X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Frt_transition%2Flpr_lpr.ma;h=58f5d5843da77f90f4d6ac01dc2315c60e6e7873;hb=bd53c4e895203eb049e75434f638f26b5a161a2b;hp=fad8b481a731d9b4873c58dc7ef36e2ce6cbf8cd;hpb=e9f96fa56226dfd74de214c89d827de0c5018ac7;p=helm.git

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 fad8b481a..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
@@ -12,346 +12,318 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "basic_2/substitution/lpx_sn_lpx_sn.ma".
-include "basic_2/multiple/fqup.ma".
-include "basic_2/reduction/lpr_drop.ma".
+include "static_2/relocation/lex_lex.ma".
+include "basic_2/rt_transition/cpm_lsubr.ma".
+include "basic_2/rt_transition/cpr.ma".
+include "basic_2/rt_transition/cpr_drops.ma".
+include "basic_2/rt_transition/lpr_drops.ma".
 
-(* SN PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS *****************************)
+(* PARALLEL R-TRANSITION FOR FULL LOCAL ENVIRONMENTS ************************)
 
-(* Main properties on context-sensitive parallel reduction for terms ********)
+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.
 
-fact cpr_conf_lpr_atom_atom:
-   ∀I,G,L1,L2. ∃∃T. ⦃G, L1⦄ ⊢ ⓪{I} ➡ T & ⦃G, L2⦄ ⊢ ⓪{I} ➡ T.
-/2 width=3 by cpr_atom, ex2_intro/ qed-.
+(* Main properties with context-sensitive parallel reduction for terms ******)
 
-fact cpr_conf_lpr_atom_delta:
-   ∀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
+fact cpr_conf_lpr_atom_atom (h):
+   ∀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
    ) →
-   ∀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_drop_conf … HLK0 … HL02) -HL02 #X2 #H2 #HLK2
-elim (lpr_inv_pair1 … H2) -H2 #K2 #W2 #HK02 #_ #H destruct
-lapply (drop_fwd_drop2 … HLK2) -W2 #HLK2
-lapply (fqup_lref … G … HLK0) -HLK0 #HLK0
+   ∀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.
+#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
+elim (lpr_drops_conf … HLK0 … HL02) -HL02 // #X2 #H2 #HLK2
+elim (lpr_inv_pair_sn … H2) -H2 #K2 #W2 #HK02 #_ #H destruct
+lapply (drops_isuni_fwd_drop2 … HLK2) -W2 // #HLK2
+lapply (fqup_lref (Ⓣ) … G0 … HLK0) -HLK0 #HLK0
 elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2
-elim (lift_total V 0 (i+1))
-/3 width=12 by cpr_lift, cpr_delta, ex2_intro/
+elim (cpm_lifts_sn … HV2 … HLK2 … HVT2) -V2 -HLK2 #T #HVT #HT2
+/3 width=6 by cpm_delta_drops, ex2_intro/
 qed-.
 
 (* Basic_1: includes: pr0_delta_delta pr2_delta_delta *)
-fact cpr_conf_lpr_delta_delta:
-   ∀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
+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
    ) →
-   ∀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
+   ∀K0,V0. ⇩*[i] L0 ≘ K0.ⓓV0 →
+   ∀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.
+#h #G0 #L0 #i #IH #K0 #V0 #HLK0 #V1 #HV01 #T1 #HVT1
 #KX #VX #H #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02
-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 (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 (drop_fwd_drop2 … HLK2) -W2 #HLK2
-lapply (fqup_lref … G … HLK0) -HLK0 #HLK0
+lapply (drops_mono … H … HLK0) -H #H destruct
+elim (lpr_drops_conf … HLK0 … HL01) -HL01 // #X1 #H1 #HLK1
+elim (lpr_inv_pair_sn … H1) -H1 #K1 #W1 #HK01 #_ #H destruct
+lapply (drops_isuni_fwd_drop2 … HLK1) -W1 // #HLK1
+elim (lpr_drops_conf … HLK0 … HL02) -HL02 // #X2 #H2 #HLK2
+elim (lpr_inv_pair_sn … H2) -H2 #K2 #W2 #HK02 #_ #H destruct
+lapply (drops_isuni_fwd_drop2 … HLK2) -W2 // #HLK2
+lapply (fqup_lref (Ⓣ) … G0 … HLK0) -HLK0 #HLK0
 elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2
-elim (lift_total V 0 (i+1)) /3 width=12 by cpr_lift, ex2_intro/
+elim (cpm_lifts_sn … HV1 … HLK1 … HVT1) -V1 -HLK1 #T #HVT #HT1
+/3 width=11 by cpm_lifts_bi, ex2_intro/
 qed-.
 
-fact cpr_conf_lpr_bind_bind:
-   ∀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
+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. ⦃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
+   ∀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
-/3 width=5 by lpr_pair, cpr_bind, ex2_intro/
+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:
-   ∀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
+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. ⦃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 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/
+   ∀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
+elim (IH … HT2 … HTX2 … HL01 … HL02) [| /2 width=1 by fqup_zeta/ ] -L0 -V0 -T0 -T2 #T2 #HT2 #HXT2
+/3 width=3 by cpm_zeta, ex2_intro/
 qed-.
 
-fact cpr_conf_lpr_zeta_zeta:
-   ∀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
+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. ⦃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 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/
+   ∀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
+elim (IH … HTX1 … HTX2 … HL01 … HL02) [| /2 width=1 by fqup_zeta/ ] -L0 -V0 -T0 -T1 #X #HX1 #HX2
+/2 width=3 by ex2_intro/
 qed-.
 
-fact cpr_conf_lpr_flat_flat:
-   ∀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
+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. ⦃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
+   ∀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) //
-elim (IH … HT01 … HT02 … HL01 … HL02) /3 width=5 by cpr_flat, ex2_intro/
+elim (IH … HT01 … HT02 … HL01 … HL02) //
+/3 width=5 by cpr_flat, ex2_intro/
 qed-.
 
-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
+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. ⦃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
+   ∀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 /3 width=3 by cpr_eps, ex2_intro/
+elim (IH … HT01 … HT02 … HL01 … HL02) // -L0 -V0 -T0
+/3 width=3 by cpm_eps, ex2_intro/
 qed-.
 
-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
+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. ⦃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
+   ∀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 /2 width=3 by ex2_intro/
+elim (IH … HT01 … HT02 … HL01 … HL02) // -L0 -V0 -T0
+/2 width=3 by ex2_intro/
 qed-.
 
-fact cpr_conf_lpr_flat_beta:
-   ∀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
+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. ⦃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
+   ∀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 (cpr_inv_abst1 … H) -H #W1 #T1 #HW01 #HT01 #H destruct
+elim (cpm_inv_abst1 … H) -H #W1 #T1 #HW01 #HT01 #H destruct
 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_beta/ (**) (* full auto not tried *)
-/4 width=5 by cpr_bind, cpr_flat, cpr_beta, ex2_intro/
+lapply (lsubr_cpm_trans … HT2 (L2.ⓓⓝW2.V2) ?) -HT2 /2 width=1 by lsubr_beta/ (**) (* full auto not tried *)
+/4 width=5 by cpm_bind, cpr_flat, cpm_beta, ex2_intro/
 qed-.
 
 (* Basic-1: includes:
             pr0_cong_upsilon_refl pr0_cong_upsilon_zeta
             pr0_cong_upsilon_cong pr0_cong_upsilon_delta
 *)
-fact cpr_conf_lpr_flat_theta:
-   ∀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
+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. ⦃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
+   ∀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
-elim (lift_total V 0 1) #U #HVU
-lapply (cpr_lift … HV2 (L2.ⓓW2) … HVU2 … HVU) -HVU2 /2 width=2 by drop_drop/ #HU2
-elim (cpr_inv_abbr1 … H) -H *
+elim (cpm_inv_abbr1 … H) -H *
 [ #W1 #T1 #HW01 #HT01 #H destruct
+  elim (cpm_lifts_sn … HV2 (Ⓣ) … (L2.ⓓW2) … HVU2) -HVU2 [| /3 width=2 by drops_refl, drops_drop/ ] #U #HVU #HU2
   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 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/
+  /4 width=7 by cpm_bind, cpm_appl, cpm_theta, ex2_intro/
+| #X0 #HXT0 #HX0 #H destruct
+  elim (cpm_inv_lifts_sn … HT02 (Ⓣ) … L0 … HXT0) -HT02 [| /3 width=2 by drops_refl, drops_drop/ ] #X2 #HXT2 #HX02
+  elim (IH … HX0 … HX02 … HL01 … HL02) [| /3 width=5 by fqup_strap1, fqu_drop/ ] -L0 -V0 -W0 -T0 #T #H1T #H2T
+  /4 width=8 by cpm_appl, cpm_zeta, lifts_flat, ex2_intro/
 ]
 qed-.
 
-fact cpr_conf_lpr_beta_beta:
-   ∀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
+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. ⦃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
+   ∀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
 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 … HT1 (L1.ⓓⓝW1.V1) ?) -HT1 /2 width=1 by lsubr_beta/
-lapply (lsubr_cpr_trans … HT2 (L2.ⓓⓝW2.V2) ?) -HT2 /2 width=1 by lsubr_beta/
-/4 width=5 by cpr_bind, cpr_flat, ex2_intro/ (**) (* full auto not tried *)
+lapply (lsubr_cpm_trans … HT1 (L1.ⓓⓝW1.V1) ?) -HT1 /2 width=1 by lsubr_beta/
+lapply (lsubr_cpm_trans … HT2 (L2.ⓓⓝW2.V2) ?) -HT2 /2 width=1 by lsubr_beta/
+/4 width=5 by cpm_bind, cpr_flat, ex2_intro/ (**) (* full auto not tried *)
 qed-.
 
 (* Basic_1: was: pr0_upsilon_upsilon *)
-fact cpr_conf_lpr_theta_theta:
-   ∀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
+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. ⦃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
+   ∀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
-elim (IH … HW01 … HW02 … HL01 … HL02) /2 width=1 by/
+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
-elim (lift_total V 0 1) #U #HVU
-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 *)
+elim (cpm_lifts_sn … HV1 (Ⓣ) … (L1.ⓓW1) … HVU1) -HVU1 /3 width=2 by drops_refl, drops_drop/ #U #HVU #HU1
+lapply (cpm_lifts_bi … HV2 (Ⓣ) … (L2.ⓓW2) … HVU2 … HVU) -HVU2 /3 width=2 by drops_refl, drops_drop/
+/4 width=7 by cpm_bind, cpm_appl, ex2_intro/ (**) (* full auto not tried *)
 qed-.
 
-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 * [| * ]
+theorem cpr_conf_lpr (h): ∀G. lex_confluent (λL.cpm h G L 0) (λL.cpm h G L 0).
+#h #G0 #L0 #T0 @(fqup_wf_ind_eq (Ⓣ) … G0 L0 T0) -G0 -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
+  elim (cpr_inv_atom1_drops … H1) -H1
+  elim (cpr_inv_atom1_drops … H2) -H2
   [ #H2 #H1 destruct
-    /2 width=1 by cpr_conf_lpr_atom_atom/
+    @cpr_conf_lpr_atom_atom
   | * #K0 #V0 #V2 #i2 #HLK0 #HV02 #HVT2 #H2 #H1 destruct
-    /3 width=10 by cpr_conf_lpr_atom_delta/
+    @(cpr_conf_lpr_atom_delta … IH) -IH /width=6 by/
   | #H2 * #K0 #V0 #V1 #i1 #HLK0 #HV01 #HVT1 #H1 destruct
-    /4 width=10 by ex2_commute, cpr_conf_lpr_atom_delta/
+    @ex2_commute @(cpr_conf_lpr_atom_delta … IH) -IH /width=6 by/
   | * #X #Y #V2 #z #H #HV02 #HVT2 #H2
     * #K0 #V0 #V1 #i #HLK0 #HV01 #HVT1 #H1 destruct
-    /3 width=17 by cpr_conf_lpr_delta_delta/
+    @(cpr_conf_lpr_delta_delta … IH) -IH /width=6 by/
   ]
-| #a #I #V0 #T0 #HG #HL #HT #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct
-  elim (cpr_inv_bind1 … H1) -H1 *
+| #p #I #V0 #T0 #HG #HL #HT #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct
+  elim (cpm_inv_bind1 … H1) -H1 *
   [ #V1 #T1 #HV01 #HT01 #H1
-  | #T1 #HT01 #HXT1 #H11 #H12
+  | #T1 #HT10 #HTX1 #H11 #H12
   ]
-  elim (cpr_inv_bind1 … H2) -H2 *
+  elim (cpm_inv_bind1 … H2) -H2 *
   [1,3: #V2 #T2 #HV02 #HT02 #H2
-  |2,4: #T2 #HT02 #HXT2 #H21 #H22
+  |2,4: #T2 #HT20 #HTX2 #H21 #H22
   ] destruct
-  [ /3 width=10 by cpr_conf_lpr_bind_bind/
-  | /4 width=11 by ex2_commute, cpr_conf_lpr_bind_zeta/
-  | /3 width=11 by cpr_conf_lpr_bind_zeta/
-  | /3 width=12 by cpr_conf_lpr_zeta_zeta/
+  [ @(cpr_conf_lpr_bind_bind … IH) -IH /width=1 by/
+  | @ex2_commute @(cpr_conf_lpr_bind_zeta … IH) -IH /width=3 by/
+  | @(cpr_conf_lpr_bind_zeta … IH) -IH /width=3 by/
+  | @(cpr_conf_lpr_zeta_zeta … IH) -IH /width=3 by/
   ]
 | #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 #W1 #Z1 #T1 #HV01 #HYW1 #HZT1 #H11 #H12 #H13
-  | #a1 #V1 #U1 #Y1 #W1 #Z1 #T1 #HV01 #HVU1 #HYW1 #HZT1 #H11 #H12 #H13
+  | #p1 #V1 #Y1 #W1 #Z1 #T1 #HV01 #HYW1 #HZT1 #H11 #H12 #H13
+  | #p1 #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 #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
+  |3,7,11,15: #p2 #V2 #Y2 #W2 #Z2 #T2 #HV02 #HYW2 #HZT2 #H21 #H22 #H23
+  |4,8,12,16: #p2 #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_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_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/
+  [ @(cpr_conf_lpr_flat_flat … IH) -IH /width=1 by/
+  | @ex2_commute @(cpr_conf_lpr_flat_eps … IH) -IH /width=1 by/
+  | @ex2_commute @(cpr_conf_lpr_flat_beta … IH) -IH /width=1 by/
+  | @ex2_commute @(cpr_conf_lpr_flat_theta … IH) -IH /width=3 by/
+  | @(cpr_conf_lpr_flat_eps … IH) -IH /width=1 by/
+  | @(cpr_conf_lpr_eps_eps … IH) -IH /width=1 by/
+  | @(cpr_conf_lpr_flat_beta … IH) -IH /width=1 by/
+  | @(cpr_conf_lpr_beta_beta … IH) -IH /width=1 by/
+  | @(cpr_conf_lpr_flat_theta … IH) -IH /width=3 by/
+  | @(cpr_conf_lpr_theta_theta … IH) -IH /width=3 by/
   ]
 ]
 qed-.
 
-(* Basic_1: includes: pr0_confluence pr2_confluence *)
-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 *************)
+(* Properties with context-sensitive parallel reduction for terms ***********)
 
-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/
+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: ∀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/
+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/
 qed-.
 
 (* Main properties **********************************************************)
 
-theorem lpr_conf: ∀G. confluent … (lpr G).
-/3 width=6 by lpx_sn_conf, cpr_conf_lpr/
+theorem lpr_conf (h) (G): confluent … (lpr h G).
+/3 width=6 by lex_conf, cpr_conf_lpr/
 qed-.