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)).
Instance decidable_E s n:
Dec (exists t, E rho D n s = Some t).

Lemma E_evaluates (s: X):
{ n: nat | exists t, E rho D n s = Some t } -> { t | evaluates R s t }.