygt: ∀h,g,L1,L2,T1,T2. ⦃L1, T1⦄ ⊃+ ⦃L2, T2⦄ → h ⊢ ⦃L1, T1⦄ >[g] ⦃L2, T2⦄.
+lemma fsupp_ygt: ∀h,g,L1,L2,T1,T2. ⦃L1, T1⦄ ⊃+ ⦃L2, T2⦄ → h ⊢ ⦃L1, T1⦄ >[h, g] ⦃L2, T2⦄.
 #h #g #L1 #L2 #T1 #T2 #H @(fsupp_ind … L2 T2 H) -L2 -T2 /3 width=1/ /3 width=4/
 qed.
 
-lemma cprs_ygt: ∀h,g,L,T1,T2. L ⊢ T1 ➡* T2 → (T1 = T2 → ⊥) →
-                h ⊢ ⦃L, T1⦄ >[g] ⦃L, T2⦄.
+lemma cprs_ygt: ∀h,g,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡* T2 → (T1 = T2 → ⊥) →
+                h ⊢ ⦃L, T1⦄ >[h, g] ⦃L, T2⦄.
 #h #g #L #T1 #T2 #H @(cprs_ind … H) -T2
 [ #H elim H //
 | #T #T2 #_ #HT2 #IHT1 #HT12
@@ -83,8 +83,8 @@ lemma cprs_ygt: ∀h,g,L,T1,T2. L ⊢ T1 ➡* T2 → (T1 = T2 → ⊥) →
 ]
 qed.
 
-lemma sstas_ygt: ∀h,g,L,T1,T2. ⦃h, L⦄ ⊢ T1 •*[g] T2 → (T1 = T2 → ⊥) →
-                 h ⊢ ⦃L, T1⦄ >[g] ⦃L, T2⦄.
+lemma sstas_ygt: ∀h,g,L,T1,T2. ⦃G, L⦄ ⊢ T1 •*[h, g] T2 → (T1 = T2 → ⊥) →
+                 h ⊢ ⦃L, T1⦄ >[h, g] ⦃L, T2⦄.
 #h #g #L #T1 #T2 #H @(sstas_ind … H) -T2
 [ #H elim H //
 | #T #T2 #l #_ #HT2 #IHT1 #HT12 -HT12
@@ -96,6 +96,6 @@ lemma sstas_ygt: ∀h,g,L,T1,T2. ⦃h, L⦄ ⊢ T1 •*[g] T2 → (T1 = T2 → 
 ]
 qed.
 
-lemma lsubsv_ygt: ∀h,g,L1,L2,T. h ⊢ L2 ¡⊑[g] L1 → (L1 = L2 → ⊥) →
-                  h ⊢ ⦃L1, T⦄ >[g] ⦃L2, T⦄.
+lemma lsubsv_ygt: ∀h,g,L1,L2,T. h ⊢ L2 ¡⊑[h, g] L1 → (L1 = L2 → ⊥) →
+                  h ⊢ ⦃L1, T⦄ >[h, g] ⦃L2, T⦄.
 /4 width=1/ qed.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/ypr.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/ypr.ma
index a471fe949..84879b10a 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/dynamic/ypr.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/ypr.ma
@@ -23,8 +23,8 @@ inductive ypr (h) (g) (L1) (T1): relation2 lenv term ≝
 | ypr_fsup  : ∀L2,T2. ⦃L1, T1⦄ ⊃ ⦃L2, T2⦄ → ypr h g L1 T1 L2 T2
 | ypr_lpr   : ∀L2. L1 ⊢ ➡ L2 → ypr h g L1 T1 L2 T1
 | ypr_cpr   : ∀T2. L1 ⊢ T1 ➡ T2 → ypr h g L1 T1 L1 T2
-| ypr_ssta  : ∀T2,l. ⦃h, L1⦄ ⊢ T1 •[g] ⦃l+1, T2⦄ → ypr h g L1 T1 L1 T2
-| ypr_lsubsv: ∀L2. h ⊢ L2 ¡⊑[g] L1 → ypr h g L1 T1 L2 T1
+| ypr_ssta  : ∀T2,l. ⦃h, L1⦄ ⊢ T1 •[h, g] ⦃l+1, T2⦄ → ypr h g L1 T1 L1 T2
+| ypr_lsubsv: ∀L2. h ⊢ L2 ¡⊑[h, g] L1 → ypr h g L1 T1 L2 T1
 .
 
 interpretation
diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/yprs.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/yprs.ma
index cc94dff44..a8be72e72 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/dynamic/yprs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/yprs.ma
@@ -28,13 +28,13 @@ interpretation "'big tree' parallel computation (closure)"
 (* Basic eliminators ********************************************************)
 
 lemma yprs_ind: ∀h,g,L1,T1. ∀R:relation2 lenv term. R L1 T1 →
-                (∀L,L2,T,T2. h ⊢ ⦃L1, T1⦄ ≥[g] ⦃L, T⦄ → h ⊢ ⦃L, T⦄ ≽[g] ⦃L2, T2⦄ → R L T → R L2 T2) →
-                ∀L2,T2. h ⊢ ⦃L1, T1⦄ ≥[g] ⦃L2, T2⦄ → R L2 T2.
+                (∀L,L2,T,T2. h ⊢ ⦃L1, T1⦄ ≥[h, g] ⦃L, T⦄ → h ⊢ ⦃L, T⦄ ≽[h, g] ⦃L2, T2⦄ → R L T → R L2 T2) →
+                ∀L2,T2. h ⊢ ⦃L1, T1⦄ ≥[h, g] ⦃L2, T2⦄ → R L2 T2.
 /3 width=7 by bi_TC_star_ind/ qed-.
 
 lemma yprs_ind_dx: ∀h,g,L2,T2. ∀R:relation2 lenv term. R L2 T2 →
-                   (∀L1,L,T1,T. h ⊢ ⦃L1, T1⦄ ≽[g] ⦃L, T⦄ → h ⊢ ⦃L, T⦄ ≥[g] ⦃L2, T2⦄ → R L T → R L1 T1) →
-                   ∀L1,T1. h ⊢ ⦃L1, T1⦄ ≥[g] ⦃L2, T2⦄ → R L1 T1.
+                   (∀L1,L,T1,T. h ⊢ ⦃L1, T1⦄ ≽[h, g] ⦃L, T⦄ → h ⊢ ⦃L, T⦄ ≥[h, g] ⦃L2, T2⦄ → R L T → R L1 T1) →
+                   ∀L1,T1. h ⊢ ⦃L1, T1⦄ ≥[h, g] ⦃L2, T2⦄ → R L1 T1.
 /3 width=7 by bi_TC_star_ind_dx/ qed-.
 
 (* Basic properties *********************************************************)
@@ -42,41 +42,41 @@ lemma yprs_ind_dx: ∀h,g,L2,T2. ∀R:relation2 lenv term. R L2 T2 →
 lemma yprs_refl: ∀h,g. bi_reflexive … (yprs h g).
 /2 width=1/ qed.
 
-lemma ypr_yprs: ∀h,g,L1,L2,T1,T2. h ⊢ ⦃L1, T1⦄ ≽[g] ⦃L2, T2⦄ →
-                h ⊢ ⦃L1, T1⦄ ≥[g] ⦃L2, T2⦄.
+lemma ypr_yprs: ∀h,g,L1,L2,T1,T2. h ⊢ ⦃L1, T1⦄ ≽[h, g] ⦃L2, T2⦄ →
+                h ⊢ ⦃L1, T1⦄ ≥[h, g] ⦃L2, T2⦄.
 /2 width=1/ qed.
 
-lemma yprs_strap1: ∀h,g,L1,L,L2,T1,T,T2. h ⊢ ⦃L1, T1⦄ ≥[g] ⦃L, T⦄ →
-                   h ⊢ ⦃L, T⦄ ≽[g] ⦃L2, T2⦄ → h ⊢ ⦃L1, T1⦄ ≥[g] ⦃L2, T2⦄.
+lemma yprs_strap1: ∀h,g,L1,L,L2,T1,T,T2. h ⊢ ⦃L1, T1⦄ ≥[h, g] ⦃L, T⦄ →
+                   h ⊢ ⦃L, T⦄ ≽[h, g] ⦃L2, T2⦄ → h ⊢ ⦃L1, T1⦄ ≥[h, g] ⦃L2, T2⦄.
 /2 width=4/ qed-.
 
-lemma yprs_strap2: ∀h,g,L1,L,L2,T1,T,T2. h ⊢ ⦃L1, T1⦄ ≽[g] ⦃L, T⦄ →
-                   h ⊢ ⦃L, T⦄ ≥[g] ⦃L2, T2⦄ → h ⊢ ⦃L1, T1⦄ ≥[g] ⦃L2, T2⦄.
+lemma yprs_strap2: ∀h,g,L1,L,L2,T1,T,T2. h ⊢ ⦃L1, T1⦄ ≽[h, g] ⦃L, T⦄ →
+                   h ⊢ ⦃L, T⦄ ≥[h, g] ⦃L2, T2⦄ → h ⊢ ⦃L1, T1⦄ ≥[h, g] ⦃L2, T2⦄.
 /2 width=4/ qed-.
 
 (* Note: this is a general property of bi_TC *)
 lemma fsupp_yprs: ∀h,g,L1,L2,T1,T2. ⦃L1, T1⦄ ⊃+ ⦃L2, T2⦄ →
-                  h ⊢ ⦃L1, T1⦄ ≥[g] ⦃L2, T2⦄.
+                  h ⊢ ⦃L1, T1⦄ ≥[h, g] ⦃L2, T2⦄.
 #h #g #L1 #L2 #T1 #T2 #H @(fsupp_ind … L2 T2 H) -L2 -T2 /3 width=1/ /3 width=4/
 qed.
 
-lemma cprs_yprs: ∀h,g,L,T1,T2. L ⊢ T1 ➡* T2 → h ⊢ ⦃L, T1⦄ ≥[g] ⦃L, T2⦄.
+lemma cprs_yprs: ∀h,g,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡* T2 → h ⊢ ⦃L, T1⦄ ≥[h, g] ⦃L, T2⦄.
 #h #g #L #T1 #T2 #H @(cprs_ind … H) -T2 // /3 width=4 by ypr_cpr, yprs_strap1/
 qed.
 
-lemma lprs_yprs: ∀h,g,L1,L2,T. L1 ⊢ ➡* L2 → h ⊢ ⦃L1, T⦄ ≥[g] ⦃L2, T⦄.
+lemma lprs_yprs: ∀h,g,L1,L2,T. L1 ⊢ ➡* L2 → h ⊢ ⦃L1, T⦄ ≥[h, g] ⦃L2, T⦄.
 #h #g #L1 #L2 #T #H @(lprs_ind … H) -L2 // /3 width=4 by ypr_lpr, yprs_strap1/
 qed.
 
-lemma sstas_yprs: ∀h,g,L,T1,T2. ⦃h, L⦄ ⊢ T1 •*[g] T2 →
-                  h ⊢ ⦃L, T1⦄ ≥[g] ⦃L, T2⦄.
+lemma sstas_yprs: ∀h,g,L,T1,T2. ⦃G, L⦄ ⊢ T1 •*[h, g] T2 →
+                  h ⊢ ⦃L, T1⦄ ≥[h, g] ⦃L, T2⦄.
 #h #g #L #T1 #T2 #H @(sstas_ind … H) -T2 // /3 width=4 by ypr_ssta, yprs_strap1/
 qed.
 
-lemma lsubsv_yprs: ∀h,g,L1,L2,T. h ⊢ L2 ¡⊑[g] L1 → h ⊢ ⦃L1, T⦄ ≥[g] ⦃L2, T⦄.
+lemma lsubsv_yprs: ∀h,g,L1,L2,T. h ⊢ L2 ¡⊑[h, g] L1 → h ⊢ ⦃L1, T⦄ ≥[h, g] ⦃L2, T⦄.
 /3 width=1/ qed.
 
 lemma cprs_lpr_yprs: ∀h,g,L1,L2,T1,T2. L1 ⊢ T1 ➡* T2 → L1 ⊢ ➡ L2 →
-                     h ⊢ ⦃L1, T1⦄ ≥[g] ⦃L2, T2⦄.
+                     h ⊢ ⦃L1, T1⦄ ≥[h, g] ⦃L2, T2⦄.
 /3 width=4 by yprs_strap1, ypr_lpr, cprs_yprs/
 qed.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/ysc.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/ysc.ma
index 87ca83fb1..09785875c 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/dynamic/ysc.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/ysc.ma
@@ -20,8 +20,8 @@ include "basic_2/dynamic/ypr.ma".
 inductive ysc (h) (g) (L1) (T1): relation2 lenv term ≝
 | ysc_fsup  : ∀L2,T2. ⦃L1, T1⦄ ⊃ ⦃L2, T2⦄ → ysc h g L1 T1 L2 T2
 | ysc_cpr   : ∀T2. L1 ⊢ T1 ➡ T2 → (T1 = T2 → ⊥) → ysc h g L1 T1 L1 T2
-| ysc_ssta  : ∀T2,l. ⦃h, L1⦄ ⊢ T1 •[g] ⦃l+1, T2⦄ → ysc h g L1 T1 L1 T2
-| ysc_lsubsv: ∀L2. h ⊢ L2 ¡⊑[g] L1 → (L1 = L2 → ⊥) → ysc h g L1 T1 L2 T1
+| ysc_ssta  : ∀T2,l. ⦃h, L1⦄ ⊢ T1 •[h, g] ⦃l+1, T2⦄ → ysc h g L1 T1 L1 T2
+| ysc_lsubsv: ∀L2. h ⊢ L2 ¡⊑[h, g] L1 → (L1 = L2 → ⊥) → ysc h g L1 T1 L2 T1
 .
 
 interpretation
@@ -30,15 +30,15 @@ interpretation
 
 (* Basic properties *********************************************************)
 
-lemma ysc_ypr: ∀h,g,L1,L2,T1,T2. h ⊢ ⦃L1, T1⦄ ≻[g] ⦃L2, T2⦄ →
-               h ⊢ ⦃L1, T1⦄ ≽[g] ⦃L2, T2⦄.
+lemma ysc_ypr: ∀h,g,L1,L2,T1,T2. h ⊢ ⦃L1, T1⦄ ≻[h, g] ⦃L2, T2⦄ →
+               h ⊢ ⦃L1, T1⦄ ≽[h, g] ⦃L2, T2⦄.
 #h #g #L1 #L2 #T1 #T2 * -L2 -T2 /2 width=1/ /2 width=2/
 qed.
 
 (* Inversion lemmas on "big tree" parallel reduction for closures ***********)
 
-lemma ypr_inv_ysc: ∀h,g,L1,L2,T1,T2. h ⊢ ⦃L1, T1⦄ ≽[g] ⦃L2, T2⦄ →
-                   h ⊢ ⦃L1, T1⦄ ≻[g] ⦃L2, T2⦄ ∨ (L1 ⊢ ➡ L2 ∧ T1 = T2).
+lemma ypr_inv_ysc: ∀h,g,L1,L2,T1,T2. h ⊢ ⦃L1, T1⦄ ≽[h, g] ⦃L2, T2⦄ →
+                   h ⊢ ⦃L1, T1⦄ ≻[h, g] ⦃L2, T2⦄ ∨ (L1 ⊢ ➡ L2 ∧ T1 = T2).
 #h #g #L1 #L2 #T1 #T2 * -L2 -T2 /3 width=1/ /3 width=2/
 [ #T2 #HT12 elim (term_eq_dec T1 T2) #H destruct /3 width=1/ /4 width=1/
 | #L2 #HL21 elim (lenv_eq_dec L1 L2) #H destruct /3 width=1/ /4 width=1/
diff --git a/matita/matita/contribs/lambdadelta/basic_2/equivalence/cpcs.ma b/matita/matita/contribs/lambdadelta/basic_2/equivalence/cpcs.ma
index d26648839..a7e5b7514 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/equivalence/cpcs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/equivalence/cpcs.ma
@@ -25,14 +25,14 @@ interpretation "context-sensitive parallel equivalence (term)"
 (* Basic eliminators ********************************************************)
 
 lemma cpcs_ind: ∀L,T1. ∀R:predicate term. R T1 →
-                (∀T,T2. L ⊢ T1 ⬌* T → L ⊢ T ⬌ T2 → R T → R T2) →
-                ∀T2. L ⊢ T1 ⬌* T2 → R T2.
+                (∀T,T2. ⦃G, L⦄ ⊢ T1 ⬌* T → ⦃G, L⦄ ⊢ T ⬌ T2 → R T → R T2) →
+                ∀T2. ⦃G, L⦄ ⊢ T1 ⬌* T2 → R T2.
 #L #T1 #R #HT1 #IHT1 #T2 #HT12 @(TC_star_ind … HT1 IHT1 … HT12) //
 qed-.
 
 lemma cpcs_ind_dx: ∀L,T2. ∀R:predicate term. R T2 →
-                   (∀T1,T. L ⊢ T1 ⬌ T → L ⊢ T ⬌* T2 → R T → R T1) →
-                   ∀T1. L ⊢ T1 ⬌* T2 → R T1.
+                   (∀T1,T. ⦃G, L⦄ ⊢ T1 ⬌ T → ⦃G, L⦄ ⊢ T ⬌* T2 → R T → R T1) →
+                   ∀T1. ⦃G, L⦄ ⊢ T1 ⬌* T2 → R T1.
 #L #T2 #R #HT2 #IHT2 #T1 #HT12
 @(TC_star_ind_dx … HT2 IHT2 … HT12) //
 qed-.
@@ -47,38 +47,38 @@ lemma cpcs_refl: ∀L. reflexive … (cpcs L).
 lemma cpcs_sym: ∀L. symmetric … (cpcs L).
 #L @TC_symmetric // qed.
 
-lemma cpc_cpcs: ∀L,T1,T2. L ⊢ T1 ⬌ T2 → L ⊢ T2 ⬌* T2.
+lemma cpc_cpcs: ∀L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬌ T2 → ⦃G, L⦄ ⊢ T2 ⬌* T2.
 /2 width=1/ qed.
 
-lemma cpcs_strap1: ∀L,T1,T,T2. L ⊢ T1 ⬌* T → L ⊢ T ⬌ T2 → L ⊢ T1 ⬌* T2.
+lemma cpcs_strap1: ∀L,T1,T,T2. ⦃G, L⦄ ⊢ T1 ⬌* T → ⦃G, L⦄ ⊢ T ⬌ T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
 #L @step qed.
 
-lemma cpcs_strap2: ∀L,T1,T,T2. L ⊢ T1 ⬌ T → L ⊢ T ⬌* T2 → L ⊢ T1 ⬌* T2.
+lemma cpcs_strap2: ∀L,T1,T,T2. ⦃G, L⦄ ⊢ T1 ⬌ T → ⦃G, L⦄ ⊢ T ⬌* T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
 #L @TC_strap qed.
 
 (* Basic_1: was: pc3_pr2_r *)
-lemma cpr_cpcs_dx: ∀L,T1,T2. L ⊢ T1 ➡ T2 → L ⊢ T1 ⬌* T2.
+lemma cpr_cpcs_dx: ∀L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡ T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
 /3 width=1/ qed.
 
 (* Basic_1: was: pc3_pr2_x *)
-lemma cpr_cpcs_sn: ∀L,T1,T2. L ⊢ T2 ➡ T1 → L ⊢ T1 ⬌* T2.
+lemma cpr_cpcs_sn: ∀L,T1,T2. ⦃G, L⦄ ⊢ T2 ➡ T1 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
 /3 width=1/ qed.
 
-lemma cpcs_cpr_strap1: ∀L,T1,T. L ⊢ T1 ⬌* T → ∀T2. L ⊢ T ➡ T2 → L ⊢ T1 ⬌* T2.
+lemma cpcs_cpr_strap1: ∀L,T1,T. ⦃G, L⦄ ⊢ T1 ⬌* T → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
 /3 width=3/ qed.
 
 (* Basic_1: was: pc3_pr2_u *)
-lemma cpcs_cpr_strap2: ∀L,T1,T. L ⊢ T1 ➡ T → ∀T2. L ⊢ T ⬌* T2 → L ⊢ T1 ⬌* T2.
+lemma cpcs_cpr_strap2: ∀L,T1,T. ⦃G, L⦄ ⊢ T1 ➡ T → ∀T2. ⦃G, L⦄ ⊢ T ⬌* T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
 /3 width=3/ qed.
 
-lemma cpcs_cpr_div: ∀L,T1,T. L ⊢ T1 ⬌* T → ∀T2. L ⊢ T2 ➡ T → L ⊢ T1 ⬌* T2.
+lemma cpcs_cpr_div: ∀L,T1,T. ⦃G, L⦄ ⊢ T1 ⬌* T → ∀T2. ⦃G, L⦄ ⊢ T2 ➡ T → ⦃G, L⦄ ⊢ T1 ⬌* T2.
 /3 width=3/ qed.
 
-lemma cpr_div: ∀L,T1,T. L ⊢ T1 ➡ T → ∀T2. L ⊢ T2 ➡ T → L ⊢ T1 ⬌* T2.
+lemma cpr_div: ∀L,T1,T. ⦃G, L⦄ ⊢ T1 ➡ T → ∀T2. ⦃G, L⦄ ⊢ T2 ➡ T → ⦃G, L⦄ ⊢ T1 ⬌* T2.
 /3 width=3/ qed-.
 
 (* Basic_1: was: pc3_pr2_u2 *)
-lemma cpcs_cpr_conf: ∀L,T1,T. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ⬌* T2 → L ⊢ T1 ⬌* T2.
+lemma cpcs_cpr_conf: ∀L,T1,T. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ⬌* T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
 /3 width=3/ qed.
 
 (* Basic_1: removed theorems 9:
diff --git a/matita/matita/contribs/lambdadelta/basic_2/equivalence/cpcs_aaa.ma b/matita/matita/contribs/lambdadelta/basic_2/equivalence/cpcs_aaa.ma
index 190092f86..1a5e48687 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/equivalence/cpcs_aaa.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/equivalence/cpcs_aaa.ma
@@ -19,8 +19,8 @@ include "basic_2/equivalence/cpcs_cpcs.ma".
 
 (* Main properties about atomic arity assignment on terms *******************)
 
-theorem aaa_cpcs_mono: ∀L,T1,T2. L ⊢ T1 ⬌* T2 →
-                       ∀A1. L ⊢ T1 ⁝ A1 → ∀A2. L ⊢ T2 ⁝ A2 →
+theorem aaa_cpcs_mono: ∀L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬌* T2 →
+                       ∀A1. ⦃G, L⦄ ⊢ T1 ⁝ A1 → ∀A2. ⦃G, L⦄ ⊢ T2 ⁝ A2 →
                        A1 = A2.
 #L #T1 #T2 #HT12 #A1 #HA1 #A2 #HA2
 elim (cpcs_inv_cprs … HT12) -HT12 #T #HT1 #HT2
diff --git a/matita/matita/contribs/lambdadelta/basic_2/equivalence/cpcs_cpcs.ma b/matita/matita/contribs/lambdadelta/basic_2/equivalence/cpcs_cpcs.ma
index 44cbd7491..125cc70fe 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/equivalence/cpcs_cpcs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/equivalence/cpcs_cpcs.ma
@@ -20,8 +20,8 @@ include "basic_2/equivalence/cpcs_cprs.ma".
 
 (* Advanced inversion lemmas ************************************************)
 
-lemma cpcs_inv_cprs: ∀L,T1,T2. L ⊢ T1 ⬌* T2 →
-                     ∃∃T. L ⊢ T1 ➡* T & L ⊢ T2 ➡* T.
+lemma cpcs_inv_cprs: ∀L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬌* T2 →
+                     ∃∃T. ⦃G, L⦄ ⊢ T1 ➡* T & ⦃G, L⦄ ⊢ T2 ➡* T.
 #L #T1 #T2 #H @(cpcs_ind … H) -T2
 [ /3 width=3/
 | #T #T2 #_ #HT2 * #T0 #HT10 elim HT2 -HT2 #HT2 #HT0
@@ -33,27 +33,27 @@ lemma cpcs_inv_cprs: ∀L,T1,T2. L ⊢ T1 ⬌* T2 →
 qed-.
 
 (* Basic_1: was: pc3_gen_sort *)
-lemma cpcs_inv_sort: ∀L,k1,k2. L ⊢ ⋆k1 ⬌* ⋆k2 → k1 = k2.
+lemma cpcs_inv_sort: ∀L,k1,k2. ⦃G, L⦄ ⊢ ⋆k1 ⬌* ⋆k2 → k1 = k2.
 #L #k1 #k2 #H
 elim (cpcs_inv_cprs … H) -H #T #H1
 >(cprs_inv_sort1 … H1) -T #H2
 lapply (cprs_inv_sort1 … H2) -L #H destruct //
 qed-.
 
-lemma cpcs_inv_abst1: ∀a,L,W1,T1,T. L ⊢ ⓛ{a}W1.T1 ⬌* T →
-                      ∃∃W2,T2. L ⊢ T ➡* ⓛ{a}W2.T2 & L ⊢ ⓛ{a}W1.T1 ➡* ⓛ{a}W2.T2.
+lemma cpcs_inv_abst1: ∀a,L,W1,T1,T. ⦃G, L⦄ ⊢ ⓛ{a}W1.T1 ⬌* T →
+                      ∃∃W2,T2. ⦃G, L⦄ ⊢ T ➡* ⓛ{a}W2.T2 & ⦃G, L⦄ ⊢ ⓛ{a}W1.T1 ➡* ⓛ{a}W2.T2.
 #a #L #W1 #T1 #T #H
 elim (cpcs_inv_cprs … H) -H #X #H1 #H2
 elim (cprs_inv_abst1 … H1) -H1 #W2 #T2 #HW12 #HT12 #H destruct
 @(ex2_2_intro … H2) -H2 /2 width=2/ (**) (* explicit constructor, /3 width=6/ is slow *)
 qed-.
 
-lemma cpcs_inv_abst2: ∀a,L,W1,T1,T. L ⊢ T ⬌* ⓛ{a}W1.T1 →
-                      ∃∃W2,T2. L ⊢ T ➡* ⓛ{a}W2.T2 & L ⊢ ⓛ{a}W1.T1 ➡* ⓛ{a}W2.T2.
+lemma cpcs_inv_abst2: ∀a,L,W1,T1,T. ⦃G, L⦄ ⊢ T ⬌* ⓛ{a}W1.T1 →
+                      ∃∃W2,T2. ⦃G, L⦄ ⊢ T ➡* ⓛ{a}W2.T2 & ⦃G, L⦄ ⊢ ⓛ{a}W1.T1 ➡* ⓛ{a}W2.T2.
 /3 width=1 by cpcs_inv_abst1, cpcs_sym/ qed-.
 
 (* Basic_1: was: pc3_gen_sort_abst *)
-lemma cpcs_inv_sort_abst: ∀a,L,W,T,k. L ⊢ ⋆k ⬌* ⓛ{a}W.T → ⊥.
+lemma cpcs_inv_sort_abst: ∀a,L,W,T,k. ⦃G, L⦄ ⊢ ⋆k ⬌* ⓛ{a}W.T → ⊥.
 #a #L #W #T #k #H
 elim (cpcs_inv_cprs … H) -H #X #H1
 >(cprs_inv_sort1 … H1) -X #H2
@@ -63,7 +63,7 @@ qed-.
 (* Basic_1: was: pc3_gen_lift *)
 lemma cpcs_inv_lift: ∀L,K,d,e. ⇩[d, e] L ≡ K →
                      ∀T1,U1. ⇧[d, e] T1 ≡ U1 → ∀T2,U2. ⇧[d, e] T2 ≡ U2 →
-                     L ⊢ U1 ⬌* U2 → K ⊢ T1 ⬌* T2.
+                     ⦃G, L⦄ ⊢ U1 ⬌* U2 → K ⊢ T1 ⬌* T2.
 #L #K #d #e #HLK #T1 #U1 #HTU1 #T2 #U2 #HTU2 #HU12
 elim (cpcs_inv_cprs … HU12) -HU12 #U #HU1 #HU2
 elim (cprs_inv_lift1 … HU1 … HLK … HTU1) -U1 #T #HTU #HT1
@@ -87,17 +87,17 @@ lapply (lprs_cprs_trans … HT1 … HL12) -HT1
 lapply (lprs_cprs_trans … HT2 … HL12) -L2 /2 width=3/
 qed-.
 
-lemma cpr_cprs_conf_cpcs: ∀L,T,T1,T2. L ⊢ T ➡* T1 → L ⊢ T ➡ T2 → L ⊢ T1 ⬌* T2.
+lemma cpr_cprs_conf_cpcs: ∀L,T,T1,T2. ⦃G, L⦄ ⊢ T ➡* T1 → ⦃G, L⦄ ⊢ T ➡ T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
 #L #T #T1 #T2 #HT1 #HT2
 elim (cprs_strip … HT1 … HT2) /2 width=3 by cpr_cprs_div/
 qed-.
 
-lemma cprs_cpr_conf_cpcs: ∀L,T,T1,T2. L ⊢ T ➡* T1 → L ⊢ T ➡ T2 → L ⊢ T2 ⬌* T1.
+lemma cprs_cpr_conf_cpcs: ∀L,T,T1,T2. ⦃G, L⦄ ⊢ T ➡* T1 → ⦃G, L⦄ ⊢ T ➡ T2 → ⦃G, L⦄ ⊢ T2 ⬌* T1.
 #L #T #T1 #T2 #HT1 #HT2
 elim (cprs_strip … HT1 … HT2) /2 width=3 by cprs_cpr_div/
 qed-.
 
-lemma cprs_conf_cpcs: ∀L,T,T1,T2. L ⊢ T ➡* T1 → L ⊢ T ➡* T2 → L ⊢ T1 ⬌* T2.
+lemma cprs_conf_cpcs: ∀L,T,T1,T2. ⦃G, L⦄ ⊢ T ➡* T1 → ⦃G, L⦄ ⊢ T ➡* T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
 #L #T #T1 #T2 #HT1 #HT2
 elim (cprs_conf … HT1 … HT2) /2 width=3/
 qed-.
@@ -118,24 +118,24 @@ lemma lpr_cpr_conf: ∀L1,L2. L1 ⊢ ➡ L2 → ∀T1,T2. L1 ⊢ T1 ➡ T2 → L
 /3 width=5 by lpr_cprs_conf, cpr_cprs/ qed-.
 
 (* Basic_1: was only: pc3_thin_dx *)
-lemma cpcs_flat: ∀L,V1,V2. L ⊢ V1 ⬌* V2 → ∀T1,T2. L ⊢ T1 ⬌* T2 →
-                 ∀I. L ⊢ ⓕ{I}V1. T1 ⬌* ⓕ{I}V2. T2.
+lemma cpcs_flat: ∀L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬌* V2 → ∀T1,T2. ⦃G, L⦄ ⊢ T1 ⬌* T2 →
+                 ∀I. ⦃G, L⦄ ⊢ ⓕ{I}V1. T1 ⬌* ⓕ{I}V2. T2.
 #L #V1 #V2 #HV12 #T1 #T2 #HT12 #I
 elim (cpcs_inv_cprs … HV12) -HV12 #V #HV1 #HV2
 elim (cpcs_inv_cprs … HT12) -HT12 /3 width=5 by cprs_flat, cprs_div/ (**) (* /3 width=5/ is too slow *)
 qed.
 
-lemma cpcs_flat_dx_cpr_rev: ∀L,V1,V2. L ⊢ V2 ➡ V1 → ∀T1,T2. L ⊢ T1 ⬌* T2 →
-                            ∀I. L ⊢ ⓕ{I}V1. T1 ⬌* ⓕ{I}V2. T2.
+lemma cpcs_flat_dx_cpr_rev: ∀L,V1,V2. ⦃G, L⦄ ⊢ V2 ➡ V1 → ∀T1,T2. ⦃G, L⦄ ⊢ T1 ⬌* T2 →
+                            ∀I. ⦃G, L⦄ ⊢ ⓕ{I}V1. T1 ⬌* ⓕ{I}V2. T2.
 /3 width=1/ qed.
 
 lemma cpcs_bind_dx: ∀a,I,L,V,T1,T2. L.ⓑ{I}V ⊢ T1 ⬌* T2 →
-                    L ⊢ ⓑ{a,I}V. T1 ⬌* ⓑ{a,I}V. T2.
+                    ⦃G, L⦄ ⊢ ⓑ{a,I}V. T1 ⬌* ⓑ{a,I}V. T2.
 #a #I #L #V #T1 #T2 #HT12
 elim (cpcs_inv_cprs … HT12) -HT12 /3 width=5 by cprs_div, cprs_bind/ (**) (* /3 width=5/ is a bit slow *)
 qed.
 
-lemma cpcs_bind_sn: ∀a,I,L,V1,V2,T. L ⊢ V1 ⬌* V2 → L ⊢ ⓑ{a,I}V1. T ⬌* ⓑ{a,I}V2. T.
+lemma cpcs_bind_sn: ∀a,I,L,V1,V2,T. ⦃G, L⦄ ⊢ V1 ⬌* V2 → ⦃G, L⦄ ⊢ ⓑ{a,I}V1. T ⬌* ⓑ{a,I}V2. T.
 #a #I #L #V1 #V2 #T #HV12
 elim (cpcs_inv_cprs … HV12) -HV12 /3 width=5 by cprs_div, cprs_bind/ (**) (* /3 width=5/ is a bit slow *)
 qed.
@@ -150,7 +150,7 @@ qed-.
 (* Basic_1: was: pc3_lift *)
 lemma cpcs_lift: ∀L,K,d,e. ⇩[d, e] L ≡ K →
                  ∀T1,U1. ⇧[d, e] T1 ≡ U1 → ∀T2,U2. ⇧[d, e] T2 ≡ U2 →
-                 K ⊢ T1 ⬌* T2 → L ⊢ U1 ⬌* U2.
+                 K ⊢ T1 ⬌* T2 → ⦃G, L⦄ ⊢ U1 ⬌* U2.
 #L #K #d #e #HLK #T1 #U1 #HTU1 #T2 #U2 #HTU2 #HT12
 elim (cpcs_inv_cprs … HT12) -HT12 #T #HT1 #HT2
 elim (lift_total T d e) #U #HTU
@@ -158,14 +158,14 @@ lapply (cprs_lift … HT1 … HLK … HTU1 … HTU) -T1 #HU1
 lapply (cprs_lift … HT2 … HLK … HTU2 … HTU) -K -T2 -T -d -e /2 width=3/
 qed.
 
-lemma cpcs_strip: ∀L,T1,T. L ⊢ T ⬌* T1 → ∀T2. L ⊢ T ⬌ T2 →
-                  ∃∃T0. L ⊢ T1 ⬌ T0 & L ⊢ T2 ⬌* T0.
+lemma cpcs_strip: ∀L,T1,T. ⦃G, L⦄ ⊢ T ⬌* T1 → ∀T2. ⦃G, L⦄ ⊢ T ⬌ T2 →
+                  ∃∃T0. ⦃G, L⦄ ⊢ T1 ⬌ T0 & ⦃G, L⦄ ⊢ T2 ⬌* T0.
 #L #T1 #T @TC_strip1 /2 width=3/ qed-.
 
 (* More inversion lemmas ****************************************************)
 
-lemma cpcs_inv_abst_sn: ∀a1,a2,L,W1,W2,T1,T2. L ⊢ ⓛ{a1}W1.T1 ⬌* ⓛ{a2}W2.T2 →
-                        ∧∧ L ⊢ W1 ⬌* W2 & L.ⓛW1 ⊢ T1 ⬌* T2 & a1 = a2.
+lemma cpcs_inv_abst_sn: ∀a1,a2,L,W1,W2,T1,T2. ⦃G, L⦄ ⊢ ⓛ{a1}W1.T1 ⬌* ⓛ{a2}W2.T2 →
+                        ∧∧ ⦃G, L⦄ ⊢ W1 ⬌* W2 & L.ⓛW1 ⊢ T1 ⬌* T2 & a1 = a2.
 #a1 #a2 #L #W1 #W2 #T1 #T2 #H
 elim (cpcs_inv_cprs … H) -H #T #H1 #H2
 elim (cprs_inv_abst1 … H1) -H1 #W0 #T0 #HW10 #HT10 #H destruct
@@ -175,8 +175,8 @@ lapply (lprs_cpcs_trans … (L.ⓛW1) … HT2) /2 width=1/ -HT2 #HT2
 /4 width=3 by and3_intro, cprs_div, cpcs_cprs_div, cpcs_sym/
 qed-.
 
-lemma cpcs_inv_abst_dx: ∀a1,a2,L,W1,W2,T1,T2. L ⊢ ⓛ{a1}W1.T1 ⬌* ⓛ{a2}W2.T2 →
-                        ∧∧ L ⊢ W1 ⬌* W2 & L. ⓛW2 ⊢ T1 ⬌* T2 & a1 = a2.
+lemma cpcs_inv_abst_dx: ∀a1,a2,L,W1,W2,T1,T2. ⦃G, L⦄ ⊢ ⓛ{a1}W1.T1 ⬌* ⓛ{a2}W2.T2 →
+                        ∧∧ ⦃G, L⦄ ⊢ W1 ⬌* W2 & L. ⓛW2 ⊢ T1 ⬌* T2 & a1 = a2.
 #a1 #a2 #L #W1 #W2 #T1 #T2 #HT12
 lapply (cpcs_sym … HT12) -HT12 #HT12
 elim (cpcs_inv_abst_sn … HT12) -HT12 /3 width=1/
@@ -185,23 +185,23 @@ qed-.
 (* Main properties **********************************************************)
 
 (* Basic_1: was pc3_t *)
-theorem cpcs_trans: ∀L,T1,T. L ⊢ T1 ⬌* T → ∀T2. L ⊢ T ⬌* T2 → L ⊢ T1 ⬌* T2.
+theorem cpcs_trans: ∀L,T1,T. ⦃G, L⦄ ⊢ T1 ⬌* T → ∀T2. ⦃G, L⦄ ⊢ T ⬌* T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
 #L #T1 #T #HT1 #T2 @(trans_TC … HT1) qed-.
 
-theorem cpcs_canc_sn: ∀L,T,T1,T2. L ⊢ T ⬌* T1 → L ⊢ T ⬌* T2 → L ⊢ T1 ⬌* T2.
+theorem cpcs_canc_sn: ∀L,T,T1,T2. ⦃G, L⦄ ⊢ T ⬌* T1 → ⦃G, L⦄ ⊢ T ⬌* T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
 /3 width=3 by cpcs_trans, cpcs_sym/ qed-. (**) (* /3 width=3/ is too slow *)
 
-theorem cpcs_canc_dx: ∀L,T,T1,T2. L ⊢ T1 ⬌* T → L ⊢ T2 ⬌* T → L ⊢ T1 ⬌* T2.
+theorem cpcs_canc_dx: ∀L,T,T1,T2. ⦃G, L⦄ ⊢ T1 ⬌* T → ⦃G, L⦄ ⊢ T2 ⬌* T → ⦃G, L⦄ ⊢ T1 ⬌* T2.
 /3 width=3 by cpcs_trans, cpcs_sym/ qed-. (**) (* /3 width=3/ is too slow *)
 
-lemma cpcs_bind1: ∀a,I,L,V1,V2. L ⊢ V1 ⬌* V2 → ∀T1,T2. L.ⓑ{I}V1 ⊢ T1 ⬌* T2 →
-                  L ⊢ ⓑ{a,I}V1. T1 ⬌* ⓑ{a,I}V2. T2.
+lemma cpcs_bind1: ∀a,I,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬌* V2 → ∀T1,T2. L.ⓑ{I}V1 ⊢ T1 ⬌* T2 →
+                  ⦃G, L⦄ ⊢ ⓑ{a,I}V1. T1 ⬌* ⓑ{a,I}V2. T2.
 #a #I #L #V1 #V2 #HV12 #T1 #T2 #HT12
 @(cpcs_trans … (ⓑ{a,I}V1.T2)) /2 width=1/
 qed.
 
-lemma cpcs_bind2: ∀a,I,L,V1,V2. L ⊢ V1 ⬌* V2 → ∀T1,T2. L.ⓑ{I}V2 ⊢ T1 ⬌* T2 →
-                  L ⊢ ⓑ{a,I}V1. T1 ⬌* ⓑ{a,I}V2. T2.
+lemma cpcs_bind2: ∀a,I,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬌* V2 → ∀T1,T2. L.ⓑ{I}V2 ⊢ T1 ⬌* T2 →
+                  ⦃G, L⦄ ⊢ ⓑ{a,I}V1. T1 ⬌* ⓑ{a,I}V2. T2.
 #a #I #L #V1 #V2 #HV12 #T1 #T2 #HT12
 @(cpcs_trans … (ⓑ{a,I}V2.T1)) /2 width=1/
 qed.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/equivalence/cpcs_cprs.ma b/matita/matita/contribs/lambdadelta/basic_2/equivalence/cpcs_cprs.ma
index 8c1fec2e1..30ee9fb9d 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/equivalence/cpcs_cprs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/equivalence/cpcs_cprs.ma
@@ -20,40 +20,40 @@ include "basic_2/equivalence/cpcs.ma".
 (* Properties about context sensitive computation on terms ******************)
 
 (* Basic_1: was: pc3_pr3_r *)
-lemma cpcs_cprs_dx: ∀L,T1,T2. L ⊢ T1 ➡* T2 → L ⊢ T1 ⬌* T2.
+lemma cpcs_cprs_dx: ∀L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡* T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
 #L #T1 #T2 #H @(cprs_ind … H) -T2 /width=1/ /3 width=3/
 qed.
 
 (* Basic_1: was: pc3_pr3_x *)
-lemma cpcs_cprs_sn: ∀L,T1,T2. L ⊢ T2 ➡* T1 → L ⊢ T1 ⬌* T2.
+lemma cpcs_cprs_sn: ∀L,T1,T2. ⦃G, L⦄ ⊢ T2 ➡* T1 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
 #L #T1 #T2 #H @(cprs_ind_dx … H) -T2 /width=1/ /3 width=3/
 qed.
 
-lemma cpcs_cprs_strap1: ∀L,T1,T. L ⊢ T1 ⬌* T → ∀T2. L ⊢ T ➡* T2 → L ⊢ T1 ⬌* T2.
+lemma cpcs_cprs_strap1: ∀L,T1,T. ⦃G, L⦄ ⊢ T1 ⬌* T → ∀T2. ⦃G, L⦄ ⊢ T ➡* T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
 #L #T1 #T #HT1 #T2 #H @(cprs_ind … H) -T2 /width=1/ /2 width=3/
 qed.
 
-lemma cpcs_cprs_strap2: ∀L,T1,T. L ⊢ T1 ➡* T → ∀T2. L ⊢ T ⬌* T2 → L ⊢ T1 ⬌* T2.
+lemma cpcs_cprs_strap2: ∀L,T1,T. ⦃G, L⦄ ⊢ T1 ➡* T → ∀T2. ⦃G, L⦄ ⊢ T ⬌* T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
 #L #T1 #T #H #T2 #HT2 @(cprs_ind_dx … H) -T1 /width=1/ /2 width=3/
 qed.
 
-lemma cpcs_cprs_div: ∀L,T1,T. L ⊢ T1 ⬌* T → ∀T2. L ⊢ T2 ➡* T → L ⊢ T1 ⬌* T2.
+lemma cpcs_cprs_div: ∀L,T1,T. ⦃G, L⦄ ⊢ T1 ⬌* T → ∀T2. ⦃G, L⦄ ⊢ T2 ➡* T → ⦃G, L⦄ ⊢ T1 ⬌* T2.
 #L #T1 #T #HT1 #T2 #H @(cprs_ind_dx … H) -T2 /width=1/ /2 width=3/
 qed.
 
 (* Basic_1: was: pc3_pr3_conf *)
-lemma cpcs_cprs_conf: ∀L,T1,T. L ⊢ T ➡* T1 → ∀T2. L ⊢ T ⬌* T2 → L ⊢ T1 ⬌* T2.
+lemma cpcs_cprs_conf: ∀L,T1,T. ⦃G, L⦄ ⊢ T ➡* T1 → ∀T2. ⦃G, L⦄ ⊢ T ⬌* T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
 #L #T1 #T #H #T2 #HT2 @(cprs_ind … H) -T1 /width=1/ /2 width=3/
 qed.
 
 (* Basic_1: was: pc3_pr3_t *)
 (* Basic_1: note: pc3_pr3_t should be renamed *)
-lemma cprs_div: ∀L,T1,T. L ⊢ T1 ➡* T → ∀T2. L ⊢ T2 ➡* T → L ⊢ T1 ⬌* T2.
+lemma cprs_div: ∀L,T1,T. ⦃G, L⦄ ⊢ T1 ➡* T → ∀T2. ⦃G, L⦄ ⊢ T2 ➡* T → ⦃G, L⦄ ⊢ T1 ⬌* T2.
 #L #T1 #T #HT1 #T2 #H @(cprs_ind_dx … H) -T2 /2 width=1/ /2 width=3/
 qed.
 
-lemma cprs_cpr_div: ∀L,T1,T. L ⊢ T1 ➡* T → ∀T2. L ⊢ T2 ➡ T → L ⊢ T1 ⬌* T2.
+lemma cprs_cpr_div: ∀L,T1,T. ⦃G, L⦄ ⊢ T1 ➡* T → ∀T2. ⦃G, L⦄ ⊢ T2 ➡ T → ⦃G, L⦄ ⊢ T1 ⬌* T2.
 /3 width=5 by cpr_cprs, cprs_div/ qed-.
 
-lemma cpr_cprs_div: ∀L,T1,T. L ⊢ T1 ➡ T → ∀T2. L ⊢ T2 ➡* T → L ⊢ T1 ⬌* T2.
+lemma cpr_cprs_div: ∀L,T1,T. ⦃G, L⦄ ⊢ T1 ➡ T → ∀T2. ⦃G, L⦄ ⊢ T2 ➡* T → ⦃G, L⦄ ⊢ T1 ⬌* T2.
 /3 width=3 by cpr_cprs, cprs_div/ qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/grammar/cl_weight.ma b/matita/matita/contribs/lambdadelta/basic_2/grammar/cl_weight.ma
index 66c8d0fb7..1188e613c 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/grammar/cl_weight.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/grammar/cl_weight.ma
@@ -12,32 +12,34 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "basic_2/notation/functions/weight_2.ma".
+include "basic_2/notation/functions/weight_3.ma".
+include "basic_2/grammar/genv.ma". (**) (* including genv after lenv shows a disambiguation bug: only the last interpretation is considered *)
 include "basic_2/grammar/lenv_weight.ma".
 include "basic_2/grammar/cl_shift.ma".
 
 (* WEIGHT OF A CLOSURE ******************************************************)
 
-definition fw: lenv → term → ? ≝ λL,T. ♯{L} + ♯{T}.
+(* activate genv *)
+definition fw: genv → lenv → term → ? ≝ λG,L,T. ♯{L} + ♯{T}.
 
-interpretation "weight (closure)" 'Weight L T = (fw L T).
+interpretation "weight (closure)" 'Weight G L T = (fw G L T).
 
 (* Basic properties *********************************************************)
 
 (* Basic_1: was: flt_shift *)
-lemma fw_shift: ∀a,K,I,V,T. ♯{K. ⓑ{I} V, T} < ♯{K, ⓑ{a,I} V. T}.
+lemma fw_shift: ∀a,I,G,K,V,T. ♯{G, K.ⓑ{I}V, T} < ♯{G, K, ⓑ{a,I}V.T}.
 normalize //
 qed.
 
-lemma fw_tpair_sn: ∀I,L,V,T. ♯{L, V} < ♯{L, ②{I}V.T}.
-normalize in ⊢ (?→?→?→?→?%%); //
+lemma fw_tpair_sn: ∀I,G,L,V,T. ♯{G, L, V} < ♯{G, L, ②{I}V.T}.
+normalize in ⊢ (?→?→?→?→?→?%%); //
 qed.
 
-lemma fw_tpair_dx: ∀I,L,V,T. ♯{L, T} < ♯{L, ②{I}V.T}.
-normalize in ⊢ (?→?→?→?→?%%); //
+lemma fw_tpair_dx: ∀I,G,L,V,T. ♯{G, L, T} < ♯{G, L, ②{I}V.T}.
+normalize in ⊢ (?→?→?→?→?→?%%); //
 qed.
 
-lemma fw_lpair_sn: ∀I,L,V,T. ♯{L, V} < ♯{L.ⓑ{I}V, T}.
+lemma fw_lpair_sn: ∀I,G,L,V,T. ♯{G, L, V} < ♯{G, L.ⓑ{I}V, T}.
 normalize /3 width=1 by monotonic_lt_plus_l, monotonic_le_plus_r/ (**) (* auto too slow without trace *)
 qed.
 
diff --git a/matita/matita/contribs/lambdadelta/basic_2/grammar/genv.ma b/matita/matita/contribs/lambdadelta/basic_2/grammar/genv.ma
index 2cf0f6d3d..38757ca16 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/grammar/genv.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/grammar/genv.ma
@@ -38,4 +38,4 @@ interpretation "abstraction (global environment)"
 
 (* Basic properties *********************************************************)
 
-axiom genv_eq_dec: ∀T1,T2:genv. Decidable (T1 = T2).
+axiom genv_eq_dec: ∀G1,G2:genv. Decidable (G1 = G2).
diff --git a/matita/matita/contribs/lambdadelta/basic_2/notation/functions/weight_2.ma b/matita/matita/contribs/lambdadelta/basic_2/notation/functions/weight_3.ma
similarity index 91%
rename from matita/matita/contribs/lambdadelta/basic_2/notation/functions/weight_2.ma
rename to matita/matita/contribs/lambdadelta/basic_2/notation/functions/weight_3.ma
index 161b66bd0..42b3c60ec 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/notation/functions/weight_2.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/notation/functions/weight_3.ma
@@ -14,6 +14,6 @@
 
 (* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
 
-notation "hvbox( ♯ { term 46 x , break term 46 y } )"
+notation "hvbox( ♯ { term 46 G , break term 46 L , break term 46 T } )"
  non associative with precedence 90
- for @{ 'Weight $x $y }.
+ for @{ 'Weight $G $L $T }.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/notation/notation.ma b/matita/matita/contribs/lambdadelta/basic_2/notation/notation.ma
index d131ad872..bc904846c 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/notation/notation.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/notation/notation.ma
@@ -14,7 +14,7 @@
 
 (* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
 
-notation "hvbox( L ⊢ break ⌘ ⦃ term 46 T ⦄ ≡ break term 46 k )"
+notation "hvbox( ⦃G, L⦄ ⊢ break ⌘ ⦃ term 46 T ⦄ ≡ break term 46 k )"
    non associative with precedence 45
    for @{ 'ICM $L $T $k }.
 
diff --git a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/atomicarity_3.ma b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/atomicarity_3.ma
index b92527537..42f9d584d 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/atomicarity_3.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/atomicarity_3.ma
@@ -14,6 +14,6 @@
 
 (* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
 
-notation "hvbox( L ⊢ break term 46 T ⁝ break term 46 A )"
+notation "hvbox( ⦃G, L⦄ ⊢ break term 46 T ⁝ break term 46 A )"
    non associative with precedence 45
    for @{ 'AtomicArity $L $T $A }.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/normal_2.ma b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/normal_2.ma
index 519b5a9d2..92cfc51d9 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/normal_2.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/normal_2.ma
@@ -14,6 +14,6 @@
 
 (* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
 
-notation "hvbox( L ⊢ 𝐍 break ⦃ term 46 T ⦄ )"
+notation "hvbox( ⦃G, L⦄ ⊢ 𝐍 break ⦃ term 46 T ⦄ )"
    non associative with precedence 45
    for @{ 'Normal $L $T }.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/notreducible_2.ma b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/notreducible_2.ma
index 2048df022..9659a6e23 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/notreducible_2.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/notreducible_2.ma
@@ -14,6 +14,6 @@
 
 (* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
 
-notation "hvbox( L ⊢ 𝐈 break ⦃ term 46 T ⦄ )"
+notation "hvbox( ⦃G, L⦄ ⊢ 𝐈 break ⦃ term 46 T ⦄ )"
    non associative with precedence 45
    for @{ 'NotReducible $L $T }.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/pconv_3.ma b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/pconv_3.ma
index 374586eb2..c4a789ab8 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/pconv_3.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/pconv_3.ma
@@ -14,6 +14,6 @@
 
 (* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
 
-notation "hvbox( L ⊢ break term 46 T1 ⬌ break term 46 T2 )"
+notation "hvbox( ⦃G, L⦄ ⊢ break term 46 T1 ⬌ break term 46 T2 )"
    non associative with precedence 45
    for @{ 'PConv $L $T1 $T2 }.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/pconvstar_3.ma b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/pconvstar_3.ma
index 8e604fa53..2088bc68d 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/pconvstar_3.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/pconvstar_3.ma
@@ -14,6 +14,6 @@
 
 (* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
 
-notation "hvbox( L ⊢ break term 46 T1 ⬌* break term 46 T2 )"
+notation "hvbox( ⦃G, L⦄ ⊢ break term 46 T1 ⬌* break term 46 T2 )"
    non associative with precedence 45
    for @{ 'PConvStar $L $T1 $T2 }.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/peval_3.ma b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/peval_3.ma
index 6740f7c03..b2d785e83 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/peval_3.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/peval_3.ma
@@ -14,6 +14,6 @@
 
 (* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
 
-notation "hvbox( L ⊢ break term 46 T1 ➡ * break 𝐍 ⦃ term 46 T2 ⦄ )"
+notation "hvbox( ⦃G, L⦄ ⊢ break term 46 T1 ➡ * break 𝐍 ⦃ term 46 T2 ⦄ )"
    non associative with precedence 45
    for @{ 'PEval $L $T1 $T2 }.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/pred_3.ma b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/pred_3.ma
index 969791be8..318edbab6 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/pred_3.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/pred_3.ma
@@ -14,6 +14,6 @@
 
 (* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
 
-notation "hvbox( L ⊢ break term 46 T1 ➡ break term 46 T2 )"
+notation "hvbox( ⦃G, L⦄ ⊢ break term 46 T1 ➡ break term 46 T2 )"
    non associative with precedence 45
    for @{ 'PRed $L $T1 $T2 }.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/predstar_3.ma b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/predstar_3.ma
index d6c5062f8..8053f7f7c 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/predstar_3.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/predstar_3.ma
@@ -14,6 +14,6 @@
 
 (* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
 
-notation "hvbox( L ⊢ break term 46 T1 ➡ * break term 46 T2 )"
+notation "hvbox( ⦃G, L⦄ ⊢ break term 46 T1 ➡ * break term 46 T2 )"
    non associative with precedence 45
    for @{ 'PRedStar $L $T1 $T2 }.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/reducible_2.ma b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/reducible_2.ma
index e6844a99f..f5cadce5f 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/reducible_2.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/reducible_2.ma
@@ -14,6 +14,6 @@
 
 (* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
 
-notation "hvbox( L ⊢ 𝐑 break ⦃ term 46 T ⦄ )"
+notation "hvbox( ⦃G, L⦄ ⊢ 𝐑 break ⦃ term 46 T ⦄ )"
    non associative with precedence 45
    for @{ 'Reducible $L $T }.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/suptermopt_4.ma b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/supterm_6.ma
similarity index 85%
rename from matita/matita/contribs/lambdadelta/basic_2/notation/relations/suptermopt_4.ma
rename to matita/matita/contribs/lambdadelta/basic_2/notation/relations/supterm_6.ma
index 3e55a3143..23c966912 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/suptermopt_4.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/supterm_6.ma
@@ -14,6 +14,6 @@
 
 (* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
 
-notation "hvbox( ⦃ term 46 L1, break term 46 T1 ⦄ ⊃⸮ break ⦃ term 46 L2 , break term 46 T2 ⦄ )"
+notation "hvbox( ⦃ term 46 G1, break term 46 L1, break term 46 T1 ⦄ ⊃ break ⦃ term 46 G2, break term 46 L2 , break term 46 T2 ⦄ )"
    non associative with precedence 45
-   for @{ 'SupTermOpt $L1 $T1 $L2 $T2 }.
+   for @{ 'SupTerm $G1 $L1 $T1 $G2 $L2 $T2 }.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/suptermoptalt_4.ma b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/suptermopt_6.ma
similarity index 85%
rename from matita/matita/contribs/lambdadelta/basic_2/notation/relations/suptermoptalt_4.ma
rename to matita/matita/contribs/lambdadelta/basic_2/notation/relations/suptermopt_6.ma
index 1035ce49a..ff1e3b551 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/suptermoptalt_4.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/suptermopt_6.ma
@@ -14,6 +14,6 @@
 
 (* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
 
-notation "hvbox( ⦃ term 46 L1, break term 46 T1 ⦄ ⊃⊃⸮ break ⦃ term 46 L2 , break term 46 T2 ⦄ )"
+notation "hvbox( ⦃ term 46 G1, break term 46 L1, break term 46 T1 ⦄ ⊃⸮ break ⦃ term 46 G2, break term 46 L2 , break term 46 T2 ⦄ )"
    non associative with precedence 45
-   for @{ 'SupTermOptAlt $L1 $T1 $L2 $T2 }.
+   for @{ 'SupTermOpt $G1 $L1 $T1 $G2 $L2 $T2 }.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/suptermoptalt_6.ma b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/suptermoptalt_6.ma
new file mode 100644
index 000000000..f5b529bc3
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/suptermoptalt_6.ma
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G1, break term 46 L1, break term 46 T1 ⦄ ⊃⊃⸮ break ⦃ term 46 G2, break term 46 L2 , break term 46 T2 ⦄ )"
+   non associative with precedence 45
+   for @{ 'SupTermOptAlt $G1 $L1 $T1 $G2 $L2 $T2 }.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/suptermplus_4.ma b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/suptermplus_6.ma
similarity index 86%
rename from matita/matita/contribs/lambdadelta/basic_2/notation/relations/suptermplus_4.ma
rename to matita/matita/contribs/lambdadelta/basic_2/notation/relations/suptermplus_6.ma
index 020d64ba3..558c62e83 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/suptermplus_4.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/suptermplus_6.ma
@@ -14,6 +14,6 @@
 
 (* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
 
-notation "hvbox( ⦃ term 46 L1, break term 46 T1 ⦄ ⊃ + break ⦃ term 46 L2 , break term 46 T2 ⦄ )"
+notation "hvbox( ⦃ term 46 G1, term 46 L1, break term 46 T1 ⦄ ⊃ + break ⦃ term 46 G2, term 46 L2 , break term 46 T2 ⦄ )"
    non associative with precedence 45
-   for @{ 'SupTermPlus $L1 $T1 $L2 $T2 }.
+   for @{ 'SupTermPlus $G1 $L1 $T1 $G2 $L2 $T2 }.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/suptermstar_4.ma b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/suptermstar_4.ma
deleted file mode 100644
index 53b3bce5e..000000000
--- a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/suptermstar_4.ma
+++ /dev/null
@@ -1,19 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⦃ term 46 L1, break term 46 T1 ⦄ ⊃ * break ⦃ term 46 L2 , break term 46 T2 ⦄ )"
-   non associative with precedence 45
-   for @{ 'SupTermStar $L1 $T1 $L2 $T2 }.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/supterm_4.ma b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/suptermstar_6.ma
similarity index 86%
rename from matita/matita/contribs/lambdadelta/basic_2/notation/relations/supterm_4.ma
rename to matita/matita/contribs/lambdadelta/basic_2/notation/relations/suptermstar_6.ma
index b2114a050..e4c64a6b6 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/supterm_4.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/suptermstar_6.ma
@@ -14,6 +14,6 @@
 
 (* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
 
-notation "hvbox( ⦃ term 46 L1, break term 46 T1 ⦄ ⊃ break ⦃ term 46 L2 , break term 46 T2 ⦄ )"
+notation "hvbox( ⦃ term 46 G1, term 46 L1, break term 46 T1 ⦄ ⊃ * break ⦃ term 46 G2, term 46 L2 , break term 46 T2 ⦄ )"
    non associative with precedence 45
-   for @{ 'SupTerm $L1 $T1 $L2 $T2 }.
+   for @{ 'SupTermStar $G1 $L1 $T1 $G2 $L2 $T2 }.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/cir.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/cir.ma
index 0aa31fe72..d9509e17a 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/reduction/cir.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/reduction/cir.ma
@@ -17,39 +17,39 @@ include "basic_2/reduction/crr.ma".
 
 (* CONTEXT-SENSITIVE IRREDUCIBLE TERMS **************************************)
 
-definition cir: lenv → predicate term ≝ λL,T. L ⊢ 𝐑⦃T⦄ → ⊥.
+definition cir: lenv → predicate term ≝ λL,T. ⦃G, L⦄ ⊢ 𝐑⦃T⦄ → ⊥.
 
 interpretation "context-sensitive irreducibility (term)"
    'NotReducible L T = (cir L T).
 
 (* Basic inversion lemmas ***************************************************)
 
-lemma cir_inv_delta: ∀L,K,V,i. ⇩[0, i] L ≡ K.ⓓV → L ⊢ 𝐈⦃#i⦄ → ⊥.
+lemma cir_inv_delta: ∀L,K,V,i. ⇩[0, i] L ≡ K.ⓓV → ⦃G, L⦄ ⊢ 𝐈⦃#i⦄ → ⊥.
 /3 width=3/ qed-.
 
-lemma cir_inv_ri2: ∀I,L,V,T. ri2 I → L ⊢ 𝐈⦃②{I}V.T⦄ → ⊥.
+lemma cir_inv_ri2: ∀I,L,V,T. ri2 I → ⦃G, L⦄ ⊢ 𝐈⦃②{I}V.T⦄ → ⊥.
 /3 width=1/ qed-.
 
-lemma cir_inv_ib2: ∀a,I,L,V,T. ib2 a I → L ⊢ 𝐈⦃ⓑ{a,I}V.T⦄ →
-                   L ⊢ 𝐈⦃V⦄ ∧ L.ⓑ{I}V ⊢ 𝐈⦃T⦄.
+lemma cir_inv_ib2: ∀a,I,L,V,T. ib2 a I → ⦃G, L⦄ ⊢ 𝐈⦃ⓑ{a,I}V.T⦄ →
+                   ⦃G, L⦄ ⊢ 𝐈⦃V⦄ ∧ L.ⓑ{I}V ⊢ 𝐈⦃T⦄.
 /4 width=1/ qed-.
 
-lemma cir_inv_bind: ∀a,I,L,V,T. L ⊢ 𝐈⦃ⓑ{a,I}V.T⦄ →
-                    ∧∧ L ⊢ 𝐈⦃V⦄ & L.ⓑ{I}V ⊢ 𝐈⦃T⦄ & ib2 a I.
+lemma cir_inv_bind: ∀a,I,L,V,T. ⦃G, L⦄ ⊢ 𝐈⦃ⓑ{a,I}V.T⦄ →
+                    ∧∧ ⦃G, L⦄ ⊢ 𝐈⦃V⦄ & L.ⓑ{I}V ⊢ 𝐈⦃T⦄ & ib2 a I.
 #a * [ elim a -a ]
 [ #L #V #T #H elim H -H /3 width=1/
 |*: #L #V #T #H elim (cir_inv_ib2 … H) -H /2 width=1/ /3 width=1/
 ]
 qed-.
 
-lemma cir_inv_appl: ∀L,V,T. L ⊢ 𝐈⦃ⓐV.T⦄ → ∧∧ L ⊢ 𝐈⦃V⦄ & L ⊢ 𝐈⦃T⦄ & 𝐒⦃T⦄.
+lemma cir_inv_appl: ∀L,V,T. ⦃G, L⦄ ⊢ 𝐈⦃ⓐV.T⦄ → ∧∧ ⦃G, L⦄ ⊢ 𝐈⦃V⦄ & ⦃G, L⦄ ⊢ 𝐈⦃T⦄ & 𝐒⦃T⦄.
 #L #V #T #HVT @and3_intro /3 width=1/
 generalize in match HVT; -HVT elim T -T //
 * // #a * #U #T #_ #_ #H elim H -H /2 width=1/
 qed-.
 
-lemma cir_inv_flat: ∀I,L,V,T. L ⊢ 𝐈⦃ⓕ{I}V.T⦄ →
-                    ∧∧ L ⊢ 𝐈⦃V⦄ & L ⊢ 𝐈⦃T⦄ & 𝐒⦃T⦄ & I = Appl.
+lemma cir_inv_flat: ∀I,L,V,T. ⦃G, L⦄ ⊢ 𝐈⦃ⓕ{I}V.T⦄ →
+                    ∧∧ ⦃G, L⦄ ⊢ 𝐈⦃V⦄ & ⦃G, L⦄ ⊢ 𝐈⦃T⦄ & 𝐒⦃T⦄ & I = Appl.
 * #L #V #T #H
 [ elim (cir_inv_appl … H) -H /2 width=1/
 | elim (cir_inv_ri2 … H) -H /2 width=1/
@@ -58,21 +58,21 @@ qed-.
 
 (* Basic properties *********************************************************)
 
-lemma cir_sort: ∀L,k. L ⊢ 𝐈⦃⋆k⦄.
+lemma cir_sort: ∀L,k. ⦃G, L⦄ ⊢ 𝐈⦃⋆k⦄.
 /2 width=3 by crr_inv_sort/ qed.
 
-lemma cir_gref: ∀L,p. L ⊢ 𝐈⦃§p⦄.
+lemma cir_gref: ∀L,p. ⦃G, L⦄ ⊢ 𝐈⦃§p⦄.
 /2 width=3 by crr_inv_gref/ qed.
 
 lemma tir_atom: ∀I. ⋆ ⊢ 𝐈⦃⓪{I}⦄.
 /2 width=2 by trr_inv_atom/ qed.
 
-lemma cir_ib2: ∀a,I,L,V,T. ib2 a I → L ⊢ 𝐈⦃V⦄ → L.ⓑ{I}V ⊢ 𝐈⦃T⦄ → L ⊢ 𝐈⦃ⓑ{a,I}V.T⦄.
+lemma cir_ib2: ∀a,I,L,V,T. ib2 a I → ⦃G, L⦄ ⊢ 𝐈⦃V⦄ → L.ⓑ{I}V ⊢ 𝐈⦃T⦄ → ⦃G, L⦄ ⊢ 𝐈⦃ⓑ{a,I}V.T⦄.
 #a #I #L #V #T #HI #HV #HT #H
 elim (crr_inv_ib2 … HI H) -HI -H /2 width=1/
 qed.
 
-lemma cir_appl: ∀L,V,T. L ⊢ 𝐈⦃V⦄ → L ⊢ 𝐈⦃T⦄ →  𝐒⦃T⦄ → L ⊢ 𝐈⦃ⓐV.T⦄.
+lemma cir_appl: ∀L,V,T. ⦃G, L⦄ ⊢ 𝐈⦃V⦄ → ⦃G, L⦄ ⊢ 𝐈⦃T⦄ →  𝐒⦃T⦄ → ⦃G, L⦄ ⊢ 𝐈⦃ⓐV.T⦄.
 #L #V #T #HV #HT #H1 #H2
 elim (crr_inv_appl … H2) -H2 /2 width=1/
 qed.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/cir_append.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/cir_append.ma
index fb88321ef..b91465fc0 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/reduction/cir_append.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/reduction/cir_append.ma
@@ -19,7 +19,7 @@ include "basic_2/reduction/cir.ma".
 
 (* Advanved properties ******************************************************)
 
-lemma cir_labst_last: ∀L,T,W. L ⊢ 𝐈⦃T⦄  → ⋆.ⓛW @@ L ⊢ 𝐈⦃T⦄.
+lemma cir_labst_last: ∀L,T,W. ⦃G, L⦄ ⊢ 𝐈⦃T⦄  → ⋆.ⓛW @@ ⦃G, L⦄ ⊢ 𝐈⦃T⦄.
 /3 width=2 by crr_inv_labst_last/ qed.
 
 lemma cir_tif: ∀T,W. ⋆ ⊢ 𝐈⦃T⦄ → ⋆.ⓛW ⊢ 𝐈⦃T⦄.
@@ -27,7 +27,7 @@ lemma cir_tif: ∀T,W. ⋆ ⊢ 𝐈⦃T⦄ → ⋆.ⓛW ⊢ 𝐈⦃T⦄.
 
 (* Advanced inversion lemmas ************************************************)
 
-lemma cir_inv_append_sn: ∀L,K,T. K @@ L ⊢ 𝐈⦃T⦄  → L ⊢ 𝐈⦃T⦄.
+lemma cir_inv_append_sn: ∀L,K,T. K @@ ⦃G, L⦄ ⊢ 𝐈⦃T⦄  → ⦃G, L⦄ ⊢ 𝐈⦃T⦄.
 /3 width=1/ qed-.
 
 lemma cir_inv_tir: ∀T,W. ⋆.ⓛW ⊢ 𝐈⦃T⦄  → ⋆ ⊢ 𝐈⦃T⦄.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/cir_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/cir_lift.ma
index 9539a6c03..0dcd1cb81 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/reduction/cir_lift.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/reduction/cir_lift.ma
@@ -20,9 +20,9 @@ include "basic_2/reduction/cir.ma".
 (* Properties on relocation *************************************************)
 
 lemma cir_lift: ∀K,T. K ⊢ 𝐈⦃T⦄ → ∀L,d,e. ⇩[d, e] L ≡ K →
-                ∀U. ⇧[d, e] T ≡ U → L ⊢ 𝐈⦃U⦄.
+                ∀U. ⇧[d, e] T ≡ U → ⦃G, L⦄ ⊢ 𝐈⦃U⦄.
 /3 width=7 by crr_inv_lift/ qed.
 
-lemma cir_inv_lift: ∀L,U. L ⊢ 𝐈⦃U⦄ → ∀K,d,e. ⇩[d, e] L ≡ K →
+lemma cir_inv_lift: ∀L,U. ⦃G, L⦄ ⊢ 𝐈⦃U⦄ → ∀K,d,e. ⇩[d, e] L ≡ K →
                     ∀T. ⇧[d, e] T ≡ U → K ⊢ 𝐈⦃T⦄.
 /3 width=7/ qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/cix.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/cix.ma
index eec73fb8e..f89a5038d 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/reduction/cix.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/reduction/cix.ma
@@ -18,43 +18,43 @@ include "basic_2/reduction/crx.ma".
 
 (* CONTEXT-SENSITIVE EXTENDED IRREDUCIBLE TERMS *****************************)
 
-definition cix: ∀h. sd h → lenv → predicate term ≝ λh,g,L,T. ⦃h, L⦄ ⊢ 𝐑[g]⦃T⦄ → ⊥.
+definition cix: ∀h. sd h → lenv → predicate term ≝ λh,g,L,T. ⦃G, L⦄ ⊢ 𝐑[h, g]⦃T⦄ → ⊥.
 
 interpretation "context-sensitive extended irreducibility (term)"
    'NotReducible h g L T = (cix h g L T).
 
 (* Basic inversion lemmas ***************************************************)
 
-lemma cix_inv_sort: ∀h,g,L,k,l. deg h g k (l+1) → ⦃h, L⦄ ⊢ 𝐈[g]⦃⋆k⦄ → ⊥.
+lemma cix_inv_sort: ∀h,g,L,k,l. deg h g k (l+1) → ⦃G, L⦄ ⊢ 𝐈[h, g]⦃⋆k⦄ → ⊥.
 /3 width=2/ qed-.
 
-lemma cix_inv_delta: ∀h,g,I,L,K,V,i. ⇩[0, i] L ≡ K.ⓑ{I}V → ⦃h, L⦄ ⊢ 𝐈[g]⦃#i⦄ → ⊥.
+lemma cix_inv_delta: ∀h,g,I,L,K,V,i. ⇩[0, i] L ≡ K.ⓑ{I}V → ⦃G, L⦄ ⊢ 𝐈[h, g]⦃#i⦄ → ⊥.
 /3 width=4/ qed-.
 
-lemma cix_inv_ri2: ∀h,g,I,L,V,T. ri2 I → ⦃h, L⦄ ⊢ 𝐈[g]⦃②{I}V.T⦄ → ⊥.
+lemma cix_inv_ri2: ∀h,g,I,L,V,T. ri2 I → ⦃G, L⦄ ⊢ 𝐈[h, g]⦃②{I}V.T⦄ → ⊥.
 /3 width=1/ qed-.
 
-lemma cix_inv_ib2: ∀h,g,a,I,L,V,T. ib2 a I → ⦃h, L⦄ ⊢ 𝐈[g]⦃ⓑ{a,I}V.T⦄ →
-                   ⦃h, L⦄ ⊢ 𝐈[g]⦃V⦄ ∧ ⦃h, L.ⓑ{I}V⦄ ⊢ 𝐈[g]⦃T⦄.
+lemma cix_inv_ib2: ∀h,g,a,I,L,V,T. ib2 a I → ⦃G, L⦄ ⊢ 𝐈[h, g]⦃ⓑ{a,I}V.T⦄ →
+                   ⦃G, L⦄ ⊢ 𝐈[h, g]⦃V⦄ ∧ ⦃h, L.ⓑ{I}V⦄ ⊢ 𝐈[h, g]⦃T⦄.
 /4 width=1/ qed-.
 
-lemma cix_inv_bind: ∀h,g,a,I,L,V,T. ⦃h, L⦄ ⊢ 𝐈[g]⦃ⓑ{a,I}V.T⦄ →
-                    ∧∧ ⦃h, L⦄ ⊢ 𝐈[g]⦃V⦄ & ⦃h, L.ⓑ{I}V⦄ ⊢ 𝐈[g]⦃T⦄ & ib2 a I.
+lemma cix_inv_bind: ∀h,g,a,I,L,V,T. ⦃G, L⦄ ⊢ 𝐈[h, g]⦃ⓑ{a,I}V.T⦄ →
+                    ∧∧ ⦃G, L⦄ ⊢ 𝐈[h, g]⦃V⦄ & ⦃h, L.ⓑ{I}V⦄ ⊢ 𝐈[h, g]⦃T⦄ & ib2 a I.
 #h #g #a * [ elim a -a ]
 [ #L #V #T #H elim H -H /3 width=1/
 |*: #L #V #T #H elim (cix_inv_ib2 … H) -H /2 width=1/ /3 width=1/
 ]
 qed-.
 
-lemma cix_inv_appl: ∀h,g,L,V,T. ⦃h, L⦄ ⊢ 𝐈[g]⦃ⓐV.T⦄ →
-                    ∧∧ ⦃h, L⦄ ⊢ 𝐈[g]⦃V⦄ & ⦃h, L⦄ ⊢ 𝐈[g]⦃T⦄ & 𝐒⦃T⦄.
+lemma cix_inv_appl: ∀h,g,L,V,T. ⦃G, L⦄ ⊢ 𝐈[h, g]⦃ⓐV.T⦄ →
+                    ∧∧ ⦃G, L⦄ ⊢ 𝐈[h, g]⦃V⦄ & ⦃G, L⦄ ⊢ 𝐈[h, g]⦃T⦄ & 𝐒⦃T⦄.
 #h #g #L #V #T #HVT @and3_intro /3 width=1/
 generalize in match HVT; -HVT elim T -T //
 * // #a * #U #T #_ #_ #H elim H -H /2 width=1/
 qed-.
 
-lemma cix_inv_flat: ∀h,g,I,L,V,T. ⦃h, L⦄ ⊢ 𝐈[g]⦃ⓕ{I}V.T⦄ →
-                    ∧∧ ⦃h, L⦄ ⊢ 𝐈[g]⦃V⦄ & ⦃h, L⦄ ⊢ 𝐈[g]⦃T⦄ & 𝐒⦃T⦄ & I = Appl.
+lemma cix_inv_flat: ∀h,g,I,L,V,T. ⦃G, L⦄ ⊢ 𝐈[h, g]⦃ⓕ{I}V.T⦄ →
+                    ∧∧ ⦃G, L⦄ ⊢ 𝐈[h, g]⦃V⦄ & ⦃G, L⦄ ⊢ 𝐈[h, g]⦃T⦄ & 𝐒⦃T⦄ & I = Appl.
 #h #g * #L #V #T #H
 [ elim (cix_inv_appl … H) -H /2 width=1/
 | elim (cix_inv_ri2 … H) -H /2 width=1/
@@ -63,31 +63,31 @@ qed-.
 
 (* Basic forward lemmas *****************************************************)
 
-lemma cix_inv_cir: ∀h,g,L,T. ⦃h, L⦄ ⊢ 𝐈[g]⦃T⦄ → L ⊢ 𝐈⦃T⦄. 
+lemma cix_inv_cir: ∀h,g,L,T. ⦃G, L⦄ ⊢ 𝐈[h, g]⦃T⦄ → ⦃G, L⦄ ⊢ 𝐈⦃T⦄. 
 /3 width=1/ qed-.
 
 (* Basic properties *********************************************************)
 
-lemma cix_sort: ∀h,g,L,k. deg h g k 0 → ⦃h, L⦄ ⊢ 𝐈[g]⦃⋆k⦄.
+lemma cix_sort: ∀h,g,L,k. deg h g k 0 → ⦃G, L⦄ ⊢ 𝐈[h, g]⦃⋆k⦄.
 #h #g #L #k #Hk #H elim (crx_inv_sort … H) -L #l #Hkl
 lapply (deg_mono … Hk Hkl) -h -k <plus_n_Sm #H destruct
 qed.
 
-lemma tix_lref: ∀h,g,i. ⦃h, ⋆⦄ ⊢ 𝐈[g]⦃#i⦄.
+lemma tix_lref: ∀h,g,i. ⦃h, ⋆⦄ ⊢ 𝐈[h, g]⦃#i⦄.
 #h #g #i #H elim (trx_inv_atom … H) -H #k #l #_ #H destruct
 qed.
 
-lemma cix_gref: ∀h,g,L,p. ⦃h, L⦄ ⊢ 𝐈[g]⦃§p⦄.
+lemma cix_gref: ∀h,g,L,p. ⦃G, L⦄ ⊢ 𝐈[h, g]⦃§p⦄.
 #h #g #L #p #H elim (crx_inv_gref … H)
 qed.
 
-lemma cix_ib2: ∀h,g,a,I,L,V,T. ib2 a I → ⦃h, L⦄ ⊢ 𝐈[g]⦃V⦄ → ⦃h, L.ⓑ{I}V⦄ ⊢ 𝐈[g]⦃T⦄ →
-                               ⦃h, L⦄ ⊢ 𝐈[g]⦃ⓑ{a,I}V.T⦄.
+lemma cix_ib2: ∀h,g,a,I,L,V,T. ib2 a I → ⦃G, L⦄ ⊢ 𝐈[h, g]⦃V⦄ → ⦃h, L.ⓑ{I}V⦄ ⊢ 𝐈[h, g]⦃T⦄ →
+                               ⦃G, L⦄ ⊢ 𝐈[h, g]⦃ⓑ{a,I}V.T⦄.
 #h #g #a #I #L #V #T #HI #HV #HT #H
 elim (crx_inv_ib2 … HI H) -HI -H /2 width=1/
 qed.
 
-lemma cix_appl: ∀h,g,L,V,T. ⦃h, L⦄ ⊢ 𝐈[g]⦃V⦄ → ⦃h, L⦄ ⊢ 𝐈[g]⦃T⦄ →  𝐒⦃T⦄ → ⦃h, L⦄ ⊢ 𝐈[g]⦃ⓐV.T⦄.
+lemma cix_appl: ∀h,g,L,V,T. ⦃G, L⦄ ⊢ 𝐈[h, g]⦃V⦄ → ⦃G, L⦄ ⊢ 𝐈[h, g]⦃T⦄ →  𝐒⦃T⦄ → ⦃G, L⦄ ⊢ 𝐈[h, g]⦃ⓐV.T⦄.
 #h #g #L #V #T #HV #HT #H1 #H2
 elim (crx_inv_appl … H2) -H2 /2 width=1/
 qed.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/cix_append.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/cix_append.ma
index 3e1d06076..27bc414dc 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/reduction/cix_append.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/reduction/cix_append.ma
@@ -19,8 +19,8 @@ include "basic_2/reduction/cix.ma".
 
 (* Advanced inversion lemmas ************************************************)
 
-lemma cix_inv_append_sn: ∀h,g,L,K,T. ⦃h, K @@ L⦄ ⊢ 𝐈[g]⦃T⦄  → ⦃h, L⦄ ⊢ 𝐈[g]⦃T⦄.
+lemma cix_inv_append_sn: ∀h,g,L,K,T. ⦃h, K @@ L⦄ ⊢ 𝐈[h, g]⦃T⦄  → ⦃G, L⦄ ⊢ 𝐈[h, g]⦃T⦄.
 /3 width=1/ qed-.
 
-lemma cix_inv_tix: ∀h,g,L,T. ⦃h, L⦄ ⊢ 𝐈[g]⦃T⦄  → ⦃h, ⋆⦄ ⊢ 𝐈[g]⦃T⦄.
+lemma cix_inv_tix: ∀h,g,L,T. ⦃G, L⦄ ⊢ 𝐈[h, g]⦃T⦄  → ⦃h, ⋆⦄ ⊢ 𝐈[h, g]⦃T⦄.
 /3 width=1/ qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/cix_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/cix_lift.ma
index f81f17541..730a66fb1 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/reduction/cix_lift.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/reduction/cix_lift.ma
@@ -19,17 +19,17 @@ include "basic_2/reduction/cix.ma".
 
 (* Advanced properties ******************************************************)
 
-lemma cix_lref: ∀h,g,L,i. ⇩[0, i] L ≡ ⋆ → ⦃h, L⦄ ⊢ 𝐈[g]⦃#i⦄.
+lemma cix_lref: ∀h,g,L,i. ⇩[0, i] L ≡ ⋆ → ⦃G, L⦄ ⊢ 𝐈[h, g]⦃#i⦄.
 #h #g #L #i #HL #H elim (crx_inv_lref … H) -h #I #K #V #HLK
 lapply (ldrop_mono … HLK … HL) -L -i #H destruct
 qed.
 
 (* Properties on relocation *************************************************)
 
-lemma cix_lift: ∀h,g,K,T. ⦃h, K⦄ ⊢ 𝐈[g]⦃T⦄ → ∀L,d,e. ⇩[d, e] L ≡ K →
-                ∀U. ⇧[d, e] T ≡ U → ⦃h, L⦄ ⊢ 𝐈[g]⦃U⦄.
+lemma cix_lift: ∀h,g,K,T. ⦃h, K⦄ ⊢ 𝐈[h, g]⦃T⦄ → ∀L,d,e. ⇩[d, e] L ≡ K →
+                ∀U. ⇧[d, e] T ≡ U → ⦃G, L⦄ ⊢ 𝐈[h, g]⦃U⦄.
 /3 width=7 by crx_inv_lift/ qed.
 
-lemma cix_inv_lift: ∀h,g,L,U. ⦃h, L⦄ ⊢ 𝐈[g]⦃U⦄ → ∀K,d,e. ⇩[d, e] L ≡ K →
-                    ∀T. ⇧[d, e] T ≡ U → ⦃h, K⦄ ⊢ 𝐈[g]⦃T⦄.
+lemma cix_inv_lift: ∀h,g,L,U. ⦃G, L⦄ ⊢ 𝐈[h, g]⦃U⦄ → ∀K,d,e. ⇩[d, e] L ≡ K →
+                    ∀T. ⇧[d, e] T ≡ U → ⦃h, K⦄ ⊢ 𝐈[h, g]⦃T⦄.
 /3 width=7/ qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/cnr.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/cnr.ma
index 821d74c7a..9d0dc0733 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/reduction/cnr.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/reduction/cnr.ma
@@ -25,28 +25,28 @@ interpretation
 
 (* Basic inversion lemmas ***************************************************)
 
-lemma cnr_inv_delta: ∀L,K,V,i. ⇩[0, i] L ≡ K.ⓓV → L ⊢ 𝐍⦃#i⦄ → ⊥.
+lemma cnr_inv_delta: ∀L,K,V,i. ⇩[0, i] L ≡ K.ⓓV → ⦃G, L⦄ ⊢ 𝐍⦃#i⦄ → ⊥.
 #L #K #V #i #HLK #H
 elim (lift_total V 0 (i+1)) #W #HVW
 lapply (H W ?) -H [ /3 width=6/ ] -HLK #H destruct
 elim (lift_inv_lref2_be … HVW) -HVW //
 qed-.
 
-lemma cnr_inv_abst: ∀a,L,V,T. L ⊢ 𝐍⦃ⓛ{a}V.T⦄ → L ⊢ 𝐍⦃V⦄ ∧ L.ⓛV ⊢ 𝐍⦃T⦄.
+lemma cnr_inv_abst: ∀a,L,V,T. ⦃G, L⦄ ⊢ 𝐍⦃ⓛ{a}V.T⦄ → ⦃G, L⦄ ⊢ 𝐍⦃V⦄ ∧ L.ⓛV ⊢ 𝐍⦃T⦄.
 #a #L #V1 #T1 #HVT1 @conj
 [ #V2 #HV2 lapply (HVT1 (ⓛ{a}V2.T1) ?) -HVT1 /2 width=2/ -HV2 #H destruct //
 | #T2 #HT2 lapply (HVT1 (ⓛ{a}V1.T2) ?) -HVT1 /2 width=2/ -HT2 #H destruct //
 ]
 qed-.
 
-lemma cnr_inv_abbr: ∀L,V,T. L ⊢ 𝐍⦃-ⓓV.T⦄ → L ⊢ 𝐍⦃V⦄ ∧ L.ⓓV ⊢ 𝐍⦃T⦄.
+lemma cnr_inv_abbr: ∀L,V,T. ⦃G, L⦄ ⊢ 𝐍⦃-ⓓV.T⦄ → ⦃G, L⦄ ⊢ 𝐍⦃V⦄ ∧ L.ⓓV ⊢ 𝐍⦃T⦄.
 #L #V1 #T1 #HVT1 @conj
 [ #V2 #HV2 lapply (HVT1 (-ⓓV2.T1) ?) -HVT1 /2 width=2/ -HV2 #H destruct //
 | #T2 #HT2 lapply (HVT1 (-ⓓV1.T2) ?) -HVT1 /2 width=2/ -HT2 #H destruct //
 ]
 qed-.
 
-lemma cnr_inv_zeta: ∀L,V,T. L ⊢ 𝐍⦃+ⓓV.T⦄ → ⊥.
+lemma cnr_inv_zeta: ∀L,V,T. ⦃G, L⦄ ⊢ 𝐍⦃+ⓓV.T⦄ → ⊥.
 #L #V #T #H elim (is_lift_dec T 0 1)
 [ * #U #HTU
   lapply (H U ?) -H /2 width=3/ #H destruct
@@ -57,7 +57,7 @@ lemma cnr_inv_zeta: ∀L,V,T. L ⊢ 𝐍⦃+ⓓV.T⦄ → ⊥.
 ]
 qed-.
 
-lemma cnr_inv_appl: ∀L,V,T. L ⊢ 𝐍⦃ⓐV.T⦄ → ∧∧ L ⊢ 𝐍⦃V⦄ & L ⊢ 𝐍⦃T⦄ & 𝐒⦃T⦄.
+lemma cnr_inv_appl: ∀L,V,T. ⦃G, L⦄ ⊢ 𝐍⦃ⓐV.T⦄ → ∧∧ ⦃G, L⦄ ⊢ 𝐍⦃V⦄ & ⦃G, L⦄ ⊢ 𝐍⦃T⦄ & 𝐒⦃T⦄.
 #L #V1 #T1 #HVT1 @and3_intro
 [ #V2 #HV2 lapply (HVT1 (ⓐV2.T1) ?) -HVT1 /2 width=1/ -HV2 #H destruct //
 | #T2 #HT2 lapply (HVT1 (ⓐV1.T2) ?) -HVT1 /2 width=1/ -HT2 #H destruct //
@@ -68,7 +68,7 @@ lemma cnr_inv_appl: ∀L,V,T. L ⊢ 𝐍⦃ⓐV.T⦄ → ∧∧ L ⊢ 𝐍⦃V
 ]
 qed-.
 
-lemma cnr_inv_tau: ∀L,V,T. L ⊢ 𝐍⦃ⓝV.T⦄ → ⊥.
+lemma cnr_inv_tau: ∀L,V,T. ⦃G, L⦄ ⊢ 𝐍⦃ⓝV.T⦄ → ⊥.
 #L #V #T #H lapply (H T ?) -H /2 width=1/ #H
 @discr_tpair_xy_y //
 qed-.
@@ -76,27 +76,27 @@ qed-.
 (* Basic properties *********************************************************)
 
 (* Basic_1: was: nf2_sort *)
-lemma cnr_sort: ∀L,k. L ⊢ 𝐍⦃⋆k⦄.
+lemma cnr_sort: ∀L,k. ⦃G, L⦄ ⊢ 𝐍⦃⋆k⦄.
 #L #k #X #H
 >(cpr_inv_sort1 … H) //
 qed.
 
 (* Basic_1: was: nf2_abst *)
-lemma cnr_abst: ∀a,L,W,T. L ⊢ 𝐍⦃W⦄ → L.ⓛW ⊢ 𝐍⦃T⦄ → L ⊢ 𝐍⦃ⓛ{a}W.T⦄.
+lemma cnr_abst: ∀a,L,W,T. ⦃G, L⦄ ⊢ 𝐍⦃W⦄ → L.ⓛW ⊢ 𝐍⦃T⦄ → ⦃G, L⦄ ⊢ 𝐍⦃ⓛ{a}W.T⦄.
 #a #L #W #T #HW #HT #X #H
 elim (cpr_inv_abst1 … H) -H #W0 #T0 #HW0 #HT0 #H destruct
 >(HW … HW0) -W0 >(HT … HT0) -T0 //
 qed.
 
 (* Basic_1: was only: nf2_appl_lref *)
-lemma cnr_appl_simple: ∀L,V,T. L ⊢ 𝐍⦃V⦄ → L ⊢ 𝐍⦃T⦄ → 𝐒⦃T⦄ → L ⊢ 𝐍⦃ⓐV.T⦄.
+lemma cnr_appl_simple: ∀L,V,T. ⦃G, L⦄ ⊢ 𝐍⦃V⦄ → ⦃G, L⦄ ⊢ 𝐍⦃T⦄ → 𝐒⦃T⦄ → ⦃G, L⦄ ⊢ 𝐍⦃ⓐV.T⦄.
 #L #V #T #HV #HT #HS #X #H
 elim (cpr_inv_appl1_simple … H) -H // #V0 #T0 #HV0 #HT0 #H destruct
 >(HV … HV0) -V0 >(HT … HT0) -T0 //
 qed.
 
 (* Basic_1: was: nf2_dec *)
-axiom cnr_dec: ∀L,T1. L ⊢ 𝐍⦃T1⦄ ∨
-               ∃∃T2. L ⊢ T1 ➡ T2 & (T1 = T2 → ⊥).
+axiom cnr_dec: ∀L,T1. ⦃G, L⦄ ⊢ 𝐍⦃T1⦄ ∨
+               ∃∃T2. ⦃G, L⦄ ⊢ T1 ➡ T2 & (T1 = T2 → ⊥).
 
 (* Basic_1: removed theorems 1: nf2_abst_shift *)
diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/cnr_cir.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/cnr_cir.ma
index 50f30ab60..269cd6561 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/reduction/cnr_cir.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/reduction/cnr_cir.ma
@@ -19,10 +19,10 @@ include "basic_2/reduction/cnr_crr.ma".
 
 (* Main properties on context-sensitive irreducible terms *******************)
 
-theorem cir_cnr: ∀L,T. L ⊢ 𝐈⦃T⦄ → L ⊢ 𝐍⦃T⦄.
+theorem cir_cnr: ∀L,T. ⦃G, L⦄ ⊢ 𝐈⦃T⦄ → ⦃G, L⦄ ⊢ 𝐍⦃T⦄.
 /2 width=3 by cpr_fwd_cir/ qed.
 
 (* Main inversion lemmas on context-sensitive irreducible terms *************)
 
-theorem cnr_inv_cir: ∀L,T. L ⊢ 𝐍⦃T⦄ → L ⊢ 𝐈⦃T⦄.
+theorem cnr_inv_cir: ∀L,T. ⦃G, L⦄ ⊢ 𝐍⦃T⦄ → ⦃G, L⦄ ⊢ 𝐈⦃T⦄.
 /2 width=4 by cnr_inv_crr/ qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/cnr_crr.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/cnr_crr.ma
index 6d68cf9e0..c52165e3e 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/reduction/cnr_crr.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/reduction/cnr_crr.ma
@@ -20,7 +20,7 @@ include "basic_2/reduction/cnr.ma".
 (* Advanced inversion lemmas on context-sensitive reducible terms ***********)
 
 (* Note: this property is unusual *)
-lemma cnr_inv_crr: ∀L,T. L ⊢ 𝐑⦃T⦄ → L ⊢ 𝐍⦃T⦄ → ⊥.
+lemma cnr_inv_crr: ∀L,T. ⦃G, L⦄ ⊢ 𝐑⦃T⦄ → ⦃G, L⦄ ⊢ 𝐍⦃T⦄ → ⊥.
 #L #T #H elim H -L -T
 [ #L #K #V #i #HLK #H
   elim (cnr_inv_delta … HLK H)
diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/cnr_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/cnr_lift.ma
index 5e3c2c419..8d0267c47 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/reduction/cnr_lift.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/reduction/cnr_lift.ma
@@ -20,7 +20,7 @@ include "basic_2/reduction/cnr.ma".
 (* Advanced properties ******************************************************)
 
 (* Basic_1: was only: nf2_csort_lref *)
-lemma cnr_lref_atom: ∀L,i. ⇩[0, i] L ≡ ⋆ → L ⊢ 𝐍⦃#i⦄.
+lemma cnr_lref_atom: ∀L,i. ⇩[0, i] L ≡ ⋆ → ⦃G, L⦄ ⊢ 𝐍⦃#i⦄.
 #L #i #HL #X #H
 elim (cpr_inv_lref1 … H) -H // *
 #K #V1 #V2 #HLK #_ #_
@@ -28,7 +28,7 @@ lapply (ldrop_mono … HL … HLK) -L #H destruct
 qed.
 
 (* Basic_1: was: nf2_lref_abst *)
-lemma cnr_lref_abst: ∀L,K,V,i. ⇩[0, i] L ≡ K. ⓛV → L ⊢ 𝐍⦃#i⦄.
+lemma cnr_lref_abst: ∀L,K,V,i. ⇩[0, i] L ≡ K. ⓛV → ⦃G, L⦄ ⊢ 𝐍⦃#i⦄.
 #L #K #V #i #HLK #X #H
 elim (cpr_inv_lref1 … H) -H // *
 #K0 #V1 #V2 #HLK0 #_ #_
@@ -39,7 +39,7 @@ qed.
 
 (* Basic_1: was: nf2_lift *)
 lemma cnr_lift: ∀L0,L,T,T0,d,e.
-                L ⊢ 𝐍⦃T⦄ → ⇩[d, e] L0 ≡ L → ⇧[d, e] T ≡ T0 → L0 ⊢ 𝐍⦃T0⦄.
+                ⦃G, L⦄ ⊢ 𝐍⦃T⦄ → ⇩[d, e] L0 ≡ L → ⇧[d, e] T ≡ T0 → L0 ⊢ 𝐍⦃T0⦄.
 #L0 #L #T #T0 #d #e #HLT #HL0 #HT0 #X #H
 elim (cpr_inv_lift1 … H … HL0 … HT0) -L0 #T1 #HT10 #HT1
 <(HLT … HT1) in HT0; -L #HT0
@@ -48,7 +48,7 @@ qed.
 
 (* Note: this was missing in basic_1 *)
 lemma cnr_inv_lift: ∀L0,L,T,T0,d,e.
-                    L0 ⊢ 𝐍⦃T0⦄ → ⇩[d, e] L0 ≡ L → ⇧[d, e] T ≡ T0 → L ⊢ 𝐍⦃T⦄.
+                    L0 ⊢ 𝐍⦃T0⦄ → ⇩[d, e] L0 ≡ L → ⇧[d, e] T ≡ T0 → ⦃G, L⦄ ⊢ 𝐍⦃T⦄.
 #L0 #L #T #T0 #d #e #HLT0 #HL0 #HT0 #X #H
 elim (lift_total X d e) #X0 #HX0
 lapply (cpr_lift … H … HL0 … HT0 … HX0) -L #HTX0
diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/cnx.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/cnx.ma
index 8e6dd66ba..5c97081f5 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/reduction/cnx.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/reduction/cnx.ma
@@ -27,37 +27,37 @@ interpretation
 
 (* Basic inversion lemmas ***************************************************)
 
-lemma cnx_inv_sort: ∀h,g,L,k. ⦃h, L⦄ ⊢ 𝐍[g]⦃⋆k⦄ → deg h g k 0.
+lemma cnx_inv_sort: ∀h,g,L,k. ⦃G, L⦄ ⊢ 𝐍[h, g]⦃⋆k⦄ → deg h g k 0.
 #h #g #L #k #H elim (deg_total h g k)
 #l @(nat_ind_plus … l) -l // #l #_ #Hkl
 lapply (H (⋆(next h k)) ?) -H /2 width=2/ -L -l #H destruct -H -e0 (**) (* destruct does not remove some premises *)
 lapply (next_lt h k) >e1 -e1 #H elim (lt_refl_false … H)
 qed-.
 
-lemma cnx_inv_delta: ∀h,g,I,L,K,V,i. ⇩[0, i] L ≡ K.ⓑ{I}V → ⦃h, L⦄ ⊢ 𝐍[g]⦃#i⦄ → ⊥.
+lemma cnx_inv_delta: ∀h,g,I,L,K,V,i. ⇩[0, i] L ≡ K.ⓑ{I}V → ⦃G, L⦄ ⊢ 𝐍[h, g]⦃#i⦄ → ⊥.
 #h #g #I #L #K #V #i #HLK #H
 elim (lift_total V 0 (i+1)) #W #HVW
 lapply (H W ?) -H [ /3 width=7/ ] -HLK #H destruct
 elim (lift_inv_lref2_be … HVW) -HVW //
 qed-.
 
-lemma cnx_inv_abst: ∀h,g,a,L,V,T. ⦃h, L⦄ ⊢ 𝐍[g]⦃ⓛ{a}V.T⦄ →
-                    ⦃h, L⦄ ⊢ 𝐍[g]⦃V⦄ ∧ ⦃h, L.ⓛV⦄ ⊢ 𝐍[g]⦃T⦄.
+lemma cnx_inv_abst: ∀h,g,a,L,V,T. ⦃G, L⦄ ⊢ 𝐍[h, g]⦃ⓛ{a}V.T⦄ →
+                    ⦃G, L⦄ ⊢ 𝐍[h, g]⦃V⦄ ∧ ⦃h, L.ⓛV⦄ ⊢ 𝐍[h, g]⦃T⦄.
 #h #g #a #L #V1 #T1 #HVT1 @conj
 [ #V2 #HV2 lapply (HVT1 (ⓛ{a}V2.T1) ?) -HVT1 /2 width=2/ -HV2 #H destruct //
 | #T2 #HT2 lapply (HVT1 (ⓛ{a}V1.T2) ?) -HVT1 /2 width=2/ -HT2 #H destruct //
 ]
 qed-.
 
-lemma cnx_inv_abbr: ∀h,g,L,V,T. ⦃h, L⦄ ⊢ 𝐍[g]⦃-ⓓV.T⦄ →
-                    ⦃h, L⦄ ⊢ 𝐍[g]⦃V⦄ ∧ ⦃h, L.ⓓV⦄ ⊢ 𝐍[g]⦃T⦄.
+lemma cnx_inv_abbr: ∀h,g,L,V,T. ⦃G, L⦄ ⊢ 𝐍[h, g]⦃-ⓓV.T⦄ →
+                    ⦃G, L⦄ ⊢ 𝐍[h, g]⦃V⦄ ∧ ⦃h, L.ⓓV⦄ ⊢ 𝐍[h, g]⦃T⦄.
 #h #g #L #V1 #T1 #HVT1 @conj
 [ #V2 #HV2 lapply (HVT1 (-ⓓV2.T1) ?) -HVT1 /2 width=2/ -HV2 #H destruct //
 | #T2 #HT2 lapply (HVT1 (-ⓓV1.T2) ?) -HVT1 /2 width=2/ -HT2 #H destruct //
 ]
 qed-.
 
-lemma cnx_inv_zeta: ∀h,g,L,V,T. ⦃h, L⦄ ⊢ 𝐍[g]⦃+ⓓV.T⦄ → ⊥.
+lemma cnx_inv_zeta: ∀h,g,L,V,T. ⦃G, L⦄ ⊢ 𝐍[h, g]⦃+ⓓV.T⦄ → ⊥.
 #h #g #L #V #T #H elim (is_lift_dec T 0 1)
 [ * #U #HTU
   lapply (H U ?) -H /2 width=3/ #H destruct
@@ -68,8 +68,8 @@ lemma cnx_inv_zeta: ∀h,g,L,V,T. ⦃h, L⦄ ⊢ 𝐍[g]⦃+ⓓV.T⦄ → ⊥.
 ]
 qed-.
 
-lemma cnx_inv_appl: ∀h,g,L,V,T. ⦃h, L⦄ ⊢ 𝐍[g]⦃ⓐV.T⦄ →
-                    ∧∧ ⦃h, L⦄ ⊢ 𝐍[g]⦃V⦄ & ⦃h, L⦄ ⊢ 𝐍[g]⦃T⦄ & 𝐒⦃T⦄.
+lemma cnx_inv_appl: ∀h,g,L,V,T. ⦃G, L⦄ ⊢ 𝐍[h, g]⦃ⓐV.T⦄ →
+                    ∧∧ ⦃G, L⦄ ⊢ 𝐍[h, g]⦃V⦄ & ⦃G, L⦄ ⊢ 𝐍[h, g]⦃T⦄ & 𝐒⦃T⦄.
 #h #g #L #V1 #T1 #HVT1 @and3_intro
 [ #V2 #HV2 lapply (HVT1 (ⓐV2.T1) ?) -HVT1 /2 width=1/ -HV2 #H destruct //
 | #T2 #HT2 lapply (HVT1 (ⓐV1.T2) ?) -HVT1 /2 width=1/ -HT2 #H destruct //
@@ -80,43 +80,43 @@ lemma cnx_inv_appl: ∀h,g,L,V,T. ⦃h, L⦄ ⊢ 𝐍[g]⦃ⓐV.T⦄ →
 ]
 qed-.
 
-lemma cnx_inv_tau: ∀h,g,L,V,T. ⦃h, L⦄ ⊢ 𝐍[g]⦃ⓝV.T⦄ → ⊥.
+lemma cnx_inv_tau: ∀h,g,L,V,T. ⦃G, L⦄ ⊢ 𝐍[h, g]⦃ⓝV.T⦄ → ⊥.
 #h #g #L #V #T #H lapply (H T ?) -H /2 width=1/ #H
 @discr_tpair_xy_y //
 qed-.
 
 (* Basic forward lemmas *****************************************************)
 
-lemma cnx_fwd_cnr: ∀h,g,L,T. ⦃h, L⦄ ⊢ 𝐍[g]⦃T⦄ → L ⊢ 𝐍⦃T⦄.
+lemma cnx_fwd_cnr: ∀h,g,L,T. ⦃G, L⦄ ⊢ 𝐍[h, g]⦃T⦄ → ⦃G, L⦄ ⊢ 𝐍⦃T⦄.
 #h #g #L #T #H #U #HTU
 @H /2 width=1/ (**) (* auto fails because a δ-expansion gets in the way *)
 qed-.
 
 (* Basic properties *********************************************************)
 
-lemma cnx_sort: ∀h,g,L,k. deg h g k 0 → ⦃h, L⦄ ⊢ 𝐍[g]⦃⋆k⦄.
+lemma cnx_sort: ∀h,g,L,k. deg h g k 0 → ⦃G, L⦄ ⊢ 𝐍[h, g]⦃⋆k⦄.
 #h #g #L #k #Hk #X #H elim (cpx_inv_sort1 … H) -H // * #l #Hkl #_
 lapply (deg_mono … Hkl Hk) -h -L <plus_n_Sm #H destruct 
 qed.
 
-lemma cnx_sort_iter: ∀h,g,L,k,l. deg h g k l → ⦃h, L⦄ ⊢ 𝐍[g]⦃⋆((next h)^l k)⦄.
+lemma cnx_sort_iter: ∀h,g,L,k,l. deg h g k l → ⦃G, L⦄ ⊢ 𝐍[h, g]⦃⋆((next h)^l k)⦄.
 #h #g #L #k #l #Hkl
 lapply (deg_iter … l Hkl) -Hkl <minus_n_n /2 width=1/
 qed.
 
-lemma cnx_abst: ∀h,g,a,L,W,T. ⦃h, L⦄ ⊢ 𝐍[g]⦃W⦄ → ⦃h, L.ⓛW⦄ ⊢ 𝐍[g]⦃T⦄ →
-                ⦃h, L⦄ ⊢ 𝐍[g]⦃ⓛ{a}W.T⦄.
+lemma cnx_abst: ∀h,g,a,L,W,T. ⦃G, L⦄ ⊢ 𝐍[h, g]⦃W⦄ → ⦃h, L.ⓛW⦄ ⊢ 𝐍[h, g]⦃T⦄ →
+                ⦃G, L⦄ ⊢ 𝐍[h, g]⦃ⓛ{a}W.T⦄.
 #h #g #a #L #W #T #HW #HT #X #H
 elim (cpx_inv_abst1 … H) -H #W0 #T0 #HW0 #HT0 #H destruct
 >(HW … HW0) -W0 >(HT … HT0) -T0 //
 qed.
 
-lemma cnx_appl_simple: ∀h,g,L,V,T. ⦃h, L⦄ ⊢ 𝐍[g]⦃V⦄ → ⦃h, L⦄ ⊢ 𝐍[g]⦃T⦄ → 𝐒⦃T⦄ →
-                       ⦃h, L⦄ ⊢ 𝐍[g]⦃ⓐV.T⦄.
+lemma cnx_appl_simple: ∀h,g,L,V,T. ⦃G, L⦄ ⊢ 𝐍[h, g]⦃V⦄ → ⦃G, L⦄ ⊢ 𝐍[h, g]⦃T⦄ → 𝐒⦃T⦄ →
+                       ⦃G, L⦄ ⊢ 𝐍[h, g]⦃ⓐV.T⦄.
 #h #g #L #V #T #HV #HT #HS #X #H
 elim (cpx_inv_appl1_simple … H) -H // #V0 #T0 #HV0 #HT0 #H destruct
 >(HV … HV0) -V0 >(HT … HT0) -T0 //
 qed.
 
-axiom cnx_dec: ∀h,g,L,T1. ⦃h, L⦄ ⊢ 𝐍[g]⦃T1⦄ ∨
-               ∃∃T2. ⦃h, L⦄ ⊢ T1 ➡[g] T2 & (T1 = T2 → ⊥).
+axiom cnx_dec: ∀h,g,L,T1. ⦃G, L⦄ ⊢ 𝐍[h, g]⦃T1⦄ ∨
+               ∃∃T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 & (T1 = T2 → ⊥).
diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/cnx_cix.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/cnx_cix.ma
index 05f1acedb..f45801a25 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/reduction/cnx_cix.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/reduction/cnx_cix.ma
@@ -19,10 +19,10 @@ include "basic_2/reduction/cnx_crx.ma".
 
 (* Main properties on context-sensitive extended irreducible terms **********)
 
-theorem cix_cnr: ∀h,g,L,T. ⦃h, L⦄ ⊢ 𝐈[g]⦃T⦄ → ⦃h, L⦄ ⊢ 𝐍[g]⦃T⦄.
+theorem cix_cnr: ∀h,g,L,T. ⦃G, L⦄ ⊢ 𝐈[h, g]⦃T⦄ → ⦃G, L⦄ ⊢ 𝐍[h, g]⦃T⦄.
 /2 width=5 by cpx_fwd_cix/ qed.
 
 (* Main inversion lemmas on context-sensitive extended irreducible terms ****)
 
-theorem cnr_inv_cix: ∀h,g,L,T. ⦃h, L⦄ ⊢ 𝐍[g]⦃T⦄ → ⦃h, L⦄ ⊢ 𝐈[g]⦃T⦄.
+theorem cnr_inv_cix: ∀h,g,L,T. ⦃G, L⦄ ⊢ 𝐍[h, g]⦃T⦄ → ⦃G, L⦄ ⊢ 𝐈[h, g]⦃T⦄.
 /2 width=6 by cnx_inv_crx/ qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/cnx_crx.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/cnx_crx.ma
index 99ca32c13..84951db34 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/reduction/cnx_crx.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/reduction/cnx_crx.ma
@@ -20,7 +20,7 @@ include "basic_2/reduction/cnx.ma".
 (* Advanced inversion lemmas on context-sensitive reducible terms ***********)
 
 (* Note: this property is unusual *)
-lemma cnx_inv_crx: ∀h,g,L,T. ⦃h, L⦄ ⊢ 𝐑[g]⦃T⦄ → ⦃h, L⦄ ⊢ 𝐍[g]⦃T⦄ → ⊥.
+lemma cnx_inv_crx: ∀h,g,L,T. ⦃G, L⦄ ⊢ 𝐑[h, g]⦃T⦄ → ⦃G, L⦄ ⊢ 𝐍[h, g]⦃T⦄ → ⊥.
 #h #g #L #T #H elim H -L -T
 [ #L #k #l #Hkl #H
   lapply (cnx_inv_sort … H) -H #H
diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/cnx_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/cnx_lift.ma
index 750d23995..d26c41cb7 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/reduction/cnx_lift.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/reduction/cnx_lift.ma
@@ -19,7 +19,7 @@ include "basic_2/reduction/cnx.ma".
 
 (* Advanced properties ******************************************************)
 
-lemma cnx_lref_atom: ∀h,g,L,i. ⇩[0, i] L ≡ ⋆ → ⦃h, L⦄ ⊢ 𝐍[g]⦃#i⦄.
+lemma cnx_lref_atom: ∀h,g,L,i. ⇩[0, i] L ≡ ⋆ → ⦃G, L⦄ ⊢ 𝐍[h, g]⦃#i⦄.
 #h #g #L #i #HL #X #H
 elim (cpx_inv_lref1 … H) -H // *
 #I #K #V1 #V2 #HLK #_ #_
@@ -28,16 +28,16 @@ qed.
 
 (* Relocation properties ****************************************************)
 
-lemma cnx_lift: ∀h,g,L0,L,T,T0,d,e. ⦃h, L⦄ ⊢ 𝐍[g]⦃T⦄ → ⇩[d, e] L0 ≡ L →
-                ⇧[d, e] T ≡ T0 → ⦃h, L0⦄ ⊢ 𝐍[g]⦃T0⦄.
+lemma cnx_lift: ∀h,g,L0,L,T,T0,d,e. ⦃G, L⦄ ⊢ 𝐍[h, g]⦃T⦄ → ⇩[d, e] L0 ≡ L →
+                ⇧[d, e] T ≡ T0 → ⦃h, L0⦄ ⊢ 𝐍[h, g]⦃T0⦄.
 #h #g #L0 #L #T #T0 #d #e #HLT #HL0 #HT0 #X #H
 elim (cpx_inv_lift1 … H … HL0 … HT0) -L0 #T1 #HT10 #HT1
 <(HLT … HT1) in HT0; -L #HT0
 >(lift_mono … HT10 … HT0) -T1 -X //
 qed.
 
-lemma cnx_inv_lift: ∀h,g,L0,L,T,T0,d,e. ⦃h, L0⦄ ⊢ 𝐍[g]⦃T0⦄ → ⇩[d, e] L0 ≡ L →
-                    ⇧[d, e] T ≡ T0 → ⦃h, L⦄ ⊢ 𝐍[g]⦃T⦄.
+lemma cnx_inv_lift: ∀h,g,L0,L,T,T0,d,e. ⦃h, L0⦄ ⊢ 𝐍[h, g]⦃T0⦄ → ⇩[d, e] L0 ≡ L →
+                    ⇧[d, e] T ≡ T0 → ⦃G, L⦄ ⊢ 𝐍[h, g]⦃T⦄.
 #h #g #L0 #L #T #T0 #d #e #HLT0 #HL0 #HT0 #X #H
 elim (lift_total X d e) #X0 #HX0
 lapply (cpx_lift … H … HL0 … HT0 … HX0) -L #HTX0
diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/cpr.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/cpr.ma
index 2f263f76c..69b591178 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/reduction/cpr.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/reduction/cpr.ma
@@ -60,23 +60,23 @@ lemma lsubr_cpr_trans: lsub_trans … cpr lsubr.
 qed-.
 
 (* Basic_1: was by definition: pr2_free *)
-lemma tpr_cpr: ∀T1,T2. ⋆ ⊢ T1 ➡ T2 → ∀L. L ⊢ T1 ➡ T2.
+lemma tpr_cpr: ∀T1,T2. ⋆ ⊢ T1 ➡ T2 → ∀L. ⦃G, L⦄ ⊢ T1 ➡ T2.
 #T1 #T2 #HT12 #L
 lapply (lsubr_cpr_trans … HT12 L ?) //
 qed.
 
 (* Basic_1: includes by definition: pr0_refl *)
-lemma cpr_refl: ∀T,L. L ⊢ T ➡ T.
+lemma cpr_refl: ∀T,L. ⦃G, L⦄ ⊢ T ➡ T.
 #T elim T -T // * /2 width=1/
 qed.
 
 (* Basic_1: was: pr2_head_1 *)
-lemma cpr_pair_sn: ∀I,L,V1,V2. L ⊢ V1 ➡ V2 →
-                   ∀T. L ⊢ ②{I}V1.T ➡ ②{I}V2.T.
+lemma cpr_pair_sn: ∀I,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡ V2 →
+                   ∀T. ⦃G, L⦄ ⊢ ②{I}V1.T ➡ ②{I}V2.T.
 * /2 width=1/ qed.
 
 lemma cpr_delift: ∀K,V,T1,L,d. ⇩[0, d] L ≡ (K.ⓓV) →
-                  ∃∃T2,T. L ⊢ T1 ➡ T2 & ⇧[d, 1] T ≡ T2.
+                  ∃∃T2,T. ⦃G, L⦄ ⊢ T1 ➡ T2 & ⇧[d, 1] T ≡ T2.
 #K #V #T1 elim T1 -T1
 [ * #i #L #d #HLK /2 width=4/
   elim (lt_or_eq_or_gt i d) #Hid [1,3: /3 width=4/ ]
@@ -101,7 +101,7 @@ qed.
 
 (* Basic inversion lemmas ***************************************************)
 
-fact cpr_inv_atom1_aux: ∀L,T1,T2. L ⊢ T1 ➡ T2 → ∀I. T1 = ⓪{I} →
+fact cpr_inv_atom1_aux: ∀L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡ T2 → ∀I. T1 = ⓪{I} →
                         T2 = ⓪{I} ∨
                         ∃∃K,V,V2,i. ⇩[O, i] L ≡ K. ⓓV &
                                     K ⊢ V ➡ V2 &
@@ -119,7 +119,7 @@ fact cpr_inv_atom1_aux: ∀L,T1,T2. L ⊢ T1 ➡ T2 → ∀I. T1 = ⓪{I} →
 ]
 qed-.
 
-lemma cpr_inv_atom1: ∀I,L,T2. L ⊢ ⓪{I} ➡ T2 →
+lemma cpr_inv_atom1: ∀I,L,T2. ⦃G, L⦄ ⊢ ⓪{I} ➡ T2 →
                      T2 = ⓪{I} ∨
                      ∃∃K,V,V2,i. ⇩[O, i] L ≡ K. ⓓV &
                                  K ⊢ V ➡ V2 &
@@ -128,14 +128,14 @@ lemma cpr_inv_atom1: ∀I,L,T2. L ⊢ ⓪{I} ➡ T2 →
 /2 width=3 by cpr_inv_atom1_aux/ qed-.
 
 (* Basic_1: includes: pr0_gen_sort pr2_gen_sort *)
-lemma cpr_inv_sort1: ∀L,T2,k. L ⊢ ⋆k ➡ T2 → T2 = ⋆k.
+lemma cpr_inv_sort1: ∀L,T2,k. ⦃G, L⦄ ⊢ ⋆k ➡ T2 → T2 = ⋆k.
 #L #T2 #k #H
 elim (cpr_inv_atom1 … H) -H //
 * #K #V #V2 #i #_ #_ #_ #H destruct
 qed-.
 
 (* Basic_1: includes: pr0_gen_lref pr2_gen_lref *)
-lemma cpr_inv_lref1: ∀L,T2,i. L ⊢ #i ➡ T2 →
+lemma cpr_inv_lref1: ∀L,T2,i. ⦃G, L⦄ ⊢ #i ➡ T2 →
                      T2 = #i ∨
                      ∃∃K,V,V2. ⇩[O, i] L ≡ K. ⓓV &
                                K ⊢ V ➡ V2 &
@@ -145,15 +145,15 @@ elim (cpr_inv_atom1 … H) -H /2 width=1/
 * #K #V #V2 #j #HLK #HV2 #HVT2 #H destruct /3 width=6/
 qed-.
 
-lemma cpr_inv_gref1: ∀L,T2,p. L ⊢ §p ➡ T2 → T2 = §p.
+lemma cpr_inv_gref1: ∀L,T2,p. ⦃G, L⦄ ⊢ §p ➡ T2 → T2 = §p.
 #L #T2 #p #H
 elim (cpr_inv_atom1 … H) -H //
 * #K #V #V2 #i #_ #_ #_ #H destruct
 qed-.
 
-fact cpr_inv_bind1_aux: ∀L,U1,U2. L ⊢ U1 ➡ U2 →
+fact cpr_inv_bind1_aux: ∀L,U1,U2. ⦃G, L⦄ ⊢ U1 ➡ U2 →
                         ∀a,I,V1,T1. U1 = ⓑ{a,I}V1. T1 → (
-                        ∃∃V2,T2. L ⊢ V1 ➡ V2 &
+                        ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 &
                                  L. ⓑ{I}V1 ⊢ T1 ➡ T2 &
                                  U2 = ⓑ{a,I}V2.T2
                         ) ∨
@@ -170,8 +170,8 @@ fact cpr_inv_bind1_aux: ∀L,U1,U2. L ⊢ U1 ➡ U2 →
 ]
 qed-.
 
-lemma cpr_inv_bind1: ∀a,I,L,V1,T1,U2. L ⊢ ⓑ{a,I}V1.T1 ➡ U2 → (
-                     ∃∃V2,T2. L ⊢ V1 ➡ V2 &
+lemma cpr_inv_bind1: ∀a,I,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡ U2 → (
+                     ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 &
                               L. ⓑ{I}V1 ⊢ T1 ➡ T2 &
                               U2 = ⓑ{a,I}V2.T2
                      ) ∨
@@ -179,8 +179,8 @@ lemma cpr_inv_bind1: ∀a,I,L,V1,T1,U2. L ⊢ ⓑ{a,I}V1.T1 ➡ U2 → (
 /2 width=3 by cpr_inv_bind1_aux/ qed-.
 
 (* Basic_1: includes: pr0_gen_abbr pr2_gen_abbr *)
-lemma cpr_inv_abbr1: ∀a,L,V1,T1,U2. L ⊢ ⓓ{a}V1.T1 ➡ U2 → (
-                     ∃∃V2,T2. L ⊢ V1 ➡ V2 &
+lemma cpr_inv_abbr1: ∀a,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓓ{a}V1.T1 ➡ U2 → (
+                     ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 &
                               L. ⓓV1 ⊢ T1 ➡ T2 &
                               U2 = ⓓ{a}V2.T2
                      ) ∨
@@ -190,8 +190,8 @@ elim (cpr_inv_bind1 … H) -H * /3 width=3/ /3 width=5/
 qed-.
 
 (* Basic_1: includes: pr0_gen_abst pr2_gen_abst *)
-lemma cpr_inv_abst1: ∀a,L,V1,T1,U2. L ⊢ ⓛ{a}V1.T1 ➡ U2 →
-                     ∃∃V2,T2. L ⊢ V1 ➡ V2 & L.ⓛV1 ⊢ T1 ➡ T2 &
+lemma cpr_inv_abst1: ∀a,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓛ{a}V1.T1 ➡ U2 →
+                     ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & L.ⓛV1 ⊢ T1 ➡ T2 &
                               U2 = ⓛ{a}V2.T2.
 #a #L #V1 #T1 #U2 #H
 elim (cpr_inv_bind1 … H) -H *
@@ -200,16 +200,16 @@ elim (cpr_inv_bind1 … H) -H *
 ]
 qed-.
 
-fact cpr_inv_flat1_aux: ∀L,U,U2. L ⊢ U ➡ U2 →
+fact cpr_inv_flat1_aux: ∀L,U,U2. ⦃G, L⦄ ⊢ U ➡ U2 →
                         ∀I,V1,U1. U = ⓕ{I}V1.U1 →
-                        ∨∨ ∃∃V2,T2. L ⊢ V1 ➡ V2 & L ⊢ U1 ➡ T2 &
+                        ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L⦄ ⊢ U1 ➡ T2 &
                                     U2 = ⓕ{I} V2. T2
-                         | (L ⊢ U1 ➡ U2 ∧ I = Cast)
-                         | ∃∃a,V2,W1,W2,T1,T2. L ⊢ V1 ➡ V2 & L ⊢ W1 ➡ W2 &
+                         | (⦃G, L⦄ ⊢ U1 ➡ U2 ∧ I = Cast)
+                         | ∃∃a,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L⦄ ⊢ W1 ➡ W2 &
                                                L.ⓛW1 ⊢ T1 ➡ T2 & U1 = ⓛ{a}W1.T1 &
                                                U2 = ⓓ{a}ⓝW2.V2.T2 & I = Appl
-                         | ∃∃a,V,V2,W1,W2,T1,T2. L ⊢ V1 ➡ V & ⇧[0,1] V ≡ V2 &
-                                                 L ⊢ W1 ➡ W2 & L.ⓓW1 ⊢ T1 ➡ T2 &
+                         | ∃∃a,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡ V & ⇧[0,1] V ≡ V2 &
+                                                 ⦃G, L⦄ ⊢ W1 ➡ W2 & L.ⓓW1 ⊢ T1 ➡ T2 &
                                                  U1 = ⓓ{a}W1.T1 &
                                                  U2 = ⓓ{a}W2.ⓐV2.T2 & I = Appl.
 #L #U #U2 * -L -U -U2
@@ -224,28 +224,28 @@ fact cpr_inv_flat1_aux: ∀L,U,U2. L ⊢ U ➡ U2 →
 ]
 qed-.
 
-lemma cpr_inv_flat1: ∀I,L,V1,U1,U2. L ⊢ ⓕ{I}V1.U1 ➡ U2 →
-                     ∨∨ ∃∃V2,T2. L ⊢ V1 ➡ V2 & L ⊢ U1 ➡ T2 &
+lemma cpr_inv_flat1: ∀I,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓕ{I}V1.U1 ➡ U2 →
+                     ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L⦄ ⊢ U1 ➡ T2 &
                                  U2 = ⓕ{I}V2.T2
-                      | (L ⊢ U1 ➡ U2 ∧ I = Cast)
-                      | ∃∃a,V2,W1,W2,T1,T2. L ⊢ V1 ➡ V2 & L ⊢ W1 ➡ W2 &
+                      | (⦃G, L⦄ ⊢ U1 ➡ U2 ∧ I = Cast)
+                      | ∃∃a,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L⦄ ⊢ W1 ➡ W2 &
                                             L.ⓛW1 ⊢ T1 ➡ T2 & U1 = ⓛ{a}W1.T1 &
                                             U2 = ⓓ{a}ⓝW2.V2.T2 & I = Appl
-                      | ∃∃a,V,V2,W1,W2,T1,T2. L ⊢ V1 ➡ V & ⇧[0,1] V ≡ V2 &
-                                              L ⊢ W1 ➡ W2 & L.ⓓW1 ⊢ T1 ➡ T2 &
+                      | ∃∃a,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡ V & ⇧[0,1] V ≡ V2 &
+                                              ⦃G, L⦄ ⊢ W1 ➡ W2 & L.ⓓW1 ⊢ T1 ➡ T2 &
                                               U1 = ⓓ{a}W1.T1 &
                                               U2 = ⓓ{a}W2.ⓐV2.T2 & I = Appl.
 /2 width=3 by cpr_inv_flat1_aux/ qed-.
 
 (* Basic_1: includes: pr0_gen_appl pr2_gen_appl *)
-lemma cpr_inv_appl1: ∀L,V1,U1,U2. L ⊢ ⓐV1.U1 ➡ U2 →
-                     ∨∨ ∃∃V2,T2. L ⊢ V1 ➡ V2 & L ⊢ U1 ➡ T2 &
+lemma cpr_inv_appl1: ∀L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓐV1.U1 ➡ U2 →
+                     ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L⦄ ⊢ U1 ➡ T2 &
                                  U2 = ⓐV2.T2
-                      | ∃∃a,V2,W1,W2,T1,T2. L ⊢ V1 ➡ V2 & L ⊢ W1 ➡ W2 &
+                      | ∃∃a,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L⦄ ⊢ W1 ➡ W2 &
                                             L.ⓛW1 ⊢ T1 ➡ T2 &
                                             U1 = ⓛ{a}W1.T1 & U2 = ⓓ{a}ⓝW2.V2.T2
-                      | ∃∃a,V,V2,W1,W2,T1,T2. L ⊢ V1 ➡ V & ⇧[0,1] V ≡ V2 &
-                                              L ⊢ W1 ➡ W2 & L.ⓓW1 ⊢ T1 ➡ T2 &
+                      | ∃∃a,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡ V & ⇧[0,1] V ≡ V2 &
+                                              ⦃G, L⦄ ⊢ W1 ➡ W2 & L.ⓓW1 ⊢ T1 ➡ T2 &
                                               U1 = ⓓ{a}W1.T1 & U2 = ⓓ{a}W2.ⓐV2.T2.
 #L #V1 #U1 #U2 #H elim (cpr_inv_flat1 … H) -H *
 [ /3 width=5/
@@ -256,8 +256,8 @@ lemma cpr_inv_appl1: ∀L,V1,U1,U2. L ⊢ ⓐV1.U1 ➡ U2 →
 qed-.
 
 (* Note: the main property of simple terms *)
-lemma cpr_inv_appl1_simple: ∀L,V1,T1,U. L ⊢ ⓐV1. T1 ➡ U → 𝐒⦃T1⦄ →
-                            ∃∃V2,T2. L ⊢ V1 ➡ V2 & L ⊢ T1 ➡ T2 &
+lemma cpr_inv_appl1_simple: ∀L,V1,T1,U. ⦃G, L⦄ ⊢ ⓐV1. T1 ➡ U → 𝐒⦃T1⦄ →
+                            ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L⦄ ⊢ T1 ➡ T2 &
                                      U = ⓐV2. T2.
 #L #V1 #T1 #U #H #HT1
 elim (cpr_inv_appl1 … H) -H *
@@ -270,11 +270,11 @@ elim (cpr_inv_appl1 … H) -H *
 qed-.
 
 (* Basic_1: includes: pr0_gen_cast pr2_gen_cast *)
-lemma cpr_inv_cast1: ∀L,V1,U1,U2. L ⊢ ⓝ V1. U1 ➡ U2 → (
-                     ∃∃V2,T2. L ⊢ V1 ➡ V2 & L ⊢ U1 ➡ T2 &
+lemma cpr_inv_cast1: ∀L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓝ V1. U1 ➡ U2 → (
+                     ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L⦄ ⊢ U1 ➡ T2 &
                               U2 = ⓝ V2. T2
                      ) ∨
-                     L ⊢ U1 ➡ U2.
+                     ⦃G, L⦄ ⊢ U1 ➡ U2.
 #L #V1 #U1 #U2 #H elim (cpr_inv_flat1 … H) -H *
 [ /3 width=5/
 | /2 width=1/
@@ -285,8 +285,8 @@ qed-.
 
 (* Basic forward lemmas *****************************************************)
 
-lemma cpr_fwd_bind1_minus: ∀I,L,V1,T1,T. L ⊢ -ⓑ{I}V1.T1 ➡ T → ∀b.
-                           ∃∃V2,T2. L ⊢ ⓑ{b,I}V1.T1 ➡ ⓑ{b,I}V2.T2 &
+lemma cpr_fwd_bind1_minus: ∀I,L,V1,T1,T. ⦃G, L⦄ ⊢ -ⓑ{I}V1.T1 ➡ T → ∀b.
+                           ∃∃V2,T2. ⦃G, L⦄ ⊢ ⓑ{b,I}V1.T1 ➡ ⓑ{b,I}V2.T2 &
                                     T = -ⓑ{I}V2.T2.
 #I #L #V1 #T1 #T #H #b
 elim (cpr_inv_bind1 … H) -H *
@@ -295,7 +295,7 @@ elim (cpr_inv_bind1 … H) -H *
 ]
 qed-.
 
-lemma cpr_fwd_shift1: ∀L1,L,T1,T. L ⊢ L1 @@ T1 ➡ T →
+lemma cpr_fwd_shift1: ∀L1,L,T1,T. ⦃G, L⦄ ⊢ L1 @@ T1 ➡ T →
                       ∃∃L2,T2. |L1| = |L2| & T = L2 @@ T2.
 #L1 @(lenv_ind_dx … L1) -L1 normalize
 [ #L #T1 #T #HT1
diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/cpr_cir.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/cpr_cir.ma
index ee49679b7..a54e9c420 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/reduction/cpr_cir.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/reduction/cpr_cir.ma
@@ -19,7 +19,7 @@ include "basic_2/reduction/cpr.ma".
 
 (* Advanced forward lemmas on context-sensitive irreducible terms ***********)
 
-lemma cpr_fwd_cir: ∀L,T1,T2. L ⊢ T1 ➡ T2 → L ⊢ 𝐈⦃T1⦄ → T2 = T1.
+lemma cpr_fwd_cir: ∀L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡ T2 → ⦃G, L⦄ ⊢ 𝐈⦃T1⦄ → T2 = T1.
 #L #T1 #T2 #H elim H -L -T1 -T2
 [ //
 | #L #K #V1 #V2 #W2 #i #HLK #_ #HVW2 #IHV12 #H
diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/cpx.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/cpx.ma
index 6e1a50a65..6225e896b 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/reduction/cpx.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/reduction/cpx.ma
@@ -63,28 +63,28 @@ lemma lsubr_cpx_trans: ∀h,g. lsub_trans … (cpx h g) lsubr.
 qed-.
 
 (* Note: this is "∀h,g,L. reflexive … (cpx h g L)" *)
-lemma cpx_refl: ∀h,g,T,L. ⦃h, L⦄ ⊢ T ➡[g] T.
+lemma cpx_refl: ∀h,g,T,L. ⦃G, L⦄ ⊢ T ➡[h, g] T.
 #h #g #T elim T -T // * /2 width=1/
 qed.
 
-lemma cpr_cpx: ∀h,g,L,T1,T2. L ⊢ T1 ➡ T2 → ⦃h, L⦄ ⊢ T1 ➡[g] T2.
+lemma cpr_cpx: ∀h,g,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡ T2 → ⦃G, L⦄ ⊢ T1 ➡[h, g] T2.
 #h #g #L #T1 #T2 #H elim H -L -T1 -T2 // /2 width=1/ /2 width=3/ /2 width=7/
 qed.
 
-fact ssta_cpx_aux: ∀h,g,L,T1,T2,l0. ⦃h, L⦄ ⊢ T1 •[g] ⦃l0, T2⦄ →
-                   ∀l. l0 = l+1 → ⦃h, L⦄ ⊢ T1 ➡[g] T2.
+fact ssta_cpx_aux: ∀h,g,L,T1,T2,l0. ⦃G, L⦄ ⊢ T1 •[h, g] ⦃l0, T2⦄ →
+                   ∀l. l0 = l+1 → ⦃G, L⦄ ⊢ T1 ➡[h, g] T2.
 #h #g #L #T1 #T2 #l0 #H elim H -L -T1 -T2 -l0 /2 width=2/ /2 width=7/ /3 width=2/ /3 width=7/
 qed-.
 
-lemma ssta_cpx: ∀h,g,L,T1,T2,l. ⦃h, L⦄ ⊢ T1 •[g] ⦃l+1, T2⦄ → ⦃h, L⦄ ⊢ T1 ➡[g] T2.
+lemma ssta_cpx: ∀h,g,L,T1,T2,l. ⦃G, L⦄ ⊢ T1 •[h, g] ⦃l+1, T2⦄ → ⦃G, L⦄ ⊢ T1 ➡[h, g] T2.
 /2 width=4 by ssta_cpx_aux/ qed.
 
-lemma cpx_pair_sn: ∀h,g,I,L,V1,V2. ⦃h, L⦄ ⊢ V1 ➡[g] V2 →
-                   ∀T. ⦃h, L⦄ ⊢ ②{I}V1.T ➡[g] ②{I}V2.T.
+lemma cpx_pair_sn: ∀h,g,I,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 →
+                   ∀T. ⦃G, L⦄ ⊢ ②{I}V1.T ➡[h, g] ②{I}V2.T.
 #h #g * /2 width=1/ qed.
 
 lemma cpx_delift: ∀h,g,I,K,V,T1,L,d. ⇩[0, d] L ≡ (K.ⓑ{I}V) →
-                  ∃∃T2,T.  ⦃h, L⦄ ⊢ T1 ➡[g] T2 & ⇧[d, 1] T ≡ T2.
+                  ∃∃T2,T.  ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 & ⇧[d, 1] T ≡ T2.
 #h #g #I #K #V #T1 elim T1 -T1
 [ * #i #L #d #HLK /2 width=4/
   elim (lt_or_eq_or_gt i d) #Hid [1,3: /3 width=4/ ]
@@ -109,10 +109,10 @@ qed.
 
 (* Basic inversion lemmas ***************************************************)
 
-fact cpx_inv_atom1_aux: ∀h,g,L,T1,T2. ⦃h, L⦄ ⊢ T1 ➡[g] T2 → ∀J. T1 = ⓪{J} →
+fact cpx_inv_atom1_aux: ∀h,g,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 → ∀J. T1 = ⓪{J} →
                         ∨∨ T2 = ⓪{J}
                          | ∃∃k,l. deg h g k (l+1) & T2 = ⋆(next h k) & J = Sort k
-                         | ∃∃I,K,V,V2,i. ⇩[O, i] L ≡ K.ⓑ{I}V & ⦃h, K⦄ ⊢ V ➡[g] V2 &
+                         | ∃∃I,K,V,V2,i. ⇩[O, i] L ≡ K.ⓑ{I}V & ⦃h, K⦄ ⊢ V ➡[h, g] V2 &
                                          ⇧[O, i + 1] V2 ≡ T2 & J = LRef i.
 #h #g #L #T1 #T2 * -L -T1 -T2
 [ #I #L #J #H destruct /2 width=1/
@@ -128,14 +128,14 @@ fact cpx_inv_atom1_aux: ∀h,g,L,T1,T2. ⦃h, L⦄ ⊢ T1 ➡[g] T2 → ∀J. T1
 ]
 qed-.
 
-lemma cpx_inv_atom1: ∀h,g,J,L,T2. ⦃h, L⦄ ⊢ ⓪{J} ➡[g] T2 →
+lemma cpx_inv_atom1: ∀h,g,J,L,T2. ⦃G, L⦄ ⊢ ⓪{J} ➡[h, g] T2 →
                      ∨∨ T2 = ⓪{J}
                       | ∃∃k,l. deg h g k (l+1) & T2 = ⋆(next h k) & J = Sort k
-                      | ∃∃I,K,V,V2,i. ⇩[O, i] L ≡ K.ⓑ{I}V & ⦃h, K⦄ ⊢ V ➡[g] V2 &
+                      | ∃∃I,K,V,V2,i. ⇩[O, i] L ≡ K.ⓑ{I}V & ⦃h, K⦄ ⊢ V ➡[h, g] V2 &
                                       ⇧[O, i + 1] V2 ≡ T2 & J = LRef i.
 /2 width=3 by cpx_inv_atom1_aux/ qed-.
 
-lemma cpx_inv_sort1: ∀h,g,L,T2,k. ⦃h, L⦄ ⊢ ⋆k ➡[g] T2 → T2 = ⋆k ∨
+lemma cpx_inv_sort1: ∀h,g,L,T2,k. ⦃G, L⦄ ⊢ ⋆k ➡[h, g] T2 → T2 = ⋆k ∨
                      ∃∃l. deg h g k (l+1) & T2 = ⋆(next h k).
 #h #g #L #T2 #k #H
 elim (cpx_inv_atom1 … H) -H /2 width=1/ *
@@ -144,9 +144,9 @@ elim (cpx_inv_atom1 … H) -H /2 width=1/ *
 ]
 qed-.
 
-lemma cpx_inv_lref1: ∀h,g,L,T2,i. ⦃h, L⦄ ⊢ #i ➡[g] T2 →
+lemma cpx_inv_lref1: ∀h,g,L,T2,i. ⦃G, L⦄ ⊢ #i ➡[h, g] T2 →
                      T2 = #i ∨
-                     ∃∃I,K,V,V2. ⇩[O, i] L ≡ K. ⓑ{I}V & ⦃h, K⦄ ⊢ V ➡[g] V2 &
+                     ∃∃I,K,V,V2. ⇩[O, i] L ≡ K. ⓑ{I}V & ⦃h, K⦄ ⊢ V ➡[h, g] V2 &
                                  ⇧[O, i + 1] V2 ≡ T2.
 #h #g #L #T2 #i #H
 elim (cpx_inv_atom1 … H) -H /2 width=1/ *
@@ -155,7 +155,7 @@ elim (cpx_inv_atom1 … H) -H /2 width=1/ *
 ]
 qed-.
 
-lemma cpx_inv_gref1: ∀h,g,L,T2,p.  ⦃h, L⦄ ⊢ §p ➡[g] T2 → T2 = §p.
+lemma cpx_inv_gref1: ∀h,g,L,T2,p.  ⦃G, L⦄ ⊢ §p ➡[h, g] T2 → T2 = §p.
 #h #g #L #T2 #p #H
 elim (cpx_inv_atom1 … H) -H // *
 [ #k #l #_ #_ #H destruct
@@ -163,12 +163,12 @@ elim (cpx_inv_atom1 … H) -H // *
 ]
 qed-.
 
-fact cpx_inv_bind1_aux: ∀h,g,L,U1,U2. ⦃h, L⦄ ⊢ U1 ➡[g] U2 →
+fact cpx_inv_bind1_aux: ∀h,g,L,U1,U2. ⦃G, L⦄ ⊢ U1 ➡[h, g] U2 →
                         ∀a,J,V1,T1. U1 = ⓑ{a,J}V1.T1 → (
-                        ∃∃V2,T2. ⦃h, L⦄ ⊢ V1 ➡[g] V2 & ⦃h, L.ⓑ{J}V1⦄ ⊢ T1 ➡[g] T2 &
+                        ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 & ⦃h, L.ⓑ{J}V1⦄ ⊢ T1 ➡[h, g] T2 &
                                  U2 = ⓑ{a,J}V2.T2
                         ) ∨
-                        ∃∃T. ⦃h, L.ⓓV1⦄ ⊢ T1 ➡[g] T & ⇧[0, 1] U2 ≡ T &
+                        ∃∃T. ⦃h, L.ⓓV1⦄ ⊢ T1 ➡[h, g] T & ⇧[0, 1] U2 ≡ T &
                              a = true & J = Abbr.
 #h #g #L #U1 #U2 * -L -U1 -U2
 [ #I #L #b #J #W #U1 #H destruct
@@ -184,25 +184,25 @@ fact cpx_inv_bind1_aux: ∀h,g,L,U1,U2. ⦃h, L⦄ ⊢ U1 ➡[g] U2 →
 ]
 qed-.
 
-lemma cpx_inv_bind1: ∀h,g,a,I,L,V1,T1,U2. ⦃h, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡[g] U2 → (
-                     ∃∃V2,T2. ⦃h, L⦄ ⊢ V1 ➡[g] V2 & ⦃h, L.ⓑ{I}V1⦄ ⊢ T1 ➡[g] T2 &
+lemma cpx_inv_bind1: ∀h,g,a,I,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡[h, g] U2 → (
+                     ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 & ⦃h, L.ⓑ{I}V1⦄ ⊢ T1 ➡[h, g] T2 &
                               U2 = ⓑ{a,I} V2. T2
                      ) ∨
-                     ∃∃T. ⦃h, L.ⓓV1⦄ ⊢ T1 ➡[g] T & ⇧[0, 1] U2 ≡ T &
+                     ∃∃T. ⦃h, L.ⓓV1⦄ ⊢ T1 ➡[h, g] T & ⇧[0, 1] U2 ≡ T &
                           a = true & I = Abbr.
 /2 width=3 by cpx_inv_bind1_aux/ qed-.
 
-lemma cpx_inv_abbr1: ∀h,g,a,L,V1,T1,U2. ⦃h, L⦄ ⊢ ⓓ{a}V1.T1 ➡[g] U2 → (
-                     ∃∃V2,T2. ⦃h, L⦄ ⊢ V1 ➡[g] V2 & ⦃h, L.ⓓV1⦄ ⊢ T1 ➡[g] T2 &
+lemma cpx_inv_abbr1: ∀h,g,a,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓓ{a}V1.T1 ➡[h, g] U2 → (
+                     ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 & ⦃h, L.ⓓV1⦄ ⊢ T1 ➡[h, g] T2 &
                               U2 = ⓓ{a} V2. T2
                      ) ∨
-                     ∃∃T. ⦃h, L.ⓓV1⦄ ⊢ T1 ➡[g] T & ⇧[0, 1] U2 ≡ T & a = true.
+                     ∃∃T. ⦃h, L.ⓓV1⦄ ⊢ T1 ➡[h, g] T & ⇧[0, 1] U2 ≡ T & a = true.
 #h #g #a #L #V1 #T1 #U2 #H
 elim (cpx_inv_bind1 … H) -H * /3 width=3/ /3 width=5/
 qed-.
 
-lemma cpx_inv_abst1: ∀h,g,a,L,V1,T1,U2.  ⦃h, L⦄ ⊢ ⓛ{a}V1.T1 ➡[g] U2 →
-                     ∃∃V2,T2. ⦃h, L⦄ ⊢ V1 ➡[g] V2 &  ⦃h, L.ⓛV1⦄ ⊢ T1 ➡[g] T2 &
+lemma cpx_inv_abst1: ∀h,g,a,L,V1,T1,U2.  ⦃G, L⦄ ⊢ ⓛ{a}V1.T1 ➡[h, g] U2 →
+                     ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 &  ⦃h, L.ⓛV1⦄ ⊢ T1 ➡[h, g] T2 &
                               U2 = ⓛ{a} V2. T2.
 #h #g #a #L #V1 #T1 #U2 #H
 elim (cpx_inv_bind1 … H) -H *
@@ -211,18 +211,18 @@ elim (cpx_inv_bind1 … H) -H *
 ]
 qed-.
 
-fact cpx_inv_flat1_aux: ∀h,g,L,U,U2. ⦃h, L⦄ ⊢ U ➡[g] U2 →
+fact cpx_inv_flat1_aux: ∀h,g,L,U,U2. ⦃G, L⦄ ⊢ U ➡[h, g] U2 →
                         ∀J,V1,U1. U = ⓕ{J}V1.U1 →
-                        ∨∨ ∃∃V2,T2. ⦃h, L⦄ ⊢ V1 ➡[g] V2 & ⦃h, L⦄ ⊢ U1 ➡[g] T2 &
+                        ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 & ⦃G, L⦄ ⊢ U1 ➡[h, g] T2 &
                                     U2 = ⓕ{J}V2.T2
-                         | (⦃h, L⦄ ⊢ U1 ➡[g] U2 ∧ J = Cast)
-                         | (⦃h, L⦄ ⊢ V1 ➡[g] U2 ∧ J = Cast)
-                         | ∃∃a,V2,W1,W2,T1,T2. ⦃h, L⦄ ⊢ V1 ➡[g] V2 & ⦃h, L⦄ ⊢ W1 ➡[g] W2 &
-                                               ⦃h, L.ⓛW1⦄ ⊢ T1 ➡[g] T2 &
+                         | (⦃G, L⦄ ⊢ U1 ➡[h, g] U2 ∧ J = Cast)
+                         | (⦃G, L⦄ ⊢ V1 ➡[h, g] U2 ∧ J = Cast)
+                         | ∃∃a,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 & ⦃G, L⦄ ⊢ W1 ➡[h, g] W2 &
+                                               ⦃h, L.ⓛW1⦄ ⊢ T1 ➡[h, g] T2 &
                                                U1 = ⓛ{a}W1.T1 &
                                                U2 = ⓓ{a}ⓝW2.V2.T2 & J = Appl
-                         | ∃∃a,V,V2,W1,W2,T1,T2. ⦃h, L⦄ ⊢ V1 ➡[g] V & ⇧[0,1] V ≡ V2 &
-                                                 ⦃h, L⦄ ⊢ W1 ➡[g] W2 & ⦃h, L.ⓓW1⦄ ⊢ T1 ➡[g] T2 &
+                         | ∃∃a,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V & ⇧[0,1] V ≡ V2 &
+                                                 ⦃G, L⦄ ⊢ W1 ➡[h, g] W2 & ⦃h, L.ⓓW1⦄ ⊢ T1 ➡[h, g] T2 &
                                                  U1 = ⓓ{a}W1.T1 &
                                                  U2 = ⓓ{a}W2.ⓐV2.T2 & J = Appl.
 #h #g #L #U #U2 * -L -U -U2
@@ -239,29 +239,29 @@ fact cpx_inv_flat1_aux: ∀h,g,L,U,U2. ⦃h, L⦄ ⊢ U ➡[g] U2 →
 ]
 qed-.
 
-lemma cpx_inv_flat1: ∀h,g,I,L,V1,U1,U2. ⦃h, L⦄ ⊢ ⓕ{I}V1.U1 ➡[g] U2 →
-                     ∨∨ ∃∃V2,T2. ⦃h, L⦄ ⊢ V1 ➡[g] V2 & ⦃h, L⦄ ⊢ U1 ➡[g] T2 &
+lemma cpx_inv_flat1: ∀h,g,I,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓕ{I}V1.U1 ➡[h, g] U2 →
+                     ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 & ⦃G, L⦄ ⊢ U1 ➡[h, g] T2 &
                                  U2 = ⓕ{I} V2. T2
-                      | (⦃h, L⦄ ⊢ U1 ➡[g] U2 ∧ I = Cast)
-                      | (⦃h, L⦄ ⊢ V1 ➡[g] U2 ∧ I = Cast)
-                      | ∃∃a,V2,W1,W2,T1,T2. ⦃h, L⦄ ⊢ V1 ➡[g] V2 & ⦃h, L⦄ ⊢ W1 ➡[g] W2 &
-                                            ⦃h, L.ⓛW1⦄ ⊢ T1 ➡[g] T2 &
+                      | (⦃G, L⦄ ⊢ U1 ➡[h, g] U2 ∧ I = Cast)
+                      | (⦃G, L⦄ ⊢ V1 ➡[h, g] U2 ∧ I = Cast)
+                      | ∃∃a,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 & ⦃G, L⦄ ⊢ W1 ➡[h, g] W2 &
+                                            ⦃h, L.ⓛW1⦄ ⊢ T1 ➡[h, g] T2 &
                                             U1 = ⓛ{a}W1.T1 &
                                             U2 = ⓓ{a}ⓝW2.V2.T2 & I = Appl
-                      | ∃∃a,V,V2,W1,W2,T1,T2. ⦃h, L⦄ ⊢ V1 ➡[g] V & ⇧[0,1] V ≡ V2 &
-                                              ⦃h, L⦄ ⊢ W1 ➡[g] W2 & ⦃h, L.ⓓW1⦄ ⊢ T1 ➡[g] T2 &
+                      | ∃∃a,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V & ⇧[0,1] V ≡ V2 &
+                                              ⦃G, L⦄ ⊢ W1 ➡[h, g] W2 & ⦃h, L.ⓓW1⦄ ⊢ T1 ➡[h, g] T2 &
                                               U1 = ⓓ{a}W1.T1 &
                                               U2 = ⓓ{a}W2.ⓐV2.T2 & I = Appl.
 /2 width=3 by cpx_inv_flat1_aux/ qed-.
 
-lemma cpx_inv_appl1: ∀h,g,L,V1,U1,U2. ⦃h, L⦄ ⊢ ⓐ V1.U1 ➡[g] U2 →
-                     ∨∨ ∃∃V2,T2. ⦃h, L⦄ ⊢ V1 ➡[g] V2 & ⦃h, L⦄ ⊢ U1 ➡[g] T2 &
+lemma cpx_inv_appl1: ∀h,g,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓐ V1.U1 ➡[h, g] U2 →
+                     ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 & ⦃G, L⦄ ⊢ U1 ➡[h, g] T2 &
                                  U2 = ⓐ V2. T2
-                      | ∃∃a,V2,W1,W2,T1,T2. ⦃h, L⦄ ⊢ V1 ➡[g] V2 & ⦃h, L⦄ ⊢ W1 ➡[g] W2 &
-                                            ⦃h, L.ⓛW1⦄ ⊢ T1 ➡[g] T2 &
+                      | ∃∃a,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 & ⦃G, L⦄ ⊢ W1 ➡[h, g] W2 &
+                                            ⦃h, L.ⓛW1⦄ ⊢ T1 ➡[h, g] T2 &
                                             U1 = ⓛ{a}W1.T1 & U2 = ⓓ{a}ⓝW2.V2.T2
-                      | ∃∃a,V,V2,W1,W2,T1,T2. ⦃h, L⦄ ⊢ V1 ➡[g] V & ⇧[0,1] V ≡ V2 &
-                                              ⦃h, L⦄ ⊢ W1 ➡[g] W2 & ⦃h, L.ⓓW1⦄ ⊢ T1 ➡[g] T2 &
+                      | ∃∃a,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V & ⇧[0,1] V ≡ V2 &
+                                              ⦃G, L⦄ ⊢ W1 ➡[h, g] W2 & ⦃h, L.ⓓW1⦄ ⊢ T1 ➡[h, g] T2 &
                                               U1 = ⓓ{a}W1.T1 & U2 = ⓓ{a}W2. ⓐV2. T2.
 #h #g #L #V1 #U1 #U2 #H elim (cpx_inv_flat1 … H) -H *
 [ /3 width=5/
@@ -272,8 +272,8 @@ lemma cpx_inv_appl1: ∀h,g,L,V1,U1,U2. ⦃h, L⦄ ⊢ ⓐ V1.U1 ➡[g] U2 →
 qed-.
 
 (* Note: the main property of simple terms *)
-lemma cpx_inv_appl1_simple: ∀h,g,L,V1,T1,U. ⦃h, L⦄ ⊢ ⓐV1.T1 ➡[g] U → 𝐒⦃T1⦄ →
-                            ∃∃V2,T2. ⦃h, L⦄ ⊢ V1 ➡[g] V2 & ⦃h, L⦄ ⊢ T1 ➡[g] T2 &
+lemma cpx_inv_appl1_simple: ∀h,g,L,V1,T1,U. ⦃G, L⦄ ⊢ ⓐV1.T1 ➡[h, g] U → 𝐒⦃T1⦄ →
+                            ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 & ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 &
                                      U = ⓐV2.T2.
 #h #g #L #V1 #T1 #U #H #HT1
 elim (cpx_inv_appl1 … H) -H *
@@ -285,11 +285,11 @@ elim (cpx_inv_appl1 … H) -H *
 ]
 qed-.
 
-lemma cpx_inv_cast1: ∀h,g,L,V1,U1,U2. ⦃h, L⦄ ⊢ ⓝV1.U1 ➡[g] U2 →
-                     ∨∨ ∃∃V2,T2. ⦃h, L⦄ ⊢ V1 ➡[g] V2 & ⦃h, L⦄ ⊢ U1 ➡[g] T2 &
+lemma cpx_inv_cast1: ∀h,g,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓝV1.U1 ➡[h, g] U2 →
+                     ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 & ⦃G, L⦄ ⊢ U1 ➡[h, g] T2 &
                                  U2 = ⓝ V2. T2
-                      | ⦃h, L⦄ ⊢ U1 ➡[g] U2
-                      | ⦃h, L⦄ ⊢ V1 ➡[g] U2.
+                      | ⦃G, L⦄ ⊢ U1 ➡[h, g] U2
+                      | ⦃G, L⦄ ⊢ V1 ➡[h, g] U2.
 #h #g #L #V1 #U1 #U2 #H elim (cpx_inv_flat1 … H) -H *
 [ /3 width=5/
 |2,3: /2 width=1/
@@ -300,8 +300,8 @@ qed-.
 
 (* Basic forward lemmas *****************************************************)
 
-lemma cpx_fwd_bind1_minus: ∀h,g,I,L,V1,T1,T. ⦃h, L⦄ ⊢ -ⓑ{I}V1.T1 ➡[g] T → ∀b.
-                           ∃∃V2,T2. ⦃h, L⦄ ⊢ ⓑ{b,I}V1.T1 ➡[g] ⓑ{b,I}V2.T2 &
+lemma cpx_fwd_bind1_minus: ∀h,g,I,L,V1,T1,T. ⦃G, L⦄ ⊢ -ⓑ{I}V1.T1 ➡[h, g] T → ∀b.
+                           ∃∃V2,T2. ⦃G, L⦄ ⊢ ⓑ{b,I}V1.T1 ➡[h, g] ⓑ{b,I}V2.T2 &
                                     T = -ⓑ{I}V2.T2.
 #h #g #I #L #V1 #T1 #T #H #b
 elim (cpx_inv_bind1 … H) -H *
@@ -310,7 +310,7 @@ elim (cpx_inv_bind1 … H) -H *
 ]
 qed-.
 
-lemma cpx_fwd_shift1: ∀h,g,L1,L,T1,T. ⦃h, L⦄ ⊢ L1 @@ T1 ➡[g] T →
+lemma cpx_fwd_shift1: ∀h,g,L1,L,T1,T. ⦃G, L⦄ ⊢ L1 @@ T1 ➡[h, g] T →
                       ∃∃L2,T2. |L1| = |L2| & T = L2 @@ T2.
 #h #g #L1 @(lenv_ind_dx … L1) -L1 normalize
 [ #L #T1 #T #HT1
diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/cpx_cix.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/cpx_cix.ma
index 7567f9f9c..9aa2d62a8 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/reduction/cpx_cix.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/reduction/cpx_cix.ma
@@ -19,7 +19,7 @@ include "basic_2/reduction/cpx.ma".
 
 (* Advanced forward lemmas on context-sensitive extended irreducible terms **)
 
-lemma cpx_fwd_cix: ∀h,g,L,T1,T2. ⦃h, L⦄ ⊢ T1 ➡[g] T2 → ⦃h, L⦄ ⊢ 𝐈[g]⦃T1⦄ → T2 = T1.
+lemma cpx_fwd_cix: ∀h,g,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 → ⦃G, L⦄ ⊢ 𝐈[h, g]⦃T1⦄ → T2 = T1.
 #h #g #L #T1 #T2 #H elim H -L -T1 -T2
 [ //
 | #L #k #l #Hkl #H elim (cix_inv_sort … Hkl H)
diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/cpx_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/cpx_lift.ma
index 30eef6262..40f6e1e97 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/reduction/cpx_lift.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/reduction/cpx_lift.ma
@@ -120,8 +120,8 @@ qed-.
 (* Properties on supclosure *************************************************)
 
 lemma fsupq_cpx_trans: ∀h,g,L1,L2,T1,T2. ⦃L1, T1⦄ ⊃⸮ ⦃L2, T2⦄ →
-                       ∀U2. ⦃h, L2⦄ ⊢ T2 ➡[g] U2 →
-                       ∃∃U1. ⦃h, L1⦄ ⊢ T1 ➡[g] U1 & ⦃L1, U1⦄ ⊃⸮ ⦃L2, U2⦄.
+                       ∀U2. ⦃h, L2⦄ ⊢ T2 ➡[h, g] U2 →
+                       ∃∃U1. ⦃h, L1⦄ ⊢ T1 ➡[h, g] U1 & ⦃L1, U1⦄ ⊃⸮ ⦃L2, U2⦄.
 #h #g #L1 #L2 #T1 #T2 #H elim H -L1 -L2 -T1 -T2 [1: /2 width=3/ |3,4,5: /3 width=3/ ]
 [ #I #L1 #V2 #U2 #HVU2
   elim (lift_total U2 0 1) /4 width=9/
@@ -133,16 +133,16 @@ lemma fsupq_cpx_trans: ∀h,g,L1,L2,T1,T2. ⦃L1, T1⦄ ⊃⸮ ⦃L2, T2⦄ →
 qed-.
 
 lemma fsupq_ssta_trans: ∀h,g,L1,L2,T1,T2. ⦃L1, T1⦄ ⊃⸮ ⦃L2, T2⦄ →
-                        ∀U2,l. ⦃h, L2⦄ ⊢ T2 •[g] ⦃l+1, U2⦄ →
-                        ∃∃U1. ⦃h, L1⦄ ⊢ T1 ➡[g] U1 & ⦃L1, U1⦄ ⊃⸮ ⦃L2, U2⦄.
+                        ∀U2,l. ⦃h, L2⦄ ⊢ T2 •[h, g] ⦃l+1, U2⦄ →
+                        ∃∃U1. ⦃h, L1⦄ ⊢ T1 ➡[h, g] U1 & ⦃L1, U1⦄ ⊃⸮ ⦃L2, U2⦄.
 /3 width=4 by fsupq_cpx_trans, ssta_cpx/ qed-.
 
 lemma fsup_cpx_trans: ∀h,g,L1,L2,T1,T2. ⦃L1, T1⦄ ⊃ ⦃L2, T2⦄ →
-                      ∀U2. ⦃h, L2⦄ ⊢ T2 ➡[g] U2 →
-                      ∃∃U1. ⦃h, L1⦄ ⊢ T1 ➡[g] U1 & ⦃L1, U1⦄ ⊃⸮ ⦃L2, U2⦄.
+                      ∀U2. ⦃h, L2⦄ ⊢ T2 ➡[h, g] U2 →
+                      ∃∃U1. ⦃h, L1⦄ ⊢ T1 ➡[h, g] U1 & ⦃L1, U1⦄ ⊃⸮ ⦃L2, U2⦄.
 /3 width=3 by fsupq_cpx_trans, fsup_fsupq/ qed-.
 
 lemma fsup_ssta_trans: ∀h,g,L1,L2,T1,T2. ⦃L1, T1⦄ ⊃ ⦃L2, T2⦄ →
-                       ∀U2,l. ⦃h, L2⦄ ⊢ T2 •[g] ⦃l+1, U2⦄ →
-                       ∃∃U1. ⦃h, L1⦄ ⊢ T1 ➡[g] U1 & ⦃L1, U1⦄ ⊃⸮ ⦃L2, U2⦄.
+                       ∀U2,l. ⦃h, L2⦄ ⊢ T2 •[h, g] ⦃l+1, U2⦄ →
+                       ∃∃U1. ⦃h, L1⦄ ⊢ T1 ➡[h, g] U1 & ⦃L1, U1⦄ ⊃⸮ ⦃L2, U2⦄.
 /3 width=4 by fsupq_ssta_trans, fsup_fsupq/ qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/crr.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/crr.ma
index b9a51d862..9fb127750 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/reduction/crr.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/reduction/crr.ma
@@ -43,7 +43,7 @@ interpretation
 
 (* Basic inversion lemmas ***************************************************)
 
-fact crr_inv_sort_aux: ∀L,T,k. L ⊢ 𝐑⦃T⦄ → T = ⋆k → ⊥.
+fact crr_inv_sort_aux: ∀L,T,k. ⦃G, L⦄ ⊢ 𝐑⦃T⦄ → T = ⋆k → ⊥.
 #L #T #k0 * -L -T
 [ #L #K #V #i #HLK #H destruct
 | #L #V #T #_ #H destruct
@@ -56,10 +56,10 @@ fact crr_inv_sort_aux: ∀L,T,k. L ⊢ 𝐑⦃T⦄ → T = ⋆k → ⊥.
 ]
 qed-.
 
-lemma crr_inv_sort: ∀L,k. L ⊢ 𝐑⦃⋆k⦄ → ⊥.
+lemma crr_inv_sort: ∀L,k. ⦃G, L⦄ ⊢ 𝐑⦃⋆k⦄ → ⊥.
 /2 width=5 by crr_inv_sort_aux/ qed-.
 
-fact crr_inv_lref_aux: ∀L,T,i. L ⊢ 𝐑⦃T⦄ → T = #i → ∃∃K,V. ⇩[0, i] L ≡ K.ⓓV.
+fact crr_inv_lref_aux: ∀L,T,i. ⦃G, L⦄ ⊢ 𝐑⦃T⦄ → T = #i → ∃∃K,V. ⇩[0, i] L ≡ K.ⓓV.
 #L #T #j * -L -T
 [ #L #K #V #i #HLK #H destruct /2 width=3/
 | #L #V #T #_ #H destruct
@@ -72,10 +72,10 @@ fact crr_inv_lref_aux: ∀L,T,i. L ⊢ 𝐑⦃T⦄ → T = #i → ∃∃K,V. ⇩
 ]
 qed-.
 
-lemma crr_inv_lref: ∀L,i. L ⊢ 𝐑⦃#i⦄ → ∃∃K,V. ⇩[0, i] L ≡ K.ⓓV.
+lemma crr_inv_lref: ∀L,i. ⦃G, L⦄ ⊢ 𝐑⦃#i⦄ → ∃∃K,V. ⇩[0, i] L ≡ K.ⓓV.
 /2 width=3 by crr_inv_lref_aux/ qed-.
 
-fact crr_inv_gref_aux: ∀L,T,p. L ⊢ 𝐑⦃T⦄ → T = §p → ⊥.
+fact crr_inv_gref_aux: ∀L,T,p. ⦃G, L⦄ ⊢ 𝐑⦃T⦄ → T = §p → ⊥.
 #L #T #q * -L -T
 [ #L #K #V #i #HLK #H destruct
 | #L #V #T #_ #H destruct
@@ -88,7 +88,7 @@ fact crr_inv_gref_aux: ∀L,T,p. L ⊢ 𝐑⦃T⦄ → T = §p → ⊥.
 ]
 qed-.
 
-lemma crr_inv_gref: ∀L,p. L ⊢ 𝐑⦃§p⦄ → ⊥.
+lemma crr_inv_gref: ∀L,p. ⦃G, L⦄ ⊢ 𝐑⦃§p⦄ → ⊥.
 /2 width=5 by crr_inv_gref_aux/ qed-.
 
 lemma trr_inv_atom: ∀I. ⋆ ⊢ 𝐑⦃⓪{I}⦄ → ⊥.
@@ -100,8 +100,8 @@ lemma trr_inv_atom: ∀I. ⋆ ⊢ 𝐑⦃⓪{I}⦄ → ⊥.
 ]
 qed-.
 
-fact crr_inv_ib2_aux: ∀a,I,L,W,U,T. ib2 a I → L ⊢ 𝐑⦃T⦄ → T = ⓑ{a,I}W.U →
-                      L ⊢ 𝐑⦃W⦄ ∨ L.ⓑ{I}W ⊢ 𝐑⦃U⦄.
+fact crr_inv_ib2_aux: ∀a,I,L,W,U,T. ib2 a I → ⦃G, L⦄ ⊢ 𝐑⦃T⦄ → T = ⓑ{a,I}W.U →
+                      ⦃G, L⦄ ⊢ 𝐑⦃W⦄ ∨ L.ⓑ{I}W ⊢ 𝐑⦃U⦄.
 #b #J #L #W0 #U #T #HI * -L -T
 [ #L #K #V #i #_ #H destruct
 | #L #V #T #_ #H destruct
@@ -116,12 +116,12 @@ fact crr_inv_ib2_aux: ∀a,I,L,W,U,T. ib2 a I → L ⊢ 𝐑⦃T⦄ → T = ⓑ{
 ]
 qed-.
 
-lemma crr_inv_ib2: ∀a,I,L,W,T. ib2 a I → L ⊢ 𝐑⦃ⓑ{a,I}W.T⦄ →
-                   L ⊢ 𝐑⦃W⦄ ∨ L.ⓑ{I}W ⊢ 𝐑⦃T⦄.
+lemma crr_inv_ib2: ∀a,I,L,W,T. ib2 a I → ⦃G, L⦄ ⊢ 𝐑⦃ⓑ{a,I}W.T⦄ →
+                   ⦃G, L⦄ ⊢ 𝐑⦃W⦄ ∨ L.ⓑ{I}W ⊢ 𝐑⦃T⦄.
 /2 width=5 by crr_inv_ib2_aux/ qed-.
 
-fact crr_inv_appl_aux: ∀L,W,U,T. L ⊢ 𝐑⦃T⦄ → T = ⓐW.U →
-                       ∨∨ L ⊢ 𝐑⦃W⦄ | L ⊢ 𝐑⦃U⦄ | (𝐒⦃U⦄ → ⊥).
+fact crr_inv_appl_aux: ∀L,W,U,T. ⦃G, L⦄ ⊢ 𝐑⦃T⦄ → T = ⓐW.U →
+                       ∨∨ ⦃G, L⦄ ⊢ 𝐑⦃W⦄ | ⦃G, L⦄ ⊢ 𝐑⦃U⦄ | (𝐒⦃U⦄ → ⊥).
 #L #W0 #U #T * -L -T
 [ #L #K #V #i #_ #H destruct
 | #L #V #T #HV #H destruct /2 width=1/
@@ -137,5 +137,5 @@ fact crr_inv_appl_aux: ∀L,W,U,T. L ⊢ 𝐑⦃T⦄ → T = ⓐW.U →
 ]
 qed-.
 
-lemma crr_inv_appl: ∀L,V,T. L ⊢ 𝐑⦃ⓐV.T⦄ → ∨∨ L ⊢ 𝐑⦃V⦄ | L ⊢ 𝐑⦃T⦄ | (𝐒⦃T⦄ → ⊥).
+lemma crr_inv_appl: ∀L,V,T. ⦃G, L⦄ ⊢ 𝐑⦃ⓐV.T⦄ → ∨∨ ⦃G, L⦄ ⊢ 𝐑⦃V⦄ | ⦃G, L⦄ ⊢ 𝐑⦃T⦄ | (𝐒⦃T⦄ → ⊥).
 /2 width=3 by crr_inv_appl_aux/ qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/crr_append.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/crr_append.ma
index fdb29e2af..e0861ae55 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/reduction/crr_append.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/reduction/crr_append.ma
@@ -19,14 +19,14 @@ include "basic_2/reduction/crr.ma".
 
 (* Advanved properties ******************************************************)
 
-lemma crr_append_sn: ∀L,K,T. L ⊢ 𝐑⦃T⦄  → K @@ L ⊢ 𝐑⦃T⦄.
+lemma crr_append_sn: ∀L,K,T. ⦃G, L⦄ ⊢ 𝐑⦃T⦄  → K @@ ⦃G, L⦄ ⊢ 𝐑⦃T⦄.
 #L #K0 #T #H elim H -L -T /2 width=1/
 #L #K #V #i #HLK
 lapply (ldrop_fwd_length_lt2 … HLK) #Hi
 lapply (ldrop_O1_append_sn_le … HLK … K0) -HLK /2 width=2/ -Hi /2 width=3/
 qed.
 
-lemma trr_crr: ∀L,T. ⋆ ⊢ 𝐑⦃T⦄ → L ⊢ 𝐑⦃T⦄.
+lemma trr_crr: ∀L,T. ⋆ ⊢ 𝐑⦃T⦄ → ⦃G, L⦄ ⊢ 𝐑⦃T⦄.
 #L #T #H lapply (crr_append_sn … H) //
 qed.
 
@@ -49,7 +49,7 @@ fact crr_inv_labst_last_aux: ∀L1,T,W. L1 ⊢ 𝐑⦃T⦄  →
 ]
 qed.
 
-lemma crr_inv_labst_last: ∀L,T,W. ⋆.ⓛW @@ L ⊢ 𝐑⦃T⦄  → L ⊢ 𝐑⦃T⦄.
+lemma crr_inv_labst_last: ∀L,T,W. ⋆.ⓛW @@ ⦃G, L⦄ ⊢ 𝐑⦃T⦄  → ⦃G, L⦄ ⊢ 𝐑⦃T⦄.
 /2 width=4/ qed-.
 
 lemma crr_inv_trr: ∀T,W. ⋆.ⓛW ⊢ 𝐑⦃T⦄  → ⋆ ⊢ 𝐑⦃T⦄.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/crr_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/crr_lift.ma
index 731d00bb2..5d923559d 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/reduction/crr_lift.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/reduction/crr_lift.ma
@@ -20,7 +20,7 @@ include "basic_2/reduction/crr.ma".
 (* Properties on relocation *************************************************)
 
 lemma crr_lift: ∀K,T. K ⊢ 𝐑⦃T⦄ → ∀L,d,e. ⇩[d, e] L ≡ K →
-                ∀U. ⇧[d, e] T ≡ U → L ⊢ 𝐑⦃U⦄.
+                ∀U. ⇧[d, e] T ≡ U → ⦃G, L⦄ ⊢ 𝐑⦃U⦄.
 #K #T #H elim H -K -T
 [ #K #K0 #V #i #HK0 #L #d #e #HLK #X #H
   elim (lift_inv_lref1 … H) -H * #Hid #H destruct
@@ -46,7 +46,7 @@ lemma crr_lift: ∀K,T. K ⊢ 𝐑⦃T⦄ → ∀L,d,e. ⇩[d, e] L ≡ K →
 ]
 qed.
 
-lemma crr_inv_lift: ∀L,U. L ⊢ 𝐑⦃U⦄ → ∀K,d,e. ⇩[d, e] L ≡ K →
+lemma crr_inv_lift: ∀L,U. ⦃G, L⦄ ⊢ 𝐑⦃U⦄ → ∀K,d,e. ⇩[d, e] L ≡ K →
                     ∀T. ⇧[d, e] T ≡ U → K ⊢ 𝐑⦃T⦄.
 #L #U #H elim H -L -U
 [ #L #L0 #W #i #HK0 #K #d #e #HLK #X #H
diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/crx.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/crx.ma
index 059e62151..922dcd4fe 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/reduction/crx.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/reduction/crx.ma
@@ -37,13 +37,13 @@ interpretation
 
 (* Basic properties *********************************************************)
 
-lemma crr_crx: ∀h,g,L,T. L ⊢ 𝐑⦃T⦄ → ⦃h, L⦄ ⊢ 𝐑[g]⦃T⦄.
+lemma crr_crx: ∀h,g,L,T. ⦃G, L⦄ ⊢ 𝐑⦃T⦄ → ⦃G, L⦄ ⊢ 𝐑[h, g]⦃T⦄.
 #h #g #L #T #H elim H -L -T // /2 width=1/ /2 width=4/
 qed.
 
 (* Basic inversion lemmas ***************************************************)
 
-fact crx_inv_sort_aux: ∀h,g,L,T,k. ⦃h, L⦄ ⊢ 𝐑[g]⦃T⦄ → T = ⋆k →
+fact crx_inv_sort_aux: ∀h,g,L,T,k. ⦃G, L⦄ ⊢ 𝐑[h, g]⦃T⦄ → T = ⋆k →
                        ∃l. deg h g k (l+1).
 #h #g #L #T #k0 * -L -T
 [ #L #k #l #Hkl #H destruct /2 width=2/
@@ -58,10 +58,10 @@ fact crx_inv_sort_aux: ∀h,g,L,T,k. ⦃h, L⦄ ⊢ 𝐑[g]⦃T⦄ → T = ⋆k
 ]
 qed-.
 
-lemma crx_inv_sort: ∀h,g,L,k. ⦃h, L⦄ ⊢ 𝐑[g]⦃⋆k⦄ → ∃l. deg h g k (l+1).
+lemma crx_inv_sort: ∀h,g,L,k. ⦃G, L⦄ ⊢ 𝐑[h, g]⦃⋆k⦄ → ∃l. deg h g k (l+1).
 /2 width=4 by crx_inv_sort_aux/ qed-.
 
-fact crx_inv_lref_aux: ∀h,g,L,T,i. ⦃h, L⦄ ⊢ 𝐑[g]⦃T⦄ → T = #i →
+fact crx_inv_lref_aux: ∀h,g,L,T,i. ⦃G, L⦄ ⊢ 𝐑[h, g]⦃T⦄ → T = #i →
                        ∃∃I,K,V. ⇩[0, i] L ≡ K.ⓑ{I}V.
 #h #g #L #T #j * -L -T
 [ #L #k #l #_ #H destruct
@@ -76,10 +76,10 @@ fact crx_inv_lref_aux: ∀h,g,L,T,i. ⦃h, L⦄ ⊢ 𝐑[g]⦃T⦄ → T = #i 
 ]
 qed-.
 
-lemma crx_inv_lref: ∀h,g,L,i. ⦃h, L⦄ ⊢ 𝐑[g]⦃#i⦄ → ∃∃I,K,V. ⇩[0, i] L ≡ K.ⓑ{I}V.
+lemma crx_inv_lref: ∀h,g,L,i. ⦃G, L⦄ ⊢ 𝐑[h, g]⦃#i⦄ → ∃∃I,K,V. ⇩[0, i] L ≡ K.ⓑ{I}V.
 /2 width=5 by crx_inv_lref_aux/ qed-.
 
-fact crx_inv_gref_aux: ∀h,g,L,T,p. ⦃h, L⦄ ⊢ 𝐑[g]⦃T⦄ → T = §p → ⊥.
+fact crx_inv_gref_aux: ∀h,g,L,T,p. ⦃G, L⦄ ⊢ 𝐑[h, g]⦃T⦄ → T = §p → ⊥.
 #h #g #L #T #q * -L -T
 [ #L #k #l #_ #H destruct
 | #I #L #K #V #i #HLK #H destruct
@@ -93,10 +93,10 @@ fact crx_inv_gref_aux: ∀h,g,L,T,p. ⦃h, L⦄ ⊢ 𝐑[g]⦃T⦄ → T = §p 
 ]
 qed-.
 
-lemma crx_inv_gref: ∀h,g,L,p. ⦃h, L⦄ ⊢ 𝐑[g]⦃§p⦄ → ⊥.
+lemma crx_inv_gref: ∀h,g,L,p. ⦃G, L⦄ ⊢ 𝐑[h, g]⦃§p⦄ → ⊥.
 /2 width=7 by crx_inv_gref_aux/ qed-.
 
-lemma trx_inv_atom: ∀h,g,I. ⦃h, ⋆⦄ ⊢ 𝐑[g]⦃⓪{I}⦄ →
+lemma trx_inv_atom: ∀h,g,I. ⦃h, ⋆⦄ ⊢ 𝐑[h, g]⦃⓪{I}⦄ →
                     ∃∃k,l. deg h g k (l+1) & I = Sort k.
 #h #g * #i #H
 [ elim (crx_inv_sort … H) -H /2 width=4/
@@ -106,8 +106,8 @@ lemma trx_inv_atom: ∀h,g,I. ⦃h, ⋆⦄ ⊢ 𝐑[g]⦃⓪{I}⦄ →
 ]
 qed-.
 
-fact crx_inv_ib2_aux: ∀h,g,a,I,L,W,U,T. ib2 a I → ⦃h, L⦄ ⊢ 𝐑[g]⦃T⦄ →
-                      T = ⓑ{a,I}W.U → ⦃h, L⦄ ⊢ 𝐑[g]⦃W⦄ ∨ ⦃h, L.ⓑ{I}W⦄ ⊢ 𝐑[g]⦃U⦄.
+fact crx_inv_ib2_aux: ∀h,g,a,I,L,W,U,T. ib2 a I → ⦃G, L⦄ ⊢ 𝐑[h, g]⦃T⦄ →
+                      T = ⓑ{a,I}W.U → ⦃G, L⦄ ⊢ 𝐑[h, g]⦃W⦄ ∨ ⦃h, L.ⓑ{I}W⦄ ⊢ 𝐑[h, g]⦃U⦄.
 #h #g #b #J #L #W0 #U #T #HI * -L -T
 [ #L #k #l #_ #H destruct
 | #I #L #K #V #i #_ #H destruct
@@ -123,12 +123,12 @@ fact crx_inv_ib2_aux: ∀h,g,a,I,L,W,U,T. ib2 a I → ⦃h, L⦄ ⊢ 𝐑[g]⦃T
 ]
 qed-.
 
-lemma crx_inv_ib2: ∀h,g,a,I,L,W,T. ib2 a I → ⦃h, L⦄ ⊢ 𝐑[g]⦃ⓑ{a,I}W.T⦄ →
-                   ⦃h, L⦄ ⊢ 𝐑[g]⦃W⦄ ∨ ⦃h, L.ⓑ{I}W⦄ ⊢ 𝐑[g]⦃T⦄.
+lemma crx_inv_ib2: ∀h,g,a,I,L,W,T. ib2 a I → ⦃G, L⦄ ⊢ 𝐑[h, g]⦃ⓑ{a,I}W.T⦄ →
+                   ⦃G, L⦄ ⊢ 𝐑[h, g]⦃W⦄ ∨ ⦃h, L.ⓑ{I}W⦄ ⊢ 𝐑[h, g]⦃T⦄.
 /2 width=5 by crx_inv_ib2_aux/ qed-.
 
-fact crx_inv_appl_aux: ∀h,g,L,W,U,T. ⦃h, L⦄ ⊢ 𝐑[g]⦃T⦄ → T = ⓐW.U →
-                       ∨∨ ⦃h, L⦄ ⊢ 𝐑[g]⦃W⦄ | ⦃h, L⦄ ⊢ 𝐑[g]⦃U⦄ | (𝐒⦃U⦄ → ⊥).
+fact crx_inv_appl_aux: ∀h,g,L,W,U,T. ⦃G, L⦄ ⊢ 𝐑[h, g]⦃T⦄ → T = ⓐW.U →
+                       ∨∨ ⦃G, L⦄ ⊢ 𝐑[h, g]⦃W⦄ | ⦃G, L⦄ ⊢ 𝐑[h, g]⦃U⦄ | (𝐒⦃U⦄ → ⊥).
 #h #g #L #W0 #U #T * -L -T
 [ #L #k #l #_ #H destruct
 | #I #L #K #V #i #_ #H destruct
@@ -145,6 +145,6 @@ fact crx_inv_appl_aux: ∀h,g,L,W,U,T. ⦃h, L⦄ ⊢ 𝐑[g]⦃T⦄ → T = ⓐ
 ]
 qed-.
 
-lemma crx_inv_appl: ∀h,g,L,V,T. ⦃h, L⦄ ⊢ 𝐑[g]⦃ⓐV.T⦄ →
-                    ∨∨ ⦃h, L⦄ ⊢ 𝐑[g]⦃V⦄ | ⦃h, L⦄ ⊢ 𝐑[g]⦃T⦄ | (𝐒⦃T⦄ → ⊥).
+lemma crx_inv_appl: ∀h,g,L,V,T. ⦃G, L⦄ ⊢ 𝐑[h, g]⦃ⓐV.T⦄ →
+                    ∨∨ ⦃G, L⦄ ⊢ 𝐑[h, g]⦃V⦄ | ⦃G, L⦄ ⊢ 𝐑[h, g]⦃T⦄ | (𝐒⦃T⦄ → ⊥).
 /2 width=3 by crx_inv_appl_aux/ qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/crx_append.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/crx_append.ma
index 9d5dca4a6..bd91ad1d5 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/reduction/crx_append.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/reduction/crx_append.ma
@@ -19,13 +19,13 @@ include "basic_2/reduction/crx.ma".
 
 (* Advanved properties ******************************************************)
 
-lemma crx_append_sn: ∀h,g,L,K,T. ⦃h, L⦄ ⊢ 𝐑[g]⦃T⦄  → ⦃h, K @@ L⦄ ⊢ 𝐑[g]⦃T⦄.
+lemma crx_append_sn: ∀h,g,L,K,T. ⦃G, L⦄ ⊢ 𝐑[h, g]⦃T⦄  → ⦃h, K @@ L⦄ ⊢ 𝐑[h, g]⦃T⦄.
 #h #g #L #K0 #T #H elim H -L -T /2 width=1/ /2 width=2/
 #I #L #K #V #i #HLK
 lapply (ldrop_fwd_length_lt2 … HLK) #Hi
 lapply (ldrop_O1_append_sn_le … HLK … K0) -HLK /2 width=2/ -Hi /2 width=4/
 qed.
 
-lemma trx_crx: ∀h,g,L,T. ⦃h, ⋆⦄ ⊢ 𝐑[g]⦃T⦄ → ⦃h, L⦄ ⊢ 𝐑[g]⦃T⦄.
+lemma trx_crx: ∀h,g,L,T. ⦃h, ⋆⦄ ⊢ 𝐑[h, g]⦃T⦄ → ⦃G, L⦄ ⊢ 𝐑[h, g]⦃T⦄.
 #h #g #L #T #H lapply (crx_append_sn … H) //
 qed.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/crx_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/crx_lift.ma
index c1989c282..5a5a286ff 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/reduction/crx_lift.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/reduction/crx_lift.ma
@@ -19,8 +19,8 @@ include "basic_2/reduction/crx.ma".
 
 (* Properties on relocation *************************************************)
 
-lemma crx_lift: ∀h,g,K,T. ⦃h, K⦄ ⊢ 𝐑[g]⦃T⦄ → ∀L,d,e. ⇩[d, e] L ≡ K →
-                ∀U. ⇧[d, e] T ≡ U → ⦃h, L⦄ ⊢ 𝐑[g]⦃U⦄.
+lemma crx_lift: ∀h,g,K,T. ⦃h, K⦄ ⊢ 𝐑[h, g]⦃T⦄ → ∀L,d,e. ⇩[d, e] L ≡ K →
+                ∀U. ⇧[d, e] T ≡ U → ⦃G, L⦄ ⊢ 𝐑[h, g]⦃U⦄.
 #h #g #K #T #H elim H -K -T
 [ #K #k #l #Hkl #L #d #e #_ #X #H
   >(lift_inv_sort1 … H) -X /2 width=2/
@@ -48,8 +48,8 @@ lemma crx_lift: ∀h,g,K,T. ⦃h, K⦄ ⊢ 𝐑[g]⦃T⦄ → ∀L,d,e. ⇩[d, e
 ]
 qed.
 
-lemma crx_inv_lift: ∀h,g,L,U. ⦃h, L⦄ ⊢ 𝐑[g]⦃U⦄ → ∀K,d,e. ⇩[d, e] L ≡ K →
-                    ∀T. ⇧[d, e] T ≡ U → ⦃h, K⦄ ⊢ 𝐑[g]⦃T⦄.
+lemma crx_inv_lift: ∀h,g,L,U. ⦃G, L⦄ ⊢ 𝐑[h, g]⦃U⦄ → ∀K,d,e. ⇩[d, e] L ≡ K →
+                    ∀T. ⇧[d, e] T ≡ U → ⦃h, K⦄ ⊢ 𝐑[h, g]⦃T⦄.
 #h #g #L #U #H elim H -L -U
 [ #L #k #l #Hkl #K #d #e #_ #X #H
   >(lift_inv_sort2 … H) -X /2 width=2/
diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/lpr.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/lpr.ma
index afab7f20f..9fa1327fa 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/reduction/lpr.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/reduction/lpr.ma
@@ -44,7 +44,7 @@ lemma lpr_inv_pair2: ∀I,L1,K2,V2. L1 ⊢ ➡ K2. ⓑ{I} V2 →
 (* Basic properties *********************************************************)
 
 (* Note: lemma 250 *)
-lemma lpr_refl: ∀L. L ⊢ ➡ L.
+lemma lpr_refl: ∀L. ⦃G, L⦄ ⊢ ➡ L.
 /2 width=1 by lpx_sn_refl/ qed.
 
 lemma lpr_pair: ∀I,K1,K2,V1,V2. K1 ⊢ ➡ K2 → K1 ⊢ V1 ➡ V2 →
@@ -66,7 +66,7 @@ lemma lpr_fwd_append1: ∀K1,L1,L. K1 @@ L1 ⊢ ➡ L →
                        ∃∃K2,L2. K1 ⊢ ➡ K2 & L = K2 @@ L2.
 /2 width=2 by lpx_sn_fwd_append1/ qed-.
 
-lemma lpr_fwd_append2: ∀L,K2,L2. L ⊢ ➡ K2 @@ L2 →
+lemma lpr_fwd_append2: ∀L,K2,L2. ⦃G, L⦄ ⊢ ➡ K2 @@ L2 →
                        ∃∃K1,L1. K1 ⊢ ➡ K2 & L = K1 @@ L1.
 /2 width=2 by lpx_sn_fwd_append2/ qed-.
 
diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/lpr_ldrop.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/lpr_ldrop.ma
index fd7b613d6..120e5e60e 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/reduction/lpr_ldrop.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/reduction/lpr_ldrop.ma
@@ -35,7 +35,7 @@ lemma lpr_ldrop_trans_O1: dropable_dx lpr.
 (* Properties on context-sensitive parallel reduction for terms *************)
 
 lemma fsup_cpr_trans: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊃ ⦃L2, T2⦄ → ∀U2. L2 ⊢ T2 ➡ U2 →
-                      ∃∃L,U1. L1 ⊢ ➡ L & L ⊢ T1 ➡ U1 & ⦃L, U1⦄ ⊃ ⦃L2, U2⦄.
+                      ∃∃L,U1. L1 ⊢ ➡ L & ⦃G, L⦄ ⊢ T1 ➡ U1 & ⦃L, U1⦄ ⊃ ⦃L2, U2⦄.
 #L1 #L2 #T1 #T2 #H elim H -L1 -L2 -T1 -T2 [1,2,3,4,5: /3 width=5/ ]
 [ #L #K #U #T #d #e #HLK #HUT #He #U2 #HU2
   elim (lift_total U2 d e) #T2 #HUT2
diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/lpr_lpr.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/lpr_lpr.ma
index d87242057..2cc08f081 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/reduction/lpr_lpr.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/reduction/lpr_lpr.ma
@@ -27,8 +27,8 @@ fact cpr_conf_lpr_atom_atom:
 fact cpr_conf_lpr_atom_delta:
    ∀L0,i. (
       ∀L,T. ⦃L0, #i⦄ ⊃+ ⦃L, T⦄ →
-      ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ➡ T2 →
-      ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ➡ L2 →
+      ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
+      ∀L1. ⦃G, L⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L⦄ ⊢ ➡ L2 →
       ∃∃T0. L1 ⊢ T1 ➡ T0 & L2 ⊢ T2 ➡ T0
    ) →
    ∀K0,V0. ⇩[O, i] L0 ≡ K0.ⓓV0 →
@@ -51,8 +51,8 @@ qed-.
 fact cpr_conf_lpr_delta_delta:
    ∀L0,i. (
       ∀L,T. ⦃L0, #i⦄ ⊃+ ⦃L, T⦄ →
-      ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ➡ T2 →
-      ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ➡ L2 →
+      ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
+      ∀L1. ⦃G, L⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L⦄ ⊢ ➡ L2 →
       ∃∃T0. L1 ⊢ T1 ➡ T0 & L2 ⊢ T2 ➡ T0
    ) →
    ∀K0,V0. ⇩[O, i] L0 ≡ K0.ⓓV0 →
@@ -80,8 +80,8 @@ qed-.
 fact cpr_conf_lpr_bind_bind:
    ∀a,I,L0,V0,T0. (
       ∀L,T. ⦃L0,ⓑ{a,I}V0.T0⦄ ⊃+ ⦃L, T⦄ →
-      ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ➡ T2 →
-      ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ➡ L2 →
+      ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
+      ∀L1. ⦃G, L⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L⦄ ⊢ ➡ L2 →
       ∃∃T0. L1 ⊢ T1 ➡ T0 & L2 ⊢ T2 ➡ T0
    ) →
    ∀V1. L0 ⊢ V0 ➡ V1 → ∀T1. L0.ⓑ{I}V0 ⊢ T0 ➡ T1 →
@@ -97,8 +97,8 @@ qed-.
 fact cpr_conf_lpr_bind_zeta:
    ∀L0,V0,T0. (
       ∀L,T. ⦃L0,+ⓓV0.T0⦄ ⊃+ ⦃L, T⦄ →
-      ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ➡ T2 →
-      ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ➡ L2 →
+      ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
+      ∀L1. ⦃G, L⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L⦄ ⊢ ➡ L2 →
       ∃∃T0. L1 ⊢ T1 ➡ T0 & L2 ⊢ T2 ➡ T0
    ) →
    ∀V1. L0 ⊢ V0 ➡ V1 → ∀T1. L0.ⓓV0 ⊢ T0 ➡ T1 →
@@ -114,8 +114,8 @@ qed-.
 fact cpr_conf_lpr_zeta_zeta:
    ∀L0,V0,T0. (
       ∀L,T. ⦃L0,+ⓓV0.T0⦄ ⊃+ ⦃L, T⦄ →
-      ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ➡ T2 →
-      ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ➡ L2 →
+      ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
+      ∀L1. ⦃G, L⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L⦄ ⊢ ➡ L2 →
       ∃∃T0. L1 ⊢ T1 ➡ T0 & L2 ⊢ T2 ➡ T0
    ) →
    ∀T1. L0.ⓓV0 ⊢ T0 ➡ T1 → ∀X1. ⇧[O, 1] X1 ≡ T1 →
@@ -133,8 +133,8 @@ qed-.
 fact cpr_conf_lpr_flat_flat:
    ∀I,L0,V0,T0. (
       ∀L,T. ⦃L0,ⓕ{I}V0.T0⦄ ⊃+ ⦃L, T⦄ →
-      ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ➡ T2 →
-      ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ➡ L2 →
+      ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
+      ∀L1. ⦃G, L⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L⦄ ⊢ ➡ L2 →
       ∃∃T0. L1 ⊢ T1 ➡ T0 & L2 ⊢ T2 ➡ T0
    ) →
    ∀V1. L0 ⊢ V0 ➡ V1 → ∀T1. L0 ⊢ T0 ➡ T1 →
@@ -150,8 +150,8 @@ qed-.
 fact cpr_conf_lpr_flat_tau:
    ∀L0,V0,T0. (
       ∀L,T. ⦃L0,ⓝV0.T0⦄ ⊃+ ⦃L, T⦄ →
-      ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ➡ T2 →
-      ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ➡ L2 →
+      ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
+      ∀L1. ⦃G, L⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L⦄ ⊢ ➡ L2 →
       ∃∃T0. L1 ⊢ T1 ➡ T0 & L2 ⊢ T2 ➡ T0
    ) →
    ∀V1,T1. L0 ⊢ T0 ➡ T1 → ∀T2. L0 ⊢ T0 ➡ T2 →
@@ -165,8 +165,8 @@ qed-.
 fact cpr_conf_lpr_tau_tau:
    ∀L0,V0,T0. (
       ∀L,T. ⦃L0,ⓝV0.T0⦄ ⊃+ ⦃L, T⦄ →
-      ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ➡ T2 →
-      ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ➡ L2 →
+      ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
+      ∀L1. ⦃G, L⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L⦄ ⊢ ➡ L2 →
       ∃∃T0. L1 ⊢ T1 ➡ T0 & L2 ⊢ T2 ➡ T0
    ) →
    ∀T1. L0 ⊢ T0 ➡ T1 → ∀T2. L0 ⊢ T0 ➡ T2 →
@@ -180,8 +180,8 @@ qed-.
 fact cpr_conf_lpr_flat_beta:
    ∀a,L0,V0,W0,T0. (
       ∀L,T. ⦃L0,ⓐV0.ⓛ{a}W0.T0⦄ ⊃+ ⦃L, T⦄ →
-      ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ➡ T2 →
-      ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ➡ L2 →
+      ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
+      ∀L1. ⦃G, L⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L⦄ ⊢ ➡ L2 →
       ∃∃T0. L1 ⊢ T1 ➡ T0 & L2 ⊢ T2 ➡ T0
    ) →
    ∀V1. L0 ⊢ V0 ➡ V1 → ∀T1. L0 ⊢ ⓛ{a}W0.T0 ➡ T1 →
@@ -205,8 +205,8 @@ qed-.
 fact cpr_conf_lpr_flat_theta:
    ∀a,L0,V0,W0,T0. (
       ∀L,T. ⦃L0,ⓐV0.ⓓ{a}W0.T0⦄ ⊃+ ⦃L, T⦄ →
-      ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ➡ T2 →
-      ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ➡ L2 →
+      ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
+      ∀L1. ⦃G, L⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L⦄ ⊢ ➡ L2 →
       ∃∃T0. L1 ⊢ T1 ➡ T0 & L2 ⊢ T2 ➡ T0
    ) →
    ∀V1. L0 ⊢ V0 ➡ V1 → ∀T1. L0 ⊢ ⓓ{a}W0.T0 ➡ T1 →
@@ -234,8 +234,8 @@ qed-.
 fact cpr_conf_lpr_beta_beta:
    ∀a,L0,V0,W0,T0. (
       ∀L,T. ⦃L0,ⓐV0.ⓛ{a}W0.T0⦄ ⊃+ ⦃L, T⦄ →
-      ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ➡ T2 →
-      ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ➡ L2 →
+      ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
+      ∀L1. ⦃G, L⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L⦄ ⊢ ➡ L2 →
       ∃∃T0. L1 ⊢ T1 ➡ T0 & L2 ⊢ T2 ➡ T0
    ) →
    ∀V1. L0 ⊢ V0 ➡ V1 → ∀W1. L0 ⊢ W0 ➡ W1 → ∀T1. L0.ⓛW0 ⊢ T0 ➡ T1 →
@@ -256,8 +256,8 @@ qed-.
 fact cpr_conf_lpr_theta_theta:
    ∀a,L0,V0,W0,T0. (
       ∀L,T. ⦃L0,ⓐV0.ⓓ{a}W0.T0⦄ ⊃+ ⦃L, T⦄ →
-      ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ➡ T2 →
-      ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ➡ L2 →
+      ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
+      ∀L1. ⦃G, L⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L⦄ ⊢ ➡ L2 →
       ∃∃T0. L1 ⊢ T1 ➡ T0 & L2 ⊢ T2 ➡ T0
    ) →
    ∀V1. L0 ⊢ V0 ➡ V1 → ∀U1. ⇧[O, 1] V1 ≡ U1 →
diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/lpx.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/lpx.ma
index d34212910..547520714 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/reduction/lpx.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/reduction/lpx.ma
@@ -25,50 +25,50 @@ interpretation "extended parallel reduction (local environment, sn variant)"
 
 (* Basic inversion lemmas ***************************************************)
 
-lemma lpx_inv_atom1: ∀h,g,L2. ⦃h, ⋆⦄ ⊢ ➡[g] L2 → L2 = ⋆.
+lemma lpx_inv_atom1: ∀h,g,L2. ⦃h, ⋆⦄ ⊢ ➡[h, g] L2 → L2 = ⋆.
 /2 width=4 by lpx_sn_inv_atom1_aux/ qed-.
 
-lemma lpx_inv_pair1: ∀h,g,I,K1,V1,L2. ⦃h, K1.ⓑ{I}V1⦄ ⊢ ➡[g] L2 →
-                     ∃∃K2,V2. ⦃h, K1⦄ ⊢ ➡[g] K2 & ⦃h, K1⦄ ⊢ V1 ➡[g] V2 &
+lemma lpx_inv_pair1: ∀h,g,I,K1,V1,L2. ⦃h, K1.ⓑ{I}V1⦄ ⊢ ➡[h, g] L2 →
+                     ∃∃K2,V2. ⦃h, K1⦄ ⊢ ➡[h, g] K2 & ⦃h, K1⦄ ⊢ V1 ➡[h, g] V2 &
                              L2 = K2. ⓑ{I} V2.
 /2 width=3 by lpx_sn_inv_pair1_aux/ qed-.
 
-lemma lpx_inv_atom2: ∀h,g,L1.  ⦃h, L1⦄ ⊢ ➡[g] ⋆ → L1 = ⋆.
+lemma lpx_inv_atom2: ∀h,g,L1.  ⦃h, L1⦄ ⊢ ➡[h, g] ⋆ → L1 = ⋆.
 /2 width=4 by lpx_sn_inv_atom2_aux/ qed-.
 
-lemma lpx_inv_pair2: ∀h,g,I,L1,K2,V2.  ⦃h, L1⦄ ⊢ ➡[g] K2.ⓑ{I}V2 →
-                     ∃∃K1,V1. ⦃h, K1⦄ ⊢ ➡[g] K2 & ⦃h, K1⦄ ⊢ V1 ➡[g] V2 &
+lemma lpx_inv_pair2: ∀h,g,I,L1,K2,V2.  ⦃h, L1⦄ ⊢ ➡[h, g] K2.ⓑ{I}V2 →
+                     ∃∃K1,V1. ⦃h, K1⦄ ⊢ ➡[h, g] K2 & ⦃h, K1⦄ ⊢ V1 ➡[h, g] V2 &
                              L1 = K1. ⓑ{I} V1.
 /2 width=3 by lpx_sn_inv_pair2_aux/ qed-.
 
 (* Basic properties *********************************************************)
 
-lemma lpx_refl: ∀h,g,L.  ⦃h, L⦄ ⊢ ➡[g] L.
+lemma lpx_refl: ∀h,g,L.  ⦃G, L⦄ ⊢ ➡[h, g] L.
 /2 width=1 by lpx_sn_refl/ qed.
 
-lemma lpx_pair: ∀h,g,I,K1,K2,V1,V2. ⦃h, K1⦄ ⊢ ➡[g] K2 → ⦃h, K1⦄ ⊢ V1 ➡[g] V2 →
-                ⦃h, K1.ⓑ{I}V1⦄ ⊢ ➡[g] K2.ⓑ{I}V2.
+lemma lpx_pair: ∀h,g,I,K1,K2,V1,V2. ⦃h, K1⦄ ⊢ ➡[h, g] K2 → ⦃h, K1⦄ ⊢ V1 ➡[h, g] V2 →
+                ⦃h, K1.ⓑ{I}V1⦄ ⊢ ➡[h, g] K2.ⓑ{I}V2.
 /2 width=1/ qed.
 
-lemma lpx_append: ∀h,g,K1,K2. ⦃h, K1⦄ ⊢ ➡[g] K2 → ∀L1,L2. ⦃h, L1⦄ ⊢ ➡[g] L2 →
-                  ⦃h, L1 @@ K1⦄ ⊢ ➡[g] L2 @@ K2.
+lemma lpx_append: ∀h,g,K1,K2. ⦃h, K1⦄ ⊢ ➡[h, g] K2 → ∀L1,L2. ⦃h, L1⦄ ⊢ ➡[h, g] L2 →
+                  ⦃h, L1 @@ K1⦄ ⊢ ➡[h, g] L2 @@ K2.
 /3 width=1 by lpx_sn_append, cpx_append/ qed.
 
-lemma lpr_lpx: ∀h,g,L1,L2. L1 ⊢ ➡ L2 → ⦃h, L1⦄ ⊢ ➡[g] L2.
+lemma lpr_lpx: ∀h,g,L1,L2. L1 ⊢ ➡ L2 → ⦃h, L1⦄ ⊢ ➡[h, g] L2.
 #h #g #L1 #L2 #H elim H -L1 -L2 // /3 width=1/
 qed.
 
 (* Basic forward lemmas *****************************************************)
 
-lemma lpx_fwd_length: ∀h,g,L1,L2. ⦃h, L1⦄ ⊢ ➡[g] L2 → |L1| = |L2|.
+lemma lpx_fwd_length: ∀h,g,L1,L2. ⦃h, L1⦄ ⊢ ➡[h, g] L2 → |L1| = |L2|.
 /2 width=2 by lpx_sn_fwd_length/ qed-.
 
 (* Advanced forward lemmas **************************************************)
 
-lemma lpx_fwd_append1: ∀h,g,K1,L1,L. ⦃h, K1 @@ L1⦄ ⊢ ➡[g] L →
-                       ∃∃K2,L2. ⦃h, K1⦄ ⊢ ➡[g] K2 & L = K2 @@ L2.
+lemma lpx_fwd_append1: ∀h,g,K1,L1,L. ⦃h, K1 @@ L1⦄ ⊢ ➡[h, g] L →
+                       ∃∃K2,L2. ⦃h, K1⦄ ⊢ ➡[h, g] K2 & L = K2 @@ L2.
 /2 width=2 by lpx_sn_fwd_append1/ qed-.
 
-lemma lpx_fwd_append2: ∀h,g,L,K2,L2. ⦃h, L⦄ ⊢ ➡[g] K2 @@ L2 →
-                       ∃∃K1,L1. ⦃h, K1⦄ ⊢ ➡[g] K2 & L = K1 @@ L1.
+lemma lpx_fwd_append2: ∀h,g,L,K2,L2. ⦃G, L⦄ ⊢ ➡[h, g] K2 @@ L2 →
+                       ∃∃K1,L1. ⦃h, K1⦄ ⊢ ➡[h, g] K2 & L = K1 @@ L1.
 /2 width=2 by lpx_sn_fwd_append2/ qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/lpx_aaa.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/lpx_aaa.ma
index 6fa24124f..41d1386dd 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/reduction/lpx_aaa.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/reduction/lpx_aaa.ma
@@ -21,8 +21,8 @@ include "basic_2/reduction/lpx_ldrop.ma".
 (* Properties on atomic arity assignment for terms **************************)
 
 (* Note: lemma 500 *)
-lemma aaa_cpx_lpx_conf: ∀h,g,L1,T1,A. L1 ⊢ T1 ⁝ A → ∀T2. ⦃h, L1⦄ ⊢ T1 ➡[g] T2 →
-                        ∀L2. ⦃h, L1⦄ ⊢ ➡[g] L2 → L2 ⊢ T2 ⁝ A.
+lemma aaa_cpx_lpx_conf: ∀h,g,L1,T1,A. L1 ⊢ T1 ⁝ A → ∀T2. ⦃h, L1⦄ ⊢ T1 ➡[h, g] T2 →
+                        ∀L2. ⦃h, L1⦄ ⊢ ➡[h, g] L2 → L2 ⊢ T2 ⁝ A.
 #h #g #L1 #T1 #A #H elim H -L1 -T1 -A
 [ #L1 #k #X #H
   elim (cpx_inv_sort1 … H) -H // * //
@@ -67,13 +67,13 @@ lemma aaa_cpx_lpx_conf: ∀h,g,L1,T1,A. L1 ⊢ T1 ⁝ A → ∀T2. ⦃h, L1⦄ 
 ]
 qed-.
 
-lemma aaa_cpx_conf: ∀h,g,L,T1,A. L ⊢ T1 ⁝ A → ∀T2. ⦃h, L⦄ ⊢ T1 ➡[g] T2 → L ⊢ T2 ⁝ A.
+lemma aaa_cpx_conf: ∀h,g,L,T1,A. ⦃G, L⦄ ⊢ T1 ⁝ A → ∀T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 → ⦃G, L⦄ ⊢ T2 ⁝ A.
 /2 width=7 by aaa_cpx_lpx_conf/ qed-.
 
-lemma aaa_lpx_conf: ∀h,g,L1,T,A. L1 ⊢ T ⁝ A → ∀L2. ⦃h, L1⦄ ⊢ ➡[g] L2 → L2 ⊢ T ⁝ A.
+lemma aaa_lpx_conf: ∀h,g,L1,T,A. L1 ⊢ T ⁝ A → ∀L2. ⦃h, L1⦄ ⊢ ➡[h, g] L2 → L2 ⊢ T ⁝ A.
 /2 width=7 by aaa_cpx_lpx_conf/ qed-.
 
-lemma aaa_cpr_conf: ∀L,T1,A. L ⊢ T1 ⁝ A → ∀T2. L ⊢ T1 ➡ T2 → L ⊢ T2 ⁝ A.
+lemma aaa_cpr_conf: ∀L,T1,A. ⦃G, L⦄ ⊢ T1 ⁝ A → ∀T2. ⦃G, L⦄ ⊢ T1 ➡ T2 → ⦃G, L⦄ ⊢ T2 ⁝ A.
 /3 width=5 by aaa_cpx_conf, cpr_cpx/ qed-.
 
 lemma aaa_lpr_conf: ∀L1,T,A. L1 ⊢ T ⁝ A → ∀L2. L1 ⊢ ➡ L2 → L2 ⊢ T ⁝ A.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/fsup.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/fsup.ma
index 53f7b5584..80ec9b0b5 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/relocation/fsup.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/fsup.ma
@@ -12,36 +12,37 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "basic_2/notation/relations/supterm_4.ma".
+include "basic_2/notation/relations/supterm_6.ma".
 include "basic_2/grammar/cl_weight.ma".
 include "basic_2/relocation/ldrop.ma".
 
 (* SUPCLOSURE ***************************************************************)
 
-inductive fsup: bi_relation lenv term ≝
-| fsup_lref_O  : ∀I,L,V. fsup (L.ⓑ{I}V) (#0) L V
-| fsup_pair_sn : ∀I,L,V,T. fsup L (②{I}V.T) L V
-| fsup_bind_dx : ∀a,I,L,V,T. fsup L (ⓑ{a,I}V.T) (L.ⓑ{I}V) T
-| fsup_flat_dx : ∀I,L,V,T.   fsup L (ⓕ{I}V.T) L T
-| fsup_ldrop_lt: ∀L,K,T,U,d,e.
-                 ⇩[d, e] L ≡ K → ⇧[d, e] T ≡ U → 0 < e → fsup L U K T
-| fsup_ldrop   : ∀L1,K1,K2,T1,T2,U1,d,e.
+(* activate genv *)
+inductive fsup: tri_relation genv lenv term ≝
+| fsup_lref_O  : ∀I,G,L,V. fsup G (L.ⓑ{I}V) (#0) G L V
+| fsup_pair_sn : ∀I,G,L,V,T. fsup G L (②{I}V.T) G L V
+| fsup_bind_dx : ∀a,I,G,L,V,T. fsup G L (ⓑ{a,I}V.T) G (L.ⓑ{I}V) T
+| fsup_flat_dx : ∀I,G,L,V,T. fsup G L (ⓕ{I}V.T) G L T
+| fsup_ldrop_lt: ∀G,L,K,T,U,d,e.
+                 ⇩[d, e] L ≡ K → ⇧[d, e] T ≡ U → 0 < e → fsup G L U G K T
+| fsup_ldrop   : ∀G1,G2,L1,K1,K2,T1,T2,U1,d,e.
                  ⇩[d, e] L1 ≡ K1 → ⇧[d, e] T1 ≡ U1 →
-                 fsup K1 T1 K2 T2 → fsup L1 U1 K2 T2
+                 fsup G1 K1 T1 G2 K2 T2 → fsup G1 L1 U1 G2 K2 T2
 .
 
 interpretation
    "structural successor (closure)"
-   'SupTerm L1 T1 L2 T2 = (fsup L1 T1 L2 T2).
+   'SupTerm G1 L1 T1 G2 L2 T2 = (fsup G1 L1 T1 G2 L2 T2).
 
 (* Basic properties *********************************************************)
 
-lemma fsup_lref_S_lt: ∀I,L,K,V,T,i. 0 < i → ⦃L, #(i-1)⦄ ⊃ ⦃K, T⦄ → ⦃L.ⓑ{I}V, #i⦄ ⊃ ⦃K, T⦄.
-#I #L #K #V #T #i #Hi #H /3 width=7 by fsup_ldrop, ldrop_ldrop, lift_lref_ge_minus/ (**) (* auto too slow without trace *)
+lemma fsup_lref_S_lt: ∀I,G1,G2,L,K,V,T,i. 0 < i → ⦃G1, L, #(i-1)⦄ ⊃ ⦃G2, K, T⦄ → ⦃G1, L.ⓑ{I}V, #i⦄ ⊃ ⦃G2, K, T⦄.
+#I #G1 #G2 #L #K #V #T #i #Hi #H /3 width=7 by fsup_ldrop, ldrop_ldrop, lift_lref_ge_minus/ (**) (* auto too slow without trace *)
 qed.
 
-lemma fsup_lref: ∀I,K,V,i,L. ⇩[0, i] L ≡ K.ⓑ{I}V → ⦃L, #i⦄ ⊃ ⦃K, V⦄.
-#I #K #V #i @(nat_elim1 i) -i #i #IH #L #H
+lemma fsup_lref: ∀I,G,K,V,i,L. ⇩[0, i] L ≡ K.ⓑ{I}V → ⦃G, L, #i⦄ ⊃ ⦃G, K, V⦄.
+#I #G #K #V #i @(nat_elim1 i) -i #i #IH #L #H
 elim (ldrop_inv_O1_pair2 … H) -H *
 [ #H1 #H2 destruct //
 | #I1 #K1 #V1 #HK1 #H #Hi destruct
@@ -51,32 +52,32 @@ qed.
 
 (* Basic forward lemmas *****************************************************)
 
-lemma fsup_fwd_fw: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊃ ⦃L2, T2⦄ → ♯{L2, T2} < ♯{L1, T1}.
-#L1 #L2 #T1 #T2 #H elim H -L1 -L2 -T1 -T2 //
-[ #L #K #T #U #d #e #HLK #HTU #HKL
+lemma fsup_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 elim H -G1 -G2 -L1 -L2 -T1 -T2 //
+[ #G #L #K #T #U #d #e #HLK #HTU #HKL
   lapply (ldrop_fwd_lw_lt … HLK HKL) -HKL -HLK #HKL
   lapply (lift_fwd_tw … HTU) -d -e #H
   normalize in ⊢ (?%%); /2 width=1/
-| #L1 #K1 #K2 #T1 #T2 #U1 #d #e #HLK1 #HTU1 #_ #IHT12
+| #G1 #G2 #L1 #K1 #K2 #T1 #T2 #U1 #d #e #HLK1 #HTU1 #_ #IHT12
   lapply (ldrop_fwd_lw … HLK1) -HLK1 #HLK1
   lapply (lift_fwd_tw … HTU1) -HTU1 #HTU1
   @(lt_to_le_to_lt … IHT12) -IHT12 /2 width=1/
 ]
 qed-.
 
-fact fsup_fwd_length_lref1_aux: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊃ ⦃L2, T2⦄ →
+fact fsup_fwd_length_lref1_aux: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃ ⦃G2, L2, T2⦄ →
                                 ∀i. T1 = #i → |L2| < |L1|.
-#L1 #L2 #T1 #T2 #H elim H -L1 -L2 -T1 -T2
+#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
 [1: normalize //
 |3: #a
 |5: /2 width=4 by ldrop_fwd_length_lt4/
-|6: #L1 #K1 #K2 #T1 #T2 #U1 #d #e #HLK1 #HTU1 #_ #IHT12 #i #H destruct
+|6: #G1 #G2 #L1 #K1 #K2 #T1 #T2 #U1 #d #e #HLK1 #HTU1 #_ #IHT12 #i #H destruct
     lapply (ldrop_fwd_length_le4 … HLK1) -HLK1 #HLK1
     elim (lift_inv_lref2 … HTU1) -HTU1 * #Hdei #H destruct
     @(lt_to_le_to_lt … HLK1) /2 width=2/
-] #I #L #V #T #j #H destruct
+] #I #G #L #V #T #j #H destruct
 qed-.
 
-lemma fsup_fwd_length_lref1: ∀L1,L2,T2,i. ⦃L1, #i⦄ ⊃ ⦃L2, T2⦄ → |L2| < |L1|.
-/2 width=5 by fsup_fwd_length_lref1_aux/
+lemma fsup_fwd_length_lref1: ∀G1,G2,L1,L2,T2,i. ⦃G1, L1, #i⦄ ⊃ ⦃G2, L2, T2⦄ → |L2| < |L1|.
+/2 width=7 by fsup_fwd_length_lref1_aux/
 qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/fsupq.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/fsupq.ma
index 4c40afe09..d45a3c83f 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/relocation/fsupq.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/fsupq.ma
@@ -12,60 +12,62 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "basic_2/notation/relations/suptermopt_4.ma".
+include "basic_2/notation/relations/suptermopt_6.ma".
 include "basic_2/relocation/fsup.ma".
 
 (* OPTIONAL SUPCLOSURE ******************************************************)
 
-inductive fsupq: bi_relation lenv term ≝
-| fsupq_refl   : ∀L,T. fsupq L T L T
-| fsupq_lref_O : ∀I,L,V. fsupq (L.ⓑ{I}V) (#0) L V
-| fsupq_pair_sn: ∀I,L,V,T. fsupq L (②{I}V.T) L V
-| fsupq_bind_dx: ∀a,I,L,V,T. fsupq L (ⓑ{a,I}V.T) (L.ⓑ{I}V) T
-| fsupq_flat_dx: ∀I,L,V,T.   fsupq L (ⓕ{I}V.T) L T
-| fsupq_ldrop  : ∀L1,K1,K2,T1,T2,U1,d,e.
-                ⇩[d, e] L1 ≡ K1 → ⇧[d, e] T1 ≡ U1 →
-                fsupq K1 T1 K2 T2 → fsupq L1 U1 K2 T2
+(* activate genv *)
+inductive fsupq: tri_relation genv lenv term ≝
+| fsupq_refl   : ∀G,L,T. fsupq G L T G L T
+| fsupq_lref_O : ∀I,G,L,V. fsupq G (L.ⓑ{I}V) (#0) G L V
+| fsupq_pair_sn: ∀I,G,L,V,T. fsupq G L (②{I}V.T) G L V
+| fsupq_bind_dx: ∀a,I,G,L,V,T. fsupq G L (ⓑ{a,I}V.T) G (L.ⓑ{I}V) T
+| fsupq_flat_dx: ∀I,G, L,V,T. fsupq G L (ⓕ{I}V.T) G L T
+| fsupq_ldrop  : ∀G1,G2,L1,K1,K2,T1,T2,U1,d,e.
+                 ⇩[d, e] L1 ≡ K1 → ⇧[d, e] T1 ≡ U1 →
+                 fsupq G1 K1 T1 G2 K2 T2 → fsupq G1 L1 U1 G2 K2 T2
 .
 
 interpretation
    "optional structural successor (closure)"
-   'SupTermOpt L1 T1 L2 T2 = (fsupq L1 T1 L2 T2).
+   'SupTermOpt G1 L1 T1 G2 L2 T2 = (fsupq G1 L1 T1 G2 L2 T2).
 
 (* Basic properties *********************************************************)
 
-lemma fsup_fsupq: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊃ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⊃⸮ ⦃L2, T2⦄.
-#L1 #L2 #T1 #T2 #H elim H -L1 -L2 -T1 -T2 // /2 width=7/ qed.
+lemma fsup_fsupq: ∀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 elim H -L1 -L2 -T1 -T2 // /2 width=7/ qed.
 
 (* Basic properties *********************************************************)
 
-lemma fsupq_lref_S_lt: ∀I,L,K,V,T,i. 0 < i → ⦃L, #(i-1)⦄ ⊃⸮ ⦃K, T⦄ → ⦃L.ⓑ{I}V, #i⦄ ⊃⸮ ⦃K, T⦄.
+lemma fsupq_lref_S_lt: ∀I,G1,G2,L,K,V,T,i.
+                       0 < i → ⦃G1, L, #(i-1)⦄ ⊃⸮ ⦃G2, K, T⦄ → ⦃G1, L.ⓑ{I}V, #i⦄ ⊃⸮ ⦃G2, K, T⦄.
 /3 width=7/ qed.
 
-lemma fsupq_lref: ∀I,K,V,i,L. ⇩[0, i] L ≡ K.ⓑ{I}V → ⦃L, #i⦄ ⊃⸮ ⦃K, V⦄.
+lemma fsupq_lref: ∀I,G,K,V,i,L. ⇩[0, i] L ≡ K.ⓑ{I}V → ⦃G, L, #i⦄ ⊃⸮ ⦃G, K, V⦄.
 /3 width=2/ qed.
 
 (* Basic forward lemmas *****************************************************)
 
-lemma fsupq_fwd_fw: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊃⸮ ⦃L2, T2⦄ → ♯{L2, T2} ≤ ♯{L1, T1}.
-#L1 #L2 #T1 #T2 #H elim H -L1 -L2 -T1 -T2 // [1,2,3: /2 width=1/ ]
-#L1 #K1 #K2 #T1 #T2 #U1 #d #e #HLK1 #HTU1 #_ #IHT12
+lemma fsupq_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 elim H -G1 -G2 -L1 -L2 -T1 -T2 // [1,2,3: /2 width=1/ ]
+#G1 #G2 #L1 #K1 #K2 #T1 #T2 #U1 #d #e #HLK1 #HTU1 #_ #IHT12
 lapply (ldrop_fwd_lw … HLK1) -HLK1 #HLK1
 lapply (lift_fwd_tw … HTU1) -HTU1 #HTU1
 @(transitive_le … IHT12) -IHT12 /2 width=1/
 qed-.
 
-fact fsupq_fwd_length_lref1_aux: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊃⸮ ⦃L2, T2⦄ →
+fact fsupq_fwd_length_lref1_aux: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃⸮ ⦃G2, L2, T2⦄ →
                                  ∀i. T1 = #i → |L2| ≤ |L1|.
-#L1 #L2 #T1 #T2 #H elim H -L1 -L2 -T1 -T2 //
-[ #a #I #L #V #T #j #H destruct
-| #L1 #K1 #K2 #T1 #T2 #U1 #d #e #HLK1 #HTU1 #_ #IHT12 #i #H destruct
+#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2 //
+[ #a #I #G #L #V #T #j #H destruct
+| #G1 #G2 #L1 #K1 #K2 #T1 #T2 #U1 #d #e #HLK1 #HTU1 #_ #IHT12 #i #H destruct
   lapply (ldrop_fwd_length_le4 … HLK1) -HLK1 #HLK1
   elim (lift_inv_lref2 … HTU1) -HTU1 * #Hdei #H destruct
   @(transitive_le … HLK1) /2 width=2/
 ]
 qed-.
 
-lemma fsupq_fwd_length_lref1: ∀L1,L2,T2,i. ⦃L1, #i⦄ ⊃⸮ ⦃L2, T2⦄ → |L2| ≤ |L1|.
-/2 width=5 by fsupq_fwd_length_lref1_aux/
+lemma fsupq_fwd_length_lref1: ∀G1,G2,L1,L2,T2,i. ⦃G1, L1, #i⦄ ⊃⸮ ⦃G2, L2, T2⦄ → |L2| ≤ |L1|.
+/2 width=7 by fsupq_fwd_length_lref1_aux/
 qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/fsupq_alt.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/fsupq_alt.ma
index 360972607..e9bc6c533 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/relocation/fsupq_alt.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/fsupq_alt.ma
@@ -12,40 +12,40 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "basic_2/notation/relations/suptermoptalt_4.ma".
+include "basic_2/notation/relations/suptermoptalt_6.ma".
 include "basic_2/relocation/fsupq.ma".
 
 (* OPTIONAL SUPCLOSURE ******************************************************)
 
 (* alternative definition of fsupq *)
-definition fsupqa: bi_relation lenv term ≝ bi_RC … fsup.
+definition fsupqa: tri_relation genv lenv term ≝ tri_RC … fsup.
 
 interpretation
    "optional structural successor (closure) alternative"
-   'SupTermOptAlt L1 T1 L2 T2 = (fsupqa L1 T1 L2 T2).
+   'SupTermOptAlt G1 L1 T1 G2 L2 T2 = (fsupqa G1 L1 T1 G2 L2 T2).
 
 (* Basic properties *********************************************************)
 
-lemma fsupqa_refl: bi_reflexive … fsupqa.
+lemma fsupqa_refl: tri_reflexive … fsupqa.
 // qed.
 
-lemma fsupqa_ldrop: ∀K1,K2,T1,T2. ⦃K1, T1⦄ ⊃⊃⸮ ⦃K2, T2⦄ →
+lemma fsupqa_ldrop: ∀G1,G2,K1,K2,T1,T2. ⦃G1, K1, T1⦄ ⊃⊃⸮ ⦃G2, K2, T2⦄ →
                     ∀L1,d,e. ⇩[d, e] L1 ≡ K1 →
-                    ∀U1. ⇧[d, e] T1 ≡ U1 → ⦃L1, U1⦄ ⊃⊃⸮ ⦃K2, T2⦄.
-#K1 #K2 #T1 #T2 * [ /3 width=7/ ] * #H1 #H2 destruct
+                    ∀U1. ⇧[d, e] T1 ≡ U1 → ⦃G1, L1, U1⦄ ⊃⊃⸮ ⦃G2, K2, T2⦄.
+#G1 #G2 #K1 #K2 #T1 #T2 * [ /3 width=7/ ] * #H1 #H2 #H3 destruct
 #L1 #d #e #HLK #U1 #HTU elim (eq_or_gt e) [2: /3 width=5/ ] #H destruct
 >(ldrop_inv_O2 … HLK) -L1 >(lift_inv_O2 … HTU) -T2 -d //
 qed.
 
 (* Main properties **********************************************************)
 
-theorem fsupq_fsupqa: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊃⸮ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⊃⊃⸮ ⦃L2, T2⦄.
-#L1 #L2 #T1 #T2 #H elim H -L1 -L2 -T1 -T2 // /2 width=1/ /2 width=7/
+theorem fsupq_fsupqa: ∀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 elim H -G1 -G2 -L1 -L2 -T1 -T2 // /2 width=1/ /2 width=7/
 qed.
 
 (* Main inversion properties ************************************************)
 
-theorem fsupqa_inv_fsupq: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊃⊃⸮ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⊃⸮ ⦃L2, T2⦄.
-#L1 #L2 #T1 #T2 #H elim H -H /2 width=1/
-* #H1 #H2 destruct //
+theorem fsupqa_inv_fsupq: ∀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 elim H -H /2 width=1/
+* #H1 #H2 #H3 destruct //
 qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/gdrop.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/gget.ma
similarity index 74%
rename from matita/matita/contribs/lambdadelta/basic_2/relocation/gdrop.ma
rename to matita/matita/contribs/lambdadelta/basic_2/relocation/gget.ma
index 277e464b7..5ef11dc4b 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/relocation/gdrop.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/gget.ma
@@ -15,20 +15,20 @@
 include "basic_2/notation/relations/rdrop_3.ma".
 include "basic_2/grammar/genv.ma".
 
-(* GLOBAL ENVIRONMENT SLICING ***********************************************)
+(* GLOBAL ENVIRONMENT READING ***********************************************)
 
-inductive gdrop (e:nat): relation genv ≝
-| gdrop_gt: ∀G. |G| ≤ e → gdrop e G (⋆)
-| gdrop_eq: ∀G. |G| = e + 1 → gdrop e G G
-| gdrop_lt: ∀I,G1,G2,V. e < |G1| → gdrop e G1 G2 → gdrop e (G1. ⓑ{I} V) G2
+inductive gget (e:nat): relation genv ≝
+| gget_gt: ∀G. |G| ≤ e → gget e G (⋆)
+| gget_eq: ∀G. |G| = e + 1 → gget e G G
+| gget_lt: ∀I,G1,G2,V. e < |G1| → gget e G1 G2 → gget e (G1. ⓑ{I} V) G2
 .
 
-interpretation "global slicing"
-   'RDrop e G1 G2 = (gdrop e G1 G2).
+interpretation "global reading"
+   'RDrop e G1 G2 = (gget e G1 G2).
 
 (* basic inversion lemmas ***************************************************)
 
-lemma gdrop_inv_gt: ∀G1,G2,e. ⇩[e] G1 ≡ G2 → |G1| ≤ e → G2 = ⋆.
+lemma gget_inv_gt: ∀G1,G2,e. ⇩[e] G1 ≡ G2 → |G1| ≤ e → G2 = ⋆.
 #G1 #G2 #e * -G1 -G2 //
 [ #G #H >H -H >commutative_plus #H
   lapply (le_plus_to_le_r … 0 H) -H #H
@@ -40,7 +40,7 @@ lemma gdrop_inv_gt: ∀G1,G2,e. ⇩[e] G1 ≡ G2 → |G1| ≤ e → G2 = ⋆.
 ]
 qed-.
 
-lemma gdrop_inv_eq: ∀G1,G2,e. ⇩[e] G1 ≡ G2 → |G1| = e + 1 → G1 = G2.
+lemma gget_inv_eq: ∀G1,G2,e. ⇩[e] G1 ≡ G2 → |G1| = e + 1 → G1 = G2.
 #G1 #G2 #e * -G1 -G2 //
 [ #G #H1 #H2 >H2 in H1; -H2 >commutative_plus #H
   lapply (le_plus_to_le_r … 0 H) -H #H
@@ -51,8 +51,8 @@ lemma gdrop_inv_eq: ∀G1,G2,e. ⇩[e] G1 ≡ G2 → |G1| = e + 1 → G1 = G2.
 ]
 qed-.
 
-fact gdrop_inv_lt_aux: ∀I,G,G1,G2,V,e. ⇩[e] G ≡ G2 → G = G1. ⓑ{I} V →
-                       e < |G1| → ⇩[e] G1 ≡ G2.
+fact gget_inv_lt_aux: ∀I,G,G1,G2,V,e. ⇩[e] G ≡ G2 → G = G1. ⓑ{I} V →
+                      e < |G1| → ⇩[e] G1 ≡ G2.
 #I #G #G1 #G2 #V #e * -G -G2
 [ #G #H1 #H destruct #H2
   lapply (le_to_lt_to_lt … H1 H2) -H1 -H2 normalize in ⊢ (? % ? → ?); >commutative_plus #H
@@ -62,20 +62,20 @@ fact gdrop_inv_lt_aux: ∀I,G,G1,G2,V,e. ⇩[e] G ≡ G2 → G = G1. ⓑ{I} V 
   elim (lt_refl_false … H)
 | #J #G #G2 #W #_ #HG2 #H destruct //
 ]
-qed.
+qed-.
 
-lemma gdrop_inv_lt: ∀I,G1,G2,V,e.
+lemma gget_inv_lt: ∀I,G1,G2,V,e.
                     ⇩[e] G1. ⓑ{I} V ≡ G2 → e < |G1| → ⇩[e] G1 ≡ G2.
-/2 width=5/ qed-.
+/2 width=5 by gget_inv_lt_aux/ qed-.
 
 (* Basic properties *********************************************************)
 
-lemma gdrop_total: ∀e,G1. ∃G2. ⇩[e] G1 ≡ G2.
+lemma gget_total: ∀e,G1. ∃G2. ⇩[e] G1 ≡ G2.
 #e #G1 elim G1 -G1 /3 width=2/
 #I #V #G1 * #G2 #HG12
 elim (lt_or_eq_or_gt e (|G1|)) #He
 [ /3 width=2/
 | destruct /3 width=2/
-| @ex_intro [2: @gdrop_gt normalize /2 width=1/ | skip ] (**) (* explicit constructor *)
+| @ex_intro [2: @gget_gt normalize /2 width=1/ | skip ] (**) (* explicit constructor *)
 ]
-qed.
+qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/gdrop_gdrop.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/gget_gget.ma
similarity index 73%
rename from matita/matita/contribs/lambdadelta/basic_2/relocation/gdrop_gdrop.ma
rename to matita/matita/contribs/lambdadelta/basic_2/relocation/gget_gget.ma
index a133e29c6..6c1001b6a 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/relocation/gdrop_gdrop.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/gget_gget.ma
@@ -12,29 +12,29 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "basic_2/relocation/gdrop.ma".
+include "basic_2/relocation/gget.ma".
 
-(* GLOBAL ENVIRONMENT SLICING ***********************************************)
+(* GLOBAL ENVIRONMENT READING ***********************************************)
 
 (* Main properties **********************************************************)
 
-theorem gdrop_mono: ∀G,G1,e. ⇩[e] G ≡ G1 → ∀G2. ⇩[e] G ≡ G2 → G1 = G2.
+theorem gget_mono: ∀G,G1,e. ⇩[e] G ≡ G1 → ∀G2. ⇩[e] G ≡ G2 → G1 = G2.
 #G #G1 #e #H elim H -G -G1
 [ #G #He #G2 #H
-  >(gdrop_inv_gt … H He) -H -He //
+  >(gget_inv_gt … H He) -H -He //
 | #G #He #G2 #H
-  >(gdrop_inv_eq … H He) -H -He //
+  >(gget_inv_eq … H He) -H -He //
 | #I #G #G1 #V #He #_ #IHG1 #G2 #H
-  lapply (gdrop_inv_lt … H He) -H -He /2 width=1/
+  lapply (gget_inv_lt … H He) -H -He /2 width=1/
 ]
 qed-.
 
-lemma gdrop_dec: ∀G1,G2,e. Decidable (⇩[e] G1 ≡ G2).
+lemma gget_dec: ∀G1,G2,e. Decidable (⇩[e] G1 ≡ G2).
 #G1 #G2 #e
-elim (gdrop_total e G1) #G #HG1
+elim (gget_total e G1) #G #HG1
 elim (genv_eq_dec G G2) #HG2
 [ destruct /2 width=1/
 | @or_intror #HG12
-  lapply (gdrop_mono … HG1 … HG12) -HG1 -HG12 /2 width=1/
+  lapply (gget_mono … HG1 … HG12) -HG1 -HG12 /2 width=1/
 ]
 qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/aaa.ma b/matita/matita/contribs/lambdadelta/basic_2/static/aaa.ma
index 63584de4a..4c4f1e572 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/static/aaa.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/static/aaa.ma
@@ -34,7 +34,7 @@ interpretation "atomic arity assignment (term)"
 
 (* Basic inversion lemmas ***************************************************)
 
-fact aaa_inv_sort_aux: ∀L,T,A. L ⊢ T ⁝ A → ∀k. T = ⋆k → A = ⓪.
+fact aaa_inv_sort_aux: ∀L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀k. T = ⋆k → A = ⓪.
 #L #T #A * -L -T -A
 [ //
 | #I #L #K #V #B #i #_ #_ #k #H destruct
@@ -45,10 +45,10 @@ fact aaa_inv_sort_aux: ∀L,T,A. L ⊢ T ⁝ A → ∀k. T = ⋆k → A = ⓪.
 ]
 qed.
 
-lemma aaa_inv_sort: ∀L,A,k. L ⊢ ⋆k ⁝ A → A = ⓪.
+lemma aaa_inv_sort: ∀L,A,k. ⦃G, L⦄ ⊢ ⋆k ⁝ A → A = ⓪.
 /2 width=5/ qed-.
 
-fact aaa_inv_lref_aux: ∀L,T,A. L ⊢ T ⁝ A → ∀i. T = #i →
+fact aaa_inv_lref_aux: ∀L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀i. T = #i →
                        ∃∃I,K,V. ⇩[0, i] L ≡ K. ⓑ{I} V & K ⊢ V ⁝ A.
 #L #T #A * -L -T -A
 [ #L #k #i #H destruct
@@ -60,11 +60,11 @@ fact aaa_inv_lref_aux: ∀L,T,A. L ⊢ T ⁝ A → ∀i. T = #i →
 ]
 qed.
 
-lemma aaa_inv_lref: ∀L,A,i. L ⊢ #i ⁝ A →
+lemma aaa_inv_lref: ∀L,A,i. ⦃G, L⦄ ⊢ #i ⁝ A →
                     ∃∃I,K,V. ⇩[0, i] L ≡ K. ⓑ{I} V & K ⊢ V ⁝ A.
 /2 width=3/ qed-.
 
-fact aaa_inv_gref_aux: ∀L,T,A. L ⊢ T ⁝ A → ∀p. T = §p → ⊥.
+fact aaa_inv_gref_aux: ∀L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀p. T = §p → ⊥.
 #L #T #A * -L -T -A
 [ #L #k #q #H destruct
 | #I #L #K #V #B #i #HLK #HB #q #H destruct
@@ -75,11 +75,11 @@ fact aaa_inv_gref_aux: ∀L,T,A. L ⊢ T ⁝ A → ∀p. T = §p → ⊥.
 ]
 qed.
 
-lemma aaa_inv_gref: ∀L,A,p. L ⊢ §p ⁝ A → ⊥.
+lemma aaa_inv_gref: ∀L,A,p. ⦃G, L⦄ ⊢ §p ⁝ A → ⊥.
 /2 width=6/ qed-.
 
-fact aaa_inv_abbr_aux: ∀L,T,A. L ⊢ T ⁝ A → ∀a,W,U. T = ⓓ{a}W. U →
-                       ∃∃B. L ⊢ W ⁝ B & L. ⓓW ⊢ U ⁝ A.
+fact aaa_inv_abbr_aux: ∀L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀a,W,U. T = ⓓ{a}W. U →
+                       ∃∃B. ⦃G, L⦄ ⊢ W ⁝ B & L. ⓓW ⊢ U ⁝ A.
 #L #T #A * -L -T -A
 [ #L #k #a #W #U #H destruct
 | #I #L #K #V #B #i #_ #_ #a #W #U #H destruct
@@ -90,12 +90,12 @@ fact aaa_inv_abbr_aux: ∀L,T,A. L ⊢ T ⁝ A → ∀a,W,U. T = ⓓ{a}W. U →
 ]
 qed.
 
-lemma aaa_inv_abbr: ∀a,L,V,T,A. L ⊢ ⓓ{a}V. T ⁝ A →
-                    ∃∃B. L ⊢ V ⁝ B & L. ⓓV ⊢ T ⁝ A.
+lemma aaa_inv_abbr: ∀a,L,V,T,A. ⦃G, L⦄ ⊢ ⓓ{a}V. T ⁝ A →
+                    ∃∃B. ⦃G, L⦄ ⊢ V ⁝ B & L. ⓓV ⊢ T ⁝ A.
 /2 width=4/ qed-.
 
-fact aaa_inv_abst_aux: ∀L,T,A. L ⊢ T ⁝ A → ∀a,W,U. T = ⓛ{a}W. U →
-                       ∃∃B1,B2. L ⊢ W ⁝ B1 & L. ⓛW ⊢ U ⁝ B2 & A = ②B1. B2.
+fact aaa_inv_abst_aux: ∀L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀a,W,U. T = ⓛ{a}W. U →
+                       ∃∃B1,B2. ⦃G, L⦄ ⊢ W ⁝ B1 & L. ⓛW ⊢ U ⁝ B2 & A = ②B1. B2.
 #L #T #A * -L -T -A
 [ #L #k #a #W #U #H destruct
 | #I #L #K #V #B #i #_ #_ #a #W #U #H destruct
@@ -106,12 +106,12 @@ fact aaa_inv_abst_aux: ∀L,T,A. L ⊢ T ⁝ A → ∀a,W,U. T = ⓛ{a}W. U →
 ]
 qed.
 
-lemma aaa_inv_abst: ∀a,L,W,T,A. L ⊢ ⓛ{a}W. T ⁝ A →
-                    ∃∃B1,B2. L ⊢ W ⁝ B1 & L. ⓛW ⊢ T ⁝ B2 & A = ②B1. B2.
+lemma aaa_inv_abst: ∀a,L,W,T,A. ⦃G, L⦄ ⊢ ⓛ{a}W. T ⁝ A →
+                    ∃∃B1,B2. ⦃G, L⦄ ⊢ W ⁝ B1 & L. ⓛW ⊢ T ⁝ B2 & A = ②B1. B2.
 /2 width=4/ qed-.
 
-fact aaa_inv_appl_aux: ∀L,T,A. L ⊢ T ⁝ A → ∀W,U. T = ⓐW. U →
-                       ∃∃B. L ⊢ W ⁝ B & L ⊢ U ⁝ ②B. A.
+fact aaa_inv_appl_aux: ∀L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀W,U. T = ⓐW. U →
+                       ∃∃B. ⦃G, L⦄ ⊢ W ⁝ B & ⦃G, L⦄ ⊢ U ⁝ ②B. A.
 #L #T #A * -L -T -A
 [ #L #k #W #U #H destruct
 | #I #L #K #V #B #i #_ #_ #W #U #H destruct
@@ -122,12 +122,12 @@ fact aaa_inv_appl_aux: ∀L,T,A. L ⊢ T ⁝ A → ∀W,U. T = ⓐW. U →
 ]
 qed.
 
-lemma aaa_inv_appl: ∀L,V,T,A. L ⊢ ⓐV. T ⁝ A →
-                    ∃∃B. L ⊢ V ⁝ B & L ⊢ T ⁝ ②B. A.
+lemma aaa_inv_appl: ∀L,V,T,A. ⦃G, L⦄ ⊢ ⓐV. T ⁝ A →
+                    ∃∃B. ⦃G, L⦄ ⊢ V ⁝ B & ⦃G, L⦄ ⊢ T ⁝ ②B. A.
 /2 width=3/ qed-.
 
-fact aaa_inv_cast_aux: ∀L,T,A. L ⊢ T ⁝ A → ∀W,U. T = ⓝW. U →
-                       L ⊢ W ⁝ A ∧ L ⊢ U ⁝ A.
+fact aaa_inv_cast_aux: ∀L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀W,U. T = ⓝW. U →
+                       ⦃G, L⦄ ⊢ W ⁝ A ∧ ⦃G, L⦄ ⊢ U ⁝ A.
 #L #T #A * -L -T -A
 [ #L #k #W #U #H destruct
 | #I #L #K #V #B #i #_ #_ #W #U #H destruct
@@ -138,6 +138,6 @@ fact aaa_inv_cast_aux: ∀L,T,A. L ⊢ T ⁝ A → ∀W,U. T = ⓝW. U →
 ]
 qed.
 
-lemma aaa_inv_cast: ∀L,W,T,A. L ⊢ ⓝW. T ⁝ A →
-                    L ⊢ W ⁝ A ∧ L ⊢ T ⁝ A.
+lemma aaa_inv_cast: ∀L,W,T,A. ⦃G, L⦄ ⊢ ⓝW. T ⁝ A →
+                    ⦃G, L⦄ ⊢ W ⁝ A ∧ ⦃G, L⦄ ⊢ T ⁝ A.
 /2 width=3/ qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/aaa_aaa.ma b/matita/matita/contribs/lambdadelta/basic_2/static/aaa_aaa.ma
index fce1c02e4..4bf7888c0 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/static/aaa_aaa.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/static/aaa_aaa.ma
@@ -19,7 +19,7 @@ include "basic_2/static/aaa.ma".
 
 (* Main properties **********************************************************)
 
-theorem aaa_mono: ∀L,T,A1. L ⊢ T ⁝ A1 → ∀A2. L ⊢ T ⁝ A2 → A1 = A2.
+theorem aaa_mono: ∀L,T,A1. ⦃G, L⦄ ⊢ T ⁝ A1 → ∀A2. ⦃G, L⦄ ⊢ T ⁝ A2 → A1 = A2.
 #L #T #A1 #H elim H -L -T -A1
 [ #L #k #A2 #H
   >(aaa_inv_sort … H) -H //
diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/ssta.ma b/matita/matita/contribs/lambdadelta/basic_2/static/ssta.ma
index 6f4f40872..9421d7436 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/static/ssta.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/static/ssta.ma
@@ -35,11 +35,11 @@ interpretation "stratified static type assignment (term)"
    'StaticType h g L T U l = (ssta h g l L T U).
 
 definition ssta_step: ∀h. sd h → lenv → relation term ≝ λh,g,L,T,U.
-                      ∃l. ⦃h, L⦄ ⊢ T •[g] ⦃l+1, U⦄.
+                      ∃l. ⦃G, L⦄ ⊢ T •[h, g] ⦃l+1, U⦄.
 
 (* Basic inversion lemmas ************************************************)
 
-fact ssta_inv_sort1_aux: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g] ⦃l, U⦄ → ∀k0. T = ⋆k0 →
+fact ssta_inv_sort1_aux: ∀h,g,L,T,U,l. ⦃G, L⦄ ⊢ T •[h, g] ⦃l, U⦄ → ∀k0. T = ⋆k0 →
                          deg h g k0 l ∧ U = ⋆(next h k0).
 #h #g #L #T #U #l * -L -T -U -l
 [ #L #k #l #Hkl #k0 #H destruct /2 width=1/
@@ -51,15 +51,15 @@ fact ssta_inv_sort1_aux: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g] ⦃l, U⦄ →
 qed.
 
 (* Basic_1: was just: sty0_gen_sort *)
-lemma ssta_inv_sort1: ∀h,g,L,U,k,l. ⦃h, L⦄ ⊢ ⋆k •[g] ⦃l, U⦄ →
+lemma ssta_inv_sort1: ∀h,g,L,U,k,l. ⦃G, L⦄ ⊢ ⋆k •[h, g] ⦃l, U⦄ →
                       deg h g k l ∧ U = ⋆(next h k).
 /2 width=4/ qed-.
 
-fact ssta_inv_lref1_aux: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g] ⦃l, U⦄ → ∀j. T = #j →
-                         (∃∃K,V,W. ⇩[0, j] L ≡ K. ⓓV & ⦃h, K⦄ ⊢ V •[g] ⦃l, W⦄ &
+fact ssta_inv_lref1_aux: ∀h,g,L,T,U,l. ⦃G, L⦄ ⊢ T •[h, g] ⦃l, U⦄ → ∀j. T = #j →
+                         (∃∃K,V,W. ⇩[0, j] L ≡ K. ⓓV & ⦃h, K⦄ ⊢ V •[h, g] ⦃l, W⦄ &
                                    ⇧[0, j + 1] W ≡ U
                          ) ∨
-                         (∃∃K,W,V,l0. ⇩[0, j] L ≡ K. ⓛW & ⦃h, K⦄ ⊢ W •[g] ⦃l0, V⦄ &
+                         (∃∃K,W,V,l0. ⇩[0, j] L ≡ K. ⓛW & ⦃h, K⦄ ⊢ W •[h, g] ⦃l0, V⦄ &
                                       ⇧[0, j + 1] W ≡ U & l = l0 + 1
                          ).
 #h #g #L #T #U #l * -L -T -U -l
@@ -73,16 +73,16 @@ fact ssta_inv_lref1_aux: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g] ⦃l, U⦄ →
 qed.
 
 (* Basic_1: was just: sty0_gen_lref *)
-lemma ssta_inv_lref1: ∀h,g,L,U,i,l. ⦃h, L⦄ ⊢ #i •[g] ⦃l, U⦄ →
-                      (∃∃K,V,W. ⇩[0, i] L ≡ K. ⓓV & ⦃h, K⦄ ⊢ V •[g] ⦃l, W⦄ &
+lemma ssta_inv_lref1: ∀h,g,L,U,i,l. ⦃G, L⦄ ⊢ #i •[h, g] ⦃l, U⦄ →
+                      (∃∃K,V,W. ⇩[0, i] L ≡ K. ⓓV & ⦃h, K⦄ ⊢ V •[h, g] ⦃l, W⦄ &
                                 ⇧[0, i + 1] W ≡ U
                       ) ∨
-                      (∃∃K,W,V,l0. ⇩[0, i] L ≡ K. ⓛW & ⦃h, K⦄ ⊢ W •[g] ⦃l0, V⦄ &
+                      (∃∃K,W,V,l0. ⇩[0, i] L ≡ K. ⓛW & ⦃h, K⦄ ⊢ W •[h, g] ⦃l0, V⦄ &
                                    ⇧[0, i + 1] W ≡ U & l = l0 + 1
                       ).
 /2 width=3/ qed-.
 
-fact ssta_inv_gref1_aux: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g] ⦃l, U⦄ → ∀p0. T = §p0 → ⊥.
+fact ssta_inv_gref1_aux: ∀h,g,L,T,U,l. ⦃G, L⦄ ⊢ T •[h, g] ⦃l, U⦄ → ∀p0. T = §p0 → ⊥.
 #h #g #L #T #U #l * -L -T -U -l
 [ #L #k #l #_ #p0 #H destruct
 | #L #K #V #W #U #i #l #_ #_ #_ #p0 #H destruct
@@ -92,12 +92,12 @@ fact ssta_inv_gref1_aux: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g] ⦃l, U⦄ →
 | #L #W #T #U #l #_ #p0 #H destruct
 qed.
 
-lemma ssta_inv_gref1: ∀h,g,L,U,p,l. ⦃h, L⦄ ⊢ §p •[g] ⦃l, U⦄ → ⊥.
+lemma ssta_inv_gref1: ∀h,g,L,U,p,l. ⦃G, L⦄ ⊢ §p •[h, g] ⦃l, U⦄ → ⊥.
 /2 width=9/ qed-.
 
-fact ssta_inv_bind1_aux: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g] ⦃l, U⦄ →
+fact ssta_inv_bind1_aux: ∀h,g,L,T,U,l. ⦃G, L⦄ ⊢ T •[h, g] ⦃l, U⦄ →
                          ∀a,I,X,Y. T = ⓑ{a,I}Y.X →
-                         ∃∃Z. ⦃h, L.ⓑ{I}Y⦄ ⊢ X •[g] ⦃l, Z⦄ & U = ⓑ{a,I}Y.Z.
+                         ∃∃Z. ⦃h, L.ⓑ{I}Y⦄ ⊢ X •[h, g] ⦃l, Z⦄ & U = ⓑ{a,I}Y.Z.
 #h #g #L #T #U #l * -L -T -U -l
 [ #L #k #l #_ #a #I #X #Y #H destruct
 | #L #K #V #W #U #i #l #_ #_ #_ #a #I #X #Y #H destruct
@@ -109,12 +109,12 @@ fact ssta_inv_bind1_aux: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g] ⦃l, U⦄ →
 qed.
 
 (* Basic_1: was just: sty0_gen_bind *)
-lemma ssta_inv_bind1: ∀h,g,a,I,L,Y,X,U,l. ⦃h, L⦄ ⊢ ⓑ{a,I}Y.X •[g] ⦃l, U⦄ →
-                      ∃∃Z. ⦃h, L.ⓑ{I}Y⦄ ⊢ X •[g] ⦃l, Z⦄ & U = ⓑ{a,I}Y.Z.
+lemma ssta_inv_bind1: ∀h,g,a,I,L,Y,X,U,l. ⦃G, L⦄ ⊢ ⓑ{a,I}Y.X •[h, g] ⦃l, U⦄ →
+                      ∃∃Z. ⦃h, L.ⓑ{I}Y⦄ ⊢ X •[h, g] ⦃l, Z⦄ & U = ⓑ{a,I}Y.Z.
 /2 width=3/ qed-.
 
-fact ssta_inv_appl1_aux: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g] ⦃l, U⦄ → ∀X,Y. T = ⓐY.X →
-                         ∃∃Z. ⦃h, L⦄ ⊢ X •[g] ⦃l, Z⦄ & U = ⓐY.Z.
+fact ssta_inv_appl1_aux: ∀h,g,L,T,U,l. ⦃G, L⦄ ⊢ T •[h, g] ⦃l, U⦄ → ∀X,Y. T = ⓐY.X →
+                         ∃∃Z. ⦃G, L⦄ ⊢ X •[h, g] ⦃l, Z⦄ & U = ⓐY.Z.
 #h #g #L #T #U #l * -L -T -U -l
 [ #L #k #l #_ #X #Y #H destruct
 | #L #K #V #W #U #i #l #_ #_ #_ #X #Y #H destruct
@@ -126,12 +126,12 @@ fact ssta_inv_appl1_aux: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g] ⦃l, U⦄ →
 qed.
 
 (* Basic_1: was just: sty0_gen_appl *)
-lemma ssta_inv_appl1: ∀h,g,L,Y,X,U,l. ⦃h, L⦄ ⊢ ⓐY.X •[g] ⦃l, U⦄ →
-                      ∃∃Z. ⦃h, L⦄ ⊢ X •[g] ⦃l, Z⦄ & U = ⓐY.Z.
+lemma ssta_inv_appl1: ∀h,g,L,Y,X,U,l. ⦃G, L⦄ ⊢ ⓐY.X •[h, g] ⦃l, U⦄ →
+                      ∃∃Z. ⦃G, L⦄ ⊢ X •[h, g] ⦃l, Z⦄ & U = ⓐY.Z.
 /2 width=3/ qed-.
 
-fact ssta_inv_cast1_aux: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g] ⦃l, U⦄ →
-                         ∀X,Y. T = ⓝY.X → ⦃h, L⦄ ⊢ X •[g] ⦃l, U⦄.
+fact ssta_inv_cast1_aux: ∀h,g,L,T,U,l. ⦃G, L⦄ ⊢ T •[h, g] ⦃l, U⦄ →
+                         ∀X,Y. T = ⓝY.X → ⦃G, L⦄ ⊢ X •[h, g] ⦃l, U⦄.
 #h #g #L #T #U #l * -L -T -U -l
 [ #L #k #l #_ #X #Y #H destruct
 | #L #K #V #W #U #l #i #_ #_ #_ #X #Y #H destruct
@@ -143,6 +143,6 @@ fact ssta_inv_cast1_aux: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g] ⦃l, U⦄ →
 qed.
 
 (* Basic_1: was just: sty0_gen_cast *)
-lemma ssta_inv_cast1: ∀h,g,L,X,Y,U,l. ⦃h, L⦄ ⊢ ⓝY.X •[g] ⦃l, U⦄ →
-                      ⦃h, L⦄ ⊢ X •[g] ⦃l, U⦄.
+lemma ssta_inv_cast1: ∀h,g,L,X,Y,U,l. ⦃G, L⦄ ⊢ ⓝY.X •[h, g] ⦃l, U⦄ →
+                      ⦃G, L⦄ ⊢ X •[h, g] ⦃l, U⦄.
 /2 width=4/ qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/ssta_aaa.ma b/matita/matita/contribs/lambdadelta/basic_2/static/ssta_aaa.ma
index eafec7430..0f035cf7c 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/static/ssta_aaa.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/static/ssta_aaa.ma
@@ -19,7 +19,7 @@ include "basic_2/static/ssta.ma".
 
 (* Properties on atomic arity assignment for terms **************************)
 
-lemma ssta_aaa: ∀h,g,L,T,A. L ⊢ T ⁝ A → ∀U,l. ⦃h, L⦄ ⊢ T •[g] ⦃l, U⦄ → L ⊢ U ⁝ A.
+lemma ssta_aaa: ∀h,g,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀U,l. ⦃G, L⦄ ⊢ T •[h, g] ⦃l, U⦄ → ⦃G, L⦄ ⊢ U ⁝ A.
 #h #g #L #T #A #H elim H -L -T -A
 [ #L #k #U #l #H
   elim (ssta_inv_sort1 … H) -H #_ #H destruct //
diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/ssta_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/static/ssta_lift.ma
index 10585d61d..00e512498 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/static/ssta_lift.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/static/ssta_lift.ma
@@ -20,9 +20,9 @@ include "basic_2/static/ssta.ma".
 (* Properties on relocation *************************************************)
 
 (* Basic_1: was just: sty0_lift *)
-lemma ssta_lift: ∀h,g,L1,T1,U1,l. ⦃h, L1⦄ ⊢ T1 •[g] ⦃l, U1⦄ →
+lemma ssta_lift: ∀h,g,L1,T1,U1,l. ⦃h, L1⦄ ⊢ T1 •[h, g] ⦃l, U1⦄ →
                  ∀L2,d,e. ⇩[d, e] L2 ≡ L1 → ∀T2. ⇧[d, e] T1 ≡ T2 →
-                 ∀U2. ⇧[d, e] U1 ≡ U2 → ⦃h, L2⦄ ⊢ T2 •[g] ⦃l, U2⦄.
+                 ∀U2. ⇧[d, e] U1 ≡ U2 → ⦃h, L2⦄ ⊢ T2 •[h, g] ⦃l, U2⦄.
 #h #g #L1 #T1 #U1 #l #H elim H -L1 -T1 -U1 -l
 [ #L1 #k #l #Hkl #L2 #d #e #HL21 #X1 #H1 #X2 #H2
   >(lift_inv_sort1 … H1) -X1
@@ -60,9 +60,9 @@ lemma ssta_lift: ∀h,g,L1,T1,U1,l. ⦃h, L1⦄ ⊢ T1 •[g] ⦃l, U1⦄ →
 qed.
 
 (* Note: apparently this was missing in basic_1 *)
-lemma ssta_inv_lift1: ∀h,g,L2,T2,U2,l. ⦃h, L2⦄ ⊢ T2 •[g] ⦃l, U2⦄ →
+lemma ssta_inv_lift1: ∀h,g,L2,T2,U2,l. ⦃h, L2⦄ ⊢ T2 •[h, g] ⦃l, U2⦄ →
                       ∀L1,d,e. ⇩[d, e] L2 ≡ L1 → ∀T1. ⇧[d, e] T1 ≡ T2 →
-                      ∃∃U1. ⦃h, L1⦄ ⊢ T1 •[g] ⦃l, U1⦄ & ⇧[d, e] U1 ≡ U2.
+                      ∃∃U1. ⦃h, L1⦄ ⊢ T1 •[h, g] ⦃l, U1⦄ & ⇧[d, e] U1 ≡ U2.
 #h #g #L2 #T2 #U2 #l #H elim H -L2 -T2 -U2 -l
 [ #L2 #k #l #Hkl #L1 #d #e #_ #X #H
   >(lift_inv_sort2 … H) -X /3 width=3/
@@ -105,8 +105,8 @@ qed.
 (* Advanced forvard lemmas **************************************************)
 
 (* Basic_1: was just: sty0_correct *)
-lemma ssta_fwd_correct: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g] ⦃l, U⦄ →
-                        ∃T0. ⦃h, L⦄ ⊢ U •[g] ⦃l-1, T0⦄.
+lemma ssta_fwd_correct: ∀h,g,L,T,U,l. ⦃G, L⦄ ⊢ T •[h, g] ⦃l, U⦄ →
+                        ∃T0. ⦃G, L⦄ ⊢ U •[h, g] ⦃l-1, T0⦄.
 #h #g #L #T #U #l #H elim H -L -T -U -l
 [ /4 width=2/
 | #L #K #V #W #W0 #i #l #HLK #_ #HW0 * #V0 #HWV0
diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/ssta_ssta.ma b/matita/matita/contribs/lambdadelta/basic_2/static/ssta_ssta.ma
index 617ef508f..87e0d2168 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/static/ssta_ssta.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/static/ssta_ssta.ma
@@ -19,8 +19,8 @@ include "basic_2/static/ssta_lift.ma".
 (* Main properties **********************************************************)
 
 (* Note: apparently this was missing in basic_1 *)
-theorem ssta_mono: ∀h,g,L,T,U1,l1. ⦃h, L⦄ ⊢ T •[g] ⦃l1, U1⦄ →
-                   ∀U2,l2. ⦃h, L⦄ ⊢ T •[g] ⦃l2, U2⦄ → l1 = l2 ∧ U1 = U2.
+theorem ssta_mono: ∀h,g,L,T,U1,l1. ⦃G, L⦄ ⊢ T •[h, g] ⦃l1, U1⦄ →
+                   ∀U2,l2. ⦃G, L⦄ ⊢ T •[h, g] ⦃l2, U2⦄ → l1 = l2 ∧ U1 = U2.
 #h #g #L #T #U1 #l1 #H elim H -L -T -U1 -l1
 [ #L #k #l #Hkl #X #l2 #H
   elim (ssta_inv_sort1 … H) -H #Hkl2 #H destruct
@@ -49,7 +49,7 @@ qed-.
 
 (* Advanced inversion lemmas ************************************************)
 
-lemma ssta_inv_refl_pos: ∀h,g,L,T,l. ⦃h, L⦄ ⊢ T •[g] ⦃l+1, T⦄ → ⊥.
+lemma ssta_inv_refl_pos: ∀h,g,L,T,l. ⦃G, L⦄ ⊢ T •[h, g] ⦃l+1, T⦄ → ⊥.
 #h #g #L #T #l #HTT
 elim (ssta_fwd_correct … HTT) <minus_plus_m_m #U #HTU
 elim (ssta_mono … HTU … HTT) -h -L #H #_ -T -U
diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/fsupp.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/fsupp.ma
index a4bd30fac..e51e5cd5a 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/substitution/fsupp.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/fsupp.ma
@@ -12,80 +12,83 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "basic_2/notation/relations/suptermplus_4.ma".
+include "basic_2/notation/relations/suptermplus_6.ma".
 include "basic_2/relocation/fsup.ma".
 
 (* PLUS-ITERATED SUPCLOSURE *************************************************)
 
-definition fsupp: bi_relation lenv term ≝ bi_TC … fsup.
+definition fsupp: tri_relation genv lenv term ≝ tri_TC … fsup.
 
 interpretation "plus-iterated structural successor (closure)"
-   'SupTermPlus L1 T1 L2 T2 = (fsupp L1 T1 L2 T2).
+   'SupTermPlus G1 L1 T1 G2 L2 T2 = (fsupp G1 L1 T1 G2 L2 T2).
 
 (* Basic properties *********************************************************)
 
-lemma fsup_fsupp: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊃ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⊃+ ⦃L2, T2⦄.
+lemma fsup_fsupp: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊃+ ⦃G2, L2, T2⦄.
 /2 width=1/ qed.
 
-lemma fsupp_strap1: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ ⊃+ ⦃L, T⦄ → ⦃L, T⦄ ⊃ ⦃L2, T2⦄ →
-                    ⦃L1, T1⦄ ⊃+ ⦃L2, T2⦄.
-/2 width=4/ qed.
+lemma fsupp_strap1: ∀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⦄.
+/2 width=5/ qed.
 
-lemma fsupp_strap2: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ ⊃ ⦃L, T⦄ → ⦃L, T⦄ ⊃+ ⦃L2, T2⦄ →
-                    ⦃L1, T1⦄ ⊃+ ⦃L2, T2⦄.
-/2 width=4/ qed.
+lemma fsupp_strap2: ∀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⦄.
+/2 width=5/ qed.
 
-lemma fsupp_lref: ∀I,K,V,i,L. ⇩[0, i] L ≡ K.ⓑ{I}V → ⦃L, #i⦄ ⊃+ ⦃K, V⦄.
+lemma fsupp_lref: ∀I,G,L,K,V,i. ⇩[0, i] L ≡ K.ⓑ{I}V → ⦃G, L, #i⦄ ⊃+ ⦃G, K, V⦄.
 /3 width=2/ qed.
 
-lemma fsupp_pair_sn: ∀I,L,V,T. ⦃L, ②{I}V.T⦄ ⊃+ ⦃L, V⦄.
+lemma fsupp_pair_sn: ∀I,G,L,V,T. ⦃G, L, ②{I}V.T⦄ ⊃+ ⦃G, L, V⦄.
 /2 width=1/ qed.
 
-lemma fsupp_bind_dx: ∀a,K,I,V,T. ⦃K, ⓑ{a,I}V.T⦄ ⊃+ ⦃K.ⓑ{I}V, T⦄.
+lemma fsupp_bind_dx: ∀a,I,G,L,V,T. ⦃G, L, ⓑ{a,I}V.T⦄ ⊃+ ⦃G, L.ⓑ{I}V, T⦄.
 /2 width=1/ qed.
 
-lemma fsupp_flat_dx: ∀I,L,V,T. ⦃L, ⓕ{I}V.T⦄ ⊃+ ⦃L, T⦄.
+lemma fsupp_flat_dx: ∀I,G,L,V,T. ⦃G, L, ⓕ{I}V.T⦄ ⊃+ ⦃G, L, T⦄.
 /2 width=1/ qed.
 
-lemma fsupp_flat_dx_pair_sn: ∀I1,I2,L,V1,V2,T. ⦃L, ⓕ{I1}V1.②{I2}V2.T⦄ ⊃+ ⦃L, V2⦄.
-/2 width=4/ qed.
+lemma fsupp_flat_dx_pair_sn: ∀I1,I2,G,L,V1,V2,T. ⦃G, L, ⓕ{I1}V1.②{I2}V2.T⦄ ⊃+ ⦃G, L, V2⦄.
+/2 width=5/ qed.
 
-lemma fsupp_bind_dx_flat_dx: ∀a,I1,I2,L,V1,V2,T. ⦃L, ⓑ{a,I1}V1.ⓕ{I2}V2.T⦄ ⊃+ ⦃L.ⓑ{I1}V1, T⦄.
-/2 width=4/ qed.
+lemma fsupp_bind_dx_flat_dx: ∀a,G,I1,I2,L,V1,V2,T. ⦃G, L, ⓑ{a,I1}V1.ⓕ{I2}V2.T⦄ ⊃+ ⦃G, L.ⓑ{I1}V1, T⦄.
+/2 width=5/ qed.
 
-lemma fsupp_flat_dx_bind_dx: ∀a,I1,I2,L,V1,V2,T. ⦃L, ⓕ{I1}V1.ⓑ{a,I2}V2.T⦄ ⊃+ ⦃L.ⓑ{I2}V2, T⦄.
-/2 width=4/ qed.
+lemma fsupp_flat_dx_bind_dx: ∀a,I1,I2,G,L,V1,V2,T. ⦃G, L, ⓕ{I1}V1.ⓑ{a,I2}V2.T⦄ ⊃+ ⦃G, L.ⓑ{I2}V2, T⦄.
+/2 width=5/ qed.
 
 (* Basic eliminators ********************************************************)
 
-lemma fsupp_ind: ∀L1,T1. ∀R:relation2 lenv term.
-                 (∀L2,T2. ⦃L1, T1⦄ ⊃ ⦃L2, T2⦄ → R L2 T2) →
-                 (∀L,T,L2,T2. ⦃L1, T1⦄ ⊃+ ⦃L, T⦄ → ⦃L, T⦄ ⊃ ⦃L2, T2⦄ → R L T → R L2 T2) →
-                 ∀L2,T2. ⦃L1, T1⦄ ⊃+ ⦃L2, T2⦄ → R L2 T2.
-#L1 #T1 #R #IH1 #IH2 #L2 #T2 #H
-@(bi_TC_ind … IH1 IH2 ? ? H)
+lemma fsupp_ind: ∀G1,L1,T1. ∀R:relation3 ….
+                 (∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊃ ⦃G2, L2, T2⦄ → R G2 L2 T2) →
+                 (∀G,G2,L,L2,T,T2. ⦃G1, L1, T1⦄ ⊃+ ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊃ ⦃G2, L2, T2⦄ → R G L T → R G2 L2 T2) →
+                 ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊃+ ⦃G2, L2, T2⦄ → R G2 L2 T2.
+#G1 #L1 #T1 #R #IH1 #IH2 #G2 #L2 #T2 #H
+@(tri_TC_ind … IH1 IH2 G2 L2 T2 H)
 qed-.
 
-lemma fsupp_ind_dx: ∀L2,T2. ∀R:relation2 lenv term.
-                    (∀L1,T1. ⦃L1, T1⦄ ⊃ ⦃L2, T2⦄ → R L1 T1) →
-                    (∀L1,L,T1,T. ⦃L1, T1⦄ ⊃ ⦃L, T⦄ → ⦃L, T⦄ ⊃+ ⦃L2, T2⦄ → R L T → R L1 T1) →
-                    ∀L1,T1. ⦃L1, T1⦄ ⊃+ ⦃L2, T2⦄ → R L1 T1.
-#L2 #T2 #R #IH1 #IH2 #L1 #T1 #H
-@(bi_TC_ind_dx … IH1 IH2 ? ? H)
+lemma fsupp_ind_dx: ∀G2,L2,T2. ∀R:relation3 ….
+                    (∀G1,L1,T1. ⦃G1, L1, T1⦄ ⊃ ⦃G2, L2, T2⦄ → R G1 L1 T1) →
+                    (∀G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ⊃ ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊃+ ⦃G2, L2, T2⦄ → R G L T → R G1 L1 T1) →
+                    ∀G1,L1,T1. ⦃G1, L1, T1⦄ ⊃+ ⦃G2, L2, T2⦄ → R G1 L1 T1.
+#G2 #L2 #T2 #R #IH1 #IH2 #G1 #L1 #T1 #H
+@(tri_TC_ind_dx … IH1 IH2 G1 L1 T1 H)
 qed-.
 
 (* Basic forward lemmas *****************************************************)
 
-lemma fsupp_fwd_fw: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊃+ ⦃L2, T2⦄ → ♯{L2, T2} < ♯{L1, T1}.
-#L1 #L2 #T1 #T2 #H @(fsupp_ind … H) -L2 -T2
+lemma fsupp_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 @(fsupp_ind … H) -G2 -L2 -T2
 /3 width=3 by fsup_fwd_fw, transitive_lt/
 qed-.
 
 (* Advanced eliminators *****************************************************)
 
-lemma fsupp_wf_ind: ∀R:relation2 lenv term. (
-                       ∀L1,T1. (∀L2,T2. ⦃L1, T1⦄ ⊃+ ⦃L2, T2⦄ → R L2 T2) →
-                       ∀L2,T2. L1 = L2 → T1 = T2 → R L2 T2
-                    ) → ∀L1,T1. R L1 T1.
-#R #HR @(f2_ind … fw) #n #IHn #L1 #T1 #H destruct /4 width=5 by fsupp_fwd_fw/
+lemma fsupp_wf_ind: ∀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) #n #IHn #G1 #L1 #T1 #H destruct /4 width=7 by fsupp_fwd_fw/
 qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/fsupp_fsupp.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/fsupp_fsupp.ma
index 9380a99df..7740b19da 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/substitution/fsupp_fsupp.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/fsupp_fsupp.ma
@@ -18,5 +18,5 @@ include "basic_2/substitution/fsupp.ma".
 
 (* Main propertis ***********************************************************)
 
-theorem fsupp_trans: bi_transitive … fsupp.
-/2 width=4/ qed.
+theorem fsupp_trans: tri_transitive … fsupp.
+/2 width=5/ qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/fsups.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/fsups.ma
index d71d52fbf..51126ebd1 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/substitution/fsups.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/fsups.ma
@@ -12,72 +12,72 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "basic_2/notation/relations/suptermstar_4.ma".
+include "basic_2/notation/relations/suptermstar_6.ma".
 include "basic_2/relocation/fsupq.ma".
 
 (* STAR-ITERATED SUPCLOSURE *************************************************)
 
-definition fsups: bi_relation lenv term ≝ bi_TC … fsupq.
+definition fsups: tri_relation genv lenv term ≝ tri_TC … fsupq.
 
 interpretation "star-iterated structural successor (closure)"
-   'SupTermStar L1 T1 L2 T2 = (fsups L1 T1 L2 T2).
+   'SupTermStar G1 L1 T1 G2 L2 T2 = (fsups G1 L1 T1 G2 L2 T2).
 
 (* Basic eliminators ********************************************************)
 
-lemma fsups_ind: ∀L1,T1. ∀R:relation2 lenv term. R L1 T1 →
-                 (∀L,L2,T,T2. ⦃L1, T1⦄ ⊃* ⦃L, T⦄ → ⦃L, T⦄ ⊃⸮ ⦃L2, T2⦄ → R L T → R L2 T2) →
-                 ∀L2,T2. ⦃L1, T1⦄ ⊃* ⦃L2, T2⦄ → R L2 T2.
-#L1 #T1 #R #IH1 #IH2 #L2 #T2 #H
-@(bi_TC_star_ind … IH1 IH2 ? ? H) //
+lemma fsups_ind: ∀G1,L1,T1. ∀R:relation3 …. R G1 L1 T1 →
+                 (∀G,G2,L,L2,T,T2. ⦃G1, L1, T1⦄ ⊃* ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊃⸮ ⦃G2, L2, T2⦄ → R G L T → R G2 L2 T2) →
+                 ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊃* ⦃G2, L2, T2⦄ → R G2 L2 T2.
+#G1 #L1 #T1 #R #IH1 #IH2 #G2 #L2 #T2 #H
+@(tri_TC_star_ind … IH1 IH2 G2 L2 T2 H) //
 qed-.
 
-lemma fsups_ind_dx: ∀L2,T2. ∀R:relation2 lenv term. R L2 T2 →
-                    (∀L1,L,T1,T. ⦃L1, T1⦄ ⊃⸮ ⦃L, T⦄ → ⦃L, T⦄ ⊃* ⦃L2, T2⦄ → R L T → R L1 T1) →
-                    ∀L1,T1. ⦃L1, T1⦄ ⊃* ⦃L2, T2⦄ → R L1 T1.
-#L2 #T2 #R #IH1 #IH2 #L1 #T1 #H
-@(bi_TC_star_ind_dx … IH1 IH2 ? ? H) //
+lemma fsups_ind_dx: ∀G2,L2,T2. ∀R:relation3 …. R G2 L2 T2 →
+                    (∀G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ⊃⸮ ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊃* ⦃G2, L2, T2⦄ → R G L T → R G1 L1 T1) →
+                    ∀G1,L1,T1. ⦃G1, L1, T1⦄ ⊃* ⦃G2, L2, T2⦄ → R G1 L1 T1.
+#G2 #L2 #T2 #R #IH1 #IH2 #G1 #L1 #T1 #H
+@(tri_TC_star_ind_dx … IH1 IH2 G1 L1 T1 H) //
 qed-.
 
 (* Basic properties *********************************************************)
 
-lemma fsups_refl: bi_reflexive … fsups.
+lemma fsups_refl: tri_reflexive … fsups.
 /2 width=1/ qed.
 
-lemma fsupq_fsups: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊃⸮ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⊃* ⦃L2, T2⦄.
+lemma fsupq_fsups: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃⸮ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊃* ⦃G2, L2, T2⦄.
 /2 width=1/ qed.
 
-lemma fsups_strap1: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ ⊃* ⦃L, T⦄ → ⦃L, T⦄ ⊃⸮ ⦃L2, T2⦄ →
-                    ⦃L1, T1⦄ ⊃* ⦃L2, T2⦄.
-/2 width=4/ qed.
+lemma fsups_strap1: ∀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⦄.
+/2 width=5/ qed.
 
-lemma fsups_strap2: ∀L1,L,L2,T1,T,T2. ⦃L1, T1⦄ ⊃⸮ ⦃L, T⦄ → ⦃L, T⦄ ⊃* ⦃L2, T2⦄ →
-                    ⦃L1, T1⦄ ⊃* ⦃L2, T2⦄.
-/2 width=4/ qed.
+lemma fsups_strap2: ∀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⦄.
+/2 width=5/ qed.
 
 (* Basic forward lemmas *****************************************************)
 
-lemma fsups_fwd_fw: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊃* ⦃L2, T2⦄ → ♯{L2, T2} ≤ ♯{L1, T1}.
-#L1 #L2 #T1 #T2 #H @(fsups_ind … H) -L2 -T2 //
+lemma fsups_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 @(fsups_ind … H) -L2 -T2 //
 /3 width=3 by fsupq_fwd_fw, transitive_le/ (**) (* slow even with trace *)
 qed-.
 (*
 (* Advanced inversion lemmas on plus-iterated supclosure ********************)
 
-lamma fsupp_inv_bind1_fsups: ∀b,J,L1,L2,W,U,T2. ⦃L1, ⓑ{b,J}W.U⦄ ⊃+ ⦃L2, T2⦄ →
-                             ⦃L1, W⦄ ⊃* ⦃L2, T2⦄ ∨ ⦃L1.ⓑ{J}W, U⦄ ⊃* ⦃L2, T2⦄.
-#b #J #L1 #L2 #W #U #T2 #H @(fsupp_ind … H) -L2 -T2
-[ #L2 #T2 #H
-  elim (fsup_inv_bind1 … H) -H * #H1 #H2 destruct /2 width=1/
-| #L #T #L2 #T2 #_ #HT2 * /3 width=4/
+lamma fsupp_inv_bind1_fsups: ∀b,J,G1,G2,L1,L2,W,U,T2. ⦃G1, L1, ⓑ{b,J}W.U⦄ ⊃+ ⦃G2, L2, T2⦄ →
+                             ⦃G1, L1, W⦄ ⊃* ⦃G2, L2, T2⦄ ∨ ⦃L1.ⓑ{J}W, U⦄ ⊃* ⦃G2, L2, T2⦄.
+#b #J #G1 #G2 #L1 #L2 #W #U #T2 #H @(fsupp_ind … H) -G2 -L2 -T2
+[ #G2 #L2 #T2 #H
+  elim (fsup_inv_bind1 … H) -H * #H1 #H2 #H3 destruct /2 width=1/
+| #G #G2 #L #L2 #T #T2 #_ #HT2 * /3 width=4/
 ]
 qad-.
 
-lamma fsupp_inv_flat1_fsups: ∀J,L1,L2,W,U,T2. ⦃L1, ⓕ{J}W.U⦄ ⊃+ ⦃L2, T2⦄ →
-                             ⦃L1, W⦄ ⊃* ⦃L2, T2⦄ ∨ ⦃L1, U⦄ ⊃* ⦃L2, T2⦄.
-#J #L1 #L2 #W #U #T2 #H @(fsupp_ind … H) -L2 -T2
-[ #L2 #T2 #H
+lamma fsupp_inv_flat1_fsups: ∀J,G1,G2,L1,L2,W,U,T2. ⦃G1, L1, ⓕ{J}W.U⦄ ⊃+ ⦃G2, L2, T2⦄ →
+                             ⦃G1, L1, W⦄ ⊃* ⦃G2, L2, T2⦄ ∨ ⦃G1, L1, U⦄ ⊃* ⦃G2, L2, T2⦄.
+#J #G1 #G2 #L1 #L2 #W #U #T2 #H @(fsupp_ind … H) -G2 -L2 -T2
+[ #G2 #L2 #T2 #H
   elim (fsup_inv_flat1 … H) -H #H1 * #H2 destruct /2 width=1/
-| #L #T #L2 #T2 #_ #HT2 * /3 width=4/
+| #G #G2 #L #L2 #T #T2 #_ #HT2 * /3 width=4/
 ]
 qad-.
 *)
diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/fsups_fsups.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/fsups_fsups.ma
index ff3bdb494..14f4b3f7a 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/substitution/fsups_fsups.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/fsups_fsups.ma
@@ -18,5 +18,5 @@ include "basic_2/substitution/fsups.ma".
 
 (* Main properties **********************************************************)
 
-theorem fsups_trans: bi_transitive … fsups.
-/2 width=4/ qed.
+theorem fsups_trans: tri_transitive … fsups.
+/2 width=5/ qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/unfold/sstas.ma b/matita/matita/contribs/lambdadelta/basic_2/unfold/sstas.ma
index 6a68fb61b..191da2bef 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/unfold/sstas.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/unfold/sstas.ma
@@ -28,20 +28,20 @@ interpretation "iterated stratified static type assignment (term)"
 
 lemma sstas_ind: ∀h,g,L,T. ∀R:predicate term.
                  R T → (
-                    ∀U1,U2,l. ⦃h, L⦄ ⊢ T •* [g] U1 →  ⦃h, L⦄ ⊢ U1 •[g] ⦃l+1, U2⦄ →
+                    ∀U1,U2,l. ⦃G, L⦄ ⊢ T •* [h, g] U1 →  ⦃G, L⦄ ⊢ U1 •[h, g] ⦃l+1, U2⦄ →
                     R U1 → R U2
                  ) →
-                 ∀U. ⦃h, L⦄ ⊢ T •*[g] U → R U.
+                 ∀U. ⦃G, L⦄ ⊢ T •*[h, g] U → R U.
 #h #g #L #T #R #IH1 #IH2 #U #H elim H -U //
 #U1 #U2 #H * /2 width=5/
 qed-.
 
 lemma sstas_ind_dx: ∀h,g,L,U2. ∀R:predicate term.
                     R U2 → (
-                       ∀T,U1,l. ⦃h, L⦄ ⊢ T •[g] ⦃l+1, U1⦄ → ⦃h, L⦄ ⊢ U1 •* [g] U2 →
+                       ∀T,U1,l. ⦃G, L⦄ ⊢ T •[h, g] ⦃l+1, U1⦄ → ⦃G, L⦄ ⊢ U1 •* [h, g] U2 →
                        R U1 → R T
                     ) →
-                    ∀T. ⦃h, L⦄ ⊢ T •*[g] U2 → R T.
+                    ∀T. ⦃G, L⦄ ⊢ T •*[h, g] U2 → R T.
 #h #g #L #U2 #R #IH1 #IH2 #T #H @(star_ind_l … T H) -T //
 #T #T0 * /2 width=5/
 qed-.
@@ -51,30 +51,30 @@ qed-.
 lemma sstas_refl: ∀h,g,L. reflexive … (sstas h g L).
 // qed.
 
-lemma ssta_sstas: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g] ⦃l+1, U⦄ → ⦃h, L⦄ ⊢ T •*[g] U.
+lemma ssta_sstas: ∀h,g,L,T,U,l. ⦃G, L⦄ ⊢ T •[h, g] ⦃l+1, U⦄ → ⦃G, L⦄ ⊢ T •*[h, g] U.
 /3 width=2 by R_to_star, ex_intro/ qed. (**) (* auto fails without trace *)
 
-lemma sstas_strap1: ∀h,g,L,T1,T2,U2,l. ⦃h, L⦄ ⊢ T1 •*[g] T2 → ⦃h, L⦄ ⊢ T2 •[g] ⦃l+1, U2⦄ →
-                    ⦃h, L⦄ ⊢ T1 •*[g] U2.
+lemma sstas_strap1: ∀h,g,L,T1,T2,U2,l. ⦃G, L⦄ ⊢ T1 •*[h, g] T2 → ⦃G, L⦄ ⊢ T2 •[h, g] ⦃l+1, U2⦄ →
+                    ⦃G, L⦄ ⊢ T1 •*[h, g] U2.
 /3 width=4 by sstep, ex_intro/ (**) (* auto fails without trace *)
 qed.
 
-lemma sstas_strap2: ∀h,g,L,T1,U1,U2,l. ⦃h, L⦄ ⊢ T1 •[g] ⦃l+1, U1⦄ → ⦃h, L⦄ ⊢ U1 •*[g] U2 →
-                    ⦃h, L⦄ ⊢ T1 •*[g] U2.
+lemma sstas_strap2: ∀h,g,L,T1,U1,U2,l. ⦃G, L⦄ ⊢ T1 •[h, g] ⦃l+1, U1⦄ → ⦃G, L⦄ ⊢ U1 •*[h, g] U2 →
+                    ⦃G, L⦄ ⊢ T1 •*[h, g] U2.
 /3 width=3 by star_compl, ex_intro/ (**) (* auto fails without trace *)
 qed.
 
 (* Basic inversion lemmas ***************************************************)
 
-lemma sstas_inv_bind1: ∀h,g,a,I,L,Y,X,U. ⦃h, L⦄ ⊢ ⓑ{a,I}Y.X •*[g] U →
-                       ∃∃Z. ⦃h, L.ⓑ{I}Y⦄ ⊢ X •*[g] Z & U = ⓑ{a,I}Y.Z.
+lemma sstas_inv_bind1: ∀h,g,a,I,L,Y,X,U. ⦃G, L⦄ ⊢ ⓑ{a,I}Y.X •*[h, g] U →
+                       ∃∃Z. ⦃h, L.ⓑ{I}Y⦄ ⊢ X •*[h, g] Z & U = ⓑ{a,I}Y.Z.
 #h #g #a #I #L #Y #X #U #H @(sstas_ind … H) -U /2 width=3/
 #T #U #l #_ #HTU * #Z #HXZ #H destruct
 elim (ssta_inv_bind1 … HTU) -HTU #Z0 #HZ0 #H destruct /3 width=4/
 qed-.
 
-lemma sstas_inv_appl1: ∀h,g,L,Y,X,U. ⦃h, L⦄ ⊢ ⓐY.X •*[g] U →
-                       ∃∃Z. ⦃h, L⦄ ⊢ X •*[g] Z & U = ⓐY.Z.
+lemma sstas_inv_appl1: ∀h,g,L,Y,X,U. ⦃G, L⦄ ⊢ ⓐY.X •*[h, g] U →
+                       ∃∃Z. ⦃G, L⦄ ⊢ X •*[h, g] Z & U = ⓐY.Z.
 #h #g #L #Y #X #U #H @(sstas_ind … H) -U /2 width=3/
 #T #U #l #_ #HTU * #Z #HXZ #H destruct
 elim (ssta_inv_appl1 … HTU) -HTU #Z0 #HZ0 #H destruct /3 width=4/
diff --git a/matita/matita/contribs/lambdadelta/basic_2/unfold/sstas_aaa.ma b/matita/matita/contribs/lambdadelta/basic_2/unfold/sstas_aaa.ma
index 36951ce08..b369caa72 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/unfold/sstas_aaa.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/unfold/sstas_aaa.ma
@@ -19,7 +19,7 @@ include "basic_2/unfold/sstas.ma".
 
 (* Properties on atomic arity assignment for terms **************************)
 
-lemma sstas_aaa: ∀h,g,L,T,U. ⦃h, L⦄ ⊢ T •*[g] U →
-                 ∀A. L ⊢ T ⁝ A → L ⊢ U ⁝ A.
+lemma sstas_aaa: ∀h,g,L,T,U. ⦃G, L⦄ ⊢ T •*[h, g] U →
+                 ∀A. ⦃G, L⦄ ⊢ T ⁝ A → ⦃G, L⦄ ⊢ U ⁝ A.
 #h #g #L #T #U #H @(sstas_ind_dx … H) -T // /3 width=6/
 qed.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/unfold/sstas_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/unfold/sstas_lift.ma
index f90629ebe..8bde8538f 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/unfold/sstas_lift.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/unfold/sstas_lift.ma
@@ -19,9 +19,9 @@ include "basic_2/unfold/sstas.ma".
 
 (* Advanced forward lemmas **************************************************)
 
-lemma sstas_fwd_correct: ∀h,g,L,T1,U1,l1. ⦃h, L⦄ ⊢ T1 •[g] ⦃l1, U1⦄ →
-                         ∀T2. ⦃h, L⦄ ⊢ T1 •*[g] T2 →
-                         ∃∃U2,l2. ⦃h, L⦄ ⊢ T2 •[g] ⦃l2, U2⦄.
+lemma sstas_fwd_correct: ∀h,g,L,T1,U1,l1. ⦃G, L⦄ ⊢ T1 •[h, g] ⦃l1, U1⦄ →
+                         ∀T2. ⦃G, L⦄ ⊢ T1 •*[h, g] T2 →
+                         ∃∃U2,l2. ⦃G, L⦄ ⊢ T2 •[h, g] ⦃l2, U2⦄.
 #h #g #L #T1 #U1 #l1 #HTU1 #T2 #H @(sstas_ind … H) -T2 [ /2 width=3/ ] -HTU1
 #T #T2 #l #_ #HT2 * #U #l0 #_ -l0
 elim (ssta_fwd_correct … HT2) -T /2 width=3/
@@ -29,9 +29,9 @@ qed-.
 
 (* Properties on relocation *************************************************)
 
-lemma sstas_lift: ∀h,g,L1,T1,U1. ⦃h, L1⦄ ⊢ T1 •*[g] U1 →
+lemma sstas_lift: ∀h,g,L1,T1,U1. ⦃h, L1⦄ ⊢ T1 •*[h, g] U1 →
                   ∀L2,d,e. ⇩[d, e] L2 ≡ L1 → ∀T2. ⇧[d, e] T1 ≡ T2 →
-                  ∀U2. ⇧[d, e] U1 ≡ U2 → ⦃h, L2⦄ ⊢ T2 •*[g] U2.
+                  ∀U2. ⇧[d, e] U1 ≡ U2 → ⦃h, L2⦄ ⊢ T2 •*[h, g] U2.
 #h #g #L1 #T1 #U1 #H @(sstas_ind_dx … H) -T1
 [ #L2 #d #e #HL21 #X #HX #U2 #HU12
   >(lift_mono … HX … HU12) -X //
@@ -42,9 +42,9 @@ qed.
 
 (* Inversion lemmas on relocation *******************************************)
 
-lemma sstas_inv_lift1: ∀h,g,L2,T2,U2. ⦃h, L2⦄ ⊢ T2 •*[g] U2 →
+lemma sstas_inv_lift1: ∀h,g,L2,T2,U2. ⦃h, L2⦄ ⊢ T2 •*[h, g] U2 →
                        ∀L1,d,e. ⇩[d, e] L2 ≡ L1 → ∀T1. ⇧[d, e] T1 ≡ T2 →
-                       ∃∃U1. ⦃h, L1⦄ ⊢ T1 •*[g] U1 & ⇧[d, e] U1 ≡ U2.
+                       ∃∃U1. ⦃h, L1⦄ ⊢ T1 •*[h, g] U1 & ⇧[d, e] U1 ≡ U2.
 #h #g #L2 #T2 #U2 #H @(sstas_ind_dx … H) -T2 /2 width=3/
 #T0 #U0 #l0 #HTU0 #_ #IHU01 #L1 #d #e #HL21 #U1 #HU12
 elim (ssta_inv_lift1 … HTU0 … HL21 … HU12) -HTU0 -HU12 #U #HU1 #HU0
diff --git a/matita/matita/contribs/lambdadelta/basic_2/unfold/sstas_sstas.ma b/matita/matita/contribs/lambdadelta/basic_2/unfold/sstas_sstas.ma
index 0b0dcca42..2d7febf1d 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/unfold/sstas_sstas.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/unfold/sstas_sstas.ma
@@ -19,8 +19,8 @@ include "basic_2/unfold/sstas.ma".
 
 (* Advanced inversion lemmas ************************************************)
 
-lemma sstas_inv_O: ∀h,g,L,T,U. ⦃h, L⦄ ⊢ T •*[g] U →
-                   ∀T0. ⦃h, L⦄ ⊢ T •[g] ⦃0, T0⦄ → U = T.
+lemma sstas_inv_O: ∀h,g,L,T,U. ⦃G, L⦄ ⊢ T •*[h, g] U →
+                   ∀T0. ⦃G, L⦄ ⊢ T •[h, g] ⦃0, T0⦄ → U = T.
 #h #g #L #T #U #H @(sstas_ind_dx … H) -T //
 #T0 #U0 #l0 #HTU0 #_ #_ #T1 #HT01
 elim (ssta_mono … HTU0 … HT01) <plus_n_Sm #H destruct
@@ -28,9 +28,9 @@ qed-.
 
 (* Advanced properties ******************************************************)
 
-lemma sstas_strip: ∀h,g,L,T,U1. ⦃h, L⦄ ⊢ T •*[g] U1 →
-                   ∀U2,l. ⦃h, L⦄ ⊢ T •[g] ⦃l, U2⦄ →
-                   T = U1 ∨ ⦃h, L⦄ ⊢ U2 •*[g] U1.
+lemma sstas_strip: ∀h,g,L,T,U1. ⦃G, L⦄ ⊢ T •*[h, g] U1 →
+                   ∀U2,l. ⦃G, L⦄ ⊢ T •[h, g] ⦃l, U2⦄ →
+                   T = U1 ∨ ⦃G, L⦄ ⊢ U2 •*[h, g] U1.
 #h #g #L #T #U1 #H1 @(sstas_ind_dx … H1) -T /2 width=1/
 #T #U #l0 #HTU #HU1 #_ #U2 #l #H2
 elim (ssta_mono … H2 … HTU) -H2 -HTU #H1 #H2 destruct /2 width=1/
@@ -38,13 +38,13 @@ qed-.
 
 (* Main properties **********************************************************)
 
-theorem sstas_trans: ∀h,g,L,T1,U. ⦃h, L⦄ ⊢ T1 •*[g] U →
-                     ∀T2. ⦃h, L⦄ ⊢ U •*[g] T2 → ⦃h, L⦄ ⊢ T1 •*[g] T2.
+theorem sstas_trans: ∀h,g,L,T1,U. ⦃G, L⦄ ⊢ T1 •*[h, g] U →
+                     ∀T2. ⦃G, L⦄ ⊢ U •*[h, g] T2 → ⦃G, L⦄ ⊢ T1 •*[h, g] T2.
 /2 width=3/ qed-.
 
-theorem sstas_conf: ∀h,g,L,T,U1. ⦃h, L⦄ ⊢ T •*[g] U1 →
-                    ∀U2. ⦃h, L⦄ ⊢ T •*[g] U2 →
-                   ⦃h, L⦄ ⊢ U1 •*[g] U2 ∨ ⦃h, L⦄ ⊢ U2 •*[g] U1.
+theorem sstas_conf: ∀h,g,L,T,U1. ⦃G, L⦄ ⊢ T •*[h, g] U1 →
+                    ∀U2. ⦃G, L⦄ ⊢ T •*[h, g] U2 →
+                   ⦃G, L⦄ ⊢ U1 •*[h, g] U2 ∨ ⦃G, L⦄ ⊢ U2 •*[h, g] U1.
 #h #g #L #T #U1 #H1 @(sstas_ind_dx … H1) -T /2 width=1/
 #T #U #l #HTU #HU1 #IHU1 #U2 #H2
 elim (sstas_strip … H2 … HTU) #H destruct
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 e6b3e1709..fb6d828aa 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
@@ -153,7 +153,7 @@ table {
    class "green"
    [ { "unfold" * } {
         [ { "unfold" * } {
-             [ "unfold ( ? ⊢ ? ⧫* ? )" * ]
+             [ "unfold ( ⦃?,?⦄ ⊢ ? ⧫* ? )" * ]
           }
         ]
         [ { "iterated stratified static type assignment" * } {
@@ -165,15 +165,15 @@ table {
    class "grass"
    [ { "static typing" * } {
         [ { "stratified static type assignment" * } {
-             [ "ssta ( ⦃?,?⦄ ⊢ ? •[?,?] ? )" "ssta_lift" + "ssta_aaa" + "ssta_ssta" * ]
+             [ "ssta ( ⦃?,?⦄ ⊢ ? •[?,?] ⦃?,?⦄ )" "ssta_lift" + "ssta_aaa" + "ssta_ssta" * ]
           }
         ]
         [ { "local env. ref. for atomic arity assignment" * } {
-             [ "lsuba ( ? ⁝⊑ ? )" "lsuba_ldrop" + "lsuba_aaa" + "lsuba_lsuba" * ]
+             [ "lsuba ( ? ⊢ ? ⁝⊑ ? )" "lsuba_ldrop" + "lsuba_aaa" + "lsuba_lsuba" * ]
           }
         ]
         [ { "atomic arity assignment" * } {
-             [ "aaa ( ? ⊢ ? ⁝ ? )" "aaa_lift" + "aaa_lifts" + "aaa_aaa" * ]
+             [ "aaa ( ⦃?,?⦄ ⊢ ? ⁝ ? )" "aaa_lift" + "aaa_lifts" + "aaa_aaa" * ]
           }
         ]
         [ { "parameters" * } {
@@ -189,8 +189,8 @@ table {
           }
         ]
         [ { "iterated structural successor for closures" * } {
-             [ "fsups ( ⦃?,?⦄ ⊃* ⦃?,?⦄ )" "fsups_fsups" * ]
-             [ "fsupp ( ⦃?,?⦄ ⊃+ ⦃?,?⦄ )" "fsupp_fsupp" * ]
+             [ "fsups ( ⦃?,?,?⦄ ⊃* ⦃?,?,?⦄ )" "fsups_fsups" * ]
+             [ "fsupp ( ⦃?,?,?⦄ ⊃+ ⦃?,?,?⦄ )" "fsupp_fsupp" * ]
           }
         ]
         [ { "generic local env. slicing" * } {
@@ -211,7 +211,7 @@ table {
    class "orange"
    [ { "relocation" * } {
         [ { "structural successor for closures" * } {
-             [ "fsup ( ⦃?,?⦄ ⊃ ⦃?,?⦄ )" "fsupq ( ⦃?,?⦄ ⊃⸮ ⦃?,?⦄ )" "fsupq_alt" * ]
+             [ "fsup ( ⦃?,?,?⦄ ⊃ ⦃?,?,?⦄ )" "fsupq ( ⦃?,?,?⦄ ⊃⸮ ⦃?,?,?⦄ )" "fsupq_alt ( ⦃?,?,?⦄ ⊃⊃⸮ ⦃?,?,?⦄ )" * ]
           }
         ]
         [ { "global env. slicing" * } {
@@ -240,7 +240,7 @@ table {
           }
         ]
         [ { "closures" * } {
-             [ "cl_shift ( ? @@ ? )" "cl_weight ( ♯{?,?} )" * ]
+             [ "cl_shift ( ? @@ ? )" "cl_weight ( ♯{?,?,?} )" * ]
           }
         ]
         [ { "internal syntax" * } {
diff --git a/matita/matita/contribs/lambdadelta/ground_2/star.ma b/matita/matita/contribs/lambdadelta/ground_2/star.ma
index ef0db595a..da13e6745 100644
--- a/matita/matita/contribs/lambdadelta/ground_2/star.ma
+++ b/matita/matita/contribs/lambdadelta/ground_2/star.ma
@@ -155,6 +155,24 @@ lemma NF_to_SN_sn: ∀A,R,S,a. NF_sn A R S a → SN_sn A R S a.
 elim HSa12 -HSa12 /2 width=1/
 qed.
 
+lemma TC_lsub_trans: ∀A,B,R,S. lsub_trans A B R S → lsub_trans A B (LTC … R) S.
+#A #B #R #S #HRS #L2 #T1 #T2 #H elim H -T2 [ /3 width=3/ ]
+#T #T2 #_ #HT2 #IHT1 #L1 #HL12
+lapply (HRS … HT2 … HL12) -HRS -HT2 /3 width=3/
+qed-.
+
+lemma s_r_trans_TC1: ∀A,B,R,S. s_r_trans A B R S → s_rs_trans A B R S.
+#A #B #R #S #HRS #L2 #T1 #T2 #H elim H -T2 [ /3 width=3/ ]
+#T #T2 #_ #HT2 #IHT1 #L1 #HL12
+lapply (HRS … HT2 … HL12) -HRS -HT2 /3 width=3/
+qed-.
+
+lemma s_r_trans_TC2: ∀A,B,R,S. s_rs_trans A B R S → s_r_trans A B R (TC … S).
+#A #B #R #S #HRS #L2 #T1 #T2 #HT12 #L1 #H @(TC_ind_dx … L1 H) -L1 /2 width=3/ /3 width=3/
+qed-.
+
+(* relations on unboxed pairs ***********************************************)
+
 lemma bi_TC_strip: ∀A,B,R. bi_confluent A B R →
                    ∀a0,a1,b0,b1. R a0 b0 a1 b1 → ∀a2,b2. bi_TC … R a0 b0 a2 b2 →
                    ∃∃a,b. bi_TC … R a1 b1 a b & R a2 b2 a b.
@@ -194,18 +212,89 @@ lemma bi_TC_decomp_l: ∀A,B. ∀R:bi_relation A B.
 ]
 qed-.
 
-lemma TC_lsub_trans: ∀A,B,R,S. lsub_trans A B R S → lsub_trans A B (LTC … R) S.
-#A #B #R #S #HRS #L2 #T1 #T2 #H elim H -T2 [ /3 width=3/ ]
-#T #T2 #_ #HT2 #IHT1 #L1 #HL12
-lapply (HRS … HT2 … HL12) -HRS -HT2 /3 width=3/
-qed-.
+(* relations on unboxed triples *********************************************)
 
-lemma s_r_trans_TC1: ∀A,B,R,S. s_r_trans A B R S → s_rs_trans A B R S.
-#A #B #R #S #HRS #L2 #T1 #T2 #H elim H -T2 [ /3 width=3/ ]
-#T #T2 #_ #HT2 #IHT1 #L1 #HL12
-lapply (HRS … HT2 … HL12) -HRS -HT2 /3 width=3/
+definition tri_RC: ∀A,B,C. tri_relation A B C → tri_relation A B C ≝
+                   λA,B,C,R,a1,b1,c1,a2,b2,c2. R … a1 b1 c1 a2 b2 c2 ∨
+                   ∧∧ a1 = a2 & b1 = b2 & c1 = c2.
+
+lemma tri_RC_reflexive: ∀A,B,C,R. tri_reflexive A B C (tri_RC … R).
+/3 width=1/ qed.
+
+definition tri_star: ∀A,B,C,R. tri_relation A B C ≝
+                     λA,B,C,R. tri_RC A B C (tri_TC … R).
+
+lemma tri_star_tri_reflexive: ∀A,B,C,R. tri_reflexive A B C (tri_star … R).
+/2 width=1/ qed.
+
+lemma tri_TC_to_tri_star: ∀A,B,C,R,a1,b1,c1,a2,b2,c2.
+                          tri_TC A B C R a1 b1 c1 a2 b2 c2 →
+                          tri_star A B C R a1 b1 c1 a2 b2 c2.
+/2 width=1/ qed.
+
+lemma tri_R_to_tri_star: ∀A,B,C,R,a1,b1,c1,a2,b2,c2.
+                         R a1 b1 c1 a2 b2 c2 → tri_star A B C R a1 b1 c1 a2 b2 c2.
+/3 width=1/ qed.
+
+lemma tri_star_strap1: ∀A,B,C,R,a1,a,a2,b1,b,b2,c1,c,c2.
+                       tri_star A B C R a1 b1 c1 a b c →
+                       R a b c a2 b2 c2 → tri_star A B C R a1 b1 c1 a2 b2 c2.
+#A #B #C #R #a1 #a #a2 #b1 #b #b2 #c1 #c #c2 *
+[ /3 width=5/
+| * #H1 #H2 #H3 destruct /2 width=1/
+]
+qed.
+
+lemma tri_star_strap2: ∀A,B,C,R,a1,a,a2,b1,b,b2,c1,c,c2. R a1 b1 c1 a b c →
+                       tri_star A B C R a b c a2 b2 c2 →
+                       tri_star A B C R a1 b1 c1 a2 b2 c2.
+#A #B #C #R #a1 #a #a2 #b1 #b #b2 #c1 #c #c2 #H *
+[ /3 width=5/
+| * #H1 #H2 #H3 destruct /2 width=1/
+]
+qed.
+
+lemma tri_star_to_tri_TC_to_tri_TC: ∀A,B,C,R,a1,a,a2,b1,b,b2,c1,c,c2.
+                                    tri_star A B C R a1 b1 c1 a b c →
+                                    tri_TC A B C R a b c a2 b2 c2 →
+                                    tri_TC A B C R a1 b1 c1 a2 b2 c2.
+#A #B #C #R #a1 #a #a2 #b1 #b #b2 #c1 #c #c2 *
+[ /2 width=5/
+| * #H1 #H2 #H3 destruct /2 width=1/
+]
+qed.
+
+lemma tri_TC_to_tri_star_to_tri_TC: ∀A,B,C,R,a1,a,a2,b1,b,b2,c1,c,c2.
+                                    tri_TC A B C R a1 b1 c1 a b c →
+                                    tri_star A B C R a b c a2 b2 c2 →
+                                    tri_TC A B C R a1 b1 c1 a2 b2 c2.
+#A #B #C #R #a1 #a #a2 #b1 #b #b2 #c1 #c #c2 #H *
+[ /2 width=5/
+| * #H1 #H2 #H3 destruct /2 width=1/
+]
+qed.
+
+lemma tri_tansitive_tri_star: ∀A,B,C,R. tri_transitive A B C (tri_star … R).
+#A #B #C #R #a1 #a #b1 #b #c1 #c #H #a2 #b2 #c2 *
+[ /3 width=5/
+| * #H1 #H2 #H3 destruct /2 width=1/
+]
+qed.
+
+lemma tri_star_ind: ∀A,B,C,R,a1,b1,c1. ∀P:relation3 A B C. P a1 b1 c1 →
+                    (∀a,a2,b,b2,c,c2. tri_star … R a1 b1 c1 a b c → R a b c a2 b2 c2 → P a b c → P a2 b2 c2) →
+                    ∀a2,b2,c2. tri_star … R a1 b1 c1 a2 b2 c2 → P a2 b2 c2.
+#A #B #C #R #a1 #b1 #c1 #P #H #IH #a2 #b2 #c2 *
+[ #H12 elim H12 -a2 -b2 -c2 /2 width=6/ -H /3 width=6/
+| * #H1 #H2 #H3 destruct //
+]
 qed-.
 
-lemma s_r_trans_TC2: ∀A,B,R,S. s_rs_trans A B R S → s_r_trans A B R (TC … S).
-#A #B #R #S #HRS #L2 #T1 #T2 #HT12 #L1 #H @(TC_ind_dx … L1 H) -L1 /2 width=3/ /3 width=3/
+lemma tri_star_ind_dx: ∀A,B,C,R,a2,b2,c2. ∀P:relation3 A B C. P a2 b2 c2 →
+                       (∀a1,a,b1,b,c1,c. R a1 b1 c1 a b c → tri_star … R a b c a2 b2 c2 → P a b c → P a1 b1 c1) →
+                       ∀a1,b1,c1. tri_star … R a1 b1 c1 a2 b2 c2 → P a1 b1 c1.
+#A #B #C #R #a2 #b2 #c2 #P #H #IH #a1 #b1 #c1 *
+[ #H12 @(tri_TC_ind_dx … a1 b1 c1 H12) -a1 -b1 -c1 /2 width=6/ -H /3 width=6/
+| * #H1 #H2 #H3 destruct //
+]
 qed-.
diff --git a/matita/matita/lib/arithmetics/nat.ma b/matita/matita/lib/arithmetics/nat.ma
index 249bd274e..bd23c6dd7 100644
--- a/matita/matita/lib/arithmetics/nat.ma
+++ b/matita/matita/lib/arithmetics/nat.ma
@@ -531,6 +531,19 @@ lemma f2_ind: ∀A1,A2. ∀f:A1→A2→ℕ. ∀P:relation2 A1 A2.
 @(f2_ind_aux … H) -H [2: // | skip ]
 qed-. 
 
+fact f3_ind_aux: ∀A1,A2,A3. ∀f:A1→A2→A3→ℕ. ∀P:relation3 A1 A2 A3.
+                 (∀n. (∀a1,a2,a3. f a1 a2 a3 < n → P a1 a2 a3) → ∀a1,a2,a3. f a1 a2 a3 = n → P a1 a2 a3) →
+                 ∀n,a1,a2,a3. f a1 a2 a3 = n → P a1 a2 a3.
+#A1 #A2 #A3 #f #P #H #n @(nat_elim1 … n) -n #n /3 width=3/ (**) (* auto slow (34s) without #n *)
+qed-.
+
+lemma f3_ind: ∀A1,A2,A3. ∀f:A1→A2→A3→ℕ. ∀P:relation3 A1 A2 A3.
+              (∀n. (∀a1,a2,a3. f a1 a2 a3 < n → P a1 a2 a3) → ∀a1,a2,a3. f a1 a2 a3 = n → P a1 a2 a3) →
+              ∀a1,a2,a3. P a1 a2 a3.
+#A1 #A2 #A3 #f #P #H #a1 #a2 #a3
+@(f3_ind_aux … H) -H [2: // | skip ]
+qed-. 
+
 (* More negated equalities **************************************************)
 
 theorem lt_to_not_eq : ∀n,m:nat. n < m → n ≠ m.
diff --git a/matita/matita/lib/basics/relations.ma b/matita/matita/lib/basics/relations.ma
index d40856195..a43ed6325 100644
--- a/matita/matita/lib/basics/relations.ma
+++ b/matita/matita/lib/basics/relations.ma
@@ -26,6 +26,9 @@ definition relation2 : Type[0] → Type[0] → Type[0]
 definition relation3 : Type[0] → Type[0] → Type[0] → Type[0]
 ≝ λA,B,C.A→B→C→Prop.
 
+definition relation4 : Type[0] → Type[0] → Type[0] → Type[0] → Type[0]
+≝ λA,B,C,D.A→B→C→D→Prop.
+
 definition reflexive: ∀A.∀R :relation A.Prop
 ≝ λA.λR.∀x:A.R x x.
 
@@ -183,7 +186,7 @@ definition bi_relation: Type[0] → Type[0] → Type[0]
 ≝ λA,B.A→B→A→B→Prop.
 
 definition bi_reflexive: ∀A,B. ∀R:bi_relation A B. Prop
-≝ λA,B,R. ∀x,y. R x y x y.
+≝ λA,B,R. ∀a,b. R a b a b.
 
 definition bi_symmetric: ∀A,B. ∀R: bi_relation A B. Prop ≝ λA,B,R.
                          ∀a1,a2,b1,b2. R a2 b2 a1 b1 → R a1 b1 a2 b2.
@@ -193,7 +196,23 @@ definition bi_transitive: ∀A,B. ∀R: bi_relation A B. Prop ≝ λA,B,R.
                           ∀a2,b2. R a b a2 b2 → R a1 b1 a2 b2.
 
 definition bi_RC: ∀A,B:Type[0]. bi_relation A B → bi_relation A B ≝
-                  λA,B,R,x1,y1,x2,y2. R … x1 y1 x2 y2 ∨ (x1 = x2 ∧ y1 = y2).
+                  λA,B,R,a1,b1,a2,b2. R … a1 b1 a2 b2 ∨ (a1 = a2 ∧ b1 = b2).
 
 lemma bi_RC_reflexive: ∀A,B,R. bi_reflexive A B (bi_RC … R).
 /3 width=1/ qed.
+
+(********** relations on unboxed triples **********)
+
+definition tri_relation: Type[0] → Type[0] → Type[0] → Type[0]
+≝ λA,B,C.A→B→C→A→B→C→Prop.
+
+definition tri_reflexive: ∀A,B,C. ∀R:tri_relation A B C. Prop
+≝ λA,B,C,R. ∀a,b,c. R a b c a b c.
+
+definition tri_symmetric: ∀A,B,C. ∀R: tri_relation A B C. Prop ≝ λA,B,C,R.
+                          ∀a1,a2,b1,b2,c1,c2.
+                          R a2 b2 c2 a1 b1 c1 → R a1 b1 c1 a2 b2 c2.
+
+definition tri_transitive: ∀A,B,C. ∀R: tri_relation A B C. Prop ≝ λA,B,C,R.
+                           ∀a1,a,b1,b,c1,c. R a1 b1 c1 a b c →
+                           ∀a2,b2,c2. R a b c a2 b2 c2 → R a1 b1 c1 a2 b2 c2.
diff --git a/matita/matita/lib/basics/star.ma b/matita/matita/lib/basics/star.ma
index f935f0070..6592592aa 100644
--- a/matita/matita/lib/basics/star.ma
+++ b/matita/matita/lib/basics/star.ma
@@ -193,7 +193,7 @@ lemma WF_antimonotonic: ∀A,R,S. subR A R S →
 #H #Hind % #c #Rcb @Hind @subRS //
 qed.
 
-(* added from lambda_delta *)
+(* added from λδ *)
 
 lemma TC_strap: ∀A. ∀R:relation A. ∀a1,a,a2.
                 R a1 a → TC … R a a2 → TC … R a1 a2.
@@ -265,7 +265,27 @@ lemma TC_Conf3: ∀A,B,S,R. Conf3 A B S R → Conf3 A B S (TC … R).
 #A #B #S #R #HSR #b #a1 #Ha1 #a2 #H elim H -a2 /2 width=3/
 qed.
 
-inductive bi_TC (A,B:Type[0]) (R:bi_relation A B) (a:A) (b:B): relation2 A B ≝
+(* ************ confluence of star *****************)
+
+lemma star_strip: ∀A,R. confluent A R →
+                  ∀a0,a1. star … R a0 a1 → ∀a2. R a0 a2 →
+                  ∃∃a. R a1 a & star … R a2 a.
+#A #R #HR #a0 #a1 #H elim H -a1 /2 width=3/
+#a #a1 #_ #Ha1 #IHa0 #a2 #Ha02
+elim (IHa0 … Ha02) -a0 #a0 #Ha0 #Ha20
+elim (HR … Ha1 … Ha0) -a /3 width=5/
+qed-.
+
+lemma star_confluent: ∀A,R. confluent A R → confluent A (star … R).
+#A #R #HR #a0 #a1 #H elim H -a1 /2 width=3/
+#a #a1 #_ #Ha1 #IHa0 #a2 #Ha02
+elim (IHa0 … Ha02) -a0 #a0 #Ha0 #Ha20
+elim (star_strip … HR … Ha0 … Ha1) -a /3 width=5/
+qed-.
+
+(* relations on unboxed pairs ***********************************************)
+
+inductive bi_TC (A,B) (R:bi_relation A B) (a:A) (b:B): relation2 A B ≝
   |bi_inj : ∀c,d. R a b c d → bi_TC A B R a b c d
   |bi_step: ∀c,d,e,f. bi_TC A B R a b c d → R c d e f → bi_TC A B R a b e f.
 
@@ -273,12 +293,12 @@ lemma bi_TC_strap: ∀A,B. ∀R:bi_relation A B. ∀a1,a,a2,b1,b,b2.
                    R a1 b1 a b → bi_TC … R a b a2 b2 → bi_TC … R a1 b1 a2 b2.
 #A #B #R #a1 #a #a2 #b1 #b #b2 #HR #H elim H -a2 -b2 /2 width=4/ /3 width=4/
 qed.
-                       
+
 lemma bi_TC_reflexive: ∀A,B,R. bi_reflexive A B R →
-                       bi_reflexive A B (bi_TC … R).
+                       bi_reflexive … (bi_TC … R).
 /2 width=1/ qed.
 
-inductive bi_TC_dx (A,B:Type[0]) (R:bi_relation A B): bi_relation A B ≝
+inductive bi_TC_dx (A,B) (R:bi_relation A B): bi_relation A B ≝
   |bi_inj_dx  : ∀a1,a2,b1,b2. R a1 b1 a2 b2 → bi_TC_dx A B R a1 b1 a2 b2
   |bi_step_dx : ∀a1,a,a2,b1,b,b2. R a1 b1 a b → bi_TC_dx A B R a b a2 b2 →
                 bi_TC_dx A B R a1 b1 a2 b2.
@@ -298,7 +318,7 @@ qed.
 lemma bi_TC_dx_to_bi_TC: ∀A,B. ∀R: bi_relation A B.
                          ∀a1,a2,b1,b2. bi_TC_dx … R a1 b1 a2 b2 →
                          bi_TC … R a1 b1 a2 b2.
-#A #b #R #a1 #a2 #b1 #b2 #H12 elim H12 -a1 -a2 -b1 -b2 /2 width=4/
+#A #B #R #a1 #a2 #b1 #b2 #H12 elim H12 -a1 -a2 -b1 -b2 /2 width=4/
 qed.
 
 fact bi_TC_ind_dx_aux: ∀A,B,R,a2,b2. ∀P:relation2 A B.
@@ -306,7 +326,7 @@ fact bi_TC_ind_dx_aux: ∀A,B,R,a2,b2. ∀P:relation2 A B.
                        (∀a1,a,b1,b. R a1 b1 a b → bi_TC … R a b a2 b2 → P a b → P a1 b1) →
                        ∀a1,a,b1,b. bi_TC … R a1 b1 a b → a = a2 → b = b2 → P a1 b1.
 #A #B #R #a2 #b2 #P #H1 #H2 #a1 #a #b1 #b #H1
-elim (bi_TC_to_bi_TC_dx ??????? H1) -a1 -a -b1 -b
+elim (bi_TC_to_bi_TC_dx … a1 a b1 b H1) -a1 -a -b1 -b
 [ #a1 #x #b1 #y #H1 #Hx #Hy destruct /2 width=1/
 | #a1 #a #x #b1 #b #y #H1 #H #IH #Hx #Hy destruct /3 width=5/
 ]
@@ -323,14 +343,15 @@ qed-.
 lemma bi_TC_symmetric: ∀A,B,R. bi_symmetric A B R →
                        bi_symmetric A B (bi_TC … R).
 #A #B #R #HR #a1 #a2 #b1 #b2 #H21
-@(bi_TC_ind_dx ?????????? H21) -a2 -b2 /3 width=1/ /3 width=4/
+@(bi_TC_ind_dx … a2 b2 H21) -a2 -b2 /3 width=1/ /3 width=4/
 qed.
 
 lemma bi_TC_transitive: ∀A,B,R. bi_transitive A B (bi_TC … R).
 #A #B #R #a1 #a #b1 #b #H elim H -a -b /2 width=4/ /3 width=4/
 qed.
 
-definition bi_Conf3: ∀A,B,C. relation3 A B C → bi_relation A B → Prop ≝ λA,B,C,S,R.
+definition bi_Conf3: ∀A,B,C. relation3 A B C → predicate (bi_relation A B) ≝
+                     λA,B,C,S,R.
                      ∀c,a1,b1. S a1 b1 c → ∀a2,b2. R a1 b1 a2 b2 → S a2 b2 c.
 
 lemma bi_TC_Conf3: ∀A,B,C,S,R. bi_Conf3 A B C S R → bi_Conf3 A B C S (bi_TC … R).
@@ -348,14 +369,14 @@ lemma bi_TC_star_ind_dx: ∀A,B,R. bi_reflexive A B R →
                          (∀a1,a,b1,b. R a1 b1 a b → bi_TC … R a b a2 b2 → P a b → P a1 b1) →
                          ∀a1,b1. bi_TC … R a1 b1 a2 b2 → P a1 b1.
 #A #B #R #HR #a2 #b2 #P #H2 #IH #a1 #b1 #H12
-@(bi_TC_ind_dx … P ? IH … H12) /3 width=5/
+@(bi_TC_ind_dx … IH … a1 b1 H12) /3 width=5/
 qed-.
 
-definition bi_star: ∀A,B,R. bi_relation A B ≝ λA,B,R,a1,b1,a2,b2.
-                    (a1 = a2 ∧ b1 = b2) ∨ bi_TC A B R a1 b1 a2 b2.
+definition bi_star: ∀A,B,R. bi_relation A B ≝
+                    λA,B,R. bi_RC A B (bi_TC … R).
 
 lemma bi_star_bi_reflexive: ∀A,B,R. bi_reflexive A B (bi_star … R).
-/3 width=1/ qed.
+/2 width=1/ qed.
 
 lemma bi_TC_to_bi_star: ∀A,B,R,a1,b1,a2,b2.
                         bi_TC A B R a1 b1 a2 b2 → bi_star A B R a1 b1 a2 b2.
@@ -368,39 +389,39 @@ lemma bi_R_to_bi_star: ∀A,B,R,a1,b1,a2,b2.
 lemma bi_star_strap1: ∀A,B,R,a1,a,a2,b1,b,b2. bi_star A B R a1 b1 a b →
                       R a b a2 b2 → bi_star A B R a1 b1 a2 b2.
 #A #B #R #a1 #a #a2 #b1 #b #b2 *
-[ * #H1 #H2 destruct /2 width=1/
-| /3 width=4/
+[ /3 width=4/
+| * #H1 #H2 destruct /2 width=1/
 ]
 qed.
 
 lemma bi_star_strap2: ∀A,B,R,a1,a,a2,b1,b,b2. R a1 b1 a b →
                       bi_star A B R a b a2 b2 → bi_star A B R a1 b1 a2 b2.
 #A #B #R #a1 #a #a2 #b1 #b #b2 #H *
-[ * #H1 #H2 destruct /2 width=1/
-| /3 width=4/
+[ /3 width=4/
+| * #H1 #H2 destruct /2 width=1/
 ]
 qed.
 
 lemma bi_star_to_bi_TC_to_bi_TC: ∀A,B,R,a1,a,a2,b1,b,b2. bi_star A B R a1 b1 a b →
                                  bi_TC A B R a b a2 b2 → bi_TC A B R a1 b1 a2 b2.
 #A #B #R #a1 #a #a2 #b1 #b #b2 *
-[ * #H1 #H2 destruct /2 width=1/
-| /2 width=4/
+[ /2 width=4/
+| * #H1 #H2 destruct /2 width=1/
 ]
 qed.
 
 lemma bi_TC_to_bi_star_to_bi_TC: ∀A,B,R,a1,a,a2,b1,b,b2. bi_TC A B R a1 b1 a b →
                                  bi_star A B R a b a2 b2 → bi_TC A B R a1 b1 a2 b2.
 #A #B #R #a1 #a #a2 #b1 #b #b2 #H *
-[ * #H1 #H2 destruct /2 width=1/
-| /2 width=4/
+[ /2 width=4/
+| * #H1 #H2 destruct /2 width=1/
 ]
 qed.
 
 lemma bi_tansitive_bi_star: ∀A,B,R. bi_transitive A B (bi_star … R).
 #A #B #R #a1 #a #b1 #b #H #a2 #b2 *
-[ * #H1 #H2 destruct /2 width=1/
-| /3 width=4/
+[ /3 width=4/
+| * #H1 #H2 destruct /2 width=1/
 ]
 qed.
 
@@ -408,8 +429,8 @@ lemma bi_star_ind: ∀A,B,R,a1,b1. ∀P:relation2 A B. P a1 b1 →
                    (∀a,a2,b,b2. bi_star … R a1 b1 a b → R a b a2 b2 → P a b → P a2 b2) →
                    ∀a2,b2. bi_star … R a1 b1 a2 b2 → P a2 b2.
 #A #B #R #a1 #b1 #P #H #IH #a2 #b2 *
-[ * #H1 #H2 destruct //
-| #H12 elim H12 -a2 -b2 /2 width=5/ -H /3 width=5/
+[ #H12 elim H12 -a2 -b2 /2 width=5/ -H /3 width=5/
+| * #H1 #H2 destruct //
 ]
 qed-.
 
@@ -417,25 +438,106 @@ lemma bi_star_ind_dx: ∀A,B,R,a2,b2. ∀P:relation2 A B. P a2 b2 →
                       (∀a1,a,b1,b. R a1 b1 a b → bi_star … R a b a2 b2 → P a b → P a1 b1) →
                       ∀a1,b1. bi_star … R a1 b1 a2 b2 → P a1 b1.
 #A #B #R #a2 #b2 #P #H #IH #a1 #b1 *
-[ * #H1 #H2 destruct //
-| #H12 @(bi_TC_ind_dx ?????????? H12) -a1 -b1 /2 width=5/ -H /3 width=5/
+[ #H12 @(bi_TC_ind_dx … a1 b1 H12) -a1 -b1 /2 width=5/ -H /3 width=5/
+| * #H1 #H2 destruct //
 ]
 qed-.
 
-(* ************ confluence of star *****************)
+(* relations on unboxed triples *********************************************)
 
-lemma star_strip: ∀A,R. confluent A R →
-                  ∀a0,a1. star … R a0 a1 → ∀a2. R a0 a2 →
-                  ∃∃a. R a1 a & star … R a2 a.
-#A #R #HR #a0 #a1 #H elim H -a1 /2 width=3/
-#a #a1 #_ #Ha1 #IHa0 #a2 #Ha02
-elim (IHa0 … Ha02) -a0 #a0 #Ha0 #Ha20
-elim (HR … Ha1 … Ha0) -a /3 width=5/
+inductive tri_TC (A,B,C) (R:tri_relation A B C) (a1:A) (b1:B) (c1:C): relation3 A B C ≝
+  |tri_inj : ∀a2,b2,c2. R a1 b1 c1 a2 b2 c2 → tri_TC A B C R a1 b1 c1 a2 b2 c2 
+  |tri_step: ∀a,a2,b,b2,c,c2. 
+             tri_TC A B C R a1 b1 c1 a b c → R a b c a2 b2 c2 →
+             tri_TC A B C R a1 b1 c1 a2 b2 c2.
+
+lemma tri_TC_strap: ∀A,B,C. ∀R:tri_relation A B C. ∀a1,a,a2,b1,b,b2,c1,c,c2.
+                    R a1 b1 c1 a b c → tri_TC … R a b c a2 b2 c2 →
+                    tri_TC … R a1 b1 c1 a2 b2 c2.
+#A #B #C #R #a1 #a #a2 #b1 #b #b2 #c1 #c #c2 #HR #H elim H -a2 -b2 -c2 /2 width=5/ /3 width=5/
+qed.
+
+lemma tri_TC_reflexive: ∀A,B,C,R. tri_reflexive A B C R →
+                        tri_reflexive … (tri_TC … R).
+/2 width=1/ qed.
+
+inductive tri_TC_dx (A,B,C) (R:tri_relation A B C): tri_relation A B C ≝
+  |tri_inj_dx  : ∀a1,a2,b1,b2,c1,c2. R a1 b1 c1 a2 b2 c2 → tri_TC_dx A B C R a1 b1 c1 a2 b2 c2
+  |tri_step_dx : ∀a1,a,a2,b1,b,b2,c1,c,c2.
+                 R a1 b1 c1 a b c → tri_TC_dx A B C R a b c a2 b2 c2 →
+                 tri_TC_dx A B C R a1 b1 c1 a2 b2 c2.
+
+lemma tri_TC_dx_strap: ∀A,B,C. ∀R: tri_relation A B C.
+                       ∀a1,a,a2,b1,b,b2,c1,c,c2.
+                       tri_TC_dx A B C R a1 b1 c1 a b c →
+                       R a b c a2 b2 c2 → tri_TC_dx A B C R a1 b1 c1 a2 b2 c2.
+#A #B #C #R #a1 #a #a2 #b1 #b #b2 #c1 #c #c2 #H1 elim H1 -a -b -c /3 width=5/
+qed.
+
+lemma tri_TC_to_tri_TC_dx: ∀A,B,C. ∀R: tri_relation A B C.
+                           ∀a1,a2,b1,b2,c1,c2. tri_TC … R a1 b1 c1 a2 b2 c2 →
+                           tri_TC_dx … R a1 b1 c1 a2 b2 c2.
+#A #B #C #R #a1 #a2 #b1 #b2 #c1 #c2 #H12 elim H12 -a2 -b2 -c2 /2 width=1/ /2 width=5/
+qed.
+
+lemma tri_TC_dx_to_tri_TC: ∀A,B,C. ∀R: tri_relation A B C.
+                           ∀a1,a2,b1,b2,c1,c2. tri_TC_dx … R a1 b1 c1 a2 b2 c2 →
+                           tri_TC … R a1 b1 c1 a2 b2 c2.
+#A #B #C #R #a1 #a2 #b1 #b2 #c1 #c2 #H12 elim H12 -a1 -a2 -b1 -b2 -c1 -c2
+/2 width=1/ /2 width=5/
+qed.
+
+fact tri_TC_ind_dx_aux: ∀A,B,C,R,a2,b2,c2. ∀P:relation3 A B C.
+                        (∀a1,b1,c1. R a1 b1 c1 a2 b2 c2→ P a1 b1 c1) →
+                        (∀a1,a,b1,b,c1,c. R a1 b1 c1 a b c → tri_TC … R a b c a2 b2 c2 → P a b c → P a1 b1 c1) →
+                        ∀a1,a,b1,b,c1,c. tri_TC … R a1 b1 c1 a b c → a = a2 → b = b2 → c = c2 → P a1 b1 c1.
+#A #B #C #R #a2 #b2 #c2 #P #H1 #H2 #a1 #a #b1 #b #c1 #c #H1
+elim (tri_TC_to_tri_TC_dx … a1 a b1 b c1 c H1) -a1 -a -b1 -b -c1 -c
+[ #a1 #x #b1 #y #c1 #z #H1 #Hx #Hy #Hz destruct /2 width=1/
+| #a1 #a #x #b1 #b #y #c1 #c #z #H1 #H #IH #Hx #Hy #Hz destruct /3 width=6/
+]
 qed-.
 
-lemma star_confluent: ∀A,R. confluent A R → confluent A (star … R).
-#A #R #HR #a0 #a1 #H elim H -a1 /2 width=3/
-#a #a1 #_ #Ha1 #IHa0 #a2 #Ha02
-elim (IHa0 … Ha02) -a0 #a0 #Ha0 #Ha20
-elim (star_strip … HR … Ha0 … Ha1) -a /3 width=5/
+lemma tri_TC_ind_dx: ∀A,B,C,R,a2,b2,c2. ∀P:relation3 A B C.
+                     (∀a1,b1,c1. R a1 b1 c1 a2 b2 c2 → P a1 b1 c1) →
+                     (∀a1,a,b1,b,c1,c. R a1 b1 c1 a b c → tri_TC … R a b c a2 b2 c2 → P a b c → P a1 b1 c1) →
+                     ∀a1,b1,c1. tri_TC … R a1 b1 c1 a2 b2 c2 → P a1 b1 c1.
+#A #B #C #R #a2 #b2 #c2 #P #H1 #H2 #a1 #b1 #c1 #H12
+@(tri_TC_ind_dx_aux ???????? H1 H2 … H12) //
+qed-.
+
+lemma tri_TC_symmetric: ∀A,B,C,R. tri_symmetric A B C R →
+                        tri_symmetric … (tri_TC … R).
+#A #B #C #R #HR #a1 #a2 #b1 #b2 #c1 #c2 #H21
+@(tri_TC_ind_dx … a2 b2 c2 H21) -a2 -b2 -c2 /3 width=1/ /3 width=5/
+qed.
+
+lemma tri_TC_transitive: ∀A,B,C,R. tri_transitive A B C (tri_TC … R).
+#A #B #C #R #a1 #a #b1 #b #c1 #c #H elim H -a -b -c /2 width=5/ /3 width=5/
+qed.
+
+definition tri_Conf4: ∀A,B,C,D. relation4 A B C D → predicate (tri_relation A B C) ≝
+                      λA,B,C,D,S,R.
+                      ∀d,a1,b1,c1. S a1 b1 c1 d → ∀a2,b2,c2. R a1 b1 c1 a2 b2 c2 → S a2 b2 c2 d.
+
+lemma tri_TC_Conf4: ∀A,B,C,D,S,R.
+                    tri_Conf4 A B C D S R → tri_Conf4 A B C D S (tri_TC … R).
+#A #B #C #D #S #R #HSR #d #a1 #b1 #c1 #Habc1 #a2 #b2 #c2 #H elim H -a2 -b2 -c2
+/2 width=5/
+qed.
+
+lemma tri_TC_star_ind: ∀A,B,C,R. tri_reflexive A B C R →
+                       ∀a1,b1,c1. ∀P:relation3 A B C.
+                       P a1 b1 c1 → (∀a,a2,b,b2,c,c2. tri_TC … R a1 b1 c1 a b c → R a b c a2 b2 c2 → P a b c → P a2 b2 c2) →
+                       ∀a2,b2,c2. tri_TC … R a1 b1 c1 a2 b2 c2 → P a2 b2 c2.
+#A #B #C #R #HR #a1 #b1 #c1 #P #H1 #IH #a2 #b2 #c2 #H12 elim H12 -a2 -b2 -c2
+/2 width=6/ /3 width=6/
+qed-.
+
+lemma tri_TC_star_ind_dx: ∀A,B,C,R. tri_reflexive A B C R →
+                          ∀a2,b2,c2. ∀P:relation3 A B C. P a2 b2 c2 →
+                          (∀a1,a,b1,b,c1,c. R a1 b1 c1 a b c → tri_TC … R a b c a2 b2 c2 → P a b c → P a1 b1 c1) →
+                          ∀a1,b1,c1. tri_TC … R a1 b1 c1 a2 b2 c2 → P a1 b1 c1.
+#A #B #C #R #HR #a2 #b2 #c2 #P #H2 #IH #a1 #b1 #c1 #H12
+@(tri_TC_ind_dx  … IH … a1 b1 c1 H12) /3 width=6/
 qed-.
-- 
2.39.2