fc718eac88
This PR adds a code action for `grind` parameters. We need to use `set_option grind.param.codeAction true` to enable the option. The PR also adds a modifier to instruct `grind` to use the "default" pattern inference strategy.
51 lines
1.9 KiB
Text
51 lines
1.9 KiB
Text
-- This is the example from the front page of the website
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-- There's no mechanism keeping it in sync with the website,
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-- but nevertheless it's better than nothing.
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/-- A prime is a number larger than 1 with no trivial divisors -/
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def IsPrime (n : Nat) := 1 < n ∧ ∀ k, 1 < k → k < n → ¬ k ∣ n
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/-- Every number larger than 1 has a prime factor -/
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theorem exists_prime_factor :
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∀ n, 1 < n → ∃ k, IsPrime k ∧ k ∣ n := by
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intro n h1
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-- Either `n` is prime...
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by_cases hprime : IsPrime n
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· grind [Nat.dvd_refl]
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-- ... or it has a non-trivial divisor with a prime factor
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· obtain ⟨k, _⟩ : ∃ k, 1 < k ∧ k < n ∧ k ∣ n := by
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simp_all [IsPrime]
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obtain ⟨p, _, _⟩ := exists_prime_factor k (by grind)
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grind [Nat.dvd_trans]
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/-- The factorial, defined recursively, with custom notation -/
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def factorial : Nat → Nat
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| 0 => 1
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| n+1 => (n + 1) * factorial n
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notation:10000 n "!" => factorial n
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/-- The factorial is always positive -/
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theorem factorial_pos : ∀ n, 0 < n ! := by
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intro n; induction n <;>
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grind [factorial, Nat.mul_pos_iff_of_pos_left]
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/-- ... and divided by its constituent factors -/
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theorem dvd_factorial : ∀ n, ∀ k ≤ n, 0 < k → k ∣ n ! := by
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intro n; induction n <;>
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grind [Nat.dvd_mul_right, Nat.dvd_mul_left_of_dvd, factorial]
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/--
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We show that we find arbitrary large (and thus infinitely
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many) prime numbers, by picking an arbitrary number `n`
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and showing that `n! + 1` has a prime factor larger than `n`.
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-/
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theorem InfinitudeOfPrimes : ∀ n, ∃ p > n, IsPrime p := by
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intro n
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have : 1 < n ! + 1 := by grind [factorial_pos]
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obtain ⟨p, hp, _⟩ := exists_prime_factor (n ! + 1) this
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suffices ¬p ≤ n by grind
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intro (_ : p ≤ n)
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have : 1 < p := hp.1
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have : p ∣ n ! := dvd_factorial n p ‹p ≤ n› (by grind)
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have := Nat.dvd_sub ‹p ∣ n ! + 1› ‹p ∣ n !›
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grind [Nat.add_sub_cancel_left, Nat.dvd_one]
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