4a36f843f7
I changed the definition of pow so that a^(succ n) reduces to a * a^n rather than a^n * a. This has the nice effect that on nat and int, where multiplication is defined by recursion on the right, a^1 reduces to a, and a^2 reduces to a * a. The change was a pain in the neck, and in retrospect maybe not worth it, but oh, well.
63 lines
1.7 KiB
Text
63 lines
1.7 KiB
Text
/-
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Copyright (c) 2015 Jeremy Avigad. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Author: Jeremy Avigad
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Properties of the power operation in an ordered ring.
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(Right now, this file is just a stub. More soon.)
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-/
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import .group_power
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open nat
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namespace algebra
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variable {A : Type}
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section linear_ordered_semiring
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variable [s : linear_ordered_semiring A]
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include s
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theorem pow_pos_of_pos {x : A} (i : ℕ) (H : x > 0) : x^i > 0 :=
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begin
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induction i with [j, ih],
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{show (1 : A) > 0, from zero_lt_one},
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{show x^(succ j) > 0, from mul_pos H ih}
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end
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theorem pow_nonneg_of_nonneg {x : A} (i : ℕ) (H : x ≥ 0) : x^i ≥ 0 :=
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begin
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induction i with [j, ih],
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{show (1 : A) ≥ 0, from le_of_lt zero_lt_one},
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{show x^(succ j) ≥ 0, from mul_nonneg H ih}
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end
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theorem pow_le_pow_of_le {x y : A} (i : ℕ) (H₁ : 0 ≤ x) (H₂ : x ≤ y) : x^i ≤ y^i :=
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begin
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induction i with [i, ih],
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{rewrite *pow_zero, apply le.refl},
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rewrite *pow_succ,
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have H : 0 ≤ x^i, from pow_nonneg_of_nonneg i H₁,
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apply mul_le_mul H₂ ih H (le.trans H₁ H₂)
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end
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theorem pow_ge_one {x : A} (i : ℕ) (xge1 : x ≥ 1) : x^i ≥ 1 :=
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assert H : x^i ≥ 1^i, from pow_le_pow_of_le i (le_of_lt zero_lt_one) xge1,
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by rewrite one_pow at H; exact H
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set_option formatter.hide_full_terms false
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theorem pow_gt_one {x : A} {i : ℕ} (xgt1 : x > 1) (ipos : i > 0) : x^i > 1 :=
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assert xpos : x > 0, from lt.trans zero_lt_one xgt1,
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begin
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induction i with [i, ih],
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{exfalso, exact !nat.lt.irrefl ipos},
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have xige1 : x^i ≥ 1, from pow_ge_one _ (le_of_lt xgt1),
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rewrite [pow_succ, -mul_one 1, ↑has_lt.gt],
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apply mul_lt_mul xgt1 xige1 zero_lt_one,
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apply le_of_lt xpos
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end
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end linear_ordered_semiring
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end algebra
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