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include "basics/star.ma".
-include "Ground_2/xoa_props.ma".
-include "Ground_2/notation.ma".
+include "ground_2/xoa_props.ma".
+include "ground_2/notation.ma".
(* PROPERTIES OF RELATIONS **************************************************)
-definition Decidable: Prop → Prop ≝ λR. R ∨ (R → False).
+definition Decidable: Prop → Prop ≝ λR. R ∨ (R → ⊥).
definition confluent2: ∀A. ∀R1,R2: relation A. Prop ≝ λA,R1,R2.
∀a0,a1. R1 a0 a1 → ∀a2. R2 a0 a2 →
qed.
definition NF: ∀A. relation A → relation A → predicate A ≝
- λA,R,S,a1. ∀a2. R a1 a2 → S a1 a2.
+ λA,R,S,a1. ∀a2. R a1 a2 → S a2 a1.
inductive SN (A) (R,S:relation A): predicate A ≝
-| SN_intro: ∀a1. (∀a2. R a1 a2 → (S a1 a2 → False) → SN A R S a2) → SN A R S a1
+| SN_intro: ∀a1. (∀a2. R a1 a2 → (S a2 a1 → ⊥) → SN A R S a2) → SN A R S a1
.
lemma NF_to_SN: ∀A,R,S,a. NF A R S a → SN A R S a.
@SN_intro #a2 #HRa12 #HSa12
elim (HSa12 ?) -HSa12 /2 width=1/
qed.
+
+definition NF_sn: ∀A. relation A → relation A → predicate A ≝
+ λA,R,S,a2. ∀a1. R a1 a2 → S a2 a1.
+
+inductive SN_sn (A) (R,S:relation A): predicate A ≝
+| SN_sn_intro: ∀a2. (∀a1. R a1 a2 → (S a2 a1 → ⊥) → SN_sn A R S a1) → SN_sn A R S a2
+.
+
+lemma NF_to_SN_sn: ∀A,R,S,a. NF_sn A R S a → SN_sn A R S a.
+#A #R #S #a2 #Ha2
+@SN_sn_intro #a1 #HRa12 #HSa12
+elim (HSa12 ?) -HSa12 /2 width=1/
+qed.