From: Ferruccio Guidi <ferruccio.guidi@unibo.it>
Date: Thu, 6 Dec 2012 16:10:57 +0000 (+0000)
Subject: - we enabled a notation for ex2
X-Git-Tag: make_still_working~1409
X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=commitdiff_plain;h=cdcfe9f97936f02dab1970ebf3911940bf0a4e29;p=helm.git

- we enabled a notation for ex2
- we improved star_ind_l that now behaves as if starl were defined
with a "left parameter" occurring on the right. this makes star_ind_l
symmetric with respect to star_ind
- we commented the non-more-compiling sections of "turing"
- some progress in "lambda"
---

diff --git a/matita/matita/contribs/lambda/delifting_substitution.ma b/matita/matita/contribs/lambda/delifting_substitution.ma
index 38087f4fe..79a65396a 100644
--- a/matita/matita/contribs/lambda/delifting_substitution.ma
+++ b/matita/matita/contribs/lambda/delifting_substitution.ma
@@ -149,3 +149,16 @@ definition dsubstable_sn: predicate (relation term) ≝ λR.
 
 definition dsubstable: predicate (relation term) ≝ λR.
                        ∀D1,D2. R D1 D2 → ∀M1,M2. R M1 M2 → ∀d. R ([d ⬐ D1] M1) ([d ⬐ D2] M2).
+
+lemma star_dsubstable_dx: ∀R. dsubstable_dx R → dsubstable_dx (star … R).
+#R #HR #D #M1 #M2 #H elim H -M2 // /3 width=3/
+qed.
+
+lemma star_dsubstable_sn: ∀R. dsubstable_sn R → dsubstable_sn (star … R).
+#R #HR #D1 #D2 #H elim H -D2 // /3 width=3/
+qed.
+
+lemma lstar_dsubstable_dx: ∀T,R. (∀t. dsubstable_dx (R t)) →
+                           ∀l. dsubstable_dx (lstar T … R l).
+#T #R #HR #l #D #M1 #M2 #H elim H -l -M1 -M2 // /3 width=3/
+qed.
diff --git a/matita/matita/contribs/lambda/hap_computation.ma b/matita/matita/contribs/lambda/hap_computation.ma
deleted file mode 100644
index 869fa94d4..000000000
--- a/matita/matita/contribs/lambda/hap_computation.ma
+++ /dev/null
@@ -1,23 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-include "hap_reduction.ma".
-
-(* KASHIMA'S "HAP" COMPUTATION **********************************************)
-
-(* Note: this is the "head in application" computation of:
-         R. Kashima: "A proof of the Standization Theorem in λ-Calculus". Typescript note, (2000).
-*)
-definition hap: relation term ≝ star … hap1.
-
diff --git a/matita/matita/contribs/lambda/hap_reduction.ma b/matita/matita/contribs/lambda/hap_reduction.ma
deleted file mode 100644
index 7b07524fc..000000000
--- a/matita/matita/contribs/lambda/hap_reduction.ma
+++ /dev/null
@@ -1,53 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||M||                                                             *)
-(*      ||A||       A project by Andrea Asperti                           *)
-(*      ||T||                                                             *)
-(*      ||I||       Developers:                                           *)
-(*      ||T||         The HELM team.                                      *)
-(*      ||A||         http://helm.cs.unibo.it                             *)
-(*      \   /                                                             *)
-(*       \ /        This file is distributed under the terms of the       *)
-(*        v         GNU General Public License Version 2                  *)
-(*                                                                        *)
-(**************************************************************************)
-
-include "labelled_sequential_reduction.ma".
-
-(* KASHIMA'S "HAP" COMPUTATION (SINGLE STEP) ********************************)
-
-(* Note: this is one step of the "head in application" computation of:
-         R. Kashima: "A proof of the Standization Theorem in λ-Calculus". Typescript note, (2000).
-*)
-inductive hap1: relation term ≝
-| hap1_beta: ∀B,A. hap1 (@B.𝛌.A) ([⬐B]A)
-| hap1_appl: ∀B,A1,A2. hap1 A1 A2 → hap1 (@B.A1) (@B.A2)
-.
-
-lemma hap1_lift: liftable hap1.
-#h #M1 #M2 #H elim H -M1 -M2 normalize /2 width=1/
-#B #A #d <dsubst_lift_le //
-qed.
-
-lemma hap1_inv_lift: deliftable hap1.
-#h #N1 #N2 #H elim H -N1 -N2
-[ #D #C #d #M1 #H
-  elim (lift_inv_appl … H) -H #B #M #H0 #HM #H destruct
-  elim (lift_inv_abst … HM) -HM #A #H0 #H destruct /3 width=3/
-| #D1 #C1 #C2 #_ #IHC12 #d #M1 #H
-  elim (lift_inv_appl … H) -H #B #A1 #H1 #H2 #H destruct
-  elim (IHC12 ???) -IHC12 [4: // |2,3: skip ] #A2 #HA12 #H destruct (**) (* simplify line *)
-  @(ex2_1_intro … (@B.A2)) // /2 width=1/
-]
-qed-.
-
-lemma hap1_dsubstable: dsubstable_dx hap1.
-#D1 #M1 #M2 #H elim H -M1 -M2 normalize /2 width=1/
-#D2 #A #d >dsubst_dsubst_ge //
-qed.
-
-lemma hap1_lsred: ∀M,N. hap1 M N →
-                  ∃∃p. in_spine p & M ⇀[p] N.
-#M #N #H elim H -M -N /2 width=3/
-#B #A1 #A2 #_ * /3 width=3/
-qed-.
diff --git a/matita/matita/contribs/lambda/labelled_hap_computation.ma b/matita/matita/contribs/lambda/labelled_hap_computation.ma
new file mode 100644
index 000000000..e7804cb7c
--- /dev/null
+++ b/matita/matita/contribs/lambda/labelled_hap_computation.ma
@@ -0,0 +1,94 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "labelled_sequential_computation.ma".
+include "labelled_hap_reduction.ma".
+
+(* KASHIMA'S "HAP" COMPUTATION (LABELLED MULTISTEP) *************************)
+
+(* Note: this is the "head in application" computation of:
+         R. Kashima: "A proof of the Standization Theorem in λ-Calculus". Typescript note, (2000).
+*)
+definition lhap: rpseq → relation term ≝ lstar … lhap1.
+
+interpretation "labelled 'hap' computation"
+   'HApStar M p N = (lhap p M N).
+
+notation "hvbox( M break ⓗ⇀* [ term 46 p ] break term 46 N )"
+   non associative with precedence 45
+   for @{ 'HApStar $M $p $N }.
+
+lemma lhap1_lhap: ∀p,M1,M2. M1 ⓗ⇀[p] M2 → M1 ⓗ⇀*[p::◊] M2.
+/2 width=1/
+qed.
+
+lemma lhap_inv_nil: ∀s,M1,M2. M1 ⓗ⇀*[s] M2 → ◊ = s → M1 = M2.
+/2 width=5 by lstar_inv_nil/
+qed-.
+
+lemma lhap_inv_cons: ∀s,M1,M2. M1 ⓗ⇀*[s] M2 → ∀q,r. q::r = s →
+                     ∃∃M. M1 ⓗ⇀[q] M & M ⓗ⇀*[r] M2.
+/2 width=3 by lstar_inv_cons/
+qed-.
+
+lemma lhap_inv_lhap1: ∀p,M1,M2. M1 ⓗ⇀*[p::◊] M2 → M1 ⓗ⇀[p] M2.
+/2 width=1 by lstar_inv_step/
+qed-.
+
+lemma lhap_lift: ∀s. liftable (lhap s).
+/2 width=1/
+qed.
+
+lemma lhap_inv_lift: ∀s. deliftable_sn (lhap s).
+/3 width=3 by lstar_deliftable_sn, lhap1_inv_lift/
+qed-.
+
+lemma lhap_dsubst: ∀s. dsubstable_dx (lhap s).
+/2 width=1/
+qed.
+(*
+theorem lhap_mono: ∀s. singlevalued … (lhap s).
+/3 width=7 by lstar_singlevalued, lhap1_mono/
+qed-.
+*)
+theorem lhap_trans: ∀s1,M1,M. M1 ⓗ⇀*[s1] M →
+                    ∀s2,M2. M ⓗ⇀*[s2] M2 → M1 ⓗ⇀*[s1@s2] M2.
+/2 width=3 by lstar_trans/
+qed-.
+(*
+lemma hap_appl: appl_compatible_dx hap.
+/3 width=1/
+qed.
+*)
+lemma lhap_spine_fwd: ∀s,M1,M2. M1 ⓗ⇀*[s] M2 → is_spine s.
+#s #M1 #M2 #H elim H -s -M1 -M2 //
+#p #M1 #M #HM1 #s #M2 #_ #IHM2
+lapply (lhap1_spine_fwd … HM1) -HM1 /2 width=1/ 
+qed-.
+
+lemma lhap_lsreds_fwd: ∀s,M1,M2. M1 ⓗ⇀*[s] M2 → M1 ⇀*[s] M2.
+#s #M1 #M2 #H elim H -s -M1 -M2 //
+#p #M1 #M #HM1 #s #M2 #_ #IHM2
+lapply (lhap1_lsred_fwd … HM1) -HM1 /2 width=3/ 
+qed-.
+
+lemma lhap_le_fwd: ∀s,M1,M2. M1 ⓗ⇀*[s] M2 → is_le s.
+(*
+#M1 #M2 #H @(star_ind_l ??????? H) -M1 /3 width=3/
+#M1 #M #HM1 #H * #s * #H1s #H2s
+generalize in match HM1; -HM1 generalize in match M1; -M1
+@(star_ind_l ??????? H) -M
+[ #M1 #HM12 elim (hap1_lsred … HM12) -HM12 /4 width=3/
+| #M0 #M1 #HM01 #HM12 #_ #M #HM0 #HM02
+*)
diff --git a/matita/matita/contribs/lambda/labelled_hap_reduction.ma b/matita/matita/contribs/lambda/labelled_hap_reduction.ma
new file mode 100644
index 000000000..705069f8b
--- /dev/null
+++ b/matita/matita/contribs/lambda/labelled_hap_reduction.ma
@@ -0,0 +1,96 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "labelled_sequential_reduction.ma".
+
+(* KASHIMA'S "HAP" COMPUTATION (LABELLED SINGLE STEP) ***********************)
+
+(* Note: this is one step of the "head in application" computation of:
+         R. Kashima: "A proof of the Standization Theorem in λ-Calculus". Typescript note, (2000).
+*)
+inductive lhap1: rptr → relation term ≝
+| hap1_beta: ∀B,A. lhap1 (◊) (@B.𝛌.A) ([⬐B]A)
+| hap1_appl: ∀p,B,A1,A2. lhap1 p A1 A2 → lhap1 (false::p) (@B.A1) (@B.A2)
+.
+
+interpretation "labelled 'hap' reduction"
+   'HAp M p N = (lhap1 p M N).
+
+notation "hvbox( M break ⓗ⇀ [ term 46 p ] break term 46 N )"
+   non associative with precedence 45
+   for @{ 'HAp $M $p $N }.
+
+lemma lhap1_inv_abst_sn: ∀p,M,N. M ⓗ⇀[p] N → ∀A. 𝛌.A = M → ⊥.
+#p #M #N * -p -M -N
+[ #B #A #A0 #H destruct
+| #p #B #A1 #A2 #_ #A0 #H destruct
+]
+qed-.
+
+lemma lhap1_inv_appl_sn: ∀p,M,N. M ⓗ⇀[p] N → ∀B,A. @B.A = M →
+                         (∃∃C. ◊ = p & 𝛌.C = A & [⬐B]C = N) ∨
+                         ∃∃q,C. A ⓗ⇀[q] C & false::q = p & @B.C = N.
+#p #M #N * -p -M -N
+[ #B #A #B0 #A0 #H destruct /3 width=3/
+| #p #B #A1 #A2 #HA12 #B0 #A0 #H destruct /3 width=5/
+]
+qed-.
+
+lemma lhap1_inv_abst_dx: ∀p,M,N. M ⓗ⇀[p] N → ∀C. 𝛌.C = N →
+                         ∃∃B,A. ◊ = p & @B.𝛌.A = M & 𝛌.C = [⬐B]A.
+#p #M #N * -p -M -N
+[ #B #A #C #H /2 width=4/
+| #p #B #A1 #A2 #_ #C #H destruct
+]
+qed-.
+
+lemma lhap1_lift: ∀p. liftable (lhap1 p).
+#p #h #M1 #M2 #H elim H -p -M1 -M2 normalize /2 width=1/
+#B #A #d <dsubst_lift_le //
+qed.
+
+lemma lhap1_inv_lift: ∀p. deliftable_sn (lhap1 p).
+#p #h #N1 #N2 #H elim H -p -N1 -N2
+[ #D #C #d #M1 #H
+  elim (lift_inv_appl … H) -H #B #M #H0 #HM #H destruct
+  elim (lift_inv_abst … HM) -HM #A #H0 #H destruct /3 width=3/
+| #p #D1 #C1 #C2 #_ #IHC12 #d #M1 #H
+  elim (lift_inv_appl … H) -H #B #A1 #H1 #H2 #H destruct
+  elim (IHC12 ???) -IHC12 [4: // |2,3: skip ] #A2 #HA12 #H destruct (**) (* simplify line *)
+  @(ex2_intro … (@B.A2)) // /2 width=1/
+]
+qed-.
+
+lemma lhap1_dsubst: ∀p. dsubstable_dx (lhap1 p).
+#p #D1 #M1 #M2 #H elim H -p -M1 -M2 normalize /2 width=1/
+#D2 #A #d >dsubst_dsubst_ge //
+qed.
+
+lemma lhap1_spine_fwd: ∀p,M,N. M ⓗ⇀[p] N → in_spine p.
+#p #M #N #H elim H -p -M -N // /2 width=1/ 
+qed-.
+
+lemma lhap1_lsred_fwd: ∀p,M,N. M ⓗ⇀[p] N → M ⇀[p] N.
+#p #M #N #H elim H -p -M -N // /2 width=1/
+qed-.
+
+lemma lhap1_le_fwd: ∀p1,M1,M. M1 ⓗ⇀[p1] M → ∀p2,M2. M ⓗ⇀[p2] M2 → p1 ≤ p2.
+#p1 #M1 #M #H elim H -p1 -M1 -M //
+#p1 #B #A1 #A2 #HA12 #IHA12 #p2 #M2 #H
+elim (lhap1_inv_appl_sn … H ???) -H [5: // |2,3: skip ] * (**) (* simplify line *)
+[ -IHA12 #C2 #Hp2 #HAC2 #_
+  elim (lhap1_inv_abst_dx … HA12 … HAC2) -A2 #B1 #C1 #Hp1 #_ #_ //
+| -HA12 /3 width=2/
+]
+qed-.
diff --git a/matita/matita/contribs/lambda/labelled_sequential_computation.ma b/matita/matita/contribs/lambda/labelled_sequential_computation.ma
index fd096c095..52fef2514 100644
--- a/matita/matita/contribs/lambda/labelled_sequential_computation.ma
+++ b/matita/matita/contribs/lambda/labelled_sequential_computation.ma
@@ -17,12 +17,7 @@ include "labelled_sequential_reduction.ma".
 
 (* LABELLED SEQUENTIAL COMPUTATION (MULTISTEP) ******************************)
 
-(* Note: this comes from "star term lsred" *)
-inductive lsreds: rpseq → relation term ≝
-| lsreds_nil : ∀M. lsreds (◊) M M
-| lsreds_cons: ∀p,M1,M. M1 ⇀[p] M →
-               ∀s,M2. lsreds s M M2 → lsreds (p::s) M1 M2
-.
+definition lsreds: rpseq → relation term ≝ lstar … lsred.
 
 interpretation "labelled sequential computation"
    'SeqRedStar M s N = (lsreds s M N).
@@ -32,62 +27,48 @@ notation "hvbox( M break ⇀* [ term 46 s ] break term 46 N )"
    for @{ 'SeqRedStar $M $s $N }.
 
 lemma lsred_lsreds: ∀p,M1,M2. M1 ⇀[p] M2 → M1 ⇀*[p::◊] M2.
-/2 width=3/
+/2 width=1/
 qed.
 
 lemma lsreds_inv_nil: ∀s,M1,M2. M1 ⇀*[s] M2 → ◊ = s → M1 = M2.
-#s #M1 #M2 * -s -M1 -M2 //
-#p #M1 #M #_ #s #M2 #_ #H destruct
+/2 width=5 by lstar_inv_nil/
 qed-.
 
 lemma lsreds_inv_cons: ∀s,M1,M2. M1 ⇀*[s] M2 → ∀q,r. q::r = s →
                        ∃∃M. M1 ⇀[q] M & M ⇀*[r] M2.
-#s #M1 #M2 * -s -M1 -M2
-[ #M #q #r #H destruct 
-| #p #M1 #M #HM1 #s #M2 #HM2 #q #r #H destruct /2 width=3/
-]
+/2 width=3 by lstar_inv_cons/
 qed-.
 
 lemma lsreds_inv_lsred: ∀p,M1,M2. M1 ⇀*[p::◊] M2 → M1 ⇀[p] M2.
-#p #M1 #M2 #H
-elim (lsreds_inv_cons … H ???) -H [4: // |2,3: skip ] #M #HM1 #H (**) (* simplify line *)
-<(lsreds_inv_nil … H ?) -H //
-qed-.
-
-(* Note: "|s|" should be unparetesized *)
-lemma lsreds_fwd_mult: ∀s,M1,M2. M1 ⇀*[s] M2 → #{M2} ≤ #{M1} ^ (2 ^ (|s|)).
-#s #M1 #M2 #H elim H -s -M1 -M2 normalize //
-#p #M1 #M #HM1 #s #M2 #_ #IHM2
-lapply (lsred_fwd_mult … HM1) -p #HM1
-@(transitive_le … IHM2) -M2
-/3 width=1 by le_exp1, lt_O_exp, lt_to_le/ (**) (* auto: slow without trace *)
+/2 width=1 by lstar_inv_step/
 qed-.
 
 lemma lsreds_lift: ∀s. liftable (lsreds s).
-#s #h #M1 #M2 #H elim H -s -M1 -M2 // /3 width=3/
+/2 width=1/
 qed.
 
-lemma lsreds_inv_lift: ∀s. deliftable (lsreds s).
-#s #h #N1 #N2 #H elim H -s -N1 -N2 /2 width=3/
-#p #N1 #N #HN1 #s #N2 #_ #IHN2 #d #M1 #HMN1
-elim (lsred_inv_lift … HN1 … HMN1) -N1 #M #HM1 #HMN
-elim (IHN2 … HMN) -N /3 width=3/
+lemma lsreds_inv_lift: ∀s. deliftable_sn (lsreds s).
+/3 width=3 by lstar_deliftable_sn, lsred_inv_lift/
 qed-.
 
 lemma lsreds_dsubst: ∀s. dsubstable_dx (lsreds s).
-#s #D #M1 #M2 #H elim H -s -M1 -M2 // /3 width=3/
+/2 width=1/
 qed.
 
 theorem lsreds_mono: ∀s. singlevalued … (lsreds s).
-#s #M #N1 #H elim H -s -M -N1
-[ /2 width=3 by lsreds_inv_nil/
-| #p #M #M1 #HM1 #s #N1 #_ #IHMN1 #N2 #H
-  elim (lsreds_inv_cons … H ???) -H [4: // |2,3: skip ] #M2 #HM2 #HMN2 (**) (* simplify line *)
-  lapply (lsred_mono … HM1 … HM2) -M #H destruct /2 width=1/
-]
+/3 width=7 by lstar_singlevalued, lsred_mono/
 qed-.
 
 theorem lsreds_trans: ∀s1,M1,M. M1 ⇀*[s1] M →
                       ∀s2,M2. M ⇀*[s2] M2 → M1 ⇀*[s1@s2] M2.
-#s1 #M1 #M #H elim H -s1 -M1 -M normalize // /3 width=3/
+/2 width=3 by lstar_trans/
+qed-.
+
+(* Note: "|s|" should be unparetesized *)
+lemma lsreds_fwd_mult: ∀s,M1,M2. M1 ⇀*[s] M2 → #{M2} ≤ #{M1} ^ (2 ^ (|s|)).
+#s #M1 #M2 #H elim H -s -M1 -M2 normalize //
+#p #M1 #M #HM1 #s #M2 #_ #IHM2
+lapply (lsred_fwd_mult … HM1) -p #HM1
+@(transitive_le … IHM2) -M2
+/3 width=1 by le_exp1, lt_O_exp, lt_to_le/ (**) (* auto: slow without trace *)
 qed-.
diff --git a/matita/matita/contribs/lambda/labelled_sequential_reduction.ma b/matita/matita/contribs/lambda/labelled_sequential_reduction.ma
index 6957484e6..208eb4023 100644
--- a/matita/matita/contribs/lambda/labelled_sequential_reduction.ma
+++ b/matita/matita/contribs/lambda/labelled_sequential_reduction.ma
@@ -104,23 +104,23 @@ lemma lsred_lift: ∀p. liftable (lsred p).
 #B #A #d <dsubst_lift_le //
 qed.
 
-lemma lsred_inv_lift: ∀p. deliftable (lsred p).
+lemma lsred_inv_lift: ∀p. deliftable_sn (lsred p).
 #p #h #N1 #N2 #H elim H -p -N1 -N2
 [ #D #C #d #M1 #H
   elim (lift_inv_appl … H) -H #B #M #H0 #HM #H destruct
   elim (lift_inv_abst … HM) -HM #A #H0 #H destruct /3 width=3/
 | #p #C1 #C2 #_ #IHC12 #d #M1 #H
-  elim (lift_inv_abst … H) -H #A1 #H0 #H destruct
-  elim (IHC12 ???) -IHC12 [4: // |2,3: skip ] #A2 #HA12 #H destruct (**) (* simplify line *)
-  @(ex2_1_intro … (𝛌.A2)) // /2 width=1/
+  elim (lift_inv_abst … H) -H #A1 #HAC1 #H
+  elim (IHC12 … HAC1) -C1 #A2 #HA12 #HAC2 destruct
+  @(ex2_intro … (𝛌.A2)) // /2 width=1/
 | #p #D1 #D2 #C1 #_ #IHD12 #d #M1 #H
-  elim (lift_inv_appl … H) -H #B1 #A #H1 #H2 #H destruct
-  elim (IHD12 ???) -IHD12 [4: // |2,3: skip ] #B2 #HB12 #H destruct (**) (* simplify line *)
-  @(ex2_1_intro … (@B2.A)) // /2 width=1/
+  elim (lift_inv_appl … H) -H #B1 #A #HBD1 #H1 #H2
+  elim (IHD12 … HBD1) -D1 #B2 #HB12 #HBD2 destruct
+  @(ex2_intro … (@B2.A)) // /2 width=1/
 | #p #D1 #C1 #C2 #_ #IHC12 #d #M1 #H
-  elim (lift_inv_appl … H) -H #B #A1 #H1 #H2 #H destruct
-  elim (IHC12 ???) -IHC12 [4: // |2,3: skip ] #A2 #HA12 #H destruct (**) (* simplify line *)
-  @(ex2_1_intro … (@B.A2)) // /2 width=1/
+  elim (lift_inv_appl … H) -H #B #A1 #H1 #HAC1 #H2
+  elim (IHC12 … HAC1) -C1 #A2 #HA12 #HAC2 destruct
+  @(ex2_intro … (@B.A2)) // /2 width=1/
 ]
 qed-.
 
diff --git a/matita/matita/contribs/lambda/lift.ma b/matita/matita/contribs/lambda/lift.ma
index 8f2199fc3..60038de76 100644
--- a/matita/matita/contribs/lambda/lift.ma
+++ b/matita/matita/contribs/lambda/lift.ma
@@ -188,7 +188,7 @@ theorem lift_inv_lift_le: ∀h1,h2,M1,M2,d1,d2. d2 ≤ d1 →
       >(lift_inv_vref_lt … Hid2 … H) -M2 /3 width=3/
     | elim (lift_inv_vref_ge … H) -H -Hd21 // -Hid2 #Hdh2i #H destruct
       elim (le_inv_plus_l … Hdh2i) -Hdh2i #Hd2i #Hh2i
-      @(ex2_1_intro … (#(i-h2))) [ /4 width=1/ ] -Hid1
+      @(ex2_intro … (#(i-h2))) [ /4 width=1/ ] -Hid1
       >lift_vref_ge // -Hd2i /3 width=1/ (**) (* auto: needs some help here *)
     ]
   | elim (le_inv_plus_l … Hid1) #Hd1i #Hh2i
@@ -196,7 +196,7 @@ theorem lift_inv_lift_le: ∀h1,h2,M1,M2,d1,d2. d2 ≤ d1 →
     elim (le_inv_plus_l … Hdh2i) #Hd2i #_
     >(lift_vref_ge … Hid1) #H -Hid1
     >(lift_inv_vref_ge_plus … H) -H /2 width=3/ -Hdh2i
-    @(ex2_1_intro … (#(i-h2))) (**) (* auto: needs some help here *)
+    @(ex2_intro … (#(i-h2))) (**) (* auto: needs some help here *)
     [ >lift_vref_ge // -Hd1i /3 width=1/
     | >lift_vref_ge // -Hd2i -Hd1i /3 width=1/
     ]
@@ -204,12 +204,12 @@ theorem lift_inv_lift_le: ∀h1,h2,M1,M2,d1,d2. d2 ≤ d1 →
 | normalize #A1 #IHA1 #M2 #d1 #d2 #Hd21 #H
   elim (lift_inv_abst … H) -H >plus_plus_comm_23 #A2 #HA12 #H destruct
   elim (IHA1 … HA12) -IHA1 -HA12 /2 width=1/ -Hd21 #A #HA2 #HA1
-  @(ex2_1_intro … (𝛌.A)) normalize //
+  @(ex2_intro … (𝛌.A)) normalize //
 | normalize #B1 #A1 #IHB1 #IHA1 #M2 #d1 #d2 #Hd21 #H
   elim (lift_inv_appl … H) -H #B2 #A2 #HB12 #HA12 #H destruct
   elim (IHB1 … HB12) -IHB1 -HB12 // #B #HB2 #HB1
   elim (IHA1 … HA12) -IHA1 -HA12 // -Hd21 #A #HA2 #HA1
-  @(ex2_1_intro … (@B.A)) normalize //
+  @(ex2_intro … (@B.A)) normalize //
 ]
 qed-.
 
@@ -243,6 +243,30 @@ qed-.
 definition liftable: predicate (relation term) ≝ λR.
                      ∀h,M1,M2. R M1 M2 → ∀d. R (↑[d, h] M1) (↑[d, h] M2).
 
-definition deliftable: predicate (relation term) ≝ λR.
-                       ∀h,N1,N2. R N1 N2 → ∀d,M1. ↑[d, h] M1 = N1 →
-                       ∃∃M2. R M1 M2 & ↑[d, h] M2 = N2.
+definition deliftable_sn: predicate (relation term) ≝ λR.
+                          ∀h,N1,N2. R N1 N2 → ∀d,M1. ↑[d, h] M1 = N1 →
+                          ∃∃M2. R M1 M2 & ↑[d, h] M2 = N2.
+
+lemma star_liftable: ∀R. liftable R → liftable (star … R).
+#R #HR #h #M1 #M2 #H elim H -M2 // /3 width=3/
+qed.
+
+lemma star_deliftable_sn: ∀R. deliftable_sn R → deliftable_sn (star … R).
+#R #HR #h #N1 #N2 #H elim H -N2 /2 width=3/
+#N #N2 #_ #HN2 #IHN1 #d #M1 #HMN1
+elim (IHN1 … HMN1) -N1 #M #HM1 #HMN
+elim (HR … HN2 … HMN) -N /3 width=3/
+qed-.
+
+lemma lstar_liftable: ∀T,R. (∀t. liftable (R t)) →
+                      ∀l. liftable (lstar T … R l).
+#T #R #HR #l #h #M1 #M2 #H elim H -l -M1 -M2 // /3 width=3/
+qed.
+
+lemma lstar_deliftable_sn: ∀T,R. (∀t. deliftable_sn (R t)) →
+                           ∀l. deliftable_sn (lstar T … R l).
+#T #R #HR #l #h #N1 #N2 #H elim H -l -N1 -N2 /2 width=3/
+#t #N1 #N #HN1 #l #N2 #_ #IHN2 #d #M1 #HMN1
+elim (HR … HN1 … HMN1) -N1 #M #HM1 #HMN
+elim (IHN2 … HMN) -N /3 width=3/
+qed-.
diff --git a/matita/matita/contribs/lambda/parallel_reduction.ma b/matita/matita/contribs/lambda/parallel_reduction.ma
index 7aa47be7f..f5285bad1 100644
--- a/matita/matita/contribs/lambda/parallel_reduction.ma
+++ b/matita/matita/contribs/lambda/parallel_reduction.ma
@@ -72,23 +72,23 @@ lemma pred_lift: liftable pred.
 #D #D #A #C #_ #_ #IHBD #IHAC #d <dsubst_lift_le // /2 width=1/
 qed.
 
-lemma pred_inv_lift: deliftable pred.
+lemma pred_inv_lift: deliftable_sn pred.
 #h #N1 #N2 #H elim H -N1 -N2 /2 width=3/
 [ #C1 #C2 #_ #IHC12 #d #M1 #H
   elim (lift_inv_abst … H) -H #A1 #HAC1 #H
   elim (IHC12 … HAC1) -C1 #A2 #HA12 #HAC2 destruct
-  @(ex2_1_intro … (𝛌.A2)) // /2 width=1/
+  @(ex2_intro … (𝛌.A2)) // /2 width=1/
 | #D1 #D2 #C1 #C2 #_ #_ #IHD12 #IHC12 #d #M1 #H
   elim (lift_inv_appl … H) -H #B1 #A1 #HBD1 #HAC1 #H
   elim (IHD12 … HBD1) -D1 #B2 #HB12 #HBD2
   elim (IHC12 … HAC1) -C1 #A2 #HA12 #HAC2 destruct
-  @(ex2_1_intro … (@B2.A2)) // /2 width=1/
+  @(ex2_intro … (@B2.A2)) // /2 width=1/
 | #D1 #D2 #C1 #C2 #_ #_ #IHD12 #IHC12 #d #M1 #H
   elim (lift_inv_appl … H) -H #B1 #M #HBD1 #HM #H1
   elim (lift_inv_abst … HM) -HM #A1 #HAC1 #H
   elim (IHD12 … HBD1) -D1 #B2 #HB12 #HBD2
   elim (IHC12 … HAC1) -C1 #A2 #HA12 #HAC2 destruct
-  @(ex2_1_intro … ([⬐B2]A2)) /2 width=1/
+  @(ex2_intro … ([⬐B2]A2)) /2 width=1/
 ]
 qed-.
 
@@ -143,7 +143,7 @@ theorem pred_conf: confluent … pred.
   | #C #B2 #C2 #HB2 #HC2 #H2 #HM2 #B1 #N #HB1 #H #HM1 destruct
     @(pred_conf1_appl_beta … IH) // (**) (* /2 width=7 by pred_conf1_appl_beta/ does not work *)
   | #B2 #N #B2 #H #HM2 #C #B1 #C1 #HB1 #HC1 #H1 #HM1 destruct
-    @ex2_1_commute @(pred_conf1_appl_beta … IH) //
+    @ex2_commute @(pred_conf1_appl_beta … IH) //
   | #C #B2 #C2 #HB2 #HC2 #H2 #HM2 #C0 #B1 #C1 #HB1 #HC1 #H1 #HM1 destruct
     elim (IH B … HB1 … HB2) -HB1 -HB2 //
     elim (IH C … HC1 … HC2) normalize // -B -C /3 width=5/
diff --git a/matita/matita/contribs/lambda/policy.txt b/matita/matita/contribs/lambda/policy.txt
index 77b0fdab7..9f74e2339 100644
--- a/matita/matita/contribs/lambda/policy.txt
+++ b/matita/matita/contribs/lambda/policy.txt
@@ -4,12 +4,16 @@ A, B, C, D: term
 H         : transient premise
 IH        : inductive premise
 M, N      : term
+R         : arbitrary relation
+T, U      : arbitrary small type
 
 a, b      : boolean
 d, e      : variable reference depth 
 h         : relocation height
 i, j      : de Bruijn index
 k         : relocation height
+l         : arbitrary list
 m, n      : measures on terms
 p, q      : redex pointer
 r, s      : redex pointer sequence
+t, u      : arbitrary element
diff --git a/matita/matita/contribs/lambda/preamble.ma b/matita/matita/contribs/lambda/preamble.ma
index eb0f409a5..c272595a2 100644
--- a/matita/matita/contribs/lambda/preamble.ma
+++ b/matita/matita/contribs/lambda/preamble.ma
@@ -28,10 +28,6 @@ notation "⊥"
   non associative with precedence 90
   for @{'false}.
 
-lemma ex2_1_commute: ∀A0. ∀P0,P1:A0→Prop. (∃∃x0. P0 x0 & P1 x0) → ∃∃x0. P1 x0 & P0 x0.
-#A0 #P0 #P1 * /2 width=3/
-qed.
-
 (* relations *)
 
 definition confluent1: ∀A. relation A → predicate A ≝ λA,R,a0.
@@ -100,9 +96,3 @@ qed.
 notation > "◊"
   non associative with precedence 90
   for @{'nil}.
-
-let rec Allr (A:Type[0]) (R:relation A) (l:list A) on l : Prop ≝
-match l with
-[ nil       ⇒ True
-| cons a1 l ⇒ match l with [ nil ⇒ True | cons a2 _ ⇒ R a1 a2 ∧ Allr A R l ]
-].
diff --git a/matita/matita/contribs/lambda/redex_pointer.ma b/matita/matita/contribs/lambda/redex_pointer.ma
index 721904437..5746787c5 100644
--- a/matita/matita/contribs/lambda/redex_pointer.ma
+++ b/matita/matita/contribs/lambda/redex_pointer.ma
@@ -36,7 +36,7 @@ inductive rpprec: relation rptr ≝
 | rpprec_nil : ∀b,q.   rpprec (◊) (b::q)
 | rppprc_cons: ∀p,q.   rpprec (false::p) (true::q)
 | rpprec_comp: ∀b,p,q. rpprec p q → rpprec (b::p) (b::q)
-| rpprec_skip: ∀p,q.   rpprec (false::p) q → rpprec p q
+| rpprec_skip:         rpprec (false::◊) ◊
 .
 
 interpretation "'precedes' on redex pointers"
@@ -58,3 +58,19 @@ definition rple: relation rptr ≝ star … rpprec.
 
 interpretation "'less or equal to' on redex pointers"
    'leq p q = (rple p q).
+
+lemma rpprec_rple: ∀p,q. p ≺ q → p ≤ q.
+/2 width=1/
+qed.
+
+lemma rple_false: false::◊ ≤ ◊.
+/2 width=1/
+qed.
+
+lemma rple_nil: ∀p. ◊ ≤ p.
+* // /2 width=1/
+qed.
+
+lemma rple_comp: ∀p1,p2. p1 ≤ p2 → ∀b. (b::p1) ≤ (b::p2).
+#p1 #p2 #H elim H -p2 // /3 width=3/
+qed.
diff --git a/matita/matita/contribs/lambda/redex_pointer_sequence.ma b/matita/matita/contribs/lambda/redex_pointer_sequence.ma
index e32c6d2ba..43b7aa1d4 100644
--- a/matita/matita/contribs/lambda/redex_pointer_sequence.ma
+++ b/matita/matita/contribs/lambda/redex_pointer_sequence.ma
@@ -27,3 +27,7 @@ definition is_spine: predicate rpseq ≝ λs.
 (* Note: to us, a "normal" computation contracts redexes in non-decreasing positions *)
 definition is_le: predicate rpseq ≝ λs.
                   Allr … rple s.
+
+(* Note: a normal spine computation *)
+definition is_spine_le: predicate rpseq ≝ λs.
+                        is_spine s ∧ is_le s.
diff --git a/matita/matita/contribs/lambda/st_computation.ma b/matita/matita/contribs/lambda/st_computation.ma
index 792ab882c..fc7667a40 100644
--- a/matita/matita/contribs/lambda/st_computation.ma
+++ b/matita/matita/contribs/lambda/st_computation.ma
@@ -12,7 +12,7 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "hap_computation.ma".
+include "labelled_hap_computation.ma".
 
 (* KASHIMA'S "ST" COMPUTATION ***********************************************)
 
@@ -20,8 +20,8 @@ include "hap_computation.ma".
          R. Kashima: "A proof of the Standization Theorem in λ-Calculus". Typescript note, (2000).
 *)
 inductive st: relation term ≝
-| st_vref: ∀M,i. hap M (#i) → st M (#i)
-| st_abst: ∀M,A,C. hap M (𝛌.A) → st A C → st M (𝛌.C)
-| st_appl: ∀M,B,D,A,C. hap M (@B.A) → st B D → st A C → st M (@D.C) 
+| st_vref: ∀s,M,i. lhap s M (#i) → st M (#i)
+| st_abst: ∀s,M,A,C. lhap s M (𝛌.A) → st A C → st M (𝛌.C)
+| st_appl: ∀s,M,B,D,A,C. lhap s M (@B.A) → st B D → st A C → st M (@D.C) 
 .
 
diff --git a/matita/matita/contribs/lambda/term.ma b/matita/matita/contribs/lambda/term.ma
index 8c688dc8e..660830d27 100644
--- a/matita/matita/contribs/lambda/term.ma
+++ b/matita/matita/contribs/lambda/term.ma
@@ -47,3 +47,11 @@ notation "hvbox( 𝛌  . term 46 A )"
 notation "hvbox( @ term 46 C . break term 46 A )"
    non associative with precedence 46
    for @{ 'Application $C $A }.
+
+definition appl_compatible_dx: predicate (relation term) ≝ λR.
+                               ∀B,A1,A2. R A1 A2 → R (@B.A1) (@B.A2).
+
+lemma star_appl_compatible_dx: ∀R. appl_compatible_dx R →
+                               appl_compatible_dx (star … R).
+#R #HR #B #A1 #A2 #H elim H -A2 // /3 width=3/
+qed.
diff --git a/matita/matita/contribs/lambda/xoa.conf.xml b/matita/matita/contribs/lambda/xoa.conf.xml
index 1343a2c1e..4d8185190 100644
--- a/matita/matita/contribs/lambda/xoa.conf.xml
+++ b/matita/matita/contribs/lambda/xoa.conf.xml
@@ -8,7 +8,7 @@
     <key name="objects">xoa</key>
     <key name="notations">xoa_notation</key>
     <key name="include">basics/pts.ma</key>
-    <key name="ex">2 1</key>
+    <key name="ex">3 1</key>    
     <key name="ex">3 2</key>
     <key name="ex">4 3</key>
     <key name="or">3</key>
diff --git a/matita/matita/contribs/lambda/xoa.ma b/matita/matita/contribs/lambda/xoa.ma
index 2d9ee54a3..5529e4560 100644
--- a/matita/matita/contribs/lambda/xoa.ma
+++ b/matita/matita/contribs/lambda/xoa.ma
@@ -16,13 +16,13 @@
 
 include "basics/pts.ma".
 
-(* multiple existental quantifier (2, 1) *)
+(* multiple existental quantifier (3, 1) *)
 
-inductive ex2_1 (A0:Type[0]) (P0,P1:A0→Prop) : Prop ≝
-   | ex2_1_intro: ∀x0. P0 x0 → P1 x0 → ex2_1 ? ? ?
+inductive ex3_1 (A0:Type[0]) (P0,P1,P2:A0→Prop) : Prop ≝
+   | ex3_1_intro: ∀x0. P0 x0 → P1 x0 → P2 x0 → ex3_1 ? ? ? ?
 .
 
-interpretation "multiple existental quantifier (2, 1)" 'Ex P0 P1 = (ex2_1 ? P0 P1).
+interpretation "multiple existental quantifier (3, 1)" 'Ex P0 P1 P2 = (ex3_1 ? P0 P1 P2).
 
 (* multiple existental quantifier (3, 2) *)
 
diff --git a/matita/matita/contribs/lambda/xoa_notation.ma b/matita/matita/contribs/lambda/xoa_notation.ma
index de716c381..fbb7ca5fa 100644
--- a/matita/matita/contribs/lambda/xoa_notation.ma
+++ b/matita/matita/contribs/lambda/xoa_notation.ma
@@ -14,15 +14,15 @@
 
 (* This file was generated by xoa.native: do not edit *********************)
 
-(* multiple existental quantifier (2, 1) *)
+(* multiple existental quantifier (3, 1) *)
 
-notation > "hvbox(∃∃ ident x0 break . term 19 P0 break & term 19 P1)"
+notation > "hvbox(∃∃ ident x0 break . term 19 P0 break & term 19 P1 break & term 19 P2)"
  non associative with precedence 20
- for @{ 'Ex (λ${ident x0}.$P0) (λ${ident x0}.$P1) }.
+ for @{ 'Ex (λ${ident x0}.$P0) (λ${ident x0}.$P1) (λ${ident x0}.$P2) }.
 
-notation < "hvbox(∃∃ ident x0 break . term 19 P0 break & term 19 P1)"
+notation < "hvbox(∃∃ ident x0 break . term 19 P0 break & term 19 P1 break & term 19 P2)"
  non associative with precedence 20
- for @{ 'Ex (λ${ident x0}:$T0.$P0) (λ${ident x0}:$T0.$P1) }.
+ for @{ 'Ex (λ${ident x0}:$T0.$P0) (λ${ident x0}:$T0.$P1) (λ${ident x0}:$T0.$P2) }.
 
 (* multiple existental quantifier (3, 2) *)
 
diff --git a/matita/matita/lib/basics/core_notation.ma b/matita/matita/lib/basics/core_notation.ma
index 904adcd52..a37b4d859 100644
--- a/matita/matita/lib/basics/core_notation.ma
+++ b/matita/matita/lib/basics/core_notation.ma
@@ -27,6 +27,14 @@ notation > "\exists list1 ident x sep , opt (: T). term 19 Px"
           @{ ${ fold right @{$Px} rec acc @{'exists (λ${ident x}.$acc)} } }
        }.
 
+notation > "hvbox(∃∃ ident x0 break . term 19 P0 break & term 19 P1)"
+ non associative with precedence 20
+ for @{ 'exists2 (λ${ident x0}.$P0) (λ${ident x0}.$P1) }.
+
+notation < "hvbox(∃∃ ident x0 break . term 19 P0 break & term 19 P1)"
+ non associative with precedence 20
+ for @{ 'exists2 (λ${ident x0}:$T0.$P0) (λ${ident x0}:$T0.$P1) }.
+
 (* sigma ********************************************************************)
 
 notation < "hvbox(\Sigma ident i : ty break . p)"
diff --git a/matita/matita/lib/basics/lists/list.ma b/matita/matita/lib/basics/lists/list.ma
index 5d000e5c1..a8a01a904 100644
--- a/matita/matita/lib/basics/lists/list.ma
+++ b/matita/matita/lib/basics/lists/list.ma
@@ -419,6 +419,12 @@ lemma All_nth : ∀A,P,n,l.
   ]
 ] qed.
 
+let rec Allr (A:Type[0]) (R:relation A) (l:list A) on l : Prop ≝
+match l with
+[ nil       ⇒ True
+| cons a1 l ⇒ match l with [ nil ⇒ True | cons a2 _ ⇒ R a1 a2 ∧ Allr A R l ]
+].
+
 (**************************** Exists *******************************)
 
 let rec Exists (A:Type[0]) (P:A → Prop) (l:list A) on l : Prop ≝
@@ -593,3 +599,50 @@ match n with
 [ O ⇒ [ ]
 | S m ⇒ a::(make_list A a m)
 ].
+
+(* ******** labelled reflexive and transitive closure ************)
+
+inductive lstar (A:Type[0]) (B:Type[0]) (R: A→relation B): list A → relation B ≝
+| lstar_nil : ∀b. lstar A B R ([]) b b
+| lstar_cons: ∀a,b1,b. R a b1 b →
+              ∀l,b2. lstar A B R l b b2 → lstar A B R (a::l) b1 b2
+.
+
+lemma lstar_step: ∀A,B,R,a,b1,b2. R a b1 b2 → lstar A B R ([a]) b1 b2.
+/2 width=3/
+qed.
+
+lemma lstar_inv_nil: ∀A,B,R,l,b1,b2. lstar A B R l b1 b2 → [] = l → b1 = b2.
+#A #B #R #l #b1 #b2 * -l -b1 -b2 //
+#a #b1 #b #_ #l #b2 #_ #H destruct
+qed-.
+
+lemma lstar_inv_cons: ∀A,B,R,l,b1,b2. lstar A B R l b1 b2 →
+                      ∀a0,l0. a0::l0 = l →
+                      ∃∃b. R a0 b1 b & lstar A B R l0 b b2.
+#A #B #R #l #b1 #b2 * -l -b1 -b2
+[ #b #a0 #l0 #H destruct
+| #a #b1 #b #Hb1 #l #b2 #Hb2 #a0 #l0 #H destruct /2 width=3/
+]
+qed-.
+
+lemma lstar_inv_step: ∀A,B,R,a,b1,b2. lstar A B R ([a]) b1 b2 → R a b1 b2.
+#A #B #R #a #b1 #b2 #H
+elim (lstar_inv_cons ?????? H ???) -H [4: // |2,3: skip ] #b #Hb1 #H (**) (* simplify line *)
+<(lstar_inv_nil ?????? H ?) -H // (**) (* simplify line *)
+qed-.
+
+theorem lstar_singlevalued: ∀A,B,R. (∀a. singlevalued ?? (R a)) →
+                            ∀l. singlevalued … (lstar A B R l).
+#A #B #R #HR #l #b #c1 #H elim H -l -b -c1
+[ /2 width=5 by lstar_inv_nil/
+| #a #b #b1 #Hb1 #l #c1 #_ #IHbc1 #c2 #H
+  elim (lstar_inv_cons ?????? H ???) -H [4: // |2,3: skip ] #b2 #Hb2 #Hbc2 (**) (* simplify line *)
+  lapply (HR … Hb1 … Hb2) -b #H destruct /2 width=1/
+]
+qed-.
+
+theorem lstar_trans: ∀A,B,R,l1,b1,b. lstar A B R l1 b1 b →
+                     ∀l2,b2. lstar A B R l2 b b2 → lstar A B R (l1@l2) b1 b2.
+#A #B #R #l1 #b1 #b #H elim H -l1 -b1 -b normalize // /3 width=3/
+qed-.
diff --git a/matita/matita/lib/basics/logic.ma b/matita/matita/lib/basics/logic.ma
index 70e743e99..76b4239d4 100644
--- a/matita/matita/lib/basics/logic.ma
+++ b/matita/matita/lib/basics/logic.ma
@@ -147,7 +147,13 @@ inductive ex (A:Type[0]) (P:A → Prop) : Prop ≝
 interpretation "exists" 'exists x = (ex ? x).
 
 inductive ex2 (A:Type[0]) (P,Q:A →Prop) : Prop ≝
-    ex_intro2: ∀ x:A. P x → Q x → ex2 A P Q.
+    ex2_intro: ∀ x:A. P x → Q x → ex2 A P Q.
+
+interpretation "exists on two predicates" 'exists2 x1 x2 = (ex2 ? x1 x2).
+
+lemma ex2_commute: ∀A0. ∀P0,P1:A0→Prop. (∃∃x0. P0 x0 & P1 x0) → ∃∃x0. P1 x0 & P0 x0.
+#A0 #P0 #P1 * /2 width=3/
+qed.
 
 (* iff *)
 definition iff :=
diff --git a/matita/matita/lib/basics/star.ma b/matita/matita/lib/basics/star.ma
index aafb3fa6b..892cff583 100644
--- a/matita/matita/lib/basics/star.ma
+++ b/matita/matita/lib/basics/star.ma
@@ -122,15 +122,25 @@ lemma starl_to_star: ∀A,R,a,b.starl A R a b → star A R a b.
 #a #b #c #Rab #sbc #sbc @(star_compl … Rab) //
 qed.
 
-lemma star_ind_l : 
-  ∀A:Type[0].∀R:relation A.∀Q:A → A → Prop.
-  (∀a.Q a a) → 
-  (∀a,b,c.R a b → star A R b c → Q b c → Q a c) → 
-  ∀a,b.star A R a b → Q a b.
-#A #R #Q #H1 #H2 #a #b #H0 
-elim (star_to_starl ???? H0) // -H0 -b -a 
-#a #b #c #Rab #slbc @H2 // @starl_to_star //
-qed.  
+fact star_ind_l_aux: ∀A,R,a2. ∀P:predicate A.
+                     P a2 →
+                     (∀a1,a. R a1 a → star … R a a2 → P a → P a1) →
+                     ∀a1,a. star … R a1 a → a = a2 → P a1.
+#A #R #a2 #P #H1 #H2 #a1 #a #Ha1
+elim (star_to_starl ???? Ha1) -a1 -a
+[ #a #b #c #Hab #Hbc #IH #H destruct /3 width=4/
+| #a #H destruct /2 width=1/
+]
+qed-.
+
+(* imporeved version of star_ind_l with "left_parameter" *)
+lemma star_ind_l: ∀A,R,a2. ∀P:predicate A.
+                  P a2 →
+                  (∀a1,a. R a1 a → star … R a a2 → P a → P a1) →
+                  ∀a1. star … R a1 a2 → P a1.
+#A #R #a2 #P #H1 #H2 #a1 #Ha12
+@(star_ind_l_aux … H1 H2 … Ha12) //
+qed.
 
 (* RC and star *)
 
diff --git a/matita/matita/lib/turing/complexity.ma b/matita/matita/lib/turing/complexity.ma
index fa9837a85..7caf6178e 100644
--- a/matita/matita/lib/turing/complexity.ma
+++ b/matita/matita/lib/turing/complexity.ma
@@ -11,7 +11,7 @@
 
 include "turing/turing.ma".
 
-
+(* this no longer works: TODO 
 (* time *) 
 let rec time_of sig (M:TM sig) a b (p:computation sig M a b) on p ≝
   match p with 
@@ -24,7 +24,7 @@ definition ComputeB_in_time ≝ λsig.λM:TM sig.λA:(list sig) → bool.λf.
   ∀l.∃c.∃p:computation sig M (init sig M l) c.
    time_of … p ≤ f (|l|) ∧
    (stop sig M c = true) ∧ (isnilb ? (out ?? c) = false).
-
+*)
 (* Le macchine di Turing sono a Type[1], che vorrebbe un nuovo 
 esistenziale.  
 
@@ -36,18 +36,18 @@ interpretation "Exists" 'exists x = (Ex ? x).
 definition DTIME ≝ λsig:FinSet. λL: list sig → bool. λf.
 ∃M:TM sig. ComputeB sig M L ∧ ∃c. Time_Bound sig M (λx.c + c*(f x)).
 *)
-
+(* this no longer works: TODO 
 inductive DTIME (sig:FinSet) (A:list sig → bool) (f:nat→nat) : Type[1] ≝ 
 | DTIME_intro: ∀M:TM sig.∀c.
     ComputeB_in_time sig M A (λx.c + c*(f x)) → DTIME sig A f.
-
+*)
 (* space complexity *) 
 
 definition size_of_tape ≝ λsig.λtp: tape sig.|left sig tp|+|right sig tp|.
 
 definition size_of_tapes ≝ λsig.λn.λtps: Vector (tape sig) n.
 foldr ?? (λtp.λacc.max (size_of_tape sig tp) acc) 0 tps.
-
+(* this no longer works: TODO 
 definition size_of_config ≝ λsig.λM.λc.
   size_of_tapes sig (S (tapes_no sig M)) (tapes sig M c).
 
@@ -62,8 +62,8 @@ definition ComputeB_in_space ≝ λsig.λM:TM sig.λA:(list sig) → bool.λf.
   ∀l.∃c.∃p:computation sig M (init sig M l) c.
    space_of … p ≤ f (|l|) ∧
    (stop sig M c = true) ∧ (isnilb ? (out ?? c) = false).
-   
+
 inductive DSPACE (sig:FinSet) (A:list sig → bool) (f:nat→nat) : Type[1] ≝ 
 | DTIME_intro: ∀M:TM sig.∀c.
     ComputeB_in_space sig M A (λx.c + c*(f x)) → DSPACE sig A f.
-
+*)
\ No newline at end of file
diff --git a/matita/matita/lib/turing/move_char.ma b/matita/matita/lib/turing/move_char.ma
index e20591c5f..297344d54 100644
--- a/matita/matita/lib/turing/move_char.ma
+++ b/matita/matita/lib/turing/move_char.ma
@@ -96,7 +96,7 @@ lemma sem_move_char_r :
 #alpha #sep #inc #i #outc #Hloop
 lapply (sem_while … (sem_mcr_step alpha sep) inc i outc Hloop) [%]
 -Hloop * #t1 * #Hstar @(star_ind_l ??????? Hstar)
-[ #tapea whd in ⊢ (% → ?); #H1 #b #a #ls #rs #Htapea
+[ whd in ⊢ (% → ?); #H1 #b #a #ls #rs #Htapea
   %
   [ #Hb >Htapea in H1; >Hb #H1 cases (H1 ??)
     [#_ #H2 >H2 % |*: % #H2 normalize in H2; destruct (H2)]
@@ -105,7 +105,7 @@ lapply (sem_while … (sem_mcr_step alpha sep) inc i outc Hloop) [%]
     [#Hfalse @False_ind @(absurd ?? Hb) normalize in Hfalse; destruct %
     |*:% #H2 normalize in H2; destruct (H2) ]
   ]
-| #tapea #tapeb #tapec #Hstar1 #HRtrue #IH #HRfalse
+| #tapeb #tapec #Hstar1 #HRtrue #IH #HRfalse
   lapply (IH HRfalse) -IH -HRfalse whd in ⊢ (%→%); #IH
   #b0 #a0 #ls #rs #Htapea cases Hstar1 #b * #bsep *
   [* #a * #ls1 * #rs1 * >Htapea #H destruct (H) #Htapeb %
@@ -256,7 +256,7 @@ lemma sem_move_char_l :
 #alpha #sep #inc #i #outc #Hloop
 lapply (sem_while … (sem_mcl_step alpha sep) inc i outc Hloop) [%]
 -Hloop * #t1 * #Hstar @(star_ind_l ??????? Hstar)
-[ #tapea whd in ⊢ (% → ?); #H1 #b #a #ls #rs #Htapea
+[ whd in ⊢ (% → ?); #H1 #b #a #ls #rs #Htapea
   %
   [ #Hb >Htapea in H1; >Hb #H1 cases (H1 ??)
     [#_ #H2 >H2 % |*: % #H2 normalize in H2; destruct (H2)]
@@ -265,7 +265,7 @@ lapply (sem_while … (sem_mcl_step alpha sep) inc i outc Hloop) [%]
     [#Hfalse @False_ind @(absurd ?? Hb) normalize in Hfalse; destruct %
     |*:% #H2 normalize in H2; destruct (H2) ]
   ]
-| #tapea #tapeb #tapec #Hstar1 #HRtrue #IH #HRfalse
+| #tapeb #tapec #Hstar1 #HRtrue #IH #HRfalse
   lapply (IH HRfalse) -IH -HRfalse whd in ⊢ (%→%); #IH
   #b0 #a0 #ls #rs #Htapea cases Hstar1 #b * #bsep *
   [* #a * #ls1 * #rs1 * >Htapea #H destruct (H) #Htapeb %
diff --git a/matita/matita/lib/turing/multi_universal/compare.ma b/matita/matita/lib/turing/multi_universal/compare.ma
index 232a4962b..2a26abb3e 100644
--- a/matita/matita/lib/turing/multi_universal/compare.ma
+++ b/matita/matita/lib/turing/multi_universal/compare.ma
@@ -234,7 +234,7 @@ lemma wsem_compare : ∀i,j,sig,n,is_endc.i ≠ j → i < S n → j < S n →
 #i #j #sig #n #is_endc #Hneq #Hi #Hj #ta #k #outc #Hloop
 lapply (sem_while … (sem_comp_step i j sig n is_endc Hneq Hi Hj) … Hloop) //
 -Hloop * #tb * #Hstar @(star_ind_l ??????? Hstar) -Hstar
-[ #tc whd in ⊢ (%→?); * * [ * [ *
+[ whd in ⊢ (%→?); * * [ * [ *
   [* #curi * #Hcuri #Hendi #Houtc %
     [ #_ @Houtc  
     | #ls #x #xs #ci #rs #ls0 #rs0 #Hnthi #Hnthj #Hnotendc 
@@ -255,7 +255,7 @@ lapply (sem_while … (sem_comp_step i j sig n is_endc Hneq Hi Hj) … Hloop) //
     [ #_ @Houtc
     | #ls #x #xs #ci #rs #ls0 #rs0 #_ #Hnthj >Hnthj in Hcj;
       normalize in ⊢ (%→?); #H destruct (H) ] ]
-  | #tc #td #te * #x * * * #Hendcx #Hci #Hcj #Hd #Hstar #IH #He lapply (IH He) -IH *
+  | #td #te * #x * * * #Hendcx #Hci #Hcj #Hd #Hstar #IH #He lapply (IH He) -IH *
     #IH1 #IH2 %
     [ >Hci >Hcj * [* #x0 * #H destruct (H) >Hendcx #H destruct (H) 
     |* [* #H @False_ind [cases H -H #H @H % | destruct (H)] | #H destruct (H)]] 
diff --git a/matita/matita/lib/turing/multi_universal/copy.ma b/matita/matita/lib/turing/multi_universal/copy.ma
index 36e33e291..1fef8d0b0 100644
--- a/matita/matita/lib/turing/multi_universal/copy.ma
+++ b/matita/matita/lib/turing/multi_universal/copy.ma
@@ -178,8 +178,8 @@ lemma wsem_copy : ∀src,dst,sig,n,is_sep.src ≠ dst → src < S n → dst < S
   copy src dst sig n is_sep ⊫ R_copy src dst sig n is_sep.
 #src #dst #sig #n #is_sep #Hneq #Hsrc #Hdst #ta #k #outc #Hloop
 lapply (sem_while … (sem_copy_step src dst sig n is_sep Hneq Hsrc Hdst) … Hloop) //
--Hloop * #tb * #Hstar @(star_ind_l ??????? Hstar) -Hstar
-[ #tc whd in ⊢ (%→?); *
+-Hloop * #tb * #Hstar @(star_ind_l ??????? Hstar) -Hstar -ta
+[ whd in ⊢ (%→?); *
   [ * #x * * #Hx #Hsep #Houtc % [ %
     [ #ls #x0 #xs #rs #sep #Hsrctc #Hnosep >Hsrctc in Hx; normalize in ⊢ (%→?);
       #Hx0 destruct (Hx0) lapply (Hnosep ? (memb_hd …)) >Hsep
@@ -192,15 +192,15 @@ lapply (sem_while … (sem_copy_step src dst sig n is_sep Hneq Hsrc Hdst) … Hl
     | #c #Hc #Hsepc @Houtc ]
     | #_ @Houtc ]
   ]
-| #tc #td #te * #c0 * * #Hc0 #Hc0nosep #Hd #Hstar #IH #He 
+| #td #te * #c0 * * #Hc0 #Hc0nosep #Hd #Hstar #IH #He 
   lapply (IH He) -IH * * #IH1 #IH2 #IH3 % [ %
   [ #ls #x #xs #rs #sep #Hsrc_tc #Hnosep #Hsep #ls0 #x0 #target
     #c #rs0 #Hlen #Hdst_tc
     >Hsrc_tc in Hc0; normalize in ⊢ (%→?); #Hc0 destruct (Hc0)
-    <(change_vec_same … tc src (niltape ?)) in Hd:(???(???(???%??)??));
-    <(change_vec_same … tc dst (niltape ?)) in ⊢(???(???(???%??)??)→?);
-    >Hdst_tc >Hsrc_tc >change_vec_change_vec >change_vec_change_vec
-    >(change_vec_commute ?? tc ?? dst src) [|@(sym_not_eq … Hneq)]
+    <(change_vec_same … td src (niltape ?)) in Hd:(???(???(???%??)??));
+    <(change_vec_same … td dst (niltape ?)) in ⊢(???(???(???%??)??)→?);
+    >Hdst_tc >Hsrc_tc >(change_vec_change_vec ?) >change_vec_change_vec
+    >(change_vec_commute ?? td ?? dst src) [|@(sym_not_eq … Hneq)]
     >change_vec_change_vec @(list_cases2 … Hlen)
     [ #Hxsnil #Htargetnil #Hd>(IH2 … Hsep)
       [ >Hd -Hd >Hxsnil >Htargetnil @(eq_vec … (niltape ?))
@@ -221,9 +221,9 @@ lapply (sem_while … (sem_copy_step src dst sig n is_sep Hneq Hsrc Hdst) … Hl
         >nth_change_vec // >nth_change_vec // >Hxsnil % ]
     |#hd1 #hd2 #tl1 #tl2 #Hxs #Htarget >Hxs >Htarget #Hd
      >(IH1 (c0::ls) hd1 tl1 rs sep ?? Hsep (c0::ls0) hd2 tl2 c rs0)
-     [ >Hd >(change_vec_commute … ?? tc ?? src dst) //
+     [ >Hd >(change_vec_commute … ?? td ?? src dst) //
        >change_vec_change_vec
-       >(change_vec_commute … ?? tc ?? dst src) [|@sym_not_eq //]
+       >(change_vec_commute … ?? td ?? dst src) [|@sym_not_eq //]
        >change_vec_change_vec
        >reverse_cons >associative_append >associative_append % 
      | >Hd >nth_change_vec // >nth_change_vec_neq // >Hdst_tc >Htarget //
diff --git a/matita/matita/lib/turing/multi_universal/match.ma b/matita/matita/lib/turing/multi_universal/match.ma
index 9c4893727..ff31367f7 100644
--- a/matita/matita/lib/turing/multi_universal/match.ma
+++ b/matita/matita/lib/turing/multi_universal/match.ma
@@ -370,7 +370,7 @@ src ≠ dst → src < S n → dst < S n →
 #src #dst #sig #n #is_startc #is_endc #Hneq #Hsrc #Hdst #ta #k #outc #Hloop
 lapply (sem_while … (sem_match_step src dst sig n is_startc is_endc Hneq Hsrc Hdst) … Hloop) //
 -Hloop * #tb * #Hstar @(star_ind_l ??????? Hstar) -Hstar
-[ #tc #Hfalse #ls #x #xs #end #rs #Hmid_src #Hnotend #Hend #Hnotstart
+[ #Hfalse #ls #x #xs #end #rs #Hmid_src #Hnotend #Hend #Hnotstart
   cases (Hfalse … Hmid_src Hnotend Hend) -Hfalse 
   [(* current dest = None *) *
     [ * #Hcur_dst #Houtc %
@@ -401,7 +401,7 @@ lapply (sem_while … (sem_match_step src dst sig n is_startc is_endc Hneq Hsrc
      >reverse_cons >associative_append @(HFalse ?? Hnotnil)
     ]
   ]
-|-ta -tb #ta #tb #tc #Htrue #Hstar #IH #Hout lapply (IH Hout) -IH -Hout #IH whd
+|-ta #ta #tc #Htrue #Hstar #IH #Hout lapply (IH Hout) -IH -Hout #IH whd
  #ls #x #xs #end #rs #Hmid_src #Hnotend #Hend #Hnotstart 
  lapply (refl ? (current ? (nth dst ? ta (niltape ?))))
  cases (current ? (nth dst ? ta (niltape ?))) in ⊢ (???%→?); 
diff --git a/matita/matita/lib/turing/multi_universal/moves.ma b/matita/matita/lib/turing/multi_universal/moves.ma
index 519d33091..23b872463 100644
--- a/matita/matita/lib/turing/multi_universal/moves.ma
+++ b/matita/matita/lib/turing/multi_universal/moves.ma
@@ -211,7 +211,7 @@ lemma wsem_parmoveL : ∀src,dst,sig,n,is_sep.src ≠ dst → src < S n → dst
 #src #dst #sig #n #is_sep #Hneq #Hsrc #Hdst #ta #k #outc #Hloop
 lapply (sem_while … (sem_parmove_step src dst sig n L is_sep Hneq Hsrc Hdst) … Hloop) //
 -Hloop * #tb * #Hstar @(star_ind_l ??????? Hstar) -Hstar
-[ #tc whd in ⊢ (%→?); * #H #Houtc % [2: #_ @Houtc ] cases H
+[ whd in ⊢ (%→?); * #H #Houtc % [2: #_ @Houtc ] cases H
   [ * [ * #x * #Hx #Hsep #ls #x0 #xs #rs #sep #Hsrctc #Hnosep >Hsrctc in Hx; normalize in ⊢ (%→?);
     #Hx0 destruct (Hx0) lapply (Hnosep ? (memb_hd …)) >Hsep
     #Hfalse destruct (Hfalse)
@@ -220,7 +220,7 @@ lapply (sem_while … (sem_parmove_step src dst sig n L is_sep Hneq Hsrc Hdst) 
   |#Hcur_dst  #ls #x0 #xs #rs #sep #Hsrctc #Hnosep #Hsep #ls0 #x1 #target 
    #c #rs0 #Hlen #Hdsttc >Hdsttc in Hcur_dst; normalize in ⊢ (%→?); #H destruct (H)
   ]  
-| #tc #td #te * #c0 * #c1 * * * #Hc0 #Hc1 #Hc0nosep #Hd #Hstar #IH #He 
+| #td #te * #c0 * #c1 * * * #Hc0 #Hc1 #Hc0nosep #Hd #Hstar #IH #He 
   lapply (IH He) -IH * #IH1 #IH2 %
   [ #ls #x #xs #rs #sep #Hsrc_tc #Hnosep #Hsep #ls0 #x0 #target
     #c #rs0 #Hlen #Hdst_tc
@@ -247,9 +247,9 @@ lapply (sem_while … (sem_parmove_step src dst sig n L is_sep Hneq Hsrc Hdst) 
       ]
     | #hd1 #hd2 #tl1 #tl2 #Hxs #Htarget >Hxs >Htarget #Hd
       >(IH1 ls hd1 tl1 (c0::rs) sep ?? Hsep ls0 hd2 tl2 c (x0::rs0))
-      [ >Hd >(change_vec_commute … ?? tc ?? src dst) //
+      [ >Hd >(change_vec_commute … ?? td ?? src dst) //
        >change_vec_change_vec
-       >(change_vec_commute … ?? tc ?? dst src) [|@sym_not_eq //]
+       >(change_vec_commute … ?? td ?? dst src) [|@sym_not_eq //]
        >change_vec_change_vec
        >reverse_cons >associative_append
        >reverse_cons >associative_append % 
diff --git a/matita/matita/lib/turing/ntm.ma b/matita/matita/lib/turing/ntm.ma
index 603a0a0da..155b4cb92 100644
--- a/matita/matita/lib/turing/ntm.ma
+++ b/matita/matita/lib/turing/ntm.ma
@@ -122,7 +122,7 @@ let rec mem A (a:A) (l:list A) on l ≝
   [ nil ⇒ False
   | cons hd tl ⇒ a=hd ∨ mem A a tl
   ]. 
-
+(* this no longer works: TODO 
 definition reach ≝ λsig.λM:NTM sig.λc,c1:config sig M.
   ∃q,l,q1,mvs. 
     state ?? c = q ∧ 
@@ -131,14 +131,14 @@ definition reach ≝ λsig.λM:NTM sig.λc,c1:config sig M.
     mem ? 〈〈q,l〉,〈q1,mvs〉〉 (trans ? M)  ∧ 
     state ?? c1 = q1 ∧
     tapes ?? c1 = (compose_vec ??? (tape_move sig) ? (tapes ?? c) mvs).
-
+*)
 (*
 definition empty_tapes ≝ λsig.λn.
 mk_Vector ? n (make_list (tape sig) (mk_tape sig [] []) n) ?.
 elim n // normalize //
 qed.
 *)
-
+(* this no longer works: TODO
 definition init ≝ λsig.λM:NTM sig.λi:(list sig).
   mk_config ??
     (start sig M)
@@ -147,4 +147,4 @@ definition init ≝ λsig.λM:NTM sig.λi:(list sig).
 definition accepted ≝ λsig.λM:NTM sig.λw:(list sig).
 ∃c. star ? (reach sig M) (init sig M w) c ∧
   accept ?? (state ?? c) = true.
-
+*)
\ No newline at end of file
diff --git a/matita/matita/lib/turing/oracle.ma b/matita/matita/lib/turing/oracle.ma
index 4b6d7b1c7..4e7495d29 100644
--- a/matita/matita/lib/turing/oracle.ma
+++ b/matita/matita/lib/turing/oracle.ma
@@ -40,14 +40,14 @@ definition option_hd ≝ λA.λl:list A.
   [nil ⇒ None ?
   |cons a _ ⇒ Some ? a
   ].
-
+(* this no longer works: TODO 
 definition tape_move ≝ λsig.λt: tape sig.λm:sig × move.
   match \snd m with
   [ R ⇒ mk_tape sig ((\fst m)::(left ? t)) (tail ? (right ? t))
   | L ⇒ mk_tape sig (tail ? (left ? t)) ((\fst m)::(right ? t))
   | N ⇒ mk_tape sig (left ? t) ((\fst m)::(tail ? (right ? t)))
   ].
-
+*)
 definition current_chars ≝ λsig.λM:TM sig.λc:config sig M.
   vec_map ?? (λt.option_hd ? (right ? t)) (S (tapes_no sig M)) (tapes ?? c).
 
@@ -56,7 +56,7 @@ definition opt_cons ≝ λA.λa:option A.λl:list A.
   [ None ⇒ l
   | Some a ⇒ a::l
   ].
-
+(* this no longer works: TODO 
 definition step ≝ λsig.λM:TM sig.λc:config sig M.
   let 〈news,mvs,outchar〉 ≝ trans sig M 〈state ?? c,current_chars ?? c〉 in
   mk_config ?? 
@@ -74,7 +74,7 @@ definition init ≝ λsig.λM:TM sig.λi:(list sig).
     (start sig M)
     (vec_cons ? (mk_tape sig [] i) ? (empty_tapes sig (tapes_no sig M)))
     [ ].
-
+*)
 definition stop ≝ λsig.λM:TM sig.λc:config sig M.
   halt sig M (state sig M c).
 
@@ -83,7 +83,7 @@ let rec loop (A:Type[0]) n (f:A→A) p a on n ≝
   [ O ⇒ None ?
   | S m ⇒ if p a then (Some ? a) else loop A m f p (f a)
   ].
-
+(* this no longer works: TODO 
 (* Compute ? M f states that f is computed by M *)
 definition Compute ≝ λsig.λM:TM sig.λf:(list sig) → (list sig).
 ∀l.∃i.∃c.
@@ -97,3 +97,4 @@ definition ComputeB ≝ λsig.λM:TM sig.λf:(list sig) → bool.
 ∀l.∃i.∃c.
   loop ? i (step sig M) (stop sig M) (init sig M l) = Some ? c ∧
   (isnilb ? (out ?? c) = false).
+*)
\ No newline at end of file
diff --git a/matita/matita/lib/turing/turing_old.ma b/matita/matita/lib/turing/turing_old.ma
index f50426985..cafe3aadc 100644
--- a/matita/matita/lib/turing/turing_old.ma
+++ b/matita/matita/lib/turing/turing_old.ma
@@ -49,7 +49,7 @@ normalize >(len A n v) //
 qed.
 
 lemma length_map: ∀A,B,l.∀f:A→B. length ? (map ?? f l) = length ? l.
-#A #B #l #f elim l // #a #tl #Hind normalize //
+#A #B #l #f elim l //
 qed.
 
 definition vec_map ≝ λA,B.λf:A→B.λn.λv:Vector A n.
diff --git a/matita/matita/lib/turing/universal.ma b/matita/matita/lib/turing/universal.ma
index 295a5610f..35d7e0290 100644
--- a/matita/matita/lib/turing/universal.ma
+++ b/matita/matita/lib/turing/universal.ma
@@ -184,6 +184,7 @@ STATO 8:
 STATO 9:
   stato finale
 *)
+(* TODO
 match s with
  [ q0 ⇒ match c with
    [ bit x ⇒ 〈q1,〈mark x,L〉〉
@@ -229,7 +230,7 @@ match s with
    | mark D ⇒ 〈q9,〈bit D,L〉〉 ]
  | q9 ⇒ ? (* successo *)
  ].
-
+*)
 (*
 ==================================
 MACCHINE PER SPOSTARE LA "TESTINA"
@@ -290,7 +291,7 @@ STATO 3:
 STATO 4:
   stato finale
 *)
-
+(* TODO
 match s with
  [ q0 ⇒ match c with
    [ bit x ⇒ 〈q1 x,〈c,L〉〉
@@ -301,7 +302,7 @@ match s with
  | q2 d ⇒ 〈q0,〈d,L〉〉
  | q3 ⇒ 〈q4,〈grid,L〉〉
  | q4 ⇒ (* finale *) ].
-
+*)
 (*
 MACCHINA B
 ----------
@@ -334,7 +335,7 @@ STATO 3:
 STATO 4:
   stato finale
 *)
-
+(* TODO
 match s with
 [ q0 ⇒ match c with
   [ bit x ⇒ 〈q1 x,〈c,R〉〉
@@ -345,7 +346,7 @@ match s with
 | q2 d ⇒ 〈q0,〈d,R〉〉
 | q3 ⇒ 〈q4,〈grid,L〉〉 
 | q4 ⇒ (* finale *) ].
-
+*)
 (*
 MACCHINA C
 ----------
@@ -375,7 +376,7 @@ STATO 3:
   stato finale
 
 *)
-
+(* TODO
 match s with 
 [ q0 ⇒ match c with
   [ bit x ⇒ 〈q1 x,〈c,R〉〉
@@ -385,3 +386,4 @@ match s with
   | _ ⇒ 〈q2 c,〈bit x,L〉〉
 | q2 d ⇒ 〈q0,〈c,R〉〉
 | q3 ⇒ (* finale *) ].
+*)
\ No newline at end of file
diff --git a/matita/matita/lib/turing/universal/copy.ma b/matita/matita/lib/turing/universal/copy.ma
index 8693f1f1d..b2fcf29f1 100644
--- a/matita/matita/lib/turing/universal/copy.ma
+++ b/matita/matita/lib/turing/universal/copy.ma
@@ -292,7 +292,7 @@ lemma wsem_copy0 : WRealize ? copy0 R_copy0.
 #intape #k #outc #Hloop 
 lapply (sem_while … sem_copy_step intape k outc Hloop) [%] -Hloop
 * #ta * #Hstar @(star_ind_l ??????? Hstar)
-[ #tb whd in ⊢ (%→?); #Hleft
+[ whd in ⊢ (%→?); #Hleft
   #ls #c #c0 #rs #l1 #l3 #l4 #Htb #Hl1nomarks #Hl3l4nomarks #Hlen #l1' #bv
   #Hl1 #Hl1bits #l4' #bg #Hl4 #Hl4bits
   cases (Hleft … Htb) -Hleft #Hc #Houtc % %
@@ -302,7 +302,7 @@ lapply (sem_while … sem_copy_step intape k outc Hloop) [%] -Hloop
       #Hl1bits lapply (Hl1bits 〈c',false〉 ?) [ @memb_hd ] 
       >Hc #Hfalse destruct ]
   | @Houtc ]
-| #tb #tc #td whd in ⊢ (%→?→(?→%)→%→?); #Htc #Hstar1 #Hind #Htd
+| #tc #td whd in ⊢ (%→?→(?→%)→%→?); #Htc #Hstar1 #Hind #Htd
   lapply (Hind Htd) -Hind #Hind
   #ls #c #c0 #rs #l1 #l3 #l4 #Htb #Hl1nomarks #Hl3l4nomarks #Hlen #l1' #bv
   #Hl1 #Hl1bits #l4' #bg #Hl4 #Hl4bits %2
diff --git a/matita/matita/lib/turing/universal/marks.ma b/matita/matita/lib/turing/universal/marks.ma
index e82d577bc..8f1e1f129 100644
--- a/matita/matita/lib/turing/universal/marks.ma
+++ b/matita/matita/lib/turing/universal/marks.ma
@@ -146,7 +146,7 @@ lemma wsem_adv_to_mark_r :
 #alpha #test #t #i #outc #Hloop
 lapply (sem_while … (sem_atmr_step alpha test) t i outc Hloop) [%]
 -Hloop * #t1 * #Hstar @(star_ind_l ??????? Hstar)
-[ #tapea * #Htapea *
+[ * #Htapea *
   [ #H1 %
     [#_ @Htapea 
     |#ls #c #rs #H2 >H2 in H1; whd in ⊢ (??%? → ?);
@@ -159,7 +159,7 @@ lapply (sem_while … (sem_atmr_step alpha test) t i outc Hloop) [%]
      <Htapea //
     ]
   ]
-| #tapea #tapeb #tapec #Hleft #Hright #IH #HRfalse
+| #tapeb #tapec #Hleft #Hright #IH #HRfalse
   lapply (IH HRfalse) -IH #IH %
   [cases Hleft #ls * #a * #rs * * #Htapea #_ #_ >Htapea
    whd in ⊢((??%?)→?); #H destruct (H);
@@ -501,7 +501,7 @@ lemma wsem_adv_to_mark_l :
 #alpha #test #t #i #outc #Hloop
 lapply (sem_while … (sem_atml_step alpha test) t i outc Hloop) [%]
 -Hloop * #t1 * #Hstar @(star_ind_l ??????? Hstar)
-[ #tapea * #Htapea *
+[ * #Htapea *
   [ #H1 %
     [#_ @Htapea
     |#ls #c #rs #H2 >H2 in H1; whd in ⊢ (??%? → ?);
@@ -516,7 +516,7 @@ lapply (sem_while … (sem_atml_step alpha test) t i outc Hloop) [%]
       ]
     ]
   ]
-| #tapea #tapeb #tapec #Hleft #Hright #IH #HRfalse
+| #tapeb #tapec #Hleft #Hright #IH #HRfalse
   lapply (IH HRfalse) -IH #IH %
   [cases Hleft #ls0 * #a0 * #rs0 * * #Htapea #_ #_ >Htapea
    whd in ⊢ ((??%?)→?); #H destruct (H)
@@ -1200,7 +1200,7 @@ lemma wsem_compare : WRealize ? compare R_compare.
 #t #i #outc #Hloop
 lapply (sem_while ?????? sem_comp_step t i outc Hloop) [%]
 -Hloop * #t1 * #Hstar @(star_ind_l ??????? Hstar)
-[ #tapea whd in ⊢ (%→?); #Rfalse #ls #c #rs #Htapea %
+[ whd in ⊢ (%→?); #Rfalse #ls #c #rs #Htapea %
   [ %
     [ #c' #Hc' #Hc lapply (Rfalse … Htapea) -Rfalse * >Hc
       whd in ⊢ (??%?→?); #Hfalse destruct (Hfalse)
@@ -1209,7 +1209,7 @@ lapply (sem_while ?????? sem_comp_step t i outc Hloop) [%]
   | #b #b0 #bs #b0s #l1 #l2 #Hlen #Hbs1 #Hb0s1 #Hbs2 #Hb0s2 #Hl1 #Hc
     cases (Rfalse … Htapea) -Rfalse >Hc whd in ⊢ (??%?→?);#Hfalse destruct (Hfalse)
   ]
-| #tapea #tapeb #tapec #Hleft #Hright #IH #Htapec lapply (IH Htapec) -Htapec -IH #IH
+| #tapeb #tapec #Hleft #Hright #IH #Htapec lapply (IH Htapec) -Htapec -IH #IH
   whd in Hleft; #ls #c #rs #Htapea cases Hleft -Hleft
   #ls0 * #c' * #rs0 * >Htapea #Hdes destruct (Hdes) * * 
   cases (true_or_false (bit_or_null c')) #Hc'
diff --git a/matita/matita/lib/turing/universal/match_machines.ma b/matita/matita/lib/turing/universal/match_machines.ma
index 49f1b6923..eef43f14a 100644
--- a/matita/matita/lib/turing/universal/match_machines.ma
+++ b/matita/matita/lib/turing/universal/match_machines.ma
@@ -628,7 +628,7 @@ lemma wsem_match_tuple : WRealize ? match_tuple R_match_tuple0.
 #intape #k #outc #Hloop 
 lapply (sem_while … sem_match_tuple_step intape k outc Hloop) [%] -Hloop
 * #ta * #Hstar @(star_ind_l ??????? Hstar)
-[ #tb whd in ⊢ (%→?); #Hleft
+[ whd in ⊢ (%→?); #Hleft
   #ls #cur #rs #Htb cases (Hleft … Htb) #Hgrid #Houtc %
   [ #_ @Houtc 
   | #c #l1 #c1 #l2 #l3 #ls0 #rs0 #n #Hls #Hcur #Hrs 
@@ -637,7 +637,7 @@ lapply (sem_while … sem_match_tuple_step intape k outc Hloop) [%] -Hloop
   ]
 | (* in the interesting case, we execute a true iteration, then we restart the
      while cycle, finally we end with a false iteration *)
-  #tb #tc #td whd in ⊢ (%→?); #Htc
+  #tc #td whd in ⊢ (%→?); #Htc
   #Hstar1 #IH whd in ⊢ (%→?); #Hright lapply (IH Hright) -IH whd in ⊢ (%→?); #IH
   #ls #cur #rs #Htb %
   [ (* cur can't be true because we assume at least one iteration *)
@@ -710,16 +710,20 @@ lapply (sem_while … sem_match_tuple_step intape k outc Hloop) [%] -Hloop
    | (* match failed and there is no next tuple: the next while cycle will just exit *)
      * * #Hdiff #Hnobars generalize in match (refl ? tc);
      cases tc in ⊢ (???% → %);
-     [ #_ normalize in ⊢ (??%?→?); #Hfalse destruct (Hfalse)
-     |2,3: #x #xs #_ normalize in ⊢ (??%?→?); #Hfalse destruct (Hfalse) ]
-     #ls1 #cur1 #rs1 #Htc normalize in ⊢ (??%?→?); #Hcur1
+     [ #_ @daemon (* long normalize *) (* 
+          normalize in ⊢ (??%?→?); #Hfalse destruct (Hfalse)
+     *)
+     |2,3: #x #xs #_ @daemon (* long normalize *) (* 
+                     normalize in ⊢ (??%?→?); #Hfalse destruct (Hfalse) *) ]
+     #ls1 #cur1 #rs1 #Htc @daemon (* long normalize *) (* 
+                          normalize in ⊢ (??%?→?); #Hcur1
      cases (IH … Htc) -IH #IH #_ %2 %
      [ destruct (Hcur1) >IH [ >Htc % | % ]
      | #l4 #newc #mv0 #l5
        (* no_bars except the first one, where the tuple does not match ⇒ 
           no match *)
         @daemon
-     ]
+     ] *)
    ]
  ]
 qed.
diff --git a/matita/matita/lib/turing/universal/uni_step.ma b/matita/matita/lib/turing/universal/uni_step.ma
index 043ee6dd4..eabab3e83 100644
--- a/matita/matita/lib/turing/universal/uni_step.ma
+++ b/matita/matita/lib/turing/universal/uni_step.ma
@@ -540,6 +540,7 @@ definition Pre_uni_step ≝ λt1.
 lemma sem_uni_step :
   accGRealize ? uni_step us_acc Pre_uni_step
      R_uni_step_true R_uni_step_false. 
+@daemon (* this no longer works: TODO *) (*
 @(acc_sem_if_app_guarded STape … (sem_test_char ? (λc:STape.\fst c == bit false)) 
   ? (test_char_inv …) (sem_nop …) …)
 [| @(sem_seq_app_guarded … (Realize_to_GRealize … sem_init_match) ???)
@@ -679,6 +680,7 @@ lemma sem_uni_step :
   #b #Hb >Hb in Ht1; * #Hc #Ht1 lapply (Hc ? (refl ??)) -Hc #Hb' %
   // cases b in Hb'; normalize #H1 //
 ]
+*)
 qed.
 
 (*
diff --git a/matita/matita/lib/turing/universal/universal.ma b/matita/matita/lib/turing/universal/universal.ma
index 2b8618213..20e35ca3f 100644
--- a/matita/matita/lib/turing/universal/universal.ma
+++ b/matita/matita/lib/turing/universal/universal.ma
@@ -337,14 +337,17 @@ definition low_R ≝ λM,qstart,R,t1,t2.
 
 lemma sem_uni_step1: 
   uni_step ⊨ [us_acc: low_step_R_true, low_step_R_false].
+@daemon (* this no longer works: TODO *) (*
 @(acc_Realize_to_acc_Realize … sem_uni_step) 
   [@unistep_true_to_low_step | @unistep_false_to_low_step ]
+*)
 qed. 
 
 definition universalTM ≝ whileTM ? uni_step us_acc.
 
 theorem sem_universal: ∀M:normalTM. ∀qstart.
   universalTM ⊫ (low_R M qstart (R_TM FinBool M qstart)).
+@daemon (* this no longer works: TODO *) (*
 #M #q #intape #i #outc #Hloop
 lapply (sem_while … sem_uni_step1 intape i outc Hloop)
   [@daemon] -Hloop 
@@ -379,6 +382,7 @@ lapply (sem_while … sem_uni_step1 intape i outc Hloop)
     |@Houtc
     ]
   ]
+*)  
 qed.
 
 theorem sem_universal2: ∀M:normalTM. ∀R.
diff --git a/matita/matita/lib/turing/wmono.ma b/matita/matita/lib/turing/wmono.ma
index 6b7849709..f5a8dfe1b 100644
--- a/matita/matita/lib/turing/wmono.ma
+++ b/matita/matita/lib/turing/wmono.ma
@@ -363,9 +363,12 @@ lemma eq_ctape_lift_conf_R : ∀sig,S1,S2,outc.
 #sig #S1 #S2 #outc cases outc #s #t %
 qed.
 
+axiom daemon :∀P:Prop.P.
+
 theorem sem_seq: ∀sig,M1,M2,R1,R2.
   Realize sig M1 R1 → Realize sig M2 R2 → 
     Realize sig (seq sig M1 M2) (R1 ∘ R2).
+@daemon (* this no longer works: TODO *) (*
 #sig #M1 #M2 #R1 #R2 #HR1 #HR2 #t #i #outc #Hloop
 cases (HR1 t) #k1 * #outc1 * #Hloop1 #HM1
 cases (HR2 (ctape sig (states ? M1) outc1)) #k2 * #outc2 * #Hloop2 #HM2
@@ -395,5 +398,6 @@ cases (HR2 (ctape sig (states ? M1) outc1)) #k2 * #outc2 * #Hloop2 #HM2
 | @(ex_intro … (ctape ? (FinSum (states ? M1) (states ? M2)) (lift_confL … outc1)))
   % // >eq_ctape_lift_conf_L >eq_ctape_lift_conf_R //
 ]
+*)
 qed.