147 lines
6.5 KiB
Text
147 lines
6.5 KiB
Text
/-
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Copyright (c) 2016 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: Jeremy Avigad, Leonardo de Moura
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-/
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prelude
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import init.ordered_group init.ring
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/- Make sure instances defined in this file have lower priority than the ones
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defined for concrete structures -/
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set_option default_priority 100
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universe variable u
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structure ordered_semiring (α : Type u)
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extends semiring α, ordered_mul_cancel_comm_monoid α renaming
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mul→add mul_assoc→add_assoc one→zero one_mul→zero_add mul_one→add_zero mul_comm→add_comm
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mul_left_cancel→add_left_cancel mul_right_cancel→add_right_cancel
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mul_le_mul_left→add_le_add_left mul_lt_mul_left→add_lt_add_left
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le_of_mul_le_mul_left→le_of_add_le_add_left
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lt_of_mul_lt_mul_left→lt_of_add_lt_add_left :=
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(mul_le_mul_of_nonneg_left: ∀ a b c : α, a ≤ b → 0 ≤ c → c * a ≤ c * b)
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(mul_le_mul_of_nonneg_right: ∀ a b c : α, a ≤ b → 0 ≤ c → a * c ≤ b * c)
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(mul_lt_mul_of_pos_left: ∀ a b c : α, a < b → 0 < c → c * a < c * b)
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(mul_lt_mul_of_pos_right: ∀ a b c : α, a < b → 0 < c → a * c < b * c)
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/- we make it a class now (and not as part of the structure) to avoid
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ordered_semiring.to_ordered_mul_cancel_comm_monoid to be an instance -/
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attribute [class] ordered_semiring
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variable {α : Type u}
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instance add_comm_group_of_ordered_semiring (α : Type u) [s : ordered_semiring α] : semiring α :=
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@ordered_semiring.to_semiring α s
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instance monoid_of_ordered_semiring (α : Type u) [s : ordered_semiring α] : ordered_cancel_comm_monoid α :=
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@ordered_semiring.to_ordered_mul_cancel_comm_monoid α s
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section
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variable [ordered_semiring α]
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lemma mul_le_mul_of_nonneg_left {a b c : α} (h₁ : a ≤ b) (h₂ : 0 ≤ c) : c * a ≤ c * b :=
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ordered_semiring.mul_le_mul_of_nonneg_left _ a b c h₁ h₂
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lemma mul_le_mul_of_nonneg_right {a b c : α} (h₁ : a ≤ b) (h₂ : 0 ≤ c) : a * c ≤ b * c :=
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ordered_semiring.mul_le_mul_of_nonneg_right _ a b c h₁ h₂
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lemma mul_lt_mul_of_pos_left {a b c : α} (h₁ : a < b) (h₂ : 0 < c) : c * a < c * b :=
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ordered_semiring.mul_lt_mul_of_pos_left _ a b c h₁ h₂
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lemma mul_lt_mul_of_pos_right {a b c : α} (h₁ : a < b) (h₂ : 0 < c) : a * c < b * c :=
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ordered_semiring.mul_lt_mul_of_pos_right _ a b c h₁ h₂
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end
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class linear_ordered_semiring (α : Type u) extends ordered_semiring α, linear_strong_order_pair α :=
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(zero_lt_one : zero < one)
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lemma zero_lt_one [linear_ordered_semiring α] : 0 < (1:α) :=
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linear_ordered_semiring.zero_lt_one α
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class decidable_linear_ordered_semiring (α : Type u) extends linear_ordered_semiring α, decidable_linear_order α
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structure ordered_ring (α : Type u) extends ring α, ordered_mul_comm_group α renaming
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mul→add mul_assoc→add_assoc one→zero one_mul→zero_add mul_one→add_zero inv→neg
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mul_left_inv→add_left_inv mul_comm→add_comm mul_le_mul_left→add_le_add_left
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mul_lt_mul_left→add_lt_add_left, zero_ne_one_class α :=
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(mul_nonneg : ∀ a b : α, 0 ≤ a → 0 ≤ b → 0 ≤ a * b)
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(mul_pos : ∀ a b : α, 0 < a → 0 < b → 0 < a * b)
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/- we make it a class now (and not as part of the structure) to avoid
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ordered_ring.to_ordered_mul_comm_group to be an instance -/
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attribute [class] ordered_ring
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instance add_comm_group_of_ordered_ring (α : Type u) [s : ordered_ring α] : ring α :=
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@ordered_ring.to_ring α s
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instance monoid_of_ordered_ring (α : Type u) [s : ordered_ring α] : ordered_comm_group α :=
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@ordered_ring.to_ordered_mul_comm_group α s
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instance zero_ne_one_class_of_ordered_ring (α : Type u) [s : ordered_ring α] : zero_ne_one_class α :=
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@ordered_ring.to_zero_ne_one_class α s
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lemma ordered_ring.mul_le_mul_of_nonneg_left [s : ordered_ring α] {a b c : α}
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(h₁ : a ≤ b) (h₂ : 0 ≤ c) : c * a ≤ c * b :=
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have 0 ≤ b - a, from sub_nonneg_of_le h₁,
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have 0 ≤ c * (b - a), from ordered_ring.mul_nonneg s c (b - a) h₂ this,
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begin
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rw mul_sub_left_distrib at this,
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apply le_of_sub_nonneg this
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end
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lemma ordered_ring.mul_le_mul_of_nonneg_right [s : ordered_ring α] {a b c : α}
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(h₁ : a ≤ b) (h₂ : 0 ≤ c) : a * c ≤ b * c :=
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have 0 ≤ b - a, from sub_nonneg_of_le h₁,
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have 0 ≤ (b - a) * c, from ordered_ring.mul_nonneg s (b - a) c this h₂,
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begin
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rw mul_sub_right_distrib at this,
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apply le_of_sub_nonneg this
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end
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lemma ordered_ring.mul_lt_mul_of_pos_left [s : ordered_ring α] {a b c : α}
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(h₁ : a < b) (h₂ : 0 < c) : c * a < c * b :=
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have 0 < b - a, from sub_pos_of_lt h₁,
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have 0 < c * (b - a), from ordered_ring.mul_pos s c (b - a) h₂ this,
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begin
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rw mul_sub_left_distrib at this,
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apply lt_of_sub_pos this
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end
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lemma ordered_ring.mul_lt_mul_of_pos_right [s : ordered_ring α] {a b c : α}
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(h₁ : a < b) (h₂ : 0 < c) : a * c < b * c :=
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have 0 < b - a, from sub_pos_of_lt h₁,
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have 0 < (b - a) * c, from ordered_ring.mul_pos s (b - a) c this h₂,
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begin
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rw mul_sub_right_distrib at this,
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apply lt_of_sub_pos this
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end
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instance ordered_ring.to_ordered_semiring [s : ordered_ring α] : ordered_semiring α :=
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{ s with
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mul_zero := mul_zero,
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zero_mul := zero_mul,
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add_left_cancel := @add_left_cancel α _,
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add_right_cancel := @add_right_cancel α _,
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le_of_add_le_add_left := @le_of_add_le_add_left α _,
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mul_le_mul_of_nonneg_left := @ordered_ring.mul_le_mul_of_nonneg_left α _,
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mul_le_mul_of_nonneg_right := @ordered_ring.mul_le_mul_of_nonneg_right α _,
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mul_lt_mul_of_pos_left := @ordered_ring.mul_lt_mul_of_pos_left α _,
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mul_lt_mul_of_pos_right := @ordered_ring.mul_lt_mul_of_pos_right α _,
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lt_of_add_lt_add_left := @lt_of_add_lt_add_left α _}
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class linear_ordered_ring (α : Type u) extends ordered_ring α, linear_strong_order_pair α :=
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(zero_lt_one : lt zero one)
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instance linear_ordered_ring.to_linear_ordered_semiring [s : linear_ordered_ring α] : linear_ordered_semiring α :=
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{ s with
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mul_zero := mul_zero,
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zero_mul := zero_mul,
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add_left_cancel := @add_left_cancel α _,
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add_right_cancel := @add_right_cancel α _,
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le_of_add_le_add_left := @le_of_add_le_add_left α _,
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mul_le_mul_of_nonneg_left := @mul_le_mul_of_nonneg_left α _,
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mul_le_mul_of_nonneg_right := @mul_le_mul_of_nonneg_right α _,
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mul_lt_mul_of_pos_left := @mul_lt_mul_of_pos_left α _,
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mul_lt_mul_of_pos_right := @mul_lt_mul_of_pos_right α _,
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le_total := linear_ordered_ring.le_total,
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lt_of_add_lt_add_left := @lt_of_add_lt_add_left α _ }
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