Section EvaluatorTakahashi.
Variable (X: Type) (R S: X -> X -> Prop).
Hypothesis (D: Dec1 (Normal R)).
Variable (rho: X -> X).
Hypotheses (tf: tak_fun S rho).
Hypotheses (refl: Reflexive S) (H1: subrelation R S)
(H2: subrelation S (star R)).

Lemma rho_evaluates:
forall x y : X, evaluates R x y -> exists n : nat, it n rho x = y.