55 lines
1.8 KiB
Text
55 lines
1.8 KiB
Text
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/-
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Copyright (c) 2015 Microsoft Corporation. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Leonardo de Moura
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Prime numbers
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-/
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import data.nat.fact data.nat.bquant data.nat.power logic.identities
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open bool
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namespace nat
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open decidable
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definition is_prime [reducible] (p : nat) := p ≥ 2 ∧ ∀ m, m ∣ p → m = 1 ∨ m = p
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definition is_prime_ext (p : nat) := p ≥ 2 ∧ ∀ m, m ≤ p → m ∣ p → m = 1 ∨ m = p
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local attribute is_prime_ext [reducible]
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lemma is_prime_ext_iff_is_prime (p : nat) : is_prime_ext p ↔ is_prime p :=
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iff.intro
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begin
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intro h, cases h with h₁ h₂, constructor, assumption,
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intro m d, exact h₂ m (le_of_dvd (lt_of_succ_le (le_of_succ_le h₁)) d) d
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end
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begin
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intro h, cases h with h₁ h₂, constructor, assumption,
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intro m l d, exact h₂ m d
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end
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definition decidable_is_prime [instance] (p : nat) : decidable (is_prime p) :=
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decidable_of_decidable_of_iff _ (is_prime_ext_iff_is_prime p)
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lemma ge_two_of_is_prime {p : nat} : is_prime p → p ≥ 2 :=
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assume h, obtain h₁ h₂, from h, h₁
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lemma pred_prime_pos {p : nat} : is_prime p → pred p > 0 :=
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assume h,
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have h₁ : p ≥ 2, from ge_two_of_is_prime h,
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lt_of_succ_le (pred_le_pred h₁)
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lemma succ_pred_prime {p : nat} : is_prime p → succ (pred p) = p :=
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assume h, succ_pred_of_pos (lt_of_succ_le (le_of_succ_le (ge_two_of_is_prime h)))
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lemma divisor_of_prime {p m : nat} : is_prime p → m ∣ p → m = 1 ∨ m = p :=
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assume h d, obtain h₁ h₂, from h, h₂ m d
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lemma gt_one_of_pos_of_prime_dvd {i p : nat} : is_prime p → 0 < i → i mod p = 0 → 1 < i :=
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assume ipp pos h,
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have h₁ : p ∣ i, from dvd_of_mod_eq_zero h,
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have h₂ : p ≥ 2, from ge_two_of_is_prime ipp,
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have h₃ : p ≤ i, from le_of_dvd pos h₁,
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lt_of_succ_le (le.trans h₂ h₃)
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end nat
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