{"kind":"basis_map_theorem","report":{"basis":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"basis_name":"a(n) = 1 if n is a triangular number, otherwise 0.","conclusions":[{"hypotheses":["initial_row_nonzero","initial_row_real_rooted","initial_row_nonpositive_roots"],"input_interval":{"upper":{"closed":true,"value":{"lower_bound":"1","radicand":"1","rational_part":"1","sqrt_coefficient":"0","text":"1","upper_bound":"1"}}},"nonnegative_coefficients":true,"output_interval":{"upper":{"closed":true,"value":{"lower_bound":"0","radicand":"1","rational_part":"0","sqrt_coefficient":"0","text":"0","upper_bound":"0"}}},"sharp":false,"statement":"T(p) = p(a) g; an input with every root <= a has p(a) of the sign of its leading coefficient"}],"hypotheses":[{"evidence":"exact","name":"initial_row_nonzero","statement":"g = P_0 is nonzero","status":{"kind":"verified"}},{"evidence":"exact root isolation","name":"initial_row_real_rooted","statement":"g is real-rooted","status":{"kind":"verified"}},{"evidence":"exact root isolation","name":"initial_row_nonpositive_roots","statement":"every root of g is <= 0","status":{"kind":"verified"}}],"lean_replay":"not_attempted","schema":"polytool.recurrence-basis-map-theorem.v1","shape":{"constant":"1","initial_row":["1"],"kind":"geometric","slope":"0"},"status":"proved","theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)"}}