(* 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,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):
- â\88\80I,G,L1,L2. â\88\83â\88\83T. â¦\83G, L1â¦\84 â\8a¢ â\93ª{I} â\9e¡[h] T & â¦\83G, L2â¦\84 â\8a¢ â\93ª{I} â\9e¡[h] T.
+ â\88\80I,G,L1,L2. â\88\83â\88\83T. â\9dªG,L1â\9d« â\8a¢ â\93ª[I] â\9e¡[h,0] T & â\9dªG,L2â\9d« â\8a¢ â\93ª[I] â\9e¡[h,0] 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
+ ∀G0,L0,i. (
+ ∀G,L,T. ❪G0,L0,#i❫ ⬂+ ❪G,L,T❫ → IH_cpr_conf_lpr h G L T
) →
- â\88\80K0,V0. â¬\87*[i] L0 ≘ K0.ⓓV0 →
- â\88\80V2. â¦\83G, K0â¦\84 â\8a¢ V0 â\9e¡[h] V2 â\86\92 â\88\80T2. â¬\86*[↑i] V2 ≘ T2 →
- â\88\80L1. â¦\83G, L0â¦\84 â\8a¢ â\9e¡[h] L1 â\86\92 â\88\80L2. â¦\83G, L0â¦\84 â\8a¢ â\9e¡[h] L2 →
- â\88\83â\88\83T. â¦\83G, L1â¦\84 â\8a¢ #i â\9e¡[h] T & â¦\83G, L2â¦\84 â\8a¢ T2 â\9e¡[h] T.
+ â\88\80K0,V0. â\87©[i] L0 ≘ K0.ⓓV0 →
+ â\88\80V2. â\9dªG0,K0â\9d« â\8a¢ V0 â\9e¡[h,0] V2 â\86\92 â\88\80T2. â\87§[↑i] V2 ≘ T2 →
+ â\88\80L1. â\9dªG0,L0â\9d« â\8a¢ â\9e¡[h,0] L1 â\86\92 â\88\80L2. â\9dªG0,L0â\9d« â\8a¢ â\9e¡[h,0] L2 →
+ â\88\83â\88\83T. â\9dªG0,L1â\9d« â\8a¢ #i â\9e¡[h,0] T & â\9dªG0,L2â\9d« â\8a¢ T2 â\9e¡[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
(* 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
+ ∀G0,L0,i. (
+ ∀G,L,T. ❪G0,L0,#i❫ ⬂+ ❪G,L,T❫ → IH_cpr_conf_lpr h G L T
) →
- â\88\80K0,V0. â¬\87*[i] L0 ≘ K0.ⓓV0 →
- â\88\80V1. â¦\83G, K0â¦\84 â\8a¢ V0 â\9e¡[h] V1 â\86\92 â\88\80T1. â¬\86*[↑i] V1 ≘ T1 →
- â\88\80KX,VX. â¬\87*[i] L0 ≘ KX.ⓓVX →
- â\88\80V2. â¦\83G, KXâ¦\84 â\8a¢ VX â\9e¡[h] V2 â\86\92 â\88\80T2. â¬\86*[↑i] V2 ≘ T2 →
- â\88\80L1. â¦\83G, L0â¦\84 â\8a¢ â\9e¡[h] L1 â\86\92 â\88\80L2. â¦\83G, L0â¦\84 â\8a¢ â\9e¡[h] L2 →
- â\88\83â\88\83T. â¦\83G, L1â¦\84 â\8a¢ T1 â\9e¡[h] T & â¦\83G, L2â¦\84 â\8a¢ T2 â\9e¡[h] T.
+ â\88\80K0,V0. â\87©[i] L0 ≘ K0.ⓓV0 →
+ â\88\80V1. â\9dªG0,K0â\9d« â\8a¢ V0 â\9e¡[h,0] V1 â\86\92 â\88\80T1. â\87§[↑i] V1 ≘ T1 →
+ â\88\80KX,VX. â\87©[i] L0 ≘ KX.ⓓVX →
+ â\88\80V2. â\9dªG0,KXâ\9d« â\8a¢ VX â\9e¡[h,0] V2 â\86\92 â\88\80T2. â\87§[↑i] V2 ≘ T2 →
+ â\88\80L1. â\9dªG0,L0â\9d« â\8a¢ â\9e¡[h,0] L1 â\86\92 â\88\80L2. â\9dªG0,L0â\9d« â\8a¢ â\9e¡[h,0] L2 →
+ â\88\83â\88\83T. â\9dªG0,L1â\9d« â\8a¢ T1 â\9e¡[h,0] T & â\9dªG0,L2â\9d« â\8a¢ T2 â\9e¡[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
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
+ ∀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
) →
- â\88\80V1. â¦\83G, L0â¦\84 â\8a¢ V0 â\9e¡[h] V1 â\86\92 â\88\80T1. â¦\83G, L0.â\93\91{I}V0â¦\84 â\8a¢ T0 â\9e¡[h] T1 →
- â\88\80V2. â¦\83G, L0â¦\84 â\8a¢ V0 â\9e¡[h] V2 â\86\92 â\88\80T2. â¦\83G, L0.â\93\91{I}V0â¦\84 â\8a¢ T0 â\9e¡[h] T2 →
- â\88\80L1. â¦\83G, L0â¦\84 â\8a¢ â\9e¡[h] L1 â\86\92 â\88\80L2. â¦\83G, L0â¦\84 â\8a¢ â\9e¡[h] L2 →
- â\88\83â\88\83T. â¦\83G, L1â¦\84 â\8a¢ â\93\91{p,I}V1.T1 â\9e¡[h] T & â¦\83G, L2â¦\84 â\8a¢ â\93\91{p,I}V2.T2 â\9e¡[h] T.
+ â\88\80V1. â\9dªG0,L0â\9d« â\8a¢ V0 â\9e¡[h,0] V1 â\86\92 â\88\80T1. â\9dªG0,L0.â\93\91[I]V0â\9d« â\8a¢ T0 â\9e¡[h,0] T1 →
+ â\88\80V2. â\9dªG0,L0â\9d« â\8a¢ V0 â\9e¡[h,0] V2 â\86\92 â\88\80T2. â\9dªG0,L0.â\93\91[I]V0â\9d« â\8a¢ T0 â\9e¡[h,0] T2 →
+ â\88\80L1. â\9dªG0,L0â\9d« â\8a¢ â\9e¡[h,0] L1 â\86\92 â\88\80L2. â\9dªG0,L0â\9d« â\8a¢ â\9e¡[h,0] L2 →
+ â\88\83â\88\83T. â\9dªG0,L1â\9d« â\8a¢ â\93\91[p,I]V1.T1 â\9e¡[h,0] T & â\9dªG0,L2â\9d« â\8a¢ â\93\91[p,I]V2.T2 â\9e¡[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) //
-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):
- ∀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
+ ∀G0,L0,V0,T0. (
+ ∀G,L,T. ❪G0,L0,+ⓓV0.T0❫ ⬂+ ❪G,L,T❫ → IH_cpr_conf_lpr h G L T
) →
- â\88\80V1. â¦\83G, L0â¦\84 â\8a¢ V0 â\9e¡[h] V1 â\86\92 â\88\80T1. â¦\83G, L0.â\93\93V0â¦\84 â\8a¢ T0 â\9e¡[h] T1 →
- â\88\80T2. â¦\83G, L0.â\93\93V0â¦\84 â\8a¢ T0 â\9e¡[h] T2 â\86\92 â\88\80X2. â¬\86*[1] X2 â\89\98 T2 →
- â\88\80L1. â¦\83G, L0â¦\84 â\8a¢ â\9e¡[h] L1 â\86\92 â\88\80L2. â¦\83G, L0â¦\84 â\8a¢ â\9e¡[h] L2 →
- â\88\83â\88\83T. â¦\83G, L1â¦\84 â\8a¢ +â\93\93V1.T1 â\9e¡[h] T & â¦\83G, L2â¦\84 â\8a¢ X2 â\9e¡[h] T.
+ â\88\80V1. â\9dªG0,L0â\9d« â\8a¢ V0 â\9e¡[h,0] V1 â\86\92 â\88\80T1. â\9dªG0,L0.â\93\93V0â\9d« â\8a¢ T0 â\9e¡[h,0] T1 →
+ â\88\80T2. â\87§[1]T2 â\89\98 T0 â\86\92 â\88\80X2. â\9dªG0,L0â\9d« â\8a¢ T2 â\9e¡[h,0] X2 →
+ â\88\80L1. â\9dªG0,L0â\9d« â\8a¢ â\9e¡[h,0] L1 â\86\92 â\88\80L2. â\9dªG0,L0â\9d« â\8a¢ â\9e¡[h,0] L2 →
+ â\88\83â\88\83T. â\9dªG0,L1â\9d« â\8a¢ +â\93\93V1.T1 â\9e¡[h,0] T & â\9dªG0,L2â\9d« â\8a¢ X2 â\9e¡[h,0] 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/
+#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 (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
+ ∀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 ➡[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
+ ∀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
+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 (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
+ ∀I,G0,L0,V0,T0. (
+ ∀G,L,T. ❪G0,L0,ⓕ[I]V0.T0❫ ⬂+ ❪G,L,T❫ → IH_cpr_conf_lpr h G L T
) →
- â\88\80V1. â¦\83G, L0â¦\84 â\8a¢ V0 â\9e¡[h] V1 â\86\92 â\88\80T1. â¦\83G, L0â¦\84 â\8a¢ T0 â\9e¡[h] T1 →
- â\88\80V2. â¦\83G, L0â¦\84 â\8a¢ V0 â\9e¡[h] V2 â\86\92 â\88\80T2. â¦\83G, L0â¦\84 â\8a¢ T0 â\9e¡[h] T2 →
- â\88\80L1. â¦\83G, L0â¦\84 â\8a¢ â\9e¡[h] L1 â\86\92 â\88\80L2. â¦\83G, L0â¦\84 â\8a¢ â\9e¡[h] L2 →
- â\88\83â\88\83T. â¦\83G, L1â¦\84 â\8a¢ â\93\95{I}V1.T1 â\9e¡[h] T & â¦\83G, L2â¦\84 â\8a¢ â\93\95{I}V2.T2 â\9e¡[h] T.
+ â\88\80V1. â\9dªG0,L0â\9d« â\8a¢ V0 â\9e¡[h,0] V1 â\86\92 â\88\80T1. â\9dªG0,L0â\9d« â\8a¢ T0 â\9e¡[h,0] T1 →
+ â\88\80V2. â\9dªG0,L0â\9d« â\8a¢ V0 â\9e¡[h,0] V2 â\86\92 â\88\80T2. â\9dªG0,L0â\9d« â\8a¢ T0 â\9e¡[h,0] T2 →
+ â\88\80L1. â\9dªG0,L0â\9d« â\8a¢ â\9e¡[h,0] L1 â\86\92 â\88\80L2. â\9dªG0,L0â\9d« â\8a¢ â\9e¡[h,0] L2 →
+ â\88\83â\88\83T. â\9dªG0,L1â\9d« â\8a¢ â\93\95[I]V1.T1 â\9e¡[h,0] T & â\9dªG0,L2â\9d« â\8a¢ â\93\95[I]V2.T2 â\9e¡[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) //
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
+ ∀G0,L0,V0,T0. (
+ ∀G,L,T. ❪G0,L0,ⓝV0.T0❫ ⬂+ ❪G,L,T❫ → IH_cpr_conf_lpr h G L T
) →
- â\88\80V1,T1. â¦\83G, L0â¦\84 â\8a¢ T0 â\9e¡[h] T1 â\86\92 â\88\80T2. â¦\83G, L0â¦\84 â\8a¢ T0 â\9e¡[h] T2 →
- â\88\80L1. â¦\83G, L0â¦\84 â\8a¢ â\9e¡[h] L1 â\86\92 â\88\80L2. â¦\83G, L0â¦\84 â\8a¢ â\9e¡[h] L2 →
- â\88\83â\88\83T. â¦\83G, L1â¦\84 â\8a¢ â\93\9dV1.T1 â\9e¡[h] T & â¦\83G, L2â¦\84 â\8a¢ T2 â\9e¡[h] T.
+ â\88\80V1,T1. â\9dªG0,L0â\9d« â\8a¢ T0 â\9e¡[h,0] T1 â\86\92 â\88\80T2. â\9dªG0,L0â\9d« â\8a¢ T0 â\9e¡[h,0] T2 →
+ â\88\80L1. â\9dªG0,L0â\9d« â\8a¢ â\9e¡[h,0] L1 â\86\92 â\88\80L2. â\9dªG0,L0â\9d« â\8a¢ â\9e¡[h,0] L2 →
+ â\88\83â\88\83T. â\9dªG0,L1â\9d« â\8a¢ â\93\9dV1.T1 â\9e¡[h,0] T & â\9dªG0,L2â\9d« â\8a¢ T2 â\9e¡[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
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
+ ∀G0,L0,V0,T0. (
+ ∀G,L,T. ❪G0,L0,ⓝV0.T0❫ ⬂+ ❪G,L,T❫ → IH_cpr_conf_lpr h G L T
) →
- â\88\80T1. â¦\83G, L0â¦\84 â\8a¢ T0 â\9e¡[h] T1 â\86\92 â\88\80T2. â¦\83G, L0â¦\84 â\8a¢ T0 â\9e¡[h] T2 →
- â\88\80L1. â¦\83G, L0â¦\84 â\8a¢ â\9e¡[h] L1 â\86\92 â\88\80L2. â¦\83G, L0â¦\84 â\8a¢ â\9e¡[h] L2 →
- â\88\83â\88\83T. â¦\83G, L1â¦\84 â\8a¢ T1 â\9e¡[h] T & â¦\83G, L2â¦\84 â\8a¢ T2 â\9e¡[h] T.
+ â\88\80T1. â\9dªG0,L0â\9d« â\8a¢ T0 â\9e¡[h,0] T1 â\86\92 â\88\80T2. â\9dªG0,L0â\9d« â\8a¢ T0 â\9e¡[h,0] T2 →
+ â\88\80L1. â\9dªG0,L0â\9d« â\8a¢ â\9e¡[h,0] L1 â\86\92 â\88\80L2. â\9dªG0,L0â\9d« â\8a¢ â\9e¡[h,0] L2 →
+ â\88\83â\88\83T. â\9dªG0,L1â\9d« â\8a¢ T1 â\9e¡[h,0] T & â\9dªG0,L2â\9d« â\8a¢ T2 â\9e¡[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
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
+ ∀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
) →
- â\88\80V1. â¦\83G, L0â¦\84 â\8a¢ V0 â\9e¡[h] V1 â\86\92 â\88\80T1. â¦\83G, L0â¦\84 â\8a¢ â\93\9b{p}W0.T0 â\9e¡[h] T1 →
- â\88\80V2. â¦\83G, L0â¦\84 â\8a¢ V0 â\9e¡[h] V2 â\86\92 â\88\80W2. â¦\83G, L0â¦\84 â\8a¢ W0 â\9e¡[h] W2 â\86\92 â\88\80T2. â¦\83G, L0.â\93\9bW0â¦\84 â\8a¢ T0 â\9e¡[h] T2 →
- â\88\80L1. â¦\83G, L0â¦\84 â\8a¢ â\9e¡[h] L1 â\86\92 â\88\80L2. â¦\83G, L0â¦\84 â\8a¢ â\9e¡[h] L2 →
- â\88\83â\88\83T. â¦\83G, L1â¦\84 â\8a¢ â\93\90V1.T1 â\9e¡[h] T & â¦\83G, L2â¦\84 â\8a¢ â\93\93{p}â\93\9dW2.V2.T2 â\9e¡[h] T.
+ â\88\80V1. â\9dªG0,L0â\9d« â\8a¢ V0 â\9e¡[h,0] V1 â\86\92 â\88\80T1. â\9dªG0,L0â\9d« â\8a¢ â\93\9b[p]W0.T0 â\9e¡[h,0] T1 →
+ â\88\80V2. â\9dªG0,L0â\9d« â\8a¢ V0 â\9e¡[h,0] V2 â\86\92 â\88\80W2. â\9dªG0,L0â\9d« â\8a¢ W0 â\9e¡[h,0] W2 â\86\92 â\88\80T2. â\9dªG0,L0.â\93\9bW0â\9d« â\8a¢ T0 â\9e¡[h,0] T2 →
+ â\88\80L1. â\9dªG0,L0â\9d« â\8a¢ â\9e¡[h,0] L1 â\86\92 â\88\80L2. â\9dªG0,L0â\9d« â\8a¢ â\9e¡[h,0] L2 →
+ â\88\83â\88\83T. â\9dªG0,L1â\9d« â\8a¢ â\93\90V1.T1 â\9e¡[h,0] T & â\9dªG0,L2â\9d« â\8a¢ â\93\93[p]â\93\9dW2.V2.T2 â\9e¡[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
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
+ ∀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
) →
- â\88\80V1. â¦\83G, L0â¦\84 â\8a¢ V0 â\9e¡[h] V1 â\86\92 â\88\80T1. â¦\83G, L0â¦\84 â\8a¢ â\93\93{p}W0.T0 â\9e¡[h] T1 →
- â\88\80V2. â¦\83G, L0â¦\84 â\8a¢ V0 â\9e¡[h] V2 â\86\92 â\88\80U2. â¬\86*[1] V2 ≘ U2 →
- â\88\80W2. â¦\83G, L0â¦\84 â\8a¢ W0 â\9e¡[h] W2 â\86\92 â\88\80T2. â¦\83G, L0.â\93\93W0â¦\84 â\8a¢ T0 â\9e¡[h] T2 →
- â\88\80L1. â¦\83G, L0â¦\84 â\8a¢ â\9e¡[h] L1 â\86\92 â\88\80L2. â¦\83G, L0â¦\84 â\8a¢ â\9e¡[h] L2 →
- â\88\83â\88\83T. â¦\83G, L1â¦\84 â\8a¢ â\93\90V1.T1 â\9e¡[h] T & â¦\83G, L2â¦\84 â\8a¢ â\93\93{p}W2.â\93\90U2.T2 â\9e¡[h] T.
+ â\88\80V1. â\9dªG0,L0â\9d« â\8a¢ V0 â\9e¡[h,0] V1 â\86\92 â\88\80T1. â\9dªG0,L0â\9d« â\8a¢ â\93\93[p]W0.T0 â\9e¡[h,0] T1 →
+ â\88\80V2. â\9dªG0,L0â\9d« â\8a¢ V0 â\9e¡[h,0] V2 â\86\92 â\88\80U2. â\87§[1] V2 ≘ U2 →
+ â\88\80W2. â\9dªG0,L0â\9d« â\8a¢ W0 â\9e¡[h,0] W2 â\86\92 â\88\80T2. â\9dªG0,L0.â\93\93W0â\9d« â\8a¢ T0 â\9e¡[h,0] T2 →
+ â\88\80L1. â\9dªG0,L0â\9d« â\8a¢ â\9e¡[h,0] L1 â\86\92 â\88\80L2. â\9dªG0,L0â\9d« â\8a¢ â\9e¡[h,0] L2 →
+ â\88\83â\88\83T. â\9dªG0,L1â\9d« â\8a¢ â\93\90V1.T1 â\9e¡[h,0] T & â\9dªG0,L2â\9d« â\8a¢ â\93\93[p]W2.â\93\90U2.T2 â\9e¡[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
-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 (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 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/
+| #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 (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
+ ∀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
) →
- â\88\80V1. â¦\83G, L0â¦\84 â\8a¢ V0 â\9e¡[h] V1 â\86\92 â\88\80W1. â¦\83G, L0â¦\84 â\8a¢ W0 â\9e¡[h] W1 â\86\92 â\88\80T1. â¦\83G, L0.â\93\9bW0â¦\84 â\8a¢ T0 â\9e¡[h] T1 →
- â\88\80V2. â¦\83G, L0â¦\84 â\8a¢ V0 â\9e¡[h] V2 â\86\92 â\88\80W2. â¦\83G, L0â¦\84 â\8a¢ W0 â\9e¡[h] W2 â\86\92 â\88\80T2. â¦\83G, L0.â\93\9bW0â¦\84 â\8a¢ T0 â\9e¡[h] T2 →
- â\88\80L1. â¦\83G, L0â¦\84 â\8a¢ â\9e¡[h] L1 â\86\92 â\88\80L2. â¦\83G, L0â¦\84 â\8a¢ â\9e¡[h] L2 →
- â\88\83â\88\83T. â¦\83G, L1â¦\84 â\8a¢ â\93\93{p}â\93\9dW1.V1.T1 â\9e¡[h] T & â¦\83G, L2â¦\84 â\8a¢ â\93\93{p}â\93\9dW2.V2.T2 â\9e¡[h] T.
+ â\88\80V1. â\9dªG0,L0â\9d« â\8a¢ V0 â\9e¡[h,0] V1 â\86\92 â\88\80W1. â\9dªG0,L0â\9d« â\8a¢ W0 â\9e¡[h,0] W1 â\86\92 â\88\80T1. â\9dªG0,L0.â\93\9bW0â\9d« â\8a¢ T0 â\9e¡[h,0] T1 →
+ â\88\80V2. â\9dªG0,L0â\9d« â\8a¢ V0 â\9e¡[h,0] V2 â\86\92 â\88\80W2. â\9dªG0,L0â\9d« â\8a¢ W0 â\9e¡[h,0] W2 â\86\92 â\88\80T2. â\9dªG0,L0.â\93\9bW0â\9d« â\8a¢ T0 â\9e¡[h,0] T2 →
+ â\88\80L1. â\9dªG0,L0â\9d« â\8a¢ â\9e¡[h,0] L1 â\86\92 â\88\80L2. â\9dªG0,L0â\9d« â\8a¢ â\9e¡[h,0] L2 →
+ â\88\83â\88\83T. â\9dªG0,L1â\9d« â\8a¢ â\93\93[p]â\93\9dW1.V1.T1 â\9e¡[h,0] T & â\9dªG0,L2â\9d« â\8a¢ â\93\93[p]â\93\9dW2.V2.T2 â\9e¡[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
(* 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
+ ∀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
) →
- â\88\80V1. â¦\83G, L0â¦\84 â\8a¢ V0 â\9e¡[h] V1 â\86\92 â\88\80U1. â¬\86*[1] V1 ≘ U1 →
- â\88\80W1. â¦\83G, L0â¦\84 â\8a¢ W0 â\9e¡[h] W1 â\86\92 â\88\80T1. â¦\83G, L0.â\93\93W0â¦\84 â\8a¢ T0 â\9e¡[h] T1 →
- â\88\80V2. â¦\83G, L0â¦\84 â\8a¢ V0 â\9e¡[h] V2 â\86\92 â\88\80U2. â¬\86*[1] V2 ≘ U2 →
- â\88\80W2. â¦\83G, L0â¦\84 â\8a¢ W0 â\9e¡[h] W2 â\86\92 â\88\80T2. â¦\83G, L0.â\93\93W0â¦\84 â\8a¢ T0 â\9e¡[h] T2 →
- â\88\80L1. â¦\83G, L0â¦\84 â\8a¢ â\9e¡[h] L1 â\86\92 â\88\80L2. â¦\83G, L0â¦\84 â\8a¢ â\9e¡[h] L2 →
- â\88\83â\88\83T. â¦\83G, L1â¦\84 â\8a¢ â\93\93{p}W1.â\93\90U1.T1 â\9e¡[h] T & â¦\83G, L2â¦\84 â\8a¢ â\93\93{p}W2.â\93\90U2.T2 â\9e¡[h] T.
+ â\88\80V1. â\9dªG0,L0â\9d« â\8a¢ V0 â\9e¡[h,0] V1 â\86\92 â\88\80U1. â\87§[1] V1 ≘ U1 →
+ â\88\80W1. â\9dªG0,L0â\9d« â\8a¢ W0 â\9e¡[h,0] W1 â\86\92 â\88\80T1. â\9dªG0,L0.â\93\93W0â\9d« â\8a¢ T0 â\9e¡[h,0] T1 →
+ â\88\80V2. â\9dªG0,L0â\9d« â\8a¢ V0 â\9e¡[h,0] V2 â\86\92 â\88\80U2. â\87§[1] V2 ≘ U2 →
+ â\88\80W2. â\9dªG0,L0â\9d« â\8a¢ W0 â\9e¡[h,0] W2 â\86\92 â\88\80T2. â\9dªG0,L0.â\93\93W0â\9d« â\8a¢ T0 â\9e¡[h,0] T2 →
+ â\88\80L1. â\9dªG0,L0â\9d« â\8a¢ â\9e¡[h,0] L1 â\86\92 â\88\80L2. â\9dªG0,L0â\9d« â\8a¢ â\9e¡[h,0] L2 →
+ â\88\83â\88\83T. â\9dªG0,L1â\9d« â\8a¢ â\93\93[p]W1.â\93\90U1.T1 â\9e¡[h,0] T & â\9dªG0,L2â\9d« â\8a¢ â\93\93[p]W2.â\93\90U2.T2 â\9e¡[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
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/
]
| #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 (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 *
|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-.
(* Properties with context-sensitive parallel reduction for terms ***********)
-lemma lpr_cpr_conf_dx (h) (G): â\88\80L0. â\88\80T0,T1:term. â¦\83G, L0â¦\84 â\8a¢ T0 â\9e¡[h] T1 â\86\92 â\88\80L1. â¦\83G, L0â¦\84 â\8a¢ â\9e¡[h] L1 →
- â\88\83â\88\83T. â¦\83G, L1â¦\84 â\8a¢ T0 â\9e¡[h] T & â¦\83G, L1â¦\84 â\8a¢ T1 â\9e¡[h] T.
+lemma lpr_cpr_conf_dx (h) (G): â\88\80L0. â\88\80T0,T1:term. â\9dªG,L0â\9d« â\8a¢ T0 â\9e¡[h,0] T1 â\86\92 â\88\80L1. â\9dªG,L0â\9d« â\8a¢ â\9e¡[h,0] L1 →
+ â\88\83â\88\83T. â\9dªG,L1â\9d« â\8a¢ T0 â\9e¡[h,0] T & â\9dªG,L1â\9d« â\8a¢ T1 â\9e¡[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): â\88\80L0. â\88\80T0,T1:term. â¦\83G, L0â¦\84 â\8a¢ T0 â\9e¡[h] T1 â\86\92 â\88\80L1. â¦\83G, L0â¦\84 â\8a¢ â\9e¡[h] L1 →
- â\88\83â\88\83T. â¦\83G, L1â¦\84 â\8a¢ T0 â\9e¡[h] T & â¦\83G, L0â¦\84 â\8a¢ T1 â\9e¡[h] T.
+lemma lpr_cpr_conf_sn (h) (G): â\88\80L0. â\88\80T0,T1:term. â\9dªG,L0â\9d« â\8a¢ T0 â\9e¡[h,0] T1 â\86\92 â\88\80L1. â\9dªG,L0â\9d« â\8a¢ â\9e¡[h,0] L1 →
+ â\88\83â\88\83T. â\9dªG,L1â\9d« â\8a¢ T0 â\9e¡[h,0] T & â\9dªG,L0â\9d« â\8a¢ T1 â\9e¡[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/
(* 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-.