63 lines
2.3 KiB
Text
63 lines
2.3 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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Module: data.nat.power
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Authors: Leonardo de Moura
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Power
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-/
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import data.nat.basic data.nat.div
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namespace nat
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definition pow : nat → nat → nat
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| a 0 := 1
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| a (succ b) := a * pow a b
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theorem pow_zero (a : nat) : pow a 0 = 1 :=
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rfl
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theorem pow_succ (a b : nat) : pow a (succ b) = a * pow a b :=
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rfl
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theorem one_pow : ∀ (a : nat), pow 1 a = 1
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| 0 := rfl
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| (succ a) := by rewrite [pow_succ, one_pow]
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theorem pow_one : ∀ {a : nat}, a ≠ 0 → pow a 1 = a
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| 0 h := absurd rfl h
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| (succ a) h := by rewrite [pow_succ, pow_zero, mul_one]
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theorem zero_pow : ∀ {a : nat}, a ≠ 0 → pow 0 a = 0
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| 0 h := absurd rfl h
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| (succ a) h := by rewrite [pow_succ, zero_mul]
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theorem pow_add : ∀ (a b c : nat), pow a (b + c) = pow a b * pow a c
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| a b 0 := by rewrite [add_zero, pow_zero, mul_one]
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| a b (succ c) := by rewrite [add_succ, *pow_succ, pow_add a b c, mul.left_comm]
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theorem mul_self_eq_pow_2 (a : nat) : a * a = pow a 2 :=
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show a * a = pow a (succ (succ zero)), from
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by rewrite [*pow_succ, *pow_zero, mul_one]
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theorem pow_cancel_left : ∀ {a b c : nat}, a > 1 → pow a b = pow a c → b = c
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| a 0 0 h₁ h₂ := rfl
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| a (succ b) 0 h₁ h₂ :=
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assert aeq1 : a = 1, by rewrite [pow_succ at h₂, pow_zero at h₂]; exact (eq_one_of_mul_eq_one_right h₂),
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assert h₁ : 1 < 1, by rewrite [aeq1 at h₁]; exact h₁,
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absurd h₁ !lt.irrefl
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| a 0 (succ c) h₁ h₂ :=
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assert aeq1 : a = 1, by rewrite [pow_succ at h₂, pow_zero at h₂]; exact (eq_one_of_mul_eq_one_right (eq.symm h₂)),
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assert h₁ : 1 < 1, by rewrite [aeq1 at h₁]; exact h₁,
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absurd h₁ !lt.irrefl
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| a (succ b) (succ c) h₁ h₂ :=
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assert ane0 : a ≠ 0, from assume aeq0, by rewrite [aeq0 at h₁]; exact (absurd h₁ dec_trivial),
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assert beqc : pow a b = pow a c, by rewrite [*pow_succ at h₂]; exact (mul_cancel_left_of_ne_zero ane0 h₂),
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by rewrite [pow_cancel_left h₁ beqc]
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theorem pow_div_cancel : ∀ {a b : nat}, a ≠ 0 → pow a (succ b) div a = pow a b
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| a 0 h := by rewrite [pow_succ, pow_zero, mul_one, div_self (pos_of_ne_zero h)]
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| a (succ b) h := by rewrite [pow_succ, mul_div_cancel_left _ (pos_of_ne_zero h)]
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end nat
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