X-Git-Url: http://matita.cs.unibo.it/gitweb/?p=helm.git;a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Frt_transition%2Flpr_lpr.ma;h=58f5d5843da77f90f4d6ac01dc2315c60e6e7873;hp=dcbe44401f1a76ba9f3f7c41e0b80c30ea7b4948;hb=bd53c4e895203eb049e75434f638f26b5a161a2b;hpb=3b7b8afcb429a60d716d5226a5b6ab0d003228b1

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 dcbe44401..58f5d5843 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lpr_lpr.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lpr_lpr.ma
@@ -21,24 +21,24 @@ 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] 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.
 
 (* 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] T & ❪G,L2❫ ⊢ ⓪[I] ➡[h] T.
 /2 width=3 by cpr_refl, ex2_intro/ qed-.
 
 fact cpr_conf_lpr_atom_delta (h):
    ∀G0,L0,i. (
-      ∀G,L,T. ⦃G0,L0,#i⦄ ⬂+ ⦃G,L,T⦄ → IH_cpr_conf_lpr h G L T
+      ∀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] 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.
 #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
@@ -54,14 +54,14 @@ qed-.
 (* Basic_1: includes: pr0_delta_delta pr2_delta_delta *)
 fact cpr_conf_lpr_delta_delta (h):
    ∀G0,L0,i. (
-      ∀G,L,T. ⦃G0,L0,#i⦄ ⬂+ ⦃G,L,T⦄ → IH_cpr_conf_lpr h G L T
+      ∀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] 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] 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.
 #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
@@ -79,27 +79,27 @@ qed-.
 
 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
+      ∀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] 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.
 #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) //
-elim (IH … HT01 … HT02 (L1.ⓑ{I}V1) … (L2.ⓑ{I}V2)) -IH
+elim (IH … HT01 … HT02 (L1.ⓑ[I]V1) … (L2.ⓑ[I]V2)) -IH
 /3 width=5 by lpr_pair, cpm_bind, ex2_intro/
 qed-.
 
 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
+      ∀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] 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.
 #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
@@ -109,12 +109,12 @@ qed-.
 
 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
+      ∀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] 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.
 #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
@@ -124,12 +124,12 @@ qed-.
 
 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
+      ∀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] 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.
 #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) //
@@ -139,11 +139,11 @@ qed-.
 
 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
+      ∀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] 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.
 #h #G0 #L0 #V0 #T0 #IH #V1 #T1 #HT01
 #T2 #HT02 #L1 #HL01 #L2 #HL02
 elim (IH … HT01 … HT02 … HL01 … HL02) // -L0 -V0 -T0
@@ -152,11 +152,11 @@ qed-.
 
 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
+      ∀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] 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.
 #h #G0 #L0 #V0 #T0 #IH #T1 #HT01
 #T2 #HT02 #L1 #HL01 #L2 #HL02
 elim (IH … HT01 … HT02 … HL01 … HL02) // -L0 -V0 -T0
@@ -165,12 +165,12 @@ qed-.
 
 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
+      ∀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] 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.
 #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
@@ -187,13 +187,13 @@ qed-.
 *)
 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
+      ∀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] 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.
 #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
@@ -212,12 +212,12 @@ qed-.
 
 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
+      ∀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] 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.
 #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
@@ -231,14 +231,14 @@ qed-.
 (* Basic_1: was: pr0_upsilon_upsilon *)
 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
+      ∀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] 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.
 #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] T1 → ∀L1. ❪G,L0❫ ⊢ ➡[h] L1 →
+                               ∃∃T. ❪G,L1❫ ⊢ T0 ➡[h] T & ❪G,L1❫ ⊢ T1 ➡[h] 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] T1 → ∀L1. ❪G,L0❫ ⊢ ➡[h] L1 →
+                               ∃∃T. ❪G,L1❫ ⊢ T0 ➡[h] T & ❪G,L0❫ ⊢ T1 ➡[h] 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/