feat(library/algebra/ordered_group): add theorems for max and min
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Jeremy Avigad
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2 changed files with 69 additions and 31 deletions
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@ -3,10 +3,9 @@ Copyright (c) 2014 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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Authors: Jeremy Avigad
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Partially ordered additive groups, modeled on Isabelle's library. We could refine the structures,
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but we would have to declare more inheritance paths.
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Partially ordered additive groups, modeled on Isabelle's library. These classes can be refined
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if necessary.
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-/
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import logic.eq data.unit data.sigma data.prod
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import algebra.function algebra.binary
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import algebra.group algebra.order
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@ -48,16 +47,6 @@ section
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theorem add_le_add (Hab : a ≤ b) (Hcd : c ≤ d) : a + c ≤ b + d :=
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le.trans (add_le_add_right Hab c) (add_le_add_left Hcd b)
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/- theorem add_lt_add_left (H : a < b) (c : A) : c + a < c + b :=
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have H1 : c + a ≤ c + b, from add_le_add_left (le_of_lt H) c,
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have H2 : c + a ≠ c + b, from
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take H3 : c + a = c + b,
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have H4 : a = b, from add.left_cancel H3,
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ne_of_lt H H4,
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sorry--lt_of_le_of_ne H1 H2-/
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theorem le_add_of_nonneg_right (H : b ≥ 0) : a ≤ a + b :=
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begin
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have H1 : a + b ≥ a + 0, from add_le_add_left H a,
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@ -94,10 +83,6 @@ section
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theorem lt_of_add_lt_add_left (H : a + b < a + c) : b < c :=
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!ordered_cancel_comm_monoid.lt_of_add_lt_add_left H
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/-have H1 : b ≤ c, from le_of_add_le_add_left (le_of_lt H),
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have H2 : b ≠ c, from
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assume H3 : b = c, lt.irrefl _ (H3 ▸ H),
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sorry --lt_of_le_of_ne H1 H2-/
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theorem lt_of_add_lt_add_right (H : a + b < c + b) : a < c :=
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lt_of_add_lt_add_left ((add.comm a b) ▸ (add.comm c b) ▸ H)
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@ -210,16 +195,11 @@ section
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!add_zero ▸ add_lt_add Hbc Ha
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end
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-- TODO: add properties of max and min
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/- partially ordered groups -/
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structure ordered_comm_group [class] (A : Type) extends add_comm_group A, order_pair A :=
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(add_le_add_left : ∀a b, le a b → ∀c, le (add c a) (add c b))
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(add_lt_add_left : ∀a b, lt a b → ∀ c, lt (add c a) (add c b))
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--(le_of_add_le_add_left : ∀a b c, le (add a b) (add a c) → le b c)
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--(lt_of_add_lt_add_left : ∀a b c, lt (add a b) (add a c) → lt b c)
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theorem ordered_comm_group.le_of_add_le_add_left [s : ordered_comm_group A] {a b c : A} (H : a + b ≤ a + c) : b ≤ c :=
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assert H' : -a + (a + b) ≤ -a + (a + c), from ordered_comm_group.add_le_add_left _ _ H _,
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@ -441,6 +421,65 @@ section
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add_le_of_le_of_nonpos (le.refl a) (neg_nonpos_of_nonneg H)
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end
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/- partially ordered groups with min and max -/
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structure lattice_ordered_comm_group [class] (A : Type)
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extends ordered_comm_group A, lattice A
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section
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variables [s : lattice_ordered_comm_group A]
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variables (a b c : A)
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include s
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theorem min_add_add_left : min (a + b) (a + c) = a + min b c :=
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eq.symm (eq_min
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(show a + min b c ≤ a + b, from add_le_add_left !min_le_left _)
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(show a + min b c ≤ a + c, from add_le_add_left !min_le_right _)
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(take d,
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assume H₁ : d ≤ a + b,
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assume H₂ : d ≤ a + c,
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have H : d - a ≤ min b c,
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from le_min (iff.mp !le_add_iff_sub_left_le H₁) (iff.mp !le_add_iff_sub_left_le H₂),
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show d ≤ a + min b c, from iff.mp' !le_add_iff_sub_left_le H))
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theorem min_add_add_right : min (a + c) (b + c) = min a b + c :=
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by rewrite [add.comm a c, add.comm b c, add.comm _ c]; apply min_add_add_left
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theorem max_add_add_left : max (a + b) (a + c) = a + max b c :=
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eq.symm (eq_max
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(add_le_add_left !le_max_left _)
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(add_le_add_left !le_max_right _)
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(λ d H₁ H₂,
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have H : max b c ≤ d - a,
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from max_le (iff.mp !add_le_iff_le_sub_left H₁) (iff.mp !add_le_iff_le_sub_left H₂),
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show a + max b c ≤ d, from iff.mp' !add_le_iff_le_sub_left H))
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theorem max_add_add_right : max (a + c) (b + c) = max a b + c :=
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by rewrite [add.comm a c, add.comm b c, add.comm _ c]; apply max_add_add_left
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theorem max_neg_neg : max (-a) (-b) = - min a b :=
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eq.symm (eq_max
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(show -a ≤ -(min a b), from neg_le_neg !min_le_left)
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(show -b ≤ -(min a b), from neg_le_neg !min_le_right)
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(take d,
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assume H₁ : -a ≤ d,
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assume H₂ : -b ≤ d,
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have H : -d ≤ min a b,
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from le_min (!iff.mp !neg_le_iff_neg_le H₁) (!iff.mp !neg_le_iff_neg_le H₂),
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show -(min a b) ≤ d, from !iff.mp !neg_le_iff_neg_le H))
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theorem min_eq_neg_max_neg_neg : min a b = - max (-a) (-b) :=
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by rewrite [max_neg_neg, neg_neg]
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theorem min_neg_neg : min (-a) (-b) = - max a b :=
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by rewrite [min_eq_neg_max_neg_neg, *neg_neg]
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theorem max_eq_neg_min_neg_neg : max a b = - min (-a) (-b) :=
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by rewrite [min_neg_neg, neg_neg]
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end
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/- linear ordered group with decidable order -/
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structure decidable_linear_ordered_comm_group [class] (A : Type)
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extends add_comm_group A, decidable_linear_order A :=
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(add_le_add_left : ∀ a b, le a b → ∀ c, le (add c a) (add c b))
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@ -450,9 +489,10 @@ private theorem add_le_add_left' (A : Type) (s : decidable_linear_ordered_comm_g
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a ≤ b → (∀ c : A, c + a ≤ c + b) :=
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decidable_linear_ordered_comm_group.add_le_add_left a b
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definition decidable_linear_ordered_comm_group.to_ordered_comm_group [trans-instance] [reducible] [coercion]
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(A : Type) [s : decidable_linear_ordered_comm_group A] : ordered_comm_group A :=
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⦃ordered_comm_group, s,
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definition decidable_linear_ordered_comm_group.to_lattice_ordered_comm_group
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[trans-instance] [reducible] [coercion]
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(A : Type) [s : decidable_linear_ordered_comm_group A] : lattice_ordered_comm_group A :=
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⦃ lattice_ordered_comm_group, s, decidable_linear_order.to_lattice,
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le_of_lt := @le_of_lt A s,
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add_le_add_left := add_le_add_left' A s,
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lt_of_le_of_lt := @lt_of_le_of_lt A s,
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@ -281,7 +281,6 @@ section migrate_algebra
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show decidable (b ≤ a), from _
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definition decidable_gt [instance] (a b : ℤ) : decidable (a > b) :=
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show decidable (b < a), from _
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definition min : ℤ → ℤ → ℤ := algebra.min
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definition max : ℤ → ℤ → ℤ := algebra.max
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definition abs : ℤ → ℤ := algebra.abs
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@ -295,7 +294,6 @@ section migrate_algebra
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attribute lt_of_lt_of_le lt_of_le_of_lt gt_of_gt_of_ge gt_of_ge_of_gt [trans]
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end migrate_algebra
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/- more facts specific to int -/
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theorem of_nat_nonneg (n : ℕ) : 0 ≤ of_nat n := trivial
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