lpx_sn R (K1. ⓑ{I} V1) (K2. ⓑ{I} V2)
.
-definition lpx_sn_confluent: predicate (lenv→relation term) ≝ λR.
- ∀L0,T0,T1. R L0 T0 T1 → ∀T2. R L0 T0 T2 →
- ∀L1. lpx_sn R L0 L1 → ∀L2. lpx_sn R L0 L2 →
- ∃∃T. R L1 T1 T & R L2 T2 T.
-
-definition lpx_sn_transitive: predicate (lenv→relation term) ≝ λR.
- ∀L1,T1,T. R L1 T1 T → ∀L2. lpx_sn R L1 L2 →
- ∀T2. R L2 T T2 → R L1 T1 T2.
+definition lpx_sn_confluent: relation (lenv→relation term) ≝ λR1,R2.
+ ∀L0,T0,T1. R1 L0 T0 T1 → ∀T2. R2 L0 T0 T2 →
+ ∀L1. lpx_sn R1 L0 L1 → ∀L2. lpx_sn R2 L0 L2 →
+ ∃∃T. R2 L1 T1 T & R1 L2 T2 T.
+definition lpx_sn_transitive: relation (lenv→relation term) ≝ λR1,R2.
+ ∀L1,T1,T. R1 L1 T1 T → ∀L2. lpx_sn R1 L1 L2 →
+ ∀T2. R2 L2 T T2 → R1 L1 T1 T2.
(* Basic inversion lemmas ***************************************************)
#R #L1 #L2 #H elim H -L1 -L2 normalize //
qed-.
+(* Advanced forward lemmas **************************************************)
+
+lemma lpx_sn_fwd_append1: ∀R,L1,K1,L. lpx_sn R (K1 @@ L1) L →
+ ∃∃K2,L2. lpx_sn R K1 K2 & L = K2 @@ L2.
+#R #L1 elim L1 -L1
+[ #K1 #K2 #HK12
+ @(ex2_2_intro … K2 (⋆)) // (* explicit constructor, /2 width=4/ does not work *)
+| #L1 #I #V1 #IH #K1 #X #H
+ elim (lpx_sn_inv_pair1 … H) -H #L #V2 #H1 #HV12 #H destruct
+ elim (IH … H1) -IH -H1 #K2 #L2 #HK12 #H destruct
+ @(ex2_2_intro … (L2.ⓑ{I}V2) HK12) // (* explicit constructor, /2 width=4/ does not work *)
+]
+qed-.
+
+lemma lpx_sn_fwd_append2: ∀R,L2,K2,L. lpx_sn R L (K2 @@ L2) →
+ ∃∃K1,L1. lpx_sn R K1 K2 & L = K1 @@ L1.
+#R #L2 elim L2 -L2
+[ #K2 #K1 #HK12
+ @(ex2_2_intro … K1 (⋆)) // (**) (* explicit constructor, /2 width=4/ does not work *)
+| #L2 #I #V2 #IH #K2 #X #H
+ elim (lpx_sn_inv_pair2 … H) -H #L #V1 #H1 #HV12 #H destruct
+ elim (IH … H1) -IH -H1 #K1 #L1 #HK12 #H destruct
+ @(ex2_2_intro … (L1.ⓑ{I}V1) HK12) // (* explicit constructor, /2 width=4/ does not work *)
+]
+qed-.
+
(* Basic properties *********************************************************)
lemma lpx_sn_refl: ∀R. (∀L. reflexive ? (R L)) → reflexive … (lpx_sn R).
#R #HR #K1 #K2 #H elim H -K1 -K2 // /3 width=1/
qed-.
-lemma lpx_sn_trans: ∀R. lpx_sn_transitive R → Transitive … (lpx_sn R).
+(* Advanced properties ******************************************************)
+
+lemma lpx_sn_trans: ∀R. lpx_sn_transitive R R → Transitive … (lpx_sn R).
#R #HR #L1 #L #H elim H -L1 -L //
#I #L1 #L #V1 #V #HL1 #HV1 #IHL1 #X #H
elim (lpx_sn_inv_pair1 … H) -H #L2 #V2 #HL2 #HV2 #H destruct /3 width=5/
qed-.
-lemma lpx_sn_conf: ∀R. lpx_sn_confluent R → confluent … (lpx_sn R).
-#R #HR #L0 @(f_ind … length … L0) -L0 #n #IH *
+lemma lpx_sn_conf: ∀R1,R2. lpx_sn_confluent R1 R2 →
+ confluent2 … (lpx_sn R1) (lpx_sn R2).
+#R1 #R2 #HR12 #L0 @(f_ind … length … L0) -L0 #n #IH *
[ #_ #X1 #H1 #X2 #H2 -n
>(lpx_sn_inv_atom1 … H1) -X1
>(lpx_sn_inv_atom1 … H2) -X2 /2 width=3/
elim (lpx_sn_inv_pair1 … H1) -H1 #L1 #V1 #HL01 #HV01 #H destruct
elim (lpx_sn_inv_pair1 … H2) -H2 #L2 #V2 #HL02 #HV02 #H destruct
elim (IH … HL01 … HL02) -IH normalize // #L #HL1 #HL2
- elim (HR … HV01 … HV02 … HL01 … HL02) -L0 -V0 /3 width=5/
+ elim (HR12 … HV01 … HV02 … HL01 … HL02) -L0 -V0 /3 width=5/
]
qed-.