X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Freduction%2Flpr_lpr.ma;h=723ed5c5944fa38ea158f0c2980c4f3a32f183e5;hb=52e675f555f559c047d5449db7fc89a51b977d35;hp=f9a63718a5d6ed6f04e18d26d0fb2d04b9855de3;hpb=e02bd4f3df78b5cc374d49d0ddf48b311188f514;p=helm.git

diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/lpr_lpr.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/lpr_lpr.ma
index f9a63718a..723ed5c59 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/reduction/lpr_lpr.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/reduction/lpr_lpr.ma
@@ -12,188 +12,189 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "basic_2/grammar/lpx_sn_lpx_sn.ma".
-include "basic_2/relocation/fsup.ma".
-include "basic_2/reduction/lpr_ldrop.ma".
+include "basic_2/substitution/lpx_sn_lpx_sn.ma".
+include "basic_2/multiple/fqup.ma".
+include "basic_2/reduction/lpr_drop.ma".
 
 (* SN PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS *****************************)
 
 (* Main properties on context-sensitive parallel reduction for terms ********)
 
 fact cpr_conf_lpr_atom_atom:
-   ∀I,L1,L2. ∃∃T. L1 ⊢ ⓪{I} ➡ T & L2 ⊢ ⓪{I} ➡ T.
-/2 width=3/ qed-.
+   ∀I,G,L1,L2. ∃∃T. ⦃G, L1⦄ ⊢ ⓪{I} ➡ T & ⦃G, L2⦄ ⊢ ⓪{I} ➡ T.
+/2 width=3 by cpr_atom, ex2_intro/ qed-.
 
 fact cpr_conf_lpr_atom_delta:
-   ∀L0,i. (
-      ∀L,T.♯{L, T} < ♯{L0, #i} →
-      ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ➡ T2 →
-      ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ➡ L2 →
-      ∃∃T0. L1 ⊢ T1 ➡ T0 & L2 ⊢ T2 ➡ T0
+   ∀G,L0,i. (
+      ∀L,T. ⦃G, L0, #i⦄ ⊐+ ⦃G, L, T⦄ →
+      ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
+      ∀L1. ⦃G, L⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L⦄ ⊢ ➡ L2 →
+      ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
    ) →
-   ∀K0,V0. ⇩[O, i] L0 ≡ K0.ⓓV0 →
-   ∀V2. K0 ⊢ V0 ➡ V2 → ∀T2. ⇧[O, i + 1] V2 ≡ T2 →
-   ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ➡ L2 →
-   ∃∃T. L1 ⊢ #i ➡ T & L2 ⊢ T2 ➡ T.
-#L0 #i #IH #K0 #V0 #HLK0 #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02
-elim (lpr_ldrop_conf … HLK0 … HL01) -HL01 #X1 #H1 #HLK1
+   ∀K0,V0. ⇩[i] L0 ≡ K0.ⓓV0 →
+   ∀V2. ⦃G, K0⦄ ⊢ V0 ➡ V2 → ∀T2. ⇧[O, i + 1] V2 ≡ T2 →
+   ∀L1. ⦃G, L0⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡ L2 →
+   ∃∃T. ⦃G, L1⦄ ⊢ #i ➡ T & ⦃G, L2⦄ ⊢ T2 ➡ T.
+#G #L0 #i #IH #K0 #V0 #HLK0 #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02
+elim (lpr_drop_conf … HLK0 … HL01) -HL01 #X1 #H1 #HLK1
 elim (lpr_inv_pair1 … H1) -H1 #K1 #V1 #HK01 #HV01 #H destruct
-elim (lpr_ldrop_conf … HLK0 … HL02) -HL02 #X2 #H2 #HLK2
+elim (lpr_drop_conf … HLK0 … HL02) -HL02 #X2 #H2 #HLK2
 elim (lpr_inv_pair1 … H2) -H2 #K2 #W2 #HK02 #_ #H destruct
-lapply (ldrop_fwd_ldrop2 … HLK2) -W2 #HLK2
-lapply (ldrop_pair2_fwd_fw … HLK0 (#i)) -HLK0 #HLK0
+lapply (drop_fwd_drop2 … HLK2) -W2 #HLK2
+lapply (fqup_lref … G … HLK0) -HLK0 #HLK0
 elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2
-elim (lift_total V 0 (i+1)) #T #HVT
-lapply (cpr_lift … HV2 … HLK2 … HVT2 … HVT) -K2 -V2 /3 width=6/
+elim (lift_total V 0 (i+1))
+/3 width=12 by cpr_lift, cpr_delta, ex2_intro/
 qed-.
 
 (* Basic_1: includes: pr0_delta_delta pr2_delta_delta *)
 fact cpr_conf_lpr_delta_delta:
-   ∀L0,i. (
-      ∀L,T.♯{L, T} < ♯{L0, #i} →
-      ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ➡ T2 →
-      ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ➡ L2 →
-      ∃∃T0. L1 ⊢ T1 ➡ T0 & L2 ⊢ T2 ➡ T0
+   ∀G,L0,i. (
+      ∀L,T. ⦃G, L0, #i⦄ ⊐+ ⦃G, L, T⦄ →
+      ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
+      ∀L1. ⦃G, L⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L⦄ ⊢ ➡ L2 →
+      ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
    ) →
-   ∀K0,V0. ⇩[O, i] L0 ≡ K0.ⓓV0 →
-   ∀V1. K0 ⊢ V0 ➡ V1 → ∀T1. ⇧[O, i + 1] V1 ≡ T1 →
-   ∀KX,VX. ⇩[O, i] L0 ≡ KX.ⓓVX →
-   ∀V2. KX ⊢ VX ➡ V2 → ∀T2. ⇧[O, i + 1] V2 ≡ T2 →
-   ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ➡ L2 →
-   ∃∃T. L1 ⊢ T1 ➡ T & L2 ⊢ T2 ➡ T.
-#L0 #i #IH #K0 #V0 #HLK0 #V1 #HV01 #T1 #HVT1
+   ∀K0,V0. ⇩[i] L0 ≡ K0.ⓓV0 →
+   ∀V1. ⦃G, K0⦄ ⊢ V0 ➡ V1 → ∀T1. ⇧[O, i + 1] V1 ≡ T1 →
+   ∀KX,VX. ⇩[i] L0 ≡ KX.ⓓVX →
+   ∀V2. ⦃G, KX⦄ ⊢ VX ➡ V2 → ∀T2. ⇧[O, i + 1] V2 ≡ T2 →
+   ∀L1. ⦃G, L0⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡ L2 →
+   ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡ T & ⦃G, L2⦄ ⊢ T2 ➡ T.
+#G #L0 #i #IH #K0 #V0 #HLK0 #V1 #HV01 #T1 #HVT1
 #KX #VX #H #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02
-lapply (ldrop_mono … H … HLK0) -H #H destruct
-elim (lpr_ldrop_conf … HLK0 … HL01) -HL01 #X1 #H1 #HLK1
+lapply (drop_mono … H … HLK0) -H #H destruct
+elim (lpr_drop_conf … HLK0 … HL01) -HL01 #X1 #H1 #HLK1
 elim (lpr_inv_pair1 … H1) -H1 #K1 #W1 #HK01 #_ #H destruct
-lapply (ldrop_fwd_ldrop2 … HLK1) -W1 #HLK1
-elim (lpr_ldrop_conf … HLK0 … HL02) -HL02 #X2 #H2 #HLK2
+lapply (drop_fwd_drop2 … HLK1) -W1 #HLK1
+elim (lpr_drop_conf … HLK0 … HL02) -HL02 #X2 #H2 #HLK2
 elim (lpr_inv_pair1 … H2) -H2 #K2 #W2 #HK02 #_ #H destruct
-lapply (ldrop_fwd_ldrop2 … HLK2) -W2 #HLK2
-lapply (ldrop_pair2_fwd_fw … HLK0 (#i)) -HLK0 #HLK0
+lapply (drop_fwd_drop2 … HLK2) -W2 #HLK2
+lapply (fqup_lref … G … HLK0) -HLK0 #HLK0
 elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2
-elim (lift_total V 0 (i+1)) #T #HVT
-lapply (cpr_lift … HV1 … HLK1 … HVT1 … HVT) -K1 -V1
-lapply (cpr_lift … HV2 … HLK2 … HVT2 … HVT) -K2 -V2 -V /2 width=3/
+elim (lift_total V 0 (i+1)) /3 width=12 by cpr_lift, ex2_intro/
 qed-.
 
 fact cpr_conf_lpr_bind_bind:
-   ∀a,I,L0,V0,T0. (
-      ∀L,T.♯{L,T} < ♯{L0,ⓑ{a,I}V0.T0} →
-      ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ➡ T2 →
-      ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ➡ L2 →
-      ∃∃T0. L1 ⊢ T1 ➡ T0 & L2 ⊢ T2 ➡ T0
+   ∀a,I,G,L0,V0,T0. (
+      ∀L,T. ⦃G, L0, ⓑ{a,I}V0.T0⦄ ⊐+ ⦃G, L, T⦄ →
+      ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
+      ∀L1. ⦃G, L⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L⦄ ⊢ ➡ L2 →
+      ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
    ) →
-   ∀V1. L0 ⊢ V0 ➡ V1 → ∀T1. L0.ⓑ{I}V0 ⊢ T0 ➡ T1 →
-   ∀V2. L0 ⊢ V0 ➡ V2 → ∀T2. L0.ⓑ{I}V0 ⊢ T0 ➡ T2 →
-   ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ➡ L2 →
-   ∃∃T. L1 ⊢ ⓑ{a,I}V1.T1 ➡ T & L2 ⊢ ⓑ{a,I}V2.T2 ➡ T.
-#a #I #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
+   ∀V1. ⦃G, L0⦄ ⊢ V0 ➡ V1 → ∀T1. ⦃G, L0.ⓑ{I}V0⦄ ⊢ T0 ➡ T1 →
+   ∀V2. ⦃G, L0⦄ ⊢ V0 ➡ V2 → ∀T2. ⦃G, L0.ⓑ{I}V0⦄ ⊢ T0 ➡ T2 →
+   ∀L1. ⦃G, L0⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡ L2 →
+   ∃∃T. ⦃G, L1⦄ ⊢ ⓑ{a,I}V1.T1 ➡ T & ⦃G, L2⦄ ⊢ ⓑ{a,I}V2.T2 ➡ T.
+#a #I #G #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 // /2 width=1/ /3 width=5/
+elim (IH … HT01 … HT02 (L1.ⓑ{I}V1) … (L2.ⓑ{I}V2)) -IH
+/3 width=5 by lpr_pair, cpr_bind, ex2_intro/
 qed-.
 
 fact cpr_conf_lpr_bind_zeta:
-   ∀L0,V0,T0. (
-      ∀L,T.♯{L,T} < ♯{L0,+ⓓV0.T0} →
-      ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ➡ T2 →
-      ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ➡ L2 →
-      ∃∃T0. L1 ⊢ T1 ➡ T0 & L2 ⊢ T2 ➡ T0
+   ∀G,L0,V0,T0. (
+      ∀L,T. ⦃G, L0, +ⓓV0.T0⦄ ⊐+ ⦃G, L, T⦄ →
+      ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
+      ∀L1. ⦃G, L⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L⦄ ⊢ ➡ L2 →
+      ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
    ) →
-   ∀V1. L0 ⊢ V0 ➡ V1 → ∀T1. L0.ⓓV0 ⊢ T0 ➡ T1 →
-   ∀T2. L0.ⓓV0 ⊢ T0 ➡ T2 → ∀X2. ⇧[O, 1] X2 ≡ T2 →
-   ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ➡ L2 →
-   ∃∃T. L1 ⊢ +ⓓV1.T1 ➡ T & L2 ⊢ X2 ➡ T.
-#L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
+   ∀V1. ⦃G, L0⦄ ⊢ V0 ➡ V1 → ∀T1. ⦃G, L0.ⓓV0⦄ ⊢ T0 ➡ T1 →
+   ∀T2. ⦃G, L0.ⓓV0⦄ ⊢ T0 ➡ T2 → ∀X2. ⇧[O, 1] X2 ≡ T2 →
+   ∀L1. ⦃G, L0⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡ L2 →
+   ∃∃T. ⦃G, L1⦄ ⊢ +ⓓV1.T1 ➡ T & ⦃G, L2⦄ ⊢ X2 ➡ T.
+#G #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
 #T2 #HT02 #X2 #HXT2 #L1 #HL01 #L2 #HL02
-elim (IH … HT01 … HT02 (L1.ⓓV1) … (L2.ⓓV1)) -IH -HT01 -HT02 // /2 width=1/ -L0 -V0 -T0 #T #HT1 #HT2
-elim (cpr_inv_lift1 … HT2 L2 … HXT2) -T2 /2 width=1/ /3 width=3/
+elim (IH … HT01 … HT02 (L1.ⓓV1) … (L2.ⓓV1)) -IH -HT01 -HT02 /2 width=1 by lpr_pair/ -L0 -V0 -T0 #T #HT1 #HT2
+elim (cpr_inv_lift1 … HT2 L2 … HXT2) -T2 /3 width=3 by cpr_zeta, drop_drop, ex2_intro/
 qed-.
 
 fact cpr_conf_lpr_zeta_zeta:
-   ∀L0,V0,T0. (
-      ∀L,T.♯{L,T} < ♯{L0,+ⓓV0.T0} →
-      ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ➡ T2 →
-      ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ➡ L2 →
-      ∃∃T0. L1 ⊢ T1 ➡ T0 & L2 ⊢ T2 ➡ T0
+   ∀G,L0,V0,T0. (
+      ∀L,T. ⦃G, L0, +ⓓV0.T0⦄ ⊐+ ⦃G, L, T⦄ →
+      ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
+      ∀L1. ⦃G, L⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L⦄ ⊢ ➡ L2 →
+      ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
    ) →
-   ∀T1. L0.ⓓV0 ⊢ T0 ➡ T1 → ∀X1. ⇧[O, 1] X1 ≡ T1 →
-   ∀T2. L0.ⓓV0 ⊢ T0 ➡ T2 → ∀X2. ⇧[O, 1] X2 ≡ T2 →
-   ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ➡ L2 →
-   ∃∃T. L1 ⊢ X1 ➡ T & L2 ⊢ X2 ➡ T.
-#L0 #V0 #T0 #IH #T1 #HT01 #X1 #HXT1
+   ∀T1. ⦃G, L0.ⓓV0⦄ ⊢ T0 ➡ T1 → ∀X1. ⇧[O, 1] X1 ≡ T1 →
+   ∀T2. ⦃G, L0.ⓓV0⦄ ⊢ T0 ➡ T2 → ∀X2. ⇧[O, 1] X2 ≡ T2 →
+   ∀L1. ⦃G, L0⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡ L2 →
+   ∃∃T. ⦃G, L1⦄ ⊢ X1 ➡ T & ⦃G, L2⦄ ⊢ X2 ➡ T.
+#G #L0 #V0 #T0 #IH #T1 #HT01 #X1 #HXT1
 #T2 #HT02 #X2 #HXT2 #L1 #HL01 #L2 #HL02
-elim (IH … HT01 … HT02 (L1.ⓓV0) … (L2.ⓓV0)) -IH -HT01 -HT02 // /2 width=1/ -L0 -T0 #T #HT1 #HT2
-elim (cpr_inv_lift1 … HT1 L1 … HXT1) -T1 /2 width=1/ #T1 #HT1 #HXT1
-elim (cpr_inv_lift1 … HT2 L2 … HXT2) -T2 /2 width=1/ #T2 #HT2 #HXT2 
-lapply (lift_inj … HT2 … HT1) -T #H destruct /2 width=3/
+elim (IH … HT01 … HT02 (L1.ⓓV0) … (L2.ⓓV0)) -IH -HT01 -HT02 /2 width=1 by lpr_pair/ -L0 -T0 #T #HT1 #HT2
+elim (cpr_inv_lift1 … HT1 L1 … HXT1) -T1 /2 width=2 by drop_drop/ #T1 #HT1 #HXT1
+elim (cpr_inv_lift1 … HT2 L2 … HXT2) -T2 /2 width=2 by drop_drop/ #T2 #HT2 #HXT2
+lapply (lift_inj … HT2 … HT1) -T #H destruct /2 width=3 by ex2_intro/
 qed-.
 
 fact cpr_conf_lpr_flat_flat:
-   ∀I,L0,V0,T0. (
-      ∀L,T.♯{L,T} < ♯{L0,ⓕ{I}V0.T0} →
-      ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ➡ T2 →
-      ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ➡ L2 →
-      ∃∃T0. L1 ⊢ T1 ➡ T0 & L2 ⊢ T2 ➡ T0
+   ∀I,G,L0,V0,T0. (
+      ∀L,T. ⦃G, L0, ⓕ{I}V0.T0⦄ ⊐+ ⦃G, L, T⦄ →
+      ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
+      ∀L1. ⦃G, L⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L⦄ ⊢ ➡ L2 →
+      ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
    ) →
-   ∀V1. L0 ⊢ V0 ➡ V1 → ∀T1. L0 ⊢ T0 ➡ T1 →
-   ∀V2. L0 ⊢ V0 ➡ V2 → ∀T2. L0 ⊢ T0 ➡ T2 →
-   ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ➡ L2 →
-   ∃∃T. L1 ⊢ ⓕ{I}V1.T1 ➡ T & L2 ⊢ ⓕ{I}V2.T2 ➡ T.
-#I #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01
+   ∀V1. ⦃G, L0⦄ ⊢ V0 ➡ V1 → ∀T1. ⦃G, L0⦄ ⊢ T0 ➡ T1 →
+   ∀V2. ⦃G, L0⦄ ⊢ V0 ➡ V2 → ∀T2. ⦃G, L0⦄ ⊢ T0 ➡ T2 →
+   ∀L1. ⦃G, L0⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡ L2 →
+   ∃∃T. ⦃G, L1⦄ ⊢ ⓕ{I}V1.T1 ➡ T & ⦃G, L2⦄ ⊢ ⓕ{I}V2.T2 ➡ T.
+#I #G #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 … HL01 … HL02) // /3 width=5/
+elim (IH … HT01 … HT02 … HL01 … HL02) /3 width=5 by cpr_flat, ex2_intro/
 qed-.
 
-fact cpr_conf_lpr_flat_tau:
-   ∀L0,V0,T0. (
-      ∀L,T.♯{L,T} < ♯{L0,ⓝV0.T0} →
-      ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ➡ T2 →
-      ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ➡ L2 →
-      ∃∃T0. L1 ⊢ T1 ➡ T0 & L2 ⊢ T2 ➡ T0
+fact cpr_conf_lpr_flat_eps:
+   ∀G,L0,V0,T0. (
+      ∀L,T. ⦃G, L0, ⓝV0.T0⦄ ⊐+ ⦃G, L, T⦄ →
+      ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
+      ∀L1. ⦃G, L⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L⦄ ⊢ ➡ L2 →
+      ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
    ) →
-   ∀V1,T1. L0 ⊢ T0 ➡ T1 → ∀T2. L0 ⊢ T0 ➡ T2 →
-   ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ➡ L2 →
-   ∃∃T. L1 ⊢ ⓝV1.T1 ➡ T & L2 ⊢ T2 ➡ T.
-#L0 #V0 #T0 #IH #V1 #T1 #HT01
+   ∀V1,T1. ⦃G, L0⦄ ⊢ T0 ➡ T1 → ∀T2. ⦃G, L0⦄ ⊢ T0 ➡ T2 →
+   ∀L1. ⦃G, L0⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡ L2 →
+   ∃∃T. ⦃G, L1⦄ ⊢ ⓝV1.T1 ➡ T & ⦃G, L2⦄ ⊢ T2 ➡ T.
+#G #L0 #V0 #T0 #IH #V1 #T1 #HT01
 #T2 #HT02 #L1 #HL01 #L2 #HL02
-elim (IH … HT01 … HT02 … HL01 … HL02) // -L0 -V0 -T0 /3 width=3/
+elim (IH … HT01 … HT02 … HL01 … HL02) // -L0 -V0 -T0 /3 width=3 by cpr_eps, ex2_intro/
 qed-.
 
-fact cpr_conf_lpr_tau_tau:
-   ∀L0,V0,T0. (
-      ∀L,T.♯{L,T} < ♯{L0,ⓝV0.T0} →
-      ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ➡ T2 →
-      ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ➡ L2 →
-      ∃∃T0. L1 ⊢ T1 ➡ T0 & L2 ⊢ T2 ➡ T0
+fact cpr_conf_lpr_eps_eps:
+   ∀G,L0,V0,T0. (
+      ∀L,T. ⦃G, L0, ⓝV0.T0⦄ ⊐+ ⦃G, L, T⦄ →
+      ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
+      ∀L1. ⦃G, L⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L⦄ ⊢ ➡ L2 →
+      ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
    ) →
-   ∀T1. L0 ⊢ T0 ➡ T1 → ∀T2. L0 ⊢ T0 ➡ T2 →
-   ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ➡ L2 →
-   ∃∃T. L1 ⊢ T1 ➡ T & L2 ⊢ T2 ➡ T.
-#L0 #V0 #T0 #IH #T1 #HT01
+   ∀T1. ⦃G, L0⦄ ⊢ T0 ➡ T1 → ∀T2. ⦃G, L0⦄ ⊢ T0 ➡ T2 →
+   ∀L1. ⦃G, L0⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡ L2 →
+   ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡ T & ⦃G, L2⦄ ⊢ T2 ➡ T.
+#G #L0 #V0 #T0 #IH #T1 #HT01
 #T2 #HT02 #L1 #HL01 #L2 #HL02
-elim (IH … HT01 … HT02 … HL01 … HL02) // -L0 -V0 -T0 /2 width=3/
+elim (IH … HT01 … HT02 … HL01 … HL02) // -L0 -V0 -T0 /2 width=3 by ex2_intro/
 qed-.
 
 fact cpr_conf_lpr_flat_beta:
-   ∀a,L0,V0,W0,T0. (
-      ∀L,T.♯{L,T} < ♯{L0,ⓐV0.ⓛ{a}W0.T0} →
-      ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ➡ T2 →
-      ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ➡ L2 →
-      ∃∃T0. L1 ⊢ T1 ➡ T0 & L2 ⊢ T2 ➡ T0
+   ∀a,G,L0,V0,W0,T0. (
+      ∀L,T. ⦃G, L0, ⓐV0.ⓛ{a}W0.T0⦄ ⊐+ ⦃G, L, T⦄ →
+      ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
+      ∀L1. ⦃G, L⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L⦄ ⊢ ➡ L2 →
+      ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
    ) →
-   ∀V1. L0 ⊢ V0 ➡ V1 → ∀T1. L0 ⊢ ⓛ{a}W0.T0 ➡ T1 →
-   ∀V2. L0 ⊢ V0 ➡ V2 → ∀T2. L0.ⓛW0 ⊢ T0 ➡ T2 →
-   ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ➡ L2 →
-   ∃∃T. L1 ⊢ ⓐV1.T1 ➡ T & L2 ⊢ ⓓ{a}V2.T2 ➡ T.
-#a #L0 #V0 #W0 #T0 #IH #V1 #HV01 #X #H
-#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02
+   ∀V1. ⦃G, L0⦄ ⊢ V0 ➡ V1 → ∀T1. ⦃G, L0⦄ ⊢ ⓛ{a}W0.T0 ➡ T1 →
+   ∀V2. ⦃G, L0⦄ ⊢ V0 ➡ V2 → ∀W2. ⦃G, L0⦄ ⊢ W0 ➡ W2 → ∀T2. ⦃G, L0.ⓛW0⦄ ⊢ T0 ➡ T2 →
+   ∀L1. ⦃G, L0⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡ L2 →
+   ∃∃T. ⦃G, L1⦄ ⊢ ⓐV1.T1 ➡ T & ⦃G, L2⦄ ⊢ ⓓ{a}ⓝW2.V2.T2 ➡ T.
+#a #G #L0 #V0 #W0 #T0 #IH #V1 #HV01 #X #H
+#V2 #HV02 #W2 #HW02 #T2 #HT02 #L1 #HL01 #L2 #HL02
 elim (cpr_inv_abst1 … H) -H #W1 #T1 #HW01 #HT01 #H destruct
-elim (IH … HV01 … HV02 … HL01 … HL02) -HV01 -HV02 /2 width=1/ #V #HV1 #HV2
-elim (IH … HT01 … HT02 (L1.ⓛW1) … (L2.ⓛW1)) /2 width=1/ -L0 -V0 -W0 -T0 #T #HT1 #HT2
-lapply (cpr_lsubr_trans … HT2 (L2.ⓓV2) ?) -HT2 /2 width=1/ /3 width=5/
+elim (IH … HV01 … HV02 … HL01 … HL02) -HV01 -HV02 /2 width=1 by/ #V #HV1 #HV2
+elim (IH … HW01 … HW02 … HL01 … HL02) /2 width=1 by/ #W #HW1 #HW2
+elim (IH … HT01 … HT02 (L1.ⓛW1) … (L2.ⓛW2)) /2 width=1 by lpr_pair/ -L0 -V0 -W0 -T0 #T #HT1 #HT2
+lapply (lsubr_cpr_trans … HT2 (L2.ⓓⓝW2.V2) ?) -HT2 /2 width=1 by lsubr_abst/ (**) (* full auto not tried *)
+/4 width=5 by cpr_bind, cpr_flat, cpr_beta, ex2_intro/
 qed-.
 
 (* Basic-1: includes:
@@ -201,81 +202,83 @@ qed-.
             pr0_cong_upsilon_cong pr0_cong_upsilon_delta
 *)
 fact cpr_conf_lpr_flat_theta:
-   ∀a,L0,V0,W0,T0. (
-      ∀L,T.♯{L,T} < ♯{L0,ⓐV0.ⓓ{a}W0.T0} →
-      ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ➡ T2 →
-      ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ➡ L2 →
-      ∃∃T0. L1 ⊢ T1 ➡ T0 & L2 ⊢ T2 ➡ T0
+   ∀a,G,L0,V0,W0,T0. (
+      ∀L,T. ⦃G, L0, ⓐV0.ⓓ{a}W0.T0⦄ ⊐+ ⦃G, L, T⦄ →
+      ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
+      ∀L1. ⦃G, L⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L⦄ ⊢ ➡ L2 →
+      ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
    ) →
-   ∀V1. L0 ⊢ V0 ➡ V1 → ∀T1. L0 ⊢ ⓓ{a}W0.T0 ➡ T1 →
-   ∀V2. L0 ⊢ V0 ➡ V2 → ∀U2. ⇧[O, 1] V2 ≡ U2 →
-   ∀W2. L0 ⊢ W0 ➡ W2 → ∀T2. L0.ⓓW0 ⊢ T0 ➡ T2 →
-   ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ➡ L2 →
-   ∃∃T. L1 ⊢ ⓐV1.T1 ➡ T & L2 ⊢ ⓓ{a}W2.ⓐU2.T2 ➡ T.
-#a #L0 #V0 #W0 #T0 #IH #V1 #HV01 #X #H
+   ∀V1. ⦃G, L0⦄ ⊢ V0 ➡ V1 → ∀T1. ⦃G, L0⦄ ⊢ ⓓ{a}W0.T0 ➡ T1 →
+   ∀V2. ⦃G, L0⦄ ⊢ V0 ➡ V2 → ∀U2. ⇧[O, 1] V2 ≡ U2 →
+   ∀W2. ⦃G, L0⦄ ⊢ W0 ➡ W2 → ∀T2. ⦃G, L0.ⓓW0⦄ ⊢ T0 ➡ T2 →
+   ∀L1. ⦃G, L0⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡ L2 →
+   ∃∃T. ⦃G, L1⦄ ⊢ ⓐV1.T1 ➡ T & ⦃G, L2⦄ ⊢ ⓓ{a}W2.ⓐU2.T2 ➡ T.
+#a #G #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/ #V #HV1 #HV2
+elim (IH … HV01 … HV02 … HL01 … HL02) -HV01 -HV02 /2 width=1 by/ #V #HV1 #HV2
 elim (lift_total V 0 1) #U #HVU
-lapply (cpr_lift … HV2 (L2.ⓓW2) … HVU2 … HVU) -HVU2 /2 width=1/ #HU2
+lapply (cpr_lift … HV2 (L2.ⓓW2) … HVU2 … HVU) -HVU2 /2 width=2 by drop_drop/ #HU2
 elim (cpr_inv_abbr1 … H) -H *
 [ #W1 #T1 #HW01 #HT01 #H destruct
-  elim (IH … HW01 … HW02 … HL01 … HL02) /2 width=1/
-  elim (IH … HT01 … HT02 (L1.ⓓW1) … (L2.ⓓW2)) /2 width=1/ -L0 -V0 -W0 -T0
-  /4 width=7 by cpr_bind, cpr_flat, cpr_theta, ex2_intro/ (**) (* timeout=35 *)
+  elim (IH … HW01 … HW02 … HL01 … HL02) /2 width=1 by/
+  elim (IH … HT01 … HT02 (L1.ⓓW1) … (L2.ⓓW2)) /2 width=1 by lpr_pair/ -L0 -V0 -W0 -T0
+  /4 width=7 by cpr_bind, cpr_flat, cpr_theta, ex2_intro/
 | #T1 #HT01 #HXT1 #H destruct
-  elim (IH … HT01 … HT02 (L1.ⓓW2) … (L2.ⓓW2)) /2 width=1/ -L0 -V0 -W0 -T0 #T #HT1 #HT2
-  elim (cpr_inv_lift1 … HT1 L1 … HXT1) -HXT1 /2 width=1/ #Y #HYT #HXY
-  @(ex2_intro … (ⓐV.Y)) /2 width=1/ /3 width=5/ (**) (* auto /4 width=9/ is too slow *)
-] 
+  elim (IH … HT01 … HT02 (L1.ⓓW2) … (L2.ⓓW2)) /2 width=1 by lpr_pair/ -L0 -V0 -W0 -T0 #T #HT1 #HT2
+  elim (cpr_inv_lift1 … HT1 L1 … HXT1) -HXT1
+  /4 width=9 by cpr_flat, cpr_zeta, drop_drop, lift_flat, ex2_intro/
+]
 qed-.
 
 fact cpr_conf_lpr_beta_beta:
-   ∀a,L0,V0,W0,T0. (
-      ∀L,T.♯{L,T} < ♯{L0,ⓐV0.ⓛ{a}W0.T0} →
-      ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ➡ T2 →
-      ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ➡ L2 →
-      ∃∃T0. L1 ⊢ T1 ➡ T0 & L2 ⊢ T2 ➡ T0
+   ∀a,G,L0,V0,W0,T0. (
+      ∀L,T. ⦃G, L0, ⓐV0.ⓛ{a}W0.T0⦄ ⊐+ ⦃G, L, T⦄ →
+      ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
+      ∀L1. ⦃G, L⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L⦄ ⊢ ➡ L2 →
+      ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
    ) →
-   ∀V1. L0 ⊢ V0 ➡ V1 → ∀T1. L0.ⓛW0 ⊢ T0 ➡ T1 →
-   ∀V2. L0 ⊢ V0 ➡ V2 → ∀T2. L0.ⓛW0 ⊢ T0 ➡ T2 →
-   ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ➡ L2 →
-   ∃∃T. L1 ⊢ ⓓ{a}V1.T1 ➡ T & L2 ⊢ ⓓ{a}V2.T2 ➡ T.
-#a #L0 #V0 #W0 #T0 #IH #V1 #HV01 #T1 #HT01
-#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02
-elim (IH … HV01 … HV02 … HL01 … HL02) -HV01 -HV02 /2 width=1/ #V #HV1 #HV2
-elim (IH … HT01 … HT02 (L1.ⓛW0) … (L2.ⓛW0)) /2 width=1/ -L0 -V0 -T0 #T #HT1 #HT2
-lapply (cpr_lsubr_trans … HT1 (L1.ⓓV1) ?) -HT1 /2 width=1/
-lapply (cpr_lsubr_trans … HT2 (L2.ⓓV2) ?) -HT2 /2 width=1/ /3 width=5/
+   ∀V1. ⦃G, L0⦄ ⊢ V0 ➡ V1 → ∀W1. ⦃G, L0⦄ ⊢ W0 ➡ W1 → ∀T1. ⦃G, L0.ⓛW0⦄ ⊢ T0 ➡ T1 →
+   ∀V2. ⦃G, L0⦄ ⊢ V0 ➡ V2 → ∀W2. ⦃G, L0⦄ ⊢ W0 ➡ W2 → ∀T2. ⦃G, L0.ⓛW0⦄ ⊢ T0 ➡ T2 →
+   ∀L1. ⦃G, L0⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡ L2 →
+   ∃∃T. ⦃G, L1⦄ ⊢ ⓓ{a}ⓝW1.V1.T1 ➡ T & ⦃G, L2⦄ ⊢ ⓓ{a}ⓝW2.V2.T2 ➡ T.
+#a #G #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
+elim (IH … HW01 … HW02 … HL01 … HL02) /2 width=1/ #W #HW1 #HW2
+elim (IH … HT01 … HT02 (L1.ⓛW1) … (L2.ⓛW2)) /2 width=1 by lpr_pair/ -L0 -V0 -W0 -T0 #T #HT1 #HT2
+lapply (lsubr_cpr_trans … HT1 (L1.ⓓⓝW1.V1) ?) -HT1 /2 width=1 by lsubr_abst/
+lapply (lsubr_cpr_trans … HT2 (L2.ⓓⓝW2.V2) ?) -HT2 /2 width=1 by lsubr_abst/
+/4 width=5 by cpr_bind, cpr_flat, ex2_intro/ (**) (* full auto not tried *)
 qed-.
 
 (* Basic_1: was: pr0_upsilon_upsilon *)
 fact cpr_conf_lpr_theta_theta:
-   ∀a,L0,V0,W0,T0. (
-      ∀L,T.♯{L,T} < ♯{L0,ⓐV0.ⓓ{a}W0.T0} →
-      ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ➡ T2 →
-      ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ➡ L2 →
-      ∃∃T0. L1 ⊢ T1 ➡ T0 & L2 ⊢ T2 ➡ T0
+   ∀a,G,L0,V0,W0,T0. (
+      ∀L,T. ⦃G, L0, ⓐV0.ⓓ{a}W0.T0⦄ ⊐+ ⦃G, L, T⦄ →
+      ∀T1. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 →
+      ∀L1. ⦃G, L⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L⦄ ⊢ ➡ L2 →
+      ∃∃T0. ⦃G, L1⦄ ⊢ T1 ➡ T0 & ⦃G, L2⦄ ⊢ T2 ➡ T0
    ) →
-   ∀V1. L0 ⊢ V0 ➡ V1 → ∀U1. ⇧[O, 1] V1 ≡ U1 →
-   ∀W1. L0 ⊢ W0 ➡ W1 → ∀T1. L0.ⓓW0 ⊢ T0 ➡ T1 →
-   ∀V2. L0 ⊢ V0 ➡ V2 → ∀U2. ⇧[O, 1] V2 ≡ U2 →
-   ∀W2. L0 ⊢ W0 ➡ W2 → ∀T2. L0.ⓓW0 ⊢ T0 ➡ T2 →
-   ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ➡ L2 →
-   ∃∃T. L1 ⊢ ⓓ{a}W1.ⓐU1.T1 ➡ T & L2 ⊢ ⓓ{a}W2.ⓐU2.T2 ➡ T.
-#a #L0 #V0 #W0 #T0 #IH #V1 #HV01 #U1 #HVU1 #W1 #HW01 #T1 #HT01
+   ∀V1. ⦃G, L0⦄ ⊢ V0 ➡ V1 → ∀U1. ⇧[O, 1] V1 ≡ U1 →
+   ∀W1. ⦃G, L0⦄ ⊢ W0 ➡ W1 → ∀T1. ⦃G, L0.ⓓW0⦄ ⊢ T0 ➡ T1 →
+   ∀V2. ⦃G, L0⦄ ⊢ V0 ➡ V2 → ∀U2. ⇧[O, 1] V2 ≡ U2 →
+   ∀W2. ⦃G, L0⦄ ⊢ W0 ➡ W2 → ∀T2. ⦃G, L0.ⓓW0⦄ ⊢ T0 ➡ T2 →
+   ∀L1. ⦃G, L0⦄ ⊢ ➡ L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡ L2 →
+   ∃∃T. ⦃G, L1⦄ ⊢ ⓓ{a}W1.ⓐU1.T1 ➡ T & ⦃G, L2⦄ ⊢ ⓓ{a}W2.ⓐU2.T2 ➡ T.
+#a #G #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/ #V #HV1 #HV2
-elim (IH … HW01 … HW02 … HL01 … HL02) /2 width=1/
-elim (IH … HT01 … HT02 (L1.ⓓW1) … (L2.ⓓW2)) /2 width=1/ -L0 -V0 -W0 -T0
+elim (IH … HV01 … HV02 … HL01 … HL02) -HV01 -HV02 /2 width=1 by/ #V #HV1 #HV2
+elim (IH … HW01 … HW02 … HL01 … HL02) /2 width=1 by/
+elim (IH … HT01 … HT02 (L1.ⓓW1) … (L2.ⓓW2)) /2 width=1 by lpr_pair/ -L0 -V0 -W0 -T0
 elim (lift_total V 0 1) #U #HVU
-lapply (cpr_lift … HV1 (L1.ⓓW1) … HVU1 … HVU) -HVU1 /2 width=1/
-lapply (cpr_lift … HV2 (L2.ⓓW2) … HVU2 … HVU) -HVU2 /2 width=1/
-/4 width=7 by cpr_bind, cpr_flat, ex2_intro/ (**) (* timeout 40 *)
+lapply (cpr_lift … HV1 (L1.ⓓW1) … HVU1 … HVU) -HVU1 /2 width=2 by drop_drop/
+lapply (cpr_lift … HV2 (L2.ⓓW2) … HVU2 … HVU) -HVU2 /2 width=2 by drop_drop/
+/4 width=7 by cpr_bind, cpr_flat, ex2_intro/ (**) (* full auto not tried *)
 qed-.
 
-theorem cpr_conf_lpr: lpx_sn_confluent cpr cpr.
-#L0 #T0 @(f2_ind … fw … L0 T0) -L0 -T0 #n #IH #L0 * [|*]
-[ #I0 #Hn #T1 #H1 #T2 #H2 #L1 #HL01 #L2 #HL02 destruct
+theorem cpr_conf_lpr: ∀G. lpx_sn_confluent (cpr G) (cpr G).
+#G #L0 #T0 @(fqup_wf_ind_eq … G L0 T0) -G -L0 -T0 #G #L #T #IH #G0 #L0 * [| * ]
+[ #I0 #HG #HL #HT #T1 #H1 #T2 #H2 #L1 #HL01 #L2 #HL02 destruct
   elim (cpr_inv_atom1 … H1) -H1
   elim (cpr_inv_atom1 … H2) -H2
   [ #H2 #H1 destruct
@@ -288,7 +291,7 @@ theorem cpr_conf_lpr: lpx_sn_confluent cpr cpr.
     * #K0 #V0 #V1 #i #HLK0 #HV01 #HVT1 #H1 destruct
     /3 width=17 by cpr_conf_lpr_delta_delta/
   ]
-| #a #I #V0 #T0 #Hn #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct
+| #a #I #V0 #T0 #HG #HL #HT #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct
   elim (cpr_inv_bind1 … H1) -H1 *
   [ #V1 #T1 #HV01 #HT01 #H1
   | #T1 #HT01 #HXT1 #H11 #H12
@@ -302,27 +305,27 @@ theorem cpr_conf_lpr: lpx_sn_confluent cpr cpr.
   | /3 width=11 by cpr_conf_lpr_bind_zeta/
   | /3 width=12 by cpr_conf_lpr_zeta_zeta/
   ]
-| #I #V0 #T0 #Hn #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct
+| #I #V0 #T0 #HG #HL #HT #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct
   elim (cpr_inv_flat1 … H1) -H1 *
   [ #V1 #T1 #HV01 #HT01 #H1
   | #HX1 #H1
-  | #a1 #V1 #Y1 #Z1 #T1 #HV01 #HZT1 #H11 #H12 #H13
+  | #a1 #V1 #Y1 #W1 #Z1 #T1 #HV01 #HYW1 #HZT1 #H11 #H12 #H13
   | #a1 #V1 #U1 #Y1 #W1 #Z1 #T1 #HV01 #HVU1 #HYW1 #HZT1 #H11 #H12 #H13
   ]
   elim (cpr_inv_flat1 … H2) -H2 *
   [1,5,9,13: #V2 #T2 #HV02 #HT02 #H2
   |2,6,10,14: #HX2 #H2
-  |3,7,11,15: #a2 #V2 #Y2 #Z2 #T2 #HV02 #HZT2 #H21 #H22 #H23
+  |3,7,11,15: #a2 #V2 #Y2 #W2 #Z2 #T2 #HV02 #HYW2 #HZT2 #H21 #H22 #H23
   |4,8,12,16: #a2 #V2 #U2 #Y2 #W2 #Z2 #T2 #HV02 #HVU2 #HYW2 #HZT2 #H21 #H22 #H23
   ] destruct
   [ /3 width=10 by cpr_conf_lpr_flat_flat/
-  | /4 width=8 by ex2_commute, cpr_conf_lpr_flat_tau/
-  | /4 width=11 by ex2_commute, cpr_conf_lpr_flat_beta/
+  | /4 width=8 by ex2_commute, cpr_conf_lpr_flat_eps/
+  | /4 width=12 by ex2_commute, cpr_conf_lpr_flat_beta/
   | /4 width=14 by ex2_commute, cpr_conf_lpr_flat_theta/
-  | /3 width=8 by cpr_conf_lpr_flat_tau/
-  | /3 width=7 by cpr_conf_lpr_tau_tau/
-  | /3 width=11 by cpr_conf_lpr_flat_beta/
-  | /3 width=11 by cpr_conf_lpr_beta_beta/
+  | /3 width=8 by cpr_conf_lpr_flat_eps/
+  | /3 width=7 by cpr_conf_lpr_eps_eps/
+  | /3 width=12 by cpr_conf_lpr_flat_beta/
+  | /3 width=13 by cpr_conf_lpr_beta_beta/
   | /3 width=14 by cpr_conf_lpr_flat_theta/
   | /3 width=17 by cpr_conf_lpr_theta_theta/
   ]
@@ -330,25 +333,25 @@ theorem cpr_conf_lpr: lpx_sn_confluent cpr cpr.
 qed-.
 
 (* Basic_1: includes: pr0_confluence pr2_confluence *)
-theorem cpr_conf: ∀L. confluent … (cpr L).
+theorem cpr_conf: ∀G,L. confluent … (cpr G L).
 /2 width=6 by cpr_conf_lpr/ qed-.
 
 (* Properties on context-sensitive parallel reduction for terms *************)
 
-lemma lpr_cpr_conf_dx: ∀L0,T0,T1. L0 ⊢ T0 ➡ T1 → ∀L1. L0 ⊢ ➡ L1 →
-                       ∃∃T. L1 ⊢ T0 ➡ T & L1 ⊢ T1 ➡ T.
-#L0 #T0 #T1 #HT01 #L1 #HL01
-elim (cpr_conf_lpr … HT01 T0 … HL01 … HL01) // -L0 /2 width=3/
+lemma lpr_cpr_conf_dx: ∀G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡ T1 → ∀L1. ⦃G, L0⦄ ⊢ ➡ L1 →
+                       ∃∃T. ⦃G, L1⦄ ⊢ T0 ➡ T & ⦃G, L1⦄ ⊢ T1 ➡ T.
+#G #L0 #T0 #T1 #HT01 #L1 #HL01
+elim (cpr_conf_lpr … HT01 T0 … HL01 … HL01) /2 width=3 by ex2_intro/
 qed-.
 
-lemma lpr_cpr_conf_sn: ∀L0,T0,T1. L0 ⊢ T0 ➡ T1 → ∀L1. L0 ⊢ ➡ L1 →
-                       ∃∃T. L1 ⊢ T0 ➡ T & L0 ⊢ T1 ➡ T.
-#L0 #T0 #T1 #HT01 #L1 #HL01
-elim (cpr_conf_lpr … HT01 T0 … L0 … HL01) // -HT01 -HL01 /2 width=3/
+lemma lpr_cpr_conf_sn: ∀G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡ T1 → ∀L1. ⦃G, L0⦄ ⊢ ➡ L1 →
+                       ∃∃T. ⦃G, L1⦄ ⊢ T0 ➡ T & ⦃G, L0⦄ ⊢ T1 ➡ T.
+#G #L0 #T0 #T1 #HT01 #L1 #HL01
+elim (cpr_conf_lpr … HT01 T0 … L0 … HL01) /2 width=3 by ex2_intro/
 qed-.
 
 (* Main properties **********************************************************)
 
-theorem lpr_conf: confluent … lpr.
+theorem lpr_conf: ∀G. confluent … (lpr G).
 /3 width=6 by lpx_sn_conf, cpr_conf_lpr/
 qed-.