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Mathlib.Algebra.Order.Field.Canonical.Defs

Canonically ordered semifields #

A canonically linear ordered field is a linear ordered field in which a ≤ b iff there exists c with b = a + c.

  • add : ααα
  • add_assoc : ∀ (a b c : α), a + b + c = a + (b + c)
  • zero : α
  • zero_add : ∀ (a : α), 0 + a = a
  • add_zero : ∀ (a : α), a + 0 = a
  • nsmul : αα
  • nsmul_zero : ∀ (x : α), AddMonoid.nsmul 0 x = 0
  • nsmul_succ : ∀ (n : ) (x : α), AddMonoid.nsmul (n + 1) x = AddMonoid.nsmul n x + x
  • add_comm : ∀ (a b : α), a + b = b + a
  • le : ααProp
  • lt : ααProp
  • le_refl : ∀ (a : α), a a
  • le_trans : ∀ (a b c : α), a bb ca c
  • lt_iff_le_not_le : ∀ (a b : α), a < b a b ¬b a
  • le_antisymm : ∀ (a b : α), a bb aa = b
  • add_le_add_left : ∀ (a b : α), a b∀ (c : α), c + a c + b
  • bot : α
  • bot_le : ∀ (a : α), a
  • exists_add_of_le : ∀ {a b : α}, a b∃ (c : α), b = a + c
  • le_self_add : ∀ (a b : α), a a + b
  • mul : ααα
  • left_distrib : ∀ (a b c : α), a * (b + c) = a * b + a * c
  • right_distrib : ∀ (a b c : α), (a + b) * c = a * c + b * c
  • zero_mul : ∀ (a : α), 0 * a = 0
  • mul_zero : ∀ (a : α), a * 0 = 0
  • mul_assoc : ∀ (a b c : α), a * b * c = a * (b * c)
  • one : α
  • one_mul : ∀ (a : α), 1 * a = a
  • mul_one : ∀ (a : α), a * 1 = a
  • natCast : α
  • natCast_zero : NatCast.natCast 0 = 0
  • natCast_succ : ∀ (n : ), NatCast.natCast (n + 1) = NatCast.natCast n + 1
  • npow : αα
  • npow_zero : ∀ (x : α), CanonicallyOrderedCommSemiring.npow 0 x = 1
  • npow_succ : ∀ (n : ) (x : α), CanonicallyOrderedCommSemiring.npow (n + 1) x = CanonicallyOrderedCommSemiring.npow n x * x
  • mul_comm : ∀ (a b : α), a * b = b * a
  • eq_zero_or_eq_zero_of_mul_eq_zero : ∀ {a b : α}, a * b = 0a = 0 b = 0
  • le_of_add_le_add_left : ∀ (a b c : α), a + b a + cb c
  • exists_pair_ne : ∃ (x : α), ∃ (y : α), x y
  • zero_le_one : 0 1

    In a strict ordered semiring, 0 ≤ 1.

  • mul_lt_mul_of_pos_left : ∀ (a b c : α), a < b0 < cc * a < c * b

    Left multiplication by a positive element is strictly monotone.

  • mul_lt_mul_of_pos_right : ∀ (a b c : α), a < b0 < ca * c < b * c

    Right multiplication by a positive element is strictly monotone.

  • min : ααα
  • max : ααα
  • compare : ααOrdering
  • le_total : ∀ (a b : α), a b b a

    A linear order is total.

  • decidableLE : DecidableRel fun (x1 x2 : α) => x1 x2

    In a linearly ordered type, we assume the order relations are all decidable.

  • decidableEq : DecidableEq α

    In a linearly ordered type, we assume the order relations are all decidable.

  • decidableLT : DecidableRel fun (x1 x2 : α) => x1 < x2

    In a linearly ordered type, we assume the order relations are all decidable.

  • min_def : ∀ (a b : α), min a b = if a b then a else b

    The minimum function is equivalent to the one you get from minOfLe.

  • max_def : ∀ (a b : α), max a b = if a b then b else a

    The minimum function is equivalent to the one you get from maxOfLe.

  • compare_eq_compareOfLessAndEq : ∀ (a b : α), compare a b = compareOfLessAndEq a b

    Comparison via compare is equal to the canonical comparison given decidable < and =.

  • inv : αα
  • div : ααα
  • div_eq_mul_inv : ∀ (a b : α), a / b = a * b⁻¹

    a / b := a * b⁻¹

  • zpow : αα

    The power operation: a ^ n = a * ··· * a; a ^ (-n) = a⁻¹ * ··· a⁻¹ (n times)

  • zpow_zero' : ∀ (a : α), CanonicallyLinearOrderedSemifield.zpow 0 a = 1

    a ^ 0 = 1

  • zpow_succ' : ∀ (n : ) (a : α), CanonicallyLinearOrderedSemifield.zpow (↑n.succ) a = CanonicallyLinearOrderedSemifield.zpow (↑n) a * a

    a ^ (n + 1) = a ^ n * a

  • a ^ -(n + 1) = (a ^ (n + 1))⁻¹

  • inv_zero : 0⁻¹ = 0

    The inverse of 0 in a group with zero is 0.

  • mul_inv_cancel : ∀ (a : α), a 0a * a⁻¹ = 1

    Every nonzero element of a group with zero is invertible.

  • nnratCast : ℚ≥0α
  • nnratCast_def : ∀ (q : ℚ≥0), q = q.num / q.den

    However NNRat.cast is defined, it must be propositionally equal to a / b.

    Do not use this lemma directly. Use NNRat.cast_def instead.

  • nnqsmul : ℚ≥0αα

    Scalar multiplication by a nonnegative rational number.

    Unless there is a risk of a Module ℚ≥0 _ instance diamond, write nnqsmul := _. This will set nnqsmul to (NNRat.cast · * ·) thanks to unification in the default proof of nnqsmul_def.

    Do not use directly. Instead use the notation.

  • nnqsmul_def : ∀ (q : ℚ≥0) (a : α), CanonicallyLinearOrderedSemifield.nnqsmul q a = q * a

    However qsmul is defined, it must be propositionally equal to multiplication by Rat.cast.

    Do not use this lemma directly. Use NNRat.smul_def instead.

Instances
    @[instance 100]
    Equations
    @[instance 100]
    Equations
    • One or more equations did not get rendered due to their size.