YES

proof:
The input problem is infeasible because

[1] the following set of Horn clauses is satisfiable:

	le(x, min(xs)) = false ==> min(cons(x, xs)) = min(xs)
	le(x, min(xs)) = true ==> min(cons(x, xs)) = x
	min(cons(x, nil)) = x
	le(s(x), s(y)) = le(x, y)
	le(0, s(y)) = true
	le(x, 0) = false
	le(x1, min(nil)) = true ==> true__ = false__
	true__ = false__ ==> \bottom

This holds because

[2] the following E does not entail the following G (Claessen-Smallbone's transformation (2018)):

E:
	g1(false, x, xs) = min(xs)
	g1(le(x, min(xs)), x, xs) = min(cons(x, xs))
	g1(true, x, xs) = x
	le(0, s(y)) = true
	le(s(x), s(y)) = le(x, y)
	le(x, 0) = false
	min(cons(x, nil)) = x
	t1(le(x1, min(nil))) = true__
	t1(true) = false__
	t2(false__) = true__
	t2(true__) = false__
G:
	true__ = false__

This holds because

[3] the following ground-complete ordered TRS entails E but does not entail G:

	g1(false, x, xs) -> min(xs)
	g1(le(Y0, min(nil)), Y0, nil) -> Y0
	g1(true, x, xs) -> x
	le(0, s(y)) -> true
	le(s(x), s(y)) -> le(x, y)
	le(x, 0) -> false
	min(cons(x, xs)) -> g1(le(x, min(xs)), x, xs)
	t1(le(x1, min(nil))) -> true__
	t1(true) -> false__
	t2(false__) -> true__
	t2(true__) -> false__
with the LPO induced by
	nil > 0 > true > s > false > cons > g1 > le > min > t2 > t1 > true__ > false__