awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.GroupTheory.ArchimedeanDensely (goal)
Goal:
G : Type u_1
inst : CommGroup G
inst_1 : LinearOrder G
inst_2 : IsOrderedMonoid G
inst_3 : MulArchimedean G
e : G ≃o Additive G := OrderIso.refl G
rw : Nonempty.{max 1 (u_1 + 1)} (Additive G ≃+o ℤ) ↔ ¬DenselyOrdered G
rw₁ : ¬DenselyOrdered.{u_1} (Additive G) ↔ ¬DenselyOrdered G
⊢ Nonempty.{max 1 (u_1 + 1)} (G ≃*o Multiplicative ℤ) ↔ ¬DenselyOrdered G
have rw₂ : ¬DenselyOrdered.{u_1} (Additive G) ↔ ¬DenselyOrdered.{u_1} (Additive G)
Tactic: rw [← OrderAddMonoidIso.toMultiplicativeRight.nonempty_congr,
    LinearOrderedAddCommGroup.discrete_iff_not_denselyOrdered,
    denselyOrdered_iff_of_orderIsoClass e]
-- Lean 4 proof step from Mathlib.GroupTheory.ArchimedeanDensely (goal)
Goal:
G : Type u_1
inst : CommGroup G
inst_1 : LinearOrder G
inst_2 : IsOrderedMonoid G
inst_3 : MulArchimedean G
inst_4 : Nontrivial G
⊢ IsCyclic G ↔ ¬DenselyOrdered G
have rw : ¬DenselyOrdered.{u_1} (Additive G) ↔ ¬DenselyOrdered G
Tactic: rw [← isAddCyclic_additive_iff, LinearOrderedAddCommGroup.isAddCyclic_iff_not_denselyOrdered]
-- Lean 4 proof step from Mathlib.GroupTheory.ArchimedeanDensely (goal)
Goal:
G : Type u_2
inst : LinearOrderedCommGroupWithZero G
inst_1 : Nontrivial.{u_2} Gˣ
inst_2 : MulArchimedean G
⊢ Nonempty.{max 1 (u_2 + 1)} (G ≃*o WithZero (Multiplicative ℤ)) ∨ DenselyOrdered G
have rw : Nonempty.{max 1 (u_2 + 1)} (G ≃*o WithZero (Multiplicative ℤ)) ∨ DenselyOrdered.{u_2} Gˣ
Tactic: rw [← denselyOrdered_units_iff]
-- Lean 4 proof step from Mathlib.GroupTheory.ArchimedeanDensely (goal)
Goal:
G : Type u_2
inst : LinearOrderedCommGroupWithZero G
inst_1 : Nontrivial.{u_2} Gˣ
inst_2 : MulArchimedean G
⊢ Nonempty.{max 1 (u_2 + 1)} (G ≃*o WithZero (Multiplicative ℤ)) ∨ DenselyOrdered.{u_2} Gˣ
have refine : Nonempty.{u_2 + 1} (Gˣ ≃*o Multiplicative ℤ) → Nonempty.{max 1 (u_2 + 1)} (G ≃*o WithZero (Multiplicative ℤ))
Tactic: refine (LinearOrderedCommGroup.discrete_or_denselyOrdered Gˣ).imp_left ?_
-- Lean 4 proof step from Mathlib.GroupTheory.ArchimedeanDensely (goal)
Goal:
G : Type u_2
inst : LinearOrderedCommGroupWithZero G
inst_1 : Nontrivial.{u_2} Gˣ
inst_2 : MulArchimedean G
⊢ Nonempty.{max 1 (u_2 + 1)} (G ≃*o WithZero (Multiplicative ℤ)) ∨ DenselyOrdered G
have rw : Nonempty.{max 1 (u_2 + 1)} (G ≃*o WithZero (Multiplicative ℤ)) ∨ DenselyOrdered.{u_2} Gˣ
Tactic: classical
  rw [← denselyOrdered_units_iff]
  refine (LinearOrderedCommGroup.discrete_or_denselyOrdered Gˣ).imp_left ?_
  intro ⟨f⟩
  exact ⟨OrderMonoidIso.withZeroUnits.symm.trans f.withZero⟩
-- Lean 4 proof step from Mathlib.GroupTheory.ArchimedeanDensely (goal)
Goal:
G : Type u_2
inst : LinearOrderedCommGroupWithZero G
inst_1 : Nontrivial.{u_2} Gˣ
inst_2 : MulArchimedean G
rw rw₁ rw₂ : Nonempty.{max 1 (u_2 + 1)} (G ≃*o WithZero (Multiplicative ℤ)) ∨ DenselyOrdered.{u_2} Gˣ
⊢ Nonempty.{max 1 (u_2 + 1)} (G ≃*o WithZero (Multiplicative ℤ)) ∨ DenselyOrdered G
have rw₃ : Nonempty.{u_2 + 1} (Gˣ ≃*o Multiplicative ℤ) → Nonempty.{max 1 (u_2 + 1)} (G ≃*o WithZero (Multiplicative ℤ))
Tactic: classical
  rw [← denselyOrdered_units_iff]
  refine (LinearOrderedCommGroup.discrete_or_denselyOrdered Gˣ).imp_left ?_
  intro ⟨f⟩
  exact ⟨OrderMonoidIso.withZeroUnits.symm.trans f.withZero⟩
-- Lean 4 proof step from Mathlib.GroupTheory.ArchimedeanDensely (goal)
Goal:
G : Type u_2
inst : LinearOrderedCommGroupWithZero G
inst_1 : Nontrivial.{u_2} Gˣ
inst_2 : MulArchimedean G
rw rw₁ rw₂ : Nonempty.{max 1 (u_2 + 1)} (G ≃*o WithZero (Multiplicative ℤ)) ∨ DenselyOrdered.{u_2} Gˣ
rw₃ : Nonempty.{u_2 + 1} (Gˣ ≃*o Multiplicative ℤ) → Nonempty.{max 1 (u_2 + 1)} (G ≃*o WithZero (Multiplicative ℤ))
⊢ Nonempty.{max 1 (u_2 + 1)} (G ≃*o WithZero (Multiplicative ℤ)) ∨ DenselyOrdered G
have rw₄ : ∀ (f : Gˣ ≃*o Multiplicative ℤ), Nonempty.{max 1 (u_2 + 1)} (G ≃*o WithZero (Multiplicative ℤ))
Tactic: classical
  rw [← denselyOrdered_units_iff]
  refine (LinearOrderedCommGroup.discrete_or_denselyOrdered Gˣ).imp_left ?_
  intro ⟨f⟩
  exact ⟨OrderMonoidIso.withZeroUnits.symm.trans f.withZero⟩
-- Lean 4 proof step from Mathlib.GroupTheory.ArchimedeanDensely (goal)
Goal:
G : Type u_2
inst : LinearOrderedCommGroupWithZero G
inst_1 : Nontrivial.{u_2} Gˣ
inst_2 : MulArchimedean G
⊢ Nonempty.{max 1 (u_2 + 1)} (G ≃*o WithZero (Multiplicative ℤ)) ↔ ¬DenselyOrdered G
have rw : Nonempty.{max 1 (u_2 + 1)} (G ≃*o WithZero (Multiplicative ℤ)) ↔ Nonempty.{u_2 + 1} (Gˣ ≃*o Multiplicative ℤ)
Tactic: rw [← denselyOrdered_units_iff,
      ← LinearOrderedCommGroup.discrete_iff_not_denselyOrdered]
-- Lean 4 proof step from Mathlib.GroupTheory.ArchimedeanDensely (goal)
Goal:
G : Type u_2
inst : LinearOrderedCommGroupWithZero G
inst_1 : Nontrivial.{u_2} Gˣ
inst_2 : MulArchimedean G
⊢ Nonempty.{max 1 (u_2 + 1)} (G ≃*o WithZero (Multiplicative ℤ)) ↔ Nonempty.{u_2 + 1} (Gˣ ≃*o Multiplicative ℤ)
have refine_1 : (G ≃*o WithZero (Multiplicative ℤ)) → Gˣ ≃*o Multiplicative ℤ
Tactic: refine Nonempty.congr ?_ ?_
-- Lean 4 proof step from Mathlib.GroupTheory.ArchimedeanDensely (goal)
Goal:
G : Type u_2
inst : LinearOrderedCommGroupWithZero G
inst_1 : Nontrivial.{u_2} Gˣ
inst_2 : MulArchimedean G
refine_1 : (G ≃*o WithZero (Multiplicative ℤ)) → Gˣ ≃*o Multiplicative ℤ
⊢ Nonempty.{max 1 (u_2 + 1)} (G ≃*o WithZero (Multiplicative ℤ)) ↔ Nonempty.{u_2 + 1} (Gˣ ≃*o Multiplicative ℤ)
have refine_2 : (Gˣ ≃*o Multiplicative ℤ) → G ≃*o WithZero (Multiplicative ℤ)
Tactic: refine Nonempty.congr ?_ ?_
-- Lean 4 proof step from Mathlib.GroupTheory.ArchimedeanDensely (goal)
Goal:
G : Type u_2
inst : LinearOrderedCommGroupWithZero G
inst_1 : Nontrivial.{u_2} Gˣ
inst_2 : MulArchimedean G
f : G ≃*o WithZero (Multiplicative ℤ)
⊢ Gˣ ≃*o Multiplicative ℤ
have refine_1 : ∀ {a b : Gˣ},
  LE.le (α := Multiplicative ℤ)
      (Equiv.toFun
        (MulEquiv.toEquiv
          ((Equiv.symm MulEquiv.withZero : (a : WithZero Gˣ ≃* WithZero (Multiplicative ℤ)) → Gˣ ≃* Multiplicative ℤ)
            (MulEquiv.trans withZeroUnitsEquiv (↑f : G ≃* WithZero (Multiplicative ℤ)))))
        a)
      (Equiv.toFun
        (MulEquiv.toEquiv
          ((Equiv.symm MulEquiv.withZero : (a : WithZero Gˣ ≃* WithZero (Multiplicative ℤ)) → Gˣ ≃* Multiplicative ℤ)
            (MulEquiv.trans withZeroUnitsEquiv (↑f : G ≃* WithZero (Multiplicative ℤ)))))
        b) ↔
    a ≤ b
Tactic: refine ⟨MulEquiv.withZero.symm (withZeroUnitsEquiv.trans f), ?_⟩
-- Lean 4 proof step from Mathlib.GroupTheory.ArchimedeanDensely (goal)
Goal:
G : Type u_2
inst : LinearOrderedCommGroupWithZero G
inst_1 : Nontrivial.{u_2} Gˣ
inst_2 : MulArchimedean G
f : G ≃*o WithZero (Multiplicative ℤ)
a b : Gˣ
⊢ LE.le (α := Multiplicative ℤ)
      (Equiv.toFun
        (MulEquiv.toEquiv
          ((Equiv.symm MulEquiv.withZero : (a : WithZero Gˣ ≃* WithZero (Multiplicative ℤ)) → Gˣ ≃* Multiplicative ℤ)
            (MulEquiv.trans withZeroUnitsEquiv (↑f : G ≃* WithZero (Multiplicative ℤ)))))
        a)
      (Equiv.toFun
        (MulEquiv.toEquiv
          ((Equiv.symm MulEquiv.withZero : (a : WithZero Gˣ ≃* WithZero (Multiplicative ℤ)) → Gˣ ≃* Multiplicative ℤ)
            (MulEquiv.trans withZeroUnitsEquiv (↑f : G ≃* WithZero (Multiplicative ℤ)))))
        b) ↔
    a ≤ b
have refine_1 : unzero (x := (f : (a : G) → WithZero (Multiplicative ℤ)) (↑a : G)) sorry ≤
    unzero (x := (f : (a : G) → WithZero (Multiplicative ℤ)) (↑b : G)) sorry ↔
  a ≤ b
Tactic: simp only [MulEquiv.withZero, withZeroUnitsEquiv, MulEquiv.trans_apply,
      MulEquiv.coe_mk, Equiv.coe_fn_mk, recZeroCoe_coe, OrderMonoidIso.coe_mulEquiv,
      MulEquiv.symm_trans_apply, MulEquiv.symm_mk, Equiv.coe_fn_symm_mk, map_eq_zero, coe_ne_zero,
      ↓reduceDIte, unzero_coe, MulEquiv.toEquiv_eq_coe, Equiv.toFun_as_coe, EquivLike.coe_coe]
-- Lean 4 proof step from Mathlib.GroupTheory.ArchimedeanDensely (goal)
Goal:
G : Type u_2
inst : LinearOrderedCommGroupWithZero G
inst_1 : Nontrivial.{u_2} Gˣ
inst_2 : MulArchimedean G
f : G ≃*o WithZero (Multiplicative ℤ)
a b : Gˣ
⊢ unzero (x := (f : (a : G) → WithZero (Multiplicative ℤ)) (↑a : G)) ⋯ ≤
      unzero (x := (f : (a : G) → WithZero (Multiplicative ℤ)) (↑b : G)) ⋯ ↔
    a ≤ b
have refine_1 : unzero (x := (f : (a : G) → WithZero (Multiplicative ℤ)) (↑a : G)) sorry ≤
    unzero (x := (f : (a : G) → WithZero (Multiplicative ℤ)) (↑b : G)) sorry ↔
  (↑a : G) ≤ (↑b : G)
Tactic: rw [← Units.val_le_val, ← map_le_map_iff f, ← coe_le_coe, coe_unzero, coe_unzero]
-- Lean 4 proof step from Mathlib.GroupTheory.ArchimedeanDensely (goal)
Goal:
G : Type u_2
inst : LinearOrderedCommGroupWithZero G
inst_1 : Nontrivial.{u_2} Gˣ
inst_2 : MulArchimedean G
f : G ≃*o WithZero (Multiplicative ℤ)
a b : Gˣ
refine_1 :
  unzero (x := (f : (a : G) → WithZero (Multiplicative ℤ)) (↑a : G)) ⋯ ≤
      unzero (x := (f : (a : G) → WithZero (Multiplicative ℤ)) (↑b : G)) ⋯ ↔
    (↑a : G) ≤ (↑b : G)
⊢ unzero (x := (f : (a : G) → WithZero (Multiplicative ℤ)) (↑a : G)) ⋯ ≤
      unzero (x := (f : (a : G) → WithZero (Multiplicative ℤ)) (↑b : G)) ⋯ ↔
    a ≤ b
have refine_1₁ : unzero (x := (f : (a : G) → WithZero (Multiplicative ℤ)) (↑a : G)) sorry ≤
    unzero (x := (f : (a : G) → WithZero (Multiplicative ℤ)) (↑b : G)) sorry ↔
  LE.le (α := WithZero (Multiplicative ℤ))
    ((f : (a : G) → WithZero (Multiplicative ℤ)) (↑a : G) : WithZero (Multiplicative ℤ))
    ((f : (a : G) → WithZero (Multiplicative ℤ)) (↑b : G) : WithZero (Multiplicative ℤ))
Tactic: rw [← Units.val_le_val, ← map_le_map_iff f, ← coe_le_coe, coe_unzero, coe_unzero]
-- Lean 4 proof step from Mathlib.GroupTheory.ArchimedeanDensely (goal)
Goal:
G : Type u_2
inst : LinearOrderedCommGroupWithZero G
inst_1 : Nontrivial.{u_2} Gˣ
inst_2 : MulArchimedean G
f : G ≃*o WithZero (Multiplicative ℤ)
a b : Gˣ
refine_1 :
  unzero (x := (f : (a : G) → WithZero (Multiplicative ℤ)) (↑a : G)) ⋯ ≤
      unzero (x := (f : (a : G) → WithZero (Multiplicative ℤ)) (↑b : G)) ⋯ ↔
    (↑a : G) ≤ (↑b : G)
refine_1₁ :
  unzero (x := (f : (a : G) → WithZero (Multiplicative ℤ)) (↑a : G)) ⋯ ≤
      unzero (x := (f : (a : G) → WithZero (Multiplicative ℤ)) (↑b : G)) ⋯ ↔
    LE.le (α := WithZero (Multiplicative ℤ))
      ((f : (a : G) → WithZero (Multiplicative ℤ)) (↑a : G) : WithZero (Multiplicative ℤ))
      ((f : (a : G) → WithZero (Multiplicative ℤ)) (↑b : G) : WithZero (Multiplicative ℤ))
⊢ unzero (x := (f : (a : G) → WithZero (Multiplicative ℤ)) (↑a : G)) ⋯ ≤
      unzero (x := (f : (a : G) → WithZero (Multiplicative ℤ)) (↑b : G)) ⋯ ↔
    a ≤ b
have refine_1₂ : (↑(unzero (x := (f : (a : G) → WithZero (Multiplicative ℤ)) (↑a : G)) sorry) : WithZero (Multiplicative ℤ)) ≤
    (↑(unzero (x := (f : (a : G) → WithZero (Multiplicative ℤ)) (↑b : G)) sorry) : WithZero (Multiplicative ℤ)) ↔
  LE.le (α := WithZero (Multiplicative ℤ))
    ((f : (a : G) → WithZero (Multiplicative ℤ)) (↑a : G) : WithZero (Multiplicative ℤ))
    ((f : (a : G) → WithZero (Multiplicative ℤ)) (↑b : G) : WithZero (Multiplicative ℤ))
Tactic: rw [← Units.val_le_val, ← map_le_map_iff f, ← coe_le_coe, coe_unzero, coe_unzero]
-- Lean 4 proof step from Mathlib.GroupTheory.ArchimedeanDensely (goal)
Goal:
G : Type u_2
inst : LinearOrderedCommGroupWithZero G
inst_1 : Nontrivial.{u_2} Gˣ
inst_2 : MulArchimedean G
f : G ≃*o WithZero (Multiplicative ℤ)
a b : Gˣ
refine_1 :
  unzero (x := (f : (a : G) → WithZero (Multiplicative ℤ)) (↑a : G)) ⋯ ≤
      unzero (x := (f : (a : G) → WithZero (Multiplicative ℤ)) (↑b : G)) ⋯ ↔
    (↑a : G) ≤ (↑b : G)
refine_1₁ :
  unzero (x := (f : (a : G) → WithZero (Multiplicative ℤ)) (↑a : G)) ⋯ ≤
      unzero (x := (f : (a : G) → WithZero (Multiplicative ℤ)) (↑b : G)) ⋯ ↔
    LE.le (α := WithZero (Multiplicative ℤ))
      ((f : (a : G) → WithZero (Multiplicative ℤ)) (↑a : G) : WithZero (Multiplicative ℤ))
      ((f : (a : G) → WithZero (Multiplicative ℤ)) (↑b : G) : WithZero (Multiplicative ℤ))
refine_1₂ :
  (↑(unzero (x := (f : (a : G) → WithZero (Multiplicative ℤ)) (↑a : G)) ⋯) : WithZero (Multiplicative ℤ)) ≤
      (↑(unzero (x := (f : (a : G) → WithZero (Multiplicative ℤ)) (↑b : G)) ⋯) : WithZero (Multiplicative ℤ)) ↔
    LE.le (α := WithZero (Multiplicative ℤ))
      ((f : (a : G) → WithZero (Multiplicative ℤ)) (↑a : G) : WithZero (Multiplicative ℤ))
      ((f : (a : G) → WithZero (Multiplicative ℤ)) (↑b : G) : WithZero (Multiplicative ℤ))
⊢ unzero (x := (f : (a : G) → WithZero (Multiplicative ℤ)) (↑a : G)) ⋯ ≤
      unzero (x := (f : (a : G) → WithZero (Multiplicative ℤ)) (↑b : G)) ⋯ ↔
    a ≤ b
have refine_1₃ : (f : (a : G) → WithZero (Multiplicative ℤ)) (↑a : G) ≤
    (↑(unzero (x := (f : (a : G) → WithZero (Multiplicative ℤ)) (↑b : G)) sorry) : WithZero (Multiplicative ℤ)) ↔
  LE.le (α := WithZero (Multiplicative ℤ))
    ((f : (a : G) → WithZero (Multiplicative ℤ)) (↑a : G) : WithZero (Multiplicative ℤ))
    ((f : (a : G) → WithZero (Multiplicative ℤ)) (↑b : G) : WithZero (Multiplicative ℤ))
Tactic: rw [← Units.val_le_val, ← map_le_map_iff f, ← coe_le_coe, coe_unzero, coe_unzero]
-- Lean 4 proof step from Mathlib.GroupTheory.ArchimedeanDensely (goal)
Goal:
G : Type u_2
inst : LinearOrderedCommGroupWithZero G
inst_1 : Nontrivial.{u_2} Gˣ
inst_2 : MulArchimedean G
f : G ≃*o WithZero (Multiplicative ℤ)
a b : Gˣ
refine_1 :
  unzero (x := (f : (a : G) → WithZero (Multiplicative ℤ)) (↑a : G)) ⋯ ≤
      unzero (x := (f : (a : G) → WithZero (Multiplicative ℤ)) (↑b : G)) ⋯ ↔
    (↑a : G) ≤ (↑b : G)
refine_1₁ :
  unzero (x := (f : (a : G) → WithZero (Multiplicative ℤ)) (↑a : G)) ⋯ ≤
      unzero (x := (f : (a : G) → WithZero (Multiplicative ℤ)) (↑b : G)) ⋯ ↔
    LE.le (α := WithZero (Multiplicative ℤ))
      ((f : (a : G) → WithZero (Multiplicative ℤ)) (↑a : G) : WithZero (Multiplicative ℤ))
      ((f : (a : G) → WithZero (Multiplicative ℤ)) (↑b : G) : WithZero (Multiplicative ℤ))
refine_1₂ :
  (↑(unzero (x := (f : (a : G) → WithZero (Multiplicative ℤ)) (↑a : G)) ⋯) : WithZero (Multiplicative ℤ)) ≤
      (↑(unzero (x := (f : (a : G) → WithZero (Multiplicative ℤ)) (↑b : G)) ⋯) : WithZero (Multiplicative ℤ)) ↔
    LE.le (α := WithZero (Multiplicative ℤ))
      ((f : (a : G) → WithZero (Multiplicative ℤ)) (↑a : G) : WithZero (Multiplicative ℤ))
      ((f : (a : G) → WithZero (Multiplicative ℤ)) (↑b : G) : WithZero (Multiplicative ℤ))
refine_1₃ :
  (f : (a : G) → WithZero (Multiplicative ℤ)) (↑a : G) ≤
      (↑(unzero (x := (f : (a : G) → WithZero (Multiplicative ℤ)) (↑b : G)) ⋯) : WithZero (Multiplicative ℤ)) ↔
    LE.le (α := WithZero (Multiplicative ℤ))
      ((f : (a : G) → WithZero (Multiplicative ℤ)) (↑a : G) : WithZero (Multiplicative ℤ))
      ((f : (a : G) → WithZero (Multiplicative ℤ)) (↑b : G) : WithZero (Multiplicative ℤ))
⊢ unzero (x := (f : (a : G) → WithZero (Multiplicative ℤ)) (↑a : G)) ⋯ ≤
      unzero (x := (f : (a : G) → WithZero (Multiplicative ℤ)) (↑b : G)) ⋯ ↔
    a ≤ b
have refine_1₄ : LE.le (α := WithZero (Multiplicative ℤ))
    ((f : (a : G) → WithZero (Multiplicative ℤ)) (↑a : G) : WithZero (Multiplicative ℤ))
    ((f : (a : G) → WithZero (Multiplicative ℤ)) (↑b : G) : WithZero (Multiplicative ℤ)) ↔
  LE.le (α := WithZero (Multiplicative ℤ))
    ((f : (a : G) → WithZero (Multiplicative ℤ)) (↑a : G) : WithZero (Multiplicative ℤ))
    ((f : (a : G) → WithZero (Multiplicative ℤ)) (↑b : G) : WithZero (Multiplicative ℤ))
Tactic: rw [← Units.val_le_val, ← map_le_map_iff f, ← coe_le_coe, coe_unzero, coe_unzero]
-- Lean 4 proof step from Mathlib.GroupTheory.ArchimedeanDensely (goal)
Goal:
G : Type u_2
inst : LinearOrderedCommGroupWithZero G
inst_1 : Nontrivial.{u_2} Gˣ
inst_2 : MulArchimedean G
f : Gˣ ≃*o Multiplicative ℤ
⊢ G ≃*o WithZero (Multiplicative ℤ)
have refine_2 : ∀ {a b : G},
  LE.le (α := WithZero (Multiplicative ℤ))
      (Equiv.toFun
        (MulEquiv.toEquiv
          (MulEquiv.trans (MulEquiv.symm withZeroUnitsEquiv)
            ((MulEquiv.withZero : (a : Gˣ ≃* Multiplicative ℤ) → WithZero Gˣ ≃* WithZero (Multiplicative ℤ))
              (↑f : Gˣ ≃* Multiplicative ℤ))))
        a)
      (Equiv.toFun
        (MulEquiv.toEquiv
          (MulEquiv.trans (MulEquiv.symm withZeroUnitsEquiv)
            ((MulEquiv.withZero : (a : Gˣ ≃* Multiplicative ℤ) → WithZero Gˣ ≃* WithZero (Multiplicative ℤ))
              (↑f : Gˣ ≃* Multiplicative ℤ))))
        b) ↔
    a ≤ b
Tactic: refine ⟨withZeroUnitsEquiv.symm.trans (MulEquiv.withZero f), ?_⟩
-- Lean 4 proof step from Mathlib.GroupTheory.ArchimedeanDensely (goal)
Goal:
G : Type u_2
inst : LinearOrderedCommGroupWithZero G
inst_1 : Nontrivial.{u_2} Gˣ
inst_2 : MulArchimedean G
f : Gˣ ≃*o Multiplicative ℤ
a b : G
⊢ LE.le (α := WithZero (Multiplicative ℤ))
      (Equiv.toFun
        (MulEquiv.toEquiv
          (MulEquiv.trans (MulEquiv.symm withZeroUnitsEquiv)
            ((MulEquiv.withZero : (a : Gˣ ≃* Multiplicative ℤ) → WithZero Gˣ ≃* WithZero (Multiplicative ℤ))
              (↑f : Gˣ ≃* Multiplicative ℤ))))
        a)
      (Equiv.toFun
        (MulEquiv.toEquiv
          (MulEquiv.trans (MulEquiv.symm withZeroUnitsEquiv)
            ((MulEquiv.withZero : (a : Gˣ ≃* Multiplicative ℤ) → WithZero Gˣ ≃* WithZero (Multiplicative ℤ))
              (↑f : Gˣ ≃* Multiplicative ℤ))))
        b) ↔
    a ≤ b
have refine_2 : LE.le (α := WithZero (Multiplicative ℤ))
    (((map' (↑(↑f : Gˣ ≃* Multiplicative ℤ) : Gˣ →* Multiplicative ℤ) : WithZero Gˣ →*₀ WithZero (Multiplicative ℤ)) :
          (a : WithZero Gˣ) → WithZero (Multiplicative ℤ))
        (if h : a = 0 then 0 else (↑(Units.mk0 a h) : WithZero Gˣ)) :
      WithZero (Multiplicative ℤ))
    (((map' (↑(↑f : Gˣ ≃* Multiplicative ℤ) : Gˣ →* Multiplicative ℤ) : WithZero Gˣ →*₀ WithZero (Multiplicative ℤ)) :
          (a : WithZero Gˣ) → WithZero (Multiplicative ℤ))
        (if h : b = 0 then 0 else (↑(Units.mk0 b h) : WithZero Gˣ)) :
      WithZero (Multiplicative ℤ)) ↔
  a ≤ b
Tactic: simp only [withZeroUnitsEquiv, MulEquiv.symm_mk, MulEquiv.withZero,
      MulEquiv.toEquiv_eq_coe, Equiv.toFun_as_coe, EquivLike.coe_coe,
      MulEquiv.trans_apply, MulEquiv.coe_mk, Equiv.coe_fn_symm_mk, Equiv.coe_fn_mk]
-- Lean 4 proof step from Mathlib.GroupTheory.ArchimedeanDensely (goal)
Goal:
G : Type u_2
inst : LinearOrderedCommGroupWithZero G
inst_1 : Nontrivial.{u_2} Gˣ
inst_2 : MulArchimedean G
f : Gˣ ≃*o Multiplicative ℤ
a b : G
⊢ LE.le (α := WithZero (Multiplicative ℤ))
      (((map' (↑(↑f : Gˣ ≃* Multiplicative ℤ) : Gˣ →* Multiplicative ℤ) : WithZero Gˣ →*₀ WithZero (Multiplicative ℤ)) :
            (a : WithZero Gˣ) → WithZero (Multiplicative ℤ))
          (if h : a = 0 then 0 else (↑(Units.mk0 a h) : WithZero Gˣ)) :
        WithZero (Multiplicative ℤ))
      (((map' (↑(↑f : Gˣ ≃* Multiplicative ℤ) : Gˣ →* Multiplicative ℤ) : WithZero Gˣ →*₀ WithZero (Multiplicative ℤ)) :
            (a : WithZero Gˣ) → WithZero (Multiplicative ℤ))
          (if h : b = 0 then 0 else (↑(Units.mk0 b h) : WithZero Gˣ)) :
        WithZero (Multiplicative ℤ)) ↔
    a ≤ b
have pos : a = 0 →
  b = 0 →
    (LE.le (α := WithZero (Multiplicative ℤ))
        (((map' (↑(↑f : Gˣ ≃* Multiplicative ℤ) : Gˣ →* Multiplicative ℤ) :
                WithZero Gˣ →*₀ WithZero (Multiplicative ℤ)) :
              (a : WithZero Gˣ) → WithZero (Multiplicative ℤ))
            0 :
          WithZero (Multiplicative ℤ))
        (((map' (↑(↑f : Gˣ ≃* Multiplicative ℤ) : Gˣ →* Multiplicative ℤ) :
                WithZero Gˣ →*₀ WithZero (Multiplicative ℤ)) :
              (a : WithZero Gˣ) → WithZero (Multiplicative ℤ))
            0 :
          WithZero (Multiplicative ℤ)) ↔
      a ≤ b)
Tactic: split_ifs
-- Lean 4 proof step from Mathlib.GroupTheory.ArchimedeanDensely (goal)
Goal:
G : Type u_2
inst : LinearOrderedCommGroupWithZero G
inst_1 : Nontrivial.{u_2} Gˣ
inst_2 : MulArchimedean G
f : Gˣ ≃*o Multiplicative ℤ
a b : G
pos :
  a = 0 →
    b = 0 →
      (LE.le (α := WithZero (Multiplicative ℤ))
          (((map' (↑(↑f : Gˣ ≃* Multiplicative ℤ) : Gˣ →* Multiplicative ℤ) :
                  WithZero Gˣ →*₀ WithZero (Multiplicative ℤ)) :
                (a : WithZero Gˣ) → WithZero (Multiplicative ℤ))
              0 :
            WithZero (Multiplicative ℤ))
          (((map' (↑(↑f : Gˣ ≃* Multiplicative ℤ) : Gˣ →* Multiplicative ℤ) :
                  WithZero Gˣ →*₀ WithZero (Multiplicative ℤ)) :
                (a : WithZero Gˣ) → WithZero (Multiplicative ℤ))
              0 :
            WithZero (Multiplicative ℤ)) ↔
        a ≤ b)
⊢ LE.le (α := WithZero (Multiplicative ℤ))
      (((map' (↑(↑f : Gˣ ≃* Multiplicative ℤ) : Gˣ →* Multiplicative ℤ) : WithZero Gˣ →*₀ WithZero (Multiplicative ℤ)) :
            (a : WithZero Gˣ) → WithZero (Multiplicative ℤ))
          (if h : a = 0 then 0 else (↑(Units.mk0 a h) : WithZero Gˣ)) :
        WithZero (Multiplicative ℤ))
      (((map' (↑(↑f : Gˣ ≃* Multiplicative ℤ) : Gˣ →* Multiplicative ℤ) : WithZero Gˣ →*₀ WithZero (Multiplicative ℤ)) :
            (a : WithZero Gˣ) → WithZero (Multiplicative ℤ))
          (if h : b = 0 then 0 else (↑(Units.mk0 b h) : WithZero Gˣ)) :
        WithZero (Multiplicative ℤ)) ↔
    a ≤ b
have neg : a = 0 →
  ∀ (h : ¬b = 0),
    LE.le (α := WithZero (Multiplicative ℤ))
        (((map' (↑(↑f : Gˣ ≃* Multiplicative ℤ) : Gˣ →* Multiplicative ℤ) :
                WithZero Gˣ →*₀ WithZero (Multiplicative ℤ)) :
              (a : WithZero Gˣ) → WithZero (Multiplicative ℤ))
            0 :
          WithZero (Multiplicative ℤ))
        (((map' (↑(↑f : Gˣ ≃* Multiplicative ℤ) : Gˣ →* Multiplicative ℤ) :
                WithZero Gˣ →*₀ WithZero (Multiplicative ℤ)) :
              (a : WithZero Gˣ) → WithZero (Multiplicative ℤ))
            (↑(Units.mk0 b h) : WithZero Gˣ) :
          WithZero (Multiplicative ℤ)) ↔
      a ≤ b
Tactic: split_ifs
-- Lean 4 proof step from Mathlib.GroupTheory.ArchimedeanDensely (goal)
Goal:
G : Type u_2
inst : LinearOrderedCommGroupWithZero G
inst_1 : Nontrivial.{u_2} Gˣ
inst_2 : MulArchimedean G
f : Gˣ ≃*o Multiplicative ℤ
a b : G
pos :
  a = 0 →
    b = 0 →
      (LE.le (α := WithZero (Multiplicative ℤ))
          (((map' (↑(↑f : Gˣ ≃* Multiplicative ℤ) : Gˣ →* Multiplicative ℤ) :
                  WithZero Gˣ →*₀ WithZero (Multiplicative ℤ)) :
                (a : WithZero Gˣ) → WithZero (Multiplicative ℤ))
              0 :
            WithZero (Multiplicative ℤ))
          (((map' (↑(↑f : Gˣ ≃* Multiplicative ℤ) : Gˣ →* Multiplicative ℤ) :
                  WithZero Gˣ →*₀ WithZero (Multiplicative ℤ)) :
                (a : WithZero Gˣ) → WithZero (Multiplicative ℤ))
              0 :
            WithZero (Multiplicative ℤ)) ↔
        a ≤ b)
neg :
  a = 0 →
    ∀ (h : ¬b = 0),
      LE.le (α := WithZero (Multiplicative ℤ))
          (((map' (↑(↑f : Gˣ ≃* Multiplicative ℤ) : Gˣ →* Multiplicative ℤ) :
                  WithZero Gˣ →*₀ WithZero (Multiplicative ℤ)) :
                (a : WithZero Gˣ) → WithZero (Multiplicative ℤ))
              0 :
            WithZero (Multiplicative ℤ))
          (((map' (↑(↑f : Gˣ ≃* Multiplicative ℤ) : Gˣ →* Multiplicative ℤ) :
                  WithZero Gˣ →*₀ WithZero (Multiplicative ℤ)) :
                (a : WithZero Gˣ) → WithZero (Multiplicative ℤ))
              (↑(Units.mk0 b h) : WithZero Gˣ) :
            WithZero (Multiplicative ℤ)) ↔
        a ≤ b
⊢ LE.le (α := WithZero (Multiplicative ℤ))
      (((map' (↑(↑f : Gˣ ≃* Multiplicative ℤ) : Gˣ →* Multiplicative ℤ) : WithZero Gˣ →*₀ WithZero (Multiplicative ℤ)) :
            (a : WithZero Gˣ) → WithZero (Multiplicative ℤ))
          (if h : a = 0 then 0 else (↑(Units.mk0 a h) : WithZero Gˣ)) :
        WithZero (Multiplicative ℤ))
      (((map' (↑(↑f : Gˣ ≃* Multiplicative ℤ) : Gˣ →* Multiplicative ℤ) : WithZero Gˣ →*₀ WithZero (Multiplicative ℤ)) :
            (a : WithZero Gˣ) → WithZero (Multiplicative ℤ))
          (if h : b = 0 then 0 else (↑(Units.mk0 b h) : WithZero Gˣ)) :
        WithZero (Multiplicative ℤ)) ↔
    a ≤ b
have pos₁ : ∀ (h : ¬a = 0),
  b = 0 →
    (LE.le (α := WithZero (Multiplicative ℤ))
        (((map' (↑(↑f : Gˣ ≃* Multiplicative ℤ) : Gˣ →* Multiplicative ℤ) :
                WithZero Gˣ →*₀ WithZero (Multiplicative ℤ)) :
              (a : WithZero Gˣ) → WithZero (Multiplicative ℤ))
            (↑(Units.mk0 a h) : WithZero Gˣ) :
          WithZero (Multiplicative ℤ))
        (((map' (↑(↑f : Gˣ ≃* Multiplicative ℤ) : Gˣ →* Multiplicative ℤ) :
                WithZero Gˣ →*₀ WithZero (Multiplicative ℤ)) :
              (a : WithZero Gˣ) → WithZero (Multiplicative ℤ))
            0 :
          WithZero (Multiplicative ℤ)) ↔
      a ≤ b)
Tactic: split_ifs