theory Approximation_Quickcheck_Ex
imports "../Approximation"
begin

lemma
  fixes x::real and y::real
  shows "sin x ≤ tan x"
  using [[quickcheck_approximation_custom_seed = 1]]
  quickcheck[approximation, expect=counterexample]
  oops

lemma "x ≤ y ==> arctan y / y ≤ arctan x / x"
  using [[quickcheck_approximation_custom_seed = 1]]
  quickcheck[approximation, expect=counterexample]
  oops

lemma "0 < x ==> x ≤ y ==> arctan y / y ≤ arctan x / x"
  using [[quickcheck_approximation_custom_seed = 1]]
  quickcheck[approximation, expect=no_counterexample]
  by (rule arctan_divide_mono)

lemma
  fixes x::real
  shows "exp (exp x + exp y + sin x * sin y) - 0.4 > 0 ∨ 0.98 - sin x / (sin x * sin y + 2)^2 > 0"
  using [[quickcheck_approximation_custom_seed = 1]]
  quickcheck[approximation, expect=counterexample, size=10, iterations=1000, verbose]
  oops

lemma
  fixes x::real
  shows "x > 1 ==> x ≤ 2 ^ 20 * log 2 x + 1 ∧ (sin x)² + (cos x)² = 1"
  using [[quickcheck_approximation_custom_seed = 1]]
  using [[quickcheck_approximation_epsilon = 0.00000001]]
    ― ‹avoids spurious counterexamples in approximate computation of term‹(sin x)² + (cos x)²›
 and therefore avoids expensive failing attempts for certification›
  quickcheck[approximation, expect=counterexample, size=20]
  oops

end