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feat: CanonicallyOrderedAddCancelCommMonoid #9307

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feat: CanonicallyOrderedAddCancelCommMonoid #9307

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ee70404
feat: CanonicallyOrderedAddCancelCommMonoid
madvorak Dec 27, 2023
e757a46
Nat instance fix
madvorak Dec 27, 2023
08990c3
bugfix attempt
madvorak Dec 28, 2023
015754c
missing colon
madvorak Dec 28, 2023
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21 changes: 16 additions & 5 deletions Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean
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Expand Up @@ -92,7 +92,7 @@ theorem le_iff_forall_one_lt_lt_mul' : a ≤ b ↔ ∀ ε, 1 < ε → a < b * ε
end ExistsMulOfLE


/-- A canonically ordered additive monoid is an ordered commutative additive monoid
/-- A canonically ordered additive commutative monoid is an ordered commutative additive monoid
in which the ordering coincides with the subtractibility relation,
which is to say, `a ≤ b` iff there exists `c` with `b = a + c`.
This is satisfied by the natural numbers, for example, but not
Expand All @@ -108,7 +108,7 @@ class CanonicallyOrderedAddCommMonoid (α : Type*) extends OrderedAddCommMonoid
-- see Note [lower instance priority]
attribute [instance 100] CanonicallyOrderedAddCommMonoid.toOrderBot

/-- A canonically ordered monoid is an ordered commutative monoid
/-- A canonically ordered commutative monoid is an ordered commutative monoid
in which the ordering coincides with the divisibility relation,
which is to say, `a ≤ b` iff there exists `c` with `b = a * c`.
Examples seem rare; it seems more likely that the `OrderDual`
Expand Down Expand Up @@ -320,13 +320,24 @@ instance bit0 {M} [CanonicallyOrderedAddCommMonoid M] {x : M} [NeZero x] : NeZer

end NeZero

/-- A canonically linear-ordered additive monoid is a canonically ordered additive monoid
whose ordering is a linear order. -/
/-- A canonically ordered additive cancellative commutative monoid is
a canonically ordered additive commutative monoid in which addition is cancellative. -/
class CanonicallyOrderedAddCancelCommMonoid (α : Type*)
extends CanonicallyOrderedAddCommMonoid α, OrderedCancelAddCommMonoid α

/-- A canonically ordered cancellative commutative monoid is
a canonically ordered commutative monoid in which multiplication is cancellative. -/
@[to_additive]
class CanonicallyOrderedCancelCommMonoid (α : Type*)
extends CanonicallyOrderedCommMonoid α, OrderedCancelCommMonoid α

/-- A canonically linear-ordered additive commutative monoid is
a canonically ordered additive commutative monoid whose ordering is a linear order. -/
class CanonicallyLinearOrderedAddCommMonoid (α : Type*)
extends CanonicallyOrderedAddCommMonoid α, LinearOrderedAddCommMonoid α
#align canonically_linear_ordered_add_monoid CanonicallyLinearOrderedAddCommMonoid

/-- A canonically linear-ordered monoid is a canonically ordered monoid
/-- A canonically linear-ordered commutative monoid is a canonically ordered commutative monoid
whose ordering is a linear order. -/
@[to_additive]
class CanonicallyLinearOrderedCommMonoid (α : Type*)
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3 changes: 3 additions & 0 deletions Mathlib/Data/Nat/Order/Basic.lean
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Expand Up @@ -76,6 +76,9 @@ instance canonicallyOrderedCommSemiring : CanonicallyOrderedCommSemiring ℕ :=
instance canonicallyLinearOrderedAddCommMonoid : CanonicallyLinearOrderedAddCommMonoid ℕ :=
{ (inferInstance : CanonicallyOrderedAddCommMonoid ℕ), Nat.linearOrder with }

instance canonicallyOrderedAddCancelCommMonoid : CanonicallyOrderedAddCancelCommMonoid ℕ :=
{ (inferInstance : CanonicallyOrderedAddCommMonoid ℕ), Nat.linearOrderedCommSemiring with }

variable {m n k l : ℕ}

/-! ### Equalities and inequalities involving zero and one -/
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4 changes: 4 additions & 0 deletions Mathlib/Data/Rat/NNRat.lean
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Expand Up @@ -233,6 +233,10 @@ instance [AddCommMonoid α] [DistribMulAction ℚ α] : DistribMulAction ℚ≥0
instance [AddCommMonoid α] [Module ℚ α] : Module ℚ≥0 α :=
Module.compHom α coeHom

instance canonicallyOrderedAddCancelCommMonoid : CanonicallyOrderedAddCancelCommMonoid ℚ≥0 :=
{ (inferInstance : CanonicallyOrderedAddCommMonoid ℚ≥0),
(inferInstance : OrderedCancelAddCommMonoid ℚ≥0) with }

@[simp]
theorem coe_coeHom : ⇑coeHom = ((↑) : ℚ≥0 → ℚ) :=
rfl
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4 changes: 4 additions & 0 deletions Mathlib/Data/Real/NNReal.lean
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Expand Up @@ -563,6 +563,10 @@ instance contravariant_add : ContravariantClass ℝ≥0 ℝ≥0 (· + ·) (· <
instance covariant_mul : CovariantClass ℝ≥0 ℝ≥0 (· * ·) (· ≤ ·) := inferInstance
#align nnreal.covariant_mul NNReal.covariant_mul

instance canonicallyOrderedAddCancelCommMonoid : CanonicallyOrderedAddCancelCommMonoid ℝ≥0 :=
{ (inferInstance : CanonicallyOrderedAddCommMonoid ℝ≥0),
(inferInstance : OrderedCancelAddCommMonoid ℝ≥0) with }

-- porting note: TODO: delete?
nonrec theorem le_of_forall_pos_le_add {a b : ℝ≥0} (h : ∀ ε, 0 < ε → a ≤ b + ε) : a ≤ b :=
le_of_forall_pos_le_add h
Expand Down