YES

proof:
The input problem is infeasible because

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

	min(xs) = y & le(x, y) = false ==> min(cons(x, xs)) = y
	min(xs) = y & le(x, y) = 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
	min(nil) = x3 & le(x1, x3) = false ==> 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(false) = false__
	t1(le(x1, min(nil))) = true__
	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(false) -> false__
	t1(le(x1, min(nil))) -> true__
	t2(false__) -> true__
	t2(true__) -> false__
with the LPO induced by
	s > true > cons > le > false > nil > g1 > min > t1 > 0 > t2 > true__ > false__