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refactor(library/data/nat/primes): rename is_prime to prime

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Leonardo de Moura 2015-07-02 23:21:10 -07:00
parent d76edf331b
commit 372d17ab96
1 changed files with 14 additions and 14 deletions
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28
library/data/nat/primes.lean
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@ -11,12 +11,12 @@ open bool
namespace nat
open decidable
definition is_prime [reducible] (p : nat) := p ≥ 2 ∧ ∀ m, m ∣ p → m = 1 ∨ m = p
definition prime [reducible] (p : nat) := p ≥ 2 ∧ ∀ m, m ∣ p → m = 1 ∨ m = p
definition is_prime_ext (p : nat) := p ≥ 2 ∧ ∀ m, m ≤ p → m ∣ p → m = 1 ∨ m = p
local attribute is_prime_ext [reducible]
definition prime_ext (p : nat) := p ≥ 2 ∧ ∀ m, m ≤ p → m ∣ p → m = 1 ∨ m = p
local attribute prime_ext [reducible]
lemma is_prime_ext_iff_is_prime (p : nat) : is_prime_ext p ↔ is_prime p :=
lemma prime_ext_iff_prime (p : nat) : prime_ext p ↔ prime p :=
iff.intro
begin
intro h, cases h with h₁ h₂, constructor, assumption,
@ -27,27 +27,27 @@ iff.intro
intro m l d, exact h₂ m d
end
definition decidable_is_prime [instance] (p : nat) : decidable (is_prime p) :=
decidable_of_decidable_of_iff _ (is_prime_ext_iff_is_prime p)
definition decidable_prime [instance] (p : nat) : decidable (prime p) :=
decidable_of_decidable_of_iff _ (prime_ext_iff_prime p)
lemma ge_two_of_is_prime {p : nat} : is_prime p → p ≥ 2 :=
lemma ge_two_of_prime {p : nat} : prime p → p ≥ 2 :=
assume h, obtain h₁ h₂, from h, h₁
lemma pred_prime_pos {p : nat} : is_prime p → pred p > 0 :=
lemma pred_prime_pos {p : nat} : prime p → pred p > 0 :=
assume h,
have h₁ : p ≥ 2, from ge_two_of_is_prime h,
have h₁ : p ≥ 2, from ge_two_of_prime h,
lt_of_succ_le (pred_le_pred h₁)
lemma succ_pred_prime {p : nat} : is_prime p → succ (pred p) = p :=
assume h, succ_pred_of_pos (lt_of_succ_le (le_of_succ_le (ge_two_of_is_prime h)))
lemma succ_pred_prime {p : nat} : prime p → succ (pred p) = p :=
assume h, succ_pred_of_pos (lt_of_succ_le (le_of_succ_le (ge_two_of_prime h)))
lemma divisor_of_prime {p m : nat} : is_prime p → m ∣ p → m = 1 ∨ m = p :=
lemma divisor_of_prime {p m : nat} : prime p → m ∣ p → m = 1 ∨ m = p :=
assume h d, obtain h₁ h₂, from h, h₂ m d
lemma gt_one_of_pos_of_prime_dvd {i p : nat} : is_prime p → 0 < i → i mod p = 0 → 1 < i :=
lemma gt_one_of_pos_of_prime_dvd {i p : nat} : prime p → 0 < i → i mod p = 0 → 1 < i :=
assume ipp pos h,
have h₁ : p ∣ i, from dvd_of_mod_eq_zero h,
have h₂ : p ≥ 2, from ge_two_of_is_prime ipp,
have h₂ : p ≥ 2, from ge_two_of_prime ipp,
have h₃ : p ≤ i, from le_of_dvd pos h₁,
lt_of_succ_le (le.trans h₂ h₃)