File: Mathlib/Algebra/Order/Group/PosPart.lean

full_name: oneLePart
code: /-- The *positive part* of an element `a` in a lattice ordered group is `a ⊔ 1`, denoted `a⁺ᵐ`. -/
@[to_additive
"The *positive part* of an element `a` in a lattice ordered group is `a ⊔ 0`, denoted `a⁺`."]
def oneLePart (a : α) : α 
goal: α

full_name: posPart
code: /-- The *positive part* of an element `a` in a lattice ordered group is `a ⊔ 1`, denoted `a⁺ᵐ`. -/
@[to_additive
"The *positive part* of an element `a` in a lattice ordered group is `a ⊔ 0`, denoted `a⁺`."]
def oneLePart (a : α) : α 
goal: α

full_name: negPart
code: /-- The *negative part* of an element `a` in a lattice ordered group is `a⁻¹ ⊔ 1`, denoted `a⁻ᵐ `.
-/
@[to_additive
"The *negative part* of an element `a` in a lattice ordered group is `(-a) ⊔ 0`, denoted `a⁻`."]
def leOnePart (a : α) : α 
goal: α

full_name: leOnePart
code: /-- The *negative part* of an element `a` in a lattice ordered group is `a⁻¹ ⊔ 1`, denoted `a⁻ᵐ `.
-/
@[to_additive
"The *negative part* of an element `a` in a lattice ordered group is `(-a) ⊔ 0`, denoted `a⁻`."]
def leOnePart (a : α) : α 
goal: α

full_name: «_aux_Mathlib_Algebra_Order_Group_PosPart___macroRules_term_⁺ᵐ_1»
code: @[inherit_doc] postfix:max "⁺ᵐ " => oneLePart
goal: Lean.Macro

full_name: _aux_Mathlib_Algebra_Order_Group_PosPart___unexpand_oneLePart_1
code: @[inherit_doc] postfix:max "⁺ᵐ " => oneLePart
goal: Lean.PrettyPrinter.Unexpander

full_name: «_aux_Mathlib_Algebra_Order_Group_PosPart___macroRules_term_⁻ᵐ_1»
code: @[inherit_doc] postfix:max "⁻ᵐ" => leOnePart
goal: Lean.Macro

full_name: _aux_Mathlib_Algebra_Order_Group_PosPart___unexpand_leOnePart_1
code: @[inherit_doc] postfix:max "⁻ᵐ" => leOnePart
goal: Lean.PrettyPrinter.Unexpander

full_name: _aux_Mathlib_Algebra_Order_Group_PosPart___unexpand_posPart_1
code: @[inherit_doc] postfix:max "⁺" => posPart
goal: Lean.PrettyPrinter.Unexpander

full_name: «_aux_Mathlib_Algebra_Order_Group_PosPart___macroRules_term_⁺_1»
code: @[inherit_doc] postfix:max "⁺" => posPart
goal: Lean.Macro

full_name: _aux_Mathlib_Algebra_Order_Group_PosPart___unexpand_negPart_1
code: @[inherit_doc] postfix:max "⁻" => negPart
goal: Lean.PrettyPrinter.Unexpander

full_name: «_aux_Mathlib_Algebra_Order_Group_PosPart___macroRules_term_⁻_1»
code: @[inherit_doc] postfix:max "⁻" => negPart
goal: Lean.Macro

full_name: oneLePart_mono
code: @[to_additive] lemma oneLePart_mono : Monotone (oneLePart : α → α) 
goal: Monotone oneLePart

full_name: posPart_mono
code: @[to_additive] lemma oneLePart_mono : Monotone (oneLePart : α → α) 
goal: Monotone posPart

full_name: oneLePart_one
code: @[to_additive (attr := simp)] lemma oneLePart_one : (1 : α)⁺ᵐ = 1 
goal: 1⁺ᵐ = 1

full_name: posPart_zero
code: @[to_additive (attr := simp)] lemma oneLePart_one : (1 : α)⁺ᵐ = 1 
goal: 0⁺ = 0

full_name: negPart_zero
code: @[to_additive (attr := simp)] lemma leOnePart_one : (1 : α)⁻ᵐ = 1 
goal: 0⁻ = 0

full_name: leOnePart_one
code: @[to_additive (attr := simp)] lemma leOnePart_one : (1 : α)⁻ᵐ = 1 
goal: 1⁻ᵐ = 1

full_name: one_le_oneLePart
code: @[to_additive posPart_nonneg] lemma one_le_oneLePart (a : α) : 1 ≤ a⁺ᵐ 
goal: 1 ≤ a⁺ᵐ

full_name: posPart_nonneg
code: @[to_additive posPart_nonneg] lemma one_le_oneLePart (a : α) : 1 ≤ a⁺ᵐ 
goal: 0 ≤ a⁺

full_name: one_le_leOnePart
code: @[to_additive negPart_nonneg] lemma one_le_leOnePart (a : α) : 1 ≤ a⁻ᵐ 
goal: 1 ≤ a⁻ᵐ

full_name: negPart_nonneg
code: @[to_additive negPart_nonneg] lemma one_le_leOnePart (a : α) : 1 ≤ a⁻ᵐ 
goal: 0 ≤ a⁻

full_name: le_posPart
code: @[to_additive le_posPart] lemma le_oneLePart (a : α) : a ≤ a⁺ᵐ 
goal: a ≤ a⁺

full_name: le_oneLePart
code: @[to_additive le_posPart] lemma le_oneLePart (a : α) : a ≤ a⁺ᵐ 
goal: a ≤ a⁺ᵐ

full_name: neg_le_negPart
code: @[to_additive] lemma inv_le_leOnePart (a : α) : a⁻¹ ≤ a⁻ᵐ 
goal: -a ≤ a⁻

full_name: inv_le_leOnePart
code: @[to_additive] lemma inv_le_leOnePart (a : α) : a⁻¹ ≤ a⁻ᵐ 
goal: a⁻¹ ≤ a⁻ᵐ

full_name: oneLePart_eq_self
code: @[to_additive (attr := simp)] lemma oneLePart_eq_self : a⁺ᵐ = a ↔ 1 ≤ a 
goal: a⁺ᵐ = a ↔ 1 ≤ a

full_name: posPart_eq_self
code: @[to_additive (attr := simp)] lemma oneLePart_eq_self : a⁺ᵐ = a ↔ 1 ≤ a 
goal: a⁺ = a ↔ 0 ≤ a

full_name: oneLePart_eq_one
code: @[to_additive] lemma oneLePart_eq_one : a⁺ᵐ = 1 ↔ a ≤ 1 
goal: a⁺ᵐ = 1 ↔ a ≤ 1

full_name: posPart_eq_zero
code: @[to_additive] lemma oneLePart_eq_one : a⁺ᵐ = 1 ↔ a ≤ 1 
goal: a⁺ = 0 ↔ a ≤ 0

full_name: leOnePart_eq_inv'
code: /-- See also `leOnePart_eq_inv`. -/
@[to_additive "See also `negPart_eq_neg`."]
lemma leOnePart_eq_inv' : a⁻ᵐ = a⁻¹ ↔ 1 ≤ a⁻¹ 
goal: a⁻ᵐ = a⁻¹ ↔ 1 ≤ a⁻¹

full_name: negPart_eq_neg'
code: /-- See also `leOnePart_eq_inv`. -/
@[to_additive "See also `negPart_eq_neg`."]
lemma leOnePart_eq_inv' : a⁻ᵐ = a⁻¹ ↔ 1 ≤ a⁻¹ 
goal: a⁻ = -a ↔ 0 ≤ -a

full_name: leOnePart_eq_one'
code: /-- See also `leOnePart_eq_one`. -/
@[to_additive "See also `negPart_eq_zero`."]
lemma leOnePart_eq_one' : a⁻ᵐ = 1 ↔ a⁻¹ ≤ 1 
goal: a⁻ᵐ = 1 ↔ a⁻¹ ≤ 1

full_name: negPart_eq_zero'
code: /-- See also `leOnePart_eq_one`. -/
@[to_additive "See also `negPart_eq_zero`."]
lemma leOnePart_eq_one' : a⁻ᵐ = 1 ↔ a⁻¹ ≤ 1 
goal: a⁻ = 0 ↔ -a ≤ 0

full_name: oneLePart_le_one
code: @[to_additive] lemma oneLePart_le_one : a⁺ᵐ ≤ 1 ↔ a ≤ 1 
goal: a⁺ᵐ ≤ 1 ↔ a ≤ 1

full_name: posPart_nonpos
code: @[to_additive] lemma oneLePart_le_one : a⁺ᵐ ≤ 1 ↔ a ≤ 1 
goal: a⁺ ≤ 0 ↔ a ≤ 0

full_name: leOnePart_le_one'
code: /-- See also `leOnePart_le_one`. -/
@[to_additive "See also `negPart_nonpos`."]
lemma leOnePart_le_one' : a⁻ᵐ ≤ 1 ↔ a⁻¹ ≤ 1 
goal: a⁻ᵐ ≤ 1 ↔ a⁻¹ ≤ 1

full_name: negPart_nonpos'
code: /-- See also `leOnePart_le_one`. -/
@[to_additive "See also `negPart_nonpos`."]
lemma leOnePart_le_one' : a⁻ᵐ ≤ 1 ↔ a⁻¹ ≤ 1 
goal: a⁻ ≤ 0 ↔ -a ≤ 0

full_name: leOnePart_le_one
code: @[to_additive] lemma leOnePart_le_one : a⁻ᵐ ≤ 1 ↔ a⁻¹ ≤ 1 
goal: a⁻ᵐ ≤ 1 ↔ a⁻¹ ≤ 1

full_name: negPart_nonpos
code: @[to_additive] lemma leOnePart_le_one : a⁻ᵐ ≤ 1 ↔ a⁻¹ ≤ 1 
goal: a⁻ ≤ 0 ↔ -a ≤ 0

full_name: posPart_pos
code: @[to_additive (attr := simp) posPart_pos] lemma one_lt_oneLePart (ha : 1 < a) : 1 < a⁺ᵐ 
goal: 0 < a⁺

full_name: one_lt_oneLePart
code: @[to_additive (attr := simp) posPart_pos] lemma one_lt_oneLePart (ha : 1 < a) : 1 < a⁺ᵐ 
goal: 1 < a⁺ᵐ

full_name: oneLePart_inv
code: @[to_additive (attr := simp)] lemma oneLePart_inv (a : α) : a⁻¹⁺ᵐ = a⁻ᵐ 
goal: a⁻¹⁺ᵐ = a⁻ᵐ

full_name: posPart_neg
code: @[to_additive (attr := simp)] lemma oneLePart_inv (a : α) : a⁻¹⁺ᵐ = a⁻ᵐ 
goal: (-a)⁺ = a⁻

full_name: negPart_neg
code: @[to_additive (attr := simp)] lemma leOnePart_inv (a : α) : a⁻¹⁻ᵐ = a⁺ᵐ 
goal: (-a)⁻ = a⁺

full_name: leOnePart_inv
code: @[to_additive (attr := simp)] lemma leOnePart_inv (a : α) : a⁻¹⁻ᵐ = a⁺ᵐ 
goal: a⁻¹⁻ᵐ = a⁺ᵐ

full_name: leOnePart_eq_inv
code: @[to_additive (attr := simp)] lemma leOnePart_eq_inv : a⁻ᵐ = a⁻¹ ↔ a ≤ 1 
goal: a⁻ᵐ = a⁻¹ ↔ a ≤ 1

full_name: negPart_eq_neg
code: @[to_additive (attr := simp)] lemma leOnePart_eq_inv : a⁻ᵐ = a⁻¹ ↔ a ≤ 1 
goal: a⁻ = -a ↔ a ≤ 0

full_name: negPart_eq_zero
code: @[to_additive (attr := simp)]
lemma leOnePart_eq_one : a⁻ᵐ = 1 ↔ 1 ≤ a 
goal: a⁻ = 0 ↔ 0 ≤ a

full_name: leOnePart_eq_one
code: @[to_additive (attr := simp)]
lemma leOnePart_eq_one : a⁻ᵐ = 1 ↔ 1 ≤ a 
goal: a⁻ᵐ = 1 ↔ 1 ≤ a

full_name: negPart_pos
code: @[to_additive (attr := simp) negPart_pos] lemma one_lt_ltOnePart (ha : a < 1) : 1 < a⁻ᵐ 
goal: 0 < a⁻

full_name: one_lt_ltOnePart
code: @[to_additive (attr := simp) negPart_pos] lemma one_lt_ltOnePart (ha : a < 1) : 1 < a⁻ᵐ 
goal: 1 < a⁻ᵐ

full_name: posPart_sub_negPart
code: @[to_additive (attr := simp)] lemma oneLePart_div_leOnePart (a : α) : a⁺ᵐ / a⁻ᵐ = a 
goal: a⁺ - a⁻ = a

full_name: oneLePart_div_leOnePart
code: @[to_additive (attr := simp)] lemma oneLePart_div_leOnePart (a : α) : a⁺ᵐ / a⁻ᵐ = a 
goal: a⁺ᵐ / a⁻ᵐ = a

full_name: leOnePart_div_oneLePart
code: @[to_additive (attr := simp)] lemma leOnePart_div_oneLePart (a : α) : a⁻ᵐ / a⁺ᵐ = a⁻¹ 
goal: a⁻ᵐ / a⁺ᵐ = a⁻¹

full_name: negPart_sub_posPart
code: @[to_additive (attr := simp)] lemma leOnePart_div_oneLePart (a : α) : a⁻ᵐ / a⁺ᵐ = a⁻¹ 
goal: a⁻ - a⁺ = -a

full_name: negPart_anti
code: @[to_additive] lemma leOnePart_anti : Antitone (leOnePart : α → α) 
goal: Antitone negPart

full_name: leOnePart_anti
code: @[to_additive] lemma leOnePart_anti : Antitone (leOnePart : α → α) 
goal: Antitone leOnePart

full_name: negPart_eq_neg_inf_zero
code: @[to_additive]
lemma leOnePart_eq_inv_inf_one (a : α) : a⁻ᵐ = (a ⊓ 1)⁻¹ 
goal: a⁻ = -(a ⊓ 0)

full_name: leOnePart_eq_inv_inf_one
code: @[to_additive]
lemma leOnePart_eq_inv_inf_one (a : α) : a⁻ᵐ = (a ⊓ 1)⁻¹ 
goal: a⁻ᵐ = (a ⊓ 1)⁻¹

full_name: posPart_add_negPart
code: @[to_additive] lemma oneLePart_mul_leOnePart (a : α) : a⁺ᵐ * a⁻ᵐ = |a|ₘ 
goal: a⁺ + a⁻ = |a|

full_name: oneLePart_mul_leOnePart
code: @[to_additive] lemma oneLePart_mul_leOnePart (a : α) : a⁺ᵐ * a⁻ᵐ = |a|ₘ 
goal: a⁺ᵐ * a⁻ᵐ = mabs a

full_name: leOnePart_mul_oneLePart
code: @[to_additive] lemma leOnePart_mul_oneLePart (a : α) : a⁻ᵐ * a⁺ᵐ = |a|ₘ 
goal: a⁻ᵐ * a⁺ᵐ = mabs a

full_name: negPart_add_posPart
code: @[to_additive] lemma leOnePart_mul_oneLePart (a : α) : a⁻ᵐ * a⁺ᵐ = |a|ₘ 
goal: a⁻ + a⁺ = |a|

full_name: oneLePart_inf_leOnePart_eq_one
code: @[to_additive] lemma oneLePart_inf_leOnePart_eq_one (a : α) : a⁺ᵐ ⊓ a⁻ᵐ = 1 
goal: a⁺ᵐ ⊓ a⁻ᵐ = 1

full_name: posPart_inf_negPart_eq_zero
code: @[to_additive] lemma oneLePart_inf_leOnePart_eq_one (a : α) : a⁺ᵐ ⊓ a⁻ᵐ = 1 
goal: a⁺ ⊓ a⁻ = 0

full_name: sup_eq_mul_oneLePart_div
code: @[to_additive] lemma sup_eq_mul_oneLePart_div (a b : α) : a ⊔ b = b * (a / b)⁺ᵐ 
goal: a ⊔ b = b * (a / b)⁺ᵐ

full_name: sup_eq_add_posPart_sub
code: @[to_additive] lemma sup_eq_mul_oneLePart_div (a b : α) : a ⊔ b = b * (a / b)⁺ᵐ 
goal: a ⊔ b = b + (a - b)⁺

full_name: inf_eq_sub_posPart_sub
code: @[to_additive] lemma inf_eq_div_oneLePart_div (a b : α) : a ⊓ b = a / (a / b)⁺ᵐ 
goal: a ⊓ b = a - (a - b)⁺

full_name: inf_eq_div_oneLePart_div
code: @[to_additive] lemma inf_eq_div_oneLePart_div (a b : α) : a ⊓ b = a / (a / b)⁺ᵐ 
goal: a ⊓ b = a / (a / b)⁺ᵐ

full_name: le_iff_oneLePart_leOnePart
code: @[to_additive] lemma le_iff_oneLePart_leOnePart (a b : α) : a ≤ b ↔ a⁺ᵐ ≤ b⁺ᵐ ∧ b⁻ᵐ ≤ a⁻ᵐ 
goal: a ≤ b ↔ a⁺ᵐ ≤ b⁺ᵐ ∧ b⁻ᵐ ≤ a⁻ᵐ

full_name: le_iff_posPart_negPart
code: @[to_additive] lemma le_iff_oneLePart_leOnePart (a b : α) : a ≤ b ↔ a⁺ᵐ ≤ b⁺ᵐ ∧ b⁻ᵐ ≤ a⁻ᵐ 
goal: a ≤ b ↔ a⁺ ≤ b⁺ ∧ b⁻ ≤ a⁻

full_name: mabs_mul_eq_oneLePart_sq
code: @[to_additive abs_add_eq_two_nsmul_posPart]
lemma mabs_mul_eq_oneLePart_sq (a : α) : |a|ₘ * a = a⁺ᵐ ^ 2 
goal: mabs a * a = a⁺ᵐ ^ 2

full_name: abs_add_eq_two_nsmul_posPart
code: @[to_additive abs_add_eq_two_nsmul_posPart]
lemma mabs_mul_eq_oneLePart_sq (a : α) : |a|ₘ * a = a⁺ᵐ ^ 2 
goal: |a| + a = 2 • a⁺

full_name: add_abs_eq_two_nsmul_posPart
code: @[to_additive add_abs_eq_two_nsmul_posPart]
lemma mul_mabs_eq_oneLePart_sq (a : α) : a * |a|ₘ = a⁺ᵐ ^ 2 
goal: a + |a| = 2 • a⁺

full_name: mul_mabs_eq_oneLePart_sq
code: @[to_additive add_abs_eq_two_nsmul_posPart]
lemma mul_mabs_eq_oneLePart_sq (a : α) : a * |a|ₘ = a⁺ᵐ ^ 2 
goal: a * mabs a = a⁺ᵐ ^ 2

full_name: mabs_div_eq_leOnePart_sq
code: @[to_additive abs_sub_eq_two_nsmul_negPart]
lemma mabs_div_eq_leOnePart_sq (a : α) : |a|ₘ / a = a⁻ᵐ ^ 2 
goal: mabs a / a = a⁻ᵐ ^ 2

full_name: abs_sub_eq_two_nsmul_negPart
code: @[to_additive abs_sub_eq_two_nsmul_negPart]
lemma mabs_div_eq_leOnePart_sq (a : α) : |a|ₘ / a = a⁻ᵐ ^ 2 
goal: |a| - a = 2 • a⁻

full_name: div_mabs_eq_inv_leOnePart_sq
code: @[to_additive sub_abs_eq_neg_two_nsmul_negPart]
lemma div_mabs_eq_inv_leOnePart_sq (a : α) : a / |a|ₘ = (a⁻ᵐ ^ 2)⁻¹ 
goal: a / mabs a = (a⁻ᵐ ^ 2)⁻¹

full_name: sub_abs_eq_neg_two_nsmul_negPart
code: @[to_additive sub_abs_eq_neg_two_nsmul_negPart]
lemma div_mabs_eq_inv_leOnePart_sq (a : α) : a / |a|ₘ = (a⁻ᵐ ^ 2)⁻¹ 
goal: a - |a| = -(2 • a⁻)

full_name: posPart_eq_ite
code: @[to_additive] lemma oneLePart_eq_ite : a⁺ᵐ = if 1 ≤ a then a else 1 
goal: a⁺ = if 0 ≤ a then a else 0

full_name: oneLePart_eq_ite
code: @[to_additive] lemma oneLePart_eq_ite : a⁺ᵐ = if 1 ≤ a then a else 1 
goal: a⁺ᵐ = if 1 ≤ a then a else 1

full_name: one_lt_oneLePart_iff
code: @[to_additive (attr := simp) posPart_pos_iff] lemma one_lt_oneLePart_iff : 1 < a⁺ᵐ ↔ 1 < a 
goal: 1 < a⁺ᵐ ↔ 1 < a

full_name: posPart_pos_iff
code: @[to_additive (attr := simp) posPart_pos_iff] lemma one_lt_oneLePart_iff : 1 < a⁺ᵐ ↔ 1 < a 
goal: 0 < a⁺ ↔ 0 < a

full_name: posPart_eq_of_posPart_pos
code: @[to_additive posPart_eq_of_posPart_pos]
lemma oneLePart_of_one_lt_oneLePart (ha : 1 < a⁺ᵐ) : a⁺ᵐ = a 
goal: a⁺ = a

full_name: oneLePart_of_one_lt_oneLePart
code: @[to_additive posPart_eq_of_posPart_pos]
lemma oneLePart_of_one_lt_oneLePart (ha : 1 < a⁺ᵐ) : a⁺ᵐ = a 
goal: a⁺ᵐ = a

full_name: leOnePart_eq_ite
code: @[to_additive] lemma leOnePart_eq_ite : a⁻ᵐ = if a ≤ 1 then a⁻¹ else 1 
goal: a⁻ᵐ = if a ≤ 1 then a⁻¹ else 1

full_name: negPart_eq_ite
code: @[to_additive] lemma leOnePart_eq_ite : a⁻ᵐ = if a ≤ 1 then a⁻¹ else 1 
goal: a⁻ = if a ≤ 0 then -a else 0

full_name: negPart_pos_iff
code: @[to_additive (attr := simp) negPart_pos_iff] lemma one_lt_ltOnePart_iff : 1 < a⁻ᵐ ↔ a < 1 
goal: 0 < a⁻ ↔ a < 0

full_name: one_lt_ltOnePart_iff
code: @[to_additive (attr := simp) negPart_pos_iff] lemma one_lt_ltOnePart_iff : 1 < a⁻ᵐ ↔ a < 1 
goal: 1 < a⁻ᵐ ↔ a < 1

full_name: Pi.posPart_apply
code: @[to_additive (attr := simp)] lemma oneLePart_apply (f : ∀ i, α i) (i : ι) : f⁺ i = (f i)⁺ 
goal: f⁺ i = (f i)⁺

full_name: Pi.oneLePart_apply
code: @[to_additive (attr := simp)] lemma oneLePart_apply (f : ∀ i, α i) (i : ι) : f⁺ i = (f i)⁺ 
goal: f⁺ i = (f i)⁺

full_name: Pi.leOnePart_apply
code: @[to_additive (attr := simp)] lemma leOnePart_apply (f : ∀ i, α i) (i : ι) : f⁻ i = (f i)⁻ 
goal: f⁻ i = (f i)⁻

full_name: Pi.negPart_apply
code: @[to_additive (attr := simp)] lemma leOnePart_apply (f : ∀ i, α i) (i : ι) : f⁻ i = (f i)⁻ 
goal: f⁻ i = (f i)⁻

full_name: Pi.posPart_def
code: @[to_additive] lemma oneLePart_def (f : ∀ i, α i) : f⁺ = fun i ↦ (f i)⁺ 
goal: f⁺ = fun i => (f i)⁺

full_name: Pi.oneLePart_def
code: @[to_additive] lemma oneLePart_def (f : ∀ i, α i) : f⁺ = fun i ↦ (f i)⁺ 
goal: f⁺ = fun i => (f i)⁺

full_name: Pi.leOnePart_def
code: @[to_additive] lemma leOnePart_def (f : ∀ i, α i) : f⁻ = fun i ↦ (f i)⁻ 
goal: f⁻ = fun i => (f i)⁻

full_name: Pi.negPart_def
code: @[to_additive] lemma leOnePart_def (f : ∀ i, α i) : f⁻ = fun i ↦ (f i)⁻ 
goal: f⁻ = fun i => (f i)⁻