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Mathlib.Algebra.Order.Group.Instances

Additional instances for ordered commutative groups. #

theorem OrderDual.orderedAddCommGroup.proof_1 {α : Type u_1} [OrderedAddCommGroup α] (a : αᵒᵈ) (b : αᵒᵈ) :
a - b = a + -b
Equations
  • OrderDual.orderedAddCommGroup = let __src := OrderDual.orderedAddCommMonoid; let __src_1 := OrderDual.instAddGroup; OrderedAddCommGroup.mk
theorem OrderDual.orderedAddCommGroup.proof_7 {α : Type u_1} [OrderedAddCommGroup α] (a : αᵒᵈ) (b : αᵒᵈ) :
a b∀ (c : αᵒᵈ), c + a c + b
theorem OrderDual.orderedAddCommGroup.proof_6 {α : Type u_1} [OrderedAddCommGroup α] (a : αᵒᵈ) (b : αᵒᵈ) :
a + b = b + a
Equations
  • OrderDual.orderedCommGroup = let __src := OrderDual.orderedCommMonoid; let __src_1 := OrderDual.instGroup; OrderedCommGroup.mk
Equations
  • One or more equations did not get rendered due to their size.
theorem OrderDual.linearOrderedAddCommGroup.proof_3 {α : Type u_1} [LinearOrderedAddCommGroup α] (a : αᵒᵈ) (b : αᵒᵈ) :
max a b = if a b then b else a
theorem OrderDual.linearOrderedAddCommGroup.proof_2 {α : Type u_1} [LinearOrderedAddCommGroup α] (a : αᵒᵈ) (b : αᵒᵈ) :
min a b = if a b then a else b
Equations
  • One or more equations did not get rendered due to their size.