Canonically ordered monoids

class ExistsMulOfLE (α : Type u) [Mul α] [LE α] : Prop

An OrderedCommMonoid with one-sided 'division' in the sense that if a ≤ b, there is some c for which a * c = b. This is a weaker version of the condition on canonical orderings defined by CanonicallyOrderedCommMonoid.

• exists_mul_of_le : ∀ {a b : α}, a ≤ b → ∃ (c : α), b = a * c

  For a ≤ b, a left divides b

class ExistsAddOfLE (α : Type u) [Add α] [LE α] : Prop

An OrderedAddCommMonoid with one-sided 'subtraction' in the sense that if a ≤ b, then there is some c for which a + c = b. This is a weaker version of the condition on canonical orderings defined by CanonicallyOrderedAddCommMonoid.

• exists_add_of_le : ∀ {a b : α}, a ≤ b → ∃ (c : α), b = a + c

  For a ≤ b, there is a c so b = a + c.

instance AddGroup.existsAddOfLE (α : Type u) [AddGroup α] [LE α] : ExistsAddOfLE α

Equations

• ⋯ = ⋯

instance Group.existsMulOfLE (α : Type u) [Group α] [LE α] : ExistsMulOfLE α

Equations

• ⋯ = ⋯

theorem exists_pos_add_of_lt' {α : Type u} [AddZeroClass α] [Preorder α] [ContravariantClass α α (fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x < x_1] [ExistsAddOfLE α] {a : α} {b : α} (h : a < b) : ∃ (c : α), 0 < c ∧ a + c = b

theorem exists_one_lt_mul_of_lt' {α : Type u} [MulOneClass α] [Preorder α] [ContravariantClass α α (fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x < x_1] [ExistsMulOfLE α] {a : α} {b : α} (h : a < b) : ∃ (c : α), 1 < c ∧ a * c = b

theorem le_of_forall_pos_le_add {α : Type u} [LinearOrder α] [DenselyOrdered α] [AddMonoid α] [ExistsAddOfLE α] [CovariantClass α α (fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x < x_1] [ContravariantClass α α (fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x < x_1] {a : α} {b : α} (h : ∀ (ε : α), 0 < ε → a ≤ b + ε) : a ≤ b

theorem le_of_forall_one_lt_le_mul {α : Type u} [LinearOrder α] [DenselyOrdered α] [Monoid α] [ExistsMulOfLE α] [CovariantClass α α (fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x < x_1] [ContravariantClass α α (fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x < x_1] {a : α} {b : α} (h : ∀ (ε : α), 1 < ε → a ≤ b * ε) : a ≤ b

theorem le_of_forall_pos_lt_add' {α : Type u} [LinearOrder α] [DenselyOrdered α] [AddMonoid α] [ExistsAddOfLE α] [CovariantClass α α (fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x < x_1] [ContravariantClass α α (fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x < x_1] {a : α} {b : α} (h : ∀ (ε : α), 0 < ε → a < b + ε) : a ≤ b

theorem le_of_forall_one_lt_lt_mul' {α : Type u} [LinearOrder α] [DenselyOrdered α] [Monoid α] [ExistsMulOfLE α] [CovariantClass α α (fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x < x_1] [ContravariantClass α α (fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x < x_1] {a : α} {b : α} (h : ∀ (ε : α), 1 < ε → a < b * ε) : a ≤ b

theorem le_iff_forall_pos_lt_add' {α : Type u} [LinearOrder α] [DenselyOrdered α] [AddMonoid α] [ExistsAddOfLE α] [CovariantClass α α (fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x < x_1] [ContravariantClass α α (fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x < x_1] {a : α} {b : α} : a ≤ b ↔ ∀ (ε : α), 0 < ε → a < b + ε

theorem le_iff_forall_one_lt_lt_mul' {α : Type u} [LinearOrder α] [DenselyOrdered α] [Monoid α] [ExistsMulOfLE α] [CovariantClass α α (fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x < x_1] [ContravariantClass α α (fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x < x_1] {a : α} {b : α} : a ≤ b ↔ ∀ (ε : α), 1 < ε → a < b * ε

class CanonicallyOrderedAddCommMonoid (α : Type u_1) extends OrderedAddCommMonoid , OrderBot : Type u_1

A canonically ordered additive 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 the integers or other nontrivial OrderedAddCommGroups.

• add : α → α → α
• 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 : ℕ → α → α
• nsmul_zero : ∀ (x : α), AddMonoid.nsmul 0 x = 0
• nsmul_succ : ∀ (n : ℕ) (x : α), AddMonoid.nsmul (n + 1) x = x + AddMonoid.nsmul n x
• add_comm : ∀ (a b : α), a + b = b + a
• le : α → α → Prop
• lt : α → α → Prop
• le_refl : ∀ (a : α), a ≤ a
• le_trans : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c
• lt_iff_le_not_le : ∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a
• le_antisymm : ∀ (a b : α), a ≤ b → b ≤ a → a = 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

  For a ≤ b, there is a c so b = a + c.

• le_self_add : ∀ (a b : α), a ≤ a + b

  For any a and b, a ≤ a + b

class CanonicallyOrderedCommMonoid (α : Type u_1) extends OrderedCommMonoid , OrderBot : Type u_1

A canonically ordered 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 of a naturally-occurring lattice satisfies this than the lattice itself (for example, dual of the lattice of ideals of a PID or Dedekind domain satisfy this; collections of all things ≤ 1 seem to be more natural that collections of all things ≥ 1).

• mul : α → α → α
• 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
• npow : ℕ → α → α
• npow_zero : ∀ (x : α), Monoid.npow 0 x = 1
• npow_succ : ∀ (n : ℕ) (x : α), Monoid.npow (n + 1) x = x * Monoid.npow n x
• mul_comm : ∀ (a b : α), a * b = b * a
• le : α → α → Prop
• lt : α → α → Prop
• le_refl : ∀ (a : α), a ≤ a
• le_trans : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c
• lt_iff_le_not_le : ∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a
• le_antisymm : ∀ (a b : α), a ≤ b → b ≤ a → a = b
• mul_le_mul_left : ∀ (a b : α), a ≤ b → ∀ (c : α), c * a ≤ c * b
• bot : α
• bot_le : ∀ (a : α), ⊥ ≤ a
• exists_mul_of_le : ∀ {a b : α}, a ≤ b → ∃ (c : α), b = a * c

  For a ≤ b, there is a c so b = a * c.

• le_self_mul : ∀ (a b : α), a ≤ a * b

  For any a and b, a ≤ a * b

instance CanonicallyOrderedAddCommMonoid.existsAddOfLE (α : Type u) [h : CanonicallyOrderedAddCommMonoid α] : ExistsAddOfLE α

Equations

• ⋯ = ⋯

instance CanonicallyOrderedCommMonoid.existsMulOfLE (α : Type u) [h : CanonicallyOrderedCommMonoid α] : ExistsMulOfLE α

Equations

• ⋯ = ⋯

theorem le_self_add {α : Type u} [CanonicallyOrderedAddCommMonoid α] {a : α} {c : α} : a ≤ a + c

theorem le_self_mul {α : Type u} [CanonicallyOrderedCommMonoid α] {a : α} {c : α} : a ≤ a * c

theorem le_add_self {α : Type u} [CanonicallyOrderedAddCommMonoid α] {a : α} {b : α} : a ≤ b + a

theorem le_mul_self {α : Type u} [CanonicallyOrderedCommMonoid α] {a : α} {b : α} : a ≤ b * a

@[simp]

theorem self_le_add_right {α : Type u} [CanonicallyOrderedAddCommMonoid α] (a : α) (b : α) : a ≤ a + b

@[simp]

theorem self_le_mul_right {α : Type u} [CanonicallyOrderedCommMonoid α] (a : α) (b : α) : a ≤ a * b

@[simp]

theorem self_le_add_left {α : Type u} [CanonicallyOrderedAddCommMonoid α] (a : α) (b : α) : a ≤ b + a

@[simp]

theorem self_le_mul_left {α : Type u} [CanonicallyOrderedCommMonoid α] (a : α) (b : α) : a ≤ b * a

theorem le_of_add_le_left {α : Type u} [CanonicallyOrderedAddCommMonoid α] {a : α} {b : α} {c : α} : a + b ≤ c → a ≤ c

theorem le_of_mul_le_left {α : Type u} [CanonicallyOrderedCommMonoid α] {a : α} {b : α} {c : α} : a * b ≤ c → a ≤ c

theorem le_of_add_le_right {α : Type u} [CanonicallyOrderedAddCommMonoid α] {a : α} {b : α} {c : α} : a + b ≤ c → b ≤ c

theorem le_of_mul_le_right {α : Type u} [CanonicallyOrderedCommMonoid α] {a : α} {b : α} {c : α} : a * b ≤ c → b ≤ c

theorem le_add_of_le_left {α : Type u} [CanonicallyOrderedAddCommMonoid α] {a : α} {b : α} {c : α} : a ≤ b → a ≤ b + c

theorem le_mul_of_le_left {α : Type u} [CanonicallyOrderedCommMonoid α] {a : α} {b : α} {c : α} : a ≤ b → a ≤ b * c

theorem le_add_of_le_right {α : Type u} [CanonicallyOrderedAddCommMonoid α] {a : α} {b : α} {c : α} : a ≤ c → a ≤ b + c

theorem le_mul_of_le_right {α : Type u} [CanonicallyOrderedCommMonoid α] {a : α} {b : α} {c : α} : a ≤ c → a ≤ b * c

theorem le_iff_exists_add {α : Type u} [CanonicallyOrderedAddCommMonoid α] {a : α} {b : α} : a ≤ b ↔ ∃ (c : α), b = a + c

theorem le_iff_exists_mul {α : Type u} [CanonicallyOrderedCommMonoid α] {a : α} {b : α} : a ≤ b ↔ ∃ (c : α), b = a * c

theorem le_iff_exists_add' {α : Type u} [CanonicallyOrderedAddCommMonoid α] {a : α} {b : α} : a ≤ b ↔ ∃ (c : α), b = c + a

theorem le_iff_exists_mul' {α : Type u} [CanonicallyOrderedCommMonoid α] {a : α} {b : α} : a ≤ b ↔ ∃ (c : α), b = c * a

@[simp]

theorem zero_le {α : Type u} [CanonicallyOrderedAddCommMonoid α] (a : α) : 0 ≤ a

@[simp]

theorem one_le {α : Type u} [CanonicallyOrderedCommMonoid α] (a : α) : 1 ≤ a

theorem bot_eq_zero {α : Type u} [CanonicallyOrderedAddCommMonoid α] : ⊥ = 0

theorem bot_eq_one {α : Type u} [CanonicallyOrderedCommMonoid α] : ⊥ = 1

@[simp]

theorem add_eq_zero_iff {α : Type u} [CanonicallyOrderedAddCommMonoid α] {a : α} {b : α} : a + b = 0 ↔ a = 0 ∧ b = 0

@[simp]

theorem mul_eq_one_iff {α : Type u} [CanonicallyOrderedCommMonoid α] {a : α} {b : α} : a * b = 1 ↔ a = 1 ∧ b = 1

@[simp]

theorem nonpos_iff_eq_zero {α : Type u} [CanonicallyOrderedAddCommMonoid α] {a : α} : a ≤ 0 ↔ a = 0

@[simp]

theorem le_one_iff_eq_one {α : Type u} [CanonicallyOrderedCommMonoid α] {a : α} : a ≤ 1 ↔ a = 1

theorem pos_iff_ne_zero {α : Type u} [CanonicallyOrderedAddCommMonoid α] {a : α} : 0 < a ↔ a ≠ 0

theorem one_lt_iff_ne_one {α : Type u} [CanonicallyOrderedCommMonoid α] {a : α} : 1 < a ↔ a ≠ 1

theorem eq_zero_or_pos {α : Type u} [CanonicallyOrderedAddCommMonoid α] (a : α) : a = 0 ∨ 0 < a

theorem eq_one_or_one_lt {α : Type u} [CanonicallyOrderedCommMonoid α] (a : α) : a = 1 ∨ 1 < a

@[simp]

theorem add_pos_iff {α : Type u} [CanonicallyOrderedAddCommMonoid α] {a : α} {b : α} : 0 < a + b ↔ 0 < a ∨ 0 < b

@[simp]

theorem one_lt_mul_iff {α : Type u} [CanonicallyOrderedCommMonoid α] {a : α} {b : α} : 1 < a * b ↔ 1 < a ∨ 1 < b

theorem exists_pos_add_of_lt {α : Type u} [CanonicallyOrderedAddCommMonoid α] {a : α} {b : α} (h : a < b) : ∃ (c : α), ∃ (x : 0 < c), a + c = b

theorem exists_one_lt_mul_of_lt {α : Type u} [CanonicallyOrderedCommMonoid α] {a : α} {b : α} (h : a < b) : ∃ (c : α), ∃ (x : 1 < c), a * c = b

theorem le_add_left {α : Type u} [CanonicallyOrderedAddCommMonoid α] {a : α} {b : α} {c : α} (h : a ≤ c) : a ≤ b + c

theorem le_mul_left {α : Type u} [CanonicallyOrderedCommMonoid α] {a : α} {b : α} {c : α} (h : a ≤ c) : a ≤ b * c

theorem le_add_right {α : Type u} [CanonicallyOrderedAddCommMonoid α] {a : α} {b : α} {c : α} (h : a ≤ b) : a ≤ b + c

theorem le_mul_right {α : Type u} [CanonicallyOrderedCommMonoid α] {a : α} {b : α} {c : α} (h : a ≤ b) : a ≤ b * c

theorem lt_iff_exists_add {α : Type u} [CanonicallyOrderedAddCommMonoid α] {a : α} {b : α} [CovariantClass α α (fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x < x_1] : a < b ↔ ∃ (c : α), c > 0 ∧ b = a + c

theorem lt_iff_exists_mul {α : Type u} [CanonicallyOrderedCommMonoid α] {a : α} {b : α} [CovariantClass α α (fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x < x_1] : a < b ↔ ∃ (c : α), c > 1 ∧ b = a * c

theorem pos_of_gt {M : Type u_1} [CanonicallyOrderedAddCommMonoid M] {n : M} {m : M} (h : n < m) : 0 < m

theorem NeZero.pos {M : Type u_1} (a : M) [CanonicallyOrderedAddCommMonoid M] [NeZero a] : 0 < a

theorem NeZero.of_gt {M : Type u_1} [CanonicallyOrderedAddCommMonoid M] {x : M} {y : M} (h : x < y) : NeZero y

instance NeZero.of_gt' {M : Type u_1} [CanonicallyOrderedAddCommMonoid M] [One M] {y : M} [Fact (1 < y)] : NeZero y

Equations

• ⋯ = ⋯

instance NeZero.bit0 {M : Type u_1} [CanonicallyOrderedAddCommMonoid M] {x : M} [NeZero x] : NeZero (bit0 x)

Equations

• ⋯ = ⋯

class CanonicallyLinearOrderedAddCommMonoid (α : Type u_1) extends CanonicallyOrderedAddCommMonoid , Min , Max , Ord : Type u_1

A canonically linear-ordered additive monoid is a canonically ordered additive monoid whose ordering is a linear order.

• add : α → α → α
• 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 : ℕ → α → α
• nsmul_zero : ∀ (x : α), AddMonoid.nsmul 0 x = 0
• nsmul_succ : ∀ (n : ℕ) (x : α), AddMonoid.nsmul (n + 1) x = x + AddMonoid.nsmul n x
• add_comm : ∀ (a b : α), a + b = b + a
• le : α → α → Prop
• lt : α → α → Prop
• le_refl : ∀ (a : α), a ≤ a
• le_trans : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c
• lt_iff_le_not_le : ∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a
• le_antisymm : ∀ (a b : α), a ≤ b → b ≤ a → a = 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
• min : α → α → α
• max : α → α → α
• compare : α → α → Ordering
• le_total : ∀ (a b : α), a ≤ b ∨ b ≤ a

  A linear order is total.

• decidableLE : DecidableRel fun (x x_1 : α) => x ≤ x_1

  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 (x x_1 : α) => x < x_1

  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 =.

class CanonicallyLinearOrderedCommMonoid (α : Type u_1) extends CanonicallyOrderedCommMonoid , Min , Max , Ord : Type u_1

A canonically linear-ordered monoid is a canonically ordered monoid whose ordering is a linear order.

• mul : α → α → α
• 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
• npow : ℕ → α → α
• npow_zero : ∀ (x : α), Monoid.npow 0 x = 1
• npow_succ : ∀ (n : ℕ) (x : α), Monoid.npow (n + 1) x = x * Monoid.npow n x
• mul_comm : ∀ (a b : α), a * b = b * a
• le : α → α → Prop
• lt : α → α → Prop
• le_refl : ∀ (a : α), a ≤ a
• le_trans : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c
• lt_iff_le_not_le : ∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a
• le_antisymm : ∀ (a b : α), a ≤ b → b ≤ a → a = b
• mul_le_mul_left : ∀ (a b : α), a ≤ b → ∀ (c : α), c * a ≤ c * b
• bot : α
• bot_le : ∀ (a : α), ⊥ ≤ a
• exists_mul_of_le : ∀ {a b : α}, a ≤ b → ∃ (c : α), b = a * c
• le_self_mul : ∀ (a b : α), a ≤ a * b
• min : α → α → α
• max : α → α → α
• compare : α → α → Ordering
• le_total : ∀ (a b : α), a ≤ b ∨ b ≤ a

  A linear order is total.

• decidableLE : DecidableRel fun (x x_1 : α) => x ≤ x_1

  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 (x x_1 : α) => x < x_1

  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 =.

instance CanonicallyLinearOrderedAddCommMonoid.semilatticeSup {α : Type u} [CanonicallyLinearOrderedAddCommMonoid α] : SemilatticeSup α

Equations

• CanonicallyLinearOrderedAddCommMonoid.semilatticeSup = let __src := LinearOrder.toLattice; SemilatticeSup.mk ⋯ ⋯ ⋯

theorem CanonicallyLinearOrderedAddCommMonoid.semilatticeSup.proof_3 {α : Type u_1} [CanonicallyLinearOrderedAddCommMonoid α] (a : α) (b : α) (c : α) : a ≤ c → b ≤ c → a ⊔ b ≤ c

theorem CanonicallyLinearOrderedAddCommMonoid.semilatticeSup.proof_1 {α : Type u_1} [CanonicallyLinearOrderedAddCommMonoid α] (a : α) (b : α) : a ≤ a ⊔ b

theorem CanonicallyLinearOrderedAddCommMonoid.semilatticeSup.proof_2 {α : Type u_1} [CanonicallyLinearOrderedAddCommMonoid α] (a : α) (b : α) : b ≤ a ⊔ b

instance CanonicallyLinearOrderedCommMonoid.semilatticeSup {α : Type u} [CanonicallyLinearOrderedCommMonoid α] : SemilatticeSup α

Equations

• CanonicallyLinearOrderedCommMonoid.semilatticeSup = let __src := LinearOrder.toLattice; SemilatticeSup.mk ⋯ ⋯ ⋯

theorem min_add_distrib {α : Type u} [CanonicallyLinearOrderedAddCommMonoid α] (a : α) (b : α) (c : α) : min a (b + c) = min a (min a b + min a c)

theorem min_mul_distrib {α : Type u} [CanonicallyLinearOrderedCommMonoid α] (a : α) (b : α) (c : α) : min a (b * c) = min a (min a b * min a c)

theorem min_add_distrib' {α : Type u} [CanonicallyLinearOrderedAddCommMonoid α] (a : α) (b : α) (c : α) : min (a + b) c = min (min a c + min b c) c

theorem min_mul_distrib' {α : Type u} [CanonicallyLinearOrderedCommMonoid α] (a : α) (b : α) (c : α) : min (a * b) c = min (min a c * min b c) c

theorem zero_min {α : Type u} [CanonicallyLinearOrderedAddCommMonoid α] (a : α) : min 0 a = 0

theorem one_min {α : Type u} [CanonicallyLinearOrderedCommMonoid α] (a : α) : min 1 a = 1

theorem min_zero {α : Type u} [CanonicallyLinearOrderedAddCommMonoid α] (a : α) : min a 0 = 0

theorem min_one {α : Type u} [CanonicallyLinearOrderedCommMonoid α] (a : α) : min a 1 = 1

@[simp]

theorem bot_eq_zero' {α : Type u} [CanonicallyLinearOrderedAddCommMonoid α] : ⊥ = 0

In a linearly ordered monoid, we are happy for bot_eq_zero to be a @[simp] lemma

@[simp]

theorem bot_eq_one' {α : Type u} [CanonicallyLinearOrderedCommMonoid α] : ⊥ = 1

In a linearly ordered monoid, we are happy for bot_eq_one to be a @[simp] lemma.