Section Prop_induction.
Variable A : Set.
Variable R : A -> A -> Prop.
Hypothesis W : well_founded R.
Hint Resolve W: arith.
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End Prop_induction.
Definition coarser (A : Set) (R S : A -> A -> Prop) :=
forall x y : A, R x y -> S x y.

Lemma wf_coarser :
forall (A : Set) (R S : A -> A -> Prop),
coarser A R S -> well_founded S -> well_founded R.