update in basic_2 and apps_2

[helm.git] / matita / matita / contribs / lambdadelta / basic_2 / rt_transition / lpr_lpr.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lpr_lpr.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lpr_lpr.ma
index 2b92825d3514708c5a3727cfd534e5781e310b01..73509adecf4ba65ec0423b515f9b02987f28cd25 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lpr_lpr.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lpr_lpr.ma
@@ -21,14 +21,14 @@ include "basic_2/rt_transition/lpr_drops.ma".
 (* PARALLEL R-TRANSITION FOR FULL LOCAL ENVIRONMENTS ************************)
 
 definition IH_cpr_conf_lpr (h): relation3 genv lenv term ≝ λG,L,T.
-                           ∀T1. ❪G,L❫ ⊢ T ➡[h] T1 → ∀T2. ❪G,L❫ ⊢ T ➡[h] T2 →
-                           ∀L1. ❪G,L❫ ⊢ ➡[h] L1 → ∀L2. ❪G,L❫ ⊢ ➡[h] L2 →
-                           ∃∃T0. ❪G,L1❫ ⊢ T1 ➡[h] T0 & ❪G,L2❫ ⊢ T2 ➡[h] T0.
+                           ∀T1. ❪G,L❫ ⊢ T ➡[h,0] T1 → ∀T2. ❪G,L❫ ⊢ T ➡[h,0] T2 →
+                           ∀L1. ❪G,L❫ ⊢ ➡[h,0] L1 → ∀L2. ❪G,L❫ ⊢ ➡[h,0] L2 →
+                           ∃∃T0. ❪G,L1❫ ⊢ T1 ➡[h,0] T0 & ❪G,L2❫ ⊢ T2 ➡[h,0] T0.
 
 (* Main properties with context-sensitive parallel reduction for terms ******)
 
 fact cpr_conf_lpr_atom_atom (h):
-   ∀I,G,L1,L2. ∃∃T. ❪G,L1❫ ⊢ ⓪[I] ➡[h] T & ❪G,L2❫ ⊢ ⓪[I] ➡[h] T.
+   ∀I,G,L1,L2. ∃∃T. ❪G,L1❫ ⊢ ⓪[I] ➡[h,0] T & ❪G,L2❫ ⊢ ⓪[I] ➡[h,0] T.
 /2 width=3 by cpr_refl, ex2_intro/ qed-.
 
 fact cpr_conf_lpr_atom_delta (h):
@@ -36,9 +36,9 @@ fact cpr_conf_lpr_atom_delta (h):
       ∀G,L,T. ❪G0,L0,#i❫ ⬂+ ❪G,L,T❫ → IH_cpr_conf_lpr h G L T
    ) →
    ∀K0,V0. ⇩[i] L0 ≘ K0.ⓓV0 →
-   ∀V2. ❪G0,K0❫ ⊢ V0 ➡[h] V2 → ∀T2. ⇧[↑i] V2 ≘ T2 →
-   ∀L1. ❪G0,L0❫ ⊢ ➡[h] L1 → ∀L2. ❪G0,L0❫ ⊢ ➡[h] L2 →
-   ∃∃T. ❪G0,L1❫ ⊢ #i ➡[h] T & ❪G0,L2❫ ⊢ T2 ➡[h] T.
+   ∀V2. ❪G0,K0❫ ⊢ V0 ➡[h,0] V2 → ∀T2. ⇧[↑i] V2 ≘ T2 →
+   ∀L1. ❪G0,L0❫ ⊢ ➡[h,0] L1 → ∀L2. ❪G0,L0❫ ⊢ ➡[h,0] L2 →
+   ∃∃T. ❪G0,L1❫ ⊢ #i ➡[h,0] T & ❪G0,L2❫ ⊢ T2 ➡[h,0] T.
 #h #G0 #L0 #i #IH #K0 #V0 #HLK0 #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02
 elim (lpr_drops_conf … HLK0 … HL01) -HL01 // #X1 #H1 #HLK1
 elim (lpr_inv_pair_sn … H1) -H1 #K1 #V1 #HK01 #HV01 #H destruct
@@ -57,11 +57,11 @@ fact cpr_conf_lpr_delta_delta (h):
       ∀G,L,T. ❪G0,L0,#i❫ ⬂+ ❪G,L,T❫ → IH_cpr_conf_lpr h G L T
    ) →
    ∀K0,V0. ⇩[i] L0 ≘ K0.ⓓV0 →
-   ∀V1. ❪G0,K0❫ ⊢ V0 ➡[h] V1 → ∀T1. ⇧[↑i] V1 ≘ T1 →
+   ∀V1. ❪G0,K0❫ ⊢ V0 ➡[h,0] V1 → ∀T1. ⇧[↑i] V1 ≘ T1 →
    ∀KX,VX. ⇩[i] L0 ≘ KX.ⓓVX →
-   ∀V2. ❪G0,KX❫ ⊢ VX ➡[h] V2 → ∀T2. ⇧[↑i] V2 ≘ T2 →
-   ∀L1. ❪G0,L0❫ ⊢ ➡[h] L1 → ∀L2. ❪G0,L0❫ ⊢ ➡[h] L2 →
-   ∃∃T. ❪G0,L1❫ ⊢ T1 ➡[h] T & ❪G0,L2❫ ⊢ T2 ➡[h] T.
+   ∀V2. ❪G0,KX❫ ⊢ VX ➡[h,0] V2 → ∀T2. ⇧[↑i] V2 ≘ T2 →
+   ∀L1. ❪G0,L0❫ ⊢ ➡[h,0] L1 → ∀L2. ❪G0,L0❫ ⊢ ➡[h,0] L2 →
+   ∃∃T. ❪G0,L1❫ ⊢ T1 ➡[h,0] T & ❪G0,L2❫ ⊢ T2 ➡[h,0] T.
 #h #G0 #L0 #i #IH #K0 #V0 #HLK0 #V1 #HV01 #T1 #HVT1
 #KX #VX #H #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02
 lapply (drops_mono … H … HLK0) -H #H destruct
@@ -81,10 +81,10 @@ fact cpr_conf_lpr_bind_bind (h):
    ∀p,I,G0,L0,V0,T0. (
       ∀G,L,T. ❪G0,L0,ⓑ[p,I]V0.T0❫ ⬂+ ❪G,L,T❫ → IH_cpr_conf_lpr h G L T
    ) →
-   ∀V1. ❪G0,L0❫ ⊢ V0 ➡[h] V1 → ∀T1. ❪G0,L0.ⓑ[I]V0❫ ⊢ T0 ➡[h] T1 →
-   ∀V2. ❪G0,L0❫ ⊢ V0 ➡[h] V2 → ∀T2. ❪G0,L0.ⓑ[I]V0❫ ⊢ T0 ➡[h] T2 →
-   ∀L1. ❪G0,L0❫ ⊢ ➡[h] L1 → ∀L2. ❪G0,L0❫ ⊢ ➡[h] L2 →
-   ∃∃T. ❪G0,L1❫ ⊢ ⓑ[p,I]V1.T1 ➡[h] T & ❪G0,L2❫ ⊢ ⓑ[p,I]V2.T2 ➡[h] T.
+   ∀V1. ❪G0,L0❫ ⊢ V0 ➡[h,0] V1 → ∀T1. ❪G0,L0.ⓑ[I]V0❫ ⊢ T0 ➡[h,0] T1 →
+   ∀V2. ❪G0,L0❫ ⊢ V0 ➡[h,0] V2 → ∀T2. ❪G0,L0.ⓑ[I]V0❫ ⊢ T0 ➡[h,0] T2 →
+   ∀L1. ❪G0,L0❫ ⊢ ➡[h,0] L1 → ∀L2. ❪G0,L0❫ ⊢ ➡[h,0] L2 →
+   ∃∃T. ❪G0,L1❫ ⊢ ⓑ[p,I]V1.T1 ➡[h,0] T & ❪G0,L2❫ ⊢ ⓑ[p,I]V2.T2 ➡[h,0] T.
 #h #p #I #G0 #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
 #V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02
 elim (IH … HV01 … HV02 … HL01 … HL02) //
@@ -96,10 +96,10 @@ fact cpr_conf_lpr_bind_zeta (h):
    ∀G0,L0,V0,T0. (
       ∀G,L,T. ❪G0,L0,+ⓓV0.T0❫ ⬂+ ❪G,L,T❫ → IH_cpr_conf_lpr h G L T
    ) →
-   ∀V1. ❪G0,L0❫ ⊢ V0 ➡[h] V1 → ∀T1. ❪G0,L0.ⓓV0❫ ⊢ T0 ➡[h] T1 →
-   ∀T2. ⇧[1]T2 ≘ T0 → ∀X2. ❪G0,L0❫ ⊢ T2 ➡[h] X2 →
-   ∀L1. ❪G0,L0❫ ⊢ ➡[h] L1 → ∀L2. ❪G0,L0❫ ⊢ ➡[h] L2 →
-   ∃∃T. ❪G0,L1❫ ⊢ +ⓓV1.T1 ➡[h] T & ❪G0,L2❫ ⊢ X2 ➡[h] T.
+   ∀V1. ❪G0,L0❫ ⊢ V0 ➡[h,0] V1 → ∀T1. ❪G0,L0.ⓓV0❫ ⊢ T0 ➡[h,0] T1 →
+   ∀T2. ⇧[1]T2 ≘ T0 → ∀X2. ❪G0,L0❫ ⊢ T2 ➡[h,0] X2 →
+   ∀L1. ❪G0,L0❫ ⊢ ➡[h,0] L1 → ∀L2. ❪G0,L0❫ ⊢ ➡[h,0] L2 →
+   ∃∃T. ❪G0,L1❫ ⊢ +ⓓV1.T1 ➡[h,0] T & ❪G0,L2❫ ⊢ X2 ➡[h,0] T.
 #h #G0 #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
 #T2 #HT20 #X2 #HTX2 #L1 #HL01 #L2 #HL02
 elim (cpm_inv_lifts_sn … HT01 (Ⓣ) … L0 … HT20) -HT01 [| /3 width=1 by drops_refl, drops_drop/ ] #T #HT1 #HT2
@@ -111,10 +111,10 @@ fact cpr_conf_lpr_zeta_zeta (h):
    ∀G0,L0,V0,T0. (
       ∀G,L,T. ❪G0,L0,+ⓓV0.T0❫ ⬂+ ❪G,L,T❫ → IH_cpr_conf_lpr h G L T
    ) →
-   ∀T1. ⇧[1] T1 ≘ T0 → ∀X1. ❪G0,L0❫ ⊢ T1 ➡[h] X1 →
-   ∀T2. ⇧[1] T2 ≘ T0 → ∀X2. ❪G0,L0❫ ⊢ T2 ➡[h] X2 →
-   ∀L1. ❪G0,L0❫ ⊢ ➡[h] L1 → ∀L2. ❪G0,L0❫ ⊢ ➡[h] L2 →
-   ∃∃T. ❪G0,L1❫ ⊢ X1 ➡[h] T & ❪G0,L2❫ ⊢ X2 ➡[h] T.
+   ∀T1. ⇧[1] T1 ≘ T0 → ∀X1. ❪G0,L0❫ ⊢ T1 ➡[h,0] X1 →
+   ∀T2. ⇧[1] T2 ≘ T0 → ∀X2. ❪G0,L0❫ ⊢ T2 ➡[h,0] X2 →
+   ∀L1. ❪G0,L0❫ ⊢ ➡[h,0] L1 → ∀L2. ❪G0,L0❫ ⊢ ➡[h,0] L2 →
+   ∃∃T. ❪G0,L1❫ ⊢ X1 ➡[h,0] T & ❪G0,L2❫ ⊢ X2 ➡[h,0] T.
 #h #G0 #L0 #V0 #T0 #IH #T1 #HT10 #X1 #HTX1
 #T2 #HT20 #X2 #HTX2 #L1 #HL01 #L2 #HL02
 lapply (lifts_inj … HT20 … HT10) -HT20 #H destruct
@@ -126,10 +126,10 @@ fact cpr_conf_lpr_flat_flat (h):
    ∀I,G0,L0,V0,T0. (
       ∀G,L,T. ❪G0,L0,ⓕ[I]V0.T0❫ ⬂+ ❪G,L,T❫ → IH_cpr_conf_lpr h G L T
    ) →
-   ∀V1. ❪G0,L0❫ ⊢ V0 ➡[h] V1 → ∀T1. ❪G0,L0❫ ⊢ T0 ➡[h] T1 →
-   ∀V2. ❪G0,L0❫ ⊢ V0 ➡[h] V2 → ∀T2. ❪G0,L0❫ ⊢ T0 ➡[h] T2 →
-   ∀L1. ❪G0,L0❫ ⊢ ➡[h] L1 → ∀L2. ❪G0,L0❫ ⊢ ➡[h] L2 →
-   ∃∃T. ❪G0,L1❫ ⊢ ⓕ[I]V1.T1 ➡[h] T & ❪G0,L2❫ ⊢ ⓕ[I]V2.T2 ➡[h] T.
+   ∀V1. ❪G0,L0❫ ⊢ V0 ➡[h,0] V1 → ∀T1. ❪G0,L0❫ ⊢ T0 ➡[h,0] T1 →
+   ∀V2. ❪G0,L0❫ ⊢ V0 ➡[h,0] V2 → ∀T2. ❪G0,L0❫ ⊢ T0 ➡[h,0] T2 →
+   ∀L1. ❪G0,L0❫ ⊢ ➡[h,0] L1 → ∀L2. ❪G0,L0❫ ⊢ ➡[h,0] L2 →
+   ∃∃T. ❪G0,L1❫ ⊢ ⓕ[I]V1.T1 ➡[h,0] T & ❪G0,L2❫ ⊢ ⓕ[I]V2.T2 ➡[h,0] T.
 #h #I #G0 #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
 #V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02
 elim (IH … HV01 … HV02 … HL01 … HL02) //
@@ -141,9 +141,9 @@ fact cpr_conf_lpr_flat_eps (h):
    ∀G0,L0,V0,T0. (
       ∀G,L,T. ❪G0,L0,ⓝV0.T0❫ ⬂+ ❪G,L,T❫ → IH_cpr_conf_lpr h G L T
    ) →
-   ∀V1,T1. ❪G0,L0❫ ⊢ T0 ➡[h] T1 → ∀T2. ❪G0,L0❫ ⊢ T0 ➡[h] T2 →
-   ∀L1. ❪G0,L0❫ ⊢ ➡[h] L1 → ∀L2. ❪G0,L0❫ ⊢ ➡[h] L2 →
-   ∃∃T. ❪G0,L1❫ ⊢ ⓝV1.T1 ➡[h] T & ❪G0,L2❫ ⊢ T2 ➡[h] T.
+   ∀V1,T1. ❪G0,L0❫ ⊢ T0 ➡[h,0] T1 → ∀T2. ❪G0,L0❫ ⊢ T0 ➡[h,0] T2 →
+   ∀L1. ❪G0,L0❫ ⊢ ➡[h,0] L1 → ∀L2. ❪G0,L0❫ ⊢ ➡[h,0] L2 →
+   ∃∃T. ❪G0,L1❫ ⊢ ⓝV1.T1 ➡[h,0] T & ❪G0,L2❫ ⊢ T2 ➡[h,0] T.
 #h #G0 #L0 #V0 #T0 #IH #V1 #T1 #HT01
 #T2 #HT02 #L1 #HL01 #L2 #HL02
 elim (IH … HT01 … HT02 … HL01 … HL02) // -L0 -V0 -T0
@@ -154,9 +154,9 @@ fact cpr_conf_lpr_eps_eps (h):
    ∀G0,L0,V0,T0. (
       ∀G,L,T. ❪G0,L0,ⓝV0.T0❫ ⬂+ ❪G,L,T❫ → IH_cpr_conf_lpr h G L T
    ) →
-   ∀T1. ❪G0,L0❫ ⊢ T0 ➡[h] T1 → ∀T2. ❪G0,L0❫ ⊢ T0 ➡[h] T2 →
-   ∀L1. ❪G0,L0❫ ⊢ ➡[h] L1 → ∀L2. ❪G0,L0❫ ⊢ ➡[h] L2 →
-   ∃∃T. ❪G0,L1❫ ⊢ T1 ➡[h] T & ❪G0,L2❫ ⊢ T2 ➡[h] T.
+   ∀T1. ❪G0,L0❫ ⊢ T0 ➡[h,0] T1 → ∀T2. ❪G0,L0❫ ⊢ T0 ➡[h,0] T2 →
+   ∀L1. ❪G0,L0❫ ⊢ ➡[h,0] L1 → ∀L2. ❪G0,L0❫ ⊢ ➡[h,0] L2 →
+   ∃∃T. ❪G0,L1❫ ⊢ T1 ➡[h,0] T & ❪G0,L2❫ ⊢ T2 ➡[h,0] T.
 #h #G0 #L0 #V0 #T0 #IH #T1 #HT01
 #T2 #HT02 #L1 #HL01 #L2 #HL02
 elim (IH … HT01 … HT02 … HL01 … HL02) // -L0 -V0 -T0
@@ -167,10 +167,10 @@ fact cpr_conf_lpr_flat_beta (h):
    ∀p,G0,L0,V0,W0,T0. (
       ∀G,L,T. ❪G0,L0,ⓐV0.ⓛ[p]W0.T0❫ ⬂+ ❪G,L,T❫ → IH_cpr_conf_lpr h G L T
    ) →
-   ∀V1. ❪G0,L0❫ ⊢ V0 ➡[h] V1 → ∀T1. ❪G0,L0❫ ⊢ ⓛ[p]W0.T0 ➡[h] T1 →
-   ∀V2. ❪G0,L0❫ ⊢ V0 ➡[h] V2 → ∀W2. ❪G0,L0❫ ⊢ W0 ➡[h] W2 → ∀T2. ❪G0,L0.ⓛW0❫ ⊢ T0 ➡[h] T2 →
-   ∀L1. ❪G0,L0❫ ⊢ ➡[h] L1 → ∀L2. ❪G0,L0❫ ⊢ ➡[h] L2 →
-   ∃∃T. ❪G0,L1❫ ⊢ ⓐV1.T1 ➡[h] T & ❪G0,L2❫ ⊢ ⓓ[p]ⓝW2.V2.T2 ➡[h] T.
+   ∀V1. ❪G0,L0❫ ⊢ V0 ➡[h,0] V1 → ∀T1. ❪G0,L0❫ ⊢ ⓛ[p]W0.T0 ➡[h,0] T1 →
+   ∀V2. ❪G0,L0❫ ⊢ V0 ➡[h,0] V2 → ∀W2. ❪G0,L0❫ ⊢ W0 ➡[h,0] W2 → ∀T2. ❪G0,L0.ⓛW0❫ ⊢ T0 ➡[h,0] T2 →
+   ∀L1. ❪G0,L0❫ ⊢ ➡[h,0] L1 → ∀L2. ❪G0,L0❫ ⊢ ➡[h,0] L2 →
+   ∃∃T. ❪G0,L1❫ ⊢ ⓐV1.T1 ➡[h,0] T & ❪G0,L2❫ ⊢ ⓓ[p]ⓝW2.V2.T2 ➡[h,0] T.
 #h #p #G0 #L0 #V0 #W0 #T0 #IH #V1 #HV01 #X #H
 #V2 #HV02 #W2 #HW02 #T2 #HT02 #L1 #HL01 #L2 #HL02
 elim (cpm_inv_abst1 … H) -H #W1 #T1 #HW01 #HT01 #H destruct
@@ -189,11 +189,11 @@ fact cpr_conf_lpr_flat_theta (h):
    ∀p,G0,L0,V0,W0,T0. (
       ∀G,L,T. ❪G0,L0,ⓐV0.ⓓ[p]W0.T0❫ ⬂+ ❪G,L,T❫ → IH_cpr_conf_lpr h G L T
    ) →
-   ∀V1. ❪G0,L0❫ ⊢ V0 ➡[h] V1 → ∀T1. ❪G0,L0❫ ⊢ ⓓ[p]W0.T0 ➡[h] T1 →
-   ∀V2. ❪G0,L0❫ ⊢ V0 ➡[h] V2 → ∀U2. ⇧[1] V2 ≘ U2 →
-   ∀W2. ❪G0,L0❫ ⊢ W0 ➡[h] W2 → ∀T2. ❪G0,L0.ⓓW0❫ ⊢ T0 ➡[h] T2 →
-   ∀L1. ❪G0,L0❫ ⊢ ➡[h] L1 → ∀L2. ❪G0,L0❫ ⊢ ➡[h] L2 →
-   ∃∃T. ❪G0,L1❫ ⊢ ⓐV1.T1 ➡[h] T & ❪G0,L2❫ ⊢ ⓓ[p]W2.ⓐU2.T2 ➡[h] T.
+   ∀V1. ❪G0,L0❫ ⊢ V0 ➡[h,0] V1 → ∀T1. ❪G0,L0❫ ⊢ ⓓ[p]W0.T0 ➡[h,0] T1 →
+   ∀V2. ❪G0,L0❫ ⊢ V0 ➡[h,0] V2 → ∀U2. ⇧[1] V2 ≘ U2 →
+   ∀W2. ❪G0,L0❫ ⊢ W0 ➡[h,0] W2 → ∀T2. ❪G0,L0.ⓓW0❫ ⊢ T0 ➡[h,0] T2 →
+   ∀L1. ❪G0,L0❫ ⊢ ➡[h,0] L1 → ∀L2. ❪G0,L0❫ ⊢ ➡[h,0] L2 →
+   ∃∃T. ❪G0,L1❫ ⊢ ⓐV1.T1 ➡[h,0] T & ❪G0,L2❫ ⊢ ⓓ[p]W2.ⓐU2.T2 ➡[h,0] T.
 #h #p #G0 #L0 #V0 #W0 #T0 #IH #V1 #HV01 #X #H
 #V2 #HV02 #U2 #HVU2 #W2 #HW02 #T2 #HT02 #L1 #HL01 #L2 #HL02
 elim (IH … HV01 … HV02 … HL01 … HL02) -HV01 -HV02 /2 width=1 by/ #V #HV1 #HV2
@@ -214,10 +214,10 @@ fact cpr_conf_lpr_beta_beta (h):
    ∀p,G0,L0,V0,W0,T0. (
       ∀G,L,T. ❪G0,L0,ⓐV0.ⓛ[p]W0.T0❫ ⬂+ ❪G,L,T❫ → IH_cpr_conf_lpr h G L T
    ) →
-   ∀V1. ❪G0,L0❫ ⊢ V0 ➡[h] V1 → ∀W1. ❪G0,L0❫ ⊢ W0 ➡[h] W1 → ∀T1. ❪G0,L0.ⓛW0❫ ⊢ T0 ➡[h] T1 →
-   ∀V2. ❪G0,L0❫ ⊢ V0 ➡[h] V2 → ∀W2. ❪G0,L0❫ ⊢ W0 ➡[h] W2 → ∀T2. ❪G0,L0.ⓛW0❫ ⊢ T0 ➡[h] T2 →
-   ∀L1. ❪G0,L0❫ ⊢ ➡[h] L1 → ∀L2. ❪G0,L0❫ ⊢ ➡[h] L2 →
-   ∃∃T. ❪G0,L1❫ ⊢ ⓓ[p]ⓝW1.V1.T1 ➡[h] T & ❪G0,L2❫ ⊢ ⓓ[p]ⓝW2.V2.T2 ➡[h] T.
+   ∀V1. ❪G0,L0❫ ⊢ V0 ➡[h,0] V1 → ∀W1. ❪G0,L0❫ ⊢ W0 ➡[h,0] W1 → ∀T1. ❪G0,L0.ⓛW0❫ ⊢ T0 ➡[h,0] T1 →
+   ∀V2. ❪G0,L0❫ ⊢ V0 ➡[h,0] V2 → ∀W2. ❪G0,L0❫ ⊢ W0 ➡[h,0] W2 → ∀T2. ❪G0,L0.ⓛW0❫ ⊢ T0 ➡[h,0] T2 →
+   ∀L1. ❪G0,L0❫ ⊢ ➡[h,0] L1 → ∀L2. ❪G0,L0❫ ⊢ ➡[h,0] L2 →
+   ∃∃T. ❪G0,L1❫ ⊢ ⓓ[p]ⓝW1.V1.T1 ➡[h,0] T & ❪G0,L2❫ ⊢ ⓓ[p]ⓝW2.V2.T2 ➡[h,0] T.
 #h #p #G0 #L0 #V0 #W0 #T0 #IH #V1 #HV01 #W1 #HW01 #T1 #HT01
 #V2 #HV02 #W2 #HW02 #T2 #HT02 #L1 #HL01 #L2 #HL02
 elim (IH … HV01 … HV02 … HL01 … HL02) -HV01 -HV02 /2 width=1 by/ #V #HV1 #HV2
@@ -233,12 +233,12 @@ fact cpr_conf_lpr_theta_theta (h):
    ∀p,G0,L0,V0,W0,T0. (
       ∀G,L,T. ❪G0,L0,ⓐV0.ⓓ[p]W0.T0❫ ⬂+ ❪G,L,T❫ → IH_cpr_conf_lpr h G L T
    ) →
-   ∀V1. ❪G0,L0❫ ⊢ V0 ➡[h] V1 → ∀U1. ⇧[1] V1 ≘ U1 →
-   ∀W1. ❪G0,L0❫ ⊢ W0 ➡[h] W1 → ∀T1. ❪G0,L0.ⓓW0❫ ⊢ T0 ➡[h] T1 →
-   ∀V2. ❪G0,L0❫ ⊢ V0 ➡[h] V2 → ∀U2. ⇧[1] V2 ≘ U2 →
-   ∀W2. ❪G0,L0❫ ⊢ W0 ➡[h] W2 → ∀T2. ❪G0,L0.ⓓW0❫ ⊢ T0 ➡[h] T2 →
-   ∀L1. ❪G0,L0❫ ⊢ ➡[h] L1 → ∀L2. ❪G0,L0❫ ⊢ ➡[h] L2 →
-   ∃∃T. ❪G0,L1❫ ⊢ ⓓ[p]W1.ⓐU1.T1 ➡[h] T & ❪G0,L2❫ ⊢ ⓓ[p]W2.ⓐU2.T2 ➡[h] T.
+   ∀V1. ❪G0,L0❫ ⊢ V0 ➡[h,0] V1 → ∀U1. ⇧[1] V1 ≘ U1 →
+   ∀W1. ❪G0,L0❫ ⊢ W0 ➡[h,0] W1 → ∀T1. ❪G0,L0.ⓓW0❫ ⊢ T0 ➡[h,0] T1 →
+   ∀V2. ❪G0,L0❫ ⊢ V0 ➡[h,0] V2 → ∀U2. ⇧[1] V2 ≘ U2 →
+   ∀W2. ❪G0,L0❫ ⊢ W0 ➡[h,0] W2 → ∀T2. ❪G0,L0.ⓓW0❫ ⊢ T0 ➡[h,0] T2 →
+   ∀L1. ❪G0,L0❫ ⊢ ➡[h,0] L1 → ∀L2. ❪G0,L0❫ ⊢ ➡[h,0] L2 →
+   ∃∃T. ❪G0,L1❫ ⊢ ⓓ[p]W1.ⓐU1.T1 ➡[h,0] T & ❪G0,L2❫ ⊢ ⓓ[p]W2.ⓐU2.T2 ➡[h,0] T.
 #h #p #G0 #L0 #V0 #W0 #T0 #IH #V1 #HV01 #U1 #HVU1 #W1 #HW01 #T1 #HT01
 #V2 #HV02 #U2 #HVU2 #W2 #HW02 #T2 #HT02 #L1 #HL01 #L2 #HL02
 elim (IH … HV01 … HV02 … HL01 … HL02) -HV01 -HV02 /2 width=1 by/ #V #HV1 #HV2
@@ -308,15 +308,15 @@ qed-.
 
 (* Properties with context-sensitive parallel reduction for terms ***********)
 
-lemma lpr_cpr_conf_dx (h) (G): ∀L0. ∀T0,T1:term. ❪G,L0❫ ⊢ T0 ➡[h] T1 → ∀L1. ❪G,L0❫ ⊢ ➡[h] L1 →
-                               ∃∃T. ❪G,L1❫ ⊢ T0 ➡[h] T & ❪G,L1❫ ⊢ T1 ➡[h] T.
+lemma lpr_cpr_conf_dx (h) (G): ∀L0. ∀T0,T1:term. ❪G,L0❫ ⊢ T0 ➡[h,0] T1 → ∀L1. ❪G,L0❫ ⊢ ➡[h,0] L1 →
+                               ∃∃T. ❪G,L1❫ ⊢ T0 ➡[h,0] T & ❪G,L1❫ ⊢ T1 ➡[h,0] T.
 #h #G #L0 #T0 #T1 #HT01 #L1 #HL01
 elim (cpr_conf_lpr … HT01 T0 … HL01 … HL01) -HT01 -HL01
 /2 width=3 by ex2_intro/
 qed-.
 
-lemma lpr_cpr_conf_sn (h) (G): ∀L0. ∀T0,T1:term. ❪G,L0❫ ⊢ T0 ➡[h] T1 → ∀L1. ❪G,L0❫ ⊢ ➡[h] L1 →
-                               ∃∃T. ❪G,L1❫ ⊢ T0 ➡[h] T & ❪G,L0❫ ⊢ T1 ➡[h] T.
+lemma lpr_cpr_conf_sn (h) (G): ∀L0. ∀T0,T1:term. ❪G,L0❫ ⊢ T0 ➡[h,0] T1 → ∀L1. ❪G,L0❫ ⊢ ➡[h,0] L1 →
+                               ∃∃T. ❪G,L1❫ ⊢ T0 ➡[h,0] T & ❪G,L0❫ ⊢ T1 ➡[h,0] T.
 #h #G #L0 #T0 #T1 #HT01 #L1 #HL01
 elim (cpr_conf_lpr … HT01 T0 … L0 … HL01) -HT01 -HL01
 /2 width=3 by ex2_intro/
@@ -324,6 +324,6 @@ qed-.
 
 (* Main properties **********************************************************)
 
-theorem lpr_conf (h) (G): confluent … (lpr h G).
+theorem lpr_conf (h) (G): confluent … (lpr h 0 G).
 /3 width=6 by lex_conf, cpr_conf_lpr/
 qed-.