From b6e1db4f1b0f1d5121f2b214562f96c5b0fa544e Mon Sep 17 00:00:00 2001
From: Ferruccio Guidi <ferruccio.guidi@unibo.it>
Date: Mon, 4 Jul 2016 21:06:25 +0000
Subject: [PATCH] basic properties of cpr ...

---
 .../lambdadelta/basic_2/rt_transition/cpm.ma  |  18 +
 .../lambdadelta/basic_2/rt_transition/cpr.etc |  87 +++++
 .../lambdadelta/basic_2/rt_transition/cpr.ma  | 325 +++---------------
 .../basic_2/rt_transition/partial.txt         |   1 +
 .../lambdadelta/ground_2/lib/arith.ma         |   4 +
 .../lambdadelta/ground_2/steps/rtc_max.ma     |  84 +++++
 .../lambdadelta/ground_2/steps/rtc_plus.ma    |   9 +-
 7 files changed, 252 insertions(+), 276 deletions(-)
 create mode 100644 matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpr.etc
 create mode 100644 matita/matita/contribs/lambdadelta/ground_2/steps/rtc_max.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpm.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpm.ma
index 3201a5b8f..7138b3210 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpm.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpm.ma
@@ -53,6 +53,7 @@ lemma cpm_lref: ∀n,h,I,G,K,V,T,U,i. ⦃G, K⦄ ⊢ #i ➡[n, h] T →
 /3 width=5 by cpg_lref, ex2_intro/
 qed.
 
+(* Basic_2A1: includes: cpr_bind *)
 lemma cpm_bind: ∀n,h,p,I,G,L,V1,V2,T1,T2.
                 ⦃G, L⦄ ⊢ V1 ➡[h] V2 → ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ➡[n, h] T2 →
                 ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ➡[n, h] ⓑ{p,I}V2.T2.
@@ -60,6 +61,9 @@ lemma cpm_bind: ∀n,h,p,I,G,L,V1,V2,T1,T2.
 /5 width=5 by cpg_bind, isrt_plus_O1, isr_shift, ex2_intro/
 qed.
 
+(* Note: cpr_flat: does not hold in basic_1 *)
+(* Basic_1: includes: pr2_thin_dx *)
+(* Basic_2A1: includes: cpr_flat *)
 lemma cpm_flat: ∀n,h,I,G,L,V1,V2,T1,T2.
                 ⦃G, L⦄ ⊢ V1 ➡[h] V2 → ⦃G, L⦄ ⊢ T1 ➡[n, h] T2 →
                 ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ➡[n, h] ⓕ{I}V2.T2.
@@ -67,12 +71,14 @@ lemma cpm_flat: ∀n,h,I,G,L,V1,V2,T1,T2.
 /5 width=5 by isrt_plus_O1, isr_shift, cpg_flat, ex2_intro/
 qed.
 
+(* Basic_2A1: includes: cpr_zeta *)
 lemma cpm_zeta: ∀n,h,G,L,V,T1,T,T2. ⦃G, L.ⓓV⦄ ⊢ T1 ➡[n, h] T →
                 ⬆*[1] T2 ≡ T → ⦃G, L⦄ ⊢ +ⓓV.T1 ➡[n, h] T2.
 #n #h #G #L #V #T1 #T #T2 *
 /3 width=5 by cpg_zeta, isrt_plus_O2, ex2_intro/
 qed.
 
+(* Basic_2A1: includes: cpr_eps *)
 lemma cpm_eps: ∀n,h,G,L,V,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[n, h] T2 → ⦃G, L⦄ ⊢ ⓝV.T1 ➡[n, h] T2.
 #n #h #G #L #V #T1 #T2 *
 /3 width=3 by cpg_eps, isrt_plus_O2, ex2_intro/
@@ -83,6 +89,7 @@ lemma cpm_ee: ∀n,h,G,L,V1,V2,T. ⦃G, L⦄ ⊢ V1 ➡[n, h] V2 → ⦃G, L⦄
 /3 width=3 by cpg_ee, isrt_succ, ex2_intro/
 qed.
 
+(* Basic_2A1: includes: cpr_beta *)
 lemma cpm_beta: ∀n,h,p,G,L,V1,V2,W1,W2,T1,T2.
                 ⦃G, L⦄ ⊢ V1 ➡[h] V2 → ⦃G, L⦄ ⊢ W1 ➡[h] W2 → ⦃G, L.ⓛW1⦄ ⊢ T1 ➡[n, h] T2 →
                 ⦃G, L⦄ ⊢ ⓐV1.ⓛ{p}W1.T1 ➡[n, h] ⓓ{p}ⓝW2.V2.T2.
@@ -90,6 +97,7 @@ lemma cpm_beta: ∀n,h,p,G,L,V1,V2,W1,W2,T1,T2.
 /6 width=7 by cpg_beta, isrt_plus_O2, isrt_plus, isr_shift, ex2_intro/
 qed.
 
+(* Basic_2A1: includes: cpr_theta *)
 lemma cpm_theta: ∀n,h,p,G,L,V1,V,V2,W1,W2,T1,T2.
                  ⦃G, L⦄ ⊢ V1 ➡[h] V → ⬆*[1] V ≡ V2 → ⦃G, L⦄ ⊢ W1 ➡[h] W2 →
                  ⦃G, L.ⓓW1⦄ ⊢ T1 ➡[n, h] T2 →
@@ -100,10 +108,12 @@ qed.
 
 (* Basic properties on r-transition *****************************************)
 
+(* Basic_1: includes by definition: pr0_refl *)
 (* Basic_2A1: includes: cpr_atom *)
 lemma cpr_refl: ∀h,G,L. reflexive … (cpm 0 h G L).
 /2 width=3 by ex2_intro/ qed.
 
+(* Basic_1: was: pr2_head_1 *)
 lemma cpr_pair_sn: ∀h,I,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 →
                    ∀T. ⦃G, L⦄ ⊢ ②{I}V1.T ➡[h] ②{I}V2.T.
 #h #I #G #L #V1 #V2 *
@@ -173,6 +183,7 @@ lemma cpm_inv_gref1: ∀n,h,G,L,T2,l. ⦃G, L⦄ ⊢ §l ➡[n, h] T2 → T2 = 
 #H1 #H2 destruct /3 width=1 by isrt_inv_00, conj/ 
 qed-.
 
+(* Basic_2A1: includes: cpr_inv_bind1 *)
 lemma cpm_inv_bind1: ∀n,h,p,I,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ➡[n, h] U2 → (
                      ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 & ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ➡[n, h] T2 &
                               U2 = ⓑ{p,I}V2.T2
@@ -189,6 +200,8 @@ lemma cpm_inv_bind1: ∀n,h,p,I,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ➡[n
 ]
 qed-.
 
+(* Basic_1: includes: pr0_gen_abbr pr2_gen_abbr *)
+(* Basic_2A1: includes: cpr_inv_abbr1 *)
 lemma cpm_inv_abbr1: ∀n,h,p,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓓ{p}V1.T1 ➡[n, h] U2 → (
                      ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 & ⦃G, L.ⓓV1⦄ ⊢ T1 ➡[n, h] T2 &
                               U2 = ⓓ{p}V2.T2
@@ -204,6 +217,8 @@ lemma cpm_inv_abbr1: ∀n,h,p,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓓ{p}V1.T1 ➡[n, h]
 ]
 qed-.
 
+(* Basic_1: includes: pr0_gen_abst pr2_gen_abst *)
+(* Basic_2A1: includes: cpr_inv_abst1 *)
 lemma cpm_inv_abst1: ∀n,h,p,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓛ{p}V1.T1 ➡[n, h] U2 →
                      ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 & ⦃G, L.ⓛV1⦄ ⊢ T1 ➡[n, h] T2 &
                               U2 = ⓛ{p}V2.T2.
@@ -254,6 +269,8 @@ lemma cpm_inv_flat1: ∀n,h,I,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓕ{I}V1.U1 ➡[n, h]
 ]
 qed-.
 
+(* Basic_1: includes: pr0_gen_appl pr2_gen_appl *)
+(* Basic_2A1: includes: cpr_inv_appl1 *)
 lemma cpm_inv_appl1: ∀n,h,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓐ V1.U1 ➡[n, h] U2 →
                      ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 & ⦃G, L⦄ ⊢ U1 ➡[n, h] T2 &
                                  U2 = ⓐV2.T2
@@ -305,6 +322,7 @@ qed-.
 
 (* Basic forward lemmas *****************************************************)
 
+(* Basic_2A1: includes: cpr_fwd_bind1_minus *)
 lemma cpm_fwd_bind1_minus: ∀n,h,I,G,L,V1,T1,T. ⦃G, L⦄ ⊢ -ⓑ{I}V1.T1 ➡[n, h] T → ∀p.
                            ∃∃V2,T2. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ➡[n, h] ⓑ{p,I}V2.T2 &
                                     T = -ⓑ{I}V2.T2.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpr.etc b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpr.etc
new file mode 100644
index 000000000..dee79d193
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpr.etc
@@ -0,0 +1,87 @@
+(* Basic_1: includes: pr2_delta1 *)
+| cpr_delta: ∀G,L,K,V,V2,W2,i.
+             ⬇[i] L ≡ K. ⓓV → cpr G K V V2 →
+             ⬆[0, i + 1] V2 ≡ W2 → cpr G L (#i) W2
+
+lemma cpr_cpx: ∀h,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡ T2 → ⦃G, L⦄ ⊢ T1 ➡[h] T2.
+#h #o #G #L #T1 #T2 #H elim H -L -T1 -T2
+/2 width=7 by cpx_delta, cpx_bind, cpx_flat, cpx_zeta, cpx_eps, cpx_beta, cpx_theta/
+qed.
+
+lemma lsubr_cpr_trans: ∀G. lsub_trans … (cpr G) lsubr.
+#G #L1 #T1 #T2 #H elim H -G -L1 -T1 -T2
+[ //
+| #G #L1 #K1 #V1 #V2 #W2 #i #HLK1 #_ #HVW2 #IHV12 #L2 #HL12
+  elim (lsubr_fwd_drop2_abbr … HL12 … HLK1) -L1 *
+  /3 width=6 by cpr_delta/
+|3,7: /4 width=1 by lsubr_pair, cpr_bind, cpr_beta/
+|4,6: /3 width=1 by cpr_flat, cpr_eps/
+|5,8: /4 width=3 by lsubr_pair, cpr_zeta, cpr_theta/
+]
+qed-.
+
+(* Basic_1: was by definition: pr2_free *)
+lemma tpr_cpr: ∀G,T1,T2. ⦃G, ⋆⦄ ⊢ T1 ➡ T2 → ∀L. ⦃G, L⦄ ⊢ T1 ➡ T2.
+#G #T1 #T2 #HT12 #L
+lapply (lsubr_cpr_trans … HT12 L ?) //
+qed.
+
+lemma cpr_delift: ∀G,K,V,T1,L,l. ⬇[l] L ≡ (K.ⓓV) →
+                  ∃∃T2,T. ⦃G, L⦄ ⊢ T1 ➡ T2 & ⬆[l, 1] T ≡ T2.
+#G #K #V #T1 elim T1 -T1
+[ * /2 width=4 by cpr_atom, lift_sort, lift_gref, ex2_2_intro/
+  #i #L #l #HLK elim (lt_or_eq_or_gt i l)
+  #Hil [1,3: /4 width=4 by lift_lref_ge_minus, lift_lref_lt, ylt_inj, yle_inj, ex2_2_intro/ ]
+  destruct
+  elim (lift_total V 0 (i+1)) #W #HVW
+  elim (lift_split … HVW i i) /3 width=6 by cpr_delta, ex2_2_intro/
+| * [ #a ] #I #W1 #U1 #IHW1 #IHU1 #L #l #HLK
+  elim (IHW1 … HLK) -IHW1 #W2 #W #HW12 #HW2
+  [ elim (IHU1 (L. ⓑ{I}W1) (l+1)) -IHU1 /3 width=9 by drop_drop, cpr_bind, lift_bind, ex2_2_intro/
+  | elim (IHU1 … HLK) -IHU1 -HLK /3 width=8 by cpr_flat, lift_flat, ex2_2_intro/
+  ]
+]
+qed-.
+
+fact lstas_cpr_aux: ∀h,G,L,T1,T2,d. ⦃G, L⦄ ⊢ T1 •*[h, d] T2 →
+                    d = 0 → ⦃G, L⦄ ⊢ T1 ➡ T2.
+#h #G #L #T1 #T2 #d #H elim H -G -L -T1 -T2 -d
+/3 width=1 by cpr_eps, cpr_flat, cpr_bind/
+[ #G #L #K #V1 #V2 #W2 #i #d #HLK #_ #HVW2 #IHV12 #H destruct
+  /3 width=6 by cpr_delta/
+| #G #L #K #V1 #V2 #W2 #i #d #_ #_ #_ #_ <plus_n_Sm #H destruct
+]
+qed-.
+
+lemma lstas_cpr: ∀h,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 •*[h, 0] T2 → ⦃G, L⦄ ⊢ T1 ➡ T2.
+/2 width=4 by lstas_cpr_aux/ qed.
+
+lemma cpr_inv_atom1: ∀I,G,L,T2. ⦃G, L⦄ ⊢ ⓪{I} ➡ T2 →
+                     T2 = ⓪{I} ∨
+                     ∃∃K,V,V2,i. ⬇[i] L ≡ K. ⓓV & ⦃G, K⦄ ⊢ V ➡ V2 &
+                                 ⬆[O, i + 1] V2 ≡ T2 & I = LRef i.
+/2 width=3 by cpr_inv_atom1_aux/ qed-.
+
+(* Basic_1: includes: pr0_gen_lref pr2_gen_lref *)
+lemma cpr_inv_lref1: ∀G,L,T2,i. ⦃G, L⦄ ⊢ #i ➡ T2 →
+                     T2 = #i ∨
+                     ∃∃K,V,V2. ⬇[i] L ≡ K. ⓓV & ⦃G, K⦄ ⊢ V ➡ V2 &
+                               ⬆[O, i + 1] V2 ≡ T2.
+#G #L #T2 #i #H
+elim (cpr_inv_atom1 … H) -H /2 width=1 by or_introl/
+* #K #V #V2 #j #HLK #HV2 #HVT2 #H destruct /3 width=6 by ex3_3_intro, or_intror/
+qed-.
+
+(* Note: the main property of simple terms *)
+lemma cpr_inv_appl1_simple: ∀G,L,V1,T1,U. ⦃G, L⦄ ⊢ ⓐV1. T1 ➡ U → 𝐒⦃T1⦄ →
+                            ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L⦄ ⊢ T1 ➡ T2 &
+                                     U = ⓐV2. T2.
+#G #L #V1 #T1 #U #H #HT1
+elim (cpr_inv_appl1 … H) -H *
+[ /2 width=5 by ex3_2_intro/
+| #a #V2 #W1 #W2 #U1 #U2 #_ #_ #_ #H #_ destruct
+  elim (simple_inv_bind … HT1)
+| #a #V #V2 #W1 #W2 #U1 #U2 #_ #_ #_ #_ #H #_ destruct
+  elim (simple_inv_bind … HT1)
+]
+qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpr.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpr.ma
index c05a2585d..382bb119b 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpr.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpr.ma
@@ -12,297 +12,78 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "basic_2/notation/relations/pred_4.ma".
-include "basic_2/static/lsubr.ma".
-include "basic_2/unfold/lstas.ma".
-
-(* CONTEXT-SENSITIVE PARALLEL REDUCTION FOR TERMS ***************************)
-
-(* activate genv *)
-(* Basic_1: includes: pr0_delta1 pr2_delta1 pr2_thin_dx *)
-(* Note: cpr_flat: does not hold in basic_1 *)
-inductive cpr: relation4 genv lenv term term ≝
-| cpr_atom : ∀I,G,L. cpr G L (⓪{I}) (⓪{I})
-| cpr_delta: ∀G,L,K,V,V2,W2,i.
-             ⬇[i] L ≡ K. ⓓV → cpr G K V V2 →
-             ⬆[0, i + 1] V2 ≡ W2 → cpr G L (#i) W2
-| cpr_bind : ∀a,I,G,L,V1,V2,T1,T2.
-             cpr G L V1 V2 → cpr G (L.ⓑ{I}V1) T1 T2 →
-             cpr G L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2)
-| cpr_flat : ∀I,G,L,V1,V2,T1,T2.
-             cpr G L V1 V2 → cpr G L T1 T2 →
-             cpr G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
-| cpr_zeta : ∀G,L,V,T1,T,T2. cpr G (L.ⓓV) T1 T →
-             ⬆[0, 1] T2 ≡ T → cpr G L (+ⓓV.T1) T2
-| cpr_eps  : ∀G,L,V,T1,T2. cpr G L T1 T2 → cpr G L (ⓝV.T1) T2
-| cpr_beta : ∀a,G,L,V1,V2,W1,W2,T1,T2.
-             cpr G L V1 V2 → cpr G L W1 W2 → cpr G (L.ⓛW1) T1 T2 →
-             cpr G L (ⓐV1.ⓛ{a}W1.T1) (ⓓ{a}ⓝW2.V2.T2)
-| cpr_theta: ∀a,G,L,V1,V,V2,W1,W2,T1,T2.
-             cpr G L V1 V → ⬆[0, 1] V ≡ V2 → cpr G L W1 W2 → cpr G (L.ⓓW1) T1 T2 →
-             cpr G L (ⓐV1.ⓓ{a}W1.T1) (ⓓ{a}W2.ⓐV2.T2)
-.
-
-interpretation "context-sensitive parallel reduction (term)"
-   'PRed G L T1 T2 = (cpr G L T1 T2).
-
-(* Basic properties *********************************************************)
-
-lemma cpr_cpx: ∀h,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡ T2 → ⦃G, L⦄ ⊢ T1 ➡[h] T2.
-#h #o #G #L #T1 #T2 #H elim H -L -T1 -T2
-/2 width=7 by cpx_delta, cpx_bind, cpx_flat, cpx_zeta, cpx_eps, cpx_beta, cpx_theta/
-qed.
-
-lemma lsubr_cpr_trans: ∀G. lsub_trans … (cpr G) lsubr.
-#G #L1 #T1 #T2 #H elim H -G -L1 -T1 -T2
-[ //
-| #G #L1 #K1 #V1 #V2 #W2 #i #HLK1 #_ #HVW2 #IHV12 #L2 #HL12
-  elim (lsubr_fwd_drop2_abbr … HL12 … HLK1) -L1 *
-  /3 width=6 by cpr_delta/
-|3,7: /4 width=1 by lsubr_pair, cpr_bind, cpr_beta/
-|4,6: /3 width=1 by cpr_flat, cpr_eps/
-|5,8: /4 width=3 by lsubr_pair, cpr_zeta, cpr_theta/
+include "basic_2/rt_transition/cpm.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL R-TRANSITION FOR TERMS ************************)
+
+(* Basic inversion properties ***********************************************)
+
+lemma cpr_inv_atom1: ∀h,J,G,L,T2. ⦃G, L⦄ ⊢ ⓪{J} ➡[h] T2 →
+                     ∨∨ T2 = ⓪{J}
+                      | ∃∃K,V1,V2. ⦃G, K⦄ ⊢ V1 ➡[h] V2 & ⬆*[1] V2 ≡ T2 &
+                                   L = K.ⓓV1 & J = LRef 0
+                      | ∃∃I,K,V,T,i. ⦃G, K⦄ ⊢ #i ➡[h] T & ⬆*[1] T ≡ T2 &
+                                     L = K.ⓑ{I}V & J = LRef (⫯i).
+#h #J #G #L #T2 #H elim (cpm_inv_atom1 … H) -H *
+/3 width=9 by or3_intro0, or3_intro1, or3_intro2, ex4_5_intro, ex4_3_intro/
+[ #n #_ #_ #H destruct
+| #n #K #V1 #V2 #_ #_ #_ #_ #H destruct
 ]
 qed-.
 
-(* Basic_1: was by definition: pr2_free *)
-lemma tpr_cpr: ∀G,T1,T2. ⦃G, ⋆⦄ ⊢ T1 ➡ T2 → ∀L. ⦃G, L⦄ ⊢ T1 ➡ T2.
-#G #T1 #T2 #HT12 #L
-lapply (lsubr_cpr_trans … HT12 L ?) //
-qed.
-
-(* Basic_1: includes by definition: pr0_refl *)
-lemma cpr_refl: ∀G,T,L. ⦃G, L⦄ ⊢ T ➡ T.
-#G #T elim T -T // * /2 width=1 by cpr_bind, cpr_flat/
-qed.
-
-(* Basic_1: was: pr2_head_1 *)
-lemma cpr_pair_sn: ∀I,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡ V2 →
-                   ∀T. ⦃G, L⦄ ⊢ ②{I}V1.T ➡ ②{I}V2.T.
-* /2 width=1 by cpr_bind, cpr_flat/ qed.
-
-lemma cpr_delift: ∀G,K,V,T1,L,l. ⬇[l] L ≡ (K.ⓓV) →
-                  ∃∃T2,T. ⦃G, L⦄ ⊢ T1 ➡ T2 & ⬆[l, 1] T ≡ T2.
-#G #K #V #T1 elim T1 -T1
-[ * /2 width=4 by cpr_atom, lift_sort, lift_gref, ex2_2_intro/
-  #i #L #l #HLK elim (lt_or_eq_or_gt i l)
-  #Hil [1,3: /4 width=4 by lift_lref_ge_minus, lift_lref_lt, ylt_inj, yle_inj, ex2_2_intro/ ]
-  destruct
-  elim (lift_total V 0 (i+1)) #W #HVW
-  elim (lift_split … HVW i i) /3 width=6 by cpr_delta, ex2_2_intro/
-| * [ #a ] #I #W1 #U1 #IHW1 #IHU1 #L #l #HLK
-  elim (IHW1 … HLK) -IHW1 #W2 #W #HW12 #HW2
-  [ elim (IHU1 (L. ⓑ{I}W1) (l+1)) -IHU1 /3 width=9 by drop_drop, cpr_bind, lift_bind, ex2_2_intro/
-  | elim (IHU1 … HLK) -IHU1 -HLK /3 width=8 by cpr_flat, lift_flat, ex2_2_intro/
-  ]
-]
-qed-.
-
-fact lstas_cpr_aux: ∀h,G,L,T1,T2,d. ⦃G, L⦄ ⊢ T1 •*[h, d] T2 →
-                    d = 0 → ⦃G, L⦄ ⊢ T1 ➡ T2.
-#h #G #L #T1 #T2 #d #H elim H -G -L -T1 -T2 -d
-/3 width=1 by cpr_eps, cpr_flat, cpr_bind/
-[ #G #L #K #V1 #V2 #W2 #i #d #HLK #_ #HVW2 #IHV12 #H destruct
-  /3 width=6 by cpr_delta/
-| #G #L #K #V1 #V2 #W2 #i #d #_ #_ #_ #_ <plus_n_Sm #H destruct
-]
-qed-.
-
-lemma lstas_cpr: ∀h,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 •*[h, 0] T2 → ⦃G, L⦄ ⊢ T1 ➡ T2.
-/2 width=4 by lstas_cpr_aux/ qed.
-
-(* Basic inversion lemmas ***************************************************)
-
-fact cpr_inv_atom1_aux: ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡ T2 → ∀I. T1 = ⓪{I} →
-                        T2 = ⓪{I} ∨
-                        ∃∃K,V,V2,i. ⬇[i] L ≡ K. ⓓV & ⦃G, K⦄ ⊢ V ➡ V2 &
-                                    ⬆[O, i + 1] V2 ≡ T2 & I = LRef i.
-#G #L #T1 #T2 * -G -L -T1 -T2
-[ #I #G #L #J #H destruct /2 width=1 by or_introl/
-| #L #G #K #V #V2 #T2 #i #HLK #HV2 #HVT2 #J #H destruct /3 width=8 by ex4_4_intro, or_intror/
-| #a #I #G #L #V1 #V2 #T1 #T2 #_ #_ #J #H destruct
-| #I #G #L #V1 #V2 #T1 #T2 #_ #_ #J #H destruct
-| #G #L #V #T1 #T #T2 #_ #_ #J #H destruct
-| #G #L #V #T1 #T2 #_ #J #H destruct
-| #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #J #H destruct
-| #a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #J #H destruct
-]
-qed-.
-
-lemma cpr_inv_atom1: ∀I,G,L,T2. ⦃G, L⦄ ⊢ ⓪{I} ➡ T2 →
-                     T2 = ⓪{I} ∨
-                     ∃∃K,V,V2,i. ⬇[i] L ≡ K. ⓓV & ⦃G, K⦄ ⊢ V ➡ V2 &
-                                 ⬆[O, i + 1] V2 ≡ T2 & I = LRef i.
-/2 width=3 by cpr_inv_atom1_aux/ qed-.
-
 (* Basic_1: includes: pr0_gen_sort pr2_gen_sort *)
-lemma cpr_inv_sort1: ∀G,L,T2,s. ⦃G, L⦄ ⊢ ⋆s ➡ T2 → T2 = ⋆s.
-#G #L #T2 #s #H
-elim (cpr_inv_atom1 … H) -H //
-* #K #V #V2 #i #_ #_ #_ #H destruct
+lemma cpr_inv_sort1: ∀h,G,L,T2,s. ⦃G, L⦄ ⊢ ⋆s ➡[h] T2 → T2 = ⋆s.
+#h #G #L #T2 #s #H elim (cpm_inv_sort1 … H) -H * // #_ #H destruct
 qed-.
 
-(* Basic_1: includes: pr0_gen_lref pr2_gen_lref *)
-lemma cpr_inv_lref1: ∀G,L,T2,i. ⦃G, L⦄ ⊢ #i ➡ T2 →
-                     T2 = #i ∨
-                     ∃∃K,V,V2. ⬇[i] L ≡ K. ⓓV & ⦃G, K⦄ ⊢ V ➡ V2 &
-                               ⬆[O, i + 1] V2 ≡ T2.
-#G #L #T2 #i #H
-elim (cpr_inv_atom1 … H) -H /2 width=1 by or_introl/
-* #K #V #V2 #j #HLK #HV2 #HVT2 #H destruct /3 width=6 by ex3_3_intro, or_intror/
+lemma cpr_inv_zero1: ∀h,G,L,T2. ⦃G, L⦄ ⊢ #0 ➡[h] T2 →
+                     T2 = #0 ∨
+                     ∃∃K,V1,V2. ⦃G, K⦄ ⊢ V1 ➡[h] V2 & ⬆*[1] V2 ≡ T2 &
+                                L = K.ⓓV1.
+#h #G #L #T2 #H elim (cpm_inv_zero1 … H) -H *
+/3 width=6 by ex3_3_intro, or_introl, or_intror/
+#n #K #V1 #V2 #_ #_ #_ #H destruct
 qed-.
 
-lemma cpr_inv_gref1: ∀G,L,T2,p. ⦃G, L⦄ ⊢ §p ➡ T2 → T2 = §p.
-#G #L #T2 #p #H
-elim (cpr_inv_atom1 … H) -H //
-* #K #V #V2 #i #_ #_ #_ #H destruct
+lemma cpr_inv_lref1: ∀h,G,L,T2,i. ⦃G, L⦄ ⊢ #⫯i ➡[h] T2 →
+                     T2 = #(⫯i) ∨
+                     ∃∃I,K,V,T. ⦃G, K⦄ ⊢ #i ➡[h] T & ⬆*[1] T ≡ T2 & L = K.ⓑ{I}V.
+#h #G #L #T2 #i #H elim (cpm_inv_lref1 … H) -H *
+/3 width=7 by ex3_4_intro, or_introl, or_intror/
 qed-.
 
-fact cpr_inv_bind1_aux: ∀G,L,U1,U2. ⦃G, L⦄ ⊢ U1 ➡ U2 →
-                        ∀a,I,V1,T1. U1 = ⓑ{a,I}V1. T1 → (
-                        ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ➡ T2 &
-                                 U2 = ⓑ{a,I}V2.T2
-                        ) ∨
-                        ∃∃T. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡ T & ⬆[0, 1] U2 ≡ T &
-                             a = true & I = Abbr.
-#G #L #U1 #U2 * -L -U1 -U2
-[ #I #G #L #b #J #W1 #U1 #H destruct
-| #L #G #K #V #V2 #W2 #i #_ #_ #_ #b #J #W #U1 #H destruct
-| #a #I #G #L #V1 #V2 #T1 #T2 #HV12 #HT12 #b #J #W #U1 #H destruct /3 width=5 by ex3_2_intro, or_introl/
-| #I #G #L #V1 #V2 #T1 #T2 #_ #_ #b #J #W #U1 #H destruct
-| #G #L #V #T1 #T #T2 #HT1 #HT2 #b #J #W #U1 #H destruct /3 width=3 by ex4_intro, or_intror/
-| #G #L #V #T1 #T2 #_ #b #J #W #U1 #H destruct
-| #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #b #J #W #U1 #H destruct
-| #a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #b #J #W #U1 #H destruct
-]
-qed-.
-
-lemma cpr_inv_bind1: ∀a,I,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡ U2 → (
-                     ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ➡ T2 &
-                              U2 = ⓑ{a,I}V2.T2
-                     ) ∨
-                     ∃∃T. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡ T & ⬆[0, 1] U2 ≡ T &
-                          a = true & I = Abbr.
-/2 width=3 by cpr_inv_bind1_aux/ qed-.
-
-(* Basic_1: includes: pr0_gen_abbr pr2_gen_abbr *)
-lemma cpr_inv_abbr1: ∀a,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓓ{a}V1.T1 ➡ U2 → (
-                     ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L. ⓓV1⦄ ⊢ T1 ➡ T2 &
-                              U2 = ⓓ{a}V2.T2
-                     ) ∨
-                     ∃∃T. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡ T & ⬆[0, 1] U2 ≡ T & a = true.
-#a #G #L #V1 #T1 #U2 #H
-elim (cpr_inv_bind1 … H) -H *
-/3 width=5 by ex3_2_intro, ex3_intro, or_introl, or_intror/
-qed-.
-
-(* Basic_1: includes: pr0_gen_abst pr2_gen_abst *)
-lemma cpr_inv_abst1: ∀a,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓛ{a}V1.T1 ➡ U2 →
-                     ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L.ⓛV1⦄ ⊢ T1 ➡ T2 &
-                              U2 = ⓛ{a}V2.T2.
-#a #G #L #V1 #T1 #U2 #H
-elim (cpr_inv_bind1 … H) -H *
-[ /3 width=5 by ex3_2_intro/
-| #T #_ #_ #_ #H destruct
-]
-qed-.
-
-fact cpr_inv_flat1_aux: ∀G,L,U,U2. ⦃G, L⦄ ⊢ U ➡ U2 →
-                        ∀I,V1,U1. U = ⓕ{I}V1.U1 →
-                        ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L⦄ ⊢ U1 ➡ T2 &
-                                    U2 = ⓕ{I} V2. T2
-                         | (⦃G, L⦄ ⊢ U1 ➡ U2 ∧ I = Cast)
-                         | ∃∃a,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L⦄ ⊢ W1 ➡ W2 &
-                                               ⦃G, L.ⓛW1⦄ ⊢ T1 ➡ T2 & U1 = ⓛ{a}W1.T1 &
-                                               U2 = ⓓ{a}ⓝW2.V2.T2 & I = Appl
-                         | ∃∃a,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡ V & ⬆[0,1] V ≡ V2 &
-                                                 ⦃G, L⦄ ⊢ W1 ➡ W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ➡ T2 &
-                                                 U1 = ⓓ{a}W1.T1 &
-                                                 U2 = ⓓ{a}W2.ⓐV2.T2 & I = Appl.
-#G #L #U #U2 * -L -U -U2
-[ #I #G #L #J #W1 #U1 #H destruct
-| #G #L #K #V #V2 #W2 #i #_ #_ #_ #J #W #U1 #H destruct
-| #a #I #G #L #V1 #V2 #T1 #T2 #_ #_ #J #W #U1 #H destruct
-| #I #G #L #V1 #V2 #T1 #T2 #HV12 #HT12 #J #W #U1 #H destruct /3 width=5 by or4_intro0, ex3_2_intro/
-| #G #L #V #T1 #T #T2 #_ #_ #J #W #U1 #H destruct
-| #G #L #V #T1 #T2 #HT12 #J #W #U1 #H destruct /3 width=1 by or4_intro1, conj/
-| #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 #HW12 #HT12 #J #W #U1 #H destruct /3 width=11 by or4_intro2, ex6_6_intro/
-| #a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 #HW12 #HT12 #J #W #U1 #H destruct /3 width=13 by or4_intro3, ex7_7_intro/
-]
+lemma cpr_inv_gref1: ∀h,G,L,T2,l. ⦃G, L⦄ ⊢ §l ➡[h] T2 → T2 = §l.
+#h #G #L #T2 #l #H elim (cpm_inv_gref1 … H) -H //
 qed-.
 
-lemma cpr_inv_flat1: ∀I,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓕ{I}V1.U1 ➡ U2 →
-                     ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L⦄ ⊢ U1 ➡ T2 &
+lemma cpr_inv_flat1: ∀h,I,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓕ{I}V1.U1 ➡[h] U2 →
+                     ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 & ⦃G, L⦄ ⊢ U1 ➡[h] T2 &
                                  U2 = ⓕ{I}V2.T2
-                      | (⦃G, L⦄ ⊢ U1 ➡ U2 ∧ I = Cast)
-                      | ∃∃a,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L⦄ ⊢ W1 ➡ W2 &
-                                            ⦃G, L.ⓛW1⦄ ⊢ T1 ➡ T2 & U1 = ⓛ{a}W1.T1 &
-                                            U2 = ⓓ{a}ⓝW2.V2.T2 & I = Appl
-                      | ∃∃a,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡ V & ⬆[0,1] V ≡ V2 &
-                                              ⦃G, L⦄ ⊢ W1 ➡ W2 & ⦃G, 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: ∀G,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. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L⦄ ⊢ W1 ➡ W2 &
-                                            ⦃G, L.ⓛW1⦄ ⊢ T1 ➡ T2 &
-                                            U1 = ⓛ{a}W1.T1 & U2 = ⓓ{a}ⓝW2.V2.T2
-                      | ∃∃a,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡ V & ⬆[0,1] V ≡ V2 &
-                                              ⦃G, L⦄ ⊢ W1 ➡ W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ➡ T2 &
-                                              U1 = ⓓ{a}W1.T1 & U2 = ⓓ{a}W2.ⓐV2.T2.
-#G #L #V1 #U1 #U2 #H elim (cpr_inv_flat1 … H) -H *
-[ /3 width=5 by or3_intro0, ex3_2_intro/
-| #_ #H destruct
-| /3 width=11 by or3_intro1, ex5_6_intro/
-| /3 width=13 by or3_intro2, ex6_7_intro/
-]
-qed-.
-
-(* Note: the main property of simple terms *)
-lemma cpr_inv_appl1_simple: ∀G,L,V1,T1,U. ⦃G, L⦄ ⊢ ⓐV1. T1 ➡ U → 𝐒⦃T1⦄ →
-                            ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L⦄ ⊢ T1 ➡ T2 &
-                                     U = ⓐV2. T2.
-#G #L #V1 #T1 #U #H #HT1
-elim (cpr_inv_appl1 … H) -H *
-[ /2 width=5 by ex3_2_intro/
-| #a #V2 #W1 #W2 #U1 #U2 #_ #_ #_ #H #_ destruct
-  elim (simple_inv_bind … HT1)
-| #a #V #V2 #W1 #W2 #U1 #U2 #_ #_ #_ #_ #H #_ destruct
-  elim (simple_inv_bind … HT1)
-]
+                      | (⦃G, L⦄ ⊢ U1 ➡[h] U2 ∧ I = Cast)
+                      | ∃∃p,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 & ⦃G, L⦄ ⊢ W1 ➡[h] W2 &
+                                            ⦃G, L.ⓛW1⦄ ⊢ T1 ➡[h] T2 & U1 = ⓛ{p}W1.T1 &
+                                            U2 = ⓓ{p}ⓝW2.V2.T2 & I = Appl
+                      | ∃∃p,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V & ⬆*[1] V ≡ V2 &
+                                              ⦃G, L⦄ ⊢ W1 ➡[h] W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ➡[h] T2 &
+                                              U1 = ⓓ{p}W1.T1 &
+                                              U2 = ⓓ{p}W2.ⓐV2.T2 & I = Appl.
+#h #I #G #L #V1 #U1 #U2 #H elim (cpm_inv_flat1 … H) -H *
+/3 width=13 by or4_intro0, or4_intro1, or4_intro2, or4_intro3, ex7_7_intro, ex6_6_intro, ex3_2_intro, conj/
+#n #_ #_ #H destruct
 qed-.
 
 (* Basic_1: includes: pr0_gen_cast pr2_gen_cast *)
-lemma cpr_inv_cast1: ∀G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓝ V1. U1 ➡ U2 → (
-                     ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡ V2 & ⦃G, L⦄ ⊢ U1 ➡ T2 &
-                              U2 = ⓝ V2. T2
-                     ) ∨ ⦃G, L⦄ ⊢ U1 ➡ U2.
-#G #L #V1 #U1 #U2 #H elim (cpr_inv_flat1 … H) -H *
-[ /3 width=5 by ex3_2_intro, or_introl/
-| /2 width=1 by or_intror/
-| #a #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #H destruct
-| #a #V #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #_ #H destruct
-]
-qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma cpr_fwd_bind1_minus: ∀I,G,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 #G #L #V1 #T1 #T #H #b
-elim (cpr_inv_bind1 … H) -H *
-[ #V2 #T2 #HV12 #HT12 #H destruct /3 width=4 by cpr_bind, ex2_2_intro/
-| #T2 #_ #_ #H destruct
-]
+lemma cpr_inv_cast1: ∀h,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓝ V1. U1 ➡[h] U2 → (
+                     ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 & ⦃G, L⦄ ⊢ U1 ➡[h] T2 &
+                              U2 = ⓝV2.T2
+                     ) ∨ ⦃G, L⦄ ⊢ U1 ➡[h] U2.
+#h #G #L #V1 #U1 #U2 #H elim (cpm_inv_cast1 … H) -H
+/2 width=1 by or_introl, or_intror/ * #n #_ #H destruct
 qed-.
 
-(* Basic_1: removed theorems 11:
+(* Basic_1: removed theorems 12:
             pr0_subst0_back pr0_subst0_fwd pr0_subst0
+            pr0_delta1
             pr2_head_2 pr2_cflat clear_pr2_trans
             pr2_gen_csort pr2_gen_cflat pr2_gen_cbind
             pr2_gen_ctail pr2_ctail
diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/partial.txt b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/partial.txt
index 8658ca5cf..642ca8aa2 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/partial.txt
+++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/partial.txt
@@ -2,3 +2,4 @@ cpg.ma cpg_simple.ma cpg_drops.ma cpg_lsubr.ma
 cpx.ma cpx_simple.ma cpx_drops.ma cpx_lsubr.ma
 lfpx.ma lfpx_length.ma lfpx_drops.ma lfpx_fqup.ma 
 cpm.ma
+cpr.ma
diff --git a/matita/matita/contribs/lambdadelta/ground_2/lib/arith.ma b/matita/matita/contribs/lambdadelta/ground_2/lib/arith.ma
index 9fedf60c0..83322f4d8 100644
--- a/matita/matita/contribs/lambdadelta/ground_2/lib/arith.ma
+++ b/matita/matita/contribs/lambdadelta/ground_2/lib/arith.ma
@@ -158,6 +158,10 @@ qed.
 
 (* Inversion & forward lemmas ***********************************************)
 
+lemma max_inv_O3: ∀x,y. (x ∨ y) = 0 → 0 = x ∧ 0 = y.
+/4 width=2 by le_maxr, le_maxl, le_n_O_to_eq, conj/
+qed-.
+
 lemma plus_inv_O3: ∀x,y. x + y = 0 → x = 0 ∧ y = 0.
 /2 width=1 by plus_le_0/ qed-.
 
diff --git a/matita/matita/contribs/lambdadelta/ground_2/steps/rtc_max.ma b/matita/matita/contribs/lambdadelta/ground_2/steps/rtc_max.ma
new file mode 100644
index 000000000..1c8861fa8
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/ground_2/steps/rtc_max.ma
@@ -0,0 +1,84 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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 "ground_2/steps/rtc_shift.ma".
+
+(* RT-TRANSITION COUNTER ****************************************************)
+
+definition max (c1:rtc) (c2:rtc): rtc ≝ match c1 with [
+   mk_rtc ri1 rs1 ti1 ts1 ⇒ match c2 with [
+      mk_rtc ri2 rs2 ti2 ts2 ⇒ 〈ri1∨ri2, rs1∨rs2, ti1∨ti2, ts1∨ts2〉
+   ]
+].
+
+interpretation "maximum (rtc)"
+   'or c1 c2 = (max c1 c2).
+
+(* Basic properties *********************************************************)
+
+lemma max_rew: ∀ri1,ri2,rs1,rs2,ti1,ti2,ts1,ts2.
+                 〈ri1∨ri2, rs1∨rs2, ti1∨ti2, ts1∨ts2〉 =
+                 (〈ri1,rs1,ti1,ts1〉 ∨ 〈ri2,rs2,ti2,ts2〉).
+// qed.
+
+lemma max_O_dx: ∀c. c = (c ∨ 𝟘𝟘).
+* #ri #rs #ti #ts <max_rew //
+qed.
+
+(* Basic inversion properties ***********************************************)
+
+lemma max_inv_dx: ∀ri,rs,ti,ts,c1,c2. 〈ri,rs,ti,ts〉 = (c1 ∨ c2) →
+                  ∃∃ri1,rs1,ti1,ts1,ri2,rs2,ti2,ts2.
+                  (ri1∨ri2) = ri & (rs1∨rs2) = rs & (ti1∨ti2) = ti & (ts1∨ts2) = ts &
+                  〈ri1,rs1,ti1,ts1〉 = c1 & 〈ri2,rs2,ti2,ts2〉 = c2.
+#ri #rs #ti #ts * #ri1 #rs1 #ti1 #ts1 * #ri2 #rs2 #ti2 #ts2
+<max_rew #H destruct /2 width=14 by ex6_8_intro/
+qed-.
+
+(* Properties with test for constrained rt-transition counter ***************)
+
+lemma isrt_max: ∀n1,n2,c1,c2. 𝐑𝐓⦃n1, c1⦄ → 𝐑𝐓⦃n2, c2⦄ → 𝐑𝐓⦃n1∨n2, c1∨c2⦄.
+#n1 #n2 #c1 #c2 * #ri1 #rs1 #H1 * #ri2 #rs2 #H2 destruct
+/2 width=3 by ex1_2_intro/
+qed.
+
+lemma isrt_max_O1: ∀n,c1,c2. 𝐑𝐓⦃0, c1⦄ → 𝐑𝐓⦃n, c2⦄ → 𝐑𝐓⦃n, c1∨c2⦄.
+/2 width=1 by isrt_max/ qed.
+
+lemma isrt_max_O2: ∀n,c1,c2. 𝐑𝐓⦃n, c1⦄ → 𝐑𝐓⦃0, c2⦄ → 𝐑𝐓⦃n, c1∨c2⦄.
+#n #c1 #c2 #H1 #H2 >(max_O2 n) /2 width=1 by isrt_max/
+qed.
+
+(* Inversion properties with test for constrained rt-transition counter *****)
+
+lemma isrt_inv_max: ∀n,c1,c2. 𝐑𝐓⦃n, c1 ∨ c2⦄ →
+                    ∃∃n1,n2. 𝐑𝐓⦃n1, c1⦄ & 𝐑𝐓⦃n2, c2⦄ & (n1 ∨ n2) = n.
+#n #c1 #c2 * #ri #rs #H
+elim (max_inv_dx … H) -H #ri1 #rs1 #ti1 #ts1 #ri2 #rs2 #ti2 #ts2 #_ #_ #H1 #H2 #H3 #H4
+elim (max_inv_O3 … H1) -H1 /3 width=5 by ex3_2_intro, ex1_2_intro/
+qed-.
+
+lemma isrt_inv_max_O_dx: ∀n,c1,c2. 𝐑𝐓⦃n, c1 ∨ c2⦄ → 𝐑𝐓⦃0, c2⦄ → 𝐑𝐓⦃n, c1⦄.
+#n #c1 #c2 #H #H2
+elim (isrt_inv_max … H) -H #n1 #n2 #Hn1 #Hn2 #H destruct
+lapply (isrt_mono … Hn2 H2) -c2 #H destruct //
+qed-.
+
+(* Properties with shift ****************************************************)
+(*
+lemma max_shift: ∀c1,c2. (↓c1) ∨ (↓c2) = ↓(c1∨c2).
+* #ri1 #rs1 #ti1 #ts1 * #ri2 #rs2 #ti2 #ts2
+<shift_rew <shift_rew <shift_rew <max_rew //
+qed.
+*)
diff --git a/matita/matita/contribs/lambdadelta/ground_2/steps/rtc_plus.ma b/matita/matita/contribs/lambdadelta/ground_2/steps/rtc_plus.ma
index 900e6b483..87adfee3a 100644
--- a/matita/matita/contribs/lambdadelta/ground_2/steps/rtc_plus.ma
+++ b/matita/matita/contribs/lambdadelta/ground_2/steps/rtc_plus.ma
@@ -27,10 +27,11 @@ interpretation "plus (rtc)"
 
 (* Basic properties *********************************************************)
 
-lemma plus_rew: ∀ri1,ri2,rs1,rs2,ti1,ti2,ts1,ts2. 
-                〈ri1+ri2, rs1+rs2, ti1+ti2, ts1+ts2〉 =
-                plus (〈ri1,rs1,ti1,ts1〉) (〈ri2,rs2,ti2,ts2〉).
-// qed. (**) (* disambiguation of plus fails *)
+(**) (* plus is not disambiguated parentheses *) 
+lemma plus_rew: ∀ri1,ri2,rs1,rs2,ti1,ti2,ts1,ts2.
+                 〈ri1+ri2, rs1+rs2, ti1+ti2, ts1+ts2〉 =
+                 (〈ri1,rs1,ti1,ts1〉) + (〈ri2,rs2,ti2,ts2〉).
+// qed.
 
 lemma plus_O_dx: ∀c. c = c + 𝟘𝟘.
 * #ri #rs #ti #ts <plus_rew //
-- 
2.39.2