--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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 "basic_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".
+
+(* PARALLEL R-TRANSITION FOR FULL LOCAL ENVIRONMENTS ************************)
+
+(* 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.
+/2 width=3 by cpr_refl, ex2_intro/ qed-.
+
+fact cpr_conf_lpr_atom_delta (h):
+ ∀G,L0,i. (
+ ∀L,T. ⦃G, L0, #i⦄ ⊐+ ⦃G, 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
+ ) →
+ ∀K0,V0. ⬇*[i] L0 ≘ K0.ⓓV0 →
+ ∀V2. ⦃G, K0⦄ ⊢ V0 ➡[h] V2 → ∀T2. ⬆*[↑i] V2 ≘ T2 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h] L2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ #i ➡[h] T & ⦃G, 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 (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 (h):
+ ∀G,L0,i. (
+ ∀L,T. ⦃G, L0, #i⦄ ⊐+ ⦃G, 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
+ ) →
+ ∀K0,V0. ⬇*[i] L0 ≘ K0.ⓓV0 →
+ ∀V1. ⦃G, K0⦄ ⊢ V0 ➡[h] V1 → ∀T1. ⬆*[↑i] V1 ≘ T1 →
+ ∀KX,VX. ⬇*[i] L0 ≘ KX.ⓓVX →
+ ∀V2. ⦃G, KX⦄ ⊢ VX ➡[h] V2 → ∀T2. ⬆*[↑i] V2 ≘ T2 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h] L2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡[h] T & ⦃G, 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
+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 (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 (h):
+ ∀p,I,G,L0,V0,T0. (
+ ∀L,T. ⦃G, L0, ⓑ{p,I}V0.T0⦄ ⊐+ ⦃G, 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
+ ) →
+ ∀V1. ⦃G, L0⦄ ⊢ V0 ➡[h] V1 → ∀T1. ⦃G, L0.ⓑ{I}V0⦄ ⊢ T0 ➡[h] T1 →
+ ∀V2. ⦃G, L0⦄ ⊢ V0 ➡[h] V2 → ∀T2. ⦃G, L0.ⓑ{I}V0⦄ ⊢ T0 ➡[h] T2 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h] L2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ ⓑ{p,I}V1.T1 ➡[h] T & ⦃G, 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, cpm_bind, ex2_intro/
+qed-.
+
+fact cpr_conf_lpr_bind_zeta (h):
+ ∀G,L0,V0,T0. (
+ ∀L,T. ⦃G, L0, +ⓓV0.T0⦄ ⊐+ ⦃G, 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
+ ) →
+ ∀V1. ⦃G, L0⦄ ⊢ V0 ➡[h] V1 → ∀T1. ⦃G, L0.ⓓV0⦄ ⊢ T0 ➡[h] T1 →
+ ∀T2. ⦃G, L0.ⓓV0⦄ ⊢ T0 ➡[h] T2 → ∀X2. ⬆*[1] X2 ≘ T2 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h] L2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ +ⓓV1.T1 ➡[h] T & ⦃G, L2⦄ ⊢ X2 ➡[h] T.
+#h #G0 #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 (cpm_inv_lifts_sn … HT2 (Ⓣ) … L2 … HXT2) -T2
+/3 width=3 by cpm_zeta, drops_refl, drops_drop, ex2_intro/
+qed-.
+
+fact cpr_conf_lpr_zeta_zeta (h):
+ ∀G,L0,V0,T0. (
+ ∀L,T. ⦃G, L0, +ⓓV0.T0⦄ ⊐+ ⦃G, 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, L0.ⓓV0⦄ ⊢ T0 ➡[h] T1 → ∀X1. ⬆*[1] X1 ≘ T1 →
+ ∀T2. ⦃G, L0.ⓓV0⦄ ⊢ T0 ➡[h] T2 → ∀X2. ⬆*[1] X2 ≘ T2 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h] L2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ X1 ➡[h] T & ⦃G, L2⦄ ⊢ X2 ➡[h] T.
+#h #G0 #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 (cpm_inv_lifts_sn … HT1 (Ⓣ) … L1 … HXT1) -T1 /3 width=2 by drops_refl, drops_drop/ #T1 #HT1 #HXT1
+elim (cpm_inv_lifts_sn … HT2 (Ⓣ) … L2 … HXT2) -T2 /3 width=2 by drops_refl, drops_drop/ #T2 #HT2 #HXT2
+lapply (lifts_inj … HT2 … HT1) -T #H destruct
+/2 width=3 by ex2_intro/
+qed-.
+
+fact cpr_conf_lpr_flat_flat (h):
+ ∀I,G,L0,V0,T0. (
+ ∀L,T. ⦃G, L0, ⓕ{I}V0.T0⦄ ⊐+ ⦃G, 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
+ ) →
+ ∀V1. ⦃G, L0⦄ ⊢ V0 ➡[h] V1 → ∀T1. ⦃G, L0⦄ ⊢ T0 ➡[h] T1 →
+ ∀V2. ⦃G, L0⦄ ⊢ V0 ➡[h] V2 → ∀T2. ⦃G, L0⦄ ⊢ T0 ➡[h] T2 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h] L2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ ⓕ{I}V1.T1 ➡[h] T & ⦃G, 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/
+qed-.
+
+fact cpr_conf_lpr_flat_eps (h):
+ ∀G,L0,V0,T0. (
+ ∀L,T. ⦃G, L0, ⓝV0.T0⦄ ⊐+ ⦃G, 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
+ ) →
+ ∀V1,T1. ⦃G, L0⦄ ⊢ T0 ➡[h] T1 → ∀T2. ⦃G, L0⦄ ⊢ T0 ➡[h] T2 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h] L2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ ⓝV1.T1 ➡[h] T & ⦃G, 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 cpm_eps, ex2_intro/
+qed-.
+
+fact cpr_conf_lpr_eps_eps (h):
+ ∀G,L0,V0,T0. (
+ ∀L,T. ⦃G, L0, ⓝV0.T0⦄ ⊐+ ⦃G, 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, L0⦄ ⊢ T0 ➡[h] T1 → ∀T2. ⦃G, L0⦄ ⊢ T0 ➡[h] T2 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h] L2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡[h] T & ⦃G, 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/
+qed-.
+
+fact cpr_conf_lpr_flat_beta (h):
+ ∀p,G,L0,V0,W0,T0. (
+ ∀L,T. ⦃G, L0, ⓐV0.ⓛ{p}W0.T0⦄ ⊐+ ⦃G, 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
+ ) →
+ ∀V1. ⦃G, L0⦄ ⊢ V0 ➡[h] V1 → ∀T1. ⦃G, L0⦄ ⊢ ⓛ{p}W0.T0 ➡[h] T1 →
+ ∀V2. ⦃G, L0⦄ ⊢ V0 ➡[h] V2 → ∀W2. ⦃G, L0⦄ ⊢ W0 ➡[h] W2 → ∀T2. ⦃G, L0.ⓛW0⦄ ⊢ T0 ➡[h] T2 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h] L2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ ⓐV1.T1 ➡[h] T & ⦃G, 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
+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_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 (h):
+ ∀p,G,L0,V0,W0,T0. (
+ ∀L,T. ⦃G, L0, ⓐV0.ⓓ{p}W0.T0⦄ ⊐+ ⦃G, 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
+ ) →
+ ∀V1. ⦃G, L0⦄ ⊢ V0 ➡[h] V1 → ∀T1. ⦃G, L0⦄ ⊢ ⓓ{p}W0.T0 ➡[h] T1 →
+ ∀V2. ⦃G, L0⦄ ⊢ V0 ➡[h] V2 → ∀U2. ⬆*[1] V2 ≘ U2 →
+ ∀W2. ⦃G, L0⦄ ⊢ W0 ➡[h] W2 → ∀T2. ⦃G, L0.ⓓW0⦄ ⊢ T0 ➡[h] T2 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h] L2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ ⓐV1.T1 ➡[h] T & ⦃G, 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 (cpm_lifts_sn … HV2 (Ⓣ) … (L2.ⓓW2) … HVU2) -HVU2 /3 width=2 by drops_refl, drops_drop/ #U #HVU #HU2
+elim (cpm_inv_abbr1 … H) -H *
+[ #W1 #T1 #HW01 #HT01 #H destruct
+ 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 cpm_bind, cpm_appl, cpm_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 (cpm_inv_lifts_sn … HT1 (Ⓣ) … L1 … HXT1) -HXT1 /3 width=2 by drops_refl, drops_drop/
+ /4 width=9 by cpm_appl, cpm_zeta, lifts_flat, ex2_intro/
+]
+qed-.
+
+fact cpr_conf_lpr_beta_beta (h):
+ ∀p,G,L0,V0,W0,T0. (
+ ∀L,T. ⦃G, L0, ⓐV0.ⓛ{p}W0.T0⦄ ⊐+ ⦃G, 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
+ ) →
+ ∀V1. ⦃G, L0⦄ ⊢ V0 ➡[h] V1 → ∀W1. ⦃G, L0⦄ ⊢ W0 ➡[h] W1 → ∀T1. ⦃G, L0.ⓛW0⦄ ⊢ T0 ➡[h] T1 →
+ ∀V2. ⦃G, L0⦄ ⊢ V0 ➡[h] V2 → ∀W2. ⦃G, L0⦄ ⊢ W0 ➡[h] W2 → ∀T2. ⦃G, L0.ⓛW0⦄ ⊢ T0 ➡[h] T2 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h] L2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ ⓓ{p}ⓝW1.V1.T1 ➡[h] T & ⦃G, 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_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 (h):
+ ∀p,G,L0,V0,W0,T0. (
+ ∀L,T. ⦃G, L0, ⓐV0.ⓓ{p}W0.T0⦄ ⊐+ ⦃G, 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
+ ) →
+ ∀V1. ⦃G, L0⦄ ⊢ V0 ➡[h] V1 → ∀U1. ⬆*[1] V1 ≘ U1 →
+ ∀W1. ⦃G, L0⦄ ⊢ W0 ➡[h] W1 → ∀T1. ⦃G, L0.ⓓW0⦄ ⊢ T0 ➡[h] T1 →
+ ∀V2. ⦃G, L0⦄ ⊢ V0 ➡[h] V2 → ∀U2. ⬆*[1] V2 ≘ U2 →
+ ∀W2. ⦃G, L0⦄ ⊢ W0 ➡[h] W2 → ∀T2. ⦃G, L0.ⓓW0⦄ ⊢ T0 ➡[h] T2 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h] L2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ ⓓ{p}W1.ⓐU1.T1 ➡[h] T & ⦃G, 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/ #W #HW1 #HW2
+elim (IH … HT01 … HT02 (L1.ⓓW1) … (L2.ⓓW2)) /2 width=1 by lpr_pair/ -L0 -V0 -W0 -T0
+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 (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_drops … H1) -H1
+ elim (cpr_inv_atom1_drops … H2) -H2
+ [ #H2 #H1 destruct
+ /2 width=1 by cpr_conf_lpr_atom_atom/
+ | * #K0 #V0 #V2 #i2 #HLK0 #HV02 #HVT2 #H2 #H1 destruct
+ /3 width=10 by cpr_conf_lpr_atom_delta/
+ | #H2 * #K0 #V0 #V1 #i1 #HLK0 #HV01 #HVT1 #H1 destruct
+ /4 width=10 by ex2_commute, cpr_conf_lpr_atom_delta/
+ | * #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/
+ ]
+| #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
+ ]
+ elim (cpm_inv_bind1 … H2) -H2 *
+ [1,3: #V2 #T2 #HV02 #HT02 #H2
+ |2,4: #T2 #HT02 #HXT2 #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/
+ ]
+| #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
+ | #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: #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/
+ ]
+]
+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.
+#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.
+#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 (h) (G): confluent … (lpr h G).
+/3 width=6 by lex_conf, cpr_conf_lpr/
+qed-.
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