interpretation "generic extension on referred entries (local environment)"
'RelationStar R T L1 L2 = (lfxs R T L1 L2).
-definition R_frees_confluent: predicate (relation3 …) ≝
- λRN.
- ∀f1,L,T1. L ⊢ 𝐅*⦃T1⦄ ≡ f1 → ∀T2. RN L T1 T2 →
- ∃∃f2. L ⊢ 𝐅*⦃T2⦄ ≡ f2 & f2 ⊆ f1.
-
-definition lexs_frees_confluent: relation (relation3 …) ≝
- λRN,RP.
- ∀f1,L1,T. L1 ⊢ 𝐅*⦃T⦄ ≡ f1 →
- ∀L2. L1 ⪤*[RN, RP, f1] L2 →
- ∃∃f2. L2 ⊢ 𝐅*⦃T⦄ ≡ f2 & f2 ⊆ f1.
-
definition R_confluent2_lfxs: relation4 (relation3 lenv term term)
(relation3 lenv term term) … ≝
λR1,R2,RP1,RP2.
∀K1,K,V1. K1 ⪤*[R1, V1] K → ∀V. R1 K1 V1 V →
∀K2. K ⪤*[R2, V] K2 → K ⪤*[R2, V1] K2.
-definition lfxs_transitive: relation3 ? (relation3 ?? term) ? ≝
+definition lfxs_transitive: relation3 ? (relation3 ?? term) … ≝
λR1,R2,R3.
∀K1,K,V1. K1 ⪤*[R1, V1] K →
∀V. R1 K1 V1 V → ∀V2. R2 K V V2 → R3 K1 V1 V2.
/3 width=5 by lexs_pair_repl, ex2_intro/
qed-.
-lemma lfxs_sym: ∀R. lexs_frees_confluent (cext2 R) cfull →
- (∀L1,L2,T1,T2. R L1 T1 T2 → R L2 T2 T1) →
- ∀T. symmetric … (lfxs R T).
-#R #H1R #H2R #T #L1 #L2 * #f1 #Hf1 #HL12 elim (H1R … Hf1 … HL12) -Hf1
-/5 width=5 by sle_lexs_trans, lexs_sym, cext2_sym, ex2_intro/
-qed-.
-
(* Basic_2A1: uses: llpx_sn_co *)
lemma lfxs_co: ∀R1,R2. (∀L,T1,T2. R1 L T1 T2 → R2 L T1 T2) →
∀L1,L2,T. L1 ⪤*[R1, T] L2 → L1 ⪤*[R2, T] L2.